Data-Driven Model-Free Adaptive Control of Particle...
18
Research Article Data-Driven Model-Free Adaptive Control of Particle Quality in Drug Development Phase of Spray Fluidized-Bed Granulation Process Zhengsong Wang, 1 Dakuo He, 1,2 Xu Zhu, 1 Jiahuan Luo, 1 Yu Liang, 1 and Xu Wang 3 1 College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning 110004, China 2 State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, China 3 State Key Laboratory of Process Automation in Mining & Metallurgy, Beijing 100160, China Correspondence should be addressed to Dakuo He; [email protected] Received 28 September 2017; Accepted 19 November 2017; Published 12 December 2017 Academic Editor: Julio Blanco-Fern´ andez Copyright © 2017 Zhengsong Wang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A novel data-driven model-free adaptive control (DDMFAC) approach is first proposed by combining the advantages of model- free adaptive control (MFAC) and data-driven optimal iterative learning control (DDOILC), and then its stability and convergence analysis is given to prove algorithm stability and asymptotical convergence of tracking error. Besides, the parameters of presented approach are adaptively adjusted with fuzzy logic to determine the occupied proportions of MFAC and DDOILC according to their different control performances in different control stages. Lastly, the proposed fuzzy DDMFAC (FDDMFAC) approach is applied to the control of particle quality in drug development phase of spray fluidized-bed granulation process (SFBGP), and its control effect is compared with MFAC and DDOILC and their fuzzy forms, in which the parameters of MFAC and DDOILC are adaptively adjusted with fuzzy logic. e effectiveness of the presented FDDMFAC approach is verified by a series of simulations. 1. Introduction Spray fluidized-bed granulation process (SFBGP) is a process that forms small particles into larger granules using the liquid binding solution sprayed onto fluidized particles by a spray nozzle above the powder bed [1]. Because of advantage of sin- gle unit operation, SFBGP has been widely applied to produce granules aiming at improving power flowability and physico- chemical properties of drugs [2, 3]. As a key technique in the production of pharmaceutical solid dosage forms of tablets and capsules, the primary target of SFBGP is to produce granules with consistent product quality for the following pharmaceutical processes. erefore, particle quality control of SFBGP is of great theoretical and practical significance. For a SFBGP, the quality of particles can be evaluated by many factors such as production yield, drug content, size, density, friability, flowability, and compressibility [2, 4], and among them granule size is the key characteristic to control [5–8]. To achieve the desired granule size, model-based control (MBC) schemes, which require a priori physical and mathematical knowledge of the process, may be the effective techniques. Previous works have introduced MBC frameworks and examined the implementation of MBC strategies on conventional granulation design [3, 9–13]. It is well known that SFBGP is a complicated process that is significantly influenced by both the material-related factors such as the nature and characteristics of powder particles and binding agents and the process factors associated with granulation such as fluidizing air velocity and binder feed rate [2, 3, 5, 7, 14, 15]. In the drug development phase, however, the frequent adjustment of prescription will give rise to the variation of material attributes. So for a brand-new prescrip- tion, the operating condition space should be redesigned in order to accurately and rapidly achieve the desired particle quality. But for such a SFBGP whose material attributes have already changed and that we have never encountered before, the absence of process operating experience and historical data leads to great difficulty in developing accurate Hindawi Complexity Volume 2017, Article ID 4960106, 17 pages https://doi.org/10.1155/2017/4960106
Transcript of Data-Driven Model-Free Adaptive Control of Particle...
Research ArticleData-Driven Model-Free Adaptive Control ofParticle Quality in Drug Development Phase ofSpray Fluidized-Bed Granulation Process
Zhengsong Wang1 Dakuo He12 Xu Zhu1 Jiahuan Luo1 Yu Liang1 and Xu Wang3
1College of Information Science and Engineering Northeastern University Shenyang Liaoning 110004 China2State Key Laboratory of Synthetical Automation for Process Industries Northeastern University Shenyang 110004 China3State Key Laboratory of Process Automation in Mining amp Metallurgy Beijing 100160 China
Correspondence should be addressed to Dakuo He hedakuoiseneueducn
Received 28 September 2017 Accepted 19 November 2017 Published 12 December 2017
Academic Editor Julio Blanco-Fernandez
Copyright copy 2017 Zhengsong Wang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
A novel data-driven model-free adaptive control (DDMFAC) approach is first proposed by combining the advantages of model-free adaptive control (MFAC) and data-driven optimal iterative learning control (DDOILC) and then its stability and convergenceanalysis is given to prove algorithm stability and asymptotical convergence of tracking error Besides the parameters of presentedapproach are adaptively adjusted with fuzzy logic to determine the occupied proportions of MFAC andDDOILC according to theirdifferent control performances in different control stages Lastly the proposed fuzzy DDMFAC (FDDMFAC) approach is appliedto the control of particle quality in drug development phase of spray fluidized-bed granulation process (SFBGP) and its controleffect is compared withMFAC and DDOILC and their fuzzy forms in which the parameters of MFAC and DDOILC are adaptivelyadjusted with fuzzy logic The effectiveness of the presented FDDMFAC approach is verified by a series of simulations
1 Introduction
Spray fluidized-bed granulation process (SFBGP) is a processthat forms small particles into larger granules using the liquidbinding solution sprayed onto fluidized particles by a spraynozzle above the powder bed [1] Because of advantage of sin-gle unit operation SFBGPhas beenwidely applied to producegranules aiming at improving power flowability and physico-chemical properties of drugs [2 3] As a key technique in theproduction of pharmaceutical solid dosage forms of tabletsand capsules the primary target of SFBGP is to producegranules with consistent product quality for the followingpharmaceutical processes Therefore particle quality controlof SFBGP is of great theoretical and practical significance
For a SFBGP the quality of particles can be evaluated bymany factors such as production yield drug content sizedensity friability flowability and compressibility [2 4] andamong them granule size is the key characteristic to control[5ndash8] To achieve the desired granule size model-based
control (MBC) schemes which require a priori physicaland mathematical knowledge of the process may be theeffective techniques Previous works have introduced MBCframeworks and examined the implementation of MBCstrategies on conventional granulation design [3 9ndash13]
It is well known that SFBGP is a complicated process thatis significantly influenced by both thematerial-related factorssuch as the nature and characteristics of powder particlesand binding agents and the process factors associated withgranulation such as fluidizing air velocity and binder feed rate[2 3 5 7 14 15] In the drug development phase howeverthe frequent adjustment of prescription will give rise to thevariation of material attributes So for a brand-new prescrip-tion the operating condition space should be redesigned inorder to accurately and rapidly achieve the desired particlequality But for such a SFBGP whose material attributeshave already changed and that we have never encounteredbefore the absence of process operating experience andhistorical data leads to great difficulty in developing accurate
HindawiComplexityVolume 2017 Article ID 4960106 17 pageshttpsdoiorg10115520174960106
2 Complexity
process models If traditional MBC schemes are used lots ofexperiments in the actual process should first be implementedas long as the prescription is changed to get enough processdata used for developing precisemathematicalmodels Such away is laborious time-consuming and resource-wasting sothat the MBC techniques are no more applicable to qualitycontrol task in this work
Data-driven control (DDC) [16 17] means that only theinputoutput (IO) measurement data of controlled plant areused in controller design DDC approaches do not require amodel of a plant and the modeling process the unmodeleddynamics and the theoretical assumptions all disappear [16ndash19] Therefore DDC has attracted considerable attentionin recent years [18] and there are many DDC approachestogether with their practical applications in many fields thatcould be found in the literature like the following model-free adaptive control (MFAC) [16 17 20ndash24] data-drivenoptimal iterative learning control (DDOILC) [25ndash27] virtualreference feedback tuning [28ndash30] lazy learning control [31]dynamic programming methods [32] and others [33ndash36] Inspite of this to the best of authorsrsquo knowledge there is noreport about the research on application of DDC to SFBGPthat has been published in the literature In this work a studyon particle quality control based on DDC approaches in thedrug development phase of a SFBGP is conducted to resolvethe practical difficulty encountered in redesigning operatingcondition when prescription and material attributes are allchanged
The mechanism model of SFBGP should first be intro-duced to conduct such quality control research Such aprocess model plays several significant roles first modelinganalyzes the mechanism of SFBGP identifies the importantmanipulated variables and quality indices and establishesthe relationship between them Second such a model can beused as a simulator of a real SFBGP to generate the requiredprocess data used for simulation and analysis In this worka widely accepted mechanistic model for SFBGP [37 38] isintroduced to simulate the actual SFBGP and produce therequired process data
In this work we select two classical and representativeDDC approaches MFAC and DDOILC to study data-drivenmodel-free adaptive control (DDMFAC) of average particlesize (APS) for SFBGP with simulation experiment researchMFAC algorithm is proposed by Hou and Jin [16 17] basedon a new dynamic linearization technique (DLT) and thenapplied in several areas [18] The main feature of MFACis that the controller design depends merely on the IOmeasurement data of the controlled plant Instead of iden-tifying a nonlinear process model of a plant an equivalentlocal dynamical linearization model is constructed alongthe dynamic operation points of the system using the DLTwith a novel concept called pseudo-partial derivative (PPD)The time-varying PPD could be estimated merely using theIO measurement data of the controlled plant DDOILC isdeveloped and applied to both linear and nonlinear systemsby Chi et al [25] For this approach the only requiredknowledge of a controlled system is that theMarkovmatricesof linear systems or the partial derivatives of nonlinearsystems with respect to control inputs are bounded DDOILC
is actually the extension of MFAC and these two approacheshave the same systematic framework Compared with otherDDC methods MFAC and DDOILC have several attractiveadvantages that make them more suitable for many practicalcontrol applications [16] First they do not require anyprocess model and structural information of the controlledplant and merely depend on the real time measurement IOdata of the controlled plant which indicates that it is feasibleto independently design a generic controller for a certain classof practical industrial processes Second they are lower costcontrollers because they do not need any training processThird they are simple and easily implemented with smallcomputational burden and have strong robustness Finallythey have been successfully implemented in many practicalfields such as chemical industry [20 21 25 26] linearmotor control and injectionmoulding process [22] PH valuecontrol [23] and robotic welding process [24]
The controllers of MFAC and DDOILC all consist ofa control input iterative learning law and a PPD matrixiterative updating law From this respect the prime differencebetween them is whether or not to use the predicted dataof system output of current iteration in controller designIn MFAC the predicted system output of current iterationis utilized but merely the output measurement data ofprevious iterations are used in DDOILC which makes theirdifferent characteristicsThrough theory analysis MFAC andDDOILC both have advantages with regard to different initialvalues (IVs) for particle quality When IV is far away fromdesired particle quality that is the initial tracking error islarger DDOILC has quicker convergence speed The reasonfor this is that the utilization of actual output of previousiterations in controller design of DDOILC will give largerupdating step size for control input and PPD vector Howeveras IV gets closer to desired outputMFACwill gradually showits advantages since the prediction accuracy of its dynamiclinearization model (DLM) is getting higher and higher Toverify the conclusion obtained through theoretical analysiswe conducted several comparative experimental studies byconstructing a series of IVs and the simulation resultsvalidate our conclusion
In the drug development phase the frequent variationsof prescription will lead to diversity of material propertiesFor different material properties the same group of empiricaloperating conditionswill give entirely different IVs of particlequality It is demanded that the DDC approaches mustaccommodate various IVs in the practice Therefore tofurther improve the overall control effect a novel hybridDDC approach which is namedDDMFAC and combines theadvantages ofMFAC andDDOILC is presented in this paperThe newly designed controller is able to exhibit the featuresof MFAC and DDOILC in different extent by adjusting theirweighted factors Compared withMFAC andDDOILC how-ever evenDDMFACdoes not have overwhelming superiorityfor all IVs Simulation results have revealed that DDMFACmay be inferior to DDOILC or MFAC in particle qualitycontrol if there is no reasonable adjustment strategy ofweighted parameters By the previous theoretical analysisfor MFAC and DDOILC and the subsequent simulationverification we can sum up weighted parameters adjustment
Complexity 3
FI Compressed air unit
Liquid supply unit
Exhaust blower
BlowerFilterHeater
Dust-arresterinstallation
Spray nozzle
Fluidizing agent
Figure 1 The schematic diagram of SFBGP
rules for DDMFACThen fuzzy adaptive adjustment of targetweighted parameters is implemented with the aim of rapidlyconverging DDMFAC by making full use of the advantageof MFAC or DDOILC at different control stages We namedthe final DDC approach presented in this paper as fuzzyDDMFAC (FDDMFAC) and a series of simulations validateits effectiveness
2 Preliminaries
21 Mathematical Mechanism Model of Spray Fluidized-BedGranulation Process SFBGP is a commonly used unit oper-ation in the pharmaceutical industry The powders can bemixed granulated and dried in the same equipment whichminimizes the equipment costs loss of product transfers andpossibility of the cross-contamination [3] The rationale ofspray fluidized-bed granulation can be briefly described asfollows (1) Particles circulate within granulator by pumpinghot air from the bottom of granulator and meanwhile binderliquid is sprayed on the fluidized particles in the forms ofsmall droplets [38] (2) As a result of collisions and coales-cence between the surface-wetted particles liquid bridges areformed and aggregation of particles occurs leading to thegrowth of granules [1 39]
The schematic diagram of SFBGP is shown in Figure 1The overall granulation process in fluidized-bed spray gran-ulator is divided into three stages powder mixing granula-tion and drying Firstly mixed raw materials for drug andingredients are put in granulator Passing through the filterand heater hot air pumped in by blower is pumped intogranulator from the bottom of granulator which forces thefluidization of powder to sufficiently blend materials After-wards liquid binder is pumped into spray nozzle from liquidsupply unit and atomized into droplets by compressed air thatis simultaneously pumped into spray nozzle from compressedair unit The droplets are dispersed over the surface offluidized particles which contributes to the agglomerationof colliding and surface-wetted powder particles to formgranules Finally the granules are dried to a predeterminedmoisture content level The dust-arrester installation is usedto prevent powder particles that are not in contact withdroplets from being carried out by off-gas
PBMs have been widely used for modeling SFBGP inwhich the granule size density distribution evolves as afunction of timeThe population balance is a number balancearound each size fraction of the size distribution based onnumber conservation law The rate of change of the numberof particles in a given size interval is equal to the rate at which
4 Complexity
the granules enter and leave that size interval The mathe-matical mechanism model of SFBGP is developed based onfollowing assumptions (1) Each particle is spherical (2)Thetotal volume of the particles is conserved (3) The particlesize in a certain particular size interval is represented by theleft boundary of the interval (4) The relative growth rate isuniform for each particle within the same volume intervalA one-dimensional PBM that describes the rate of change inparticle number density function 119899(119905 119897) is given by
120597119899 (119905 119897)120597119905 = 12sdot int1198970
120573(119905 (1198973 minus 1205833)13 120583) 119899 (119905 (1198973 minus 1205833)13) 119899 (119905 120583)(1198973 minus 1205833)23 119889120583
minus 119899 (119905 119897) intinfin0
120573 (119905 119897 120583) 119899 (119905 120583) 119889120583
(1)
where 119899(119905 119897) is the particle number density function in termsof particle diameter 119897 and120573 is the aggregationmodel [37 38]120573(119905 119897 120583) can generally be partitioned into size dependent andsize independent parts [3 37 38 40 41]
120573 (119905 119897 119906) = 1205730 (119905) 120573lowast (119897 120583) (2)
where 1205730(119905) is the granulation rate constant that can bethought of as an aggregation efficiency and depends onall parameters involved in the aggregation process exceptparticle size such as the operating condition and binderproperties The latter term 120573lowast(119897 120583) reflects the influence ofparticle size on the likelihood of aggregation
In order to solve a PBM in (1) the discretized approach[42 43] is applied to determine the change in number ofparticles119873119894 in interval 119894 as
d119873119894d119905 = 119894minus2sum
119895=1
2119895minus119894+1120573119894minus1119895119873119894minus1119873119895 + 12120573119894minus1119894minus11198732119894minus1
minus 119873119894 119894minus1sum119895=1
2119895minus119894120573119894119895119873119895 minus 119873119894119899maxsum119895=119894
120573119894119895119873119895(3)
where119873119894 is the number of particles within particle size range(119871 119894 119871 119894+1) 119871 119894 and 119871 119894+1 are the lower and upper limits of 119894thparticle size interval 119894 = 1 2 119899max and 119899max is the totalnumber of particle size intervals with a value of 12 in thispaper In the discretization scheme the length domain ofparticles is divided into geometric intervals in the way thatthe upper and lower limits of each size interval are in a ratioof 119903 = 119871 119894+1119871 119894 = 3radic2 The size range used in modeling isfrom 50 120583m to 800120583mwhich is divided into 12 intervals Thefunction ldquoode45rdquo inMATLAB is used to solve (3) to obtain thefinal number of particles in each size interval Then the APSof final granules119863119898 is calculated by the following formula
119863119898 = 119899maxsum119894=1
119881(119873119891119894) 119889119901119894 (4)
where 119881(119873119891119894) is the volume fraction of end granules at sizeinterval 119894 = 1 2 119899max119873119891119894 is the number of end granulesin the 119894th size interval and 119889119901119894 is the geometric mean of lowerlimit and upper limit of 119894th size interval 119894 = 1 2 119899max
We have just briefly introduced the general framework ofPBM-basedmechanismmodel of SFBGPDetailed derivationprocess and setting of model parameters are given in [37 38]and they will not be covered hereThe developed mechanismmodel is capable of not only linking APS of particles withthe operating variable (binder spray rate) but also reflectingthe relationship between APS and material attributes suchas viscosity of binder and particle density Hence thismechanism model can be used for simulating the practicalchallenges in particle quality control of SFBGP For examplea series of IVs can be constructed by changing the materialattributes while keeping the operating condition unchangedTo veritably simulate the actual granulation process the inputcontrol curve should be time-varying But to simplify thingsthe input control curve is divided into three stages in thewhole granulation process in this work
22 The Data-Driven Control Approaches Used in This WorkAs the classic and representative DDC methods MFAC andDDOILChave somemerits thatmake themmore competitivein practical control applications Therefore in view of thesemerits MFAC and DDOILC are selected and applied toperform data-driven particle quality control for SFBGP inthis work Now we first briefly introduce their controllerdesign procedures For details please refer to [16 25]
A repeatable batch process system to be controlled isgiven as follows
119910 (119896) = 119891 (u (119896)) (5)
where 119910(119896) and u(119896) are the system output and input at 119896thiteration respectively 119896 is the number of iterations and 119891(∙)is an unknown nonlinear functionThis nonlinear system (5)satisfies the following assumptions
(1) The derivatives of 119891(∙) with respect to control inputu(119896) are continuous
(2) System (5) is generalized Lipschitz that is |Δ119910(119896 +1)| le 119887Δu(119896) for any 119896 and Δu(119896) = 0 whereΔ119910(119896 + 1) = 119910(119896 + 1) minus 119910(119896) Δu(119896) = u(119896) minus u(119896 minus 1)and 119887 is a positive constant
221 MFAC Design For the one-step-ahead controller [44]the control goal is to seek a control input sequence u(119896) thatbrings 119910(119896 + 1) to 119910119889 where 119910119889 is the desired system outputsignal In general the weighted one-step-ahead controllermay lead to steady-state tracking error [16] So the followingcontrol input criterion function is used to design the controllaw
119869 (u (119896)) = 1003816100381610038161003816119910119889 minus 1199101 (119896 + 1)10038161003816100381610038162 + 120582 Δu (119896)2 (6)
where 120582 gt 0 is a weighting constantSubstituting the dynamic linearization form
1199101 (119896 + 1) = 119910 (119896) + 120593119879 (119896) Δu (119896) (7)
Complexity 5
into (6) as well as letting 120597119869120597u(119896) be zero givesu (119896) = u (119896 minus 1) + 1205881120593 (119896) (119910119889 minus 119910 (119896))
120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (8)
where 1205881 isin (0 1] is a step-size constant that is added to make(8) more general and 120593(119896) is PPD vector
Because PPD vector 120593(119896) is unknown it needs to beestimated Define criterion function for the unknown PPDvector as follows
119869 ( (119896)) = 10038161003816100381610038161003816Δ119910 (119896) minus 119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(9)
where 120583 gt 0 is a weighting factor and (119896) is the estimationof 120593(119896)
Using optimal condition 120597119869120597(119896) = 0 then gives
(119896)= (119896 minus 1)
+ 120578Δu (119896 minus 1) (Δ119910 (119896) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2
(10)
where 120578 isin (0 2) is a step-size constantDefining the tracking error 119890(119896) = 119910119889 minus 119910(119896) and
combining PPD estimation and input control law MFACscheme is designed as follows
(119896)= (119896 minus 1)
+ 120578Δu (119896 minus 1) (Δ119910 (119896) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2
u (119896) = u (119896 minus 1) + 1205881 (119896)120582 + 1003817100381710038171003817 (119896)10038171003817100381710038172 119890 (119896)
(11)
222 DDOILC Design The control objective is to find anappropriate control input sequence u(119896) such that the systemoutput 119910(119896) follows 119910119889 Consider the control input criterionfunction as
119869 (u (119896)) = 1003816100381610038161003816119910119889 minus 1199102 (119896)10038161003816100381610038162 + 120582 Δu (119896)2 (12)
Substituting the dynamic linearization equation
1199102 (119896) = 119910 (119896 minus 1) + 120593119879 (119896) Δu (119896) (13)
into (12) and the optimal condition leads to
u (119896) = u (119896 minus 1) + 1205882120593 (119896)120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 119890 (119896 minus 1) (14)
where 1205882 isin (0 1] determines the step size and 119890(119896 minus 1) =119910119889 minus 119910(119896 minus 1)
Let (119896) denote its estimation at 119896th iterationThendefinethe criterion function as
119869 ( (119896)) = 10038161003816100381610038161003816Δ119910 (119896 minus 1) minus 119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(15)
According to optimal condition the iterative updating law ofestimate (119896) is developed as
(119896) = (119896 minus 1)+ 120578Δu (119896 minus 1) (Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))
120583 + Δu (119896 minus 1)2 (16)
where 120583 gt 0 0 lt 120578 lt 2The DDOILC scheme is designed as follows
(119896) = (119896 minus 1)+ 120578Δu (119896 minus 1) (Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))
120583 + Δu (119896 minus 1)2 u (119896) = u (119896 minus 1) + 1205882 (119896)120582 + 1003817100381710038171003817 (119896)10038171003817100381710038172 119890 (119896 minus 1)
(17)
3 Theoretical Analysis of MFAC and DDOILCand Simulations in SFBGP
31 Theoretical Analysis of MFAC and DDOILC The con-troller design procedures of MFAC and DDOILC have beenintroduced in previous sections Although they have the samesystemic framework and similar expression form of con-troller there are still differences between them which makesthem two different methods Comparing the controllers ofMFAC and DDOILC the main differences can be listed asfollows
(1) Δ119910(119896) and Δ119910(119896 minus 1) are respectively used in PPDiterative updating laws
(2) 119890(119896) and 119890(119896minus1) are separately utilized in control inputiterative learning laws
The expressions of Δ119910(119896) Δ119910(119896minus1) 119890(119896) and 119890(119896minus1) arelisted as follows
MFACΔ119910 (119896) = 119910 (119896) minus 119910 (119896 minus 1)119890 (119896) = 119910119889 minus 119910 (119896)
DDOILCΔ119910 (119896 minus 1) = 119910 (119896 minus 1) minus 119910 (119896 minus 2)119890 (119896 minus 1) = 119910119889 minus 119910 (119896 minus 1)
(18)
Obviously the main differences between MFAC andDDOILC can be described as follows system output 119910(119896) at119896th iteration is used in MFAC but is not used in DDOILC Inpractical application all the system outputs used inDDOILC119910(119896minus1) and 119910(119896minus2) are the actual values at (119896minus1)th and (119896minus2)th iterations In MFAC however the actual value of system
6 Complexity
y
yd
y (k + 1)y(k)
y(k)y (k minus 1)
y (k minus 1)
y(k minus 2)
IV = y0
u(k minus 2) u(k minus 1) u(k) u(k + 1) u(k + 2) Input
e (k)
e (k minus 1)ep (k)
ep (k minus 1)
Δy (k)
Δy(k minus 1)
Tracking errorPredicted error of DLM
Δy
Figure 2 The variable relations during control process of MFAC and DDOILC
output 119910(119896) is obtained only after the calculation of u(119896) So119910(119896) does not have actual value when computing u(119896) anda predicted system output 119910(119896) will be used to replace 119910(119896)in practical application of MFAC 119910(119896) is estimated by thefollowing DLM
119910 (119896) = 119910 (119896 minus 1) + 120593119879 (119896 minus 1) Δu (119896 minus 1) (19)
and 119890(119896) is replaced by 119890(119896) = 119910119889 minus 119910(119896)Figure 2 gives the variable relations during control pro-
cess of MFAC and DDOILC In terms of a single controlprocess curve when system output 119910 is greatly different from119910119889 that is the tracking error 119890 is relatively great 119890(119896 minus 1)is much larger than 119890(119896) and DDOILC will achieve a biggerupdating step size for control input which gives DDOILC afaster convergence speed As 119910 keeps getting closer to 119910119889 itis found that DLM prediction error 119890119901 is getting smaller andsmaller until 119890119901 = 0 The difference between 119890(119896 minus 1) and 119890(119896)will also be smaller and smaller DDOILC will not only loseits advantage but also perform worse than MFAC when 119910 isvery close to119910119889 because a larger updating step size for controlinput will bring difficulty in convergence while at this timeMFAC will achieve a fast and stable control effect due to theimprovement of prediction accuracy of DLM From anotherpoint of view if IV of system output 1199100 is far away from 119910119889the initial tracking error and DLM prediction error will beboth at a high level and DDOILC will perform better thanMFAC because a greater updating step size for control inputis necessary to give a faster convergence speed But as1199100 keepsgetting closer to 119910119889 119890119901 will gradually decrease and MFACwill gradually show its advantage Therefore the conclusions
obtained from theoretical analysis can be summarized asfollows
(1) When tracking error is large and 1199100 is relatively faraway from119910119889 DDOILCwill have a faster convergencespeed and a better control performance
(2) When tracking error gradually decreases and 1199100gradually gets closer to 119910119889 MFAC will graduallyreveal its advantage and achieve better control effect
32 SimulationVerification To validate conclusions obtainedfrom theoretical analysis simulation verification in SFBGP isconducted In this simulation particle quality to be controlledis APS and the desired quality is 119910119889 = 120 120583mThe viscosityof binder is treated as material attribute to be changed andthen a series of IVs (1199100) are constructed by adjusting viscosityof binder while keeping the operating condition unchangedFour sets of IVs are selected as representatives in this workand the detailed establishing information of them is shownin Table 1 Besides the parameter settings of MFAC andDDOILC for different IVs are listed in Table 2
For different IVs we conduct particle quality control inSFBGP based on MFAC and DDOILC and the control per-formances are plotted in Figure 3Moreover DLMpredictionerror profiles of MFAC are shown in Figure 4 Then theexplanation and analysis of simulation results will be givenAs is shown in Figure 3(a) DDOILC performs better thanMFAC when 1199100 = 60 that is 1199100 is relatively far away from119910119889 When 1199100 = 70 as is described in Figure 3(b) DDOILChas a faster convergence speed when APS is controlled from70 120583m to about 110 120583m but MFAC achieves a relatively better
Complexity 7
MFACDDOILC
5 10 150Iterations
60
80
100
120A
PS (
m)
(a) 1199100 = 60
MFACDDOILC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
(c) 1199100 = 80
MFACDDOILC
5 10 150Iterations
95
100
105
110
115
120
125
APS
(m
)
(d) 1199100 = 100
Figure 3 The control profiles of APS in SFBGP based on MFAC and DDOILC
Table 1The detailed establishing information of representative IVs
Info IV1199100 = 60 1199100 = 70 1199100 = 80 1199100 = 100
Operatingconditions(gs)
First stage 03 second stage 01 third stage 015
Viscosity ofbinder (Pasdots) 00214 00250 00304 00625
control performance when it is controlled from 110 120583m to 119910119889which indicates that both DDOILC and MFAC have certainadvantages in such case However from Figure 3(c) it can beseen that MFAC is slightly superior to DDOILC in controleffect when 1199100 = 80 and this just manifests the advantage ofMFAC As 1199100 gets closer and closer to 119910119889 such as 1199100 = 100
Table 2The parameter settings ofMFAC andDDOILC for differentIVs
IV Parameter1205881 1205882 120582 120578 1205831199100 = 60 00505 00712
00010 00750 100001199100 = 70 00047 005031199100 = 80 00032 001991199100 = 100 00020 149119864 minus 4
plotted in Figure 3(d)MFAC has a huge advantage comparedwith DDOILC It is obvious that the simulation results haveverified the conclusions obtained from theoretical analysisThe reason for increasing superiority of MFAC is attributedto the smaller and smaller prediction error 119890119901 of DLM Asis shown in Figure 4 on the one hand the curves of 119890119901 for
8 Complexity
IV = 60IV = 70
IV = 80IV = 100
IV = 60 14320IV = 70 08061IV = 80 04718IV = 100 04000
5 10 150Iterations
0
5
10
15
20
25
e p(
m)
Average of ep
Figure 4 The DLM prediction error profiles of MFAC for differentIVs
different IVs all show a decreasing trend along with iterationuntil 119890119901 = 0 as APS keeps getting closer to 119910119889 which is inaccordance with our obtained conclusion On the other handfor different IVs 119890119901 will also decrease with the increase of IVwhich is proved by comparing the curves of 119890119901 under differentIVs To further illustrate this point the averages of 119890119901 fordifferent IVs are computed and then listed in Figure 4 andthe decreasing averages of 119890119901 along with the increase of IValso validate the rationality of our analysis
4 The Proposed Hybrid DDC Approach andSimulation Study in SFBGP
Through theoretical analysis and simulation study we haveproved that MFAC and DDOILC both have strengths inparticle quality control under different IVs In real-worldapplications the same operating condition may give entirelydifferent IVs due to the changes in material properties whichbrings great difficulties in the selection of control methodsThus the uncertainties of IV have to be considered duringcontrol application In order to achieve favourable overallcontrol effect it is demanded that the DDC approaches mustaccommodate various IVs Therefore we try to combineadvantages of MFAC and DDOILC to present a hybridapproach that is named DDMFAC The basic idea is that thefeatures of MFAC and DDOILC are both reflected in thecriterion functions when designing controller of DDMFACThe expected features of hybrid approach can be summarizedas the following three points
(1) When tracking error is large DDMFACmanifests thefeature of DDOILC to acquire a faster convergencespeed
(2) When tracking error is small enough DDMFAC hasthe ability ofMFAC to achieve a fast and stable controleffect
(3) DDMFAC is able to achieve better control perfor-mance for different IVs
Through such combination the shortcomings of MFACand DDOILC will be complemented and their respec-tive advantages will be highlighted in DDMFAC Next wedescribe the controller design for DDMFAC in Section 41
41 DDMFAC Design Consider the following control inputcriterion function
119869 (u (119896)) = 120572119872 1003816100381610038161003816119910119889 minus 1199101 (119896 + 1)10038161003816100381610038162 + 120572119863 1003816100381610038161003816119910119889 minus 1199102 (119896)10038161003816100381610038162+ 1205821 Δu (119896)2 (20)
where |119910119889 minus 1199101(119896 + 1)|2 and |119910119889 minus 1199102(119896)|2 respectively comefrom MFAC and DDOILC and 120572119872 120572119863 are the respectiveweighted factors and 120572119872 + 120572119863 = 0
Substituting (7) and (13) into (20) gives
119869 (u (119896)) = 120572119872 10038171003817100381710038171003817119890 (119896) minus 120593119879 (119896) Δu (119896)100381710038171003817100381710038172+ 120572119863 10038171003817100381710038171003817119890 (119896 minus 1) minus 120593119879 (119896) Δu (119896)100381710038171003817100381710038172+ 1205821 Δu (119896)2
(21)
Using optimal condition 120597119869120597u(119896) = 0 then gives
Δu (119896) = 120593 (119896) (120572119872119890 (119896) + 120572119863119890 (119896 minus 1))1205821 + (120572119872 + 120572119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (22)
Dividing numerator and denominator by 120572119872+120572119863 at the righthand of (22) we haveΔu (119896)= 120593 (119896) ((120572119872 (120572119872 + 120572119863)) 119890 (119896) + (120572119863 (120572119872 + 120572119863)) 119890 (119896 minus 1))
(1205821 (120572119872 + 120572119863)) + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (23)
Letting 120588119872 = 120572119872(120572119872 + 120572119863) 120588119863 = 120572119863(120572119872 + 120572119863) and 120582 =1205821(120572119872 + 120572119863) thenu (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (24)
where 120582 gt 0 is the weighted factor and 120588119872 120588119863 isin [0 1] are thestep-size factors
Define the following PPD vector criterion function
119869 ( (119896))= 10038171003817100381710038171003817120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896) Δu (119896 minus 1)100381710038171003817100381710038172
+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172 (25)
where 120573119872 120573119863 isin [0 1] are the weighted factors and 120573119872+120573119863 =1We rewrite (25) as
119869 ( (119896)) = 10038161003816100381610038161003816120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1)minus 119879 (119896 minus 1) Δu (119896 minus 1) minus Δ119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(26)
Complexity 9
Utilizing optimal condition 120597119869120597(119896) = 0 we have
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (27)
Thus the controller of DDMFAC is designed as follows
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (28)
u (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (29)
42 Stability and Convergence Analysis of DDMFAC
(1)The Boundedness of (119896) Define the parameter estimationerror of 120593(119896) as follows
(119896) = 120593 (119896) minus (119896) (30)
Substituting (28) into (30) gives
(119896) = 120593 (119896) minus (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 (31)
Similarly define (119896 minus 1) = 120593(119896 minus 1) minus (119896 minus 1) and then (119896 minus 1) = 120593 (119896 minus 1) minus (119896 minus 1) (32)
Combining (31) and (32) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1) (33)
According to (7) and (13) it can be obtained that
Δ119910 (119896) = 120593119879 (119896 minus 1) Δu (119896 minus 1)
Δ119910 (119896 minus 1) = 120593119879 (119896 minus 1) Δu (119896 minus 1) (34)
Substituting (34) into (33) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) ((120573119872 + 120573119863)120593119879 (119896 minus 1) Δu (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578Δu (119896 minus 1) ((120593 (119896 minus 1) minus (119896 minus 1))119879 Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578 (119896 minus 1) Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2) + 120593 (119896) minus 120593 (119896 minus 1)
(35)
10 Complexity
Because |120593(119896)| le 119887 is bounded [16] |120593(119896)minus120593(119896minus1)| le 2119887Taking the modulus of both sides of (35) we can obtain
1003817100381710038171003817 (119896)1003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2)100381710038171003817100381710038171003817100381710038171003817 + 2119887 (36)
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
= 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 + (minus2 + 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
times 1205781003817100381710038171003817 (119896 minus 1) Δu (119896 minus 1)10038171003817100381710038172120583 + Δu (119896 minus 1)2
(37)
Given that 0 lt 120578 lt 2 120583 gt 0(minus2 + 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2) lt 0 (38)
Synthesizing (37) and (38) we have
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
le 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 (39)
According to (36) and (39) there is obviously a positivenumber 1198891 isin (0 1) which makes
1003817100381710038171003817 (119896)1003817100381710038171003817 le 1198891 1003817100381710038171003817 (119896 minus 1)1003817100381710038171003817 + 2119887 le sdot sdot sdotle 1198891198961 1003817100381710038171003817 (0)1003817100381710038171003817 + 2119887 (1 minus 1198891198961)1 minus 1198891 (40)
Therefore (119896) is bounded(2)TheConvergence of Tracking Error 119890(119896) Tracking error 119890(119896)is derived as
119890 (119896) = 119910119889 minus 119910 (119896) = 119910119889 minus 119910 (119896 minus 1) minus 120593119879 (119896) Δu (119896)= 119890 (119896 minus 1) minus 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
(120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)= (1 minus 120588119863 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)
minus 120588119872 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 119890 (119896)
(41)
Transposition gives
119890 (119896) = [1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172] 119890 (119896 minus 1) (42)
So as long as appropriate parameters 120588119872 120588119863 and 120582 areselected there must be a positive number 1198892 isin (0 1) thatmakes
0 lt 10038171003817100381710038171003817100381710038171003817100381710038171 minus(120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172
1003817100381710038171003817100381710038171003817100381710038171003817 le 1198892 lt 1 (43)
By (42) and (43) it can be obtained that
119890 (119896) le 1003817100381710038171003817100381710038171003817100381710038171003817(1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)1003817100381710038171003817100381710038171003817100381710038171003817le 1198892 119890 (119896 minus 1) le sdot sdot sdot le 1198891198962 119890 (0)
(44)
Therefore lim119896rarrinfin119890(119896) = 0 that is 119890(119896) is proved toconverge to zero
(3)The Boundedness of System Input and Output By referringto 1198862 + 1198872 ge 2119886119887 the relational expression can be obtained asfollows
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
120582 + 1003817100381710038171003817120593 (119896)100381710038171003817100381721003817100381710038171003817100381710038171003817100381710038171003817 le
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
2radic120582 1003817100381710038171003817120593 (119896)10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 =
12radic120582 (45)
And we have
u (119896) = u (119896) minus u (119896 minus 1) + u (119896 minus 1) minus sdot sdot sdot + u (1)minus u (0) + u (0) = u (0) + 119896sum
119895=1
u (119895) (46)
Combining (24) (44) (45) and (46) as well as utilizingSchwartzrsquos inequality gives
u (119896) le u (0) + 119896sum119895=1
1003817100381710038171003817Δu (119895)1003817100381710038171003817
le u (0) + 12radic120582
119896sum119895=1
1003817100381710038171003817120588119863119890 (119895 minus 1) + 120588119872119890 (119895)1003817100381710038171003817
le u (0) + 1205881198632radic120582119896sum119895=1
119889119895minus12 119890 (0)
+ 1205881198722radic120582119896sum119895=1
1198891198952 119890 (0)
le u (0) + 1205881198632radic1205821 minus 119889119895minus121 minus 1198892 119890 (0)
+ 1205881198722radic1205821198892 (1 minus 1198891198952)1 minus 1198892 119890 (0)
le u (0) + 120588119863(1 minus 1198892) 2radic120582 119890 (0)
Complexity 11
Table 3 The parameter settings of DDMFAC for different IVs
IV Parameter120588119872 120588119863 120573119872 120573119863 120582 120578 1205831199100 = 60 00275 00275 05000 05000
00010 00750 100001199100 = 70 50119864 minus 4 50119864 minus 4 05000 050001199100 = 80 46119864 minus 4 46119864 minus 4 05000 050001199100 = 100 18119864 minus 4 18119864 minus 4 05000 05000
+ 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)le u (0) + 120588119863 + 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)
(47)
Because the initial tracking error 119890(0) and initial inputu(0) are given bounded the system input u(119896) is boundedaccording to (47) Besides due to the boundedness of 119910119889 and119890(119896) the system output 119910(119896) is obviously bounded43 Fuzzy DDMFAC The controller design and stability andconvergence analysis for DDMFAC have been respectivelydiscussed in detail in the above sections Although thedesigned controller has the characters of both MFAC andDDOILC weighted factors 120588119872 120588119863 120573119872 and 120573119863 are allconstant throughout control process DDMFAC is incapableof adaptively selecting the dominance of MFAC or DDOILCto duly show the expected features in each stage of controlprocessThe aforementioned features of DDMFACwe expectcannot be reflected in this case Such DDMFAC approachwith fixed weighting factors has no advantage comparedwith MFAC or DDOILC which appeals to a reasonable andadaptive adjustment strategy for 120588119872 120588119863 120573119872 and 120573119863 There-fore the adaptive adjustment strategy based on fuzzy logicfor weighting factors is considered because its advantageousperformance in parameter adjustment has been proven inour previous work [45] The formulation of fuzzy rules offersus the opportunity to adjust weighting factors in the waywe expected Such DDMFAC approach with fuzzy adaptiveadjustment for parameters is namedFDDMFAC in this paperConcretely the fuzzy rules can be formulated by referringto the expected features of DDMFAC The input variables offuzzy rules are the absolute values of tracking error 119890(119896 minus 1)and DLM prediction error 119890119901(119896minus1) of MFAC and the outputvariables are 120588119872 120588119863 and 120573119863 Because of the relationshipbetween 120573119872 and 120573119863 there is no need to use 120573119872 as an outputand it is calculated by 120573119872 = 1 minus 120573119863 The fuzzy rules can bedetailedly described as follows
(1) If |119890(119896 minus 1)| and |119890119901(119896 minus 1)| are great then thedominance of DDOILC is preferred and that ofMFAC is restrained in DDMFAC So 120588119863 and 120573119863 haverelatively large values and 120588119872 is a relatively smallnumber
(2) 120588119863 and 120573119863 will show a decreasing trend with thegradual decrease of |119890(119896 minus 1)| and |119890119901(119896 minus 1)| while120588119872 will present an increasing trend
44 Simulations and Result Analysis The proposed DDM-FAC approach is firstly applied to the particle quality controlin SFBGP and then its control effect is compared withMFACand DDOILC under different IVs In the simulation justlike MFAC 119910(119896) is replaced by a predicted value 119910(119896) thatis estimated using (19) See Table 3 for parameter settings ofDDMFAC It is not hard to find 120588119872 = 120588119863 and 120573119872 = 120573119863for a certain IV which means that MFAC and DDOILC areset to have the same dominance in DDMFAC By the wayMFAC and DDOILC can be regarded as the extreme cases ofDDMFAC described as follows
MFAC 120588119872 = 120573119872 = 1120588119863 = 120573119863 = 0
DDOILC 120588119872 = 120573119872 = 0120588119863 = 120573119863 = 1
(48)
The control effects of DDMFAC with the parametersettings in Table 3 together with control curves of MFACand DDOILC are plotted in Figure 5 The simulation resultsactually present the control effects of DDMFAC with threesets of parameter settings From simulation it is indicatedthat all three approaches have gained some certain advantagesunder different IV conditions Concretely as is shown inFigure 5(a) DDMFAC has advantage in control performancecompared with DDOILC and MFAC when 1199100 = 60 butMFAC has absolute superiority when 1199100 = 100 plottedin Figure 5(d) In addition from the simulation results inFigures 5(b) and 5(c) it can be seen that DDOILC andMFAC have achieved certain but not absolute advantagesrespectively This demonstrates that parameter setting hasimportant influence on the control effect of DDMFACTherefore the way to achieve satisfactory control effect is toadaptively adjust weighted parameters of DDMFAC insteadof keeping them constant
Then the idea of fuzzy adjustment is introduced to adjustthe weights 120588119872 120588119863 and 120573119863 The fuzzy adjustment rules ofthree weights are respectively shown in Tables 4 and 5 Therespective membership functions when 1199100 = 60 are shownin Figure 6 and membership functions for the other IVs areno longer displayed Please see Figure 7 for control effectof FDDMFAC and comparisons with MFAC and DDOILCunder different IVs Obviously DDMFAC approach withfuzzy adjustment for parameters that is FDDMFAC has thebest performance in particle quality control for all IVs This
12 Complexity
Table 4 The fuzzy rules of 120588119872120588119872 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB PB PB PB PB PB PB PBNM PB PM PM PM PM PS PSNS PM PM PM PS PS Z ZZ Z Z Z NS NS NS NSPS NS NS NM NM NB NB NBPM NM NM NM NB NB NB NBPB NB NB NB NB NB NB NB
MFACDDOILCDDMFAC
60
80
100
120
APS
(m
)
5 10 150Iterations
(a) 1199100 = 60
MFACDDOILCDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILCDDMFAC
80
90
100
110
120
APS
(m
)
2 4 60Iterations
(c) 1199100 = 80
95
100
105
110
115
120
125
APS
(m
)
5 10 150Iterations
MFACDDOILCDDMFAC
(d) 1199100 = 100
Figure 5 The comparison of control effects of MFAC DDOILC and DDMFAC
Complexity 13
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
02 04 06 08 10D
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
005 01 0150D
NB NM NS Z PS PM PB
times10minus3
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
05 1 15 2 25 3 35 4 450M
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
10 20 30 40 50 600|e(k minus 1)| (m)
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
5 10 15 20 250|ep (k minus 1)| (m)
Figure 6 The membership functions of FDDMFAC when 1199100 = 60
Table 5 The fuzzy rules of 120588119863 120573119863120588119863 120573119863 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB NB NB NB NB NB NB NBNM NB NM NM NM NM NS NSNS NM NM NM NS NS Z ZZ Z Z Z PS PM PM PMPS PS PS PM PM PB PB PBPM PM PM PM PB PB PB PBPB PB PB PB PB PB PB PB
validates the effectiveness of FDDMFAC proposed in thispaper
Although FDDMFAC approach has been proven to havea better control performance than MFAC and DDOILCwe still have a question of whether it is possible that thesuperiority of FDDMFAC is entirely attributed to fuzzyadjustment of parameters and has nothing to do with thestructure of DDMFAC itself In order to clarify this problemanother set of simulations are conducted inwhich the controlperformance of FDDMFAC is compared with fuzzy MFAC(FMFAC) and fuzzy DDOILC (FDDOILC) The absolutevalue of tracking error |119890(119896 minus 1)| is regarded as the inputvariable of fuzzy rules and the step-size factors 1205881 1205882 ofMFAC andDDOILC are the output variablesThe fuzzy rulesof FMFAC and FDDOILC are shown in Table 6 and themembership functions when 1199100 = 60 are shown in Figure 8For the simulation results please see Figure 9 From thesimulation it is concluded that FDDMFAC achieves the best
Table 6 The fuzzy rules of FMFAC and FDDOILC
|119890(119896 minus 1)| NB NM NS Z PS PM PB1205881 1205882 NB NM NS Z PS PM PB
control performance which demonstrates the effectiveness ofthe proposed approach in this paper
5 Conclusions
This paper aimed at solving the problem of particle qualitycontrol in the drug development phase of SFBGP Themodel-based control approaches are incapable of handlingsuch quality control because the accurate process modelcannot be acquired Thus the data-driven and model-freecontrol approaches are considered in this work We firstly
14 Complexity
MFACDDOILCFDDMFAC
5 10 150Iterations
60
80
100
120
APS
(m
)y0 = 60
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
y0 = 70
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
MFACDDOILCFDDMFAC
95
100
105
110
115
120
125A
PS (
m)
5 10 150Iterations
y0 = 100
Figure 7 The comparison of control effects of MFAC DDOILC and FDDMFAC
analyzed the features of MFAC and DDOILC and thenconducted experimental research to compare the perfor-mances of MFAC and DDOILC in particle quality controlunder different IVs The simulation results have verifiedour theoretical analysis and concluded that MFAC andDDOILC both have advantages under different IVs To beused in an actual production environment the expectedDDCapproaches must accommodate various IVs because of thechanges in material attributes Thus the hybrid approachFDDMFAC which combines the advantages of MFAC andDDOILC by adaptively adjusting their respective weightedfactors with fuzzy logic was proposed in this work tocomplement their respective shortcomings Through a series
of simulation results we conclude that FDDMFAC achievesbetter control performance than other approaches whichvalidates the effectiveness of FDDMFAC
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thanks are due to National Natural Science Foundation ofChina under Grant nos 61773105 61374147 61703085 and61004083 and Fundamental Research Funds for the Central
Complexity 15NB NM NS Z PS PM PB
001 002 00302
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
002 00401
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
20 40 600|e(k minus 1)| (m)
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
Figure 8 The membership functions of FDDMFAC when 1199100 = 60
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
60
80
100
120
APS
(m
)
y0 = 60
FMFACFDDOILCFDDMFAC
70
80
90
100
110
120
APS
(m
)
2 4 6 8 100Iterations
y0 = 70
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
100
105
110
115
120
125
APS
(m
)
y0 = 100
Figure 9 The comparison of control effects of FMFAC FDDOILC and FDDMFAC
16 Complexity
Universities under Grant nos N150402001 and N120404014for supporting this research work
References
[1] J Bouffard M Kaster and H Dumont ldquoInfluence of processvariable and physicochemical properties on the granulationmechanism of mannitol in a fluid bed top spray granulatorrdquoDrug Development and Industrial Pharmacy vol 31 no 9 pp923ndash933 2005
