Research Article Modeling, Parameters...

Research ArticleModeling Parameters Identification and Control ofHigh Pressure Fuel Cell Back-Pressure Valve

Fengxiang Chen Ling Liu Shiguang Liu and Tong Zhang

Clean Energy Automotive Engineering Center Tongji University Shanghai 201804 China

Correspondence should be addressed to Fengxiang Chen fxchentongjieducn

Received 5 June 2014 Revised 5 September 2014 Accepted 19 September 2014 Published 16 November 2014

Academic Editor Yan Liang

Copyright copy 2014 Fengxiang Chen et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The reactant pressure is crucial to the efficiency and lifespan of a high pressure PEMFC engineThis paper analyses a regulated back-pressure valve (BPV) for the cathode outlet flow in a high pressure PEMFC engine which can achieve precisely pressure controlThemodeling parameters identification and nonlinear controller design of a BPV system are consideredThe identified parametersare used in designing active disturbance rejection controller (ADRC) Simulations and extensive experiments are conducted withthe xPC Target and show that the proposed controller can not only achieve good dynamic and static performance but also havestrong robustness against parametersrsquo disturbance and external disturbance

1 Introduction

High pressure proton exchangemembrane fuel cell (PEMFC)engines offer clean and efficient energy production andare advantaged of high stack power density quick systemresponse and easy humidification One of the dominantdesign parameters for high pressure PEMFC system is theoperating pressure [1] In the case of stored compressedhydrogen the pressure of the cathode air supply systembecomes an important optimized parameter [2] Com-pared with the low-pressure system high operating pressureimproves the performance of the fuel cell stack but consumesmore power due to the flow device such as the compressor(nearly consuming up to 20 percent of the net power [3])In order to maintain a high efficiency during operation itis important to control air pressure effectively In this studyan electronic throttle (ET) body is served as a back-pressurevalve (BPV) the cathode air supply system can operate in aproper pressure mainly by auto-tuning the angle of the BPVfor cathode outlet flow at a given power Therefore the fastand accurate control of the angle of the BPV is a key issue toregulate the back pressure for a high pressure PEMFC engine

The analysis of BPV model-based control system maybegin with the mathematical model of a BPV system TheBPV is different from the traditional ET in that it has to

consider the influence of the flow resistance In additionthe BPV has many unknown parameters and there is nosystematic method to estimate its parameters so far Songhas estimated some parameters [4] The simulations showthat the model and real plant have unacceptable bias dueto excess approximation Grep and Lee have designed somesimple algorithms [5] and modified them to identify the ETparameters in [6] But there is no mention of the parameterverification Based on the previous studies several algorithmsare developed to estimate the unknown parameters in thispaper Parameter identification uses simple equipment suchas current voltage sensor and obtained parameters areoptimized through the optimization toolbox The model andacquired parameters can be used for the model-based con-troller designWhat is more they can also be used to find outthe optimal opening of the BPV at a given power by merginginto a PEMFC model At present set-points are usuallyobtained by repeating experiments on PEMFC engine

In the aspect of control various control design techniqueshave been studied in the previous publications such asproportional-integral-derivative (PID) control [7] and slidingmode control [8] In [9] the application of receptive fieldweighted regression (RFWR)method is demonstrated for thecomposite control of ET In [10] an ET controller based onthe idea of back-stepping method is proposed which meets

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 246015 10 pageshttpdxdoiorg1011552014246015

2 Mathematical Problems in Engineering

the general requirements of the state stability Then in [11]the back-stepping controller combined with state observeris designed for the real-time system In [12] a variablestructure concept is utilized based on the sliding modeobserver technique and feedback back-stepping techniqueHowever the above methods have strongmodel dependenceParameters-varying or external disturbances can usuallydeteriorate the control performance Most importantly thesystem parameters used in the simulations are not verifiedwith actual ET model

In this paper a detailed nonlinear dynamic model ofBPV is provided and several algorithms are designed toestimate the unknown parameters The methods used inthe identification process consist of extensive measurementsexperiments and verifications The back-pressure valve usedin this paper is a commercial available ET system made byBosch in type of DV-E5 The rest of the paper is organizedas follows in Section 2 a detailed nonlinear dynamic modelof BPV is provided Next in Section 3 the identification algo-rithms are explained Finally the active disturbance rejectioncontroller (ADRC) is designed based on the mathematicalmodel and the identified parameters in Section 4 After thatthe model controlling effect and real-time controlling effectare presented

