DESIGN OF ROBUST CONTROLLERS FOR A MILK OF LIME BLENDING TANK

8/7/2019 DESIGN OF ROBUST CONTROLLERS FOR A MILK OF LIME BLENDING TANK

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DESIGN OF ROBUST CONTROLLERS FO R A MILK OF LIME BLENDING TA NK

F. Tadeol and P, Vega2lun ivers idad d e Walladolid, Spain and 2Universidad de Salamanca, Spain

Keywords: Process Control, Com puter Aided Control System Design, Multivariable Control Systems.

ABSTRACT.

This paper presents the use of an industrial Milk of LimeBlending Tank (MLBT) for the design and evaluation ofdifferent robust c,o ntrollers. Th e plant is a multi-input,multi-output non-linear system with high interactionbetween loops and large disturbance actions. The IMC,H, and p. methods are used to design controllers whichprovide robust stability as well as good nominalperformance in the control of the density at the outputand the level of juice in the tank in the face ofuncertainties due 00 non-linearities within the system anddisturbance actioos. A non-linear model and operatingrecords of disturbances taken from the real process areused to develop the weighting functions to include theplant uncertainties in the design procedure. Finally, acomparison between the performance of the designedrobust controllers and a conventional PID is carried out.

INTRODUCTION

One of the main stages in the sugar manufacturingprocess takes place in the purification station where theimpurities of the juice are removed by means of twocarbonation proce ses See McGinnis ( 6 ) ) .An auxiliarycomponent of the carbonation section is the Milk ofLime Blending Tank (MLBT) which provides a uniformflow of milk of lime (calcium hydroxide). One of themost important aim of the control system is to keep thedensity at the output at prescribed values despite thedisturbance effects. The level of liquid in the tank ismaintained within a specified range to avoid flowobstruction due to the high density of the dissolution.The interaction bebeen the control loops, the presenceof non-linearities and disturbances just ify the use ofadvanced control techniques.

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Currently, most of the advanced controllers are usuallydesigned by using a linear model of the process based onfixed information 1 of the plant that is imperfect andincomplete. Control quality may deteriorate when the

This work was supported in part by the CICYT undergrant BI094-0679-C 02-02

working conditions change. The essence of robustcontrol is to model the uncertainties themselves and toincorporate them into the design procedure o f the controlsystem, with the aim of ensuring stability. By usingsimulation techniques, this paper presents a comparativestudy of different Robust Control techniques and theclassical PID control for a MLBT of a particular sugarfactory placed in Spain.The paper begins with a description o f the process to becontrolled and the n on-linear mathematical model of theplant. Next, the control problem is stated and, after ashort review of the basis of Robust Control, the designof the controllers by means of the p (Doyle (3)) and theIMC techniques (Morari and Zafirou (7)) is described.Th e design of the H,-optimal controller was describedin (Tadeo et al. (8)), therefore it is no t repeated in thispaper for lack of space. Finally the performance of theregulators is compared o n a simulation of the real plant,

THE MILK OF LIME BLENDING TANK

The system dealt with is a stirred tank (Figure 1) wherea suspension of lime is mixed with w ater to decrease thedissolution density. The product leaves the tank and it isdissolved with the raw juice (obtained by a diffusionprocess from sliced sug ar beet and water) during the firststeps of the purification process, in which the non-sugarspresent in the juice form insoluble products which arethen easily eliminated by filtering.

essure

Mllk of Lime Concentrahon,cFigure 1: Milk of Lime Blending TankThe mathematical model of the MLBT comprises twonon-linear state space equations corresponding to massbalances (Vega et al. (9)):

U w c C InternationAI Conference on CONTROL 96,2-5 September 1996, Conference Publication NO.427 0 IEE 1996


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where ul, u2, u3 are input signals to manipulate thevalves, x1 is the concentration of the mixture at theoutput, x2 is the level of liquid in the tank, p1 is theconcentration of input lime suspension, p2 the backpressure, The parameters h l = 5 , h2 = 3.6, h3 = 9.285are the valve constants, a = 10m2 the tank area and a =0.573 relates the concentration of the lime solution to thedensity of dry lime. Finally, the output variables are thespecific gravity of the mixture at the output, y1 = l / ( l -a x l) and the level of liquid, y2 = x2.We used ACSL (Advanced Continuous SimulationLanguage) to implement the equations together withrecords of data from the real plant fo r the study.

THE CONTROL PROBLEMThe main control aim is to maintain steady constantdensity at the output in the light of large disturbanceseffects. The level of liquid in the tank must be keptwithin a specified range to avoid flow obstruction asmentioned above. As it is shown in figure I , there arecontrol valves already in place at the input and outputthat allows the control of such variables, although thereare are some effect which must be taken into accountwhen designing the controller:Non-linearities: From the ma thematical description of theplant, it can be seen that the plant is non-linear, Thesystem may be op erating far from the nominal value, andthe non-linearities effects will become observable.Perturbations: The most severe disturbances are theconcentration of the input lime mixture and the backpressure.The use of Robust Control techniques seems to be a goodchoice in order to design a concentration and levelcontroller providing good control performance in everyworking condition.