[2] H Liu K Wang W Schlindwein and M Li ldquoUsing the Box-Behnken experimental design to optimise operating parametersin pulsed spray fluidised bed granulationrdquo International Journalof Pharmaceutics vol 448 no 2 pp 329ndash338 2013
[3] H Liu and M Li ldquoPopulation balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granula-tionrdquo International Journal of Pharmaceutics vol 468 no 1-2pp 223ndash233 2014
[4] T Albuquerque V H Dias N Poellinger and J F PintoldquoConstruction of a quality index for granules produced byfluidized bed technology and application of the correspondenceanalysis as a discriminant procedurerdquo European Journal ofPharmaceutics and Biopharmaceutics vol 75 no 3 pp 418ndash4242010
[5] H Ehlers A Liu H Raikkonen et al ldquoGranule size control andtargeting in pulsed spray fluid bed granulationrdquo InternationalJournal of Pharmaceutics vol 377 no 1-2 pp 9ndash15 2009
[6] C F W Sanders M J Hounslow and F J Doyle III ldquoIden-tification of models for control of wet granulationrdquo PowderTechnology vol 188 no 3 pp 255ndash263 2009
[7] T Narvanen T Lipsanen O Antikainen H Raikkonen andJ Yliruusi ldquoControlling granule size by granulation liquid feedpulsingrdquo International Journal of Pharmaceutics vol 357 no 1-2pp 132ndash138 2008
[8] A Faure P York and R C Rowe ldquoProcess control and scale-up of pharmaceutical wet granulation processes a reviewrdquoEuropean Journal of Pharmaceutics and Biopharmaceutics vol52 no 3 pp 269ndash277 2001
[9] M Pottmann B A Ogunnaike A A Adetayo and B JEnnis ldquoModel-based control of a granulation systemrdquo PowderTechnology vol 108 no 2-3 pp 192ndash201 2000
[10] E P Gatzke and F J Doyle III ldquoModel predictive control of agranulation system using soft output constraints and prioritizedcontrol objectivesrdquo Powder Technology vol 121 no 2-3 pp 149ndash158 2001
[11] F Y Wang and I T Cameron ldquoReview and future directionsin the modelling and control of continuous drum granulationrdquoPowder Technology vol 124 no 3 pp 238ndash253 2002
[12] T Glaser C F W Sanders F Y Wang et al ldquoModel predictivecontrol of continuous drum granulationrdquo Journal of ProcessControl vol 19 no 4 pp 615ndash622 2009
[13] R Ramachandran andA Chaudhury ldquoModel-based design andcontrol of a continuous drum granulation processrdquo ChemicalEngineering Research and Design vol 90 no 8 pp 1063ndash10732012
[14] Z H Loh D Z L Er L W Chan C V Liew and PW S HengldquoSpray granulation for drug formulationrdquo Expert Opinion onDrug Delivery vol 8 no 12 pp 1645ndash1661 2011
[15] S H Schaafsma N W F Kossen M T Mos L Blauw andA C Hoffmann ldquoEffects and control of humidity and particlemixing in fluid-bed granulationrdquo AIChE Journal vol 45 no 6pp 1202ndash1210 1999
[16] Z-S Hou and S-T Jin ldquoA novel data-driven control approachfor a class of discrete-time nonlinear systemsrdquo IEEE Transac-tions onControl Systems Technology vol 19 no 6 pp 1549ndash15582011
[17] Z Hou and S Jin ldquoData-driven model-free adaptive controlfor a class of MIMO nonlinear discrete-time systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 22no 12 pp 2173ndash2188 2011
[18] D Xu B Jiang and P Shi ldquoA novel model-free adaptivecontrol design for multivariable industrial processesrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6391ndash6398 2014
[19] X Wang X Li J Wang X Fang and X Zhu ldquoData-drivenmodel-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeletonrdquo Information Sciences vol 327pp 246ndash257 2016
[20] L dos Santos Coelho andA A R Coelho ldquoModel-free adaptivecontrol optimization using a chaotic particle swarm approachrdquoChaos Solitons amp Fractals vol 41 no 4 pp 2001ndash2009 2009
[21] L D S Coelho M W Pessoa R Rodrigues Sumar andA Augusto Rodrigues Coelho ldquoModel-free adaptive controldesign using evolutionary-neural compensatorrdquo Expert Systemswith Applications vol 37 no 1 pp 499ndash508 2010
[22] K K Tan T H Lee S N Huang and F M Leu ldquoAdaptive-predictive control of a class of SISOnonlinear systemsrdquoDynam-ics and Control vol 11 no 2 pp 151ndash174 2001
[23] B Zhang andW Zhang ldquoAdaptive predictive functional controlof a class of nonlinear systemsrdquo ISA Transactions vol 45 no2 pp 175ndash183 2006
[24] F L Lu J F Wang C J Fan and S B Chen ldquoAn improvedmodel-free adaptive control with G function fuzzy reasoningregulation design and its applicationsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems andControl Engineering vol 222 no 8 pp 817ndash828 2008
[25] R Chi D Wang Z Hou and S Jin ldquoData-driven optimalterminal iterative learning controlrdquo Journal of Process Controlvol 22 no 10 pp 2026ndash2037 2012
[26] R Chi Z Hou S Jin and D Wang ldquoImproved data-drivenoptimal TILC using time-varying input signalsrdquo Journal ofProcess Control vol 24 no 12 pp 78ndash85 2014
[27] X Bu Z Hou and R Chi ldquoModel free adaptive iterative learn-ing control for farm vehicle path trackingrdquo IFAC ProceedingsVolumes vol 46 no 20 pp 153ndash158 2013
[28] M C Campi A Lecchini and SM Savaresi ldquoVirtual referencefeedback tuning a direct method for the design of feedbackcontrollersrdquo Automatica vol 38 no 8 pp 1337ndash1346 2002
[29] R-C Roman M-B Radac R-E Precup and E M PetriuldquoData-driven model-free adaptive control tuned by virtualreference feedback tuningrdquoActa PolytechnicaHungarica vol 13no 1 pp 83ndash96 2016
[30] P Yan D Liu D Wang and H Ma ldquoData-driven controllerdesign for general MIMO nonlinear systems via virtual refer-ence feedback tuning and neural networksrdquo Neurocomputingvol 171 pp 815ndash825 2016
[31] T Pan and S Li ldquoA hierarchical search and updating databasestrategy for lazy learningrdquo International Journal of InnovativeComputing Information andControl vol 4 no 6 pp 1383ndash13922008
[32] H Zhang R Song Q Wei and T Zhang ldquoOptimal trackingcontrol for a class of nonlinear discrete-time systems withtime delays based on heuristic dynamic programmingrdquo IEEE
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Complexity
process models If traditional MBC schemes are used lots ofexperiments in the actual process should first be implementedas long as the prescription is changed to get enough processdata used for developing precisemathematicalmodels Such away is laborious time-consuming and resource-wasting sothat the MBC techniques are no more applicable to qualitycontrol task in this work
Data-driven control (DDC) [16 17] means that only theinputoutput (IO) measurement data of controlled plant areused in controller design DDC approaches do not require amodel of a plant and the modeling process the unmodeleddynamics and the theoretical assumptions all disappear [16ndash19] Therefore DDC has attracted considerable attentionin recent years [18] and there are many DDC approachestogether with their practical applications in many fields thatcould be found in the literature like the following model-free adaptive control (MFAC) [16 17 20ndash24] data-drivenoptimal iterative learning control (DDOILC) [25ndash27] virtualreference feedback tuning [28ndash30] lazy learning control [31]dynamic programming methods [32] and others [33ndash36] Inspite of this to the best of authorsrsquo knowledge there is noreport about the research on application of DDC to SFBGPthat has been published in the literature In this work a studyon particle quality control based on DDC approaches in thedrug development phase of a SFBGP is conducted to resolvethe practical difficulty encountered in redesigning operatingcondition when prescription and material attributes are allchanged
The mechanism model of SFBGP should first be intro-duced to conduct such quality control research Such aprocess model plays several significant roles first modelinganalyzes the mechanism of SFBGP identifies the importantmanipulated variables and quality indices and establishesthe relationship between them Second such a model can beused as a simulator of a real SFBGP to generate the requiredprocess data used for simulation and analysis In this worka widely accepted mechanistic model for SFBGP [37 38] isintroduced to simulate the actual SFBGP and produce therequired process data
In this work we select two classical and representativeDDC approaches MFAC and DDOILC to study data-drivenmodel-free adaptive control (DDMFAC) of average particlesize (APS) for SFBGP with simulation experiment researchMFAC algorithm is proposed by Hou and Jin [16 17] basedon a new dynamic linearization technique (DLT) and thenapplied in several areas [18] The main feature of MFACis that the controller design depends merely on the IOmeasurement data of the controlled plant Instead of iden-tifying a nonlinear process model of a plant an equivalentlocal dynamical linearization model is constructed alongthe dynamic operation points of the system using the DLTwith a novel concept called pseudo-partial derivative (PPD)The time-varying PPD could be estimated merely using theIO measurement data of the controlled plant DDOILC isdeveloped and applied to both linear and nonlinear systemsby Chi et al [25] For this approach the only requiredknowledge of a controlled system is that theMarkovmatricesof linear systems or the partial derivatives of nonlinearsystems with respect to control inputs are bounded DDOILC
is actually the extension of MFAC and these two approacheshave the same systematic framework Compared with otherDDC methods MFAC and DDOILC have several attractiveadvantages that make them more suitable for many practicalcontrol applications [16] First they do not require anyprocess model and structural information of the controlledplant and merely depend on the real time measurement IOdata of the controlled plant which indicates that it is feasibleto independently design a generic controller for a certain classof practical industrial processes Second they are lower costcontrollers because they do not need any training processThird they are simple and easily implemented with smallcomputational burden and have strong robustness Finallythey have been successfully implemented in many practicalfields such as chemical industry [20 21 25 26] linearmotor control and injectionmoulding process [22] PH valuecontrol [23] and robotic welding process [24]
The controllers of MFAC and DDOILC all consist ofa control input iterative learning law and a PPD matrixiterative updating law From this respect the prime differencebetween them is whether or not to use the predicted dataof system output of current iteration in controller designIn MFAC the predicted system output of current iterationis utilized but merely the output measurement data ofprevious iterations are used in DDOILC which makes theirdifferent characteristicsThrough theory analysis MFAC andDDOILC both have advantages with regard to different initialvalues (IVs) for particle quality When IV is far away fromdesired particle quality that is the initial tracking error islarger DDOILC has quicker convergence speed The reasonfor this is that the utilization of actual output of previousiterations in controller design of DDOILC will give largerupdating step size for control input and PPD vector Howeveras IV gets closer to desired outputMFACwill gradually showits advantages since the prediction accuracy of its dynamiclinearization model (DLM) is getting higher and higher Toverify the conclusion obtained through theoretical analysiswe conducted several comparative experimental studies byconstructing a series of IVs and the simulation resultsvalidate our conclusion
In the drug development phase the frequent variationsof prescription will lead to diversity of material propertiesFor different material properties the same group of empiricaloperating conditionswill give entirely different IVs of particlequality It is demanded that the DDC approaches mustaccommodate various IVs in the practice Therefore tofurther improve the overall control effect a novel hybridDDC approach which is namedDDMFAC and combines theadvantages ofMFAC andDDOILC is presented in this paperThe newly designed controller is able to exhibit the featuresof MFAC and DDOILC in different extent by adjusting theirweighted factors Compared withMFAC andDDOILC how-ever evenDDMFACdoes not have overwhelming superiorityfor all IVs Simulation results have revealed that DDMFACmay be inferior to DDOILC or MFAC in particle qualitycontrol if there is no reasonable adjustment strategy ofweighted parameters By the previous theoretical analysisfor MFAC and DDOILC and the subsequent simulationverification we can sum up weighted parameters adjustment
Complexity 3
FI Compressed air unit
Liquid supply unit
Exhaust blower
BlowerFilterHeater
Dust-arresterinstallation
Spray nozzle
Fluidizing agent
Figure 1 The schematic diagram of SFBGP
rules for DDMFACThen fuzzy adaptive adjustment of targetweighted parameters is implemented with the aim of rapidlyconverging DDMFAC by making full use of the advantageof MFAC or DDOILC at different control stages We namedthe final DDC approach presented in this paper as fuzzyDDMFAC (FDDMFAC) and a series of simulations validateits effectiveness
2 Preliminaries
21 Mathematical Mechanism Model of Spray Fluidized-BedGranulation Process SFBGP is a commonly used unit oper-ation in the pharmaceutical industry The powders can bemixed granulated and dried in the same equipment whichminimizes the equipment costs loss of product transfers andpossibility of the cross-contamination [3] The rationale ofspray fluidized-bed granulation can be briefly described asfollows (1) Particles circulate within granulator by pumpinghot air from the bottom of granulator and meanwhile binderliquid is sprayed on the fluidized particles in the forms ofsmall droplets [38] (2) As a result of collisions and coales-cence between the surface-wetted particles liquid bridges areformed and aggregation of particles occurs leading to thegrowth of granules [1 39]
The schematic diagram of SFBGP is shown in Figure 1The overall granulation process in fluidized-bed spray gran-ulator is divided into three stages powder mixing granula-tion and drying Firstly mixed raw materials for drug andingredients are put in granulator Passing through the filterand heater hot air pumped in by blower is pumped intogranulator from the bottom of granulator which forces thefluidization of powder to sufficiently blend materials After-wards liquid binder is pumped into spray nozzle from liquidsupply unit and atomized into droplets by compressed air thatis simultaneously pumped into spray nozzle from compressedair unit The droplets are dispersed over the surface offluidized particles which contributes to the agglomerationof colliding and surface-wetted powder particles to formgranules Finally the granules are dried to a predeterminedmoisture content level The dust-arrester installation is usedto prevent powder particles that are not in contact withdroplets from being carried out by off-gas
PBMs have been widely used for modeling SFBGP inwhich the granule size density distribution evolves as afunction of timeThe population balance is a number balancearound each size fraction of the size distribution based onnumber conservation law The rate of change of the numberof particles in a given size interval is equal to the rate at which
4 Complexity
the granules enter and leave that size interval The mathe-matical mechanism model of SFBGP is developed based onfollowing assumptions (1) Each particle is spherical (2)Thetotal volume of the particles is conserved (3) The particlesize in a certain particular size interval is represented by theleft boundary of the interval (4) The relative growth rate isuniform for each particle within the same volume intervalA one-dimensional PBM that describes the rate of change inparticle number density function 119899(119905 119897) is given by
120597119899 (119905 119897)120597119905 = 12sdot int1198970
120573(119905 (1198973 minus 1205833)13 120583) 119899 (119905 (1198973 minus 1205833)13) 119899 (119905 120583)(1198973 minus 1205833)23 119889120583
minus 119899 (119905 119897) intinfin0
120573 (119905 119897 120583) 119899 (119905 120583) 119889120583
(1)
where 119899(119905 119897) is the particle number density function in termsof particle diameter 119897 and120573 is the aggregationmodel [37 38]120573(119905 119897 120583) can generally be partitioned into size dependent andsize independent parts [3 37 38 40 41]
120573 (119905 119897 119906) = 1205730 (119905) 120573lowast (119897 120583) (2)
where 1205730(119905) is the granulation rate constant that can bethought of as an aggregation efficiency and depends onall parameters involved in the aggregation process exceptparticle size such as the operating condition and binderproperties The latter term 120573lowast(119897 120583) reflects the influence ofparticle size on the likelihood of aggregation
In order to solve a PBM in (1) the discretized approach[42 43] is applied to determine the change in number ofparticles119873119894 in interval 119894 as
d119873119894d119905 = 119894minus2sum
119895=1
2119895minus119894+1120573119894minus1119895119873119894minus1119873119895 + 12120573119894minus1119894minus11198732119894minus1
minus 119873119894 119894minus1sum119895=1
2119895minus119894120573119894119895119873119895 minus 119873119894119899maxsum119895=119894
120573119894119895119873119895(3)
where119873119894 is the number of particles within particle size range(119871 119894 119871 119894+1) 119871 119894 and 119871 119894+1 are the lower and upper limits of 119894thparticle size interval 119894 = 1 2 119899max and 119899max is the totalnumber of particle size intervals with a value of 12 in thispaper In the discretization scheme the length domain ofparticles is divided into geometric intervals in the way thatthe upper and lower limits of each size interval are in a ratioof 119903 = 119871 119894+1119871 119894 = 3radic2 The size range used in modeling isfrom 50 120583m to 800120583mwhich is divided into 12 intervals Thefunction ldquoode45rdquo inMATLAB is used to solve (3) to obtain thefinal number of particles in each size interval Then the APSof final granules119863119898 is calculated by the following formula
119863119898 = 119899maxsum119894=1
119881(119873119891119894) 119889119901119894 (4)
where 119881(119873119891119894) is the volume fraction of end granules at sizeinterval 119894 = 1 2 119899max119873119891119894 is the number of end granulesin the 119894th size interval and 119889119901119894 is the geometric mean of lowerlimit and upper limit of 119894th size interval 119894 = 1 2 119899max
We have just briefly introduced the general framework ofPBM-basedmechanismmodel of SFBGPDetailed derivationprocess and setting of model parameters are given in [37 38]and they will not be covered hereThe developed mechanismmodel is capable of not only linking APS of particles withthe operating variable (binder spray rate) but also reflectingthe relationship between APS and material attributes suchas viscosity of binder and particle density Hence thismechanism model can be used for simulating the practicalchallenges in particle quality control of SFBGP For examplea series of IVs can be constructed by changing the materialattributes while keeping the operating condition unchangedTo veritably simulate the actual granulation process the inputcontrol curve should be time-varying But to simplify thingsthe input control curve is divided into three stages in thewhole granulation process in this work
22 The Data-Driven Control Approaches Used in This WorkAs the classic and representative DDC methods MFAC andDDOILChave somemerits thatmake themmore competitivein practical control applications Therefore in view of thesemerits MFAC and DDOILC are selected and applied toperform data-driven particle quality control for SFBGP inthis work Now we first briefly introduce their controllerdesign procedures For details please refer to [16 25]
A repeatable batch process system to be controlled isgiven as follows
119910 (119896) = 119891 (u (119896)) (5)
where 119910(119896) and u(119896) are the system output and input at 119896thiteration respectively 119896 is the number of iterations and 119891(∙)is an unknown nonlinear functionThis nonlinear system (5)satisfies the following assumptions
(1) The derivatives of 119891(∙) with respect to control inputu(119896) are continuous
(2) System (5) is generalized Lipschitz that is |Δ119910(119896 +1)| le 119887Δu(119896) for any 119896 and Δu(119896) = 0 whereΔ119910(119896 + 1) = 119910(119896 + 1) minus 119910(119896) Δu(119896) = u(119896) minus u(119896 minus 1)and 119887 is a positive constant
221 MFAC Design For the one-step-ahead controller [44]the control goal is to seek a control input sequence u(119896) thatbrings 119910(119896 + 1) to 119910119889 where 119910119889 is the desired system outputsignal In general the weighted one-step-ahead controllermay lead to steady-state tracking error [16] So the followingcontrol input criterion function is used to design the controllaw
119869 (u (119896)) = 1003816100381610038161003816119910119889 minus 1199101 (119896 + 1)10038161003816100381610038162 + 120582 Δu (119896)2 (6)
where 120582 gt 0 is a weighting constantSubstituting the dynamic linearization form
1199101 (119896 + 1) = 119910 (119896) + 120593119879 (119896) Δu (119896) (7)
Complexity 5
into (6) as well as letting 120597119869120597u(119896) be zero givesu (119896) = u (119896 minus 1) + 1205881120593 (119896) (119910119889 minus 119910 (119896))
120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (8)
where 1205881 isin (0 1] is a step-size constant that is added to make(8) more general and 120593(119896) is PPD vector
Because PPD vector 120593(119896) is unknown it needs to beestimated Define criterion function for the unknown PPDvector as follows
119869 ( (119896)) = 10038161003816100381610038161003816Δ119910 (119896) minus 119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(9)
where 120583 gt 0 is a weighting factor and (119896) is the estimationof 120593(119896)
Using optimal condition 120597119869120597(119896) = 0 then gives
(119896)= (119896 minus 1)
+ 120578Δu (119896 minus 1) (Δ119910 (119896) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2
(10)
where 120578 isin (0 2) is a step-size constantDefining the tracking error 119890(119896) = 119910119889 minus 119910(119896) and
combining PPD estimation and input control law MFACscheme is designed as follows
(119896)= (119896 minus 1)
+ 120578Δu (119896 minus 1) (Δ119910 (119896) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2
u (119896) = u (119896 minus 1) + 1205881 (119896)120582 + 1003817100381710038171003817 (119896)10038171003817100381710038172 119890 (119896)
(11)
222 DDOILC Design The control objective is to find anappropriate control input sequence u(119896) such that the systemoutput 119910(119896) follows 119910119889 Consider the control input criterionfunction as
119869 (u (119896)) = 1003816100381610038161003816119910119889 minus 1199102 (119896)10038161003816100381610038162 + 120582 Δu (119896)2 (12)
Substituting the dynamic linearization equation
1199102 (119896) = 119910 (119896 minus 1) + 120593119879 (119896) Δu (119896) (13)
into (12) and the optimal condition leads to
u (119896) = u (119896 minus 1) + 1205882120593 (119896)120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 119890 (119896 minus 1) (14)
where 1205882 isin (0 1] determines the step size and 119890(119896 minus 1) =119910119889 minus 119910(119896 minus 1)
Let (119896) denote its estimation at 119896th iterationThendefinethe criterion function as
119869 ( (119896)) = 10038161003816100381610038161003816Δ119910 (119896 minus 1) minus 119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(15)
According to optimal condition the iterative updating law ofestimate (119896) is developed as
(119896) = (119896 minus 1)+ 120578Δu (119896 minus 1) (Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))
120583 + Δu (119896 minus 1)2 (16)
where 120583 gt 0 0 lt 120578 lt 2The DDOILC scheme is designed as follows
(119896) = (119896 minus 1)+ 120578Δu (119896 minus 1) (Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))
120583 + Δu (119896 minus 1)2 u (119896) = u (119896 minus 1) + 1205882 (119896)120582 + 1003817100381710038171003817 (119896)10038171003817100381710038172 119890 (119896 minus 1)
(17)
3 Theoretical Analysis of MFAC and DDOILCand Simulations in SFBGP
31 Theoretical Analysis of MFAC and DDOILC The con-troller design procedures of MFAC and DDOILC have beenintroduced in previous sections Although they have the samesystemic framework and similar expression form of con-troller there are still differences between them which makesthem two different methods Comparing the controllers ofMFAC and DDOILC the main differences can be listed asfollows
(1) Δ119910(119896) and Δ119910(119896 minus 1) are respectively used in PPDiterative updating laws
(2) 119890(119896) and 119890(119896minus1) are separately utilized in control inputiterative learning laws
The expressions of Δ119910(119896) Δ119910(119896minus1) 119890(119896) and 119890(119896minus1) arelisted as follows
MFACΔ119910 (119896) = 119910 (119896) minus 119910 (119896 minus 1)119890 (119896) = 119910119889 minus 119910 (119896)
DDOILCΔ119910 (119896 minus 1) = 119910 (119896 minus 1) minus 119910 (119896 minus 2)119890 (119896 minus 1) = 119910119889 minus 119910 (119896 minus 1)
(18)
Obviously the main differences between MFAC andDDOILC can be described as follows system output 119910(119896) at119896th iteration is used in MFAC but is not used in DDOILC Inpractical application all the system outputs used inDDOILC119910(119896minus1) and 119910(119896minus2) are the actual values at (119896minus1)th and (119896minus2)th iterations In MFAC however the actual value of system
6 Complexity
y
yd
y (k + 1)y(k)
y(k)y (k minus 1)
y (k minus 1)
y(k minus 2)
IV = y0
u(k minus 2) u(k minus 1) u(k) u(k + 1) u(k + 2) Input
e (k)
e (k minus 1)ep (k)
ep (k minus 1)
Δy (k)
Δy(k minus 1)
Tracking errorPredicted error of DLM
Δy
Figure 2 The variable relations during control process of MFAC and DDOILC
output 119910(119896) is obtained only after the calculation of u(119896) So119910(119896) does not have actual value when computing u(119896) anda predicted system output 119910(119896) will be used to replace 119910(119896)in practical application of MFAC 119910(119896) is estimated by thefollowing DLM
119910 (119896) = 119910 (119896 minus 1) + 120593119879 (119896 minus 1) Δu (119896 minus 1) (19)
and 119890(119896) is replaced by 119890(119896) = 119910119889 minus 119910(119896)Figure 2 gives the variable relations during control pro-
cess of MFAC and DDOILC In terms of a single controlprocess curve when system output 119910 is greatly different from119910119889 that is the tracking error 119890 is relatively great 119890(119896 minus 1)is much larger than 119890(119896) and DDOILC will achieve a biggerupdating step size for control input which gives DDOILC afaster convergence speed As 119910 keeps getting closer to 119910119889 itis found that DLM prediction error 119890119901 is getting smaller andsmaller until 119890119901 = 0 The difference between 119890(119896 minus 1) and 119890(119896)will also be smaller and smaller DDOILC will not only loseits advantage but also perform worse than MFAC when 119910 isvery close to119910119889 because a larger updating step size for controlinput will bring difficulty in convergence while at this timeMFAC will achieve a fast and stable control effect due to theimprovement of prediction accuracy of DLM From anotherpoint of view if IV of system output 1199100 is far away from 119910119889the initial tracking error and DLM prediction error will beboth at a high level and DDOILC will perform better thanMFAC because a greater updating step size for control inputis necessary to give a faster convergence speed But as1199100 keepsgetting closer to 119910119889 119890119901 will gradually decrease and MFACwill gradually show its advantage Therefore the conclusions
obtained from theoretical analysis can be summarized asfollows
(1) When tracking error is large and 1199100 is relatively faraway from119910119889 DDOILCwill have a faster convergencespeed and a better control performance
(2) When tracking error gradually decreases and 1199100gradually gets closer to 119910119889 MFAC will graduallyreveal its advantage and achieve better control effect
32 SimulationVerification To validate conclusions obtainedfrom theoretical analysis simulation verification in SFBGP isconducted In this simulation particle quality to be controlledis APS and the desired quality is 119910119889 = 120 120583mThe viscosityof binder is treated as material attribute to be changed andthen a series of IVs (1199100) are constructed by adjusting viscosityof binder while keeping the operating condition unchangedFour sets of IVs are selected as representatives in this workand the detailed establishing information of them is shownin Table 1 Besides the parameter settings of MFAC andDDOILC for different IVs are listed in Table 2
For different IVs we conduct particle quality control inSFBGP based on MFAC and DDOILC and the control per-formances are plotted in Figure 3Moreover DLMpredictionerror profiles of MFAC are shown in Figure 4 Then theexplanation and analysis of simulation results will be givenAs is shown in Figure 3(a) DDOILC performs better thanMFAC when 1199100 = 60 that is 1199100 is relatively far away from119910119889 When 1199100 = 70 as is described in Figure 3(b) DDOILChas a faster convergence speed when APS is controlled from70 120583m to about 110 120583m but MFAC achieves a relatively better
Complexity 7
MFACDDOILC
5 10 150Iterations
60
80
100
120A
PS (
m)
(a) 1199100 = 60
MFACDDOILC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
(c) 1199100 = 80
MFACDDOILC
5 10 150Iterations
95
100
105
110
115
120
125
APS
(m
)
(d) 1199100 = 100
Figure 3 The control profiles of APS in SFBGP based on MFAC and DDOILC
Table 1The detailed establishing information of representative IVs
Info IV1199100 = 60 1199100 = 70 1199100 = 80 1199100 = 100
Operatingconditions(gs)
First stage 03 second stage 01 third stage 015
Viscosity ofbinder (Pasdots) 00214 00250 00304 00625
control performance when it is controlled from 110 120583m to 119910119889which indicates that both DDOILC and MFAC have certainadvantages in such case However from Figure 3(c) it can beseen that MFAC is slightly superior to DDOILC in controleffect when 1199100 = 80 and this just manifests the advantage ofMFAC As 1199100 gets closer and closer to 119910119889 such as 1199100 = 100
Table 2The parameter settings ofMFAC andDDOILC for differentIVs
IV Parameter1205881 1205882 120582 120578 1205831199100 = 60 00505 00712
00010 00750 100001199100 = 70 00047 005031199100 = 80 00032 001991199100 = 100 00020 149119864 minus 4
plotted in Figure 3(d)MFAC has a huge advantage comparedwith DDOILC It is obvious that the simulation results haveverified the conclusions obtained from theoretical analysisThe reason for increasing superiority of MFAC is attributedto the smaller and smaller prediction error 119890119901 of DLM Asis shown in Figure 4 on the one hand the curves of 119890119901 for
8 Complexity
IV = 60IV = 70
IV = 80IV = 100
IV = 60 14320IV = 70 08061IV = 80 04718IV = 100 04000
5 10 150Iterations
0
5
10
15
20
25
e p(
m)
Average of ep
Figure 4 The DLM prediction error profiles of MFAC for differentIVs
different IVs all show a decreasing trend along with iterationuntil 119890119901 = 0 as APS keeps getting closer to 119910119889 which is inaccordance with our obtained conclusion On the other handfor different IVs 119890119901 will also decrease with the increase of IVwhich is proved by comparing the curves of 119890119901 under differentIVs To further illustrate this point the averages of 119890119901 fordifferent IVs are computed and then listed in Figure 4 andthe decreasing averages of 119890119901 along with the increase of IValso validate the rationality of our analysis
4 The Proposed Hybrid DDC Approach andSimulation Study in SFBGP
Through theoretical analysis and simulation study we haveproved that MFAC and DDOILC both have strengths inparticle quality control under different IVs In real-worldapplications the same operating condition may give entirelydifferent IVs due to the changes in material properties whichbrings great difficulties in the selection of control methodsThus the uncertainties of IV have to be considered duringcontrol application In order to achieve favourable overallcontrol effect it is demanded that the DDC approaches mustaccommodate various IVs Therefore we try to combineadvantages of MFAC and DDOILC to present a hybridapproach that is named DDMFAC The basic idea is that thefeatures of MFAC and DDOILC are both reflected in thecriterion functions when designing controller of DDMFACThe expected features of hybrid approach can be summarizedas the following three points
(1) When tracking error is large DDMFACmanifests thefeature of DDOILC to acquire a faster convergencespeed
(2) When tracking error is small enough DDMFAC hasthe ability ofMFAC to achieve a fast and stable controleffect
(3) DDMFAC is able to achieve better control perfor-mance for different IVs
Through such combination the shortcomings of MFACand DDOILC will be complemented and their respec-tive advantages will be highlighted in DDMFAC Next wedescribe the controller design for DDMFAC in Section 41
41 DDMFAC Design Consider the following control inputcriterion function
119869 (u (119896)) = 120572119872 1003816100381610038161003816119910119889 minus 1199101 (119896 + 1)10038161003816100381610038162 + 120572119863 1003816100381610038161003816119910119889 minus 1199102 (119896)10038161003816100381610038162+ 1205821 Δu (119896)2 (20)
where |119910119889 minus 1199101(119896 + 1)|2 and |119910119889 minus 1199102(119896)|2 respectively comefrom MFAC and DDOILC and 120572119872 120572119863 are the respectiveweighted factors and 120572119872 + 120572119863 = 0
Substituting (7) and (13) into (20) gives
119869 (u (119896)) = 120572119872 10038171003817100381710038171003817119890 (119896) minus 120593119879 (119896) Δu (119896)100381710038171003817100381710038172+ 120572119863 10038171003817100381710038171003817119890 (119896 minus 1) minus 120593119879 (119896) Δu (119896)100381710038171003817100381710038172+ 1205821 Δu (119896)2
(21)
Using optimal condition 120597119869120597u(119896) = 0 then gives
Δu (119896) = 120593 (119896) (120572119872119890 (119896) + 120572119863119890 (119896 minus 1))1205821 + (120572119872 + 120572119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (22)
Dividing numerator and denominator by 120572119872+120572119863 at the righthand of (22) we haveΔu (119896)= 120593 (119896) ((120572119872 (120572119872 + 120572119863)) 119890 (119896) + (120572119863 (120572119872 + 120572119863)) 119890 (119896 minus 1))
(1205821 (120572119872 + 120572119863)) + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (23)
Letting 120588119872 = 120572119872(120572119872 + 120572119863) 120588119863 = 120572119863(120572119872 + 120572119863) and 120582 =1205821(120572119872 + 120572119863) thenu (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (24)
where 120582 gt 0 is the weighted factor and 120588119872 120588119863 isin [0 1] are thestep-size factors
Define the following PPD vector criterion function
119869 ( (119896))= 10038171003817100381710038171003817120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896) Δu (119896 minus 1)100381710038171003817100381710038172
+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172 (25)
where 120573119872 120573119863 isin [0 1] are the weighted factors and 120573119872+120573119863 =1We rewrite (25) as
119869 ( (119896)) = 10038161003816100381610038161003816120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1)minus 119879 (119896 minus 1) Δu (119896 minus 1) minus Δ119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(26)
Complexity 9
Utilizing optimal condition 120597119869120597(119896) = 0 we have
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (27)
Thus the controller of DDMFAC is designed as follows
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (28)
u (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (29)
42 Stability and Convergence Analysis of DDMFAC
(1)The Boundedness of (119896) Define the parameter estimationerror of 120593(119896) as follows
(119896) = 120593 (119896) minus (119896) (30)
Substituting (28) into (30) gives
(119896) = 120593 (119896) minus (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 (31)
Similarly define (119896 minus 1) = 120593(119896 minus 1) minus (119896 minus 1) and then (119896 minus 1) = 120593 (119896 minus 1) minus (119896 minus 1) (32)
Combining (31) and (32) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1) (33)
According to (7) and (13) it can be obtained that
Δ119910 (119896) = 120593119879 (119896 minus 1) Δu (119896 minus 1)
Δ119910 (119896 minus 1) = 120593119879 (119896 minus 1) Δu (119896 minus 1) (34)
Substituting (34) into (33) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) ((120573119872 + 120573119863)120593119879 (119896 minus 1) Δu (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578Δu (119896 minus 1) ((120593 (119896 minus 1) minus (119896 minus 1))119879 Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578 (119896 minus 1) Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2) + 120593 (119896) minus 120593 (119896 minus 1)
(35)
10 Complexity
Because |120593(119896)| le 119887 is bounded [16] |120593(119896)minus120593(119896minus1)| le 2119887Taking the modulus of both sides of (35) we can obtain
1003817100381710038171003817 (119896)1003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2)100381710038171003817100381710038171003817100381710038171003817 + 2119887 (36)
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
= 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 + (minus2 + 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
times 1205781003817100381710038171003817 (119896 minus 1) Δu (119896 minus 1)10038171003817100381710038172120583 + Δu (119896 minus 1)2
(37)
Given that 0 lt 120578 lt 2 120583 gt 0(minus2 + 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2) lt 0 (38)
Synthesizing (37) and (38) we have
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
le 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 (39)
According to (36) and (39) there is obviously a positivenumber 1198891 isin (0 1) which makes
1003817100381710038171003817 (119896)1003817100381710038171003817 le 1198891 1003817100381710038171003817 (119896 minus 1)1003817100381710038171003817 + 2119887 le sdot sdot sdotle 1198891198961 1003817100381710038171003817 (0)1003817100381710038171003817 + 2119887 (1 minus 1198891198961)1 minus 1198891 (40)
Therefore (119896) is bounded(2)TheConvergence of Tracking Error 119890(119896) Tracking error 119890(119896)is derived as
119890 (119896) = 119910119889 minus 119910 (119896) = 119910119889 minus 119910 (119896 minus 1) minus 120593119879 (119896) Δu (119896)= 119890 (119896 minus 1) minus 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
(120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)= (1 minus 120588119863 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)
minus 120588119872 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 119890 (119896)
(41)
Transposition gives
119890 (119896) = [1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172] 119890 (119896 minus 1) (42)
So as long as appropriate parameters 120588119872 120588119863 and 120582 areselected there must be a positive number 1198892 isin (0 1) thatmakes
0 lt 10038171003817100381710038171003817100381710038171003817100381710038171 minus(120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172
1003817100381710038171003817100381710038171003817100381710038171003817 le 1198892 lt 1 (43)
By (42) and (43) it can be obtained that
119890 (119896) le 1003817100381710038171003817100381710038171003817100381710038171003817(1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)1003817100381710038171003817100381710038171003817100381710038171003817le 1198892 119890 (119896 minus 1) le sdot sdot sdot le 1198891198962 119890 (0)
(44)
Therefore lim119896rarrinfin119890(119896) = 0 that is 119890(119896) is proved toconverge to zero
(3)The Boundedness of System Input and Output By referringto 1198862 + 1198872 ge 2119886119887 the relational expression can be obtained asfollows
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
120582 + 1003817100381710038171003817120593 (119896)100381710038171003817100381721003817100381710038171003817100381710038171003817100381710038171003817 le
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
2radic120582 1003817100381710038171003817120593 (119896)10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 =
12radic120582 (45)
And we have
u (119896) = u (119896) minus u (119896 minus 1) + u (119896 minus 1) minus sdot sdot sdot + u (1)minus u (0) + u (0) = u (0) + 119896sum
119895=1
u (119895) (46)
Combining (24) (44) (45) and (46) as well as utilizingSchwartzrsquos inequality gives
u (119896) le u (0) + 119896sum119895=1
1003817100381710038171003817Δu (119895)1003817100381710038171003817
le u (0) + 12radic120582
119896sum119895=1
1003817100381710038171003817120588119863119890 (119895 minus 1) + 120588119872119890 (119895)1003817100381710038171003817
le u (0) + 1205881198632radic120582119896sum119895=1
119889119895minus12 119890 (0)
+ 1205881198722radic120582119896sum119895=1
1198891198952 119890 (0)
le u (0) + 1205881198632radic1205821 minus 119889119895minus121 minus 1198892 119890 (0)
+ 1205881198722radic1205821198892 (1 minus 1198891198952)1 minus 1198892 119890 (0)
le u (0) + 120588119863(1 minus 1198892) 2radic120582 119890 (0)
Complexity 11
Table 3 The parameter settings of DDMFAC for different IVs
IV Parameter120588119872 120588119863 120573119872 120573119863 120582 120578 1205831199100 = 60 00275 00275 05000 05000
00010 00750 100001199100 = 70 50119864 minus 4 50119864 minus 4 05000 050001199100 = 80 46119864 minus 4 46119864 minus 4 05000 050001199100 = 100 18119864 minus 4 18119864 minus 4 05000 05000
+ 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)le u (0) + 120588119863 + 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)
(47)
Because the initial tracking error 119890(0) and initial inputu(0) are given bounded the system input u(119896) is boundedaccording to (47) Besides due to the boundedness of 119910119889 and119890(119896) the system output 119910(119896) is obviously bounded43 Fuzzy DDMFAC The controller design and stability andconvergence analysis for DDMFAC have been respectivelydiscussed in detail in the above sections Although thedesigned controller has the characters of both MFAC andDDOILC weighted factors 120588119872 120588119863 120573119872 and 120573119863 are allconstant throughout control process DDMFAC is incapableof adaptively selecting the dominance of MFAC or DDOILCto duly show the expected features in each stage of controlprocessThe aforementioned features of DDMFACwe expectcannot be reflected in this case Such DDMFAC approachwith fixed weighting factors has no advantage comparedwith MFAC or DDOILC which appeals to a reasonable andadaptive adjustment strategy for 120588119872 120588119863 120573119872 and 120573119863 There-fore the adaptive adjustment strategy based on fuzzy logicfor weighting factors is considered because its advantageousperformance in parameter adjustment has been proven inour previous work [45] The formulation of fuzzy rules offersus the opportunity to adjust weighting factors in the waywe expected Such DDMFAC approach with fuzzy adaptiveadjustment for parameters is namedFDDMFAC in this paperConcretely the fuzzy rules can be formulated by referringto the expected features of DDMFAC The input variables offuzzy rules are the absolute values of tracking error 119890(119896 minus 1)and DLM prediction error 119890119901(119896minus1) of MFAC and the outputvariables are 120588119872 120588119863 and 120573119863 Because of the relationshipbetween 120573119872 and 120573119863 there is no need to use 120573119872 as an outputand it is calculated by 120573119872 = 1 minus 120573119863 The fuzzy rules can bedetailedly described as follows
(1) If |119890(119896 minus 1)| and |119890119901(119896 minus 1)| are great then thedominance of DDOILC is preferred and that ofMFAC is restrained in DDMFAC So 120588119863 and 120573119863 haverelatively large values and 120588119872 is a relatively smallnumber
(2) 120588119863 and 120573119863 will show a decreasing trend with thegradual decrease of |119890(119896 minus 1)| and |119890119901(119896 minus 1)| while120588119872 will present an increasing trend
44 Simulations and Result Analysis The proposed DDM-FAC approach is firstly applied to the particle quality controlin SFBGP and then its control effect is compared withMFACand DDOILC under different IVs In the simulation justlike MFAC 119910(119896) is replaced by a predicted value 119910(119896) thatis estimated using (19) See Table 3 for parameter settings ofDDMFAC It is not hard to find 120588119872 = 120588119863 and 120573119872 = 120573119863for a certain IV which means that MFAC and DDOILC areset to have the same dominance in DDMFAC By the wayMFAC and DDOILC can be regarded as the extreme cases ofDDMFAC described as follows
MFAC 120588119872 = 120573119872 = 1120588119863 = 120573119863 = 0
DDOILC 120588119872 = 120573119872 = 0120588119863 = 120573119863 = 1
(48)
The control effects of DDMFAC with the parametersettings in Table 3 together with control curves of MFACand DDOILC are plotted in Figure 5 The simulation resultsactually present the control effects of DDMFAC with threesets of parameter settings From simulation it is indicatedthat all three approaches have gained some certain advantagesunder different IV conditions Concretely as is shown inFigure 5(a) DDMFAC has advantage in control performancecompared with DDOILC and MFAC when 1199100 = 60 butMFAC has absolute superiority when 1199100 = 100 plottedin Figure 5(d) In addition from the simulation results inFigures 5(b) and 5(c) it can be seen that DDOILC andMFAC have achieved certain but not absolute advantagesrespectively This demonstrates that parameter setting hasimportant influence on the control effect of DDMFACTherefore the way to achieve satisfactory control effect is toadaptively adjust weighted parameters of DDMFAC insteadof keeping them constant
Then the idea of fuzzy adjustment is introduced to adjustthe weights 120588119872 120588119863 and 120573119863 The fuzzy adjustment rules ofthree weights are respectively shown in Tables 4 and 5 Therespective membership functions when 1199100 = 60 are shownin Figure 6 and membership functions for the other IVs areno longer displayed Please see Figure 7 for control effectof FDDMFAC and comparisons with MFAC and DDOILCunder different IVs Obviously DDMFAC approach withfuzzy adjustment for parameters that is FDDMFAC has thebest performance in particle quality control for all IVs This
12 Complexity
Table 4 The fuzzy rules of 120588119872120588119872 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB PB PB PB PB PB PB PBNM PB PM PM PM PM PS PSNS PM PM PM PS PS Z ZZ Z Z Z NS NS NS NSPS NS NS NM NM NB NB NBPM NM NM NM NB NB NB NBPB NB NB NB NB NB NB NB
MFACDDOILCDDMFAC
60
80
100
120
APS
(m
)
5 10 150Iterations
(a) 1199100 = 60
MFACDDOILCDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILCDDMFAC
80
90
100
110
120
APS
(m
)
2 4 60Iterations
(c) 1199100 = 80
95
100
105
110
115
120
125
APS
(m
)
5 10 150Iterations
MFACDDOILCDDMFAC
(d) 1199100 = 100
Figure 5 The comparison of control effects of MFAC DDOILC and DDMFAC
Complexity 13
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
02 04 06 08 10D
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
005 01 0150D
NB NM NS Z PS PM PB
times10minus3
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
05 1 15 2 25 3 35 4 450M
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
10 20 30 40 50 600|e(k minus 1)| (m)
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
5 10 15 20 250|ep (k minus 1)| (m)
Figure 6 The membership functions of FDDMFAC when 1199100 = 60
Table 5 The fuzzy rules of 120588119863 120573119863120588119863 120573119863 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB NB NB NB NB NB NB NBNM NB NM NM NM NM NS NSNS NM NM NM NS NS Z ZZ Z Z Z PS PM PM PMPS PS PS PM PM PB PB PBPM PM PM PM PB PB PB PBPB PB PB PB PB PB PB PB
validates the effectiveness of FDDMFAC proposed in thispaper
Although FDDMFAC approach has been proven to havea better control performance than MFAC and DDOILCwe still have a question of whether it is possible that thesuperiority of FDDMFAC is entirely attributed to fuzzyadjustment of parameters and has nothing to do with thestructure of DDMFAC itself In order to clarify this problemanother set of simulations are conducted inwhich the controlperformance of FDDMFAC is compared with fuzzy MFAC(FMFAC) and fuzzy DDOILC (FDDOILC) The absolutevalue of tracking error |119890(119896 minus 1)| is regarded as the inputvariable of fuzzy rules and the step-size factors 1205881 1205882 ofMFAC andDDOILC are the output variablesThe fuzzy rulesof FMFAC and FDDOILC are shown in Table 6 and themembership functions when 1199100 = 60 are shown in Figure 8For the simulation results please see Figure 9 From thesimulation it is concluded that FDDMFAC achieves the best
Table 6 The fuzzy rules of FMFAC and FDDOILC
|119890(119896 minus 1)| NB NM NS Z PS PM PB1205881 1205882 NB NM NS Z PS PM PB
control performance which demonstrates the effectiveness ofthe proposed approach in this paper
5 Conclusions
This paper aimed at solving the problem of particle qualitycontrol in the drug development phase of SFBGP Themodel-based control approaches are incapable of handlingsuch quality control because the accurate process modelcannot be acquired Thus the data-driven and model-freecontrol approaches are considered in this work We firstly
14 Complexity
MFACDDOILCFDDMFAC
5 10 150Iterations
60
80
100
120
APS
(m
)y0 = 60
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
y0 = 70
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
MFACDDOILCFDDMFAC
95
100
105
110
115
120
125A
PS (
m)
5 10 150Iterations
y0 = 100
Figure 7 The comparison of control effects of MFAC DDOILC and FDDMFAC
analyzed the features of MFAC and DDOILC and thenconducted experimental research to compare the perfor-mances of MFAC and DDOILC in particle quality controlunder different IVs The simulation results have verifiedour theoretical analysis and concluded that MFAC andDDOILC both have advantages under different IVs To beused in an actual production environment the expectedDDCapproaches must accommodate various IVs because of thechanges in material attributes Thus the hybrid approachFDDMFAC which combines the advantages of MFAC andDDOILC by adaptively adjusting their respective weightedfactors with fuzzy logic was proposed in this work tocomplement their respective shortcomings Through a series
of simulation results we conclude that FDDMFAC achievesbetter control performance than other approaches whichvalidates the effectiveness of FDDMFAC
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thanks are due to National Natural Science Foundation ofChina under Grant nos 61773105 61374147 61703085 and61004083 and Fundamental Research Funds for the Central
Complexity 15NB NM NS Z PS PM PB
001 002 00302
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
002 00401
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
20 40 600|e(k minus 1)| (m)
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
Figure 8 The membership functions of FDDMFAC when 1199100 = 60
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
60
80
100
120
APS
(m
)
y0 = 60
FMFACFDDOILCFDDMFAC
70
80
90
100
110
120
APS
(m
)
2 4 6 8 100Iterations
y0 = 70
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
100
105
110
115
120
125
APS
(m
)
y0 = 100
Figure 9 The comparison of control effects of FMFAC FDDOILC and FDDMFAC
16 Complexity
Universities under Grant nos N150402001 and N120404014for supporting this research work
References
[1] J Bouffard M Kaster and H Dumont ldquoInfluence of processvariable and physicochemical properties on the granulationmechanism of mannitol in a fluid bed top spray granulatorrdquoDrug Development and Industrial Pharmacy vol 31 no 9 pp923ndash933 2005
[2] H Liu K Wang W Schlindwein and M Li ldquoUsing the Box-Behnken experimental design to optimise operating parametersin pulsed spray fluidised bed granulationrdquo International Journalof Pharmaceutics vol 448 no 2 pp 329ndash338 2013
[3] H Liu and M Li ldquoPopulation balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granula-tionrdquo International Journal of Pharmaceutics vol 468 no 1-2pp 223ndash233 2014
[4] T Albuquerque V H Dias N Poellinger and J F PintoldquoConstruction of a quality index for granules produced byfluidized bed technology and application of the correspondenceanalysis as a discriminant procedurerdquo European Journal ofPharmaceutics and Biopharmaceutics vol 75 no 3 pp 418ndash4242010
[5] H Ehlers A Liu H Raikkonen et al ldquoGranule size control andtargeting in pulsed spray fluid bed granulationrdquo InternationalJournal of Pharmaceutics vol 377 no 1-2 pp 9ndash15 2009
[6] C F W Sanders M J Hounslow and F J Doyle III ldquoIden-tification of models for control of wet granulationrdquo PowderTechnology vol 188 no 3 pp 255ndash263 2009
[7] T Narvanen T Lipsanen O Antikainen H Raikkonen andJ Yliruusi ldquoControlling granule size by granulation liquid feedpulsingrdquo International Journal of Pharmaceutics vol 357 no 1-2pp 132ndash138 2008
[8] A Faure P York and R C Rowe ldquoProcess control and scale-up of pharmaceutical wet granulation processes a reviewrdquoEuropean Journal of Pharmaceutics and Biopharmaceutics vol52 no 3 pp 269ndash277 2001
[9] M Pottmann B A Ogunnaike A A Adetayo and B JEnnis ldquoModel-based control of a granulation systemrdquo PowderTechnology vol 108 no 2-3 pp 192ndash201 2000
[10] E P Gatzke and F J Doyle III ldquoModel predictive control of agranulation system using soft output constraints and prioritizedcontrol objectivesrdquo Powder Technology vol 121 no 2-3 pp 149ndash158 2001
[11] F Y Wang and I T Cameron ldquoReview and future directionsin the modelling and control of continuous drum granulationrdquoPowder Technology vol 124 no 3 pp 238ndash253 2002
[12] T Glaser C F W Sanders F Y Wang et al ldquoModel predictivecontrol of continuous drum granulationrdquo Journal of ProcessControl vol 19 no 4 pp 615ndash622 2009