2 Modeling of the Back-Pressure Valve

The ET is featured with high nonlinearity The model of EThas been approached which includes the nonlinear returnspring and friction model Yet the BPV model is differentfrom the traditional ET in that it has to consider the influenceof the flow resistance

The BPV is a variable flow resistance device whichprovides needed pressure based on the position of the valveplate It consists of a brushed DC motor with permanentmagnets linked to the valve shaft by means of a set of spurgearbox a nonlinear return spring system a valve plate andtwo redundant potentiometers (Figure 1) [13]

21 MechanismModeling Themotor voltage 119864119886is defined as

the control input and the BPV angle 120579 as the output where 120579is the angle between valve plate and vertical direction of inletflow shown in Figure 2The angle of the valve is between 105∘and 1015∘ due to the physical limit and the effective openingrange is 0ndash91∘

The BPV is driven by a brushed DC motor The motormodel can be described by the following equations [14]

119864119886= 119877119886119894119886+ 119871119886

119889119894119886

119889119905+ 119881119887 (1)

119881119887= 119896119887

120579119898 (2)

119879119898= 119896119905119894119886 (3)

where119881119887is the back electromotive force and the value of119871

119886is

so tiny that it can be ignoredThe details of other variables are

120579m

120579 Returnspring

Position sensor

Reduction gear Valve

Ea

Ra LaM

Figure 1 The structure of back-pressure valve

120579

Figure 2 The angle of the BPV

defined in the Nomenclature part The mechanical equationsof the motion are described by

119869119898

120579119898= minus119896119898

120579119898minus 119879119871+ 119879119898

119869119892

120579 = minus119879sp minus 119879tf + 119899119879119871(4)

where 119899 = 120579119898120579 and 119879sp and 119879tf are described in the

following

119879sp =

119896op (120579 minus 1205790) + 119896pre 1205790lt 120579 lt 120579op

119896cl (120579 minus 1205790) minus 119896pre 120579cl lt 120579 lt 12057900 120579 = 120579

0

119879tf = 119896119891120579 + 119896tf sgn ( 120579)

(5)

When considering previous equations (1)ndash(5) the equa-tion ofmotion of the electromechanical system can bewrittenin form

119869 120579 + 1198992

119896119898

120579 + 119879sp + 119879tf = 119899119896119905119894119886 (6)

where 119869 is the inertial moment in normalized units and 119869 =119869119892+ 1198992

119869119898

If we define1199091= 120579119909

2= 120579 as state variables 119906 isin [minus119864

119886 119864119886]

as the controlled input then the system can be described by

[1

2

] = [

[

0 1

minus

119896sp

119869minus119886

]

]

[1199091

1199092

] + [

[

0

119899119896119905

119869119877119886

]

]

119906 + [0

119887 (1199092)] + [

0

1] 119889

(7)

Mathematical Problems in Engineering 3

where 119887(1199092) = (minus119896tf sgn(1199092)+119896sp1205790 minus119896pre sgn(1199091 minus1205790))119869 119886 =

(1198992

119896119898+ 119896119891+ 1198992

119896119887119896119905119877119886)119869 and 119889 stands for the summation

of unmodeled dynamics and unknown disturbances forexample the load torque of inlet flow

22 Fluid Modeling This model is based on the PEMFCmodel proposed by Pukrushpan [15] The mass conservationprinciple is used to develop the outlet manifold model Forany manifold

119889119898

119889119905= 119882in minus119882out (8)

where 119882in and 119882out are mass flow rates in and out of themanifold respectively

The temperature of the stack has been well regulated to aconstant by the thermal management of the PEMFC There-fore the change of air temperature in the outlet manifold isnegligible and thus the pressure can be determined by

119889119901om119889119905

=

119877119892119879om

119881om

119889119898

119889119905 (9)

The outlet mass flow is governed by a BPV The BPVis a variable flow resistance device which provides neededpressure based on the position of the valve plate The flow isa function of manifold pressure 119901om the valve angle 120579 thediameter of the BPV 119863 and the downstream pressure of themanifold 119901atm Thus the air flow through the BPV satisfies[16]