ROBUST CONTROL THEORY REVIEWThe classical method for designing model basedcontrollers uses a linearised model (6)of the real plant(G ) around the working point. The aim of RobustControl is to model the Uncertainties (variations of thenominal model) and use them together with the nominalmodel in the design procedure to ensure robust stability

(the controlled system must remain stable despite thepresent uncertainties of the model and the disturbancesacting on the system).Uncertainty Description. When using Robust ControlTechniques it is useful to describe the uncertainty as amatrix A that relates G an d 8 (See Chiang and Safonov(2)). In the case of the MLBT the uncertainties weredescribed as output multiplicative inverse uncertainties.The relationship between G an d @ is given byG = [I -A]@.Robust Stability. Robust control theory uses themaximum singular value at each frequency among allthe upper singular values $Ai) for of all considereduncertainties A i as a bound to check if the system isstable when any of them is present. Suppose that atransfer function called weightingfactor W 1(s) such as

is selected, then a plant affected by an outputmultiplicative inverse uncertainty A , belonging to the setconsidered in the design, is stable ifand only if

- 1o (S( j w) )I~ Wl 1 (31where S is the closed loop sensivity function whichrelates frequency disturbances d(s) to the output signaly(s). Therefore, any controller K(s) which satisfies thiscondition ensures Robust Stability (See Chiang andSafonov (2)). The inequality (3) expresses the robuststability criterion.

ROBUST CONTROL DESIGN FOR THE MLBT

Non-linearities description.The uncertainties due to the non-linear characteristics ofthe system were described as outpu t multiplicativeinverse uncertainties. By using this description and byworking out a set of linearised models at differentworking points (GI, G2,..) , the corresponding set ofmodel uncertainties Ai was obtained and the each uppersingular value ??(Ai) over the frequency rangeevaluated. For instance, the size of the uncertaintiescaused by varying the concentration of lime in the tankagainst frequency are plotted in figure 3 .


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ao i

7 0eo5 03 - 0 -

e 2 0 -0 -

3 = O -'0

- 1 0

For the case of the M LB T the frequency spectrum of thedisturbances on the outputs can be seen in figure 5together with the transfer function that limited them thatmore precisely was foun d to be

unc-rt - "t y ma t r c s s ,or r ? *O.y,ns ," [ O 0 , 0 a,

-2 0

I- I

I K = [ I5.6755 0.3830- -7.2524 0.1274-

. _ _ and the PIS tuning for each loop was: kp = 3.2, Tl = 2.5I-i and for loop two ,k = 5.0, Ti = 2.1.

1 0 - 1 r e - 3 7 0- 1 0 . 10 . ' ' -d If Wl(s) >W,(s) the condition for the disturbances isFigure 3 . Uqcertainties due to non -linearities.f?.qu."Cy included in the non-linearity robustness condition.

CONTROLLER DESIGNFigure 4 shows thb worst case gain o f these uncertaintiestogether with thd weighting function W1 given by:i

Disturbance desciiptionThe main distl/rbances are those affecting theconcentration of the input lime mixture and the backpressure. These p$rturbations are adjusted, by the use ofa transfer function lmatrix, to equate them to disturbancesin the density outdut and the level. The transfer from on edimension to another acts as a low pass filter and so thedisturbances to thd outputs are low frequency signals. Afrequency analysis of the system outputs using the FastFourier Transform, allows the design of a controller toreduce their effekts by filtering. According to thecan be done by bounding thematrix singular values. Abe chosen such as

should be minimized.

The Internal Model Control (IMC) framework includesan intemal model of the process and it operates in openloop when the nominal model and th e real model are thesame. Moreover, the model can include the availableinformation about non-linearities, constraints, modeluncertainty, etc., so that complex nonlinear algorithmscan be implemented without complex stability issues.The structure is shown in figure 6, where Q(s) is theIMC controller, G is the real system and d is theintemal model of the system.

d

-

To design robust controllers using the IM C structure atwo-step procedure is proposed. In the first step acontroller Q is selected to get a good response for thenominal model (it could be any controller: PID, H, ...).

I10 0 'O ' In the second step this controller is augmented by a low-pass filter with parameters that can be tuned t o f d f i l l theperformance and robustness specification (See Morariand Zafiriou (2)).