[13] R Ramachandran andA Chaudhury ldquoModel-based design andcontrol of a continuous drum granulation processrdquo ChemicalEngineering Research and Design vol 90 no 8 pp 1063ndash10732012
[14] Z H Loh D Z L Er L W Chan C V Liew and PW S HengldquoSpray granulation for drug formulationrdquo Expert Opinion onDrug Delivery vol 8 no 12 pp 1645ndash1661 2011
[15] S H Schaafsma N W F Kossen M T Mos L Blauw andA C Hoffmann ldquoEffects and control of humidity and particlemixing in fluid-bed granulationrdquo AIChE Journal vol 45 no 6pp 1202ndash1210 1999
[16] Z-S Hou and S-T Jin ldquoA novel data-driven control approachfor a class of discrete-time nonlinear systemsrdquo IEEE Transac-tions onControl Systems Technology vol 19 no 6 pp 1549ndash15582011
[17] Z Hou and S Jin ldquoData-driven model-free adaptive controlfor a class of MIMO nonlinear discrete-time systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 22no 12 pp 2173ndash2188 2011
[18] D Xu B Jiang and P Shi ldquoA novel model-free adaptivecontrol design for multivariable industrial processesrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6391ndash6398 2014
[19] X Wang X Li J Wang X Fang and X Zhu ldquoData-drivenmodel-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeletonrdquo Information Sciences vol 327pp 246ndash257 2016
[20] L dos Santos Coelho andA A R Coelho ldquoModel-free adaptivecontrol optimization using a chaotic particle swarm approachrdquoChaos Solitons amp Fractals vol 41 no 4 pp 2001ndash2009 2009
[21] L D S Coelho M W Pessoa R Rodrigues Sumar andA Augusto Rodrigues Coelho ldquoModel-free adaptive controldesign using evolutionary-neural compensatorrdquo Expert Systemswith Applications vol 37 no 1 pp 499ndash508 2010
[22] K K Tan T H Lee S N Huang and F M Leu ldquoAdaptive-predictive control of a class of SISOnonlinear systemsrdquoDynam-ics and Control vol 11 no 2 pp 151ndash174 2001
[23] B Zhang andW Zhang ldquoAdaptive predictive functional controlof a class of nonlinear systemsrdquo ISA Transactions vol 45 no2 pp 175ndash183 2006
[24] F L Lu J F Wang C J Fan and S B Chen ldquoAn improvedmodel-free adaptive control with G function fuzzy reasoningregulation design and its applicationsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems andControl Engineering vol 222 no 8 pp 817ndash828 2008
[25] R Chi D Wang Z Hou and S Jin ldquoData-driven optimalterminal iterative learning controlrdquo Journal of Process Controlvol 22 no 10 pp 2026ndash2037 2012
[26] R Chi Z Hou S Jin and D Wang ldquoImproved data-drivenoptimal TILC using time-varying input signalsrdquo Journal ofProcess Control vol 24 no 12 pp 78ndash85 2014
[27] X Bu Z Hou and R Chi ldquoModel free adaptive iterative learn-ing control for farm vehicle path trackingrdquo IFAC ProceedingsVolumes vol 46 no 20 pp 153ndash158 2013
[28] M C Campi A Lecchini and SM Savaresi ldquoVirtual referencefeedback tuning a direct method for the design of feedbackcontrollersrdquo Automatica vol 38 no 8 pp 1337ndash1346 2002
[29] R-C Roman M-B Radac R-E Precup and E M PetriuldquoData-driven model-free adaptive control tuned by virtualreference feedback tuningrdquoActa PolytechnicaHungarica vol 13no 1 pp 83ndash96 2016
[30] P Yan D Liu D Wang and H Ma ldquoData-driven controllerdesign for general MIMO nonlinear systems via virtual refer-ence feedback tuning and neural networksrdquo Neurocomputingvol 171 pp 815ndash825 2016
[31] T Pan and S Li ldquoA hierarchical search and updating databasestrategy for lazy learningrdquo International Journal of InnovativeComputing Information andControl vol 4 no 6 pp 1383ndash13922008
[32] H Zhang R Song Q Wei and T Zhang ldquoOptimal trackingcontrol for a class of nonlinear discrete-time systems withtime delays based on heuristic dynamic programmingrdquo IEEE
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Complexity 3
FI Compressed air unit
Liquid supply unit
Exhaust blower
BlowerFilterHeater
Dust-arresterinstallation
Spray nozzle
Fluidizing agent
Figure 1 The schematic diagram of SFBGP
rules for DDMFACThen fuzzy adaptive adjustment of targetweighted parameters is implemented with the aim of rapidlyconverging DDMFAC by making full use of the advantageof MFAC or DDOILC at different control stages We namedthe final DDC approach presented in this paper as fuzzyDDMFAC (FDDMFAC) and a series of simulations validateits effectiveness
2 Preliminaries
21 Mathematical Mechanism Model of Spray Fluidized-BedGranulation Process SFBGP is a commonly used unit oper-ation in the pharmaceutical industry The powders can bemixed granulated and dried in the same equipment whichminimizes the equipment costs loss of product transfers andpossibility of the cross-contamination [3] The rationale ofspray fluidized-bed granulation can be briefly described asfollows (1) Particles circulate within granulator by pumpinghot air from the bottom of granulator and meanwhile binderliquid is sprayed on the fluidized particles in the forms ofsmall droplets [38] (2) As a result of collisions and coales-cence between the surface-wetted particles liquid bridges areformed and aggregation of particles occurs leading to thegrowth of granules [1 39]
The schematic diagram of SFBGP is shown in Figure 1The overall granulation process in fluidized-bed spray gran-ulator is divided into three stages powder mixing granula-tion and drying Firstly mixed raw materials for drug andingredients are put in granulator Passing through the filterand heater hot air pumped in by blower is pumped intogranulator from the bottom of granulator which forces thefluidization of powder to sufficiently blend materials After-wards liquid binder is pumped into spray nozzle from liquidsupply unit and atomized into droplets by compressed air thatis simultaneously pumped into spray nozzle from compressedair unit The droplets are dispersed over the surface offluidized particles which contributes to the agglomerationof colliding and surface-wetted powder particles to formgranules Finally the granules are dried to a predeterminedmoisture content level The dust-arrester installation is usedto prevent powder particles that are not in contact withdroplets from being carried out by off-gas
PBMs have been widely used for modeling SFBGP inwhich the granule size density distribution evolves as afunction of timeThe population balance is a number balancearound each size fraction of the size distribution based onnumber conservation law The rate of change of the numberof particles in a given size interval is equal to the rate at which
4 Complexity
the granules enter and leave that size interval The mathe-matical mechanism model of SFBGP is developed based onfollowing assumptions (1) Each particle is spherical (2)Thetotal volume of the particles is conserved (3) The particlesize in a certain particular size interval is represented by theleft boundary of the interval (4) The relative growth rate isuniform for each particle within the same volume intervalA one-dimensional PBM that describes the rate of change inparticle number density function 119899(119905 119897) is given by
120597119899 (119905 119897)120597119905 = 12sdot int1198970
120573(119905 (1198973 minus 1205833)13 120583) 119899 (119905 (1198973 minus 1205833)13) 119899 (119905 120583)(1198973 minus 1205833)23 119889120583
minus 119899 (119905 119897) intinfin0
120573 (119905 119897 120583) 119899 (119905 120583) 119889120583
(1)
where 119899(119905 119897) is the particle number density function in termsof particle diameter 119897 and120573 is the aggregationmodel [37 38]120573(119905 119897 120583) can generally be partitioned into size dependent andsize independent parts [3 37 38 40 41]
120573 (119905 119897 119906) = 1205730 (119905) 120573lowast (119897 120583) (2)
where 1205730(119905) is the granulation rate constant that can bethought of as an aggregation efficiency and depends onall parameters involved in the aggregation process exceptparticle size such as the operating condition and binderproperties The latter term 120573lowast(119897 120583) reflects the influence ofparticle size on the likelihood of aggregation
In order to solve a PBM in (1) the discretized approach[42 43] is applied to determine the change in number ofparticles119873119894 in interval 119894 as
d119873119894d119905 = 119894minus2sum
119895=1
2119895minus119894+1120573119894minus1119895119873119894minus1119873119895 + 12120573119894minus1119894minus11198732119894minus1
minus 119873119894 119894minus1sum119895=1
2119895minus119894120573119894119895119873119895 minus 119873119894119899maxsum119895=119894
120573119894119895119873119895(3)
where119873119894 is the number of particles within particle size range(119871 119894 119871 119894+1) 119871 119894 and 119871 119894+1 are the lower and upper limits of 119894thparticle size interval 119894 = 1 2 119899max and 119899max is the totalnumber of particle size intervals with a value of 12 in thispaper In the discretization scheme the length domain ofparticles is divided into geometric intervals in the way thatthe upper and lower limits of each size interval are in a ratioof 119903 = 119871 119894+1119871 119894 = 3radic2 The size range used in modeling isfrom 50 120583m to 800120583mwhich is divided into 12 intervals Thefunction ldquoode45rdquo inMATLAB is used to solve (3) to obtain thefinal number of particles in each size interval Then the APSof final granules119863119898 is calculated by the following formula
119863119898 = 119899maxsum119894=1
119881(119873119891119894) 119889119901119894 (4)
where 119881(119873119891119894) is the volume fraction of end granules at sizeinterval 119894 = 1 2 119899max119873119891119894 is the number of end granulesin the 119894th size interval and 119889119901119894 is the geometric mean of lowerlimit and upper limit of 119894th size interval 119894 = 1 2 119899max
We have just briefly introduced the general framework ofPBM-basedmechanismmodel of SFBGPDetailed derivationprocess and setting of model parameters are given in [37 38]and they will not be covered hereThe developed mechanismmodel is capable of not only linking APS of particles withthe operating variable (binder spray rate) but also reflectingthe relationship between APS and material attributes suchas viscosity of binder and particle density Hence thismechanism model can be used for simulating the practicalchallenges in particle quality control of SFBGP For examplea series of IVs can be constructed by changing the materialattributes while keeping the operating condition unchangedTo veritably simulate the actual granulation process the inputcontrol curve should be time-varying But to simplify thingsthe input control curve is divided into three stages in thewhole granulation process in this work
22 The Data-Driven Control Approaches Used in This WorkAs the classic and representative DDC methods MFAC andDDOILChave somemerits thatmake themmore competitivein practical control applications Therefore in view of thesemerits MFAC and DDOILC are selected and applied toperform data-driven particle quality control for SFBGP inthis work Now we first briefly introduce their controllerdesign procedures For details please refer to [16 25]
A repeatable batch process system to be controlled isgiven as follows
119910 (119896) = 119891 (u (119896)) (5)
where 119910(119896) and u(119896) are the system output and input at 119896thiteration respectively 119896 is the number of iterations and 119891(∙)is an unknown nonlinear functionThis nonlinear system (5)satisfies the following assumptions
(1) The derivatives of 119891(∙) with respect to control inputu(119896) are continuous
(2) System (5) is generalized Lipschitz that is |Δ119910(119896 +1)| le 119887Δu(119896) for any 119896 and Δu(119896) = 0 whereΔ119910(119896 + 1) = 119910(119896 + 1) minus 119910(119896) Δu(119896) = u(119896) minus u(119896 minus 1)and 119887 is a positive constant
221 MFAC Design For the one-step-ahead controller [44]the control goal is to seek a control input sequence u(119896) thatbrings 119910(119896 + 1) to 119910119889 where 119910119889 is the desired system outputsignal In general the weighted one-step-ahead controllermay lead to steady-state tracking error [16] So the followingcontrol input criterion function is used to design the controllaw
119869 (u (119896)) = 1003816100381610038161003816119910119889 minus 1199101 (119896 + 1)10038161003816100381610038162 + 120582 Δu (119896)2 (6)
where 120582 gt 0 is a weighting constantSubstituting the dynamic linearization form
1199101 (119896 + 1) = 119910 (119896) + 120593119879 (119896) Δu (119896) (7)
Complexity 5
into (6) as well as letting 120597119869120597u(119896) be zero givesu (119896) = u (119896 minus 1) + 1205881120593 (119896) (119910119889 minus 119910 (119896))
120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (8)
where 1205881 isin (0 1] is a step-size constant that is added to make(8) more general and 120593(119896) is PPD vector
Because PPD vector 120593(119896) is unknown it needs to beestimated Define criterion function for the unknown PPDvector as follows
119869 ( (119896)) = 10038161003816100381610038161003816Δ119910 (119896) minus 119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(9)
where 120583 gt 0 is a weighting factor and (119896) is the estimationof 120593(119896)
Using optimal condition 120597119869120597(119896) = 0 then gives
(119896)= (119896 minus 1)
+ 120578Δu (119896 minus 1) (Δ119910 (119896) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2
(10)
where 120578 isin (0 2) is a step-size constantDefining the tracking error 119890(119896) = 119910119889 minus 119910(119896) and
combining PPD estimation and input control law MFACscheme is designed as follows
(119896)= (119896 minus 1)
+ 120578Δu (119896 minus 1) (Δ119910 (119896) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2
u (119896) = u (119896 minus 1) + 1205881 (119896)120582 + 1003817100381710038171003817 (119896)10038171003817100381710038172 119890 (119896)
(11)
222 DDOILC Design The control objective is to find anappropriate control input sequence u(119896) such that the systemoutput 119910(119896) follows 119910119889 Consider the control input criterionfunction as
119869 (u (119896)) = 1003816100381610038161003816119910119889 minus 1199102 (119896)10038161003816100381610038162 + 120582 Δu (119896)2 (12)
Substituting the dynamic linearization equation
1199102 (119896) = 119910 (119896 minus 1) + 120593119879 (119896) Δu (119896) (13)
into (12) and the optimal condition leads to
u (119896) = u (119896 minus 1) + 1205882120593 (119896)120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 119890 (119896 minus 1) (14)
where 1205882 isin (0 1] determines the step size and 119890(119896 minus 1) =119910119889 minus 119910(119896 minus 1)
Let (119896) denote its estimation at 119896th iterationThendefinethe criterion function as
119869 ( (119896)) = 10038161003816100381610038161003816Δ119910 (119896 minus 1) minus 119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(15)
According to optimal condition the iterative updating law ofestimate (119896) is developed as
(119896) = (119896 minus 1)+ 120578Δu (119896 minus 1) (Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))
120583 + Δu (119896 minus 1)2 (16)
where 120583 gt 0 0 lt 120578 lt 2The DDOILC scheme is designed as follows
(119896) = (119896 minus 1)+ 120578Δu (119896 minus 1) (Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))
120583 + Δu (119896 minus 1)2 u (119896) = u (119896 minus 1) + 1205882 (119896)120582 + 1003817100381710038171003817 (119896)10038171003817100381710038172 119890 (119896 minus 1)
(17)
3 Theoretical Analysis of MFAC and DDOILCand Simulations in SFBGP
31 Theoretical Analysis of MFAC and DDOILC The con-troller design procedures of MFAC and DDOILC have beenintroduced in previous sections Although they have the samesystemic framework and similar expression form of con-troller there are still differences between them which makesthem two different methods Comparing the controllers ofMFAC and DDOILC the main differences can be listed asfollows
(1) Δ119910(119896) and Δ119910(119896 minus 1) are respectively used in PPDiterative updating laws
(2) 119890(119896) and 119890(119896minus1) are separately utilized in control inputiterative learning laws
The expressions of Δ119910(119896) Δ119910(119896minus1) 119890(119896) and 119890(119896minus1) arelisted as follows
MFACΔ119910 (119896) = 119910 (119896) minus 119910 (119896 minus 1)119890 (119896) = 119910119889 minus 119910 (119896)
DDOILCΔ119910 (119896 minus 1) = 119910 (119896 minus 1) minus 119910 (119896 minus 2)119890 (119896 minus 1) = 119910119889 minus 119910 (119896 minus 1)
(18)
Obviously the main differences between MFAC andDDOILC can be described as follows system output 119910(119896) at119896th iteration is used in MFAC but is not used in DDOILC Inpractical application all the system outputs used inDDOILC119910(119896minus1) and 119910(119896minus2) are the actual values at (119896minus1)th and (119896minus2)th iterations In MFAC however the actual value of system
6 Complexity
y
yd
y (k + 1)y(k)
y(k)y (k minus 1)
y (k minus 1)
y(k minus 2)
IV = y0
u(k minus 2) u(k minus 1) u(k) u(k + 1) u(k + 2) Input
e (k)
e (k minus 1)ep (k)
ep (k minus 1)
Δy (k)
Δy(k minus 1)
Tracking errorPredicted error of DLM
Δy
Figure 2 The variable relations during control process of MFAC and DDOILC
output 119910(119896) is obtained only after the calculation of u(119896) So119910(119896) does not have actual value when computing u(119896) anda predicted system output 119910(119896) will be used to replace 119910(119896)in practical application of MFAC 119910(119896) is estimated by thefollowing DLM
119910 (119896) = 119910 (119896 minus 1) + 120593119879 (119896 minus 1) Δu (119896 minus 1) (19)
and 119890(119896) is replaced by 119890(119896) = 119910119889 minus 119910(119896)Figure 2 gives the variable relations during control pro-
cess of MFAC and DDOILC In terms of a single controlprocess curve when system output 119910 is greatly different from119910119889 that is the tracking error 119890 is relatively great 119890(119896 minus 1)is much larger than 119890(119896) and DDOILC will achieve a biggerupdating step size for control input which gives DDOILC afaster convergence speed As 119910 keeps getting closer to 119910119889 itis found that DLM prediction error 119890119901 is getting smaller andsmaller until 119890119901 = 0 The difference between 119890(119896 minus 1) and 119890(119896)will also be smaller and smaller DDOILC will not only loseits advantage but also perform worse than MFAC when 119910 isvery close to119910119889 because a larger updating step size for controlinput will bring difficulty in convergence while at this timeMFAC will achieve a fast and stable control effect due to theimprovement of prediction accuracy of DLM From anotherpoint of view if IV of system output 1199100 is far away from 119910119889the initial tracking error and DLM prediction error will beboth at a high level and DDOILC will perform better thanMFAC because a greater updating step size for control inputis necessary to give a faster convergence speed But as1199100 keepsgetting closer to 119910119889 119890119901 will gradually decrease and MFACwill gradually show its advantage Therefore the conclusions
obtained from theoretical analysis can be summarized asfollows
(1) When tracking error is large and 1199100 is relatively faraway from119910119889 DDOILCwill have a faster convergencespeed and a better control performance
(2) When tracking error gradually decreases and 1199100gradually gets closer to 119910119889 MFAC will graduallyreveal its advantage and achieve better control effect
32 SimulationVerification To validate conclusions obtainedfrom theoretical analysis simulation verification in SFBGP isconducted In this simulation particle quality to be controlledis APS and the desired quality is 119910119889 = 120 120583mThe viscosityof binder is treated as material attribute to be changed andthen a series of IVs (1199100) are constructed by adjusting viscosityof binder while keeping the operating condition unchangedFour sets of IVs are selected as representatives in this workand the detailed establishing information of them is shownin Table 1 Besides the parameter settings of MFAC andDDOILC for different IVs are listed in Table 2
For different IVs we conduct particle quality control inSFBGP based on MFAC and DDOILC and the control per-formances are plotted in Figure 3Moreover DLMpredictionerror profiles of MFAC are shown in Figure 4 Then theexplanation and analysis of simulation results will be givenAs is shown in Figure 3(a) DDOILC performs better thanMFAC when 1199100 = 60 that is 1199100 is relatively far away from119910119889 When 1199100 = 70 as is described in Figure 3(b) DDOILChas a faster convergence speed when APS is controlled from70 120583m to about 110 120583m but MFAC achieves a relatively better
Complexity 7
MFACDDOILC
5 10 150Iterations
60
80
100
120A
PS (
m)
(a) 1199100 = 60
MFACDDOILC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
(c) 1199100 = 80
MFACDDOILC
5 10 150Iterations
95
100
105
110
115
120
125
APS
(m
)
(d) 1199100 = 100
Figure 3 The control profiles of APS in SFBGP based on MFAC and DDOILC
Table 1The detailed establishing information of representative IVs
Info IV1199100 = 60 1199100 = 70 1199100 = 80 1199100 = 100
Operatingconditions(gs)
First stage 03 second stage 01 third stage 015
Viscosity ofbinder (Pasdots) 00214 00250 00304 00625
control performance when it is controlled from 110 120583m to 119910119889which indicates that both DDOILC and MFAC have certainadvantages in such case However from Figure 3(c) it can beseen that MFAC is slightly superior to DDOILC in controleffect when 1199100 = 80 and this just manifests the advantage ofMFAC As 1199100 gets closer and closer to 119910119889 such as 1199100 = 100
Table 2The parameter settings ofMFAC andDDOILC for differentIVs
IV Parameter1205881 1205882 120582 120578 1205831199100 = 60 00505 00712
00010 00750 100001199100 = 70 00047 005031199100 = 80 00032 001991199100 = 100 00020 149119864 minus 4
plotted in Figure 3(d)MFAC has a huge advantage comparedwith DDOILC It is obvious that the simulation results haveverified the conclusions obtained from theoretical analysisThe reason for increasing superiority of MFAC is attributedto the smaller and smaller prediction error 119890119901 of DLM Asis shown in Figure 4 on the one hand the curves of 119890119901 for
8 Complexity
IV = 60IV = 70
IV = 80IV = 100
IV = 60 14320IV = 70 08061IV = 80 04718IV = 100 04000
5 10 150Iterations
0
5
10
15
20
25
e p(
m)
Average of ep
Figure 4 The DLM prediction error profiles of MFAC for differentIVs
different IVs all show a decreasing trend along with iterationuntil 119890119901 = 0 as APS keeps getting closer to 119910119889 which is inaccordance with our obtained conclusion On the other handfor different IVs 119890119901 will also decrease with the increase of IVwhich is proved by comparing the curves of 119890119901 under differentIVs To further illustrate this point the averages of 119890119901 fordifferent IVs are computed and then listed in Figure 4 andthe decreasing averages of 119890119901 along with the increase of IValso validate the rationality of our analysis
4 The Proposed Hybrid DDC Approach andSimulation Study in SFBGP
Through theoretical analysis and simulation study we haveproved that MFAC and DDOILC both have strengths inparticle quality control under different IVs In real-worldapplications the same operating condition may give entirelydifferent IVs due to the changes in material properties whichbrings great difficulties in the selection of control methodsThus the uncertainties of IV have to be considered duringcontrol application In order to achieve favourable overallcontrol effect it is demanded that the DDC approaches mustaccommodate various IVs Therefore we try to combineadvantages of MFAC and DDOILC to present a hybridapproach that is named DDMFAC The basic idea is that thefeatures of MFAC and DDOILC are both reflected in thecriterion functions when designing controller of DDMFACThe expected features of hybrid approach can be summarizedas the following three points
(1) When tracking error is large DDMFACmanifests thefeature of DDOILC to acquire a faster convergencespeed
(2) When tracking error is small enough DDMFAC hasthe ability ofMFAC to achieve a fast and stable controleffect
(3) DDMFAC is able to achieve better control perfor-mance for different IVs
Through such combination the shortcomings of MFACand DDOILC will be complemented and their respec-tive advantages will be highlighted in DDMFAC Next wedescribe the controller design for DDMFAC in Section 41
41 DDMFAC Design Consider the following control inputcriterion function
119869 (u (119896)) = 120572119872 1003816100381610038161003816119910119889 minus 1199101 (119896 + 1)10038161003816100381610038162 + 120572119863 1003816100381610038161003816119910119889 minus 1199102 (119896)10038161003816100381610038162+ 1205821 Δu (119896)2 (20)
where |119910119889 minus 1199101(119896 + 1)|2 and |119910119889 minus 1199102(119896)|2 respectively comefrom MFAC and DDOILC and 120572119872 120572119863 are the respectiveweighted factors and 120572119872 + 120572119863 = 0
Substituting (7) and (13) into (20) gives
119869 (u (119896)) = 120572119872 10038171003817100381710038171003817119890 (119896) minus 120593119879 (119896) Δu (119896)100381710038171003817100381710038172+ 120572119863 10038171003817100381710038171003817119890 (119896 minus 1) minus 120593119879 (119896) Δu (119896)100381710038171003817100381710038172+ 1205821 Δu (119896)2
(21)
Using optimal condition 120597119869120597u(119896) = 0 then gives
Δu (119896) = 120593 (119896) (120572119872119890 (119896) + 120572119863119890 (119896 minus 1))1205821 + (120572119872 + 120572119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (22)
Dividing numerator and denominator by 120572119872+120572119863 at the righthand of (22) we haveΔu (119896)= 120593 (119896) ((120572119872 (120572119872 + 120572119863)) 119890 (119896) + (120572119863 (120572119872 + 120572119863)) 119890 (119896 minus 1))
(1205821 (120572119872 + 120572119863)) + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (23)
Letting 120588119872 = 120572119872(120572119872 + 120572119863) 120588119863 = 120572119863(120572119872 + 120572119863) and 120582 =1205821(120572119872 + 120572119863) thenu (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (24)
where 120582 gt 0 is the weighted factor and 120588119872 120588119863 isin [0 1] are thestep-size factors
Define the following PPD vector criterion function
119869 ( (119896))= 10038171003817100381710038171003817120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896) Δu (119896 minus 1)100381710038171003817100381710038172
+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172 (25)
where 120573119872 120573119863 isin [0 1] are the weighted factors and 120573119872+120573119863 =1We rewrite (25) as
119869 ( (119896)) = 10038161003816100381610038161003816120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1)minus 119879 (119896 minus 1) Δu (119896 minus 1) minus Δ119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(26)
Complexity 9
Utilizing optimal condition 120597119869120597(119896) = 0 we have
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (27)
Thus the controller of DDMFAC is designed as follows
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (28)
u (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (29)
42 Stability and Convergence Analysis of DDMFAC
(1)The Boundedness of (119896) Define the parameter estimationerror of 120593(119896) as follows
(119896) = 120593 (119896) minus (119896) (30)
Substituting (28) into (30) gives
(119896) = 120593 (119896) minus (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 (31)
Similarly define (119896 minus 1) = 120593(119896 minus 1) minus (119896 minus 1) and then (119896 minus 1) = 120593 (119896 minus 1) minus (119896 minus 1) (32)
Combining (31) and (32) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1) (33)
According to (7) and (13) it can be obtained that
Δ119910 (119896) = 120593119879 (119896 minus 1) Δu (119896 minus 1)
Δ119910 (119896 minus 1) = 120593119879 (119896 minus 1) Δu (119896 minus 1) (34)
Substituting (34) into (33) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) ((120573119872 + 120573119863)120593119879 (119896 minus 1) Δu (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578Δu (119896 minus 1) ((120593 (119896 minus 1) minus (119896 minus 1))119879 Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578 (119896 minus 1) Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2) + 120593 (119896) minus 120593 (119896 minus 1)
(35)
10 Complexity
Because |120593(119896)| le 119887 is bounded [16] |120593(119896)minus120593(119896minus1)| le 2119887Taking the modulus of both sides of (35) we can obtain
1003817100381710038171003817 (119896)1003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2)100381710038171003817100381710038171003817100381710038171003817 + 2119887 (36)
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
= 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 + (minus2 + 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
times 1205781003817100381710038171003817 (119896 minus 1) Δu (119896 minus 1)10038171003817100381710038172120583 + Δu (119896 minus 1)2
(37)
Given that 0 lt 120578 lt 2 120583 gt 0(minus2 + 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2) lt 0 (38)
Synthesizing (37) and (38) we have
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
le 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 (39)
According to (36) and (39) there is obviously a positivenumber 1198891 isin (0 1) which makes
1003817100381710038171003817 (119896)1003817100381710038171003817 le 1198891 1003817100381710038171003817 (119896 minus 1)1003817100381710038171003817 + 2119887 le sdot sdot sdotle 1198891198961 1003817100381710038171003817 (0)1003817100381710038171003817 + 2119887 (1 minus 1198891198961)1 minus 1198891 (40)
Therefore (119896) is bounded(2)TheConvergence of Tracking Error 119890(119896) Tracking error 119890(119896)is derived as
119890 (119896) = 119910119889 minus 119910 (119896) = 119910119889 minus 119910 (119896 minus 1) minus 120593119879 (119896) Δu (119896)= 119890 (119896 minus 1) minus 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
(120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)= (1 minus 120588119863 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)
minus 120588119872 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 119890 (119896)
(41)
Transposition gives
119890 (119896) = [1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172] 119890 (119896 minus 1) (42)
So as long as appropriate parameters 120588119872 120588119863 and 120582 areselected there must be a positive number 1198892 isin (0 1) thatmakes
0 lt 10038171003817100381710038171003817100381710038171003817100381710038171 minus(120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172
1003817100381710038171003817100381710038171003817100381710038171003817 le 1198892 lt 1 (43)
By (42) and (43) it can be obtained that
119890 (119896) le 1003817100381710038171003817100381710038171003817100381710038171003817(1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)1003817100381710038171003817100381710038171003817100381710038171003817le 1198892 119890 (119896 minus 1) le sdot sdot sdot le 1198891198962 119890 (0)
(44)
Therefore lim119896rarrinfin119890(119896) = 0 that is 119890(119896) is proved toconverge to zero
(3)The Boundedness of System Input and Output By referringto 1198862 + 1198872 ge 2119886119887 the relational expression can be obtained asfollows
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
120582 + 1003817100381710038171003817120593 (119896)100381710038171003817100381721003817100381710038171003817100381710038171003817100381710038171003817 le
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
2radic120582 1003817100381710038171003817120593 (119896)10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 =
12radic120582 (45)
And we have
u (119896) = u (119896) minus u (119896 minus 1) + u (119896 minus 1) minus sdot sdot sdot + u (1)minus u (0) + u (0) = u (0) + 119896sum
119895=1
u (119895) (46)
Combining (24) (44) (45) and (46) as well as utilizingSchwartzrsquos inequality gives
u (119896) le u (0) + 119896sum119895=1
1003817100381710038171003817Δu (119895)1003817100381710038171003817
le u (0) + 12radic120582
119896sum119895=1
1003817100381710038171003817120588119863119890 (119895 minus 1) + 120588119872119890 (119895)1003817100381710038171003817
le u (0) + 1205881198632radic120582119896sum119895=1
119889119895minus12 119890 (0)
+ 1205881198722radic120582119896sum119895=1
1198891198952 119890 (0)
le u (0) + 1205881198632radic1205821 minus 119889119895minus121 minus 1198892 119890 (0)
+ 1205881198722radic1205821198892 (1 minus 1198891198952)1 minus 1198892 119890 (0)
le u (0) + 120588119863(1 minus 1198892) 2radic120582 119890 (0)
Complexity 11
Table 3 The parameter settings of DDMFAC for different IVs
IV Parameter120588119872 120588119863 120573119872 120573119863 120582 120578 1205831199100 = 60 00275 00275 05000 05000
00010 00750 100001199100 = 70 50119864 minus 4 50119864 minus 4 05000 050001199100 = 80 46119864 minus 4 46119864 minus 4 05000 050001199100 = 100 18119864 minus 4 18119864 minus 4 05000 05000
+ 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)le u (0) + 120588119863 + 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)
(47)
Because the initial tracking error 119890(0) and initial inputu(0) are given bounded the system input u(119896) is boundedaccording to (47) Besides due to the boundedness of 119910119889 and119890(119896) the system output 119910(119896) is obviously bounded43 Fuzzy DDMFAC The controller design and stability andconvergence analysis for DDMFAC have been respectivelydiscussed in detail in the above sections Although thedesigned controller has the characters of both MFAC andDDOILC weighted factors 120588119872 120588119863 120573119872 and 120573119863 are allconstant throughout control process DDMFAC is incapableof adaptively selecting the dominance of MFAC or DDOILCto duly show the expected features in each stage of controlprocessThe aforementioned features of DDMFACwe expectcannot be reflected in this case Such DDMFAC approachwith fixed weighting factors has no advantage comparedwith MFAC or DDOILC which appeals to a reasonable andadaptive adjustment strategy for 120588119872 120588119863 120573119872 and 120573119863 There-fore the adaptive adjustment strategy based on fuzzy logicfor weighting factors is considered because its advantageousperformance in parameter adjustment has been proven inour previous work [45] The formulation of fuzzy rules offersus the opportunity to adjust weighting factors in the waywe expected Such DDMFAC approach with fuzzy adaptiveadjustment for parameters is namedFDDMFAC in this paperConcretely the fuzzy rules can be formulated by referringto the expected features of DDMFAC The input variables offuzzy rules are the absolute values of tracking error 119890(119896 minus 1)and DLM prediction error 119890119901(119896minus1) of MFAC and the outputvariables are 120588119872 120588119863 and 120573119863 Because of the relationshipbetween 120573119872 and 120573119863 there is no need to use 120573119872 as an outputand it is calculated by 120573119872 = 1 minus 120573119863 The fuzzy rules can bedetailedly described as follows
(1) If |119890(119896 minus 1)| and |119890119901(119896 minus 1)| are great then thedominance of DDOILC is preferred and that ofMFAC is restrained in DDMFAC So 120588119863 and 120573119863 haverelatively large values and 120588119872 is a relatively smallnumber
(2) 120588119863 and 120573119863 will show a decreasing trend with thegradual decrease of |119890(119896 minus 1)| and |119890119901(119896 minus 1)| while120588119872 will present an increasing trend
44 Simulations and Result Analysis The proposed DDM-FAC approach is firstly applied to the particle quality controlin SFBGP and then its control effect is compared withMFACand DDOILC under different IVs In the simulation justlike MFAC 119910(119896) is replaced by a predicted value 119910(119896) thatis estimated using (19) See Table 3 for parameter settings ofDDMFAC It is not hard to find 120588119872 = 120588119863 and 120573119872 = 120573119863for a certain IV which means that MFAC and DDOILC areset to have the same dominance in DDMFAC By the wayMFAC and DDOILC can be regarded as the extreme cases ofDDMFAC described as follows
MFAC 120588119872 = 120573119872 = 1120588119863 = 120573119863 = 0
DDOILC 120588119872 = 120573119872 = 0120588119863 = 120573119863 = 1
(48)
The control effects of DDMFAC with the parametersettings in Table 3 together with control curves of MFACand DDOILC are plotted in Figure 5 The simulation resultsactually present the control effects of DDMFAC with threesets of parameter settings From simulation it is indicatedthat all three approaches have gained some certain advantagesunder different IV conditions Concretely as is shown inFigure 5(a) DDMFAC has advantage in control performancecompared with DDOILC and MFAC when 1199100 = 60 butMFAC has absolute superiority when 1199100 = 100 plottedin Figure 5(d) In addition from the simulation results inFigures 5(b) and 5(c) it can be seen that DDOILC andMFAC have achieved certain but not absolute advantagesrespectively This demonstrates that parameter setting hasimportant influence on the control effect of DDMFACTherefore the way to achieve satisfactory control effect is toadaptively adjust weighted parameters of DDMFAC insteadof keeping them constant
Then the idea of fuzzy adjustment is introduced to adjustthe weights 120588119872 120588119863 and 120573119863 The fuzzy adjustment rules ofthree weights are respectively shown in Tables 4 and 5 Therespective membership functions when 1199100 = 60 are shownin Figure 6 and membership functions for the other IVs areno longer displayed Please see Figure 7 for control effectof FDDMFAC and comparisons with MFAC and DDOILCunder different IVs Obviously DDMFAC approach withfuzzy adjustment for parameters that is FDDMFAC has thebest performance in particle quality control for all IVs This
12 Complexity
Table 4 The fuzzy rules of 120588119872120588119872 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB PB PB PB PB PB PB PBNM PB PM PM PM PM PS PSNS PM PM PM PS PS Z ZZ Z Z Z NS NS NS NSPS NS NS NM NM NB NB NBPM NM NM NM NB NB NB NBPB NB NB NB NB NB NB NB
MFACDDOILCDDMFAC
60
80
100
120
APS
(m
)
5 10 150Iterations
(a) 1199100 = 60
MFACDDOILCDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILCDDMFAC
80
90
100
110
120
APS
(m
)
2 4 60Iterations
(c) 1199100 = 80
95
100
105
110
115
120
125
APS
(m
)
5 10 150Iterations
MFACDDOILCDDMFAC
(d) 1199100 = 100
Figure 5 The comparison of control effects of MFAC DDOILC and DDMFAC
Complexity 13
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
02 04 06 08 10D
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
005 01 0150D
NB NM NS Z PS PM PB
times10minus3
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
05 1 15 2 25 3 35 4 450M
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
10 20 30 40 50 600|e(k minus 1)| (m)
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
5 10 15 20 250|ep (k minus 1)| (m)
Figure 6 The membership functions of FDDMFAC when 1199100 = 60
Table 5 The fuzzy rules of 120588119863 120573119863120588119863 120573119863 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB NB NB NB NB NB NB NBNM NB NM NM NM NM NS NSNS NM NM NM NS NS Z ZZ Z Z Z PS PM PM PMPS PS PS PM PM PB PB PBPM PM PM PM PB PB PB PBPB PB PB PB PB PB PB PB
validates the effectiveness of FDDMFAC proposed in thispaper
Although FDDMFAC approach has been proven to havea better control performance than MFAC and DDOILCwe still have a question of whether it is possible that thesuperiority of FDDMFAC is entirely attributed to fuzzyadjustment of parameters and has nothing to do with thestructure of DDMFAC itself In order to clarify this problemanother set of simulations are conducted inwhich the controlperformance of FDDMFAC is compared with fuzzy MFAC(FMFAC) and fuzzy DDOILC (FDDOILC) The absolutevalue of tracking error |119890(119896 minus 1)| is regarded as the inputvariable of fuzzy rules and the step-size factors 1205881 1205882 ofMFAC andDDOILC are the output variablesThe fuzzy rulesof FMFAC and FDDOILC are shown in Table 6 and themembership functions when 1199100 = 60 are shown in Figure 8For the simulation results please see Figure 9 From thesimulation it is concluded that FDDMFAC achieves the best
Table 6 The fuzzy rules of FMFAC and FDDOILC
|119890(119896 minus 1)| NB NM NS Z PS PM PB1205881 1205882 NB NM NS Z PS PM PB
control performance which demonstrates the effectiveness ofthe proposed approach in this paper
5 Conclusions
This paper aimed at solving the problem of particle qualitycontrol in the drug development phase of SFBGP Themodel-based control approaches are incapable of handlingsuch quality control because the accurate process modelcannot be acquired Thus the data-driven and model-freecontrol approaches are considered in this work We firstly
14 Complexity
MFACDDOILCFDDMFAC
5 10 150Iterations
60
80
100
120
APS
(m
)y0 = 60
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
y0 = 70
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
MFACDDOILCFDDMFAC
95
100
105
110
115
120
125A
PS (
m)
5 10 150Iterations
y0 = 100
Figure 7 The comparison of control effects of MFAC DDOILC and FDDMFAC
analyzed the features of MFAC and DDOILC and thenconducted experimental research to compare the perfor-mances of MFAC and DDOILC in particle quality controlunder different IVs The simulation results have verifiedour theoretical analysis and concluded that MFAC andDDOILC both have advantages under different IVs To beused in an actual production environment the expectedDDCapproaches must accommodate various IVs because of thechanges in material attributes Thus the hybrid approachFDDMFAC which combines the advantages of MFAC andDDOILC by adaptively adjusting their respective weightedfactors with fuzzy logic was proposed in this work tocomplement their respective shortcomings Through a series
of simulation results we conclude that FDDMFAC achievesbetter control performance than other approaches whichvalidates the effectiveness of FDDMFAC
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thanks are due to National Natural Science Foundation ofChina under Grant nos 61773105 61374147 61703085 and61004083 and Fundamental Research Funds for the Central
Complexity 15NB NM NS Z PS PM PB
001 002 00302
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
002 00401
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
20 40 600|e(k minus 1)| (m)
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
Figure 8 The membership functions of FDDMFAC when 1199100 = 60
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
60
80
100
120
APS
(m
)
y0 = 60
FMFACFDDOILCFDDMFAC
70
80
90
100
110
120
APS
(m
)
2 4 6 8 100Iterations
y0 = 70
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
100
105
110
115
120
125
APS
(m
)
y0 = 100
Figure 9 The comparison of control effects of FMFAC FDDOILC and FDDMFAC
16 Complexity
Universities under Grant nos N150402001 and N120404014for supporting this research work
References
[1] J Bouffard M Kaster and H Dumont ldquoInfluence of processvariable and physicochemical properties on the granulationmechanism of mannitol in a fluid bed top spray granulatorrdquoDrug Development and Industrial Pharmacy vol 31 no 9 pp923ndash933 2005
[2] H Liu K Wang W Schlindwein and M Li ldquoUsing the Box-Behnken experimental design to optimise operating parametersin pulsed spray fluidised bed granulationrdquo International Journalof Pharmaceutics vol 448 no 2 pp 329ndash338 2013
[3] H Liu and M Li ldquoPopulation balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granula-tionrdquo International Journal of Pharmaceutics vol 468 no 1-2pp 223ndash233 2014
[4] T Albuquerque V H Dias N Poellinger and J F PintoldquoConstruction of a quality index for granules produced byfluidized bed technology and application of the correspondenceanalysis as a discriminant procedurerdquo European Journal ofPharmaceutics and Biopharmaceutics vol 75 no 3 pp 418ndash4242010
[5] H Ehlers A Liu H Raikkonen et al ldquoGranule size control andtargeting in pulsed spray fluid bed granulationrdquo InternationalJournal of Pharmaceutics vol 377 no 1-2 pp 9ndash15 2009
[6] C F W Sanders M J Hounslow and F J Doyle III ldquoIden-tification of models for control of wet granulationrdquo PowderTechnology vol 188 no 3 pp 255ndash263 2009
[7] T Narvanen T Lipsanen O Antikainen H Raikkonen andJ Yliruusi ldquoControlling granule size by granulation liquid feedpulsingrdquo International Journal of Pharmaceutics vol 357 no 1-2pp 132ndash138 2008
[8] A Faure P York and R C Rowe ldquoProcess control and scale-up of pharmaceutical wet granulation processes a reviewrdquoEuropean Journal of Pharmaceutics and Biopharmaceutics vol52 no 3 pp 269ndash277 2001
[9] M Pottmann B A Ogunnaike A A Adetayo and B JEnnis ldquoModel-based control of a granulation systemrdquo PowderTechnology vol 108 no 2-3 pp 192ndash201 2000
[10] E P Gatzke and F J Doyle III ldquoModel predictive control of agranulation system using soft output constraints and prioritizedcontrol objectivesrdquo Powder Technology vol 121 no 2-3 pp 149ndash158 2001
[11] F Y Wang and I T Cameron ldquoReview and future directionsin the modelling and control of continuous drum granulationrdquoPowder Technology vol 124 no 3 pp 238ndash253 2002
[12] T Glaser C F W Sanders F Y Wang et al ldquoModel predictivecontrol of continuous drum granulationrdquo Journal of ProcessControl vol 19 no 4 pp 615ndash622 2009
[13] R Ramachandran andA Chaudhury ldquoModel-based design andcontrol of a continuous drum granulation processrdquo ChemicalEngineering Research and Design vol 90 no 8 pp 1063ndash10732012
[14] Z H Loh D Z L Er L W Chan C V Liew and PW S HengldquoSpray granulation for drug formulationrdquo Expert Opinion onDrug Delivery vol 8 no 12 pp 1645ndash1661 2011
[15] S H Schaafsma N W F Kossen M T Mos L Blauw andA C Hoffmann ldquoEffects and control of humidity and particlemixing in fluid-bed granulationrdquo AIChE Journal vol 45 no 6pp 1202ndash1210 1999
[16] Z-S Hou and S-T Jin ldquoA novel data-driven control approachfor a class of discrete-time nonlinear systemsrdquo IEEE Transac-tions onControl Systems Technology vol 19 no 6 pp 1549ndash15582011
[17] Z Hou and S Jin ldquoData-driven model-free adaptive controlfor a class of MIMO nonlinear discrete-time systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 22no 12 pp 2173ndash2188 2011
[18] D Xu B Jiang and P Shi ldquoA novel model-free adaptivecontrol design for multivariable industrial processesrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6391ndash6398 2014
[19] X Wang X Li J Wang X Fang and X Zhu ldquoData-drivenmodel-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeletonrdquo Information Sciences vol 327pp 246ndash257 2016
[20] L dos Santos Coelho andA A R Coelho ldquoModel-free adaptivecontrol optimization using a chaotic particle swarm approachrdquoChaos Solitons amp Fractals vol 41 no 4 pp 2001ndash2009 2009
[21] L D S Coelho M W Pessoa R Rodrigues Sumar andA Augusto Rodrigues Coelho ldquoModel-free adaptive controldesign using evolutionary-neural compensatorrdquo Expert Systemswith Applications vol 37 no 1 pp 499ndash508 2010
[22] K K Tan T H Lee S N Huang and F M Leu ldquoAdaptive-predictive control of a class of SISOnonlinear systemsrdquoDynam-ics and Control vol 11 no 2 pp 151ndash174 2001
[23] B Zhang andW Zhang ldquoAdaptive predictive functional controlof a class of nonlinear systemsrdquo ISA Transactions vol 45 no2 pp 175ndash183 2006
[24] F L Lu J F Wang C J Fan and S B Chen ldquoAn improvedmodel-free adaptive control with G function fuzzy reasoningregulation design and its applicationsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems andControl Engineering vol 222 no 8 pp 817ndash828 2008
[25] R Chi D Wang Z Hou and S Jin ldquoData-driven optimalterminal iterative learning controlrdquo Journal of Process Controlvol 22 no 10 pp 2026ndash2037 2012
[26] R Chi Z Hou S Jin and D Wang ldquoImproved data-drivenoptimal TILC using time-varying input signalsrdquo Journal ofProcess Control vol 24 no 12 pp 78ndash85 2014
[27] X Bu Z Hou and R Chi ldquoModel free adaptive iterative learn-ing control for farm vehicle path trackingrdquo IFAC ProceedingsVolumes vol 46 no 20 pp 153ndash158 2013
[28] M C Campi A Lecchini and SM Savaresi ldquoVirtual referencefeedback tuning a direct method for the design of feedbackcontrollersrdquo Automatica vol 38 no 8 pp 1337ndash1346 2002
[29] R-C Roman M-B Radac R-E Precup and E M PetriuldquoData-driven model-free adaptive control tuned by virtualreference feedback tuningrdquoActa PolytechnicaHungarica vol 13no 1 pp 83ndash96 2016
[30] P Yan D Liu D Wang and H Ma ldquoData-driven controllerdesign for general MIMO nonlinear systems via virtual refer-ence feedback tuning and neural networksrdquo Neurocomputingvol 171 pp 815ndash825 2016
[31] T Pan and S Li ldquoA hierarchical search and updating databasestrategy for lazy learningrdquo International Journal of InnovativeComputing Information andControl vol 4 no 6 pp 1383ndash13922008
[32] H Zhang R Song Q Wei and T Zhang ldquoOptimal trackingcontrol for a class of nonlinear discrete-time systems withtime delays based on heuristic dynamic programmingrdquo IEEE
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Complexity
the granules enter and leave that size interval The mathe-matical mechanism model of SFBGP is developed based onfollowing assumptions (1) Each particle is spherical (2)Thetotal volume of the particles is conserved (3) The particlesize in a certain particular size interval is represented by theleft boundary of the interval (4) The relative growth rate isuniform for each particle within the same volume intervalA one-dimensional PBM that describes the rate of change inparticle number density function 119899(119905 119897) is given by
120597119899 (119905 119897)120597119905 = 12sdot int1198970
120573(119905 (1198973 minus 1205833)13 120583) 119899 (119905 (1198973 minus 1205833)13) 119899 (119905 120583)(1198973 minus 1205833)23 119889120583
minus 119899 (119905 119897) intinfin0
120573 (119905 119897 120583) 119899 (119905 120583) 119889120583
(1)
where 119899(119905 119897) is the particle number density function in termsof particle diameter 119897 and120573 is the aggregationmodel [37 38]120573(119905 119897 120583) can generally be partitioned into size dependent andsize independent parts [3 37 38 40 41]
120573 (119905 119897 119906) = 1205730 (119905) 120573lowast (119897 120583) (2)
where 1205730(119905) is the granulation rate constant that can bethought of as an aggregation efficiency and depends onall parameters involved in the aggregation process exceptparticle size such as the operating condition and binderproperties The latter term 120573lowast(119897 120583) reflects the influence ofparticle size on the likelihood of aggregation
In order to solve a PBM in (1) the discretized approach[42 43] is applied to determine the change in number ofparticles119873119894 in interval 119894 as
d119873119894d119905 = 119894minus2sum
119895=1
2119895minus119894+1120573119894minus1119895119873119894minus1119873119895 + 12120573119894minus1119894minus11198732119894minus1
minus 119873119894 119894minus1sum119895=1
2119895minus119894120573119894119895119873119895 minus 119873119894119899maxsum119895=119894
120573119894119895119873119895(3)
where119873119894 is the number of particles within particle size range(119871 119894 119871 119894+1) 119871 119894 and 119871 119894+1 are the lower and upper limits of 119894thparticle size interval 119894 = 1 2 119899max and 119899max is the totalnumber of particle size intervals with a value of 12 in thispaper In the discretization scheme the length domain ofparticles is divided into geometric intervals in the way thatthe upper and lower limits of each size interval are in a ratioof 119903 = 119871 119894+1119871 119894 = 3radic2 The size range used in modeling isfrom 50 120583m to 800120583mwhich is divided into 12 intervals Thefunction ldquoode45rdquo inMATLAB is used to solve (3) to obtain thefinal number of particles in each size interval Then the APSof final granules119863119898 is calculated by the following formula
119863119898 = 119899maxsum119894=1
119881(119873119891119894) 119889119901119894 (4)
where 119881(119873119891119894) is the volume fraction of end granules at sizeinterval 119894 = 1 2 119899max119873119891119894 is the number of end granulesin the 119894th size interval and 119889119901119894 is the geometric mean of lowerlimit and upper limit of 119894th size interval 119894 = 1 2 119899max
We have just briefly introduced the general framework ofPBM-basedmechanismmodel of SFBGPDetailed derivationprocess and setting of model parameters are given in [37 38]and they will not be covered hereThe developed mechanismmodel is capable of not only linking APS of particles withthe operating variable (binder spray rate) but also reflectingthe relationship between APS and material attributes suchas viscosity of binder and particle density Hence thismechanism model can be used for simulating the practicalchallenges in particle quality control of SFBGP For examplea series of IVs can be constructed by changing the materialattributes while keeping the operating condition unchangedTo veritably simulate the actual granulation process the inputcontrol curve should be time-varying But to simplify thingsthe input control curve is divided into three stages in thewhole granulation process in this work
22 The Data-Driven Control Approaches Used in This WorkAs the classic and representative DDC methods MFAC andDDOILChave somemerits thatmake themmore competitivein practical control applications Therefore in view of thesemerits MFAC and DDOILC are selected and applied toperform data-driven particle quality control for SFBGP inthis work Now we first briefly introduce their controllerdesign procedures For details please refer to [16 25]
A repeatable batch process system to be controlled isgiven as follows
119910 (119896) = 119891 (u (119896)) (5)