119876119873= 285119860119870

11988101198632radic1199012Δ119901

119878 (10)

1198701198810=1 minus cos 120579radic119890cos

2120579

(11)

Δ119901 =

1199011minus 1199012

1199012

1199012

gt 05

1199012

1199012

1199012

le 05

(12)

where 1198701198810

is the flow coefficient which is related to theopening of the BPV and 0 le 119870

1198810le 1119863 is the valve diameter

119860 is the structure constant which can be set to 62 for ourBPV [16] Δ119901 (kgfcm2) is the pressure drop after the valve(which is less than half of the inlet pressure) 119901

1= 119901om

1199012= 119901atmWhen the pressure drop exceeds the limit the flow

dynamic will be choked which may lead to the instability ofthe flow rate [17] Choked phenomenon is shown in Figure 3where Δ119901cr is the critical pressure When pressure differenceexceeds the critical pressure for example Δ119901 = Δ119901

119886 the

actual flow is 119876max where 1198761015840

max ≫ 119876max

3 Parameter Identification

The valve parameters are usually unknown or cannot beaccurately measured Advanced nonlinear control algorithmrequires modeling of the plant and good estimate of itsparametersThis section provides some techniques to identifythe parameters of the aforementioned model [18]

radicΔparadicΔp

Q

Qmax

radicΔpcr

Q998400max

Figure 3 Choked phenomenon

Monitor Plant

DAQ cardTarget computer

Host computer

Figure 4 xPC Target platform

31 xPC Target The real-time simulation of rapid controlprototyping (RCP) is that the real plant is controlled bysimulating model For our RCP we have used xPC Targetsystem (Figure 4) Various tests are conducted to estimate theparameters on the platform

32 Characteristic Curve The plant has two potentiometerssupplied by 5-volt DC The calibration curves of ThrottlePosition Sensor (TPS) 1 and TPS2 are shown in Figure 5The angle of the valve is between 105∘ and 1015∘ due to thephysical limit When there is no voltage applied to the drivenmotor the output voltage of TPS1 is 083V so the defaultposition 120579

0is 175∘

33 Identification of 119877119886 There is nearly no rotation when a

slowly changing voltage is applied to the motor at the almostclosed position [19] Since the motor current changes slowlythe voltage across the inductance can be neglected Besidesthe inner resistance of the storage can be ignored Thereforethe armature circuit equations (1) and (2) can be described by

119864119886= 119877119886119894119886 (13)

Figure 6 displays the captured curves of themotor voltageagainst the motor current Due to the good symmetry ofthe H-bridge the direction of the current and voltage is


105 1015 105

5485

Volta

ge (V

)

05

Potentiometer track angel (∘)

Lead potentiometer

Gradient plusmn 00476

TPS 1TPS 2

Figure 5 Characteristic curve of angle sensors

Curr

ent (

A)

Voltage (V)

MeasuredApproximated

1

05

005 1 15

15

2 25

Figure 6 Motor current versus voltage

neglected The inverse of the slope represents the armatureresistance which reads 119877

119886= 20Ω

34 Identification of 119896119887 The circuit equations (1) and (2) can

be rewritten as

119881119887= 119899119896119887

120579 = 119864119886minus 119877119886119894119886 (14)

where 119899 = 2068 is the total gear ratio of the two set reductiongearThe back-EMF voltage is generated when the valve shaftis rotating Figure 7 shows the captured back-EMF voltageand the platersquos angular velocity

We get the back electromotive force constant 119896119887

=

00184V(rads) at the time when the platersquos angular velocityis steady Usually 119896

119887and 119896119905are almost the same for onemotor

35 Static Load Test Figure 8 shows the typical systemresponse when inputting sinusoidal signal If the frequency

2

1

0

minus1

minus2

minus3

minus4

minus51211510510 1195985

Angle (rad)

Back-EMF voltage (V)

Angular velocity (rads)

Time (s)

Figure 7 Velocity versus back-EMF voltage

10 20 30 40 50 60 70 80

0

5

10 20 30 40 50 60 70 80

0

2

4

10 20 30 40 50 60 70 800

1

2

minus5

minus2

minus4

Inpu

t vol

tage

(V)

Curr

ent (

A)

Ang

ular

(rad

)