-.oL- ' '10..Ir.q".ncyFigure 5 : Frequ ,ncy spectra and weighting function

i


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For the MLBT the chosen nominal controller a was theone which ensures perfect cancellation, i.e., Q = 6-1:

I3.0259(s + 1.3064) 0.1613(s+ 8.7278)

-4.2033(~+ 1.2019) 0.0537(~+ 8.7278)Q(s>=This controller was augmented with a low pass filter Fsuch that F = ( l + h s ) - l l a nd Q = Q F . Then thesensitivity is given by S = ___" I , so the singularvalues of S were: 1+ hs

Is 11-o(S) = o(S) =-- 1 + h s s=j,As the robustness condition given by the uncertainties

was E(s)


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controller K (solid) and theKred (dashed).

1 0 -1o2 10' 10-' lo o 'C *'?"*,,cy10-2 Fi$ure 10: Model Reduction

SIMULATION^

compared with the classicalcontrolled systed in different situations, for most of thesimulations they mproved the system response in termsof robustness andistability.Nominal perfor ance (Output concentration change)First of all, th controlled system performed in thewas studied. Wi hout any disturbances and with thenominal paramet rs a step in the desired output limeconcentration of 6% (from yl=1.16 to yl=l.lO) wassimulated to stu:.y its nominal performance. figure 1 1shows that the robust controllers need slightly more time

by the PID controlled varyones have sharp peaks.

operation point!osen to calculate the nominal model

the performance of the wholeoutput valve opening fromwas simulated. The same outputto yl=1.10:6%) as

was demanded.the PID controlled systemperformance when thissystem controlled withmodified. This isthis parameterthe uncertainties considered duringThe robust control only presents alevel, that is soon corrected. Th e

and 0.8 (p).

Disturbance rejection: Input lime concentration. Byusing the nominal model with the available data of theinput lime concentration disturbance, the controlledsystem was simulated. The robust controlled systemsperformed better than the classical designed controller,as expected, because the disturbance frequency spectrumwas considered in the robust control design procedure.The results are illustrated in figure 13. They demonstratethat the robust controllers improve the disturbancerejection, compared with the PID controller. Table 1summarizes the standard deviation and mean value ofthe system outputs with the different controllers: Thebest ones are in bold. The worst ones are underlined.Recall that y1 0= l.l 6 to y20=1.2.

CONCLUSIONSThis paper has shown that the Robust Control Design isan efficient altemative to the Classical Control Designwhen dealing with MIMO systems that suffer fromimportant interactions and disturbances. Moreover, thedifficulty of the Robust Control D esign does not increasewith the number of control loops. This is because it isbased on the maximum singular value.The improved performance of the MLBT using thedesigned IMC, H, and p controllers was shown: Thedisturbance rejection is increased, and the interactionbetween loops is decreased when it is working far fromthe nominal point. They were considered feasible toimplement in the real plant.REFERENCES( 1 ) Balas, G.J and Doyle, J.C., 1991 p-Analysis andSvnthesis Toolbox, Th e Mathworks Inc.(2).Chiang, and Safonov, M.G., 1992, Robust ControlToolbox, The Mathw orks Inc.(3) Doyle, J.C., 1982, "Analysis of Feedback Systemswith Structured Uncertainty", IEE Proc. 1982;(4) Doyle, J.C., and Stein, G., 1981, "MultivariableFeedback Design: Concepts for a Classical/ModemSynthesis", IEEE Trans. Automat.. Contr.. AC-26: 4-16.(5 ) Green, M, Limebeer, D., 1995, Linear RobustControl, Prentice Hall(6 ) McGinnis, R.A., 1971, Beet-Sugar Te chn olo m, BeetSugar Development Fundation.(7) Morari, M, Zafiriou, E,. 1989 Robust ProcessControl, Prentice-Hall(8) Tadeo, F., 1994, "The Robust Control of a Milk ofLime Blending Tank", AMST'94, MechanicalEngineering Publications Ltd, London(9) Vega, P., Cameron, R.G. and de Prada, C. 1987, AModel of a Milk of Lime Mixing Tank, Research ReportR381, School of Con trol Eng., Univ. of Bradford, UK

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PID Hinf IM C pv19

4/ Tank Level ,

0d

PID Hinf MC p9 ,I 6.6667 8.3333 17.0.0 1.6667 3.3333Figure 11:Tank Level Setpoint Change in the nominal working point

O u t pu t C o n c e n t r a t i o n

I IPIDp 0 lc H i

0 6 . 6 6 6 7 E. 3333T 5 . O

i.6667 i .3333 Figure 12:Output valve opening changed from 50% to 20%

02. Tank L e v e l 0I - O u t p u t C o n c e n t r a t i o n

? ! , , , , , I 9d 0.0 3.3333 6.6667 f O . 0 13.333 16.667 20.0 0 3.3333 6.6667 j O . 0 13.333 16.6670 3.3333 6.6667 j O . 0 13.333 16.667

D. 0

0.0

0. 0Figure 13 : Disturbance rejection: Input Lime Concentration

DESIGN OF ROBUST CONTROLLERS FOR A MILK OF LIME BLENDING TANK

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