where 119910(119896) and u(119896) are the system output and input at 119896thiteration respectively 119896 is the number of iterations and 119891(∙)is an unknown nonlinear functionThis nonlinear system (5)satisfies the following assumptions
(1) The derivatives of 119891(∙) with respect to control inputu(119896) are continuous
(2) System (5) is generalized Lipschitz that is |Δ119910(119896 +1)| le 119887Δu(119896) for any 119896 and Δu(119896) = 0 whereΔ119910(119896 + 1) = 119910(119896 + 1) minus 119910(119896) Δu(119896) = u(119896) minus u(119896 minus 1)and 119887 is a positive constant
221 MFAC Design For the one-step-ahead controller [44]the control goal is to seek a control input sequence u(119896) thatbrings 119910(119896 + 1) to 119910119889 where 119910119889 is the desired system outputsignal In general the weighted one-step-ahead controllermay lead to steady-state tracking error [16] So the followingcontrol input criterion function is used to design the controllaw
119869 (u (119896)) = 1003816100381610038161003816119910119889 minus 1199101 (119896 + 1)10038161003816100381610038162 + 120582 Δu (119896)2 (6)
where 120582 gt 0 is a weighting constantSubstituting the dynamic linearization form
1199101 (119896 + 1) = 119910 (119896) + 120593119879 (119896) Δu (119896) (7)
Complexity 5
into (6) as well as letting 120597119869120597u(119896) be zero givesu (119896) = u (119896 minus 1) + 1205881120593 (119896) (119910119889 minus 119910 (119896))
120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (8)
where 1205881 isin (0 1] is a step-size constant that is added to make(8) more general and 120593(119896) is PPD vector
Because PPD vector 120593(119896) is unknown it needs to beestimated Define criterion function for the unknown PPDvector as follows
119869 ( (119896)) = 10038161003816100381610038161003816Δ119910 (119896) minus 119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(9)
where 120583 gt 0 is a weighting factor and (119896) is the estimationof 120593(119896)
Using optimal condition 120597119869120597(119896) = 0 then gives
(119896)= (119896 minus 1)
+ 120578Δu (119896 minus 1) (Δ119910 (119896) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2
(10)
where 120578 isin (0 2) is a step-size constantDefining the tracking error 119890(119896) = 119910119889 minus 119910(119896) and
combining PPD estimation and input control law MFACscheme is designed as follows
(119896)= (119896 minus 1)
+ 120578Δu (119896 minus 1) (Δ119910 (119896) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2
u (119896) = u (119896 minus 1) + 1205881 (119896)120582 + 1003817100381710038171003817 (119896)10038171003817100381710038172 119890 (119896)
(11)
222 DDOILC Design The control objective is to find anappropriate control input sequence u(119896) such that the systemoutput 119910(119896) follows 119910119889 Consider the control input criterionfunction as
119869 (u (119896)) = 1003816100381610038161003816119910119889 minus 1199102 (119896)10038161003816100381610038162 + 120582 Δu (119896)2 (12)
Substituting the dynamic linearization equation
1199102 (119896) = 119910 (119896 minus 1) + 120593119879 (119896) Δu (119896) (13)
into (12) and the optimal condition leads to
u (119896) = u (119896 minus 1) + 1205882120593 (119896)120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 119890 (119896 minus 1) (14)
where 1205882 isin (0 1] determines the step size and 119890(119896 minus 1) =119910119889 minus 119910(119896 minus 1)
Let (119896) denote its estimation at 119896th iterationThendefinethe criterion function as
119869 ( (119896)) = 10038161003816100381610038161003816Δ119910 (119896 minus 1) minus 119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(15)
According to optimal condition the iterative updating law ofestimate (119896) is developed as
(119896) = (119896 minus 1)+ 120578Δu (119896 minus 1) (Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))
120583 + Δu (119896 minus 1)2 (16)
where 120583 gt 0 0 lt 120578 lt 2The DDOILC scheme is designed as follows
(119896) = (119896 minus 1)+ 120578Δu (119896 minus 1) (Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))
120583 + Δu (119896 minus 1)2 u (119896) = u (119896 minus 1) + 1205882 (119896)120582 + 1003817100381710038171003817 (119896)10038171003817100381710038172 119890 (119896 minus 1)
(17)
3 Theoretical Analysis of MFAC and DDOILCand Simulations in SFBGP
31 Theoretical Analysis of MFAC and DDOILC The con-troller design procedures of MFAC and DDOILC have beenintroduced in previous sections Although they have the samesystemic framework and similar expression form of con-troller there are still differences between them which makesthem two different methods Comparing the controllers ofMFAC and DDOILC the main differences can be listed asfollows
(1) Δ119910(119896) and Δ119910(119896 minus 1) are respectively used in PPDiterative updating laws
(2) 119890(119896) and 119890(119896minus1) are separately utilized in control inputiterative learning laws
The expressions of Δ119910(119896) Δ119910(119896minus1) 119890(119896) and 119890(119896minus1) arelisted as follows
MFACΔ119910 (119896) = 119910 (119896) minus 119910 (119896 minus 1)119890 (119896) = 119910119889 minus 119910 (119896)
DDOILCΔ119910 (119896 minus 1) = 119910 (119896 minus 1) minus 119910 (119896 minus 2)119890 (119896 minus 1) = 119910119889 minus 119910 (119896 minus 1)
(18)
Obviously the main differences between MFAC andDDOILC can be described as follows system output 119910(119896) at119896th iteration is used in MFAC but is not used in DDOILC Inpractical application all the system outputs used inDDOILC119910(119896minus1) and 119910(119896minus2) are the actual values at (119896minus1)th and (119896minus2)th iterations In MFAC however the actual value of system
6 Complexity
y
yd
y (k + 1)y(k)
y(k)y (k minus 1)
y (k minus 1)
y(k minus 2)
IV = y0
u(k minus 2) u(k minus 1) u(k) u(k + 1) u(k + 2) Input
e (k)
e (k minus 1)ep (k)
ep (k minus 1)
Δy (k)
Δy(k minus 1)
Tracking errorPredicted error of DLM
Δy
Figure 2 The variable relations during control process of MFAC and DDOILC
output 119910(119896) is obtained only after the calculation of u(119896) So119910(119896) does not have actual value when computing u(119896) anda predicted system output 119910(119896) will be used to replace 119910(119896)in practical application of MFAC 119910(119896) is estimated by thefollowing DLM
119910 (119896) = 119910 (119896 minus 1) + 120593119879 (119896 minus 1) Δu (119896 minus 1) (19)
and 119890(119896) is replaced by 119890(119896) = 119910119889 minus 119910(119896)Figure 2 gives the variable relations during control pro-
cess of MFAC and DDOILC In terms of a single controlprocess curve when system output 119910 is greatly different from119910119889 that is the tracking error 119890 is relatively great 119890(119896 minus 1)is much larger than 119890(119896) and DDOILC will achieve a biggerupdating step size for control input which gives DDOILC afaster convergence speed As 119910 keeps getting closer to 119910119889 itis found that DLM prediction error 119890119901 is getting smaller andsmaller until 119890119901 = 0 The difference between 119890(119896 minus 1) and 119890(119896)will also be smaller and smaller DDOILC will not only loseits advantage but also perform worse than MFAC when 119910 isvery close to119910119889 because a larger updating step size for controlinput will bring difficulty in convergence while at this timeMFAC will achieve a fast and stable control effect due to theimprovement of prediction accuracy of DLM From anotherpoint of view if IV of system output 1199100 is far away from 119910119889the initial tracking error and DLM prediction error will beboth at a high level and DDOILC will perform better thanMFAC because a greater updating step size for control inputis necessary to give a faster convergence speed But as1199100 keepsgetting closer to 119910119889 119890119901 will gradually decrease and MFACwill gradually show its advantage Therefore the conclusions
obtained from theoretical analysis can be summarized asfollows
(1) When tracking error is large and 1199100 is relatively faraway from119910119889 DDOILCwill have a faster convergencespeed and a better control performance
(2) When tracking error gradually decreases and 1199100gradually gets closer to 119910119889 MFAC will graduallyreveal its advantage and achieve better control effect
32 SimulationVerification To validate conclusions obtainedfrom theoretical analysis simulation verification in SFBGP isconducted In this simulation particle quality to be controlledis APS and the desired quality is 119910119889 = 120 120583mThe viscosityof binder is treated as material attribute to be changed andthen a series of IVs (1199100) are constructed by adjusting viscosityof binder while keeping the operating condition unchangedFour sets of IVs are selected as representatives in this workand the detailed establishing information of them is shownin Table 1 Besides the parameter settings of MFAC andDDOILC for different IVs are listed in Table 2
For different IVs we conduct particle quality control inSFBGP based on MFAC and DDOILC and the control per-formances are plotted in Figure 3Moreover DLMpredictionerror profiles of MFAC are shown in Figure 4 Then theexplanation and analysis of simulation results will be givenAs is shown in Figure 3(a) DDOILC performs better thanMFAC when 1199100 = 60 that is 1199100 is relatively far away from119910119889 When 1199100 = 70 as is described in Figure 3(b) DDOILChas a faster convergence speed when APS is controlled from70 120583m to about 110 120583m but MFAC achieves a relatively better
Complexity 7
MFACDDOILC
5 10 150Iterations
60
80
100
120A
PS (
m)
(a) 1199100 = 60
MFACDDOILC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
(c) 1199100 = 80
MFACDDOILC
5 10 150Iterations
95
100
105
110
115
120
125
APS
(m
)
(d) 1199100 = 100
Figure 3 The control profiles of APS in SFBGP based on MFAC and DDOILC
Table 1The detailed establishing information of representative IVs
Info IV1199100 = 60 1199100 = 70 1199100 = 80 1199100 = 100
Operatingconditions(gs)
First stage 03 second stage 01 third stage 015
Viscosity ofbinder (Pasdots) 00214 00250 00304 00625
control performance when it is controlled from 110 120583m to 119910119889which indicates that both DDOILC and MFAC have certainadvantages in such case However from Figure 3(c) it can beseen that MFAC is slightly superior to DDOILC in controleffect when 1199100 = 80 and this just manifests the advantage ofMFAC As 1199100 gets closer and closer to 119910119889 such as 1199100 = 100
Table 2The parameter settings ofMFAC andDDOILC for differentIVs
IV Parameter1205881 1205882 120582 120578 1205831199100 = 60 00505 00712
00010 00750 100001199100 = 70 00047 005031199100 = 80 00032 001991199100 = 100 00020 149119864 minus 4
plotted in Figure 3(d)MFAC has a huge advantage comparedwith DDOILC It is obvious that the simulation results haveverified the conclusions obtained from theoretical analysisThe reason for increasing superiority of MFAC is attributedto the smaller and smaller prediction error 119890119901 of DLM Asis shown in Figure 4 on the one hand the curves of 119890119901 for
8 Complexity
IV = 60IV = 70
IV = 80IV = 100
IV = 60 14320IV = 70 08061IV = 80 04718IV = 100 04000
5 10 150Iterations
0
5
10
15
20
25
e p(
m)
Average of ep
Figure 4 The DLM prediction error profiles of MFAC for differentIVs
different IVs all show a decreasing trend along with iterationuntil 119890119901 = 0 as APS keeps getting closer to 119910119889 which is inaccordance with our obtained conclusion On the other handfor different IVs 119890119901 will also decrease with the increase of IVwhich is proved by comparing the curves of 119890119901 under differentIVs To further illustrate this point the averages of 119890119901 fordifferent IVs are computed and then listed in Figure 4 andthe decreasing averages of 119890119901 along with the increase of IValso validate the rationality of our analysis
4 The Proposed Hybrid DDC Approach andSimulation Study in SFBGP
Through theoretical analysis and simulation study we haveproved that MFAC and DDOILC both have strengths inparticle quality control under different IVs In real-worldapplications the same operating condition may give entirelydifferent IVs due to the changes in material properties whichbrings great difficulties in the selection of control methodsThus the uncertainties of IV have to be considered duringcontrol application In order to achieve favourable overallcontrol effect it is demanded that the DDC approaches mustaccommodate various IVs Therefore we try to combineadvantages of MFAC and DDOILC to present a hybridapproach that is named DDMFAC The basic idea is that thefeatures of MFAC and DDOILC are both reflected in thecriterion functions when designing controller of DDMFACThe expected features of hybrid approach can be summarizedas the following three points
(1) When tracking error is large DDMFACmanifests thefeature of DDOILC to acquire a faster convergencespeed
(2) When tracking error is small enough DDMFAC hasthe ability ofMFAC to achieve a fast and stable controleffect
(3) DDMFAC is able to achieve better control perfor-mance for different IVs
Through such combination the shortcomings of MFACand DDOILC will be complemented and their respec-tive advantages will be highlighted in DDMFAC Next wedescribe the controller design for DDMFAC in Section 41
41 DDMFAC Design Consider the following control inputcriterion function
119869 (u (119896)) = 120572119872 1003816100381610038161003816119910119889 minus 1199101 (119896 + 1)10038161003816100381610038162 + 120572119863 1003816100381610038161003816119910119889 minus 1199102 (119896)10038161003816100381610038162+ 1205821 Δu (119896)2 (20)
where |119910119889 minus 1199101(119896 + 1)|2 and |119910119889 minus 1199102(119896)|2 respectively comefrom MFAC and DDOILC and 120572119872 120572119863 are the respectiveweighted factors and 120572119872 + 120572119863 = 0
Substituting (7) and (13) into (20) gives
119869 (u (119896)) = 120572119872 10038171003817100381710038171003817119890 (119896) minus 120593119879 (119896) Δu (119896)100381710038171003817100381710038172+ 120572119863 10038171003817100381710038171003817119890 (119896 minus 1) minus 120593119879 (119896) Δu (119896)100381710038171003817100381710038172+ 1205821 Δu (119896)2
(21)
Using optimal condition 120597119869120597u(119896) = 0 then gives
Δu (119896) = 120593 (119896) (120572119872119890 (119896) + 120572119863119890 (119896 minus 1))1205821 + (120572119872 + 120572119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (22)
Dividing numerator and denominator by 120572119872+120572119863 at the righthand of (22) we haveΔu (119896)= 120593 (119896) ((120572119872 (120572119872 + 120572119863)) 119890 (119896) + (120572119863 (120572119872 + 120572119863)) 119890 (119896 minus 1))
(1205821 (120572119872 + 120572119863)) + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (23)
Letting 120588119872 = 120572119872(120572119872 + 120572119863) 120588119863 = 120572119863(120572119872 + 120572119863) and 120582 =1205821(120572119872 + 120572119863) thenu (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (24)
where 120582 gt 0 is the weighted factor and 120588119872 120588119863 isin [0 1] are thestep-size factors
Define the following PPD vector criterion function
119869 ( (119896))= 10038171003817100381710038171003817120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896) Δu (119896 minus 1)100381710038171003817100381710038172
+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172 (25)
where 120573119872 120573119863 isin [0 1] are the weighted factors and 120573119872+120573119863 =1We rewrite (25) as
119869 ( (119896)) = 10038161003816100381610038161003816120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1)minus 119879 (119896 minus 1) Δu (119896 minus 1) minus Δ119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(26)
Complexity 9
Utilizing optimal condition 120597119869120597(119896) = 0 we have
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (27)
Thus the controller of DDMFAC is designed as follows
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (28)
u (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (29)
42 Stability and Convergence Analysis of DDMFAC
(1)The Boundedness of (119896) Define the parameter estimationerror of 120593(119896) as follows
(119896) = 120593 (119896) minus (119896) (30)
Substituting (28) into (30) gives
(119896) = 120593 (119896) minus (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 (31)
Similarly define (119896 minus 1) = 120593(119896 minus 1) minus (119896 minus 1) and then (119896 minus 1) = 120593 (119896 minus 1) minus (119896 minus 1) (32)
Combining (31) and (32) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1) (33)
According to (7) and (13) it can be obtained that
Δ119910 (119896) = 120593119879 (119896 minus 1) Δu (119896 minus 1)
Δ119910 (119896 minus 1) = 120593119879 (119896 minus 1) Δu (119896 minus 1) (34)
Substituting (34) into (33) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) ((120573119872 + 120573119863)120593119879 (119896 minus 1) Δu (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578Δu (119896 minus 1) ((120593 (119896 minus 1) minus (119896 minus 1))119879 Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578 (119896 minus 1) Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2) + 120593 (119896) minus 120593 (119896 minus 1)
(35)
10 Complexity
Because |120593(119896)| le 119887 is bounded [16] |120593(119896)minus120593(119896minus1)| le 2119887Taking the modulus of both sides of (35) we can obtain
1003817100381710038171003817 (119896)1003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2)100381710038171003817100381710038171003817100381710038171003817 + 2119887 (36)
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
= 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 + (minus2 + 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
times 1205781003817100381710038171003817 (119896 minus 1) Δu (119896 minus 1)10038171003817100381710038172120583 + Δu (119896 minus 1)2
(37)
Given that 0 lt 120578 lt 2 120583 gt 0(minus2 + 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2) lt 0 (38)
Synthesizing (37) and (38) we have
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
le 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 (39)
According to (36) and (39) there is obviously a positivenumber 1198891 isin (0 1) which makes
1003817100381710038171003817 (119896)1003817100381710038171003817 le 1198891 1003817100381710038171003817 (119896 minus 1)1003817100381710038171003817 + 2119887 le sdot sdot sdotle 1198891198961 1003817100381710038171003817 (0)1003817100381710038171003817 + 2119887 (1 minus 1198891198961)1 minus 1198891 (40)
Therefore (119896) is bounded(2)TheConvergence of Tracking Error 119890(119896) Tracking error 119890(119896)is derived as
119890 (119896) = 119910119889 minus 119910 (119896) = 119910119889 minus 119910 (119896 minus 1) minus 120593119879 (119896) Δu (119896)= 119890 (119896 minus 1) minus 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
(120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)= (1 minus 120588119863 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)
minus 120588119872 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 119890 (119896)
(41)
Transposition gives
119890 (119896) = [1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172] 119890 (119896 minus 1) (42)
So as long as appropriate parameters 120588119872 120588119863 and 120582 areselected there must be a positive number 1198892 isin (0 1) thatmakes
0 lt 10038171003817100381710038171003817100381710038171003817100381710038171 minus(120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172
1003817100381710038171003817100381710038171003817100381710038171003817 le 1198892 lt 1 (43)
By (42) and (43) it can be obtained that
119890 (119896) le 1003817100381710038171003817100381710038171003817100381710038171003817(1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)1003817100381710038171003817100381710038171003817100381710038171003817le 1198892 119890 (119896 minus 1) le sdot sdot sdot le 1198891198962 119890 (0)
(44)
Therefore lim119896rarrinfin119890(119896) = 0 that is 119890(119896) is proved toconverge to zero
(3)The Boundedness of System Input and Output By referringto 1198862 + 1198872 ge 2119886119887 the relational expression can be obtained asfollows
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
120582 + 1003817100381710038171003817120593 (119896)100381710038171003817100381721003817100381710038171003817100381710038171003817100381710038171003817 le
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
2radic120582 1003817100381710038171003817120593 (119896)10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 =
12radic120582 (45)
And we have
u (119896) = u (119896) minus u (119896 minus 1) + u (119896 minus 1) minus sdot sdot sdot + u (1)minus u (0) + u (0) = u (0) + 119896sum
119895=1
u (119895) (46)
Combining (24) (44) (45) and (46) as well as utilizingSchwartzrsquos inequality gives
u (119896) le u (0) + 119896sum119895=1
1003817100381710038171003817Δu (119895)1003817100381710038171003817
le u (0) + 12radic120582
119896sum119895=1
1003817100381710038171003817120588119863119890 (119895 minus 1) + 120588119872119890 (119895)1003817100381710038171003817
le u (0) + 1205881198632radic120582119896sum119895=1
119889119895minus12 119890 (0)
+ 1205881198722radic120582119896sum119895=1
1198891198952 119890 (0)
le u (0) + 1205881198632radic1205821 minus 119889119895minus121 minus 1198892 119890 (0)
+ 1205881198722radic1205821198892 (1 minus 1198891198952)1 minus 1198892 119890 (0)
le u (0) + 120588119863(1 minus 1198892) 2radic120582 119890 (0)
Complexity 11
Table 3 The parameter settings of DDMFAC for different IVs
IV Parameter120588119872 120588119863 120573119872 120573119863 120582 120578 1205831199100 = 60 00275 00275 05000 05000
00010 00750 100001199100 = 70 50119864 minus 4 50119864 minus 4 05000 050001199100 = 80 46119864 minus 4 46119864 minus 4 05000 050001199100 = 100 18119864 minus 4 18119864 minus 4 05000 05000
+ 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)le u (0) + 120588119863 + 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)
(47)
Because the initial tracking error 119890(0) and initial inputu(0) are given bounded the system input u(119896) is boundedaccording to (47) Besides due to the boundedness of 119910119889 and119890(119896) the system output 119910(119896) is obviously bounded43 Fuzzy DDMFAC The controller design and stability andconvergence analysis for DDMFAC have been respectivelydiscussed in detail in the above sections Although thedesigned controller has the characters of both MFAC andDDOILC weighted factors 120588119872 120588119863 120573119872 and 120573119863 are allconstant throughout control process DDMFAC is incapableof adaptively selecting the dominance of MFAC or DDOILCto duly show the expected features in each stage of controlprocessThe aforementioned features of DDMFACwe expectcannot be reflected in this case Such DDMFAC approachwith fixed weighting factors has no advantage comparedwith MFAC or DDOILC which appeals to a reasonable andadaptive adjustment strategy for 120588119872 120588119863 120573119872 and 120573119863 There-fore the adaptive adjustment strategy based on fuzzy logicfor weighting factors is considered because its advantageousperformance in parameter adjustment has been proven inour previous work [45] The formulation of fuzzy rules offersus the opportunity to adjust weighting factors in the waywe expected Such DDMFAC approach with fuzzy adaptiveadjustment for parameters is namedFDDMFAC in this paperConcretely the fuzzy rules can be formulated by referringto the expected features of DDMFAC The input variables offuzzy rules are the absolute values of tracking error 119890(119896 minus 1)and DLM prediction error 119890119901(119896minus1) of MFAC and the outputvariables are 120588119872 120588119863 and 120573119863 Because of the relationshipbetween 120573119872 and 120573119863 there is no need to use 120573119872 as an outputand it is calculated by 120573119872 = 1 minus 120573119863 The fuzzy rules can bedetailedly described as follows
(1) If |119890(119896 minus 1)| and |119890119901(119896 minus 1)| are great then thedominance of DDOILC is preferred and that ofMFAC is restrained in DDMFAC So 120588119863 and 120573119863 haverelatively large values and 120588119872 is a relatively smallnumber
(2) 120588119863 and 120573119863 will show a decreasing trend with thegradual decrease of |119890(119896 minus 1)| and |119890119901(119896 minus 1)| while120588119872 will present an increasing trend
44 Simulations and Result Analysis The proposed DDM-FAC approach is firstly applied to the particle quality controlin SFBGP and then its control effect is compared withMFACand DDOILC under different IVs In the simulation justlike MFAC 119910(119896) is replaced by a predicted value 119910(119896) thatis estimated using (19) See Table 3 for parameter settings ofDDMFAC It is not hard to find 120588119872 = 120588119863 and 120573119872 = 120573119863for a certain IV which means that MFAC and DDOILC areset to have the same dominance in DDMFAC By the wayMFAC and DDOILC can be regarded as the extreme cases ofDDMFAC described as follows
MFAC 120588119872 = 120573119872 = 1120588119863 = 120573119863 = 0
DDOILC 120588119872 = 120573119872 = 0120588119863 = 120573119863 = 1
(48)
The control effects of DDMFAC with the parametersettings in Table 3 together with control curves of MFACand DDOILC are plotted in Figure 5 The simulation resultsactually present the control effects of DDMFAC with threesets of parameter settings From simulation it is indicatedthat all three approaches have gained some certain advantagesunder different IV conditions Concretely as is shown inFigure 5(a) DDMFAC has advantage in control performancecompared with DDOILC and MFAC when 1199100 = 60 butMFAC has absolute superiority when 1199100 = 100 plottedin Figure 5(d) In addition from the simulation results inFigures 5(b) and 5(c) it can be seen that DDOILC andMFAC have achieved certain but not absolute advantagesrespectively This demonstrates that parameter setting hasimportant influence on the control effect of DDMFACTherefore the way to achieve satisfactory control effect is toadaptively adjust weighted parameters of DDMFAC insteadof keeping them constant
Then the idea of fuzzy adjustment is introduced to adjustthe weights 120588119872 120588119863 and 120573119863 The fuzzy adjustment rules ofthree weights are respectively shown in Tables 4 and 5 Therespective membership functions when 1199100 = 60 are shownin Figure 6 and membership functions for the other IVs areno longer displayed Please see Figure 7 for control effectof FDDMFAC and comparisons with MFAC and DDOILCunder different IVs Obviously DDMFAC approach withfuzzy adjustment for parameters that is FDDMFAC has thebest performance in particle quality control for all IVs This
12 Complexity
Table 4 The fuzzy rules of 120588119872120588119872 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB PB PB PB PB PB PB PBNM PB PM PM PM PM PS PSNS PM PM PM PS PS Z ZZ Z Z Z NS NS NS NSPS NS NS NM NM NB NB NBPM NM NM NM NB NB NB NBPB NB NB NB NB NB NB NB
MFACDDOILCDDMFAC
60
80
100
120
APS
(m
)
5 10 150Iterations
(a) 1199100 = 60
MFACDDOILCDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILCDDMFAC
80
90
100
110
120
APS
(m
)
2 4 60Iterations
(c) 1199100 = 80
95
100
105
110
115
120
125
APS
(m
)
5 10 150Iterations
MFACDDOILCDDMFAC
(d) 1199100 = 100
Figure 5 The comparison of control effects of MFAC DDOILC and DDMFAC
Complexity 13
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
02 04 06 08 10D
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
005 01 0150D
NB NM NS Z PS PM PB
times10minus3
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
05 1 15 2 25 3 35 4 450M
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
10 20 30 40 50 600|e(k minus 1)| (m)
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
5 10 15 20 250|ep (k minus 1)| (m)
Figure 6 The membership functions of FDDMFAC when 1199100 = 60
Table 5 The fuzzy rules of 120588119863 120573119863120588119863 120573119863 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB NB NB NB NB NB NB NBNM NB NM NM NM NM NS NSNS NM NM NM NS NS Z ZZ Z Z Z PS PM PM PMPS PS PS PM PM PB PB PBPM PM PM PM PB PB PB PBPB PB PB PB PB PB PB PB
validates the effectiveness of FDDMFAC proposed in thispaper
Although FDDMFAC approach has been proven to havea better control performance than MFAC and DDOILCwe still have a question of whether it is possible that thesuperiority of FDDMFAC is entirely attributed to fuzzyadjustment of parameters and has nothing to do with thestructure of DDMFAC itself In order to clarify this problemanother set of simulations are conducted inwhich the controlperformance of FDDMFAC is compared with fuzzy MFAC(FMFAC) and fuzzy DDOILC (FDDOILC) The absolutevalue of tracking error |119890(119896 minus 1)| is regarded as the inputvariable of fuzzy rules and the step-size factors 1205881 1205882 ofMFAC andDDOILC are the output variablesThe fuzzy rulesof FMFAC and FDDOILC are shown in Table 6 and themembership functions when 1199100 = 60 are shown in Figure 8For the simulation results please see Figure 9 From thesimulation it is concluded that FDDMFAC achieves the best
Table 6 The fuzzy rules of FMFAC and FDDOILC
|119890(119896 minus 1)| NB NM NS Z PS PM PB1205881 1205882 NB NM NS Z PS PM PB
control performance which demonstrates the effectiveness ofthe proposed approach in this paper
5 Conclusions
This paper aimed at solving the problem of particle qualitycontrol in the drug development phase of SFBGP Themodel-based control approaches are incapable of handlingsuch quality control because the accurate process modelcannot be acquired Thus the data-driven and model-freecontrol approaches are considered in this work We firstly
14 Complexity
MFACDDOILCFDDMFAC
5 10 150Iterations
60
80
100
120
APS
(m
)y0 = 60
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
y0 = 70
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
MFACDDOILCFDDMFAC
95
100
105
110
115
120
125A
PS (
m)
5 10 150Iterations
y0 = 100
Figure 7 The comparison of control effects of MFAC DDOILC and FDDMFAC
analyzed the features of MFAC and DDOILC and thenconducted experimental research to compare the perfor-mances of MFAC and DDOILC in particle quality controlunder different IVs The simulation results have verifiedour theoretical analysis and concluded that MFAC andDDOILC both have advantages under different IVs To beused in an actual production environment the expectedDDCapproaches must accommodate various IVs because of thechanges in material attributes Thus the hybrid approachFDDMFAC which combines the advantages of MFAC andDDOILC by adaptively adjusting their respective weightedfactors with fuzzy logic was proposed in this work tocomplement their respective shortcomings Through a series
of simulation results we conclude that FDDMFAC achievesbetter control performance than other approaches whichvalidates the effectiveness of FDDMFAC
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thanks are due to National Natural Science Foundation ofChina under Grant nos 61773105 61374147 61703085 and61004083 and Fundamental Research Funds for the Central
Complexity 15NB NM NS Z PS PM PB
001 002 00302
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
002 00401
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
20 40 600|e(k minus 1)| (m)
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
Figure 8 The membership functions of FDDMFAC when 1199100 = 60
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
60
80
100
120
APS
(m
)
y0 = 60
FMFACFDDOILCFDDMFAC
70
80
90
100
110
120
APS
(m
)
2 4 6 8 100Iterations
y0 = 70
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
100
105
110
115
120
125
APS
(m
)
y0 = 100
Figure 9 The comparison of control effects of FMFAC FDDOILC and FDDMFAC
16 Complexity
Universities under Grant nos N150402001 and N120404014for supporting this research work
References
[1] J Bouffard M Kaster and H Dumont ldquoInfluence of processvariable and physicochemical properties on the granulationmechanism of mannitol in a fluid bed top spray granulatorrdquoDrug Development and Industrial Pharmacy vol 31 no 9 pp923ndash933 2005
[2] H Liu K Wang W Schlindwein and M Li ldquoUsing the Box-Behnken experimental design to optimise operating parametersin pulsed spray fluidised bed granulationrdquo International Journalof Pharmaceutics vol 448 no 2 pp 329ndash338 2013
[3] H Liu and M Li ldquoPopulation balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granula-tionrdquo International Journal of Pharmaceutics vol 468 no 1-2pp 223ndash233 2014
[4] T Albuquerque V H Dias N Poellinger and J F PintoldquoConstruction of a quality index for granules produced byfluidized bed technology and application of the correspondenceanalysis as a discriminant procedurerdquo European Journal ofPharmaceutics and Biopharmaceutics vol 75 no 3 pp 418ndash4242010
[5] H Ehlers A Liu H Raikkonen et al ldquoGranule size control andtargeting in pulsed spray fluid bed granulationrdquo InternationalJournal of Pharmaceutics vol 377 no 1-2 pp 9ndash15 2009
[6] C F W Sanders M J Hounslow and F J Doyle III ldquoIden-tification of models for control of wet granulationrdquo PowderTechnology vol 188 no 3 pp 255ndash263 2009
[7] T Narvanen T Lipsanen O Antikainen H Raikkonen andJ Yliruusi ldquoControlling granule size by granulation liquid feedpulsingrdquo International Journal of Pharmaceutics vol 357 no 1-2pp 132ndash138 2008
[8] A Faure P York and R C Rowe ldquoProcess control and scale-up of pharmaceutical wet granulation processes a reviewrdquoEuropean Journal of Pharmaceutics and Biopharmaceutics vol52 no 3 pp 269ndash277 2001
[9] M Pottmann B A Ogunnaike A A Adetayo and B JEnnis ldquoModel-based control of a granulation systemrdquo PowderTechnology vol 108 no 2-3 pp 192ndash201 2000
[10] E P Gatzke and F J Doyle III ldquoModel predictive control of agranulation system using soft output constraints and prioritizedcontrol objectivesrdquo Powder Technology vol 121 no 2-3 pp 149ndash158 2001
[11] F Y Wang and I T Cameron ldquoReview and future directionsin the modelling and control of continuous drum granulationrdquoPowder Technology vol 124 no 3 pp 238ndash253 2002
[12] T Glaser C F W Sanders F Y Wang et al ldquoModel predictivecontrol of continuous drum granulationrdquo Journal of ProcessControl vol 19 no 4 pp 615ndash622 2009
[13] R Ramachandran andA Chaudhury ldquoModel-based design andcontrol of a continuous drum granulation processrdquo ChemicalEngineering Research and Design vol 90 no 8 pp 1063ndash10732012
[14] Z H Loh D Z L Er L W Chan C V Liew and PW S HengldquoSpray granulation for drug formulationrdquo Expert Opinion onDrug Delivery vol 8 no 12 pp 1645ndash1661 2011
[15] S H Schaafsma N W F Kossen M T Mos L Blauw andA C Hoffmann ldquoEffects and control of humidity and particlemixing in fluid-bed granulationrdquo AIChE Journal vol 45 no 6pp 1202ndash1210 1999
[16] Z-S Hou and S-T Jin ldquoA novel data-driven control approachfor a class of discrete-time nonlinear systemsrdquo IEEE Transac-tions onControl Systems Technology vol 19 no 6 pp 1549ndash15582011
[17] Z Hou and S Jin ldquoData-driven model-free adaptive controlfor a class of MIMO nonlinear discrete-time systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 22no 12 pp 2173ndash2188 2011
[18] D Xu B Jiang and P Shi ldquoA novel model-free adaptivecontrol design for multivariable industrial processesrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6391ndash6398 2014
[19] X Wang X Li J Wang X Fang and X Zhu ldquoData-drivenmodel-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeletonrdquo Information Sciences vol 327pp 246ndash257 2016
[20] L dos Santos Coelho andA A R Coelho ldquoModel-free adaptivecontrol optimization using a chaotic particle swarm approachrdquoChaos Solitons amp Fractals vol 41 no 4 pp 2001ndash2009 2009
[21] L D S Coelho M W Pessoa R Rodrigues Sumar andA Augusto Rodrigues Coelho ldquoModel-free adaptive controldesign using evolutionary-neural compensatorrdquo Expert Systemswith Applications vol 37 no 1 pp 499ndash508 2010
[22] K K Tan T H Lee S N Huang and F M Leu ldquoAdaptive-predictive control of a class of SISOnonlinear systemsrdquoDynam-ics and Control vol 11 no 2 pp 151ndash174 2001
[23] B Zhang andW Zhang ldquoAdaptive predictive functional controlof a class of nonlinear systemsrdquo ISA Transactions vol 45 no2 pp 175ndash183 2006
[24] F L Lu J F Wang C J Fan and S B Chen ldquoAn improvedmodel-free adaptive control with G function fuzzy reasoningregulation design and its applicationsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems andControl Engineering vol 222 no 8 pp 817ndash828 2008
[25] R Chi D Wang Z Hou and S Jin ldquoData-driven optimalterminal iterative learning controlrdquo Journal of Process Controlvol 22 no 10 pp 2026ndash2037 2012
[26] R Chi Z Hou S Jin and D Wang ldquoImproved data-drivenoptimal TILC using time-varying input signalsrdquo Journal ofProcess Control vol 24 no 12 pp 78ndash85 2014
[27] X Bu Z Hou and R Chi ldquoModel free adaptive iterative learn-ing control for farm vehicle path trackingrdquo IFAC ProceedingsVolumes vol 46 no 20 pp 153ndash158 2013
[28] M C Campi A Lecchini and SM Savaresi ldquoVirtual referencefeedback tuning a direct method for the design of feedbackcontrollersrdquo Automatica vol 38 no 8 pp 1337ndash1346 2002
[29] R-C Roman M-B Radac R-E Precup and E M PetriuldquoData-driven model-free adaptive control tuned by virtualreference feedback tuningrdquoActa PolytechnicaHungarica vol 13no 1 pp 83ndash96 2016
[30] P Yan D Liu D Wang and H Ma ldquoData-driven controllerdesign for general MIMO nonlinear systems via virtual refer-ence feedback tuning and neural networksrdquo Neurocomputingvol 171 pp 815ndash825 2016
[31] T Pan and S Li ldquoA hierarchical search and updating databasestrategy for lazy learningrdquo International Journal of InnovativeComputing Information andControl vol 4 no 6 pp 1383ndash13922008
[32] H Zhang R Song Q Wei and T Zhang ldquoOptimal trackingcontrol for a class of nonlinear discrete-time systems withtime delays based on heuristic dynamic programmingrdquo IEEE
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Complexity 5
into (6) as well as letting 120597119869120597u(119896) be zero givesu (119896) = u (119896 minus 1) + 1205881120593 (119896) (119910119889 minus 119910 (119896))
120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (8)
where 1205881 isin (0 1] is a step-size constant that is added to make(8) more general and 120593(119896) is PPD vector
Because PPD vector 120593(119896) is unknown it needs to beestimated Define criterion function for the unknown PPDvector as follows
119869 ( (119896)) = 10038161003816100381610038161003816Δ119910 (119896) minus 119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(9)
where 120583 gt 0 is a weighting factor and (119896) is the estimationof 120593(119896)
Using optimal condition 120597119869120597(119896) = 0 then gives
(119896)= (119896 minus 1)
+ 120578Δu (119896 minus 1) (Δ119910 (119896) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2
(10)
where 120578 isin (0 2) is a step-size constantDefining the tracking error 119890(119896) = 119910119889 minus 119910(119896) and
combining PPD estimation and input control law MFACscheme is designed as follows
(119896)= (119896 minus 1)
+ 120578Δu (119896 minus 1) (Δ119910 (119896) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2
u (119896) = u (119896 minus 1) + 1205881 (119896)120582 + 1003817100381710038171003817 (119896)10038171003817100381710038172 119890 (119896)
(11)
222 DDOILC Design The control objective is to find anappropriate control input sequence u(119896) such that the systemoutput 119910(119896) follows 119910119889 Consider the control input criterionfunction as
119869 (u (119896)) = 1003816100381610038161003816119910119889 minus 1199102 (119896)10038161003816100381610038162 + 120582 Δu (119896)2 (12)
Substituting the dynamic linearization equation
1199102 (119896) = 119910 (119896 minus 1) + 120593119879 (119896) Δu (119896) (13)
into (12) and the optimal condition leads to
u (119896) = u (119896 minus 1) + 1205882120593 (119896)120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 119890 (119896 minus 1) (14)
where 1205882 isin (0 1] determines the step size and 119890(119896 minus 1) =119910119889 minus 119910(119896 minus 1)
Let (119896) denote its estimation at 119896th iterationThendefinethe criterion function as
119869 ( (119896)) = 10038161003816100381610038161003816Δ119910 (119896 minus 1) minus 119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(15)
According to optimal condition the iterative updating law ofestimate (119896) is developed as
(119896) = (119896 minus 1)+ 120578Δu (119896 minus 1) (Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))
120583 + Δu (119896 minus 1)2 (16)
where 120583 gt 0 0 lt 120578 lt 2The DDOILC scheme is designed as follows
(119896) = (119896 minus 1)+ 120578Δu (119896 minus 1) (Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))
120583 + Δu (119896 minus 1)2 u (119896) = u (119896 minus 1) + 1205882 (119896)120582 + 1003817100381710038171003817 (119896)10038171003817100381710038172 119890 (119896 minus 1)
(17)
3 Theoretical Analysis of MFAC and DDOILCand Simulations in SFBGP
31 Theoretical Analysis of MFAC and DDOILC The con-troller design procedures of MFAC and DDOILC have beenintroduced in previous sections Although they have the samesystemic framework and similar expression form of con-troller there are still differences between them which makesthem two different methods Comparing the controllers ofMFAC and DDOILC the main differences can be listed asfollows
(1) Δ119910(119896) and Δ119910(119896 minus 1) are respectively used in PPDiterative updating laws
(2) 119890(119896) and 119890(119896minus1) are separately utilized in control inputiterative learning laws
The expressions of Δ119910(119896) Δ119910(119896minus1) 119890(119896) and 119890(119896minus1) arelisted as follows
MFACΔ119910 (119896) = 119910 (119896) minus 119910 (119896 minus 1)119890 (119896) = 119910119889 minus 119910 (119896)
DDOILCΔ119910 (119896 minus 1) = 119910 (119896 minus 1) minus 119910 (119896 minus 2)119890 (119896 minus 1) = 119910119889 minus 119910 (119896 minus 1)
(18)
Obviously the main differences between MFAC andDDOILC can be described as follows system output 119910(119896) at119896th iteration is used in MFAC but is not used in DDOILC Inpractical application all the system outputs used inDDOILC119910(119896minus1) and 119910(119896minus2) are the actual values at (119896minus1)th and (119896minus2)th iterations In MFAC however the actual value of system
6 Complexity
y
yd
y (k + 1)y(k)
y(k)y (k minus 1)
y (k minus 1)
y(k minus 2)
IV = y0
u(k minus 2) u(k minus 1) u(k) u(k + 1) u(k + 2) Input
e (k)
e (k minus 1)ep (k)
ep (k minus 1)
Δy (k)
Δy(k minus 1)
Tracking errorPredicted error of DLM
Δy
Figure 2 The variable relations during control process of MFAC and DDOILC
output 119910(119896) is obtained only after the calculation of u(119896) So119910(119896) does not have actual value when computing u(119896) anda predicted system output 119910(119896) will be used to replace 119910(119896)in practical application of MFAC 119910(119896) is estimated by thefollowing DLM
119910 (119896) = 119910 (119896 minus 1) + 120593119879 (119896 minus 1) Δu (119896 minus 1) (19)
and 119890(119896) is replaced by 119890(119896) = 119910119889 minus 119910(119896)Figure 2 gives the variable relations during control pro-
cess of MFAC and DDOILC In terms of a single controlprocess curve when system output 119910 is greatly different from119910119889 that is the tracking error 119890 is relatively great 119890(119896 minus 1)is much larger than 119890(119896) and DDOILC will achieve a biggerupdating step size for control input which gives DDOILC afaster convergence speed As 119910 keeps getting closer to 119910119889 itis found that DLM prediction error 119890119901 is getting smaller andsmaller until 119890119901 = 0 The difference between 119890(119896 minus 1) and 119890(119896)will also be smaller and smaller DDOILC will not only loseits advantage but also perform worse than MFAC when 119910 isvery close to119910119889 because a larger updating step size for controlinput will bring difficulty in convergence while at this timeMFAC will achieve a fast and stable control effect due to theimprovement of prediction accuracy of DLM From anotherpoint of view if IV of system output 1199100 is far away from 119910119889the initial tracking error and DLM prediction error will beboth at a high level and DDOILC will perform better thanMFAC because a greater updating step size for control inputis necessary to give a faster convergence speed But as1199100 keepsgetting closer to 119910119889 119890119901 will gradually decrease and MFACwill gradually show its advantage Therefore the conclusions
obtained from theoretical analysis can be summarized asfollows
(1) When tracking error is large and 1199100 is relatively faraway from119910119889 DDOILCwill have a faster convergencespeed and a better control performance
(2) When tracking error gradually decreases and 1199100gradually gets closer to 119910119889 MFAC will graduallyreveal its advantage and achieve better control effect
32 SimulationVerification To validate conclusions obtainedfrom theoretical analysis simulation verification in SFBGP isconducted In this simulation particle quality to be controlledis APS and the desired quality is 119910119889 = 120 120583mThe viscosityof binder is treated as material attribute to be changed andthen a series of IVs (1199100) are constructed by adjusting viscosityof binder while keeping the operating condition unchangedFour sets of IVs are selected as representatives in this workand the detailed establishing information of them is shownin Table 1 Besides the parameter settings of MFAC andDDOILC for different IVs are listed in Table 2
For different IVs we conduct particle quality control inSFBGP based on MFAC and DDOILC and the control per-formances are plotted in Figure 3Moreover DLMpredictionerror profiles of MFAC are shown in Figure 4 Then theexplanation and analysis of simulation results will be givenAs is shown in Figure 3(a) DDOILC performs better thanMFAC when 1199100 = 60 that is 1199100 is relatively far away from119910119889 When 1199100 = 70 as is described in Figure 3(b) DDOILChas a faster convergence speed when APS is controlled from70 120583m to about 110 120583m but MFAC achieves a relatively better
Complexity 7
MFACDDOILC
5 10 150Iterations
60
80
100
120A
PS (
m)
(a) 1199100 = 60
MFACDDOILC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
(c) 1199100 = 80
MFACDDOILC
5 10 150Iterations
95
100
105
110
115
120
125
APS
(m
)
(d) 1199100 = 100
Figure 3 The control profiles of APS in SFBGP based on MFAC and DDOILC
Table 1The detailed establishing information of representative IVs
Info IV1199100 = 60 1199100 = 70 1199100 = 80 1199100 = 100
Operatingconditions(gs)
First stage 03 second stage 01 third stage 015
Viscosity ofbinder (Pasdots) 00214 00250 00304 00625
control performance when it is controlled from 110 120583m to 119910119889which indicates that both DDOILC and MFAC have certainadvantages in such case However from Figure 3(c) it can beseen that MFAC is slightly superior to DDOILC in controleffect when 1199100 = 80 and this just manifests the advantage ofMFAC As 1199100 gets closer and closer to 119910119889 such as 1199100 = 100
Table 2The parameter settings ofMFAC andDDOILC for differentIVs
IV Parameter1205881 1205882 120582 120578 1205831199100 = 60 00505 00712
00010 00750 100001199100 = 70 00047 005031199100 = 80 00032 001991199100 = 100 00020 149119864 minus 4
plotted in Figure 3(d)MFAC has a huge advantage comparedwith DDOILC It is obvious that the simulation results haveverified the conclusions obtained from theoretical analysisThe reason for increasing superiority of MFAC is attributedto the smaller and smaller prediction error 119890119901 of DLM Asis shown in Figure 4 on the one hand the curves of 119890119901 for
8 Complexity
IV = 60IV = 70
IV = 80IV = 100
IV = 60 14320IV = 70 08061IV = 80 04718IV = 100 04000
5 10 150Iterations
0
5
10
15
20
25
e p(
m)
Average of ep
Figure 4 The DLM prediction error profiles of MFAC for differentIVs
different IVs all show a decreasing trend along with iterationuntil 119890119901 = 0 as APS keeps getting closer to 119910119889 which is inaccordance with our obtained conclusion On the other handfor different IVs 119890119901 will also decrease with the increase of IVwhich is proved by comparing the curves of 119890119901 under differentIVs To further illustrate this point the averages of 119890119901 fordifferent IVs are computed and then listed in Figure 4 andthe decreasing averages of 119890119901 along with the increase of IValso validate the rationality of our analysis
4 The Proposed Hybrid DDC Approach andSimulation Study in SFBGP
Through theoretical analysis and simulation study we haveproved that MFAC and DDOILC both have strengths inparticle quality control under different IVs In real-worldapplications the same operating condition may give entirelydifferent IVs due to the changes in material properties whichbrings great difficulties in the selection of control methodsThus the uncertainties of IV have to be considered duringcontrol application In order to achieve favourable overallcontrol effect it is demanded that the DDC approaches mustaccommodate various IVs Therefore we try to combineadvantages of MFAC and DDOILC to present a hybridapproach that is named DDMFAC The basic idea is that thefeatures of MFAC and DDOILC are both reflected in thecriterion functions when designing controller of DDMFACThe expected features of hybrid approach can be summarizedas the following three points
(1) When tracking error is large DDMFACmanifests thefeature of DDOILC to acquire a faster convergencespeed
(2) When tracking error is small enough DDMFAC hasthe ability ofMFAC to achieve a fast and stable controleffect
(3) DDMFAC is able to achieve better control perfor-mance for different IVs
Through such combination the shortcomings of MFACand DDOILC will be complemented and their respec-tive advantages will be highlighted in DDMFAC Next wedescribe the controller design for DDMFAC in Section 41
41 DDMFAC Design Consider the following control inputcriterion function
119869 (u (119896)) = 120572119872 1003816100381610038161003816119910119889 minus 1199101 (119896 + 1)10038161003816100381610038162 + 120572119863 1003816100381610038161003816119910119889 minus 1199102 (119896)10038161003816100381610038162+ 1205821 Δu (119896)2 (20)
where |119910119889 minus 1199101(119896 + 1)|2 and |119910119889 minus 1199102(119896)|2 respectively comefrom MFAC and DDOILC and 120572119872 120572119863 are the respectiveweighted factors and 120572119872 + 120572119863 = 0
Substituting (7) and (13) into (20) gives
119869 (u (119896)) = 120572119872 10038171003817100381710038171003817119890 (119896) minus 120593119879 (119896) Δu (119896)100381710038171003817100381710038172+ 120572119863 10038171003817100381710038171003817119890 (119896 minus 1) minus 120593119879 (119896) Δu (119896)100381710038171003817100381710038172+ 1205821 Δu (119896)2
(21)
Using optimal condition 120597119869120597u(119896) = 0 then gives
Δu (119896) = 120593 (119896) (120572119872119890 (119896) + 120572119863119890 (119896 minus 1))1205821 + (120572119872 + 120572119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (22)
Dividing numerator and denominator by 120572119872+120572119863 at the righthand of (22) we haveΔu (119896)= 120593 (119896) ((120572119872 (120572119872 + 120572119863)) 119890 (119896) + (120572119863 (120572119872 + 120572119863)) 119890 (119896 minus 1))
(1205821 (120572119872 + 120572119863)) + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (23)
Letting 120588119872 = 120572119872(120572119872 + 120572119863) 120588119863 = 120572119863(120572119872 + 120572119863) and 120582 =1205821(120572119872 + 120572119863) thenu (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (24)
where 120582 gt 0 is the weighted factor and 120588119872 120588119863 isin [0 1] are thestep-size factors
Define the following PPD vector criterion function
119869 ( (119896))= 10038171003817100381710038171003817120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896) Δu (119896 minus 1)100381710038171003817100381710038172
+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172 (25)
where 120573119872 120573119863 isin [0 1] are the weighted factors and 120573119872+120573119863 =1We rewrite (25) as
119869 ( (119896)) = 10038161003816100381610038161003816120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1)minus 119879 (119896 minus 1) Δu (119896 minus 1) minus Δ119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(26)
Complexity 9
Utilizing optimal condition 120597119869120597(119896) = 0 we have
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (27)
Thus the controller of DDMFAC is designed as follows
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (28)
u (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (29)
42 Stability and Convergence Analysis of DDMFAC
(1)The Boundedness of (119896) Define the parameter estimationerror of 120593(119896) as follows
(119896) = 120593 (119896) minus (119896) (30)
Substituting (28) into (30) gives
(119896) = 120593 (119896) minus (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 (31)
Similarly define (119896 minus 1) = 120593(119896 minus 1) minus (119896 minus 1) and then (119896 minus 1) = 120593 (119896 minus 1) minus (119896 minus 1) (32)
Combining (31) and (32) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1) (33)
According to (7) and (13) it can be obtained that
Δ119910 (119896) = 120593119879 (119896 minus 1) Δu (119896 minus 1)
Δ119910 (119896 minus 1) = 120593119879 (119896 minus 1) Δu (119896 minus 1) (34)
Substituting (34) into (33) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) ((120573119872 + 120573119863)120593119879 (119896 minus 1) Δu (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578Δu (119896 minus 1) ((120593 (119896 minus 1) minus (119896 minus 1))119879 Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578 (119896 minus 1) Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2) + 120593 (119896) minus 120593 (119896 minus 1)