Time (s)

Figure 8 Static loads test

is low the angular velocity and angular acceleration can beignored Therefore the dynamic equation of the system canbe described by

119896sp (120579 minus 1205790) + 119896pre sgn (120579 minus 1205790) + 119896tf sgn 120579 = 119899119896119905119894119886= 119879 (15)

The response plotted in load torque versus angle charac-teristic is shown in Figure 9 This experiment identifies theparameters 119896sp 119896pre and 119896tf

(1) The inverse slope of the inserted straight lines rep-resented the spring stiffness which reads 119896op =

04387N sdotmrad 119896cl = 73693N sdotmrad(2) The friction torque is half of the deviation between

two inserted straight lines during the opening time orthe closing time 119896tf = 02565N sdotm

(3) The preload torque applied to the valve at the defaultposition is 03829N sdotm

36 Viscous Coefficient Identification In this experiment wetake off the middle gear then the motor is separated from


18

16

14

12

1

08

06

04

02

0minus15 minus1 minus05 0 05 1 15

Ang

le (r

ad)

Load torque (Nm)

Figure 9 Plate angle versus load torque

16

15

14

13

12

11

1

05 1 15 2 25 3 35 4 45 5

times10minus4

Ea minus Ra lowast ia

ApproximatedMeasured

ktlowastkblowasti a

Figure 10 Data fit curve

the rest of the assembly When a DC voltage is applied to themotor the motor can run freely which is given by

119864119886= 119877119886119894119886+ 119896119887

120579119898

119869119898

120579119898= 119896119905119894119886minus 119896119898

120579119898

(16)

When the motor reaches steady velocity (16) can bewritten in form

119864119886minus 119877119886119894119886

119896119905119894119886

=119896119887

119896119898

(17)

Motor voltage along with the armature current is plottedusing Matlab cftool shown in Figure 10

The slope of the line gives the spring constant which reads119896119898= 15times10

minus5NsdotmradsThe viscous coefficient of the plateis small and can be ignored in this section We will identify itin the optimization section

11 12 13 14 15 16

0

2

4

6

8

minus2

minus4

minus6

minus8

minus10

Load torque (Nm)

Angular acceleration (rads2)

t (s)

Figure 11 Angular acceleration versus load torque

Table 1 Parameters value

Parameters Identified valueInitial Optimized

119869 (kgsdotm2) 00225 000297119896119887(Vsdotsrad) 00184 001681

119896cl (Nmrad) 73653 36221119896119891(Nmsdotsrad) mdash 000134

119896119898(Nmsdotsrad) 15119890 minus 05 142130119890 minus 05

119896op (Nmrad) 04387 021249119896pre (Nm) 03829 05306119896119905(NmsdotAminus1) 00184 002505

119896tf (Nm) 02565 037039119877119886(Ω) 2 19306

1205790(rad) 03053 0302

119899 2068 2068

37 Moment of Inertia Identification When 120579 gt 0 and 120579 gt 1205790

the system can be simplified to

119869 120579 + (1198992

119896119898+ 119896119891) 120579 + 119896tf + 119896sp (120579 minus 1205790) + 119896pre = 119899119896119905119894119886 (18)

Figure 11 shows the captured load torque and angularacceleration calculated by taking derivative of angular veloc-ity and angular velocity is the derivative of the position Themoment of inertia at the steady time of 138 s is 00225 kg sdotm2

38 Optimization Simulink Design Optimization is a tool-box provided by Matlab which can estimate the unknownparameters in a model Due to the approximation of theprevious experiments there are some errors to the modelparametersTherefore we optimized the previous parametersusing parameter estimate tool In the process of optimizationthe initial values are set to the values identified above and therange for each parameter is defined The algorithm used onoptimization is nonlinear least squaremethodTheoptimizedparameters are listed in Table 1

39 Parameters Verification In this section we will com-pare the simulation results with the real measurement


2 4 6 8 10 12 1410

20

30

40

50

60

70

80

90

100

Time (s)

Ang

le (∘

)

Angle simAngle measured

(a) Subjected to sinusoidal input

1 2 3 4 510

20

30

40

50

60

70

80

90

100

110

Time (s)

Ang

le (∘

)