(35)
10 Complexity
Because |120593(119896)| le 119887 is bounded [16] |120593(119896)minus120593(119896minus1)| le 2119887Taking the modulus of both sides of (35) we can obtain
1003817100381710038171003817 (119896)1003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2)100381710038171003817100381710038171003817100381710038171003817 + 2119887 (36)
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
= 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 + (minus2 + 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
times 1205781003817100381710038171003817 (119896 minus 1) Δu (119896 minus 1)10038171003817100381710038172120583 + Δu (119896 minus 1)2
(37)
Given that 0 lt 120578 lt 2 120583 gt 0(minus2 + 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2) lt 0 (38)
Synthesizing (37) and (38) we have
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
le 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 (39)
According to (36) and (39) there is obviously a positivenumber 1198891 isin (0 1) which makes
1003817100381710038171003817 (119896)1003817100381710038171003817 le 1198891 1003817100381710038171003817 (119896 minus 1)1003817100381710038171003817 + 2119887 le sdot sdot sdotle 1198891198961 1003817100381710038171003817 (0)1003817100381710038171003817 + 2119887 (1 minus 1198891198961)1 minus 1198891 (40)
Therefore (119896) is bounded(2)TheConvergence of Tracking Error 119890(119896) Tracking error 119890(119896)is derived as
119890 (119896) = 119910119889 minus 119910 (119896) = 119910119889 minus 119910 (119896 minus 1) minus 120593119879 (119896) Δu (119896)= 119890 (119896 minus 1) minus 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
(120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)= (1 minus 120588119863 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)
minus 120588119872 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 119890 (119896)
(41)
Transposition gives
119890 (119896) = [1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172] 119890 (119896 minus 1) (42)
So as long as appropriate parameters 120588119872 120588119863 and 120582 areselected there must be a positive number 1198892 isin (0 1) thatmakes
0 lt 10038171003817100381710038171003817100381710038171003817100381710038171 minus(120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172
1003817100381710038171003817100381710038171003817100381710038171003817 le 1198892 lt 1 (43)
By (42) and (43) it can be obtained that
119890 (119896) le 1003817100381710038171003817100381710038171003817100381710038171003817(1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)1003817100381710038171003817100381710038171003817100381710038171003817le 1198892 119890 (119896 minus 1) le sdot sdot sdot le 1198891198962 119890 (0)
(44)
Therefore lim119896rarrinfin119890(119896) = 0 that is 119890(119896) is proved toconverge to zero
(3)The Boundedness of System Input and Output By referringto 1198862 + 1198872 ge 2119886119887 the relational expression can be obtained asfollows
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
120582 + 1003817100381710038171003817120593 (119896)100381710038171003817100381721003817100381710038171003817100381710038171003817100381710038171003817 le
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
2radic120582 1003817100381710038171003817120593 (119896)10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 =
12radic120582 (45)
And we have
u (119896) = u (119896) minus u (119896 minus 1) + u (119896 minus 1) minus sdot sdot sdot + u (1)minus u (0) + u (0) = u (0) + 119896sum
119895=1
u (119895) (46)
Combining (24) (44) (45) and (46) as well as utilizingSchwartzrsquos inequality gives
u (119896) le u (0) + 119896sum119895=1
1003817100381710038171003817Δu (119895)1003817100381710038171003817
le u (0) + 12radic120582
119896sum119895=1
1003817100381710038171003817120588119863119890 (119895 minus 1) + 120588119872119890 (119895)1003817100381710038171003817
le u (0) + 1205881198632radic120582119896sum119895=1
119889119895minus12 119890 (0)
+ 1205881198722radic120582119896sum119895=1
1198891198952 119890 (0)
le u (0) + 1205881198632radic1205821 minus 119889119895minus121 minus 1198892 119890 (0)
+ 1205881198722radic1205821198892 (1 minus 1198891198952)1 minus 1198892 119890 (0)
le u (0) + 120588119863(1 minus 1198892) 2radic120582 119890 (0)
Complexity 11
Table 3 The parameter settings of DDMFAC for different IVs
IV Parameter120588119872 120588119863 120573119872 120573119863 120582 120578 1205831199100 = 60 00275 00275 05000 05000
00010 00750 100001199100 = 70 50119864 minus 4 50119864 minus 4 05000 050001199100 = 80 46119864 minus 4 46119864 minus 4 05000 050001199100 = 100 18119864 minus 4 18119864 minus 4 05000 05000
+ 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)le u (0) + 120588119863 + 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)
(47)
Because the initial tracking error 119890(0) and initial inputu(0) are given bounded the system input u(119896) is boundedaccording to (47) Besides due to the boundedness of 119910119889 and119890(119896) the system output 119910(119896) is obviously bounded43 Fuzzy DDMFAC The controller design and stability andconvergence analysis for DDMFAC have been respectivelydiscussed in detail in the above sections Although thedesigned controller has the characters of both MFAC andDDOILC weighted factors 120588119872 120588119863 120573119872 and 120573119863 are allconstant throughout control process DDMFAC is incapableof adaptively selecting the dominance of MFAC or DDOILCto duly show the expected features in each stage of controlprocessThe aforementioned features of DDMFACwe expectcannot be reflected in this case Such DDMFAC approachwith fixed weighting factors has no advantage comparedwith MFAC or DDOILC which appeals to a reasonable andadaptive adjustment strategy for 120588119872 120588119863 120573119872 and 120573119863 There-fore the adaptive adjustment strategy based on fuzzy logicfor weighting factors is considered because its advantageousperformance in parameter adjustment has been proven inour previous work [45] The formulation of fuzzy rules offersus the opportunity to adjust weighting factors in the waywe expected Such DDMFAC approach with fuzzy adaptiveadjustment for parameters is namedFDDMFAC in this paperConcretely the fuzzy rules can be formulated by referringto the expected features of DDMFAC The input variables offuzzy rules are the absolute values of tracking error 119890(119896 minus 1)and DLM prediction error 119890119901(119896minus1) of MFAC and the outputvariables are 120588119872 120588119863 and 120573119863 Because of the relationshipbetween 120573119872 and 120573119863 there is no need to use 120573119872 as an outputand it is calculated by 120573119872 = 1 minus 120573119863 The fuzzy rules can bedetailedly described as follows
(1) If |119890(119896 minus 1)| and |119890119901(119896 minus 1)| are great then thedominance of DDOILC is preferred and that ofMFAC is restrained in DDMFAC So 120588119863 and 120573119863 haverelatively large values and 120588119872 is a relatively smallnumber
(2) 120588119863 and 120573119863 will show a decreasing trend with thegradual decrease of |119890(119896 minus 1)| and |119890119901(119896 minus 1)| while120588119872 will present an increasing trend
44 Simulations and Result Analysis The proposed DDM-FAC approach is firstly applied to the particle quality controlin SFBGP and then its control effect is compared withMFACand DDOILC under different IVs In the simulation justlike MFAC 119910(119896) is replaced by a predicted value 119910(119896) thatis estimated using (19) See Table 3 for parameter settings ofDDMFAC It is not hard to find 120588119872 = 120588119863 and 120573119872 = 120573119863for a certain IV which means that MFAC and DDOILC areset to have the same dominance in DDMFAC By the wayMFAC and DDOILC can be regarded as the extreme cases ofDDMFAC described as follows
MFAC 120588119872 = 120573119872 = 1120588119863 = 120573119863 = 0
DDOILC 120588119872 = 120573119872 = 0120588119863 = 120573119863 = 1
(48)
The control effects of DDMFAC with the parametersettings in Table 3 together with control curves of MFACand DDOILC are plotted in Figure 5 The simulation resultsactually present the control effects of DDMFAC with threesets of parameter settings From simulation it is indicatedthat all three approaches have gained some certain advantagesunder different IV conditions Concretely as is shown inFigure 5(a) DDMFAC has advantage in control performancecompared with DDOILC and MFAC when 1199100 = 60 butMFAC has absolute superiority when 1199100 = 100 plottedin Figure 5(d) In addition from the simulation results inFigures 5(b) and 5(c) it can be seen that DDOILC andMFAC have achieved certain but not absolute advantagesrespectively This demonstrates that parameter setting hasimportant influence on the control effect of DDMFACTherefore the way to achieve satisfactory control effect is toadaptively adjust weighted parameters of DDMFAC insteadof keeping them constant
Then the idea of fuzzy adjustment is introduced to adjustthe weights 120588119872 120588119863 and 120573119863 The fuzzy adjustment rules ofthree weights are respectively shown in Tables 4 and 5 Therespective membership functions when 1199100 = 60 are shownin Figure 6 and membership functions for the other IVs areno longer displayed Please see Figure 7 for control effectof FDDMFAC and comparisons with MFAC and DDOILCunder different IVs Obviously DDMFAC approach withfuzzy adjustment for parameters that is FDDMFAC has thebest performance in particle quality control for all IVs This
12 Complexity
Table 4 The fuzzy rules of 120588119872120588119872 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB PB PB PB PB PB PB PBNM PB PM PM PM PM PS PSNS PM PM PM PS PS Z ZZ Z Z Z NS NS NS NSPS NS NS NM NM NB NB NBPM NM NM NM NB NB NB NBPB NB NB NB NB NB NB NB
MFACDDOILCDDMFAC
60
80
100
120
APS
(m
)
5 10 150Iterations
(a) 1199100 = 60
MFACDDOILCDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILCDDMFAC
80
90
100
110
120
APS
(m
)
2 4 60Iterations
(c) 1199100 = 80
95
100
105
110
115
120
125
APS
(m
)
5 10 150Iterations
MFACDDOILCDDMFAC
(d) 1199100 = 100
Figure 5 The comparison of control effects of MFAC DDOILC and DDMFAC
Complexity 13
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
02 04 06 08 10D
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
005 01 0150D
NB NM NS Z PS PM PB
times10minus3
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
05 1 15 2 25 3 35 4 450M
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
10 20 30 40 50 600|e(k minus 1)| (m)
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
5 10 15 20 250|ep (k minus 1)| (m)
Figure 6 The membership functions of FDDMFAC when 1199100 = 60
Table 5 The fuzzy rules of 120588119863 120573119863120588119863 120573119863 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB NB NB NB NB NB NB NBNM NB NM NM NM NM NS NSNS NM NM NM NS NS Z ZZ Z Z Z PS PM PM PMPS PS PS PM PM PB PB PBPM PM PM PM PB PB PB PBPB PB PB PB PB PB PB PB
validates the effectiveness of FDDMFAC proposed in thispaper
Although FDDMFAC approach has been proven to havea better control performance than MFAC and DDOILCwe still have a question of whether it is possible that thesuperiority of FDDMFAC is entirely attributed to fuzzyadjustment of parameters and has nothing to do with thestructure of DDMFAC itself In order to clarify this problemanother set of simulations are conducted inwhich the controlperformance of FDDMFAC is compared with fuzzy MFAC(FMFAC) and fuzzy DDOILC (FDDOILC) The absolutevalue of tracking error |119890(119896 minus 1)| is regarded as the inputvariable of fuzzy rules and the step-size factors 1205881 1205882 ofMFAC andDDOILC are the output variablesThe fuzzy rulesof FMFAC and FDDOILC are shown in Table 6 and themembership functions when 1199100 = 60 are shown in Figure 8For the simulation results please see Figure 9 From thesimulation it is concluded that FDDMFAC achieves the best
Table 6 The fuzzy rules of FMFAC and FDDOILC
|119890(119896 minus 1)| NB NM NS Z PS PM PB1205881 1205882 NB NM NS Z PS PM PB
control performance which demonstrates the effectiveness ofthe proposed approach in this paper
5 Conclusions
This paper aimed at solving the problem of particle qualitycontrol in the drug development phase of SFBGP Themodel-based control approaches are incapable of handlingsuch quality control because the accurate process modelcannot be acquired Thus the data-driven and model-freecontrol approaches are considered in this work We firstly
14 Complexity
MFACDDOILCFDDMFAC
5 10 150Iterations
60
80
100
120
APS
(m
)y0 = 60
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
y0 = 70
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
MFACDDOILCFDDMFAC
95
100
105
110
115
120
125A
PS (
m)
5 10 150Iterations
y0 = 100
Figure 7 The comparison of control effects of MFAC DDOILC and FDDMFAC
analyzed the features of MFAC and DDOILC and thenconducted experimental research to compare the perfor-mances of MFAC and DDOILC in particle quality controlunder different IVs The simulation results have verifiedour theoretical analysis and concluded that MFAC andDDOILC both have advantages under different IVs To beused in an actual production environment the expectedDDCapproaches must accommodate various IVs because of thechanges in material attributes Thus the hybrid approachFDDMFAC which combines the advantages of MFAC andDDOILC by adaptively adjusting their respective weightedfactors with fuzzy logic was proposed in this work tocomplement their respective shortcomings Through a series
of simulation results we conclude that FDDMFAC achievesbetter control performance than other approaches whichvalidates the effectiveness of FDDMFAC
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thanks are due to National Natural Science Foundation ofChina under Grant nos 61773105 61374147 61703085 and61004083 and Fundamental Research Funds for the Central
Complexity 15NB NM NS Z PS PM PB
001 002 00302
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
002 00401
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
20 40 600|e(k minus 1)| (m)
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
Figure 8 The membership functions of FDDMFAC when 1199100 = 60
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
60
80
100
120
APS
(m
)
y0 = 60
FMFACFDDOILCFDDMFAC
70
80
90
100
110
120
APS
(m
)
2 4 6 8 100Iterations
y0 = 70
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
100
105
110
115
120
125
APS
(m
)
y0 = 100
Figure 9 The comparison of control effects of FMFAC FDDOILC and FDDMFAC
16 Complexity
Universities under Grant nos N150402001 and N120404014for supporting this research work
References
[1] J Bouffard M Kaster and H Dumont ldquoInfluence of processvariable and physicochemical properties on the granulationmechanism of mannitol in a fluid bed top spray granulatorrdquoDrug Development and Industrial Pharmacy vol 31 no 9 pp923ndash933 2005
[2] H Liu K Wang W Schlindwein and M Li ldquoUsing the Box-Behnken experimental design to optimise operating parametersin pulsed spray fluidised bed granulationrdquo International Journalof Pharmaceutics vol 448 no 2 pp 329ndash338 2013
[3] H Liu and M Li ldquoPopulation balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granula-tionrdquo International Journal of Pharmaceutics vol 468 no 1-2pp 223ndash233 2014
[4] T Albuquerque V H Dias N Poellinger and J F PintoldquoConstruction of a quality index for granules produced byfluidized bed technology and application of the correspondenceanalysis as a discriminant procedurerdquo European Journal ofPharmaceutics and Biopharmaceutics vol 75 no 3 pp 418ndash4242010
[5] H Ehlers A Liu H Raikkonen et al ldquoGranule size control andtargeting in pulsed spray fluid bed granulationrdquo InternationalJournal of Pharmaceutics vol 377 no 1-2 pp 9ndash15 2009
[6] C F W Sanders M J Hounslow and F J Doyle III ldquoIden-tification of models for control of wet granulationrdquo PowderTechnology vol 188 no 3 pp 255ndash263 2009
[7] T Narvanen T Lipsanen O Antikainen H Raikkonen andJ Yliruusi ldquoControlling granule size by granulation liquid feedpulsingrdquo International Journal of Pharmaceutics vol 357 no 1-2pp 132ndash138 2008
[8] A Faure P York and R C Rowe ldquoProcess control and scale-up of pharmaceutical wet granulation processes a reviewrdquoEuropean Journal of Pharmaceutics and Biopharmaceutics vol52 no 3 pp 269ndash277 2001
[9] M Pottmann B A Ogunnaike A A Adetayo and B JEnnis ldquoModel-based control of a granulation systemrdquo PowderTechnology vol 108 no 2-3 pp 192ndash201 2000
[10] E P Gatzke and F J Doyle III ldquoModel predictive control of agranulation system using soft output constraints and prioritizedcontrol objectivesrdquo Powder Technology vol 121 no 2-3 pp 149ndash158 2001
[11] F Y Wang and I T Cameron ldquoReview and future directionsin the modelling and control of continuous drum granulationrdquoPowder Technology vol 124 no 3 pp 238ndash253 2002
[12] T Glaser C F W Sanders F Y Wang et al ldquoModel predictivecontrol of continuous drum granulationrdquo Journal of ProcessControl vol 19 no 4 pp 615ndash622 2009
[13] R Ramachandran andA Chaudhury ldquoModel-based design andcontrol of a continuous drum granulation processrdquo ChemicalEngineering Research and Design vol 90 no 8 pp 1063ndash10732012
[14] Z H Loh D Z L Er L W Chan C V Liew and PW S HengldquoSpray granulation for drug formulationrdquo Expert Opinion onDrug Delivery vol 8 no 12 pp 1645ndash1661 2011
[15] S H Schaafsma N W F Kossen M T Mos L Blauw andA C Hoffmann ldquoEffects and control of humidity and particlemixing in fluid-bed granulationrdquo AIChE Journal vol 45 no 6pp 1202ndash1210 1999
[16] Z-S Hou and S-T Jin ldquoA novel data-driven control approachfor a class of discrete-time nonlinear systemsrdquo IEEE Transac-tions onControl Systems Technology vol 19 no 6 pp 1549ndash15582011
[17] Z Hou and S Jin ldquoData-driven model-free adaptive controlfor a class of MIMO nonlinear discrete-time systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 22no 12 pp 2173ndash2188 2011
[18] D Xu B Jiang and P Shi ldquoA novel model-free adaptivecontrol design for multivariable industrial processesrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6391ndash6398 2014
[19] X Wang X Li J Wang X Fang and X Zhu ldquoData-drivenmodel-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeletonrdquo Information Sciences vol 327pp 246ndash257 2016
[20] L dos Santos Coelho andA A R Coelho ldquoModel-free adaptivecontrol optimization using a chaotic particle swarm approachrdquoChaos Solitons amp Fractals vol 41 no 4 pp 2001ndash2009 2009
[21] L D S Coelho M W Pessoa R Rodrigues Sumar andA Augusto Rodrigues Coelho ldquoModel-free adaptive controldesign using evolutionary-neural compensatorrdquo Expert Systemswith Applications vol 37 no 1 pp 499ndash508 2010
[22] K K Tan T H Lee S N Huang and F M Leu ldquoAdaptive-predictive control of a class of SISOnonlinear systemsrdquoDynam-ics and Control vol 11 no 2 pp 151ndash174 2001
[23] B Zhang andW Zhang ldquoAdaptive predictive functional controlof a class of nonlinear systemsrdquo ISA Transactions vol 45 no2 pp 175ndash183 2006
[24] F L Lu J F Wang C J Fan and S B Chen ldquoAn improvedmodel-free adaptive control with G function fuzzy reasoningregulation design and its applicationsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems andControl Engineering vol 222 no 8 pp 817ndash828 2008
[25] R Chi D Wang Z Hou and S Jin ldquoData-driven optimalterminal iterative learning controlrdquo Journal of Process Controlvol 22 no 10 pp 2026ndash2037 2012
[26] R Chi Z Hou S Jin and D Wang ldquoImproved data-drivenoptimal TILC using time-varying input signalsrdquo Journal ofProcess Control vol 24 no 12 pp 78ndash85 2014
[27] X Bu Z Hou and R Chi ldquoModel free adaptive iterative learn-ing control for farm vehicle path trackingrdquo IFAC ProceedingsVolumes vol 46 no 20 pp 153ndash158 2013
[28] M C Campi A Lecchini and SM Savaresi ldquoVirtual referencefeedback tuning a direct method for the design of feedbackcontrollersrdquo Automatica vol 38 no 8 pp 1337ndash1346 2002
[29] R-C Roman M-B Radac R-E Precup and E M PetriuldquoData-driven model-free adaptive control tuned by virtualreference feedback tuningrdquoActa PolytechnicaHungarica vol 13no 1 pp 83ndash96 2016
[30] P Yan D Liu D Wang and H Ma ldquoData-driven controllerdesign for general MIMO nonlinear systems via virtual refer-ence feedback tuning and neural networksrdquo Neurocomputingvol 171 pp 815ndash825 2016
[31] T Pan and S Li ldquoA hierarchical search and updating databasestrategy for lazy learningrdquo International Journal of InnovativeComputing Information andControl vol 4 no 6 pp 1383ndash13922008
[32] H Zhang R Song Q Wei and T Zhang ldquoOptimal trackingcontrol for a class of nonlinear discrete-time systems withtime delays based on heuristic dynamic programmingrdquo IEEE
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Complexity
y
yd
y (k + 1)y(k)
y(k)y (k minus 1)
y (k minus 1)
y(k minus 2)
IV = y0
u(k minus 2) u(k minus 1) u(k) u(k + 1) u(k + 2) Input
e (k)
e (k minus 1)ep (k)
ep (k minus 1)
Δy (k)
Δy(k minus 1)
Tracking errorPredicted error of DLM
Δy
Figure 2 The variable relations during control process of MFAC and DDOILC
output 119910(119896) is obtained only after the calculation of u(119896) So119910(119896) does not have actual value when computing u(119896) anda predicted system output 119910(119896) will be used to replace 119910(119896)in practical application of MFAC 119910(119896) is estimated by thefollowing DLM
119910 (119896) = 119910 (119896 minus 1) + 120593119879 (119896 minus 1) Δu (119896 minus 1) (19)
and 119890(119896) is replaced by 119890(119896) = 119910119889 minus 119910(119896)Figure 2 gives the variable relations during control pro-
cess of MFAC and DDOILC In terms of a single controlprocess curve when system output 119910 is greatly different from119910119889 that is the tracking error 119890 is relatively great 119890(119896 minus 1)is much larger than 119890(119896) and DDOILC will achieve a biggerupdating step size for control input which gives DDOILC afaster convergence speed As 119910 keeps getting closer to 119910119889 itis found that DLM prediction error 119890119901 is getting smaller andsmaller until 119890119901 = 0 The difference between 119890(119896 minus 1) and 119890(119896)will also be smaller and smaller DDOILC will not only loseits advantage but also perform worse than MFAC when 119910 isvery close to119910119889 because a larger updating step size for controlinput will bring difficulty in convergence while at this timeMFAC will achieve a fast and stable control effect due to theimprovement of prediction accuracy of DLM From anotherpoint of view if IV of system output 1199100 is far away from 119910119889the initial tracking error and DLM prediction error will beboth at a high level and DDOILC will perform better thanMFAC because a greater updating step size for control inputis necessary to give a faster convergence speed But as1199100 keepsgetting closer to 119910119889 119890119901 will gradually decrease and MFACwill gradually show its advantage Therefore the conclusions
obtained from theoretical analysis can be summarized asfollows
(1) When tracking error is large and 1199100 is relatively faraway from119910119889 DDOILCwill have a faster convergencespeed and a better control performance
(2) When tracking error gradually decreases and 1199100gradually gets closer to 119910119889 MFAC will graduallyreveal its advantage and achieve better control effect
32 SimulationVerification To validate conclusions obtainedfrom theoretical analysis simulation verification in SFBGP isconducted In this simulation particle quality to be controlledis APS and the desired quality is 119910119889 = 120 120583mThe viscosityof binder is treated as material attribute to be changed andthen a series of IVs (1199100) are constructed by adjusting viscosityof binder while keeping the operating condition unchangedFour sets of IVs are selected as representatives in this workand the detailed establishing information of them is shownin Table 1 Besides the parameter settings of MFAC andDDOILC for different IVs are listed in Table 2
For different IVs we conduct particle quality control inSFBGP based on MFAC and DDOILC and the control per-formances are plotted in Figure 3Moreover DLMpredictionerror profiles of MFAC are shown in Figure 4 Then theexplanation and analysis of simulation results will be givenAs is shown in Figure 3(a) DDOILC performs better thanMFAC when 1199100 = 60 that is 1199100 is relatively far away from119910119889 When 1199100 = 70 as is described in Figure 3(b) DDOILChas a faster convergence speed when APS is controlled from70 120583m to about 110 120583m but MFAC achieves a relatively better
Complexity 7
MFACDDOILC
5 10 150Iterations
60
80
100
120A
PS (
m)
(a) 1199100 = 60
MFACDDOILC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
(c) 1199100 = 80
MFACDDOILC
5 10 150Iterations
95
100
105
110
115
120
125
APS
(m
)
(d) 1199100 = 100
Figure 3 The control profiles of APS in SFBGP based on MFAC and DDOILC
Table 1The detailed establishing information of representative IVs
Info IV1199100 = 60 1199100 = 70 1199100 = 80 1199100 = 100
Operatingconditions(gs)
First stage 03 second stage 01 third stage 015
Viscosity ofbinder (Pasdots) 00214 00250 00304 00625
control performance when it is controlled from 110 120583m to 119910119889which indicates that both DDOILC and MFAC have certainadvantages in such case However from Figure 3(c) it can beseen that MFAC is slightly superior to DDOILC in controleffect when 1199100 = 80 and this just manifests the advantage ofMFAC As 1199100 gets closer and closer to 119910119889 such as 1199100 = 100
Table 2The parameter settings ofMFAC andDDOILC for differentIVs
IV Parameter1205881 1205882 120582 120578 1205831199100 = 60 00505 00712
00010 00750 100001199100 = 70 00047 005031199100 = 80 00032 001991199100 = 100 00020 149119864 minus 4
plotted in Figure 3(d)MFAC has a huge advantage comparedwith DDOILC It is obvious that the simulation results haveverified the conclusions obtained from theoretical analysisThe reason for increasing superiority of MFAC is attributedto the smaller and smaller prediction error 119890119901 of DLM Asis shown in Figure 4 on the one hand the curves of 119890119901 for
8 Complexity
IV = 60IV = 70
IV = 80IV = 100
IV = 60 14320IV = 70 08061IV = 80 04718IV = 100 04000
5 10 150Iterations
0
5
10
15
20
25
e p(
m)
Average of ep
Figure 4 The DLM prediction error profiles of MFAC for differentIVs
different IVs all show a decreasing trend along with iterationuntil 119890119901 = 0 as APS keeps getting closer to 119910119889 which is inaccordance with our obtained conclusion On the other handfor different IVs 119890119901 will also decrease with the increase of IVwhich is proved by comparing the curves of 119890119901 under differentIVs To further illustrate this point the averages of 119890119901 fordifferent IVs are computed and then listed in Figure 4 andthe decreasing averages of 119890119901 along with the increase of IValso validate the rationality of our analysis
4 The Proposed Hybrid DDC Approach andSimulation Study in SFBGP
Through theoretical analysis and simulation study we haveproved that MFAC and DDOILC both have strengths inparticle quality control under different IVs In real-worldapplications the same operating condition may give entirelydifferent IVs due to the changes in material properties whichbrings great difficulties in the selection of control methodsThus the uncertainties of IV have to be considered duringcontrol application In order to achieve favourable overallcontrol effect it is demanded that the DDC approaches mustaccommodate various IVs Therefore we try to combineadvantages of MFAC and DDOILC to present a hybridapproach that is named DDMFAC The basic idea is that thefeatures of MFAC and DDOILC are both reflected in thecriterion functions when designing controller of DDMFACThe expected features of hybrid approach can be summarizedas the following three points
(1) When tracking error is large DDMFACmanifests thefeature of DDOILC to acquire a faster convergencespeed
(2) When tracking error is small enough DDMFAC hasthe ability ofMFAC to achieve a fast and stable controleffect
(3) DDMFAC is able to achieve better control perfor-mance for different IVs
Through such combination the shortcomings of MFACand DDOILC will be complemented and their respec-tive advantages will be highlighted in DDMFAC Next wedescribe the controller design for DDMFAC in Section 41
41 DDMFAC Design Consider the following control inputcriterion function
119869 (u (119896)) = 120572119872 1003816100381610038161003816119910119889 minus 1199101 (119896 + 1)10038161003816100381610038162 + 120572119863 1003816100381610038161003816119910119889 minus 1199102 (119896)10038161003816100381610038162+ 1205821 Δu (119896)2 (20)
where |119910119889 minus 1199101(119896 + 1)|2 and |119910119889 minus 1199102(119896)|2 respectively comefrom MFAC and DDOILC and 120572119872 120572119863 are the respectiveweighted factors and 120572119872 + 120572119863 = 0
Substituting (7) and (13) into (20) gives
119869 (u (119896)) = 120572119872 10038171003817100381710038171003817119890 (119896) minus 120593119879 (119896) Δu (119896)100381710038171003817100381710038172+ 120572119863 10038171003817100381710038171003817119890 (119896 minus 1) minus 120593119879 (119896) Δu (119896)100381710038171003817100381710038172+ 1205821 Δu (119896)2
(21)
Using optimal condition 120597119869120597u(119896) = 0 then gives
Δu (119896) = 120593 (119896) (120572119872119890 (119896) + 120572119863119890 (119896 minus 1))1205821 + (120572119872 + 120572119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (22)
Dividing numerator and denominator by 120572119872+120572119863 at the righthand of (22) we haveΔu (119896)= 120593 (119896) ((120572119872 (120572119872 + 120572119863)) 119890 (119896) + (120572119863 (120572119872 + 120572119863)) 119890 (119896 minus 1))
(1205821 (120572119872 + 120572119863)) + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (23)
Letting 120588119872 = 120572119872(120572119872 + 120572119863) 120588119863 = 120572119863(120572119872 + 120572119863) and 120582 =1205821(120572119872 + 120572119863) thenu (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (24)
where 120582 gt 0 is the weighted factor and 120588119872 120588119863 isin [0 1] are thestep-size factors
Define the following PPD vector criterion function
119869 ( (119896))= 10038171003817100381710038171003817120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896) Δu (119896 minus 1)100381710038171003817100381710038172
+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172 (25)
where 120573119872 120573119863 isin [0 1] are the weighted factors and 120573119872+120573119863 =1We rewrite (25) as
119869 ( (119896)) = 10038161003816100381610038161003816120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1)minus 119879 (119896 minus 1) Δu (119896 minus 1) minus Δ119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(26)
Complexity 9
Utilizing optimal condition 120597119869120597(119896) = 0 we have
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (27)
Thus the controller of DDMFAC is designed as follows
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (28)
u (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (29)
42 Stability and Convergence Analysis of DDMFAC
(1)The Boundedness of (119896) Define the parameter estimationerror of 120593(119896) as follows
(119896) = 120593 (119896) minus (119896) (30)
Substituting (28) into (30) gives
(119896) = 120593 (119896) minus (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 (31)
Similarly define (119896 minus 1) = 120593(119896 minus 1) minus (119896 minus 1) and then (119896 minus 1) = 120593 (119896 minus 1) minus (119896 minus 1) (32)
Combining (31) and (32) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1) (33)
According to (7) and (13) it can be obtained that
Δ119910 (119896) = 120593119879 (119896 minus 1) Δu (119896 minus 1)
Δ119910 (119896 minus 1) = 120593119879 (119896 minus 1) Δu (119896 minus 1) (34)
Substituting (34) into (33) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) ((120573119872 + 120573119863)120593119879 (119896 minus 1) Δu (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578Δu (119896 minus 1) ((120593 (119896 minus 1) minus (119896 minus 1))119879 Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578 (119896 minus 1) Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2) + 120593 (119896) minus 120593 (119896 minus 1)
(35)
10 Complexity
Because |120593(119896)| le 119887 is bounded [16] |120593(119896)minus120593(119896minus1)| le 2119887Taking the modulus of both sides of (35) we can obtain
1003817100381710038171003817 (119896)1003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2)100381710038171003817100381710038171003817100381710038171003817 + 2119887 (36)
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
= 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 + (minus2 + 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
times 1205781003817100381710038171003817 (119896 minus 1) Δu (119896 minus 1)10038171003817100381710038172120583 + Δu (119896 minus 1)2
(37)
Given that 0 lt 120578 lt 2 120583 gt 0(minus2 + 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2) lt 0 (38)
Synthesizing (37) and (38) we have
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
le 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 (39)
According to (36) and (39) there is obviously a positivenumber 1198891 isin (0 1) which makes
1003817100381710038171003817 (119896)1003817100381710038171003817 le 1198891 1003817100381710038171003817 (119896 minus 1)1003817100381710038171003817 + 2119887 le sdot sdot sdotle 1198891198961 1003817100381710038171003817 (0)1003817100381710038171003817 + 2119887 (1 minus 1198891198961)1 minus 1198891 (40)
Therefore (119896) is bounded(2)TheConvergence of Tracking Error 119890(119896) Tracking error 119890(119896)is derived as
119890 (119896) = 119910119889 minus 119910 (119896) = 119910119889 minus 119910 (119896 minus 1) minus 120593119879 (119896) Δu (119896)= 119890 (119896 minus 1) minus 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
(120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)= (1 minus 120588119863 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)
minus 120588119872 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 119890 (119896)
(41)
Transposition gives
119890 (119896) = [1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172] 119890 (119896 minus 1) (42)
So as long as appropriate parameters 120588119872 120588119863 and 120582 areselected there must be a positive number 1198892 isin (0 1) thatmakes
0 lt 10038171003817100381710038171003817100381710038171003817100381710038171 minus(120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172
1003817100381710038171003817100381710038171003817100381710038171003817 le 1198892 lt 1 (43)
By (42) and (43) it can be obtained that
119890 (119896) le 1003817100381710038171003817100381710038171003817100381710038171003817(1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)1003817100381710038171003817100381710038171003817100381710038171003817le 1198892 119890 (119896 minus 1) le sdot sdot sdot le 1198891198962 119890 (0)
(44)
Therefore lim119896rarrinfin119890(119896) = 0 that is 119890(119896) is proved toconverge to zero
(3)The Boundedness of System Input and Output By referringto 1198862 + 1198872 ge 2119886119887 the relational expression can be obtained asfollows
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
120582 + 1003817100381710038171003817120593 (119896)100381710038171003817100381721003817100381710038171003817100381710038171003817100381710038171003817 le
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
2radic120582 1003817100381710038171003817120593 (119896)10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 =
12radic120582 (45)
And we have
u (119896) = u (119896) minus u (119896 minus 1) + u (119896 minus 1) minus sdot sdot sdot + u (1)minus u (0) + u (0) = u (0) + 119896sum
119895=1
u (119895) (46)
Combining (24) (44) (45) and (46) as well as utilizingSchwartzrsquos inequality gives
u (119896) le u (0) + 119896sum119895=1
1003817100381710038171003817Δu (119895)1003817100381710038171003817
le u (0) + 12radic120582
119896sum119895=1
1003817100381710038171003817120588119863119890 (119895 minus 1) + 120588119872119890 (119895)1003817100381710038171003817
le u (0) + 1205881198632radic120582119896sum119895=1
119889119895minus12 119890 (0)
+ 1205881198722radic120582119896sum119895=1
1198891198952 119890 (0)
le u (0) + 1205881198632radic1205821 minus 119889119895minus121 minus 1198892 119890 (0)
+ 1205881198722radic1205821198892 (1 minus 1198891198952)1 minus 1198892 119890 (0)
le u (0) + 120588119863(1 minus 1198892) 2radic120582 119890 (0)
Complexity 11
Table 3 The parameter settings of DDMFAC for different IVs
IV Parameter120588119872 120588119863 120573119872 120573119863 120582 120578 1205831199100 = 60 00275 00275 05000 05000
00010 00750 100001199100 = 70 50119864 minus 4 50119864 minus 4 05000 050001199100 = 80 46119864 minus 4 46119864 minus 4 05000 050001199100 = 100 18119864 minus 4 18119864 minus 4 05000 05000
+ 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)le u (0) + 120588119863 + 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)
(47)
Because the initial tracking error 119890(0) and initial inputu(0) are given bounded the system input u(119896) is boundedaccording to (47) Besides due to the boundedness of 119910119889 and119890(119896) the system output 119910(119896) is obviously bounded43 Fuzzy DDMFAC The controller design and stability andconvergence analysis for DDMFAC have been respectivelydiscussed in detail in the above sections Although thedesigned controller has the characters of both MFAC andDDOILC weighted factors 120588119872 120588119863 120573119872 and 120573119863 are allconstant throughout control process DDMFAC is incapableof adaptively selecting the dominance of MFAC or DDOILCto duly show the expected features in each stage of controlprocessThe aforementioned features of DDMFACwe expectcannot be reflected in this case Such DDMFAC approachwith fixed weighting factors has no advantage comparedwith MFAC or DDOILC which appeals to a reasonable andadaptive adjustment strategy for 120588119872 120588119863 120573119872 and 120573119863 There-fore the adaptive adjustment strategy based on fuzzy logicfor weighting factors is considered because its advantageousperformance in parameter adjustment has been proven inour previous work [45] The formulation of fuzzy rules offersus the opportunity to adjust weighting factors in the waywe expected Such DDMFAC approach with fuzzy adaptiveadjustment for parameters is namedFDDMFAC in this paperConcretely the fuzzy rules can be formulated by referringto the expected features of DDMFAC The input variables offuzzy rules are the absolute values of tracking error 119890(119896 minus 1)and DLM prediction error 119890119901(119896minus1) of MFAC and the outputvariables are 120588119872 120588119863 and 120573119863 Because of the relationshipbetween 120573119872 and 120573119863 there is no need to use 120573119872 as an outputand it is calculated by 120573119872 = 1 minus 120573119863 The fuzzy rules can bedetailedly described as follows
(1) If |119890(119896 minus 1)| and |119890119901(119896 minus 1)| are great then thedominance of DDOILC is preferred and that ofMFAC is restrained in DDMFAC So 120588119863 and 120573119863 haverelatively large values and 120588119872 is a relatively smallnumber
(2) 120588119863 and 120573119863 will show a decreasing trend with thegradual decrease of |119890(119896 minus 1)| and |119890119901(119896 minus 1)| while120588119872 will present an increasing trend
44 Simulations and Result Analysis The proposed DDM-FAC approach is firstly applied to the particle quality controlin SFBGP and then its control effect is compared withMFACand DDOILC under different IVs In the simulation justlike MFAC 119910(119896) is replaced by a predicted value 119910(119896) thatis estimated using (19) See Table 3 for parameter settings ofDDMFAC It is not hard to find 120588119872 = 120588119863 and 120573119872 = 120573119863for a certain IV which means that MFAC and DDOILC areset to have the same dominance in DDMFAC By the wayMFAC and DDOILC can be regarded as the extreme cases ofDDMFAC described as follows
MFAC 120588119872 = 120573119872 = 1120588119863 = 120573119863 = 0
DDOILC 120588119872 = 120573119872 = 0120588119863 = 120573119863 = 1
(48)
The control effects of DDMFAC with the parametersettings in Table 3 together with control curves of MFACand DDOILC are plotted in Figure 5 The simulation resultsactually present the control effects of DDMFAC with threesets of parameter settings From simulation it is indicatedthat all three approaches have gained some certain advantagesunder different IV conditions Concretely as is shown inFigure 5(a) DDMFAC has advantage in control performancecompared with DDOILC and MFAC when 1199100 = 60 butMFAC has absolute superiority when 1199100 = 100 plottedin Figure 5(d) In addition from the simulation results inFigures 5(b) and 5(c) it can be seen that DDOILC andMFAC have achieved certain but not absolute advantagesrespectively This demonstrates that parameter setting hasimportant influence on the control effect of DDMFACTherefore the way to achieve satisfactory control effect is toadaptively adjust weighted parameters of DDMFAC insteadof keeping them constant
Then the idea of fuzzy adjustment is introduced to adjustthe weights 120588119872 120588119863 and 120573119863 The fuzzy adjustment rules ofthree weights are respectively shown in Tables 4 and 5 Therespective membership functions when 1199100 = 60 are shownin Figure 6 and membership functions for the other IVs areno longer displayed Please see Figure 7 for control effectof FDDMFAC and comparisons with MFAC and DDOILCunder different IVs Obviously DDMFAC approach withfuzzy adjustment for parameters that is FDDMFAC has thebest performance in particle quality control for all IVs This
12 Complexity
Table 4 The fuzzy rules of 120588119872120588119872 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB PB PB PB PB PB PB PBNM PB PM PM PM PM PS PSNS PM PM PM PS PS Z ZZ Z Z Z NS NS NS NSPS NS NS NM NM NB NB NBPM NM NM NM NB NB NB NBPB NB NB NB NB NB NB NB
MFACDDOILCDDMFAC
60
80
100
120
APS
(m
)
5 10 150Iterations
(a) 1199100 = 60
MFACDDOILCDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILCDDMFAC
80
90
100
110
120
APS
(m
)
2 4 60Iterations
(c) 1199100 = 80
95
100
105
110
115
120
125
APS
(m
)
5 10 150Iterations
MFACDDOILCDDMFAC
(d) 1199100 = 100
Figure 5 The comparison of control effects of MFAC DDOILC and DDMFAC
Complexity 13
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
02 04 06 08 10D
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
005 01 0150D
NB NM NS Z PS PM PB
times10minus3
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
05 1 15 2 25 3 35 4 450M
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
10 20 30 40 50 600|e(k minus 1)| (m)
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
5 10 15 20 250|ep (k minus 1)| (m)
Figure 6 The membership functions of FDDMFAC when 1199100 = 60
Table 5 The fuzzy rules of 120588119863 120573119863120588119863 120573119863 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB NB NB NB NB NB NB NBNM NB NM NM NM NM NS NSNS NM NM NM NS NS Z ZZ Z Z Z PS PM PM PMPS PS PS PM PM PB PB PBPM PM PM PM PB PB PB PBPB PB PB PB PB PB PB PB
validates the effectiveness of FDDMFAC proposed in thispaper
Although FDDMFAC approach has been proven to havea better control performance than MFAC and DDOILCwe still have a question of whether it is possible that thesuperiority of FDDMFAC is entirely attributed to fuzzyadjustment of parameters and has nothing to do with thestructure of DDMFAC itself In order to clarify this problemanother set of simulations are conducted inwhich the controlperformance of FDDMFAC is compared with fuzzy MFAC(FMFAC) and fuzzy DDOILC (FDDOILC) The absolutevalue of tracking error |119890(119896 minus 1)| is regarded as the inputvariable of fuzzy rules and the step-size factors 1205881 1205882 ofMFAC andDDOILC are the output variablesThe fuzzy rulesof FMFAC and FDDOILC are shown in Table 6 and themembership functions when 1199100 = 60 are shown in Figure 8For the simulation results please see Figure 9 From thesimulation it is concluded that FDDMFAC achieves the best
Table 6 The fuzzy rules of FMFAC and FDDOILC
|119890(119896 minus 1)| NB NM NS Z PS PM PB1205881 1205882 NB NM NS Z PS PM PB
control performance which demonstrates the effectiveness ofthe proposed approach in this paper
5 Conclusions
This paper aimed at solving the problem of particle qualitycontrol in the drug development phase of SFBGP Themodel-based control approaches are incapable of handlingsuch quality control because the accurate process modelcannot be acquired Thus the data-driven and model-freecontrol approaches are considered in this work We firstly
14 Complexity
MFACDDOILCFDDMFAC
5 10 150Iterations
60
80
100
120
APS
(m
)y0 = 60
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
y0 = 70
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
MFACDDOILCFDDMFAC
95
100
105
110
115
120
125A
PS (
m)
5 10 150Iterations
y0 = 100
Figure 7 The comparison of control effects of MFAC DDOILC and FDDMFAC
analyzed the features of MFAC and DDOILC and thenconducted experimental research to compare the perfor-mances of MFAC and DDOILC in particle quality controlunder different IVs The simulation results have verifiedour theoretical analysis and concluded that MFAC andDDOILC both have advantages under different IVs To beused in an actual production environment the expectedDDCapproaches must accommodate various IVs because of thechanges in material attributes Thus the hybrid approachFDDMFAC which combines the advantages of MFAC andDDOILC by adaptively adjusting their respective weightedfactors with fuzzy logic was proposed in this work tocomplement their respective shortcomings Through a series
of simulation results we conclude that FDDMFAC achievesbetter control performance than other approaches whichvalidates the effectiveness of FDDMFAC
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thanks are due to National Natural Science Foundation ofChina under Grant nos 61773105 61374147 61703085 and61004083 and Fundamental Research Funds for the Central
Complexity 15NB NM NS Z PS PM PB
001 002 00302
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
002 00401
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
20 40 600|e(k minus 1)| (m)
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
Figure 8 The membership functions of FDDMFAC when 1199100 = 60
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
60
80
100
120
APS
(m
)
y0 = 60
FMFACFDDOILCFDDMFAC
70
80
90
100
110
120
APS
(m
)
2 4 6 8 100Iterations
y0 = 70
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
100
105
110
115
120
125
APS
(m
)
y0 = 100
Figure 9 The comparison of control effects of FMFAC FDDOILC and FDDMFAC
16 Complexity
Universities under Grant nos N150402001 and N120404014for supporting this research work
References
[1] J Bouffard M Kaster and H Dumont ldquoInfluence of processvariable and physicochemical properties on the granulationmechanism of mannitol in a fluid bed top spray granulatorrdquoDrug Development and Industrial Pharmacy vol 31 no 9 pp923ndash933 2005
[2] H Liu K Wang W Schlindwein and M Li ldquoUsing the Box-Behnken experimental design to optimise operating parametersin pulsed spray fluidised bed granulationrdquo International Journalof Pharmaceutics vol 448 no 2 pp 329ndash338 2013
[3] H Liu and M Li ldquoPopulation balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granula-tionrdquo International Journal of Pharmaceutics vol 468 no 1-2pp 223ndash233 2014
[4] T Albuquerque V H Dias N Poellinger and J F PintoldquoConstruction of a quality index for granules produced byfluidized bed technology and application of the correspondenceanalysis as a discriminant procedurerdquo European Journal ofPharmaceutics and Biopharmaceutics vol 75 no 3 pp 418ndash4242010
[5] H Ehlers A Liu H Raikkonen et al ldquoGranule size control andtargeting in pulsed spray fluid bed granulationrdquo InternationalJournal of Pharmaceutics vol 377 no 1-2 pp 9ndash15 2009
[6] C F W Sanders M J Hounslow and F J Doyle III ldquoIden-tification of models for control of wet granulationrdquo PowderTechnology vol 188 no 3 pp 255ndash263 2009
[7] T Narvanen T Lipsanen O Antikainen H Raikkonen andJ Yliruusi ldquoControlling granule size by granulation liquid feedpulsingrdquo International Journal of Pharmaceutics vol 357 no 1-2pp 132ndash138 2008
[8] A Faure P York and R C Rowe ldquoProcess control and scale-up of pharmaceutical wet granulation processes a reviewrdquoEuropean Journal of Pharmaceutics and Biopharmaceutics vol52 no 3 pp 269ndash277 2001
[9] M Pottmann B A Ogunnaike A A Adetayo and B JEnnis ldquoModel-based control of a granulation systemrdquo PowderTechnology vol 108 no 2-3 pp 192ndash201 2000
[10] E P Gatzke and F J Doyle III ldquoModel predictive control of agranulation system using soft output constraints and prioritizedcontrol objectivesrdquo Powder Technology vol 121 no 2-3 pp 149ndash158 2001
[11] F Y Wang and I T Cameron ldquoReview and future directionsin the modelling and control of continuous drum granulationrdquoPowder Technology vol 124 no 3 pp 238ndash253 2002
[12] T Glaser C F W Sanders F Y Wang et al ldquoModel predictivecontrol of continuous drum granulationrdquo Journal of ProcessControl vol 19 no 4 pp 615ndash622 2009
[13] R Ramachandran andA Chaudhury ldquoModel-based design andcontrol of a continuous drum granulation processrdquo ChemicalEngineering Research and Design vol 90 no 8 pp 1063ndash10732012
[14] Z H Loh D Z L Er L W Chan C V Liew and PW S HengldquoSpray granulation for drug formulationrdquo Expert Opinion onDrug Delivery vol 8 no 12 pp 1645ndash1661 2011
[15] S H Schaafsma N W F Kossen M T Mos L Blauw andA C Hoffmann ldquoEffects and control of humidity and particlemixing in fluid-bed granulationrdquo AIChE Journal vol 45 no 6pp 1202ndash1210 1999
[16] Z-S Hou and S-T Jin ldquoA novel data-driven control approachfor a class of discrete-time nonlinear systemsrdquo IEEE Transac-tions onControl Systems Technology vol 19 no 6 pp 1549ndash15582011
[17] Z Hou and S Jin ldquoData-driven model-free adaptive controlfor a class of MIMO nonlinear discrete-time systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 22no 12 pp 2173ndash2188 2011
[18] D Xu B Jiang and P Shi ldquoA novel model-free adaptivecontrol design for multivariable industrial processesrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6391ndash6398 2014
[19] X Wang X Li J Wang X Fang and X Zhu ldquoData-drivenmodel-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeletonrdquo Information Sciences vol 327pp 246ndash257 2016
[20] L dos Santos Coelho andA A R Coelho ldquoModel-free adaptivecontrol optimization using a chaotic particle swarm approachrdquoChaos Solitons amp Fractals vol 41 no 4 pp 2001ndash2009 2009
[21] L D S Coelho M W Pessoa R Rodrigues Sumar andA Augusto Rodrigues Coelho ldquoModel-free adaptive controldesign using evolutionary-neural compensatorrdquo Expert Systemswith Applications vol 37 no 1 pp 499ndash508 2010
[22] K K Tan T H Lee S N Huang and F M Leu ldquoAdaptive-predictive control of a class of SISOnonlinear systemsrdquoDynam-ics and Control vol 11 no 2 pp 151ndash174 2001
[23] B Zhang andW Zhang ldquoAdaptive predictive functional controlof a class of nonlinear systemsrdquo ISA Transactions vol 45 no2 pp 175ndash183 2006
[24] F L Lu J F Wang C J Fan and S B Chen ldquoAn improvedmodel-free adaptive control with G function fuzzy reasoningregulation design and its applicationsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems andControl Engineering vol 222 no 8 pp 817ndash828 2008
[25] R Chi D Wang Z Hou and S Jin ldquoData-driven optimalterminal iterative learning controlrdquo Journal of Process Controlvol 22 no 10 pp 2026ndash2037 2012
[26] R Chi Z Hou S Jin and D Wang ldquoImproved data-drivenoptimal TILC using time-varying input signalsrdquo Journal ofProcess Control vol 24 no 12 pp 78ndash85 2014
[27] X Bu Z Hou and R Chi ldquoModel free adaptive iterative learn-ing control for farm vehicle path trackingrdquo IFAC ProceedingsVolumes vol 46 no 20 pp 153ndash158 2013
[28] M C Campi A Lecchini and SM Savaresi ldquoVirtual referencefeedback tuning a direct method for the design of feedbackcontrollersrdquo Automatica vol 38 no 8 pp 1337ndash1346 2002
[29] R-C Roman M-B Radac R-E Precup and E M PetriuldquoData-driven model-free adaptive control tuned by virtualreference feedback tuningrdquoActa PolytechnicaHungarica vol 13no 1 pp 83ndash96 2016
[30] P Yan D Liu D Wang and H Ma ldquoData-driven controllerdesign for general MIMO nonlinear systems via virtual refer-ence feedback tuning and neural networksrdquo Neurocomputingvol 171 pp 815ndash825 2016
[31] T Pan and S Li ldquoA hierarchical search and updating databasestrategy for lazy learningrdquo International Journal of InnovativeComputing Information andControl vol 4 no 6 pp 1383ndash13922008
[32] H Zhang R Song Q Wei and T Zhang ldquoOptimal trackingcontrol for a class of nonlinear discrete-time systems withtime delays based on heuristic dynamic programmingrdquo IEEE
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Complexity 7
MFACDDOILC
5 10 150Iterations
60
80
100
120A
PS (
m)
(a) 1199100 = 60
MFACDDOILC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
(c) 1199100 = 80
MFACDDOILC
5 10 150Iterations
95
100
105
110
115
120
125
APS
(m
)
(d) 1199100 = 100
Figure 3 The control profiles of APS in SFBGP based on MFAC and DDOILC
Table 1The detailed establishing information of representative IVs
Info IV1199100 = 60 1199100 = 70 1199100 = 80 1199100 = 100
Operatingconditions(gs)
First stage 03 second stage 01 third stage 015
Viscosity ofbinder (Pasdots) 00214 00250 00304 00625