05 15 25 35 45

(b) Subjected to a step input

Figure 12 Parameters verification subjected to sinusoidal and step input

Figure 12 shows the response of the real system versus modelwhen subjected to different control inputs Obviously thesimulations give a good fit to the real outputs

4 Controller Design

As mentioned above the BPV system is featured with strongnonlinearity and two order dynamics Thus the simpleapplication of standard PID controller may not work veryeffectively ADRC is a new control algorithm based on theactive disturbance rejection concept which can cope withthe highly nonlinear dynamics With ADRC the unknowndynamics and disturbances can be actively estimated andcompensated and it makes the feedback controller morerobust In fact systemrsquos equation (7) can be abstracted asthe following second-order nonlinear system with externaldisturbances

1= 1199092

2= 119891 (119909

1 1199092 119905) + 119908 (119905) + 119887

0119906

119910 = 1199091

(19)

where 1199091 1199092are system states 119891 is a smooth function 119908(119905)

is the external disturbance 1198870is the system parameter and 119906

is the control input signal The key of ADRC is to design anextended state observer (ESO) to estimate not only states butalso the ldquototal disturbancerdquo which contains internal uncertaindynamics and external disturbance

ADRC consists of three parts transient profile generatornonlinear state error feedback control law and extended stateobserver Taking second-order controlled object for examplethe structure of active disturbance rejection controller isshown in Figure 13 [20]

41 Transient Profile Generator Reasonable arrangement ofthe transition processmakes the set-value input to the control

Transientprofile

generator

1

2

e1

minuse2

Feedbackcontrol

z1

z2

z3Extended

stateobserver

u0

1b0 b0

uminus minus

minus Plant y

Figure 13The structure of ADRC where V1is the desired trajectory

of the input and V2is its derivative 119911

1 1199112are the estimate of the

states 1199091and 119909

2 and 119911

3is the estimate of the system disturbance 119906

is the static or dynamic feedback control law about 1198901and 1198902 where

1198901= V1minus 1199111 1198902= V2minus 1199112 Since the BPV is a second-order system

therefore the controller is designed based on the two order controlalgorithms

algorithm not mutate but changes according to a certainexpectation For example in the classical control theorypredicting filter is often adopted to process input signalTransient profile generator is a second-order system whichcan arrange smooth transition process according to theobject ability and control necessaryMeanwhile it collects thedifferential signals of each order It is an effective method tosolve the contradiction between quickness and overshoot Adiscrete-time solution for a discrete double integral plant isdescribed by

V1(119896 + 1) = V

1(119896) + ℎ lowast V

2(119896)

V2(119896 + 1) = V

2(119896) + ℎ lowastfhan (119890

1 V2 1199030ℎ0)

1198790= 2radic

119860

1199030

(20)

where 119860 is the step magnitude and 1198790is the rising time

of the input signal ℎ is the sampling period and 1199030and ℎ

0


(ℎ0= 119896ℎ 119896 isin 119885

+ or ℎ0= ℎ) are controllers parameters

which are adjusted accordingly as filter coefficients See theappendix for undefined function fhan(119890

1 V2 1199030ℎ0) (similarly

hereinafter)

42 Extended State Observer Systems are working underdifferent kinds of disturbances among which the ones thathave effects on the output signal are the most important Sowe can separate them from the output signal by defining anew stateThis is done by ESOTheESOprovides the estimateof the unmeasured systemrsquos state and the real-time action ofthe unknown disturbances and then compensates them TheESO can improve the performance of the system which is inthe form of

1198901= V1minus 1199111

1198911= fal (119890

1 05 120575)

1198912= fal (119890

1 025 120575)

1199111(119896 + 1) = 119911

1(119896) + ℎ (119911

2(119896) minus 120573

011198901)

1199112(119896 + 1) = 119911

2(119896) + ℎ (119911

3(119896) minus 120573

021198911+ 0119906)

1199113(119896 + 1) = 119911

3(119896) + ℎ (minus120573

031198912)

(21)

where 1199111 1199112are the estimate of the states 119909

1and 119909

2and 119911

3

is the estimate of the system disturbances 12057301 12057302 and 120573

03

are observer gains and 0represents the nominal value of the

system parameter 1198870 A great number of simulations show

that [20] if 120575 = ℎ 12057301= 1ℎ then 120573

02and 120573

03are functions of

ℎ

43 Feedback Control In the feedback control law 1199060is a

function of static or dynamic state errors 1198901and 1198902

1198901= V1minus 1199111

1198902= V2minus 1199112

1199060= 119896 (119890

1 1198902)