control performance when it is controlled from 110 120583m to 119910119889which indicates that both DDOILC and MFAC have certainadvantages in such case However from Figure 3(c) it can beseen that MFAC is slightly superior to DDOILC in controleffect when 1199100 = 80 and this just manifests the advantage ofMFAC As 1199100 gets closer and closer to 119910119889 such as 1199100 = 100
Table 2The parameter settings ofMFAC andDDOILC for differentIVs
IV Parameter1205881 1205882 120582 120578 1205831199100 = 60 00505 00712
00010 00750 100001199100 = 70 00047 005031199100 = 80 00032 001991199100 = 100 00020 149119864 minus 4
plotted in Figure 3(d)MFAC has a huge advantage comparedwith DDOILC It is obvious that the simulation results haveverified the conclusions obtained from theoretical analysisThe reason for increasing superiority of MFAC is attributedto the smaller and smaller prediction error 119890119901 of DLM Asis shown in Figure 4 on the one hand the curves of 119890119901 for
8 Complexity
IV = 60IV = 70
IV = 80IV = 100
IV = 60 14320IV = 70 08061IV = 80 04718IV = 100 04000
5 10 150Iterations
0
5
10
15
20
25
e p(
m)
Average of ep
Figure 4 The DLM prediction error profiles of MFAC for differentIVs
different IVs all show a decreasing trend along with iterationuntil 119890119901 = 0 as APS keeps getting closer to 119910119889 which is inaccordance with our obtained conclusion On the other handfor different IVs 119890119901 will also decrease with the increase of IVwhich is proved by comparing the curves of 119890119901 under differentIVs To further illustrate this point the averages of 119890119901 fordifferent IVs are computed and then listed in Figure 4 andthe decreasing averages of 119890119901 along with the increase of IValso validate the rationality of our analysis
4 The Proposed Hybrid DDC Approach andSimulation Study in SFBGP
Through theoretical analysis and simulation study we haveproved that MFAC and DDOILC both have strengths inparticle quality control under different IVs In real-worldapplications the same operating condition may give entirelydifferent IVs due to the changes in material properties whichbrings great difficulties in the selection of control methodsThus the uncertainties of IV have to be considered duringcontrol application In order to achieve favourable overallcontrol effect it is demanded that the DDC approaches mustaccommodate various IVs Therefore we try to combineadvantages of MFAC and DDOILC to present a hybridapproach that is named DDMFAC The basic idea is that thefeatures of MFAC and DDOILC are both reflected in thecriterion functions when designing controller of DDMFACThe expected features of hybrid approach can be summarizedas the following three points
(1) When tracking error is large DDMFACmanifests thefeature of DDOILC to acquire a faster convergencespeed
(2) When tracking error is small enough DDMFAC hasthe ability ofMFAC to achieve a fast and stable controleffect
(3) DDMFAC is able to achieve better control perfor-mance for different IVs
Through such combination the shortcomings of MFACand DDOILC will be complemented and their respec-tive advantages will be highlighted in DDMFAC Next wedescribe the controller design for DDMFAC in Section 41
41 DDMFAC Design Consider the following control inputcriterion function
119869 (u (119896)) = 120572119872 1003816100381610038161003816119910119889 minus 1199101 (119896 + 1)10038161003816100381610038162 + 120572119863 1003816100381610038161003816119910119889 minus 1199102 (119896)10038161003816100381610038162+ 1205821 Δu (119896)2 (20)
where |119910119889 minus 1199101(119896 + 1)|2 and |119910119889 minus 1199102(119896)|2 respectively comefrom MFAC and DDOILC and 120572119872 120572119863 are the respectiveweighted factors and 120572119872 + 120572119863 = 0
Substituting (7) and (13) into (20) gives
119869 (u (119896)) = 120572119872 10038171003817100381710038171003817119890 (119896) minus 120593119879 (119896) Δu (119896)100381710038171003817100381710038172+ 120572119863 10038171003817100381710038171003817119890 (119896 minus 1) minus 120593119879 (119896) Δu (119896)100381710038171003817100381710038172+ 1205821 Δu (119896)2
(21)
Using optimal condition 120597119869120597u(119896) = 0 then gives
Δu (119896) = 120593 (119896) (120572119872119890 (119896) + 120572119863119890 (119896 minus 1))1205821 + (120572119872 + 120572119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (22)
Dividing numerator and denominator by 120572119872+120572119863 at the righthand of (22) we haveΔu (119896)= 120593 (119896) ((120572119872 (120572119872 + 120572119863)) 119890 (119896) + (120572119863 (120572119872 + 120572119863)) 119890 (119896 minus 1))
(1205821 (120572119872 + 120572119863)) + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (23)
Letting 120588119872 = 120572119872(120572119872 + 120572119863) 120588119863 = 120572119863(120572119872 + 120572119863) and 120582 =1205821(120572119872 + 120572119863) thenu (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (24)
where 120582 gt 0 is the weighted factor and 120588119872 120588119863 isin [0 1] are thestep-size factors
Define the following PPD vector criterion function
119869 ( (119896))= 10038171003817100381710038171003817120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896) Δu (119896 minus 1)100381710038171003817100381710038172
+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172 (25)
where 120573119872 120573119863 isin [0 1] are the weighted factors and 120573119872+120573119863 =1We rewrite (25) as
119869 ( (119896)) = 10038161003816100381610038161003816120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1)minus 119879 (119896 minus 1) Δu (119896 minus 1) minus Δ119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(26)
Complexity 9
Utilizing optimal condition 120597119869120597(119896) = 0 we have
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (27)
Thus the controller of DDMFAC is designed as follows
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (28)
u (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (29)
42 Stability and Convergence Analysis of DDMFAC
(1)The Boundedness of (119896) Define the parameter estimationerror of 120593(119896) as follows
(119896) = 120593 (119896) minus (119896) (30)
Substituting (28) into (30) gives
(119896) = 120593 (119896) minus (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 (31)
Similarly define (119896 minus 1) = 120593(119896 minus 1) minus (119896 minus 1) and then (119896 minus 1) = 120593 (119896 minus 1) minus (119896 minus 1) (32)
Combining (31) and (32) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1) (33)
According to (7) and (13) it can be obtained that
Δ119910 (119896) = 120593119879 (119896 minus 1) Δu (119896 minus 1)
Δ119910 (119896 minus 1) = 120593119879 (119896 minus 1) Δu (119896 minus 1) (34)
Substituting (34) into (33) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) ((120573119872 + 120573119863)120593119879 (119896 minus 1) Δu (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578Δu (119896 minus 1) ((120593 (119896 minus 1) minus (119896 minus 1))119879 Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578 (119896 minus 1) Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2) + 120593 (119896) minus 120593 (119896 minus 1)
(35)
10 Complexity
Because |120593(119896)| le 119887 is bounded [16] |120593(119896)minus120593(119896minus1)| le 2119887Taking the modulus of both sides of (35) we can obtain
1003817100381710038171003817 (119896)1003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2)100381710038171003817100381710038171003817100381710038171003817 + 2119887 (36)
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
= 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 + (minus2 + 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
times 1205781003817100381710038171003817 (119896 minus 1) Δu (119896 minus 1)10038171003817100381710038172120583 + Δu (119896 minus 1)2
(37)
Given that 0 lt 120578 lt 2 120583 gt 0(minus2 + 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2) lt 0 (38)
Synthesizing (37) and (38) we have
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
le 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 (39)
According to (36) and (39) there is obviously a positivenumber 1198891 isin (0 1) which makes
1003817100381710038171003817 (119896)1003817100381710038171003817 le 1198891 1003817100381710038171003817 (119896 minus 1)1003817100381710038171003817 + 2119887 le sdot sdot sdotle 1198891198961 1003817100381710038171003817 (0)1003817100381710038171003817 + 2119887 (1 minus 1198891198961)1 minus 1198891 (40)
Therefore (119896) is bounded(2)TheConvergence of Tracking Error 119890(119896) Tracking error 119890(119896)is derived as
119890 (119896) = 119910119889 minus 119910 (119896) = 119910119889 minus 119910 (119896 minus 1) minus 120593119879 (119896) Δu (119896)= 119890 (119896 minus 1) minus 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
(120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)= (1 minus 120588119863 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)
minus 120588119872 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 119890 (119896)
(41)
Transposition gives
119890 (119896) = [1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172] 119890 (119896 minus 1) (42)
So as long as appropriate parameters 120588119872 120588119863 and 120582 areselected there must be a positive number 1198892 isin (0 1) thatmakes
0 lt 10038171003817100381710038171003817100381710038171003817100381710038171 minus(120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172
1003817100381710038171003817100381710038171003817100381710038171003817 le 1198892 lt 1 (43)
By (42) and (43) it can be obtained that
119890 (119896) le 1003817100381710038171003817100381710038171003817100381710038171003817(1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)1003817100381710038171003817100381710038171003817100381710038171003817le 1198892 119890 (119896 minus 1) le sdot sdot sdot le 1198891198962 119890 (0)
(44)
Therefore lim119896rarrinfin119890(119896) = 0 that is 119890(119896) is proved toconverge to zero
(3)The Boundedness of System Input and Output By referringto 1198862 + 1198872 ge 2119886119887 the relational expression can be obtained asfollows
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
120582 + 1003817100381710038171003817120593 (119896)100381710038171003817100381721003817100381710038171003817100381710038171003817100381710038171003817 le
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
2radic120582 1003817100381710038171003817120593 (119896)10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 =
12radic120582 (45)
And we have
u (119896) = u (119896) minus u (119896 minus 1) + u (119896 minus 1) minus sdot sdot sdot + u (1)minus u (0) + u (0) = u (0) + 119896sum
119895=1
u (119895) (46)
Combining (24) (44) (45) and (46) as well as utilizingSchwartzrsquos inequality gives
u (119896) le u (0) + 119896sum119895=1
1003817100381710038171003817Δu (119895)1003817100381710038171003817
le u (0) + 12radic120582
119896sum119895=1
1003817100381710038171003817120588119863119890 (119895 minus 1) + 120588119872119890 (119895)1003817100381710038171003817
le u (0) + 1205881198632radic120582119896sum119895=1
119889119895minus12 119890 (0)
+ 1205881198722radic120582119896sum119895=1
1198891198952 119890 (0)
le u (0) + 1205881198632radic1205821 minus 119889119895minus121 minus 1198892 119890 (0)
+ 1205881198722radic1205821198892 (1 minus 1198891198952)1 minus 1198892 119890 (0)
le u (0) + 120588119863(1 minus 1198892) 2radic120582 119890 (0)
Complexity 11
Table 3 The parameter settings of DDMFAC for different IVs
IV Parameter120588119872 120588119863 120573119872 120573119863 120582 120578 1205831199100 = 60 00275 00275 05000 05000
00010 00750 100001199100 = 70 50119864 minus 4 50119864 minus 4 05000 050001199100 = 80 46119864 minus 4 46119864 minus 4 05000 050001199100 = 100 18119864 minus 4 18119864 minus 4 05000 05000
+ 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)le u (0) + 120588119863 + 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)
(47)
Because the initial tracking error 119890(0) and initial inputu(0) are given bounded the system input u(119896) is boundedaccording to (47) Besides due to the boundedness of 119910119889 and119890(119896) the system output 119910(119896) is obviously bounded43 Fuzzy DDMFAC The controller design and stability andconvergence analysis for DDMFAC have been respectivelydiscussed in detail in the above sections Although thedesigned controller has the characters of both MFAC andDDOILC weighted factors 120588119872 120588119863 120573119872 and 120573119863 are allconstant throughout control process DDMFAC is incapableof adaptively selecting the dominance of MFAC or DDOILCto duly show the expected features in each stage of controlprocessThe aforementioned features of DDMFACwe expectcannot be reflected in this case Such DDMFAC approachwith fixed weighting factors has no advantage comparedwith MFAC or DDOILC which appeals to a reasonable andadaptive adjustment strategy for 120588119872 120588119863 120573119872 and 120573119863 There-fore the adaptive adjustment strategy based on fuzzy logicfor weighting factors is considered because its advantageousperformance in parameter adjustment has been proven inour previous work [45] The formulation of fuzzy rules offersus the opportunity to adjust weighting factors in the waywe expected Such DDMFAC approach with fuzzy adaptiveadjustment for parameters is namedFDDMFAC in this paperConcretely the fuzzy rules can be formulated by referringto the expected features of DDMFAC The input variables offuzzy rules are the absolute values of tracking error 119890(119896 minus 1)and DLM prediction error 119890119901(119896minus1) of MFAC and the outputvariables are 120588119872 120588119863 and 120573119863 Because of the relationshipbetween 120573119872 and 120573119863 there is no need to use 120573119872 as an outputand it is calculated by 120573119872 = 1 minus 120573119863 The fuzzy rules can bedetailedly described as follows
(1) If |119890(119896 minus 1)| and |119890119901(119896 minus 1)| are great then thedominance of DDOILC is preferred and that ofMFAC is restrained in DDMFAC So 120588119863 and 120573119863 haverelatively large values and 120588119872 is a relatively smallnumber
(2) 120588119863 and 120573119863 will show a decreasing trend with thegradual decrease of |119890(119896 minus 1)| and |119890119901(119896 minus 1)| while120588119872 will present an increasing trend
44 Simulations and Result Analysis The proposed DDM-FAC approach is firstly applied to the particle quality controlin SFBGP and then its control effect is compared withMFACand DDOILC under different IVs In the simulation justlike MFAC 119910(119896) is replaced by a predicted value 119910(119896) thatis estimated using (19) See Table 3 for parameter settings ofDDMFAC It is not hard to find 120588119872 = 120588119863 and 120573119872 = 120573119863for a certain IV which means that MFAC and DDOILC areset to have the same dominance in DDMFAC By the wayMFAC and DDOILC can be regarded as the extreme cases ofDDMFAC described as follows
MFAC 120588119872 = 120573119872 = 1120588119863 = 120573119863 = 0
DDOILC 120588119872 = 120573119872 = 0120588119863 = 120573119863 = 1
(48)
The control effects of DDMFAC with the parametersettings in Table 3 together with control curves of MFACand DDOILC are plotted in Figure 5 The simulation resultsactually present the control effects of DDMFAC with threesets of parameter settings From simulation it is indicatedthat all three approaches have gained some certain advantagesunder different IV conditions Concretely as is shown inFigure 5(a) DDMFAC has advantage in control performancecompared with DDOILC and MFAC when 1199100 = 60 butMFAC has absolute superiority when 1199100 = 100 plottedin Figure 5(d) In addition from the simulation results inFigures 5(b) and 5(c) it can be seen that DDOILC andMFAC have achieved certain but not absolute advantagesrespectively This demonstrates that parameter setting hasimportant influence on the control effect of DDMFACTherefore the way to achieve satisfactory control effect is toadaptively adjust weighted parameters of DDMFAC insteadof keeping them constant
Then the idea of fuzzy adjustment is introduced to adjustthe weights 120588119872 120588119863 and 120573119863 The fuzzy adjustment rules ofthree weights are respectively shown in Tables 4 and 5 Therespective membership functions when 1199100 = 60 are shownin Figure 6 and membership functions for the other IVs areno longer displayed Please see Figure 7 for control effectof FDDMFAC and comparisons with MFAC and DDOILCunder different IVs Obviously DDMFAC approach withfuzzy adjustment for parameters that is FDDMFAC has thebest performance in particle quality control for all IVs This
12 Complexity
Table 4 The fuzzy rules of 120588119872120588119872 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB PB PB PB PB PB PB PBNM PB PM PM PM PM PS PSNS PM PM PM PS PS Z ZZ Z Z Z NS NS NS NSPS NS NS NM NM NB NB NBPM NM NM NM NB NB NB NBPB NB NB NB NB NB NB NB
MFACDDOILCDDMFAC
60
80
100
120
APS
(m
)
5 10 150Iterations
(a) 1199100 = 60
MFACDDOILCDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILCDDMFAC
80
90
100
110
120
APS
(m
)
2 4 60Iterations
(c) 1199100 = 80
95
100
105
110
115
120
125
APS
(m
)
5 10 150Iterations
MFACDDOILCDDMFAC
(d) 1199100 = 100
Figure 5 The comparison of control effects of MFAC DDOILC and DDMFAC
Complexity 13
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
02 04 06 08 10D
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
005 01 0150D
NB NM NS Z PS PM PB
times10minus3
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
05 1 15 2 25 3 35 4 450M
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
10 20 30 40 50 600|e(k minus 1)| (m)
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
5 10 15 20 250|ep (k minus 1)| (m)
Figure 6 The membership functions of FDDMFAC when 1199100 = 60
Table 5 The fuzzy rules of 120588119863 120573119863120588119863 120573119863 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB NB NB NB NB NB NB NBNM NB NM NM NM NM NS NSNS NM NM NM NS NS Z ZZ Z Z Z PS PM PM PMPS PS PS PM PM PB PB PBPM PM PM PM PB PB PB PBPB PB PB PB PB PB PB PB
validates the effectiveness of FDDMFAC proposed in thispaper
Although FDDMFAC approach has been proven to havea better control performance than MFAC and DDOILCwe still have a question of whether it is possible that thesuperiority of FDDMFAC is entirely attributed to fuzzyadjustment of parameters and has nothing to do with thestructure of DDMFAC itself In order to clarify this problemanother set of simulations are conducted inwhich the controlperformance of FDDMFAC is compared with fuzzy MFAC(FMFAC) and fuzzy DDOILC (FDDOILC) The absolutevalue of tracking error |119890(119896 minus 1)| is regarded as the inputvariable of fuzzy rules and the step-size factors 1205881 1205882 ofMFAC andDDOILC are the output variablesThe fuzzy rulesof FMFAC and FDDOILC are shown in Table 6 and themembership functions when 1199100 = 60 are shown in Figure 8For the simulation results please see Figure 9 From thesimulation it is concluded that FDDMFAC achieves the best
Table 6 The fuzzy rules of FMFAC and FDDOILC
|119890(119896 minus 1)| NB NM NS Z PS PM PB1205881 1205882 NB NM NS Z PS PM PB
control performance which demonstrates the effectiveness ofthe proposed approach in this paper
5 Conclusions
This paper aimed at solving the problem of particle qualitycontrol in the drug development phase of SFBGP Themodel-based control approaches are incapable of handlingsuch quality control because the accurate process modelcannot be acquired Thus the data-driven and model-freecontrol approaches are considered in this work We firstly
14 Complexity
MFACDDOILCFDDMFAC
5 10 150Iterations
60
80
100
120
APS
(m
)y0 = 60
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
y0 = 70
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
MFACDDOILCFDDMFAC
95
100
105
110
115
120
125A
PS (
m)
5 10 150Iterations
y0 = 100
Figure 7 The comparison of control effects of MFAC DDOILC and FDDMFAC
analyzed the features of MFAC and DDOILC and thenconducted experimental research to compare the perfor-mances of MFAC and DDOILC in particle quality controlunder different IVs The simulation results have verifiedour theoretical analysis and concluded that MFAC andDDOILC both have advantages under different IVs To beused in an actual production environment the expectedDDCapproaches must accommodate various IVs because of thechanges in material attributes Thus the hybrid approachFDDMFAC which combines the advantages of MFAC andDDOILC by adaptively adjusting their respective weightedfactors with fuzzy logic was proposed in this work tocomplement their respective shortcomings Through a series
of simulation results we conclude that FDDMFAC achievesbetter control performance than other approaches whichvalidates the effectiveness of FDDMFAC
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thanks are due to National Natural Science Foundation ofChina under Grant nos 61773105 61374147 61703085 and61004083 and Fundamental Research Funds for the Central
Complexity 15NB NM NS Z PS PM PB
001 002 00302
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
002 00401
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
20 40 600|e(k minus 1)| (m)
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
Figure 8 The membership functions of FDDMFAC when 1199100 = 60
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
60
80
100
120
APS
(m
)
y0 = 60
FMFACFDDOILCFDDMFAC
70
80
90
100
110
120
APS
(m
)
2 4 6 8 100Iterations
y0 = 70
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
100
105
110
115
120
125
APS
(m
)
y0 = 100
Figure 9 The comparison of control effects of FMFAC FDDOILC and FDDMFAC
16 Complexity
Universities under Grant nos N150402001 and N120404014for supporting this research work
References
[1] J Bouffard M Kaster and H Dumont ldquoInfluence of processvariable and physicochemical properties on the granulationmechanism of mannitol in a fluid bed top spray granulatorrdquoDrug Development and Industrial Pharmacy vol 31 no 9 pp923ndash933 2005
[2] H Liu K Wang W Schlindwein and M Li ldquoUsing the Box-Behnken experimental design to optimise operating parametersin pulsed spray fluidised bed granulationrdquo International Journalof Pharmaceutics vol 448 no 2 pp 329ndash338 2013
[3] H Liu and M Li ldquoPopulation balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granula-tionrdquo International Journal of Pharmaceutics vol 468 no 1-2pp 223ndash233 2014
[4] T Albuquerque V H Dias N Poellinger and J F PintoldquoConstruction of a quality index for granules produced byfluidized bed technology and application of the correspondenceanalysis as a discriminant procedurerdquo European Journal ofPharmaceutics and Biopharmaceutics vol 75 no 3 pp 418ndash4242010
[5] H Ehlers A Liu H Raikkonen et al ldquoGranule size control andtargeting in pulsed spray fluid bed granulationrdquo InternationalJournal of Pharmaceutics vol 377 no 1-2 pp 9ndash15 2009
[6] C F W Sanders M J Hounslow and F J Doyle III ldquoIden-tification of models for control of wet granulationrdquo PowderTechnology vol 188 no 3 pp 255ndash263 2009
[7] T Narvanen T Lipsanen O Antikainen H Raikkonen andJ Yliruusi ldquoControlling granule size by granulation liquid feedpulsingrdquo International Journal of Pharmaceutics vol 357 no 1-2pp 132ndash138 2008
[8] A Faure P York and R C Rowe ldquoProcess control and scale-up of pharmaceutical wet granulation processes a reviewrdquoEuropean Journal of Pharmaceutics and Biopharmaceutics vol52 no 3 pp 269ndash277 2001
[9] M Pottmann B A Ogunnaike A A Adetayo and B JEnnis ldquoModel-based control of a granulation systemrdquo PowderTechnology vol 108 no 2-3 pp 192ndash201 2000
[10] E P Gatzke and F J Doyle III ldquoModel predictive control of agranulation system using soft output constraints and prioritizedcontrol objectivesrdquo Powder Technology vol 121 no 2-3 pp 149ndash158 2001
[11] F Y Wang and I T Cameron ldquoReview and future directionsin the modelling and control of continuous drum granulationrdquoPowder Technology vol 124 no 3 pp 238ndash253 2002
[12] T Glaser C F W Sanders F Y Wang et al ldquoModel predictivecontrol of continuous drum granulationrdquo Journal of ProcessControl vol 19 no 4 pp 615ndash622 2009
[13] R Ramachandran andA Chaudhury ldquoModel-based design andcontrol of a continuous drum granulation processrdquo ChemicalEngineering Research and Design vol 90 no 8 pp 1063ndash10732012
[14] Z H Loh D Z L Er L W Chan C V Liew and PW S HengldquoSpray granulation for drug formulationrdquo Expert Opinion onDrug Delivery vol 8 no 12 pp 1645ndash1661 2011
[15] S H Schaafsma N W F Kossen M T Mos L Blauw andA C Hoffmann ldquoEffects and control of humidity and particlemixing in fluid-bed granulationrdquo AIChE Journal vol 45 no 6pp 1202ndash1210 1999
[16] Z-S Hou and S-T Jin ldquoA novel data-driven control approachfor a class of discrete-time nonlinear systemsrdquo IEEE Transac-tions onControl Systems Technology vol 19 no 6 pp 1549ndash15582011
[17] Z Hou and S Jin ldquoData-driven model-free adaptive controlfor a class of MIMO nonlinear discrete-time systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 22no 12 pp 2173ndash2188 2011
[18] D Xu B Jiang and P Shi ldquoA novel model-free adaptivecontrol design for multivariable industrial processesrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6391ndash6398 2014
[19] X Wang X Li J Wang X Fang and X Zhu ldquoData-drivenmodel-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeletonrdquo Information Sciences vol 327pp 246ndash257 2016
[20] L dos Santos Coelho andA A R Coelho ldquoModel-free adaptivecontrol optimization using a chaotic particle swarm approachrdquoChaos Solitons amp Fractals vol 41 no 4 pp 2001ndash2009 2009
[21] L D S Coelho M W Pessoa R Rodrigues Sumar andA Augusto Rodrigues Coelho ldquoModel-free adaptive controldesign using evolutionary-neural compensatorrdquo Expert Systemswith Applications vol 37 no 1 pp 499ndash508 2010
[22] K K Tan T H Lee S N Huang and F M Leu ldquoAdaptive-predictive control of a class of SISOnonlinear systemsrdquoDynam-ics and Control vol 11 no 2 pp 151ndash174 2001
[23] B Zhang andW Zhang ldquoAdaptive predictive functional controlof a class of nonlinear systemsrdquo ISA Transactions vol 45 no2 pp 175ndash183 2006
[24] F L Lu J F Wang C J Fan and S B Chen ldquoAn improvedmodel-free adaptive control with G function fuzzy reasoningregulation design and its applicationsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems andControl Engineering vol 222 no 8 pp 817ndash828 2008
[25] R Chi D Wang Z Hou and S Jin ldquoData-driven optimalterminal iterative learning controlrdquo Journal of Process Controlvol 22 no 10 pp 2026ndash2037 2012
[26] R Chi Z Hou S Jin and D Wang ldquoImproved data-drivenoptimal TILC using time-varying input signalsrdquo Journal ofProcess Control vol 24 no 12 pp 78ndash85 2014
[27] X Bu Z Hou and R Chi ldquoModel free adaptive iterative learn-ing control for farm vehicle path trackingrdquo IFAC ProceedingsVolumes vol 46 no 20 pp 153ndash158 2013
[28] M C Campi A Lecchini and SM Savaresi ldquoVirtual referencefeedback tuning a direct method for the design of feedbackcontrollersrdquo Automatica vol 38 no 8 pp 1337ndash1346 2002
[29] R-C Roman M-B Radac R-E Precup and E M PetriuldquoData-driven model-free adaptive control tuned by virtualreference feedback tuningrdquoActa PolytechnicaHungarica vol 13no 1 pp 83ndash96 2016
[30] P Yan D Liu D Wang and H Ma ldquoData-driven controllerdesign for general MIMO nonlinear systems via virtual refer-ence feedback tuning and neural networksrdquo Neurocomputingvol 171 pp 815ndash825 2016
[31] T Pan and S Li ldquoA hierarchical search and updating databasestrategy for lazy learningrdquo International Journal of InnovativeComputing Information andControl vol 4 no 6 pp 1383ndash13922008
[32] H Zhang R Song Q Wei and T Zhang ldquoOptimal trackingcontrol for a class of nonlinear discrete-time systems withtime delays based on heuristic dynamic programmingrdquo IEEE
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Complexity
IV = 60IV = 70
IV = 80IV = 100
IV = 60 14320IV = 70 08061IV = 80 04718IV = 100 04000
5 10 150Iterations
0
5
10
15
20
25
e p(
m)
Average of ep
Figure 4 The DLM prediction error profiles of MFAC for differentIVs
different IVs all show a decreasing trend along with iterationuntil 119890119901 = 0 as APS keeps getting closer to 119910119889 which is inaccordance with our obtained conclusion On the other handfor different IVs 119890119901 will also decrease with the increase of IVwhich is proved by comparing the curves of 119890119901 under differentIVs To further illustrate this point the averages of 119890119901 fordifferent IVs are computed and then listed in Figure 4 andthe decreasing averages of 119890119901 along with the increase of IValso validate the rationality of our analysis
4 The Proposed Hybrid DDC Approach andSimulation Study in SFBGP
Through theoretical analysis and simulation study we haveproved that MFAC and DDOILC both have strengths inparticle quality control under different IVs In real-worldapplications the same operating condition may give entirelydifferent IVs due to the changes in material properties whichbrings great difficulties in the selection of control methodsThus the uncertainties of IV have to be considered duringcontrol application In order to achieve favourable overallcontrol effect it is demanded that the DDC approaches mustaccommodate various IVs Therefore we try to combineadvantages of MFAC and DDOILC to present a hybridapproach that is named DDMFAC The basic idea is that thefeatures of MFAC and DDOILC are both reflected in thecriterion functions when designing controller of DDMFACThe expected features of hybrid approach can be summarizedas the following three points
(1) When tracking error is large DDMFACmanifests thefeature of DDOILC to acquire a faster convergencespeed
(2) When tracking error is small enough DDMFAC hasthe ability ofMFAC to achieve a fast and stable controleffect
(3) DDMFAC is able to achieve better control perfor-mance for different IVs
Through such combination the shortcomings of MFACand DDOILC will be complemented and their respec-tive advantages will be highlighted in DDMFAC Next wedescribe the controller design for DDMFAC in Section 41
41 DDMFAC Design Consider the following control inputcriterion function
119869 (u (119896)) = 120572119872 1003816100381610038161003816119910119889 minus 1199101 (119896 + 1)10038161003816100381610038162 + 120572119863 1003816100381610038161003816119910119889 minus 1199102 (119896)10038161003816100381610038162+ 1205821 Δu (119896)2 (20)
where |119910119889 minus 1199101(119896 + 1)|2 and |119910119889 minus 1199102(119896)|2 respectively comefrom MFAC and DDOILC and 120572119872 120572119863 are the respectiveweighted factors and 120572119872 + 120572119863 = 0
Substituting (7) and (13) into (20) gives
119869 (u (119896)) = 120572119872 10038171003817100381710038171003817119890 (119896) minus 120593119879 (119896) Δu (119896)100381710038171003817100381710038172+ 120572119863 10038171003817100381710038171003817119890 (119896 minus 1) minus 120593119879 (119896) Δu (119896)100381710038171003817100381710038172+ 1205821 Δu (119896)2
(21)
Using optimal condition 120597119869120597u(119896) = 0 then gives
Δu (119896) = 120593 (119896) (120572119872119890 (119896) + 120572119863119890 (119896 minus 1))1205821 + (120572119872 + 120572119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (22)
Dividing numerator and denominator by 120572119872+120572119863 at the righthand of (22) we haveΔu (119896)= 120593 (119896) ((120572119872 (120572119872 + 120572119863)) 119890 (119896) + (120572119863 (120572119872 + 120572119863)) 119890 (119896 minus 1))
(1205821 (120572119872 + 120572119863)) + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (23)
Letting 120588119872 = 120572119872(120572119872 + 120572119863) 120588119863 = 120572119863(120572119872 + 120572119863) and 120582 =1205821(120572119872 + 120572119863) thenu (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (24)
where 120582 gt 0 is the weighted factor and 120588119872 120588119863 isin [0 1] are thestep-size factors
Define the following PPD vector criterion function
119869 ( (119896))= 10038171003817100381710038171003817120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896) Δu (119896 minus 1)100381710038171003817100381710038172
+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172 (25)
where 120573119872 120573119863 isin [0 1] are the weighted factors and 120573119872+120573119863 =1We rewrite (25) as
119869 ( (119896)) = 10038161003816100381610038161003816120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1)minus 119879 (119896 minus 1) Δu (119896 minus 1) minus Δ119879 (119896) Δu (119896 minus 1)100381610038161003816100381610038162+ 120583 1003817100381710038171003817Δ (119896)10038171003817100381710038172
(26)
Complexity 9
Utilizing optimal condition 120597119869120597(119896) = 0 we have
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (27)
Thus the controller of DDMFAC is designed as follows
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (28)
u (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (29)
42 Stability and Convergence Analysis of DDMFAC
(1)The Boundedness of (119896) Define the parameter estimationerror of 120593(119896) as follows
(119896) = 120593 (119896) minus (119896) (30)
Substituting (28) into (30) gives
(119896) = 120593 (119896) minus (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 (31)
Similarly define (119896 minus 1) = 120593(119896 minus 1) minus (119896 minus 1) and then (119896 minus 1) = 120593 (119896 minus 1) minus (119896 minus 1) (32)
Combining (31) and (32) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1) (33)
According to (7) and (13) it can be obtained that
Δ119910 (119896) = 120593119879 (119896 minus 1) Δu (119896 minus 1)
Δ119910 (119896 minus 1) = 120593119879 (119896 minus 1) Δu (119896 minus 1) (34)
Substituting (34) into (33) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) ((120573119872 + 120573119863)120593119879 (119896 minus 1) Δu (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578Δu (119896 minus 1) ((120593 (119896 minus 1) minus (119896 minus 1))119879 Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578 (119896 minus 1) Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2) + 120593 (119896) minus 120593 (119896 minus 1)
(35)
10 Complexity
Because |120593(119896)| le 119887 is bounded [16] |120593(119896)minus120593(119896minus1)| le 2119887Taking the modulus of both sides of (35) we can obtain
1003817100381710038171003817 (119896)1003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2)100381710038171003817100381710038171003817100381710038171003817 + 2119887 (36)
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
= 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 + (minus2 + 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
times 1205781003817100381710038171003817 (119896 minus 1) Δu (119896 minus 1)10038171003817100381710038172120583 + Δu (119896 minus 1)2
(37)
Given that 0 lt 120578 lt 2 120583 gt 0(minus2 + 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2) lt 0 (38)
Synthesizing (37) and (38) we have
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
le 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 (39)
According to (36) and (39) there is obviously a positivenumber 1198891 isin (0 1) which makes
1003817100381710038171003817 (119896)1003817100381710038171003817 le 1198891 1003817100381710038171003817 (119896 minus 1)1003817100381710038171003817 + 2119887 le sdot sdot sdotle 1198891198961 1003817100381710038171003817 (0)1003817100381710038171003817 + 2119887 (1 minus 1198891198961)1 minus 1198891 (40)
Therefore (119896) is bounded(2)TheConvergence of Tracking Error 119890(119896) Tracking error 119890(119896)is derived as
119890 (119896) = 119910119889 minus 119910 (119896) = 119910119889 minus 119910 (119896 minus 1) minus 120593119879 (119896) Δu (119896)= 119890 (119896 minus 1) minus 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
(120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)= (1 minus 120588119863 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)
minus 120588119872 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 119890 (119896)
(41)
Transposition gives
119890 (119896) = [1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172] 119890 (119896 minus 1) (42)
So as long as appropriate parameters 120588119872 120588119863 and 120582 areselected there must be a positive number 1198892 isin (0 1) thatmakes
0 lt 10038171003817100381710038171003817100381710038171003817100381710038171 minus(120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172
1003817100381710038171003817100381710038171003817100381710038171003817 le 1198892 lt 1 (43)
By (42) and (43) it can be obtained that
119890 (119896) le 1003817100381710038171003817100381710038171003817100381710038171003817(1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)1003817100381710038171003817100381710038171003817100381710038171003817le 1198892 119890 (119896 minus 1) le sdot sdot sdot le 1198891198962 119890 (0)
(44)
Therefore lim119896rarrinfin119890(119896) = 0 that is 119890(119896) is proved toconverge to zero
(3)The Boundedness of System Input and Output By referringto 1198862 + 1198872 ge 2119886119887 the relational expression can be obtained asfollows
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
120582 + 1003817100381710038171003817120593 (119896)100381710038171003817100381721003817100381710038171003817100381710038171003817100381710038171003817 le
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
2radic120582 1003817100381710038171003817120593 (119896)10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 =
12radic120582 (45)
And we have
u (119896) = u (119896) minus u (119896 minus 1) + u (119896 minus 1) minus sdot sdot sdot + u (1)minus u (0) + u (0) = u (0) + 119896sum
119895=1
u (119895) (46)
Combining (24) (44) (45) and (46) as well as utilizingSchwartzrsquos inequality gives
u (119896) le u (0) + 119896sum119895=1
1003817100381710038171003817Δu (119895)1003817100381710038171003817
le u (0) + 12radic120582
119896sum119895=1
1003817100381710038171003817120588119863119890 (119895 minus 1) + 120588119872119890 (119895)1003817100381710038171003817
le u (0) + 1205881198632radic120582119896sum119895=1
119889119895minus12 119890 (0)
+ 1205881198722radic120582119896sum119895=1
1198891198952 119890 (0)
le u (0) + 1205881198632radic1205821 minus 119889119895minus121 minus 1198892 119890 (0)
+ 1205881198722radic1205821198892 (1 minus 1198891198952)1 minus 1198892 119890 (0)
le u (0) + 120588119863(1 minus 1198892) 2radic120582 119890 (0)
Complexity 11
Table 3 The parameter settings of DDMFAC for different IVs
IV Parameter120588119872 120588119863 120573119872 120573119863 120582 120578 1205831199100 = 60 00275 00275 05000 05000
00010 00750 100001199100 = 70 50119864 minus 4 50119864 minus 4 05000 050001199100 = 80 46119864 minus 4 46119864 minus 4 05000 050001199100 = 100 18119864 minus 4 18119864 minus 4 05000 05000
+ 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)le u (0) + 120588119863 + 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)
(47)
Because the initial tracking error 119890(0) and initial inputu(0) are given bounded the system input u(119896) is boundedaccording to (47) Besides due to the boundedness of 119910119889 and119890(119896) the system output 119910(119896) is obviously bounded43 Fuzzy DDMFAC The controller design and stability andconvergence analysis for DDMFAC have been respectivelydiscussed in detail in the above sections Although thedesigned controller has the characters of both MFAC andDDOILC weighted factors 120588119872 120588119863 120573119872 and 120573119863 are allconstant throughout control process DDMFAC is incapableof adaptively selecting the dominance of MFAC or DDOILCto duly show the expected features in each stage of controlprocessThe aforementioned features of DDMFACwe expectcannot be reflected in this case Such DDMFAC approachwith fixed weighting factors has no advantage comparedwith MFAC or DDOILC which appeals to a reasonable andadaptive adjustment strategy for 120588119872 120588119863 120573119872 and 120573119863 There-fore the adaptive adjustment strategy based on fuzzy logicfor weighting factors is considered because its advantageousperformance in parameter adjustment has been proven inour previous work [45] The formulation of fuzzy rules offersus the opportunity to adjust weighting factors in the waywe expected Such DDMFAC approach with fuzzy adaptiveadjustment for parameters is namedFDDMFAC in this paperConcretely the fuzzy rules can be formulated by referringto the expected features of DDMFAC The input variables offuzzy rules are the absolute values of tracking error 119890(119896 minus 1)and DLM prediction error 119890119901(119896minus1) of MFAC and the outputvariables are 120588119872 120588119863 and 120573119863 Because of the relationshipbetween 120573119872 and 120573119863 there is no need to use 120573119872 as an outputand it is calculated by 120573119872 = 1 minus 120573119863 The fuzzy rules can bedetailedly described as follows
(1) If |119890(119896 minus 1)| and |119890119901(119896 minus 1)| are great then thedominance of DDOILC is preferred and that ofMFAC is restrained in DDMFAC So 120588119863 and 120573119863 haverelatively large values and 120588119872 is a relatively smallnumber
(2) 120588119863 and 120573119863 will show a decreasing trend with thegradual decrease of |119890(119896 minus 1)| and |119890119901(119896 minus 1)| while120588119872 will present an increasing trend
44 Simulations and Result Analysis The proposed DDM-FAC approach is firstly applied to the particle quality controlin SFBGP and then its control effect is compared withMFACand DDOILC under different IVs In the simulation justlike MFAC 119910(119896) is replaced by a predicted value 119910(119896) thatis estimated using (19) See Table 3 for parameter settings ofDDMFAC It is not hard to find 120588119872 = 120588119863 and 120573119872 = 120573119863for a certain IV which means that MFAC and DDOILC areset to have the same dominance in DDMFAC By the wayMFAC and DDOILC can be regarded as the extreme cases ofDDMFAC described as follows
MFAC 120588119872 = 120573119872 = 1120588119863 = 120573119863 = 0
DDOILC 120588119872 = 120573119872 = 0120588119863 = 120573119863 = 1
(48)
The control effects of DDMFAC with the parametersettings in Table 3 together with control curves of MFACand DDOILC are plotted in Figure 5 The simulation resultsactually present the control effects of DDMFAC with threesets of parameter settings From simulation it is indicatedthat all three approaches have gained some certain advantagesunder different IV conditions Concretely as is shown inFigure 5(a) DDMFAC has advantage in control performancecompared with DDOILC and MFAC when 1199100 = 60 butMFAC has absolute superiority when 1199100 = 100 plottedin Figure 5(d) In addition from the simulation results inFigures 5(b) and 5(c) it can be seen that DDOILC andMFAC have achieved certain but not absolute advantagesrespectively This demonstrates that parameter setting hasimportant influence on the control effect of DDMFACTherefore the way to achieve satisfactory control effect is toadaptively adjust weighted parameters of DDMFAC insteadof keeping them constant
Then the idea of fuzzy adjustment is introduced to adjustthe weights 120588119872 120588119863 and 120573119863 The fuzzy adjustment rules ofthree weights are respectively shown in Tables 4 and 5 Therespective membership functions when 1199100 = 60 are shownin Figure 6 and membership functions for the other IVs areno longer displayed Please see Figure 7 for control effectof FDDMFAC and comparisons with MFAC and DDOILCunder different IVs Obviously DDMFAC approach withfuzzy adjustment for parameters that is FDDMFAC has thebest performance in particle quality control for all IVs This
12 Complexity
Table 4 The fuzzy rules of 120588119872120588119872 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB PB PB PB PB PB PB PBNM PB PM PM PM PM PS PSNS PM PM PM PS PS Z ZZ Z Z Z NS NS NS NSPS NS NS NM NM NB NB NBPM NM NM NM NB NB NB NBPB NB NB NB NB NB NB NB
MFACDDOILCDDMFAC
60
80
100
120
APS
(m
)
5 10 150Iterations
(a) 1199100 = 60
MFACDDOILCDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILCDDMFAC
80
90
100
110
120
APS
(m
)
2 4 60Iterations
(c) 1199100 = 80
95
100
105
110
115
120
125
APS
(m
)
5 10 150Iterations
MFACDDOILCDDMFAC
(d) 1199100 = 100
Figure 5 The comparison of control effects of MFAC DDOILC and DDMFAC
Complexity 13
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
02 04 06 08 10D
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
005 01 0150D
NB NM NS Z PS PM PB
times10minus3
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
05 1 15 2 25 3 35 4 450M
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
10 20 30 40 50 600|e(k minus 1)| (m)
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
5 10 15 20 250|ep (k minus 1)| (m)
Figure 6 The membership functions of FDDMFAC when 1199100 = 60
Table 5 The fuzzy rules of 120588119863 120573119863120588119863 120573119863 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB NB NB NB NB NB NB NBNM NB NM NM NM NM NS NSNS NM NM NM NS NS Z ZZ Z Z Z PS PM PM PMPS PS PS PM PM PB PB PBPM PM PM PM PB PB PB PBPB PB PB PB PB PB PB PB
validates the effectiveness of FDDMFAC proposed in thispaper
Although FDDMFAC approach has been proven to havea better control performance than MFAC and DDOILCwe still have a question of whether it is possible that thesuperiority of FDDMFAC is entirely attributed to fuzzyadjustment of parameters and has nothing to do with thestructure of DDMFAC itself In order to clarify this problemanother set of simulations are conducted inwhich the controlperformance of FDDMFAC is compared with fuzzy MFAC(FMFAC) and fuzzy DDOILC (FDDOILC) The absolutevalue of tracking error |119890(119896 minus 1)| is regarded as the inputvariable of fuzzy rules and the step-size factors 1205881 1205882 ofMFAC andDDOILC are the output variablesThe fuzzy rulesof FMFAC and FDDOILC are shown in Table 6 and themembership functions when 1199100 = 60 are shown in Figure 8For the simulation results please see Figure 9 From thesimulation it is concluded that FDDMFAC achieves the best
Table 6 The fuzzy rules of FMFAC and FDDOILC
|119890(119896 minus 1)| NB NM NS Z PS PM PB1205881 1205882 NB NM NS Z PS PM PB
control performance which demonstrates the effectiveness ofthe proposed approach in this paper
5 Conclusions
This paper aimed at solving the problem of particle qualitycontrol in the drug development phase of SFBGP Themodel-based control approaches are incapable of handlingsuch quality control because the accurate process modelcannot be acquired Thus the data-driven and model-freecontrol approaches are considered in this work We firstly
14 Complexity
MFACDDOILCFDDMFAC
5 10 150Iterations
60
80
100
120
APS
(m
)y0 = 60
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
y0 = 70
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
MFACDDOILCFDDMFAC
95
100
105
110
115
120
125A
PS (
m)
5 10 150Iterations
y0 = 100
Figure 7 The comparison of control effects of MFAC DDOILC and FDDMFAC
analyzed the features of MFAC and DDOILC and thenconducted experimental research to compare the perfor-mances of MFAC and DDOILC in particle quality controlunder different IVs The simulation results have verifiedour theoretical analysis and concluded that MFAC andDDOILC both have advantages under different IVs To beused in an actual production environment the expectedDDCapproaches must accommodate various IVs because of thechanges in material attributes Thus the hybrid approachFDDMFAC which combines the advantages of MFAC andDDOILC by adaptively adjusting their respective weightedfactors with fuzzy logic was proposed in this work tocomplement their respective shortcomings Through a series
of simulation results we conclude that FDDMFAC achievesbetter control performance than other approaches whichvalidates the effectiveness of FDDMFAC
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thanks are due to National Natural Science Foundation ofChina under Grant nos 61773105 61374147 61703085 and61004083 and Fundamental Research Funds for the Central
Complexity 15NB NM NS Z PS PM PB
001 002 00302
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
002 00401
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
20 40 600|e(k minus 1)| (m)
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
Figure 8 The membership functions of FDDMFAC when 1199100 = 60
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
60
80
100
120
APS
(m
)
y0 = 60
FMFACFDDOILCFDDMFAC
70
80
90
100
110
120
APS
(m
)
2 4 6 8 100Iterations
y0 = 70
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
100
105
110
115
120
125
APS
(m
)
y0 = 100
Figure 9 The comparison of control effects of FMFAC FDDOILC and FDDMFAC
16 Complexity
Universities under Grant nos N150402001 and N120404014for supporting this research work
References
[1] J Bouffard M Kaster and H Dumont ldquoInfluence of processvariable and physicochemical properties on the granulationmechanism of mannitol in a fluid bed top spray granulatorrdquoDrug Development and Industrial Pharmacy vol 31 no 9 pp923ndash933 2005
[2] H Liu K Wang W Schlindwein and M Li ldquoUsing the Box-Behnken experimental design to optimise operating parametersin pulsed spray fluidised bed granulationrdquo International Journalof Pharmaceutics vol 448 no 2 pp 329ndash338 2013
[3] H Liu and M Li ldquoPopulation balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granula-tionrdquo International Journal of Pharmaceutics vol 468 no 1-2pp 223ndash233 2014
[4] T Albuquerque V H Dias N Poellinger and J F PintoldquoConstruction of a quality index for granules produced byfluidized bed technology and application of the correspondenceanalysis as a discriminant procedurerdquo European Journal ofPharmaceutics and Biopharmaceutics vol 75 no 3 pp 418ndash4242010
[5] H Ehlers A Liu H Raikkonen et al ldquoGranule size control andtargeting in pulsed spray fluid bed granulationrdquo InternationalJournal of Pharmaceutics vol 377 no 1-2 pp 9ndash15 2009
[6] C F W Sanders M J Hounslow and F J Doyle III ldquoIden-tification of models for control of wet granulationrdquo PowderTechnology vol 188 no 3 pp 255ndash263 2009
[7] T Narvanen T Lipsanen O Antikainen H Raikkonen andJ Yliruusi ldquoControlling granule size by granulation liquid feedpulsingrdquo International Journal of Pharmaceutics vol 357 no 1-2pp 132ndash138 2008
[8] A Faure P York and R C Rowe ldquoProcess control and scale-up of pharmaceutical wet granulation processes a reviewrdquoEuropean Journal of Pharmaceutics and Biopharmaceutics vol52 no 3 pp 269ndash277 2001
[9] M Pottmann B A Ogunnaike A A Adetayo and B JEnnis ldquoModel-based control of a granulation systemrdquo PowderTechnology vol 108 no 2-3 pp 192ndash201 2000
[10] E P Gatzke and F J Doyle III ldquoModel predictive control of agranulation system using soft output constraints and prioritizedcontrol objectivesrdquo Powder Technology vol 121 no 2-3 pp 149ndash158 2001
[11] F Y Wang and I T Cameron ldquoReview and future directionsin the modelling and control of continuous drum granulationrdquoPowder Technology vol 124 no 3 pp 238ndash253 2002
[12] T Glaser C F W Sanders F Y Wang et al ldquoModel predictivecontrol of continuous drum granulationrdquo Journal of ProcessControl vol 19 no 4 pp 615ndash622 2009
[13] R Ramachandran andA Chaudhury ldquoModel-based design andcontrol of a continuous drum granulation processrdquo ChemicalEngineering Research and Design vol 90 no 8 pp 1063ndash10732012
[14] Z H Loh D Z L Er L W Chan C V Liew and PW S HengldquoSpray granulation for drug formulationrdquo Expert Opinion onDrug Delivery vol 8 no 12 pp 1645ndash1661 2011
[15] S H Schaafsma N W F Kossen M T Mos L Blauw andA C Hoffmann ldquoEffects and control of humidity and particlemixing in fluid-bed granulationrdquo AIChE Journal vol 45 no 6pp 1202ndash1210 1999
[16] Z-S Hou and S-T Jin ldquoA novel data-driven control approachfor a class of discrete-time nonlinear systemsrdquo IEEE Transac-tions onControl Systems Technology vol 19 no 6 pp 1549ndash15582011
[17] Z Hou and S Jin ldquoData-driven model-free adaptive controlfor a class of MIMO nonlinear discrete-time systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 22no 12 pp 2173ndash2188 2011
[18] D Xu B Jiang and P Shi ldquoA novel model-free adaptivecontrol design for multivariable industrial processesrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6391ndash6398 2014
[19] X Wang X Li J Wang X Fang and X Zhu ldquoData-drivenmodel-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeletonrdquo Information Sciences vol 327pp 246ndash257 2016
[20] L dos Santos Coelho andA A R Coelho ldquoModel-free adaptivecontrol optimization using a chaotic particle swarm approachrdquoChaos Solitons amp Fractals vol 41 no 4 pp 2001ndash2009 2009
[21] L D S Coelho M W Pessoa R Rodrigues Sumar andA Augusto Rodrigues Coelho ldquoModel-free adaptive controldesign using evolutionary-neural compensatorrdquo Expert Systemswith Applications vol 37 no 1 pp 499ndash508 2010
[22] K K Tan T H Lee S N Huang and F M Leu ldquoAdaptive-predictive control of a class of SISOnonlinear systemsrdquoDynam-ics and Control vol 11 no 2 pp 151ndash174 2001
[23] B Zhang andW Zhang ldquoAdaptive predictive functional controlof a class of nonlinear systemsrdquo ISA Transactions vol 45 no2 pp 175ndash183 2006
[24] F L Lu J F Wang C J Fan and S B Chen ldquoAn improvedmodel-free adaptive control with G function fuzzy reasoningregulation design and its applicationsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems andControl Engineering vol 222 no 8 pp 817ndash828 2008
[25] R Chi D Wang Z Hou and S Jin ldquoData-driven optimalterminal iterative learning controlrdquo Journal of Process Controlvol 22 no 10 pp 2026ndash2037 2012
[26] R Chi Z Hou S Jin and D Wang ldquoImproved data-drivenoptimal TILC using time-varying input signalsrdquo Journal ofProcess Control vol 24 no 12 pp 78ndash85 2014
[27] X Bu Z Hou and R Chi ldquoModel free adaptive iterative learn-ing control for farm vehicle path trackingrdquo IFAC ProceedingsVolumes vol 46 no 20 pp 153ndash158 2013