(22)

Disturbance compensation is the following

119906 = 1199060minus1199113

1198870

(23)

44 Parameters Tuning According to the control targetsmade by BOSCH shown in Table 2 [21] 119905

119903lt 01 s 119860 =

[(120579op minus 120579cl) minus 1205790] lowast 92 then 2radic1198601199030lt 01 so 119903

0can be a

value of 800 In addition ℎ = 119905119904100 = 0001(119904) ℎ

0= ℎ

Curves in [20] obtained from simulation experimentsshow that if 120575 = ℎ120573

01= 1ℎ then120573

02and12057303are functions of

ℎ 12057301asymp 116ℎ

15 12057302asymp 186ℎ

22 So we can set 12057302= 16000

12057303

= 500000 Besides the nominal value of the system0= 1198870= 119899119896119905119869119877119886= 90

The PID control law is selected as the state error feedbackcontrol law for our BPV system which is described by

1199060= 1198701199011198901+ 119870119894int

119905

0

1198901119889119905 + 119870

1198891198902 (24)

Table 2 The performance of step response

Performance indicators Position angle4ndash96 96ndash4

Rise time (ms) lt100 lt100Regulation time (ms) lt300 lt300Overshoot permitted () lt06 lt35


Test type IndicatorPositionangle

Rise time(ms)

Regulationtime (ms)

Overshoot()

Modelsimulation

4ndash96 lt70 lt90 lt0296ndash4 lt80 lt100 lt05

Real-time test 4ndash96 lt80 lt130 lt0196ndash4 lt80 lt150 lt025

Based on the PID tuning method the control parametersare 119896119901= 20 119896

119894= 15 and 119896

119889= 10

45 Simulations and Analyses The controller described inprevious sections has been tested by simulation model andthe real BPV system with the identified parameters estimatedin Section 3There are two physical limitations in our systemwhich have been discussed in the previous sections Thenthe maximum opening is 1015∘ and the limp home angle is175∘ As the performance indicators shown in Table 2 theinput signals are designed as firstly 120579(119905) steps down from 98∘(96) to 20∘ (4) staying unchanged for 1 second and thensteps to 98∘ (96) We also design a tracking control input120579(119905) = 59 + 39 sin120587119905

Figures 14 and 15 show the simulation results imple-mented in Simulink The step responses from 4 to 96and from 96 to 4 have nearly the same results Both therise time and the regulation time of the responses are lessthan 100ms and with no overshoot which are fit for thetechnical targetsmade byBOSCH From the simulation resultof the sine test one can see that the tracking error is less than02∘ via ADRC Besides the estimate disturbance is in goodagreement with the real disturbance for both inputs

The real-time ADRC is implemented with xPC Targetshown in Figure 16 where 119896

119901= 20 119896

119894= 5 and 119896

119889= 035

The results show that the controller can satisfy the preconcertcontrol request

The test results of the performance are shown in Table 3

5 Conclusion

In this paper themathematicalmodeling of the BPVhas beenfurther explored and identification algorithms of proposedmodel parameters have been designed based on series ofmeasurements of the real system responses using xPC Targetplatform Due to some unique features in back-pressurevalve model structures parameters identification encounterssome challenging issues Different identification algorithms


5 60

10

20

30

40

50

60

70

80

90

100

Ang

le (d

eg)

Time (s)48 52 54 56 58 62

Input signalSimulation output

(a) Step response

5 6

0

500

1000

minus500

minus1000

minus150048 52 54 56 58 62

Time (s)

Real disturbanceEstimate disturbance

Tota

l dist

urba

nce

(b) Disturbance estimate

Figure 14 Subjected to a step input (96ndash4ndash96)

10 15 20 25 30

0

1

2

15

05

minus05

minus1

minus15

minus2

Time (s)

Dyn

amic

erro

rs (d

eg)