[28] M C Campi A Lecchini and SM Savaresi ldquoVirtual referencefeedback tuning a direct method for the design of feedbackcontrollersrdquo Automatica vol 38 no 8 pp 1337ndash1346 2002
[29] R-C Roman M-B Radac R-E Precup and E M PetriuldquoData-driven model-free adaptive control tuned by virtualreference feedback tuningrdquoActa PolytechnicaHungarica vol 13no 1 pp 83ndash96 2016
[30] P Yan D Liu D Wang and H Ma ldquoData-driven controllerdesign for general MIMO nonlinear systems via virtual refer-ence feedback tuning and neural networksrdquo Neurocomputingvol 171 pp 815ndash825 2016
[31] T Pan and S Li ldquoA hierarchical search and updating databasestrategy for lazy learningrdquo International Journal of InnovativeComputing Information andControl vol 4 no 6 pp 1383ndash13922008
[32] H Zhang R Song Q Wei and T Zhang ldquoOptimal trackingcontrol for a class of nonlinear discrete-time systems withtime delays based on heuristic dynamic programmingrdquo IEEE
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Complexity 9
Utilizing optimal condition 120597119869120597(119896) = 0 we have
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (27)
Thus the controller of DDMFAC is designed as follows
(119896) = (119896 minus 1) + 120578Δu (119896 minus 1) (120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 (28)
u (119896) = u (119896 minus 1) + 120593 (119896) (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (29)
42 Stability and Convergence Analysis of DDMFAC
(1)The Boundedness of (119896) Define the parameter estimationerror of 120593(119896) as follows
(119896) = 120593 (119896) minus (119896) (30)
Substituting (28) into (30) gives
(119896) = 120593 (119896) minus (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 (31)
Similarly define (119896 minus 1) = 120593(119896 minus 1) minus (119896 minus 1) and then (119896 minus 1) = 120593 (119896 minus 1) minus (119896 minus 1) (32)
Combining (31) and (32) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) [120573119872Δ119910 (119896) + 120573119863Δ119910 (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1)]120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1) (33)
According to (7) and (13) it can be obtained that
Δ119910 (119896) = 120593119879 (119896 minus 1) Δu (119896 minus 1)
Δ119910 (119896 minus 1) = 120593119879 (119896 minus 1) Δu (119896 minus 1) (34)
Substituting (34) into (33) we have
(119896) = (119896 minus 1) minus 120578Δu (119896 minus 1) ((120573119872 + 120573119863)120593119879 (119896 minus 1) Δu (119896 minus 1) minus 119879 (119896 minus 1) Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578Δu (119896 minus 1) ((120593 (119896 minus 1) minus (119896 minus 1))119879 Δu (119896 minus 1))120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) minus 120578 (119896 minus 1) Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2 + 120593 (119896) minus 120593 (119896 minus 1)
= (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2) + 120593 (119896) minus 120593 (119896 minus 1)
(35)
10 Complexity
Because |120593(119896)| le 119887 is bounded [16] |120593(119896)minus120593(119896minus1)| le 2119887Taking the modulus of both sides of (35) we can obtain
1003817100381710038171003817 (119896)1003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2)100381710038171003817100381710038171003817100381710038171003817 + 2119887 (36)
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
= 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 + (minus2 + 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
times 1205781003817100381710038171003817 (119896 minus 1) Δu (119896 minus 1)10038171003817100381710038172120583 + Δu (119896 minus 1)2
(37)
Given that 0 lt 120578 lt 2 120583 gt 0(minus2 + 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2) lt 0 (38)
Synthesizing (37) and (38) we have
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
le 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 (39)
According to (36) and (39) there is obviously a positivenumber 1198891 isin (0 1) which makes
1003817100381710038171003817 (119896)1003817100381710038171003817 le 1198891 1003817100381710038171003817 (119896 minus 1)1003817100381710038171003817 + 2119887 le sdot sdot sdotle 1198891198961 1003817100381710038171003817 (0)1003817100381710038171003817 + 2119887 (1 minus 1198891198961)1 minus 1198891 (40)
Therefore (119896) is bounded(2)TheConvergence of Tracking Error 119890(119896) Tracking error 119890(119896)is derived as
119890 (119896) = 119910119889 minus 119910 (119896) = 119910119889 minus 119910 (119896 minus 1) minus 120593119879 (119896) Δu (119896)= 119890 (119896 minus 1) minus 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
(120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)= (1 minus 120588119863 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)
minus 120588119872 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 119890 (119896)
(41)
Transposition gives
119890 (119896) = [1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172] 119890 (119896 minus 1) (42)
So as long as appropriate parameters 120588119872 120588119863 and 120582 areselected there must be a positive number 1198892 isin (0 1) thatmakes
0 lt 10038171003817100381710038171003817100381710038171003817100381710038171 minus(120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172
1003817100381710038171003817100381710038171003817100381710038171003817 le 1198892 lt 1 (43)
By (42) and (43) it can be obtained that
119890 (119896) le 1003817100381710038171003817100381710038171003817100381710038171003817(1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)1003817100381710038171003817100381710038171003817100381710038171003817le 1198892 119890 (119896 minus 1) le sdot sdot sdot le 1198891198962 119890 (0)
(44)
Therefore lim119896rarrinfin119890(119896) = 0 that is 119890(119896) is proved toconverge to zero
(3)The Boundedness of System Input and Output By referringto 1198862 + 1198872 ge 2119886119887 the relational expression can be obtained asfollows
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
120582 + 1003817100381710038171003817120593 (119896)100381710038171003817100381721003817100381710038171003817100381710038171003817100381710038171003817 le
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
2radic120582 1003817100381710038171003817120593 (119896)10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 =
12radic120582 (45)
And we have
u (119896) = u (119896) minus u (119896 minus 1) + u (119896 minus 1) minus sdot sdot sdot + u (1)minus u (0) + u (0) = u (0) + 119896sum
119895=1
u (119895) (46)
Combining (24) (44) (45) and (46) as well as utilizingSchwartzrsquos inequality gives
u (119896) le u (0) + 119896sum119895=1
1003817100381710038171003817Δu (119895)1003817100381710038171003817
le u (0) + 12radic120582
119896sum119895=1
1003817100381710038171003817120588119863119890 (119895 minus 1) + 120588119872119890 (119895)1003817100381710038171003817
le u (0) + 1205881198632radic120582119896sum119895=1
119889119895minus12 119890 (0)
+ 1205881198722radic120582119896sum119895=1
1198891198952 119890 (0)
le u (0) + 1205881198632radic1205821 minus 119889119895minus121 minus 1198892 119890 (0)
+ 1205881198722radic1205821198892 (1 minus 1198891198952)1 minus 1198892 119890 (0)
le u (0) + 120588119863(1 minus 1198892) 2radic120582 119890 (0)
Complexity 11
Table 3 The parameter settings of DDMFAC for different IVs
IV Parameter120588119872 120588119863 120573119872 120573119863 120582 120578 1205831199100 = 60 00275 00275 05000 05000
00010 00750 100001199100 = 70 50119864 minus 4 50119864 minus 4 05000 050001199100 = 80 46119864 minus 4 46119864 minus 4 05000 050001199100 = 100 18119864 minus 4 18119864 minus 4 05000 05000
+ 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)le u (0) + 120588119863 + 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)
(47)
Because the initial tracking error 119890(0) and initial inputu(0) are given bounded the system input u(119896) is boundedaccording to (47) Besides due to the boundedness of 119910119889 and119890(119896) the system output 119910(119896) is obviously bounded43 Fuzzy DDMFAC The controller design and stability andconvergence analysis for DDMFAC have been respectivelydiscussed in detail in the above sections Although thedesigned controller has the characters of both MFAC andDDOILC weighted factors 120588119872 120588119863 120573119872 and 120573119863 are allconstant throughout control process DDMFAC is incapableof adaptively selecting the dominance of MFAC or DDOILCto duly show the expected features in each stage of controlprocessThe aforementioned features of DDMFACwe expectcannot be reflected in this case Such DDMFAC approachwith fixed weighting factors has no advantage comparedwith MFAC or DDOILC which appeals to a reasonable andadaptive adjustment strategy for 120588119872 120588119863 120573119872 and 120573119863 There-fore the adaptive adjustment strategy based on fuzzy logicfor weighting factors is considered because its advantageousperformance in parameter adjustment has been proven inour previous work [45] The formulation of fuzzy rules offersus the opportunity to adjust weighting factors in the waywe expected Such DDMFAC approach with fuzzy adaptiveadjustment for parameters is namedFDDMFAC in this paperConcretely the fuzzy rules can be formulated by referringto the expected features of DDMFAC The input variables offuzzy rules are the absolute values of tracking error 119890(119896 minus 1)and DLM prediction error 119890119901(119896minus1) of MFAC and the outputvariables are 120588119872 120588119863 and 120573119863 Because of the relationshipbetween 120573119872 and 120573119863 there is no need to use 120573119872 as an outputand it is calculated by 120573119872 = 1 minus 120573119863 The fuzzy rules can bedetailedly described as follows
(1) If |119890(119896 minus 1)| and |119890119901(119896 minus 1)| are great then thedominance of DDOILC is preferred and that ofMFAC is restrained in DDMFAC So 120588119863 and 120573119863 haverelatively large values and 120588119872 is a relatively smallnumber
(2) 120588119863 and 120573119863 will show a decreasing trend with thegradual decrease of |119890(119896 minus 1)| and |119890119901(119896 minus 1)| while120588119872 will present an increasing trend
44 Simulations and Result Analysis The proposed DDM-FAC approach is firstly applied to the particle quality controlin SFBGP and then its control effect is compared withMFACand DDOILC under different IVs In the simulation justlike MFAC 119910(119896) is replaced by a predicted value 119910(119896) thatis estimated using (19) See Table 3 for parameter settings ofDDMFAC It is not hard to find 120588119872 = 120588119863 and 120573119872 = 120573119863for a certain IV which means that MFAC and DDOILC areset to have the same dominance in DDMFAC By the wayMFAC and DDOILC can be regarded as the extreme cases ofDDMFAC described as follows
MFAC 120588119872 = 120573119872 = 1120588119863 = 120573119863 = 0
DDOILC 120588119872 = 120573119872 = 0120588119863 = 120573119863 = 1
(48)
The control effects of DDMFAC with the parametersettings in Table 3 together with control curves of MFACand DDOILC are plotted in Figure 5 The simulation resultsactually present the control effects of DDMFAC with threesets of parameter settings From simulation it is indicatedthat all three approaches have gained some certain advantagesunder different IV conditions Concretely as is shown inFigure 5(a) DDMFAC has advantage in control performancecompared with DDOILC and MFAC when 1199100 = 60 butMFAC has absolute superiority when 1199100 = 100 plottedin Figure 5(d) In addition from the simulation results inFigures 5(b) and 5(c) it can be seen that DDOILC andMFAC have achieved certain but not absolute advantagesrespectively This demonstrates that parameter setting hasimportant influence on the control effect of DDMFACTherefore the way to achieve satisfactory control effect is toadaptively adjust weighted parameters of DDMFAC insteadof keeping them constant
Then the idea of fuzzy adjustment is introduced to adjustthe weights 120588119872 120588119863 and 120573119863 The fuzzy adjustment rules ofthree weights are respectively shown in Tables 4 and 5 Therespective membership functions when 1199100 = 60 are shownin Figure 6 and membership functions for the other IVs areno longer displayed Please see Figure 7 for control effectof FDDMFAC and comparisons with MFAC and DDOILCunder different IVs Obviously DDMFAC approach withfuzzy adjustment for parameters that is FDDMFAC has thebest performance in particle quality control for all IVs This
12 Complexity
Table 4 The fuzzy rules of 120588119872120588119872 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB PB PB PB PB PB PB PBNM PB PM PM PM PM PS PSNS PM PM PM PS PS Z ZZ Z Z Z NS NS NS NSPS NS NS NM NM NB NB NBPM NM NM NM NB NB NB NBPB NB NB NB NB NB NB NB
MFACDDOILCDDMFAC
60
80
100
120
APS
(m
)
5 10 150Iterations
(a) 1199100 = 60
MFACDDOILCDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILCDDMFAC
80
90
100
110
120
APS
(m
)
2 4 60Iterations
(c) 1199100 = 80
95
100
105
110
115
120
125
APS
(m
)
5 10 150Iterations
MFACDDOILCDDMFAC
(d) 1199100 = 100
Figure 5 The comparison of control effects of MFAC DDOILC and DDMFAC
Complexity 13
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
02 04 06 08 10D
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
005 01 0150D
NB NM NS Z PS PM PB
times10minus3
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
05 1 15 2 25 3 35 4 450M
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
10 20 30 40 50 600|e(k minus 1)| (m)
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
5 10 15 20 250|ep (k minus 1)| (m)
Figure 6 The membership functions of FDDMFAC when 1199100 = 60
Table 5 The fuzzy rules of 120588119863 120573119863120588119863 120573119863 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB NB NB NB NB NB NB NBNM NB NM NM NM NM NS NSNS NM NM NM NS NS Z ZZ Z Z Z PS PM PM PMPS PS PS PM PM PB PB PBPM PM PM PM PB PB PB PBPB PB PB PB PB PB PB PB
validates the effectiveness of FDDMFAC proposed in thispaper
Although FDDMFAC approach has been proven to havea better control performance than MFAC and DDOILCwe still have a question of whether it is possible that thesuperiority of FDDMFAC is entirely attributed to fuzzyadjustment of parameters and has nothing to do with thestructure of DDMFAC itself In order to clarify this problemanother set of simulations are conducted inwhich the controlperformance of FDDMFAC is compared with fuzzy MFAC(FMFAC) and fuzzy DDOILC (FDDOILC) The absolutevalue of tracking error |119890(119896 minus 1)| is regarded as the inputvariable of fuzzy rules and the step-size factors 1205881 1205882 ofMFAC andDDOILC are the output variablesThe fuzzy rulesof FMFAC and FDDOILC are shown in Table 6 and themembership functions when 1199100 = 60 are shown in Figure 8For the simulation results please see Figure 9 From thesimulation it is concluded that FDDMFAC achieves the best
Table 6 The fuzzy rules of FMFAC and FDDOILC
|119890(119896 minus 1)| NB NM NS Z PS PM PB1205881 1205882 NB NM NS Z PS PM PB
control performance which demonstrates the effectiveness ofthe proposed approach in this paper
5 Conclusions
This paper aimed at solving the problem of particle qualitycontrol in the drug development phase of SFBGP Themodel-based control approaches are incapable of handlingsuch quality control because the accurate process modelcannot be acquired Thus the data-driven and model-freecontrol approaches are considered in this work We firstly
14 Complexity
MFACDDOILCFDDMFAC
5 10 150Iterations
60
80
100
120
APS
(m
)y0 = 60
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
y0 = 70
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
MFACDDOILCFDDMFAC
95
100
105
110
115
120
125A
PS (
m)
5 10 150Iterations
y0 = 100
Figure 7 The comparison of control effects of MFAC DDOILC and FDDMFAC
analyzed the features of MFAC and DDOILC and thenconducted experimental research to compare the perfor-mances of MFAC and DDOILC in particle quality controlunder different IVs The simulation results have verifiedour theoretical analysis and concluded that MFAC andDDOILC both have advantages under different IVs To beused in an actual production environment the expectedDDCapproaches must accommodate various IVs because of thechanges in material attributes Thus the hybrid approachFDDMFAC which combines the advantages of MFAC andDDOILC by adaptively adjusting their respective weightedfactors with fuzzy logic was proposed in this work tocomplement their respective shortcomings Through a series
of simulation results we conclude that FDDMFAC achievesbetter control performance than other approaches whichvalidates the effectiveness of FDDMFAC
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thanks are due to National Natural Science Foundation ofChina under Grant nos 61773105 61374147 61703085 and61004083 and Fundamental Research Funds for the Central
Complexity 15NB NM NS Z PS PM PB
001 002 00302
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
002 00401
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
20 40 600|e(k minus 1)| (m)
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
Figure 8 The membership functions of FDDMFAC when 1199100 = 60
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
60
80
100
120
APS
(m
)
y0 = 60
FMFACFDDOILCFDDMFAC
70
80
90
100
110
120
APS
(m
)
2 4 6 8 100Iterations
y0 = 70
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
100
105
110
115
120
125
APS
(m
)
y0 = 100
Figure 9 The comparison of control effects of FMFAC FDDOILC and FDDMFAC
16 Complexity
Universities under Grant nos N150402001 and N120404014for supporting this research work
References
[1] J Bouffard M Kaster and H Dumont ldquoInfluence of processvariable and physicochemical properties on the granulationmechanism of mannitol in a fluid bed top spray granulatorrdquoDrug Development and Industrial Pharmacy vol 31 no 9 pp923ndash933 2005
[2] H Liu K Wang W Schlindwein and M Li ldquoUsing the Box-Behnken experimental design to optimise operating parametersin pulsed spray fluidised bed granulationrdquo International Journalof Pharmaceutics vol 448 no 2 pp 329ndash338 2013
[3] H Liu and M Li ldquoPopulation balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granula-tionrdquo International Journal of Pharmaceutics vol 468 no 1-2pp 223ndash233 2014
[4] T Albuquerque V H Dias N Poellinger and J F PintoldquoConstruction of a quality index for granules produced byfluidized bed technology and application of the correspondenceanalysis as a discriminant procedurerdquo European Journal ofPharmaceutics and Biopharmaceutics vol 75 no 3 pp 418ndash4242010
[5] H Ehlers A Liu H Raikkonen et al ldquoGranule size control andtargeting in pulsed spray fluid bed granulationrdquo InternationalJournal of Pharmaceutics vol 377 no 1-2 pp 9ndash15 2009
[6] C F W Sanders M J Hounslow and F J Doyle III ldquoIden-tification of models for control of wet granulationrdquo PowderTechnology vol 188 no 3 pp 255ndash263 2009
[7] T Narvanen T Lipsanen O Antikainen H Raikkonen andJ Yliruusi ldquoControlling granule size by granulation liquid feedpulsingrdquo International Journal of Pharmaceutics vol 357 no 1-2pp 132ndash138 2008
[8] A Faure P York and R C Rowe ldquoProcess control and scale-up of pharmaceutical wet granulation processes a reviewrdquoEuropean Journal of Pharmaceutics and Biopharmaceutics vol52 no 3 pp 269ndash277 2001
[9] M Pottmann B A Ogunnaike A A Adetayo and B JEnnis ldquoModel-based control of a granulation systemrdquo PowderTechnology vol 108 no 2-3 pp 192ndash201 2000
[10] E P Gatzke and F J Doyle III ldquoModel predictive control of agranulation system using soft output constraints and prioritizedcontrol objectivesrdquo Powder Technology vol 121 no 2-3 pp 149ndash158 2001
[11] F Y Wang and I T Cameron ldquoReview and future directionsin the modelling and control of continuous drum granulationrdquoPowder Technology vol 124 no 3 pp 238ndash253 2002
[12] T Glaser C F W Sanders F Y Wang et al ldquoModel predictivecontrol of continuous drum granulationrdquo Journal of ProcessControl vol 19 no 4 pp 615ndash622 2009
[13] R Ramachandran andA Chaudhury ldquoModel-based design andcontrol of a continuous drum granulation processrdquo ChemicalEngineering Research and Design vol 90 no 8 pp 1063ndash10732012
[14] Z H Loh D Z L Er L W Chan C V Liew and PW S HengldquoSpray granulation for drug formulationrdquo Expert Opinion onDrug Delivery vol 8 no 12 pp 1645ndash1661 2011
[15] S H Schaafsma N W F Kossen M T Mos L Blauw andA C Hoffmann ldquoEffects and control of humidity and particlemixing in fluid-bed granulationrdquo AIChE Journal vol 45 no 6pp 1202ndash1210 1999
[16] Z-S Hou and S-T Jin ldquoA novel data-driven control approachfor a class of discrete-time nonlinear systemsrdquo IEEE Transac-tions onControl Systems Technology vol 19 no 6 pp 1549ndash15582011
[17] Z Hou and S Jin ldquoData-driven model-free adaptive controlfor a class of MIMO nonlinear discrete-time systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 22no 12 pp 2173ndash2188 2011
[18] D Xu B Jiang and P Shi ldquoA novel model-free adaptivecontrol design for multivariable industrial processesrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6391ndash6398 2014
[19] X Wang X Li J Wang X Fang and X Zhu ldquoData-drivenmodel-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeletonrdquo Information Sciences vol 327pp 246ndash257 2016
[20] L dos Santos Coelho andA A R Coelho ldquoModel-free adaptivecontrol optimization using a chaotic particle swarm approachrdquoChaos Solitons amp Fractals vol 41 no 4 pp 2001ndash2009 2009
[21] L D S Coelho M W Pessoa R Rodrigues Sumar andA Augusto Rodrigues Coelho ldquoModel-free adaptive controldesign using evolutionary-neural compensatorrdquo Expert Systemswith Applications vol 37 no 1 pp 499ndash508 2010
[22] K K Tan T H Lee S N Huang and F M Leu ldquoAdaptive-predictive control of a class of SISOnonlinear systemsrdquoDynam-ics and Control vol 11 no 2 pp 151ndash174 2001
[23] B Zhang andW Zhang ldquoAdaptive predictive functional controlof a class of nonlinear systemsrdquo ISA Transactions vol 45 no2 pp 175ndash183 2006
[24] F L Lu J F Wang C J Fan and S B Chen ldquoAn improvedmodel-free adaptive control with G function fuzzy reasoningregulation design and its applicationsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems andControl Engineering vol 222 no 8 pp 817ndash828 2008
[25] R Chi D Wang Z Hou and S Jin ldquoData-driven optimalterminal iterative learning controlrdquo Journal of Process Controlvol 22 no 10 pp 2026ndash2037 2012
[26] R Chi Z Hou S Jin and D Wang ldquoImproved data-drivenoptimal TILC using time-varying input signalsrdquo Journal ofProcess Control vol 24 no 12 pp 78ndash85 2014
[27] X Bu Z Hou and R Chi ldquoModel free adaptive iterative learn-ing control for farm vehicle path trackingrdquo IFAC ProceedingsVolumes vol 46 no 20 pp 153ndash158 2013
[28] M C Campi A Lecchini and SM Savaresi ldquoVirtual referencefeedback tuning a direct method for the design of feedbackcontrollersrdquo Automatica vol 38 no 8 pp 1337ndash1346 2002
[29] R-C Roman M-B Radac R-E Precup and E M PetriuldquoData-driven model-free adaptive control tuned by virtualreference feedback tuningrdquoActa PolytechnicaHungarica vol 13no 1 pp 83ndash96 2016
[30] P Yan D Liu D Wang and H Ma ldquoData-driven controllerdesign for general MIMO nonlinear systems via virtual refer-ence feedback tuning and neural networksrdquo Neurocomputingvol 171 pp 815ndash825 2016
[31] T Pan and S Li ldquoA hierarchical search and updating databasestrategy for lazy learningrdquo International Journal of InnovativeComputing Information andControl vol 4 no 6 pp 1383ndash13922008
[32] H Zhang R Song Q Wei and T Zhang ldquoOptimal trackingcontrol for a class of nonlinear discrete-time systems withtime delays based on heuristic dynamic programmingrdquo IEEE
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Complexity
Because |120593(119896)| le 119887 is bounded [16] |120593(119896)minus120593(119896minus1)| le 2119887Taking the modulus of both sides of (35) we can obtain
1003817100381710038171003817 (119896)1003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2)100381710038171003817100381710038171003817100381710038171003817 + 2119887 (36)
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
= 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 + (minus2 + 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
times 1205781003817100381710038171003817 (119896 minus 1) Δu (119896 minus 1)10038171003817100381710038172120583 + Δu (119896 minus 1)2
(37)
Given that 0 lt 120578 lt 2 120583 gt 0(minus2 + 120578 Δu (119896 minus 1)2
120583 + Δu (119896 minus 1)2) lt 0 (38)
Synthesizing (37) and (38) we have
100381710038171003817100381710038171003817100381710038171003817 (119896 minus 1) (1 minus 120578 Δu (119896 minus 1)2120583 + Δu (119896 minus 1)2)
1003817100381710038171003817100381710038171003817100381710038172
le 1003817100381710038171003817 (119896 minus 1)10038171003817100381710038172 (39)
According to (36) and (39) there is obviously a positivenumber 1198891 isin (0 1) which makes
1003817100381710038171003817 (119896)1003817100381710038171003817 le 1198891 1003817100381710038171003817 (119896 minus 1)1003817100381710038171003817 + 2119887 le sdot sdot sdotle 1198891198961 1003817100381710038171003817 (0)1003817100381710038171003817 + 2119887 (1 minus 1198891198961)1 minus 1198891 (40)
Therefore (119896) is bounded(2)TheConvergence of Tracking Error 119890(119896) Tracking error 119890(119896)is derived as
119890 (119896) = 119910119889 minus 119910 (119896) = 119910119889 minus 119910 (119896 minus 1) minus 120593119879 (119896) Δu (119896)= 119890 (119896 minus 1) minus 1003817100381710038171003817120593 (119896)10038171003817100381710038172 (120588119872119890 (119896) + 120588119863119890 (119896 minus 1))
(120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)= (1 minus 120588119863 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)
minus 120588119872 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + 1003817100381710038171003817120593 (119896)10038171003817100381710038172 119890 (119896)
(41)
Transposition gives
119890 (119896) = [1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172] 119890 (119896 minus 1) (42)
So as long as appropriate parameters 120588119872 120588119863 and 120582 areselected there must be a positive number 1198892 isin (0 1) thatmakes
0 lt 10038171003817100381710038171003817100381710038171003817100381710038171 minus(120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172
1003817100381710038171003817100381710038171003817100381710038171003817 le 1198892 lt 1 (43)
By (42) and (43) it can be obtained that
119890 (119896) le 1003817100381710038171003817100381710038171003817100381710038171003817(1 minus (120588119872 + 120588119863) 1003817100381710038171003817120593 (119896)10038171003817100381710038172120582 + (1 + 120588119872) 1003817100381710038171003817120593 (119896)10038171003817100381710038172)119890 (119896 minus 1)1003817100381710038171003817100381710038171003817100381710038171003817le 1198892 119890 (119896 minus 1) le sdot sdot sdot le 1198891198962 119890 (0)
(44)
Therefore lim119896rarrinfin119890(119896) = 0 that is 119890(119896) is proved toconverge to zero
(3)The Boundedness of System Input and Output By referringto 1198862 + 1198872 ge 2119886119887 the relational expression can be obtained asfollows
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
120582 + 1003817100381710038171003817120593 (119896)100381710038171003817100381721003817100381710038171003817100381710038171003817100381710038171003817 le
1003817100381710038171003817100381710038171003817100381710038171003817120593 (119896)
2radic120582 1003817100381710038171003817120593 (119896)10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817 =
12radic120582 (45)
And we have
u (119896) = u (119896) minus u (119896 minus 1) + u (119896 minus 1) minus sdot sdot sdot + u (1)minus u (0) + u (0) = u (0) + 119896sum
119895=1
u (119895) (46)
Combining (24) (44) (45) and (46) as well as utilizingSchwartzrsquos inequality gives
u (119896) le u (0) + 119896sum119895=1
1003817100381710038171003817Δu (119895)1003817100381710038171003817
le u (0) + 12radic120582
119896sum119895=1
1003817100381710038171003817120588119863119890 (119895 minus 1) + 120588119872119890 (119895)1003817100381710038171003817
le u (0) + 1205881198632radic120582119896sum119895=1
119889119895minus12 119890 (0)
+ 1205881198722radic120582119896sum119895=1
1198891198952 119890 (0)
le u (0) + 1205881198632radic1205821 minus 119889119895minus121 minus 1198892 119890 (0)
+ 1205881198722radic1205821198892 (1 minus 1198891198952)1 minus 1198892 119890 (0)
le u (0) + 120588119863(1 minus 1198892) 2radic120582 119890 (0)
Complexity 11
Table 3 The parameter settings of DDMFAC for different IVs
IV Parameter120588119872 120588119863 120573119872 120573119863 120582 120578 1205831199100 = 60 00275 00275 05000 05000
00010 00750 100001199100 = 70 50119864 minus 4 50119864 minus 4 05000 050001199100 = 80 46119864 minus 4 46119864 minus 4 05000 050001199100 = 100 18119864 minus 4 18119864 minus 4 05000 05000
+ 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)le u (0) + 120588119863 + 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)
(47)
Because the initial tracking error 119890(0) and initial inputu(0) are given bounded the system input u(119896) is boundedaccording to (47) Besides due to the boundedness of 119910119889 and119890(119896) the system output 119910(119896) is obviously bounded43 Fuzzy DDMFAC The controller design and stability andconvergence analysis for DDMFAC have been respectivelydiscussed in detail in the above sections Although thedesigned controller has the characters of both MFAC andDDOILC weighted factors 120588119872 120588119863 120573119872 and 120573119863 are allconstant throughout control process DDMFAC is incapableof adaptively selecting the dominance of MFAC or DDOILCto duly show the expected features in each stage of controlprocessThe aforementioned features of DDMFACwe expectcannot be reflected in this case Such DDMFAC approachwith fixed weighting factors has no advantage comparedwith MFAC or DDOILC which appeals to a reasonable andadaptive adjustment strategy for 120588119872 120588119863 120573119872 and 120573119863 There-fore the adaptive adjustment strategy based on fuzzy logicfor weighting factors is considered because its advantageousperformance in parameter adjustment has been proven inour previous work [45] The formulation of fuzzy rules offersus the opportunity to adjust weighting factors in the waywe expected Such DDMFAC approach with fuzzy adaptiveadjustment for parameters is namedFDDMFAC in this paperConcretely the fuzzy rules can be formulated by referringto the expected features of DDMFAC The input variables offuzzy rules are the absolute values of tracking error 119890(119896 minus 1)and DLM prediction error 119890119901(119896minus1) of MFAC and the outputvariables are 120588119872 120588119863 and 120573119863 Because of the relationshipbetween 120573119872 and 120573119863 there is no need to use 120573119872 as an outputand it is calculated by 120573119872 = 1 minus 120573119863 The fuzzy rules can bedetailedly described as follows
(1) If |119890(119896 minus 1)| and |119890119901(119896 minus 1)| are great then thedominance of DDOILC is preferred and that ofMFAC is restrained in DDMFAC So 120588119863 and 120573119863 haverelatively large values and 120588119872 is a relatively smallnumber
(2) 120588119863 and 120573119863 will show a decreasing trend with thegradual decrease of |119890(119896 minus 1)| and |119890119901(119896 minus 1)| while120588119872 will present an increasing trend
44 Simulations and Result Analysis The proposed DDM-FAC approach is firstly applied to the particle quality controlin SFBGP and then its control effect is compared withMFACand DDOILC under different IVs In the simulation justlike MFAC 119910(119896) is replaced by a predicted value 119910(119896) thatis estimated using (19) See Table 3 for parameter settings ofDDMFAC It is not hard to find 120588119872 = 120588119863 and 120573119872 = 120573119863for a certain IV which means that MFAC and DDOILC areset to have the same dominance in DDMFAC By the wayMFAC and DDOILC can be regarded as the extreme cases ofDDMFAC described as follows
MFAC 120588119872 = 120573119872 = 1120588119863 = 120573119863 = 0
DDOILC 120588119872 = 120573119872 = 0120588119863 = 120573119863 = 1
(48)
The control effects of DDMFAC with the parametersettings in Table 3 together with control curves of MFACand DDOILC are plotted in Figure 5 The simulation resultsactually present the control effects of DDMFAC with threesets of parameter settings From simulation it is indicatedthat all three approaches have gained some certain advantagesunder different IV conditions Concretely as is shown inFigure 5(a) DDMFAC has advantage in control performancecompared with DDOILC and MFAC when 1199100 = 60 butMFAC has absolute superiority when 1199100 = 100 plottedin Figure 5(d) In addition from the simulation results inFigures 5(b) and 5(c) it can be seen that DDOILC andMFAC have achieved certain but not absolute advantagesrespectively This demonstrates that parameter setting hasimportant influence on the control effect of DDMFACTherefore the way to achieve satisfactory control effect is toadaptively adjust weighted parameters of DDMFAC insteadof keeping them constant
Then the idea of fuzzy adjustment is introduced to adjustthe weights 120588119872 120588119863 and 120573119863 The fuzzy adjustment rules ofthree weights are respectively shown in Tables 4 and 5 Therespective membership functions when 1199100 = 60 are shownin Figure 6 and membership functions for the other IVs areno longer displayed Please see Figure 7 for control effectof FDDMFAC and comparisons with MFAC and DDOILCunder different IVs Obviously DDMFAC approach withfuzzy adjustment for parameters that is FDDMFAC has thebest performance in particle quality control for all IVs This
12 Complexity
Table 4 The fuzzy rules of 120588119872120588119872 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB PB PB PB PB PB PB PBNM PB PM PM PM PM PS PSNS PM PM PM PS PS Z ZZ Z Z Z NS NS NS NSPS NS NS NM NM NB NB NBPM NM NM NM NB NB NB NBPB NB NB NB NB NB NB NB
MFACDDOILCDDMFAC
60
80
100
120
APS
(m
)
5 10 150Iterations
(a) 1199100 = 60
MFACDDOILCDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILCDDMFAC
80
90
100
110
120
APS
(m
)
2 4 60Iterations
(c) 1199100 = 80
95
100
105
110
115
120
125
APS
(m
)
5 10 150Iterations
MFACDDOILCDDMFAC
(d) 1199100 = 100
Figure 5 The comparison of control effects of MFAC DDOILC and DDMFAC
Complexity 13
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
02 04 06 08 10D
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
005 01 0150D
NB NM NS Z PS PM PB
times10minus3
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
05 1 15 2 25 3 35 4 450M
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
10 20 30 40 50 600|e(k minus 1)| (m)
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
5 10 15 20 250|ep (k minus 1)| (m)
Figure 6 The membership functions of FDDMFAC when 1199100 = 60
Table 5 The fuzzy rules of 120588119863 120573119863120588119863 120573119863 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB NB NB NB NB NB NB NBNM NB NM NM NM NM NS NSNS NM NM NM NS NS Z ZZ Z Z Z PS PM PM PMPS PS PS PM PM PB PB PBPM PM PM PM PB PB PB PBPB PB PB PB PB PB PB PB
validates the effectiveness of FDDMFAC proposed in thispaper
Although FDDMFAC approach has been proven to havea better control performance than MFAC and DDOILCwe still have a question of whether it is possible that thesuperiority of FDDMFAC is entirely attributed to fuzzyadjustment of parameters and has nothing to do with thestructure of DDMFAC itself In order to clarify this problemanother set of simulations are conducted inwhich the controlperformance of FDDMFAC is compared with fuzzy MFAC(FMFAC) and fuzzy DDOILC (FDDOILC) The absolutevalue of tracking error |119890(119896 minus 1)| is regarded as the inputvariable of fuzzy rules and the step-size factors 1205881 1205882 ofMFAC andDDOILC are the output variablesThe fuzzy rulesof FMFAC and FDDOILC are shown in Table 6 and themembership functions when 1199100 = 60 are shown in Figure 8For the simulation results please see Figure 9 From thesimulation it is concluded that FDDMFAC achieves the best
Table 6 The fuzzy rules of FMFAC and FDDOILC
|119890(119896 minus 1)| NB NM NS Z PS PM PB1205881 1205882 NB NM NS Z PS PM PB
control performance which demonstrates the effectiveness ofthe proposed approach in this paper
5 Conclusions
This paper aimed at solving the problem of particle qualitycontrol in the drug development phase of SFBGP Themodel-based control approaches are incapable of handlingsuch quality control because the accurate process modelcannot be acquired Thus the data-driven and model-freecontrol approaches are considered in this work We firstly
14 Complexity
MFACDDOILCFDDMFAC
5 10 150Iterations
60
80
100
120
APS
(m
)y0 = 60
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
y0 = 70
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
MFACDDOILCFDDMFAC
95
100
105
110
115
120
125A
PS (
m)
5 10 150Iterations
y0 = 100
Figure 7 The comparison of control effects of MFAC DDOILC and FDDMFAC
analyzed the features of MFAC and DDOILC and thenconducted experimental research to compare the perfor-mances of MFAC and DDOILC in particle quality controlunder different IVs The simulation results have verifiedour theoretical analysis and concluded that MFAC andDDOILC both have advantages under different IVs To beused in an actual production environment the expectedDDCapproaches must accommodate various IVs because of thechanges in material attributes Thus the hybrid approachFDDMFAC which combines the advantages of MFAC andDDOILC by adaptively adjusting their respective weightedfactors with fuzzy logic was proposed in this work tocomplement their respective shortcomings Through a series
of simulation results we conclude that FDDMFAC achievesbetter control performance than other approaches whichvalidates the effectiveness of FDDMFAC
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thanks are due to National Natural Science Foundation ofChina under Grant nos 61773105 61374147 61703085 and61004083 and Fundamental Research Funds for the Central
Complexity 15NB NM NS Z PS PM PB
001 002 00302
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
002 00401
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
20 40 600|e(k minus 1)| (m)
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
Figure 8 The membership functions of FDDMFAC when 1199100 = 60
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
60
80
100
120
APS
(m
)
y0 = 60
FMFACFDDOILCFDDMFAC
70
80
90
100
110
120
APS
(m
)
2 4 6 8 100Iterations
y0 = 70
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
100
105
110
115
120
125
APS
(m
)
y0 = 100
Figure 9 The comparison of control effects of FMFAC FDDOILC and FDDMFAC
16 Complexity
Universities under Grant nos N150402001 and N120404014for supporting this research work
References
[1] J Bouffard M Kaster and H Dumont ldquoInfluence of processvariable and physicochemical properties on the granulationmechanism of mannitol in a fluid bed top spray granulatorrdquoDrug Development and Industrial Pharmacy vol 31 no 9 pp923ndash933 2005
[2] H Liu K Wang W Schlindwein and M Li ldquoUsing the Box-Behnken experimental design to optimise operating parametersin pulsed spray fluidised bed granulationrdquo International Journalof Pharmaceutics vol 448 no 2 pp 329ndash338 2013
[3] H Liu and M Li ldquoPopulation balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granula-tionrdquo International Journal of Pharmaceutics vol 468 no 1-2pp 223ndash233 2014
[4] T Albuquerque V H Dias N Poellinger and J F PintoldquoConstruction of a quality index for granules produced byfluidized bed technology and application of the correspondenceanalysis as a discriminant procedurerdquo European Journal ofPharmaceutics and Biopharmaceutics vol 75 no 3 pp 418ndash4242010
[5] H Ehlers A Liu H Raikkonen et al ldquoGranule size control andtargeting in pulsed spray fluid bed granulationrdquo InternationalJournal of Pharmaceutics vol 377 no 1-2 pp 9ndash15 2009
[6] C F W Sanders M J Hounslow and F J Doyle III ldquoIden-tification of models for control of wet granulationrdquo PowderTechnology vol 188 no 3 pp 255ndash263 2009
[7] T Narvanen T Lipsanen O Antikainen H Raikkonen andJ Yliruusi ldquoControlling granule size by granulation liquid feedpulsingrdquo International Journal of Pharmaceutics vol 357 no 1-2pp 132ndash138 2008
[8] A Faure P York and R C Rowe ldquoProcess control and scale-up of pharmaceutical wet granulation processes a reviewrdquoEuropean Journal of Pharmaceutics and Biopharmaceutics vol52 no 3 pp 269ndash277 2001
[9] M Pottmann B A Ogunnaike A A Adetayo and B JEnnis ldquoModel-based control of a granulation systemrdquo PowderTechnology vol 108 no 2-3 pp 192ndash201 2000
[10] E P Gatzke and F J Doyle III ldquoModel predictive control of agranulation system using soft output constraints and prioritizedcontrol objectivesrdquo Powder Technology vol 121 no 2-3 pp 149ndash158 2001
[11] F Y Wang and I T Cameron ldquoReview and future directionsin the modelling and control of continuous drum granulationrdquoPowder Technology vol 124 no 3 pp 238ndash253 2002
[12] T Glaser C F W Sanders F Y Wang et al ldquoModel predictivecontrol of continuous drum granulationrdquo Journal of ProcessControl vol 19 no 4 pp 615ndash622 2009
[13] R Ramachandran andA Chaudhury ldquoModel-based design andcontrol of a continuous drum granulation processrdquo ChemicalEngineering Research and Design vol 90 no 8 pp 1063ndash10732012
[14] Z H Loh D Z L Er L W Chan C V Liew and PW S HengldquoSpray granulation for drug formulationrdquo Expert Opinion onDrug Delivery vol 8 no 12 pp 1645ndash1661 2011
[15] S H Schaafsma N W F Kossen M T Mos L Blauw andA C Hoffmann ldquoEffects and control of humidity and particlemixing in fluid-bed granulationrdquo AIChE Journal vol 45 no 6pp 1202ndash1210 1999
[16] Z-S Hou and S-T Jin ldquoA novel data-driven control approachfor a class of discrete-time nonlinear systemsrdquo IEEE Transac-tions onControl Systems Technology vol 19 no 6 pp 1549ndash15582011
[17] Z Hou and S Jin ldquoData-driven model-free adaptive controlfor a class of MIMO nonlinear discrete-time systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 22no 12 pp 2173ndash2188 2011
[18] D Xu B Jiang and P Shi ldquoA novel model-free adaptivecontrol design for multivariable industrial processesrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6391ndash6398 2014
[19] X Wang X Li J Wang X Fang and X Zhu ldquoData-drivenmodel-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeletonrdquo Information Sciences vol 327pp 246ndash257 2016
[20] L dos Santos Coelho andA A R Coelho ldquoModel-free adaptivecontrol optimization using a chaotic particle swarm approachrdquoChaos Solitons amp Fractals vol 41 no 4 pp 2001ndash2009 2009
[21] L D S Coelho M W Pessoa R Rodrigues Sumar andA Augusto Rodrigues Coelho ldquoModel-free adaptive controldesign using evolutionary-neural compensatorrdquo Expert Systemswith Applications vol 37 no 1 pp 499ndash508 2010
[22] K K Tan T H Lee S N Huang and F M Leu ldquoAdaptive-predictive control of a class of SISOnonlinear systemsrdquoDynam-ics and Control vol 11 no 2 pp 151ndash174 2001
[23] B Zhang andW Zhang ldquoAdaptive predictive functional controlof a class of nonlinear systemsrdquo ISA Transactions vol 45 no2 pp 175ndash183 2006
[24] F L Lu J F Wang C J Fan and S B Chen ldquoAn improvedmodel-free adaptive control with G function fuzzy reasoningregulation design and its applicationsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems andControl Engineering vol 222 no 8 pp 817ndash828 2008
[25] R Chi D Wang Z Hou and S Jin ldquoData-driven optimalterminal iterative learning controlrdquo Journal of Process Controlvol 22 no 10 pp 2026ndash2037 2012
[26] R Chi Z Hou S Jin and D Wang ldquoImproved data-drivenoptimal TILC using time-varying input signalsrdquo Journal ofProcess Control vol 24 no 12 pp 78ndash85 2014
[27] X Bu Z Hou and R Chi ldquoModel free adaptive iterative learn-ing control for farm vehicle path trackingrdquo IFAC ProceedingsVolumes vol 46 no 20 pp 153ndash158 2013
[28] M C Campi A Lecchini and SM Savaresi ldquoVirtual referencefeedback tuning a direct method for the design of feedbackcontrollersrdquo Automatica vol 38 no 8 pp 1337ndash1346 2002
[29] R-C Roman M-B Radac R-E Precup and E M PetriuldquoData-driven model-free adaptive control tuned by virtualreference feedback tuningrdquoActa PolytechnicaHungarica vol 13no 1 pp 83ndash96 2016
[30] P Yan D Liu D Wang and H Ma ldquoData-driven controllerdesign for general MIMO nonlinear systems via virtual refer-ence feedback tuning and neural networksrdquo Neurocomputingvol 171 pp 815ndash825 2016
[31] T Pan and S Li ldquoA hierarchical search and updating databasestrategy for lazy learningrdquo International Journal of InnovativeComputing Information andControl vol 4 no 6 pp 1383ndash13922008
[32] H Zhang R Song Q Wei and T Zhang ldquoOptimal trackingcontrol for a class of nonlinear discrete-time systems withtime delays based on heuristic dynamic programmingrdquo IEEE
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Complexity 11
Table 3 The parameter settings of DDMFAC for different IVs
IV Parameter120588119872 120588119863 120573119872 120573119863 120582 120578 1205831199100 = 60 00275 00275 05000 05000
00010 00750 100001199100 = 70 50119864 minus 4 50119864 minus 4 05000 050001199100 = 80 46119864 minus 4 46119864 minus 4 05000 050001199100 = 100 18119864 minus 4 18119864 minus 4 05000 05000
+ 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)le u (0) + 120588119863 + 1198892120588119872(1 minus 1198892) 2radic120582 119890 (0)
(47)
Because the initial tracking error 119890(0) and initial inputu(0) are given bounded the system input u(119896) is boundedaccording to (47) Besides due to the boundedness of 119910119889 and119890(119896) the system output 119910(119896) is obviously bounded43 Fuzzy DDMFAC The controller design and stability andconvergence analysis for DDMFAC have been respectivelydiscussed in detail in the above sections Although thedesigned controller has the characters of both MFAC andDDOILC weighted factors 120588119872 120588119863 120573119872 and 120573119863 are allconstant throughout control process DDMFAC is incapableof adaptively selecting the dominance of MFAC or DDOILCto duly show the expected features in each stage of controlprocessThe aforementioned features of DDMFACwe expectcannot be reflected in this case Such DDMFAC approachwith fixed weighting factors has no advantage comparedwith MFAC or DDOILC which appeals to a reasonable andadaptive adjustment strategy for 120588119872 120588119863 120573119872 and 120573119863 There-fore the adaptive adjustment strategy based on fuzzy logicfor weighting factors is considered because its advantageousperformance in parameter adjustment has been proven inour previous work [45] The formulation of fuzzy rules offersus the opportunity to adjust weighting factors in the waywe expected Such DDMFAC approach with fuzzy adaptiveadjustment for parameters is namedFDDMFAC in this paperConcretely the fuzzy rules can be formulated by referringto the expected features of DDMFAC The input variables offuzzy rules are the absolute values of tracking error 119890(119896 minus 1)and DLM prediction error 119890119901(119896minus1) of MFAC and the outputvariables are 120588119872 120588119863 and 120573119863 Because of the relationshipbetween 120573119872 and 120573119863 there is no need to use 120573119872 as an outputand it is calculated by 120573119872 = 1 minus 120573119863 The fuzzy rules can bedetailedly described as follows
(1) If |119890(119896 minus 1)| and |119890119901(119896 minus 1)| are great then thedominance of DDOILC is preferred and that ofMFAC is restrained in DDMFAC So 120588119863 and 120573119863 haverelatively large values and 120588119872 is a relatively smallnumber
(2) 120588119863 and 120573119863 will show a decreasing trend with thegradual decrease of |119890(119896 minus 1)| and |119890119901(119896 minus 1)| while120588119872 will present an increasing trend
44 Simulations and Result Analysis The proposed DDM-FAC approach is firstly applied to the particle quality controlin SFBGP and then its control effect is compared withMFACand DDOILC under different IVs In the simulation justlike MFAC 119910(119896) is replaced by a predicted value 119910(119896) thatis estimated using (19) See Table 3 for parameter settings ofDDMFAC It is not hard to find 120588119872 = 120588119863 and 120573119872 = 120573119863for a certain IV which means that MFAC and DDOILC areset to have the same dominance in DDMFAC By the wayMFAC and DDOILC can be regarded as the extreme cases ofDDMFAC described as follows
MFAC 120588119872 = 120573119872 = 1120588119863 = 120573119863 = 0
DDOILC 120588119872 = 120573119872 = 0120588119863 = 120573119863 = 1
(48)
The control effects of DDMFAC with the parametersettings in Table 3 together with control curves of MFACand DDOILC are plotted in Figure 5 The simulation resultsactually present the control effects of DDMFAC with threesets of parameter settings From simulation it is indicatedthat all three approaches have gained some certain advantagesunder different IV conditions Concretely as is shown inFigure 5(a) DDMFAC has advantage in control performancecompared with DDOILC and MFAC when 1199100 = 60 butMFAC has absolute superiority when 1199100 = 100 plottedin Figure 5(d) In addition from the simulation results inFigures 5(b) and 5(c) it can be seen that DDOILC andMFAC have achieved certain but not absolute advantagesrespectively This demonstrates that parameter setting hasimportant influence on the control effect of DDMFACTherefore the way to achieve satisfactory control effect is toadaptively adjust weighted parameters of DDMFAC insteadof keeping them constant
Then the idea of fuzzy adjustment is introduced to adjustthe weights 120588119872 120588119863 and 120573119863 The fuzzy adjustment rules ofthree weights are respectively shown in Tables 4 and 5 Therespective membership functions when 1199100 = 60 are shownin Figure 6 and membership functions for the other IVs areno longer displayed Please see Figure 7 for control effectof FDDMFAC and comparisons with MFAC and DDOILCunder different IVs Obviously DDMFAC approach withfuzzy adjustment for parameters that is FDDMFAC has thebest performance in particle quality control for all IVs This
12 Complexity
Table 4 The fuzzy rules of 120588119872120588119872 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB PB PB PB PB PB PB PBNM PB PM PM PM PM PS PSNS PM PM PM PS PS Z ZZ Z Z Z NS NS NS NSPS NS NS NM NM NB NB NBPM NM NM NM NB NB NB NBPB NB NB NB NB NB NB NB
MFACDDOILCDDMFAC
60
80
100
120
APS
(m
)
5 10 150Iterations
(a) 1199100 = 60
MFACDDOILCDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILCDDMFAC
80
90
100
110
120
APS
(m
)
2 4 60Iterations
(c) 1199100 = 80
95
100
105
110
115
120
125
APS
(m
)
5 10 150Iterations
MFACDDOILCDDMFAC
(d) 1199100 = 100
Figure 5 The comparison of control effects of MFAC DDOILC and DDMFAC
Complexity 13
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
02 04 06 08 10D
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
005 01 0150D
NB NM NS Z PS PM PB
times10minus3
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
05 1 15 2 25 3 35 4 450M
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
10 20 30 40 50 600|e(k minus 1)| (m)
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
5 10 15 20 250|ep (k minus 1)| (m)
Figure 6 The membership functions of FDDMFAC when 1199100 = 60
Table 5 The fuzzy rules of 120588119863 120573119863120588119863 120573119863 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB NB NB NB NB NB NB NBNM NB NM NM NM NM NS NSNS NM NM NM NS NS Z ZZ Z Z Z PS PM PM PMPS PS PS PM PM PB PB PBPM PM PM PM PB PB PB PBPB PB PB PB PB PB PB PB
validates the effectiveness of FDDMFAC proposed in thispaper
Although FDDMFAC approach has been proven to havea better control performance than MFAC and DDOILCwe still have a question of whether it is possible that thesuperiority of FDDMFAC is entirely attributed to fuzzyadjustment of parameters and has nothing to do with thestructure of DDMFAC itself In order to clarify this problemanother set of simulations are conducted inwhich the controlperformance of FDDMFAC is compared with fuzzy MFAC(FMFAC) and fuzzy DDOILC (FDDOILC) The absolutevalue of tracking error |119890(119896 minus 1)| is regarded as the inputvariable of fuzzy rules and the step-size factors 1205881 1205882 ofMFAC andDDOILC are the output variablesThe fuzzy rulesof FMFAC and FDDOILC are shown in Table 6 and themembership functions when 1199100 = 60 are shown in Figure 8For the simulation results please see Figure 9 From thesimulation it is concluded that FDDMFAC achieves the best
Table 6 The fuzzy rules of FMFAC and FDDOILC
|119890(119896 minus 1)| NB NM NS Z PS PM PB1205881 1205882 NB NM NS Z PS PM PB
control performance which demonstrates the effectiveness ofthe proposed approach in this paper
5 Conclusions
This paper aimed at solving the problem of particle qualitycontrol in the drug development phase of SFBGP Themodel-based control approaches are incapable of handlingsuch quality control because the accurate process modelcannot be acquired Thus the data-driven and model-freecontrol approaches are considered in this work We firstly
14 Complexity
MFACDDOILCFDDMFAC
5 10 150Iterations
60
80
100
120
APS
(m
)y0 = 60
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
y0 = 70
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
MFACDDOILCFDDMFAC
95
100
105
110
115
120
125A
PS (
m)
5 10 150Iterations
y0 = 100
Figure 7 The comparison of control effects of MFAC DDOILC and FDDMFAC
analyzed the features of MFAC and DDOILC and thenconducted experimental research to compare the perfor-mances of MFAC and DDOILC in particle quality controlunder different IVs The simulation results have verifiedour theoretical analysis and concluded that MFAC andDDOILC both have advantages under different IVs To beused in an actual production environment the expectedDDCapproaches must accommodate various IVs because of thechanges in material attributes Thus the hybrid approachFDDMFAC which combines the advantages of MFAC andDDOILC by adaptively adjusting their respective weightedfactors with fuzzy logic was proposed in this work tocomplement their respective shortcomings Through a series