(a) Dynamic errors

5 10 15

0

100

200


Tota

l dist

urba

nce

Time (s)

minus100

minus200

minus300

minus400

minus500

minus600

minus700

minus800


Figure 15 Subjected to a sinusoidal input

are introduced to resolve these issues After that the ADRCalgorithm is introduced The key component is referenceprofile generation which can avoid set point jump so thatthe output of the plant can follow reasonably The secondimportant part of the controller is the extended state observeBoth the model error and external disturbance are regardedas additional external disturbance and they are estimated andcompensated by the extended state observers in real time

Finally the controller has been implemented in the simu-lationmodel and the actual systemThe simulations show thatthe proposed controller can not only achieve a good dynamicand steady performance but also have strong robustnessagainst parametersrsquo disturbance and external disturbanceExtensive researches will be done to combine the BPVmodelwith the PEMFC system model to obtain optimized openingat a given operation


12 13 1410

20

30

40

50

60

70

80

90

100

Time (s)

Step inputxPC target output

Ang

ular

(deg

)

115 125 135 145

(a) Subjected to a step input (4ndash96ndash4)

20 25 30 35 40

0

1

2

Time (s)

15

05

minus05

minus1

minus15

minus2

Dyn

amic

erro

rs (d

eg)

(b) Subjected to a sinusoidal input

Figure 16 xPC Target test

Appendix

Consider

119889 = 119903ℎ2

1198860= ℎ1199092

119910 = 1199091+ 1198860

1198861= radic119889 (119889 + 8

10038161003816100381610038161199101003816100381610038161003816)

1198862= 1198860+sign (119910) (119886

1minus 119889)

2

119886 = (1198860+ 119910) fsg (119910 119889) + 119886

2(1 minus fsg (119910 119889))

fhan (1199091 1199092 119903 ℎ) = minus119903 (

119886

119889) fsg (119886 119889)

minus119903 sign (119886) (1 minus fsg (119886 119889))

(A1)

sign (119909) =

1 (119909 gt 0)

0 (119909 = 0)

minus1 (119909 lt 0)

(A2)

fsg (119909 119889) =(sign (119909 + 119889) minus sign (119909 minus 119889))

2 (A3)

fal (119909 120572 120575) = 119909

1205751minus120572fsg (119909 120575)

+1003816100381610038161003816fdb (119909 120575)

1003816100381610038161003816 |119909|120572 sign (119909)

(A4)

fdb (119909 119889) =sign (119909 + 119889) + sign (119909 minus 119889)

2 (A5)

Nomenclature

119871119886 Armature inductance

119894119886 Armature current119877119886 Armature resistance

119879119898 Motor drive torque

119896119905 Torque constant

119869119892 Inertia of the plate

119869119898 Inertia of the DC motor

119879119871 Load torque

119896pre Spring preload torque1205790 Default angle

119896119898 Viscous coefficient of motor

119896119887 Constant of back electromotive force

119899 Total gear ratio120579119898 Angle of motor

119879sp Spring torque119879tf Frictional torque119896cl Closing stiffness120579cl Closed angle119896op Opening stiffness120579op Opened angle119896119891 Viscous coefficient of plate

119898 Air mass in the manifold119896tf Coulomb friction constant119882 Mass flow rate119901om Manifold pressure119901atm Ambient pressure119881om Outlet manifold volume119876119873 Volume flow rate

119879om Temperature of the gas in the outlet manifold119877g Gas constant of air119878 Gas specific gravity


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the assistance of thereviewers and the support of National Nature Science Foun-dation of China (Grant no 61104077) and National SpecialFund for the Development ofMajor Research Equipment andInstruments (Grant no 2012YQ150256)

References

[1] S Gelfi A G Stefanopoulou J T Pukrushpan and H PengldquoDynamics of low-pressure and high-pressure fuel cell air sup-ply systemsrdquo in Proceedings of the American Control Conferencevol 3 pp 2049ndash2054 June 2003

[2] Y Q Sun S C Xu H S Ni C F Chang K W Ning and YZhao ldquoExperimental research on intake pressure control of highpressure fuel cell enginerdquo Journal of Vehicle Engine no 5 pp22ndash25 2008 (Chinese)

[3] J Larminie A Dicks and M S McDonald Fuel Cell SystemsExplained John Wiley amp Sons Chichester UK 2003

[4] T H Song Electronic Throttle Control Based on BacksteppingJilin University 2009