of simulation results we conclude that FDDMFAC achievesbetter control performance than other approaches whichvalidates the effectiveness of FDDMFAC
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thanks are due to National Natural Science Foundation ofChina under Grant nos 61773105 61374147 61703085 and61004083 and Fundamental Research Funds for the Central
Complexity 15NB NM NS Z PS PM PB
001 002 00302
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
002 00401
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
20 40 600|e(k minus 1)| (m)
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
Figure 8 The membership functions of FDDMFAC when 1199100 = 60
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
60
80
100
120
APS
(m
)
y0 = 60
FMFACFDDOILCFDDMFAC
70
80
90
100
110
120
APS
(m
)
2 4 6 8 100Iterations
y0 = 70
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
100
105
110
115
120
125
APS
(m
)
y0 = 100
Figure 9 The comparison of control effects of FMFAC FDDOILC and FDDMFAC
16 Complexity
Universities under Grant nos N150402001 and N120404014for supporting this research work
References
[1] J Bouffard M Kaster and H Dumont ldquoInfluence of processvariable and physicochemical properties on the granulationmechanism of mannitol in a fluid bed top spray granulatorrdquoDrug Development and Industrial Pharmacy vol 31 no 9 pp923ndash933 2005
[2] H Liu K Wang W Schlindwein and M Li ldquoUsing the Box-Behnken experimental design to optimise operating parametersin pulsed spray fluidised bed granulationrdquo International Journalof Pharmaceutics vol 448 no 2 pp 329ndash338 2013
[3] H Liu and M Li ldquoPopulation balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granula-tionrdquo International Journal of Pharmaceutics vol 468 no 1-2pp 223ndash233 2014
[4] T Albuquerque V H Dias N Poellinger and J F PintoldquoConstruction of a quality index for granules produced byfluidized bed technology and application of the correspondenceanalysis as a discriminant procedurerdquo European Journal ofPharmaceutics and Biopharmaceutics vol 75 no 3 pp 418ndash4242010
[5] H Ehlers A Liu H Raikkonen et al ldquoGranule size control andtargeting in pulsed spray fluid bed granulationrdquo InternationalJournal of Pharmaceutics vol 377 no 1-2 pp 9ndash15 2009
[6] C F W Sanders M J Hounslow and F J Doyle III ldquoIden-tification of models for control of wet granulationrdquo PowderTechnology vol 188 no 3 pp 255ndash263 2009
[7] T Narvanen T Lipsanen O Antikainen H Raikkonen andJ Yliruusi ldquoControlling granule size by granulation liquid feedpulsingrdquo International Journal of Pharmaceutics vol 357 no 1-2pp 132ndash138 2008
[8] A Faure P York and R C Rowe ldquoProcess control and scale-up of pharmaceutical wet granulation processes a reviewrdquoEuropean Journal of Pharmaceutics and Biopharmaceutics vol52 no 3 pp 269ndash277 2001
[9] M Pottmann B A Ogunnaike A A Adetayo and B JEnnis ldquoModel-based control of a granulation systemrdquo PowderTechnology vol 108 no 2-3 pp 192ndash201 2000
[10] E P Gatzke and F J Doyle III ldquoModel predictive control of agranulation system using soft output constraints and prioritizedcontrol objectivesrdquo Powder Technology vol 121 no 2-3 pp 149ndash158 2001
[11] F Y Wang and I T Cameron ldquoReview and future directionsin the modelling and control of continuous drum granulationrdquoPowder Technology vol 124 no 3 pp 238ndash253 2002
[12] T Glaser C F W Sanders F Y Wang et al ldquoModel predictivecontrol of continuous drum granulationrdquo Journal of ProcessControl vol 19 no 4 pp 615ndash622 2009
[13] R Ramachandran andA Chaudhury ldquoModel-based design andcontrol of a continuous drum granulation processrdquo ChemicalEngineering Research and Design vol 90 no 8 pp 1063ndash10732012
[14] Z H Loh D Z L Er L W Chan C V Liew and PW S HengldquoSpray granulation for drug formulationrdquo Expert Opinion onDrug Delivery vol 8 no 12 pp 1645ndash1661 2011
[15] S H Schaafsma N W F Kossen M T Mos L Blauw andA C Hoffmann ldquoEffects and control of humidity and particlemixing in fluid-bed granulationrdquo AIChE Journal vol 45 no 6pp 1202ndash1210 1999
[16] Z-S Hou and S-T Jin ldquoA novel data-driven control approachfor a class of discrete-time nonlinear systemsrdquo IEEE Transac-tions onControl Systems Technology vol 19 no 6 pp 1549ndash15582011
[17] Z Hou and S Jin ldquoData-driven model-free adaptive controlfor a class of MIMO nonlinear discrete-time systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 22no 12 pp 2173ndash2188 2011
[18] D Xu B Jiang and P Shi ldquoA novel model-free adaptivecontrol design for multivariable industrial processesrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6391ndash6398 2014
[19] X Wang X Li J Wang X Fang and X Zhu ldquoData-drivenmodel-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeletonrdquo Information Sciences vol 327pp 246ndash257 2016
[20] L dos Santos Coelho andA A R Coelho ldquoModel-free adaptivecontrol optimization using a chaotic particle swarm approachrdquoChaos Solitons amp Fractals vol 41 no 4 pp 2001ndash2009 2009
[21] L D S Coelho M W Pessoa R Rodrigues Sumar andA Augusto Rodrigues Coelho ldquoModel-free adaptive controldesign using evolutionary-neural compensatorrdquo Expert Systemswith Applications vol 37 no 1 pp 499ndash508 2010
[22] K K Tan T H Lee S N Huang and F M Leu ldquoAdaptive-predictive control of a class of SISOnonlinear systemsrdquoDynam-ics and Control vol 11 no 2 pp 151ndash174 2001
[23] B Zhang andW Zhang ldquoAdaptive predictive functional controlof a class of nonlinear systemsrdquo ISA Transactions vol 45 no2 pp 175ndash183 2006
[24] F L Lu J F Wang C J Fan and S B Chen ldquoAn improvedmodel-free adaptive control with G function fuzzy reasoningregulation design and its applicationsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems andControl Engineering vol 222 no 8 pp 817ndash828 2008
[25] R Chi D Wang Z Hou and S Jin ldquoData-driven optimalterminal iterative learning controlrdquo Journal of Process Controlvol 22 no 10 pp 2026ndash2037 2012
[26] R Chi Z Hou S Jin and D Wang ldquoImproved data-drivenoptimal TILC using time-varying input signalsrdquo Journal ofProcess Control vol 24 no 12 pp 78ndash85 2014
[27] X Bu Z Hou and R Chi ldquoModel free adaptive iterative learn-ing control for farm vehicle path trackingrdquo IFAC ProceedingsVolumes vol 46 no 20 pp 153ndash158 2013
[28] M C Campi A Lecchini and SM Savaresi ldquoVirtual referencefeedback tuning a direct method for the design of feedbackcontrollersrdquo Automatica vol 38 no 8 pp 1337ndash1346 2002
[29] R-C Roman M-B Radac R-E Precup and E M PetriuldquoData-driven model-free adaptive control tuned by virtualreference feedback tuningrdquoActa PolytechnicaHungarica vol 13no 1 pp 83ndash96 2016
[30] P Yan D Liu D Wang and H Ma ldquoData-driven controllerdesign for general MIMO nonlinear systems via virtual refer-ence feedback tuning and neural networksrdquo Neurocomputingvol 171 pp 815ndash825 2016
[31] T Pan and S Li ldquoA hierarchical search and updating databasestrategy for lazy learningrdquo International Journal of InnovativeComputing Information andControl vol 4 no 6 pp 1383ndash13922008
[32] H Zhang R Song Q Wei and T Zhang ldquoOptimal trackingcontrol for a class of nonlinear discrete-time systems withtime delays based on heuristic dynamic programmingrdquo IEEE
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Complexity
Table 4 The fuzzy rules of 120588119872120588119872 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB PB PB PB PB PB PB PBNM PB PM PM PM PM PS PSNS PM PM PM PS PS Z ZZ Z Z Z NS NS NS NSPS NS NS NM NM NB NB NBPM NM NM NM NB NB NB NBPB NB NB NB NB NB NB NB
MFACDDOILCDDMFAC
60
80
100
120
APS
(m
)
5 10 150Iterations
(a) 1199100 = 60
MFACDDOILCDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
(b) 1199100 = 70
MFACDDOILCDDMFAC
80
90
100
110
120
APS
(m
)
2 4 60Iterations
(c) 1199100 = 80
95
100
105
110
115
120
125
APS
(m
)
5 10 150Iterations
MFACDDOILCDDMFAC
(d) 1199100 = 100
Figure 5 The comparison of control effects of MFAC DDOILC and DDMFAC
Complexity 13
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
02 04 06 08 10D
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
005 01 0150D
NB NM NS Z PS PM PB
times10minus3
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
05 1 15 2 25 3 35 4 450M
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
10 20 30 40 50 600|e(k minus 1)| (m)
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
5 10 15 20 250|ep (k minus 1)| (m)
Figure 6 The membership functions of FDDMFAC when 1199100 = 60
Table 5 The fuzzy rules of 120588119863 120573119863120588119863 120573119863 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB NB NB NB NB NB NB NBNM NB NM NM NM NM NS NSNS NM NM NM NS NS Z ZZ Z Z Z PS PM PM PMPS PS PS PM PM PB PB PBPM PM PM PM PB PB PB PBPB PB PB PB PB PB PB PB
validates the effectiveness of FDDMFAC proposed in thispaper
Although FDDMFAC approach has been proven to havea better control performance than MFAC and DDOILCwe still have a question of whether it is possible that thesuperiority of FDDMFAC is entirely attributed to fuzzyadjustment of parameters and has nothing to do with thestructure of DDMFAC itself In order to clarify this problemanother set of simulations are conducted inwhich the controlperformance of FDDMFAC is compared with fuzzy MFAC(FMFAC) and fuzzy DDOILC (FDDOILC) The absolutevalue of tracking error |119890(119896 minus 1)| is regarded as the inputvariable of fuzzy rules and the step-size factors 1205881 1205882 ofMFAC andDDOILC are the output variablesThe fuzzy rulesof FMFAC and FDDOILC are shown in Table 6 and themembership functions when 1199100 = 60 are shown in Figure 8For the simulation results please see Figure 9 From thesimulation it is concluded that FDDMFAC achieves the best
Table 6 The fuzzy rules of FMFAC and FDDOILC
|119890(119896 minus 1)| NB NM NS Z PS PM PB1205881 1205882 NB NM NS Z PS PM PB
control performance which demonstrates the effectiveness ofthe proposed approach in this paper
5 Conclusions
This paper aimed at solving the problem of particle qualitycontrol in the drug development phase of SFBGP Themodel-based control approaches are incapable of handlingsuch quality control because the accurate process modelcannot be acquired Thus the data-driven and model-freecontrol approaches are considered in this work We firstly
14 Complexity
MFACDDOILCFDDMFAC
5 10 150Iterations
60
80
100
120
APS
(m
)y0 = 60
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
y0 = 70
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
MFACDDOILCFDDMFAC
95
100
105
110
115
120
125A
PS (
m)
5 10 150Iterations
y0 = 100
Figure 7 The comparison of control effects of MFAC DDOILC and FDDMFAC
analyzed the features of MFAC and DDOILC and thenconducted experimental research to compare the perfor-mances of MFAC and DDOILC in particle quality controlunder different IVs The simulation results have verifiedour theoretical analysis and concluded that MFAC andDDOILC both have advantages under different IVs To beused in an actual production environment the expectedDDCapproaches must accommodate various IVs because of thechanges in material attributes Thus the hybrid approachFDDMFAC which combines the advantages of MFAC andDDOILC by adaptively adjusting their respective weightedfactors with fuzzy logic was proposed in this work tocomplement their respective shortcomings Through a series
of simulation results we conclude that FDDMFAC achievesbetter control performance than other approaches whichvalidates the effectiveness of FDDMFAC
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thanks are due to National Natural Science Foundation ofChina under Grant nos 61773105 61374147 61703085 and61004083 and Fundamental Research Funds for the Central
Complexity 15NB NM NS Z PS PM PB
001 002 00302
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
002 00401
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
20 40 600|e(k minus 1)| (m)
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
Figure 8 The membership functions of FDDMFAC when 1199100 = 60
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
60
80
100
120
APS
(m
)
y0 = 60
FMFACFDDOILCFDDMFAC
70
80
90
100
110
120
APS
(m
)
2 4 6 8 100Iterations
y0 = 70
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
100
105
110
115
120
125
APS
(m
)
y0 = 100
Figure 9 The comparison of control effects of FMFAC FDDOILC and FDDMFAC
16 Complexity
Universities under Grant nos N150402001 and N120404014for supporting this research work
References
[1] J Bouffard M Kaster and H Dumont ldquoInfluence of processvariable and physicochemical properties on the granulationmechanism of mannitol in a fluid bed top spray granulatorrdquoDrug Development and Industrial Pharmacy vol 31 no 9 pp923ndash933 2005
[2] H Liu K Wang W Schlindwein and M Li ldquoUsing the Box-Behnken experimental design to optimise operating parametersin pulsed spray fluidised bed granulationrdquo International Journalof Pharmaceutics vol 448 no 2 pp 329ndash338 2013
[3] H Liu and M Li ldquoPopulation balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granula-tionrdquo International Journal of Pharmaceutics vol 468 no 1-2pp 223ndash233 2014
[4] T Albuquerque V H Dias N Poellinger and J F PintoldquoConstruction of a quality index for granules produced byfluidized bed technology and application of the correspondenceanalysis as a discriminant procedurerdquo European Journal ofPharmaceutics and Biopharmaceutics vol 75 no 3 pp 418ndash4242010
[5] H Ehlers A Liu H Raikkonen et al ldquoGranule size control andtargeting in pulsed spray fluid bed granulationrdquo InternationalJournal of Pharmaceutics vol 377 no 1-2 pp 9ndash15 2009
[6] C F W Sanders M J Hounslow and F J Doyle III ldquoIden-tification of models for control of wet granulationrdquo PowderTechnology vol 188 no 3 pp 255ndash263 2009
[7] T Narvanen T Lipsanen O Antikainen H Raikkonen andJ Yliruusi ldquoControlling granule size by granulation liquid feedpulsingrdquo International Journal of Pharmaceutics vol 357 no 1-2pp 132ndash138 2008
[8] A Faure P York and R C Rowe ldquoProcess control and scale-up of pharmaceutical wet granulation processes a reviewrdquoEuropean Journal of Pharmaceutics and Biopharmaceutics vol52 no 3 pp 269ndash277 2001
[9] M Pottmann B A Ogunnaike A A Adetayo and B JEnnis ldquoModel-based control of a granulation systemrdquo PowderTechnology vol 108 no 2-3 pp 192ndash201 2000
[10] E P Gatzke and F J Doyle III ldquoModel predictive control of agranulation system using soft output constraints and prioritizedcontrol objectivesrdquo Powder Technology vol 121 no 2-3 pp 149ndash158 2001
[11] F Y Wang and I T Cameron ldquoReview and future directionsin the modelling and control of continuous drum granulationrdquoPowder Technology vol 124 no 3 pp 238ndash253 2002
[12] T Glaser C F W Sanders F Y Wang et al ldquoModel predictivecontrol of continuous drum granulationrdquo Journal of ProcessControl vol 19 no 4 pp 615ndash622 2009
[13] R Ramachandran andA Chaudhury ldquoModel-based design andcontrol of a continuous drum granulation processrdquo ChemicalEngineering Research and Design vol 90 no 8 pp 1063ndash10732012
[14] Z H Loh D Z L Er L W Chan C V Liew and PW S HengldquoSpray granulation for drug formulationrdquo Expert Opinion onDrug Delivery vol 8 no 12 pp 1645ndash1661 2011
[15] S H Schaafsma N W F Kossen M T Mos L Blauw andA C Hoffmann ldquoEffects and control of humidity and particlemixing in fluid-bed granulationrdquo AIChE Journal vol 45 no 6pp 1202ndash1210 1999
[16] Z-S Hou and S-T Jin ldquoA novel data-driven control approachfor a class of discrete-time nonlinear systemsrdquo IEEE Transac-tions onControl Systems Technology vol 19 no 6 pp 1549ndash15582011
[17] Z Hou and S Jin ldquoData-driven model-free adaptive controlfor a class of MIMO nonlinear discrete-time systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 22no 12 pp 2173ndash2188 2011
[18] D Xu B Jiang and P Shi ldquoA novel model-free adaptivecontrol design for multivariable industrial processesrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6391ndash6398 2014
[19] X Wang X Li J Wang X Fang and X Zhu ldquoData-drivenmodel-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeletonrdquo Information Sciences vol 327pp 246ndash257 2016
[20] L dos Santos Coelho andA A R Coelho ldquoModel-free adaptivecontrol optimization using a chaotic particle swarm approachrdquoChaos Solitons amp Fractals vol 41 no 4 pp 2001ndash2009 2009
[21] L D S Coelho M W Pessoa R Rodrigues Sumar andA Augusto Rodrigues Coelho ldquoModel-free adaptive controldesign using evolutionary-neural compensatorrdquo Expert Systemswith Applications vol 37 no 1 pp 499ndash508 2010
[22] K K Tan T H Lee S N Huang and F M Leu ldquoAdaptive-predictive control of a class of SISOnonlinear systemsrdquoDynam-ics and Control vol 11 no 2 pp 151ndash174 2001
[23] B Zhang andW Zhang ldquoAdaptive predictive functional controlof a class of nonlinear systemsrdquo ISA Transactions vol 45 no2 pp 175ndash183 2006
[24] F L Lu J F Wang C J Fan and S B Chen ldquoAn improvedmodel-free adaptive control with G function fuzzy reasoningregulation design and its applicationsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems andControl Engineering vol 222 no 8 pp 817ndash828 2008
[25] R Chi D Wang Z Hou and S Jin ldquoData-driven optimalterminal iterative learning controlrdquo Journal of Process Controlvol 22 no 10 pp 2026ndash2037 2012
[26] R Chi Z Hou S Jin and D Wang ldquoImproved data-drivenoptimal TILC using time-varying input signalsrdquo Journal ofProcess Control vol 24 no 12 pp 78ndash85 2014
[27] X Bu Z Hou and R Chi ldquoModel free adaptive iterative learn-ing control for farm vehicle path trackingrdquo IFAC ProceedingsVolumes vol 46 no 20 pp 153ndash158 2013
[28] M C Campi A Lecchini and SM Savaresi ldquoVirtual referencefeedback tuning a direct method for the design of feedbackcontrollersrdquo Automatica vol 38 no 8 pp 1337ndash1346 2002
[29] R-C Roman M-B Radac R-E Precup and E M PetriuldquoData-driven model-free adaptive control tuned by virtualreference feedback tuningrdquoActa PolytechnicaHungarica vol 13no 1 pp 83ndash96 2016
[30] P Yan D Liu D Wang and H Ma ldquoData-driven controllerdesign for general MIMO nonlinear systems via virtual refer-ence feedback tuning and neural networksrdquo Neurocomputingvol 171 pp 815ndash825 2016
[31] T Pan and S Li ldquoA hierarchical search and updating databasestrategy for lazy learningrdquo International Journal of InnovativeComputing Information andControl vol 4 no 6 pp 1383ndash13922008
[32] H Zhang R Song Q Wei and T Zhang ldquoOptimal trackingcontrol for a class of nonlinear discrete-time systems withtime delays based on heuristic dynamic programmingrdquo IEEE
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Complexity 13
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
02 04 06 08 10D
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
005 01 0150D
NB NM NS Z PS PM PB
times10minus3
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
05 1 15 2 25 3 35 4 450M
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
10 20 30 40 50 600|e(k minus 1)| (m)
NB NM NS Z PS PM PB
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
5 10 15 20 250|ep (k minus 1)| (m)
Figure 6 The membership functions of FDDMFAC when 1199100 = 60
Table 5 The fuzzy rules of 120588119863 120573119863120588119863 120573119863 10038161003816100381610038161003816119890119901(119896 minus 1)10038161003816100381610038161003816|119890(119896 minus 1)| NB NM NS Z PS PM PBNB NB NB NB NB NB NB NBNM NB NM NM NM NM NS NSNS NM NM NM NS NS Z ZZ Z Z Z PS PM PM PMPS PS PS PM PM PB PB PBPM PM PM PM PB PB PB PBPB PB PB PB PB PB PB PB
validates the effectiveness of FDDMFAC proposed in thispaper
Although FDDMFAC approach has been proven to havea better control performance than MFAC and DDOILCwe still have a question of whether it is possible that thesuperiority of FDDMFAC is entirely attributed to fuzzyadjustment of parameters and has nothing to do with thestructure of DDMFAC itself In order to clarify this problemanother set of simulations are conducted inwhich the controlperformance of FDDMFAC is compared with fuzzy MFAC(FMFAC) and fuzzy DDOILC (FDDOILC) The absolutevalue of tracking error |119890(119896 minus 1)| is regarded as the inputvariable of fuzzy rules and the step-size factors 1205881 1205882 ofMFAC andDDOILC are the output variablesThe fuzzy rulesof FMFAC and FDDOILC are shown in Table 6 and themembership functions when 1199100 = 60 are shown in Figure 8For the simulation results please see Figure 9 From thesimulation it is concluded that FDDMFAC achieves the best
Table 6 The fuzzy rules of FMFAC and FDDOILC
|119890(119896 minus 1)| NB NM NS Z PS PM PB1205881 1205882 NB NM NS Z PS PM PB
control performance which demonstrates the effectiveness ofthe proposed approach in this paper
5 Conclusions
This paper aimed at solving the problem of particle qualitycontrol in the drug development phase of SFBGP Themodel-based control approaches are incapable of handlingsuch quality control because the accurate process modelcannot be acquired Thus the data-driven and model-freecontrol approaches are considered in this work We firstly
14 Complexity
MFACDDOILCFDDMFAC
5 10 150Iterations
60
80
100
120
APS
(m
)y0 = 60
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
y0 = 70
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
MFACDDOILCFDDMFAC
95
100
105
110
115
120
125A
PS (
m)
5 10 150Iterations
y0 = 100
Figure 7 The comparison of control effects of MFAC DDOILC and FDDMFAC
analyzed the features of MFAC and DDOILC and thenconducted experimental research to compare the perfor-mances of MFAC and DDOILC in particle quality controlunder different IVs The simulation results have verifiedour theoretical analysis and concluded that MFAC andDDOILC both have advantages under different IVs To beused in an actual production environment the expectedDDCapproaches must accommodate various IVs because of thechanges in material attributes Thus the hybrid approachFDDMFAC which combines the advantages of MFAC andDDOILC by adaptively adjusting their respective weightedfactors with fuzzy logic was proposed in this work tocomplement their respective shortcomings Through a series
of simulation results we conclude that FDDMFAC achievesbetter control performance than other approaches whichvalidates the effectiveness of FDDMFAC
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thanks are due to National Natural Science Foundation ofChina under Grant nos 61773105 61374147 61703085 and61004083 and Fundamental Research Funds for the Central
Complexity 15NB NM NS Z PS PM PB
001 002 00302
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
002 00401
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
20 40 600|e(k minus 1)| (m)
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
Figure 8 The membership functions of FDDMFAC when 1199100 = 60
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
60
80
100
120
APS
(m
)
y0 = 60
FMFACFDDOILCFDDMFAC
70
80
90
100
110
120
APS
(m
)
2 4 6 8 100Iterations
y0 = 70
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
100
105
110
115
120
125
APS
(m
)
y0 = 100
Figure 9 The comparison of control effects of FMFAC FDDOILC and FDDMFAC
16 Complexity
Universities under Grant nos N150402001 and N120404014for supporting this research work
References
[1] J Bouffard M Kaster and H Dumont ldquoInfluence of processvariable and physicochemical properties on the granulationmechanism of mannitol in a fluid bed top spray granulatorrdquoDrug Development and Industrial Pharmacy vol 31 no 9 pp923ndash933 2005
[2] H Liu K Wang W Schlindwein and M Li ldquoUsing the Box-Behnken experimental design to optimise operating parametersin pulsed spray fluidised bed granulationrdquo International Journalof Pharmaceutics vol 448 no 2 pp 329ndash338 2013
[3] H Liu and M Li ldquoPopulation balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granula-tionrdquo International Journal of Pharmaceutics vol 468 no 1-2pp 223ndash233 2014
[4] T Albuquerque V H Dias N Poellinger and J F PintoldquoConstruction of a quality index for granules produced byfluidized bed technology and application of the correspondenceanalysis as a discriminant procedurerdquo European Journal ofPharmaceutics and Biopharmaceutics vol 75 no 3 pp 418ndash4242010
[5] H Ehlers A Liu H Raikkonen et al ldquoGranule size control andtargeting in pulsed spray fluid bed granulationrdquo InternationalJournal of Pharmaceutics vol 377 no 1-2 pp 9ndash15 2009
[6] C F W Sanders M J Hounslow and F J Doyle III ldquoIden-tification of models for control of wet granulationrdquo PowderTechnology vol 188 no 3 pp 255ndash263 2009
[7] T Narvanen T Lipsanen O Antikainen H Raikkonen andJ Yliruusi ldquoControlling granule size by granulation liquid feedpulsingrdquo International Journal of Pharmaceutics vol 357 no 1-2pp 132ndash138 2008
[8] A Faure P York and R C Rowe ldquoProcess control and scale-up of pharmaceutical wet granulation processes a reviewrdquoEuropean Journal of Pharmaceutics and Biopharmaceutics vol52 no 3 pp 269ndash277 2001
[9] M Pottmann B A Ogunnaike A A Adetayo and B JEnnis ldquoModel-based control of a granulation systemrdquo PowderTechnology vol 108 no 2-3 pp 192ndash201 2000
[10] E P Gatzke and F J Doyle III ldquoModel predictive control of agranulation system using soft output constraints and prioritizedcontrol objectivesrdquo Powder Technology vol 121 no 2-3 pp 149ndash158 2001
[11] F Y Wang and I T Cameron ldquoReview and future directionsin the modelling and control of continuous drum granulationrdquoPowder Technology vol 124 no 3 pp 238ndash253 2002
[12] T Glaser C F W Sanders F Y Wang et al ldquoModel predictivecontrol of continuous drum granulationrdquo Journal of ProcessControl vol 19 no 4 pp 615ndash622 2009
[13] R Ramachandran andA Chaudhury ldquoModel-based design andcontrol of a continuous drum granulation processrdquo ChemicalEngineering Research and Design vol 90 no 8 pp 1063ndash10732012
[14] Z H Loh D Z L Er L W Chan C V Liew and PW S HengldquoSpray granulation for drug formulationrdquo Expert Opinion onDrug Delivery vol 8 no 12 pp 1645ndash1661 2011
[15] S H Schaafsma N W F Kossen M T Mos L Blauw andA C Hoffmann ldquoEffects and control of humidity and particlemixing in fluid-bed granulationrdquo AIChE Journal vol 45 no 6pp 1202ndash1210 1999
[16] Z-S Hou and S-T Jin ldquoA novel data-driven control approachfor a class of discrete-time nonlinear systemsrdquo IEEE Transac-tions onControl Systems Technology vol 19 no 6 pp 1549ndash15582011
[17] Z Hou and S Jin ldquoData-driven model-free adaptive controlfor a class of MIMO nonlinear discrete-time systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 22no 12 pp 2173ndash2188 2011
[18] D Xu B Jiang and P Shi ldquoA novel model-free adaptivecontrol design for multivariable industrial processesrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6391ndash6398 2014
[19] X Wang X Li J Wang X Fang and X Zhu ldquoData-drivenmodel-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeletonrdquo Information Sciences vol 327pp 246ndash257 2016
[20] L dos Santos Coelho andA A R Coelho ldquoModel-free adaptivecontrol optimization using a chaotic particle swarm approachrdquoChaos Solitons amp Fractals vol 41 no 4 pp 2001ndash2009 2009
[21] L D S Coelho M W Pessoa R Rodrigues Sumar andA Augusto Rodrigues Coelho ldquoModel-free adaptive controldesign using evolutionary-neural compensatorrdquo Expert Systemswith Applications vol 37 no 1 pp 499ndash508 2010
[22] K K Tan T H Lee S N Huang and F M Leu ldquoAdaptive-predictive control of a class of SISOnonlinear systemsrdquoDynam-ics and Control vol 11 no 2 pp 151ndash174 2001
[23] B Zhang andW Zhang ldquoAdaptive predictive functional controlof a class of nonlinear systemsrdquo ISA Transactions vol 45 no2 pp 175ndash183 2006
[24] F L Lu J F Wang C J Fan and S B Chen ldquoAn improvedmodel-free adaptive control with G function fuzzy reasoningregulation design and its applicationsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems andControl Engineering vol 222 no 8 pp 817ndash828 2008
[25] R Chi D Wang Z Hou and S Jin ldquoData-driven optimalterminal iterative learning controlrdquo Journal of Process Controlvol 22 no 10 pp 2026ndash2037 2012
[26] R Chi Z Hou S Jin and D Wang ldquoImproved data-drivenoptimal TILC using time-varying input signalsrdquo Journal ofProcess Control vol 24 no 12 pp 78ndash85 2014
[27] X Bu Z Hou and R Chi ldquoModel free adaptive iterative learn-ing control for farm vehicle path trackingrdquo IFAC ProceedingsVolumes vol 46 no 20 pp 153ndash158 2013
[28] M C Campi A Lecchini and SM Savaresi ldquoVirtual referencefeedback tuning a direct method for the design of feedbackcontrollersrdquo Automatica vol 38 no 8 pp 1337ndash1346 2002
[29] R-C Roman M-B Radac R-E Precup and E M PetriuldquoData-driven model-free adaptive control tuned by virtualreference feedback tuningrdquoActa PolytechnicaHungarica vol 13no 1 pp 83ndash96 2016
[30] P Yan D Liu D Wang and H Ma ldquoData-driven controllerdesign for general MIMO nonlinear systems via virtual refer-ence feedback tuning and neural networksrdquo Neurocomputingvol 171 pp 815ndash825 2016
[31] T Pan and S Li ldquoA hierarchical search and updating databasestrategy for lazy learningrdquo International Journal of InnovativeComputing Information andControl vol 4 no 6 pp 1383ndash13922008
[32] H Zhang R Song Q Wei and T Zhang ldquoOptimal trackingcontrol for a class of nonlinear discrete-time systems withtime delays based on heuristic dynamic programmingrdquo IEEE
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Complexity
MFACDDOILCFDDMFAC
5 10 150Iterations
60
80
100
120
APS
(m
)y0 = 60
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
70
80
90
100
110
120
APS
(m
)
y0 = 70
MFACDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
MFACDDOILCFDDMFAC
95
100
105
110
115
120
125A
PS (
m)
5 10 150Iterations
y0 = 100
Figure 7 The comparison of control effects of MFAC DDOILC and FDDMFAC
analyzed the features of MFAC and DDOILC and thenconducted experimental research to compare the perfor-mances of MFAC and DDOILC in particle quality controlunder different IVs The simulation results have verifiedour theoretical analysis and concluded that MFAC andDDOILC both have advantages under different IVs To beused in an actual production environment the expectedDDCapproaches must accommodate various IVs because of thechanges in material attributes Thus the hybrid approachFDDMFAC which combines the advantages of MFAC andDDOILC by adaptively adjusting their respective weightedfactors with fuzzy logic was proposed in this work tocomplement their respective shortcomings Through a series
of simulation results we conclude that FDDMFAC achievesbetter control performance than other approaches whichvalidates the effectiveness of FDDMFAC
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thanks are due to National Natural Science Foundation ofChina under Grant nos 61773105 61374147 61703085 and61004083 and Fundamental Research Funds for the Central
Complexity 15NB NM NS Z PS PM PB
001 002 00302
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
002 00401
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
20 40 600|e(k minus 1)| (m)
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
Figure 8 The membership functions of FDDMFAC when 1199100 = 60
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
60
80
100
120
APS
(m
)
y0 = 60
FMFACFDDOILCFDDMFAC
70
80
90
100
110
120
APS
(m
)
2 4 6 8 100Iterations
y0 = 70
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
100
105
110
115
120
125
APS
(m
)
y0 = 100
Figure 9 The comparison of control effects of FMFAC FDDOILC and FDDMFAC
16 Complexity
Universities under Grant nos N150402001 and N120404014for supporting this research work
References
[1] J Bouffard M Kaster and H Dumont ldquoInfluence of processvariable and physicochemical properties on the granulationmechanism of mannitol in a fluid bed top spray granulatorrdquoDrug Development and Industrial Pharmacy vol 31 no 9 pp923ndash933 2005
[2] H Liu K Wang W Schlindwein and M Li ldquoUsing the Box-Behnken experimental design to optimise operating parametersin pulsed spray fluidised bed granulationrdquo International Journalof Pharmaceutics vol 448 no 2 pp 329ndash338 2013
[3] H Liu and M Li ldquoPopulation balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granula-tionrdquo International Journal of Pharmaceutics vol 468 no 1-2pp 223ndash233 2014
[4] T Albuquerque V H Dias N Poellinger and J F PintoldquoConstruction of a quality index for granules produced byfluidized bed technology and application of the correspondenceanalysis as a discriminant procedurerdquo European Journal ofPharmaceutics and Biopharmaceutics vol 75 no 3 pp 418ndash4242010
[5] H Ehlers A Liu H Raikkonen et al ldquoGranule size control andtargeting in pulsed spray fluid bed granulationrdquo InternationalJournal of Pharmaceutics vol 377 no 1-2 pp 9ndash15 2009
[6] C F W Sanders M J Hounslow and F J Doyle III ldquoIden-tification of models for control of wet granulationrdquo PowderTechnology vol 188 no 3 pp 255ndash263 2009
[7] T Narvanen T Lipsanen O Antikainen H Raikkonen andJ Yliruusi ldquoControlling granule size by granulation liquid feedpulsingrdquo International Journal of Pharmaceutics vol 357 no 1-2pp 132ndash138 2008
[8] A Faure P York and R C Rowe ldquoProcess control and scale-up of pharmaceutical wet granulation processes a reviewrdquoEuropean Journal of Pharmaceutics and Biopharmaceutics vol52 no 3 pp 269ndash277 2001
[9] M Pottmann B A Ogunnaike A A Adetayo and B JEnnis ldquoModel-based control of a granulation systemrdquo PowderTechnology vol 108 no 2-3 pp 192ndash201 2000
[10] E P Gatzke and F J Doyle III ldquoModel predictive control of agranulation system using soft output constraints and prioritizedcontrol objectivesrdquo Powder Technology vol 121 no 2-3 pp 149ndash158 2001
[11] F Y Wang and I T Cameron ldquoReview and future directionsin the modelling and control of continuous drum granulationrdquoPowder Technology vol 124 no 3 pp 238ndash253 2002
[12] T Glaser C F W Sanders F Y Wang et al ldquoModel predictivecontrol of continuous drum granulationrdquo Journal of ProcessControl vol 19 no 4 pp 615ndash622 2009
[13] R Ramachandran andA Chaudhury ldquoModel-based design andcontrol of a continuous drum granulation processrdquo ChemicalEngineering Research and Design vol 90 no 8 pp 1063ndash10732012
[14] Z H Loh D Z L Er L W Chan C V Liew and PW S HengldquoSpray granulation for drug formulationrdquo Expert Opinion onDrug Delivery vol 8 no 12 pp 1645ndash1661 2011
[15] S H Schaafsma N W F Kossen M T Mos L Blauw andA C Hoffmann ldquoEffects and control of humidity and particlemixing in fluid-bed granulationrdquo AIChE Journal vol 45 no 6pp 1202ndash1210 1999
[16] Z-S Hou and S-T Jin ldquoA novel data-driven control approachfor a class of discrete-time nonlinear systemsrdquo IEEE Transac-tions onControl Systems Technology vol 19 no 6 pp 1549ndash15582011
[17] Z Hou and S Jin ldquoData-driven model-free adaptive controlfor a class of MIMO nonlinear discrete-time systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 22no 12 pp 2173ndash2188 2011
[18] D Xu B Jiang and P Shi ldquoA novel model-free adaptivecontrol design for multivariable industrial processesrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6391ndash6398 2014
[19] X Wang X Li J Wang X Fang and X Zhu ldquoData-drivenmodel-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeletonrdquo Information Sciences vol 327pp 246ndash257 2016
[20] L dos Santos Coelho andA A R Coelho ldquoModel-free adaptivecontrol optimization using a chaotic particle swarm approachrdquoChaos Solitons amp Fractals vol 41 no 4 pp 2001ndash2009 2009
[21] L D S Coelho M W Pessoa R Rodrigues Sumar andA Augusto Rodrigues Coelho ldquoModel-free adaptive controldesign using evolutionary-neural compensatorrdquo Expert Systemswith Applications vol 37 no 1 pp 499ndash508 2010
[22] K K Tan T H Lee S N Huang and F M Leu ldquoAdaptive-predictive control of a class of SISOnonlinear systemsrdquoDynam-ics and Control vol 11 no 2 pp 151ndash174 2001
[23] B Zhang andW Zhang ldquoAdaptive predictive functional controlof a class of nonlinear systemsrdquo ISA Transactions vol 45 no2 pp 175ndash183 2006
[24] F L Lu J F Wang C J Fan and S B Chen ldquoAn improvedmodel-free adaptive control with G function fuzzy reasoningregulation design and its applicationsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems andControl Engineering vol 222 no 8 pp 817ndash828 2008
[25] R Chi D Wang Z Hou and S Jin ldquoData-driven optimalterminal iterative learning controlrdquo Journal of Process Controlvol 22 no 10 pp 2026ndash2037 2012
[26] R Chi Z Hou S Jin and D Wang ldquoImproved data-drivenoptimal TILC using time-varying input signalsrdquo Journal ofProcess Control vol 24 no 12 pp 78ndash85 2014
[27] X Bu Z Hou and R Chi ldquoModel free adaptive iterative learn-ing control for farm vehicle path trackingrdquo IFAC ProceedingsVolumes vol 46 no 20 pp 153ndash158 2013
[28] M C Campi A Lecchini and SM Savaresi ldquoVirtual referencefeedback tuning a direct method for the design of feedbackcontrollersrdquo Automatica vol 38 no 8 pp 1337ndash1346 2002
[29] R-C Roman M-B Radac R-E Precup and E M PetriuldquoData-driven model-free adaptive control tuned by virtualreference feedback tuningrdquoActa PolytechnicaHungarica vol 13no 1 pp 83ndash96 2016
[30] P Yan D Liu D Wang and H Ma ldquoData-driven controllerdesign for general MIMO nonlinear systems via virtual refer-ence feedback tuning and neural networksrdquo Neurocomputingvol 171 pp 815ndash825 2016
[31] T Pan and S Li ldquoA hierarchical search and updating databasestrategy for lazy learningrdquo International Journal of InnovativeComputing Information andControl vol 4 no 6 pp 1383ndash13922008
[32] H Zhang R Song Q Wei and T Zhang ldquoOptimal trackingcontrol for a class of nonlinear discrete-time systems withtime delays based on heuristic dynamic programmingrdquo IEEE
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Complexity 15NB NM NS Z PS PM PB
001 002 00302
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
002 00401
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
NB NM NS Z PS PM PB
20 40 600|e(k minus 1)| (m)
0
02
04
06
08
1
Deg
ree o
f mem
bers
hip
Figure 8 The membership functions of FDDMFAC when 1199100 = 60
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
80
90
100
110
120
APS
(m
)
y0 = 80
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
60
80
100
120
APS
(m
)
y0 = 60
FMFACFDDOILCFDDMFAC
70
80
90
100
110
120
APS
(m
)
2 4 6 8 100Iterations
y0 = 70
FMFACFDDOILCFDDMFAC
2 4 6 8 100Iterations
100
105
110
115
120
125
APS
(m
)
y0 = 100
Figure 9 The comparison of control effects of FMFAC FDDOILC and FDDMFAC
16 Complexity
Universities under Grant nos N150402001 and N120404014for supporting this research work
References
[1] J Bouffard M Kaster and H Dumont ldquoInfluence of processvariable and physicochemical properties on the granulationmechanism of mannitol in a fluid bed top spray granulatorrdquoDrug Development and Industrial Pharmacy vol 31 no 9 pp923ndash933 2005
[2] H Liu K Wang W Schlindwein and M Li ldquoUsing the Box-Behnken experimental design to optimise operating parametersin pulsed spray fluidised bed granulationrdquo International Journalof Pharmaceutics vol 448 no 2 pp 329ndash338 2013
[3] H Liu and M Li ldquoPopulation balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granula-tionrdquo International Journal of Pharmaceutics vol 468 no 1-2pp 223ndash233 2014
[4] T Albuquerque V H Dias N Poellinger and J F PintoldquoConstruction of a quality index for granules produced byfluidized bed technology and application of the correspondenceanalysis as a discriminant procedurerdquo European Journal ofPharmaceutics and Biopharmaceutics vol 75 no 3 pp 418ndash4242010
[5] H Ehlers A Liu H Raikkonen et al ldquoGranule size control andtargeting in pulsed spray fluid bed granulationrdquo InternationalJournal of Pharmaceutics vol 377 no 1-2 pp 9ndash15 2009
[6] C F W Sanders M J Hounslow and F J Doyle III ldquoIden-tification of models for control of wet granulationrdquo PowderTechnology vol 188 no 3 pp 255ndash263 2009
[7] T Narvanen T Lipsanen O Antikainen H Raikkonen andJ Yliruusi ldquoControlling granule size by granulation liquid feedpulsingrdquo International Journal of Pharmaceutics vol 357 no 1-2pp 132ndash138 2008
[8] A Faure P York and R C Rowe ldquoProcess control and scale-up of pharmaceutical wet granulation processes a reviewrdquoEuropean Journal of Pharmaceutics and Biopharmaceutics vol52 no 3 pp 269ndash277 2001
[9] M Pottmann B A Ogunnaike A A Adetayo and B JEnnis ldquoModel-based control of a granulation systemrdquo PowderTechnology vol 108 no 2-3 pp 192ndash201 2000
[10] E P Gatzke and F J Doyle III ldquoModel predictive control of agranulation system using soft output constraints and prioritizedcontrol objectivesrdquo Powder Technology vol 121 no 2-3 pp 149ndash158 2001
[11] F Y Wang and I T Cameron ldquoReview and future directionsin the modelling and control of continuous drum granulationrdquoPowder Technology vol 124 no 3 pp 238ndash253 2002
[12] T Glaser C F W Sanders F Y Wang et al ldquoModel predictivecontrol of continuous drum granulationrdquo Journal of ProcessControl vol 19 no 4 pp 615ndash622 2009
[13] R Ramachandran andA Chaudhury ldquoModel-based design andcontrol of a continuous drum granulation processrdquo ChemicalEngineering Research and Design vol 90 no 8 pp 1063ndash10732012
[14] Z H Loh D Z L Er L W Chan C V Liew and PW S HengldquoSpray granulation for drug formulationrdquo Expert Opinion onDrug Delivery vol 8 no 12 pp 1645ndash1661 2011
[15] S H Schaafsma N W F Kossen M T Mos L Blauw andA C Hoffmann ldquoEffects and control of humidity and particlemixing in fluid-bed granulationrdquo AIChE Journal vol 45 no 6pp 1202ndash1210 1999
[16] Z-S Hou and S-T Jin ldquoA novel data-driven control approachfor a class of discrete-time nonlinear systemsrdquo IEEE Transac-tions onControl Systems Technology vol 19 no 6 pp 1549ndash15582011
[17] Z Hou and S Jin ldquoData-driven model-free adaptive controlfor a class of MIMO nonlinear discrete-time systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 22no 12 pp 2173ndash2188 2011
[18] D Xu B Jiang and P Shi ldquoA novel model-free adaptivecontrol design for multivariable industrial processesrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6391ndash6398 2014
[19] X Wang X Li J Wang X Fang and X Zhu ldquoData-drivenmodel-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeletonrdquo Information Sciences vol 327pp 246ndash257 2016
[20] L dos Santos Coelho andA A R Coelho ldquoModel-free adaptivecontrol optimization using a chaotic particle swarm approachrdquoChaos Solitons amp Fractals vol 41 no 4 pp 2001ndash2009 2009
[21] L D S Coelho M W Pessoa R Rodrigues Sumar andA Augusto Rodrigues Coelho ldquoModel-free adaptive controldesign using evolutionary-neural compensatorrdquo Expert Systemswith Applications vol 37 no 1 pp 499ndash508 2010
[22] K K Tan T H Lee S N Huang and F M Leu ldquoAdaptive-predictive control of a class of SISOnonlinear systemsrdquoDynam-ics and Control vol 11 no 2 pp 151ndash174 2001
[23] B Zhang andW Zhang ldquoAdaptive predictive functional controlof a class of nonlinear systemsrdquo ISA Transactions vol 45 no2 pp 175ndash183 2006
[24] F L Lu J F Wang C J Fan and S B Chen ldquoAn improvedmodel-free adaptive control with G function fuzzy reasoningregulation design and its applicationsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems andControl Engineering vol 222 no 8 pp 817ndash828 2008
[25] R Chi D Wang Z Hou and S Jin ldquoData-driven optimalterminal iterative learning controlrdquo Journal of Process Controlvol 22 no 10 pp 2026ndash2037 2012
[26] R Chi Z Hou S Jin and D Wang ldquoImproved data-drivenoptimal TILC using time-varying input signalsrdquo Journal ofProcess Control vol 24 no 12 pp 78ndash85 2014
[27] X Bu Z Hou and R Chi ldquoModel free adaptive iterative learn-ing control for farm vehicle path trackingrdquo IFAC ProceedingsVolumes vol 46 no 20 pp 153ndash158 2013
[28] M C Campi A Lecchini and SM Savaresi ldquoVirtual referencefeedback tuning a direct method for the design of feedbackcontrollersrdquo Automatica vol 38 no 8 pp 1337ndash1346 2002
[29] R-C Roman M-B Radac R-E Precup and E M PetriuldquoData-driven model-free adaptive control tuned by virtualreference feedback tuningrdquoActa PolytechnicaHungarica vol 13no 1 pp 83ndash96 2016
[30] P Yan D Liu D Wang and H Ma ldquoData-driven controllerdesign for general MIMO nonlinear systems via virtual refer-ence feedback tuning and neural networksrdquo Neurocomputingvol 171 pp 815ndash825 2016
[31] T Pan and S Li ldquoA hierarchical search and updating databasestrategy for lazy learningrdquo International Journal of InnovativeComputing Information andControl vol 4 no 6 pp 1383ndash13922008
[32] H Zhang R Song Q Wei and T Zhang ldquoOptimal trackingcontrol for a class of nonlinear discrete-time systems withtime delays based on heuristic dynamic programmingrdquo IEEE
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Complexity
Universities under Grant nos N150402001 and N120404014for supporting this research work
References
[1] J Bouffard M Kaster and H Dumont ldquoInfluence of processvariable and physicochemical properties on the granulationmechanism of mannitol in a fluid bed top spray granulatorrdquoDrug Development and Industrial Pharmacy vol 31 no 9 pp923ndash933 2005
[2] H Liu K Wang W Schlindwein and M Li ldquoUsing the Box-Behnken experimental design to optimise operating parametersin pulsed spray fluidised bed granulationrdquo International Journalof Pharmaceutics vol 448 no 2 pp 329ndash338 2013
[3] H Liu and M Li ldquoPopulation balance modelling and multi-stage optimal control of a pulsed spray fluidized bed granula-tionrdquo International Journal of Pharmaceutics vol 468 no 1-2pp 223ndash233 2014
[4] T Albuquerque V H Dias N Poellinger and J F PintoldquoConstruction of a quality index for granules produced byfluidized bed technology and application of the correspondenceanalysis as a discriminant procedurerdquo European Journal ofPharmaceutics and Biopharmaceutics vol 75 no 3 pp 418ndash4242010
[5] H Ehlers A Liu H Raikkonen et al ldquoGranule size control andtargeting in pulsed spray fluid bed granulationrdquo InternationalJournal of Pharmaceutics vol 377 no 1-2 pp 9ndash15 2009
[6] C F W Sanders M J Hounslow and F J Doyle III ldquoIden-tification of models for control of wet granulationrdquo PowderTechnology vol 188 no 3 pp 255ndash263 2009
[7] T Narvanen T Lipsanen O Antikainen H Raikkonen andJ Yliruusi ldquoControlling granule size by granulation liquid feedpulsingrdquo International Journal of Pharmaceutics vol 357 no 1-2pp 132ndash138 2008
[8] A Faure P York and R C Rowe ldquoProcess control and scale-up of pharmaceutical wet granulation processes a reviewrdquoEuropean Journal of Pharmaceutics and Biopharmaceutics vol52 no 3 pp 269ndash277 2001
[9] M Pottmann B A Ogunnaike A A Adetayo and B JEnnis ldquoModel-based control of a granulation systemrdquo PowderTechnology vol 108 no 2-3 pp 192ndash201 2000
[10] E P Gatzke and F J Doyle III ldquoModel predictive control of agranulation system using soft output constraints and prioritizedcontrol objectivesrdquo Powder Technology vol 121 no 2-3 pp 149ndash158 2001
[11] F Y Wang and I T Cameron ldquoReview and future directionsin the modelling and control of continuous drum granulationrdquoPowder Technology vol 124 no 3 pp 238ndash253 2002
[12] T Glaser C F W Sanders F Y Wang et al ldquoModel predictivecontrol of continuous drum granulationrdquo Journal of ProcessControl vol 19 no 4 pp 615ndash622 2009
[13] R Ramachandran andA Chaudhury ldquoModel-based design andcontrol of a continuous drum granulation processrdquo ChemicalEngineering Research and Design vol 90 no 8 pp 1063ndash10732012
[14] Z H Loh D Z L Er L W Chan C V Liew and PW S HengldquoSpray granulation for drug formulationrdquo Expert Opinion onDrug Delivery vol 8 no 12 pp 1645ndash1661 2011
[15] S H Schaafsma N W F Kossen M T Mos L Blauw andA C Hoffmann ldquoEffects and control of humidity and particlemixing in fluid-bed granulationrdquo AIChE Journal vol 45 no 6pp 1202ndash1210 1999
[16] Z-S Hou and S-T Jin ldquoA novel data-driven control approachfor a class of discrete-time nonlinear systemsrdquo IEEE Transac-tions onControl Systems Technology vol 19 no 6 pp 1549ndash15582011
[17] Z Hou and S Jin ldquoData-driven model-free adaptive controlfor a class of MIMO nonlinear discrete-time systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 22no 12 pp 2173ndash2188 2011
[18] D Xu B Jiang and P Shi ldquoA novel model-free adaptivecontrol design for multivariable industrial processesrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6391ndash6398 2014
[19] X Wang X Li J Wang X Fang and X Zhu ldquoData-drivenmodel-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeletonrdquo Information Sciences vol 327pp 246ndash257 2016
[20] L dos Santos Coelho andA A R Coelho ldquoModel-free adaptivecontrol optimization using a chaotic particle swarm approachrdquoChaos Solitons amp Fractals vol 41 no 4 pp 2001ndash2009 2009
[21] L D S Coelho M W Pessoa R Rodrigues Sumar andA Augusto Rodrigues Coelho ldquoModel-free adaptive controldesign using evolutionary-neural compensatorrdquo Expert Systemswith Applications vol 37 no 1 pp 499ndash508 2010
[22] K K Tan T H Lee S N Huang and F M Leu ldquoAdaptive-predictive control of a class of SISOnonlinear systemsrdquoDynam-ics and Control vol 11 no 2 pp 151ndash174 2001
[23] B Zhang andW Zhang ldquoAdaptive predictive functional controlof a class of nonlinear systemsrdquo ISA Transactions vol 45 no2 pp 175ndash183 2006
[24] F L Lu J F Wang C J Fan and S B Chen ldquoAn improvedmodel-free adaptive control with G function fuzzy reasoningregulation design and its applicationsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems andControl Engineering vol 222 no 8 pp 817ndash828 2008
[25] R Chi D Wang Z Hou and S Jin ldquoData-driven optimalterminal iterative learning controlrdquo Journal of Process Controlvol 22 no 10 pp 2026ndash2037 2012
[26] R Chi Z Hou S Jin and D Wang ldquoImproved data-drivenoptimal TILC using time-varying input signalsrdquo Journal ofProcess Control vol 24 no 12 pp 78ndash85 2014
[27] X Bu Z Hou and R Chi ldquoModel free adaptive iterative learn-ing control for farm vehicle path trackingrdquo IFAC ProceedingsVolumes vol 46 no 20 pp 153ndash158 2013
[28] M C Campi A Lecchini and SM Savaresi ldquoVirtual referencefeedback tuning a direct method for the design of feedbackcontrollersrdquo Automatica vol 38 no 8 pp 1337ndash1346 2002
[29] R-C Roman M-B Radac R-E Precup and E M PetriuldquoData-driven model-free adaptive control tuned by virtualreference feedback tuningrdquoActa PolytechnicaHungarica vol 13no 1 pp 83ndash96 2016
[30] P Yan D Liu D Wang and H Ma ldquoData-driven controllerdesign for general MIMO nonlinear systems via virtual refer-ence feedback tuning and neural networksrdquo Neurocomputingvol 171 pp 815ndash825 2016
[31] T Pan and S Li ldquoA hierarchical search and updating databasestrategy for lazy learningrdquo International Journal of InnovativeComputing Information andControl vol 4 no 6 pp 1383ndash13922008
[32] H Zhang R Song Q Wei and T Zhang ldquoOptimal trackingcontrol for a class of nonlinear discrete-time systems withtime delays based on heuristic dynamic programmingrdquo IEEE
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Complexity 17
Transactions on Neural Networks and Learning Systems vol 22no 12 pp 1851ndash1862 2011
[33] J D Rojas and R Vilanova ldquoData-driven based IMC controlrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 3 pp 1557ndash1574 2012
[34] C Li J Yi and T Wang ldquoEncoding prior knowledge intodata driven design of interval type-2 fuzzy logic systemsrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 3 pp 1133ndash1144 2012
[35] M-B Radac and R-E Precup ldquoOptimal behaviour predictionusing a primitive-based data-driven model-free iterative learn-ing control approachrdquo Computers in Industry vol 74 pp 95ndash109 2015
[36] M-B Radac R-E Precup and E Petriu ldquoModel-freeprimitive-based iterative learning control approach to trajec-tory tracking of MIMO systems with experimental validationrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 11 pp 2925ndash2938 2015
[37] M Hussain J Kumar M Peglow and E Tsotsas ldquoModelingspray fluidized bed aggregation kinetics on the basis of Monte-Carlo simulation resultsrdquoChemical Engineering Science vol 101pp 35ndash45 2013
[38] M Hussain M Peglow E Tsotsas and J Kumar ldquoModelingof aggregation kernel using Monte Carlo simulations of sprayfluidized bed agglomerationrdquo AIChE Journal vol 60 no 3 pp855ndash868 2014
[39] H Liu and M Li ldquoTwo-compartmental population balancemodeling of a pulsed spray fluidized bed granulation basedon computational fluid dynamics (CFD) analysisrdquo InternationalJournal of Pharmaceutics vol 475 no 1 pp e256ndashe269 2014
[40] G K Reynolds J S Fu Y S Cheong M J Hounslow andA D Salman ldquoBreakage in granulation a reviewrdquo ChemicalEngineering Science vol 60 no 14 pp 3969ndash3992 2005
[41] M Peglow J Kumar G Warnecke S Heinrich and L MorlldquoA new technique to determine rate constants for growth andagglomeration with size- and time-dependent nuclei forma-tionrdquo Chemical Engineering Science vol 61 no 1 pp 282ndash2922006
[42] M J Hounslow R L Ryall and V R Marshall ldquoA discretizedpopulation balance for nucleation growth and aggregationrdquoAIChE Journal vol 34 no 11 pp 1821ndash1832 1988
[43] M J Hounslow JM K Pearson and T Instone ldquoTracer studiesof high-shear granulation II Population balance modelingrdquoAIChE Journal vol 47 no 9 pp 1984ndash1999 2001
[44] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl vol 33 Prentice-Hall Upper Saddle River NJ USA1984
[45] D He Z Wang L Yang and Z Mao ldquoOptimization controlof the color-coating production process for model uncertaintyrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 9731823 12 pages 2016
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