[5] R Grepl and B Lee ldquoModeling identification and control ofelectronic throttle using dSpace toolsrdquo 2008

[6] R Grepl and B Lee ldquoModeling parameter estimation andnonlinear control of automotive electronic throttle using arapid-control prototyping techniquerdquo International Journal ofAutomotive Technology vol 11 no 4 pp 601ndash610 2010

[7] C Yang ldquoModel-based analysis and tuning of electronic throttlecontrollersrdquo SAE Paper 01-0524 2004

[8] K Nakano U Sawut K Higuchi and Y Okajima ldquoModellingand observer-based sliding-mode control of electronic throttlesystemsrdquoTransactions on Electrical Engineering Electronics andCommunications vol 4 no 1 pp 22ndash28 2006

[9] R Grepl ldquoAdaptive composite control of electronic throttleusing local learningmethodrdquo inProceedings of the IEEE Interna-tional Symposium on Industrial Electronics (ISIE rsquo10) pp 58ndash61July 2010

[10] H Chen Y-F Hu H-Z Guo and T-H Song ldquoControl ofelectronic throttle based on backstepping approachrdquo ControlTheory and Applications vol 28 no 4 pp 491ndash503 2011(Chinese)

[11] Y FHuC Li J LiH YGuo P Y Sun andHChen ldquoObserver-based output feedback control of electronic throttlesrdquo ActaAutomatica Sinica vol 37 no 6 pp 746ndash754 2011 (Chinese)

[12] Y Pan U Ozguner and O H Dagci ldquoVariable-structure con-trol of electronic throttle valverdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3899ndash3907 2008

[13] J Deur D Pavkovi andN Peri ldquoAn adaptive nonlinear strategyof electronic throttle controlrdquo in Proceedings of the 2004 SAEWorld Congress 2004

[14] Y Chang ldquoModel-based analysis and tuning of electronicthrottle controllersrdquo in Proceedings of the Society of AutomotiveEngineers World Congress SAE International Detroit MichUSA 2004

[15] J T PukrushpanModeling and Control of Fuel Cell Systems andFuel Processors University of Michigan 2003

[16] B D Fan ldquoFlow coefficient of batterfly valverdquo Fluid Mechanicsno 11 pp 25ndash28 1981 (Chinese)

[17] R Hendricks C R J Simoneau and R C Ehlers ldquoChokedflow of fluid nitrogen with emphasis on the thermodynamiccritical regionrdquo in Advances in Cryogenic Engineering vol 18 ofAdvances in Cryogenic Engineering pp 150ndash161 Springer NewYork NY USA 1973

[18] F X Chen L Liu S G Liu and Y L Fan ldquoModelingand parameters identification of high pressure fuel cell back-pressure valverdquo in Proceedings of the IEEE 9th Conference onIndustrial Electronics and Applications (ICIEA rsquo14) pp 1033ndash1038 Hangzhou China June 2014

[19] R N K Loh T Pornthanomwong J S Pyko A Lee and MN Karsiti ldquoModeling parameters identification and control ofan electronic throttle control (ETC) systemrdquo in Proceedings ofthe International Conference on Intelligent andAdvanced Systems(ICIAS rsquo07) pp 1029ndash1035 Kuala Lumpur Malaysia November2007

[20] J Q Han Active Disturbance Rejection Control TechniqueNational Defense Industry Press 2008

[21] Allgemeine Uberarbeitung Technical Customer InformationThrottle Body for ETC Systems BOSCH Stuttgart Germany2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the general requirements of the state stability Then in [11]the back-stepping controller combined with state observeris designed for the real-time system In [12] a variablestructure concept is utilized based on the sliding modeobserver technique and feedback back-stepping techniqueHowever the above methods have strongmodel dependenceParameters-varying or external disturbances can usuallydeteriorate the control performance Most importantly thesystem parameters used in the simulations are not verifiedwith actual ET model

In this paper a detailed nonlinear dynamic model ofBPV is provided and several algorithms are designed toestimate the unknown parameters The methods used inthe identification process consist of extensive measurementsexperiments and verifications The back-pressure valve usedin this paper is a commercial available ET system made byBosch in type of DV-E5 The rest of the paper is organizedas follows in Section 2 a detailed nonlinear dynamic modelof BPV is provided Next in Section 3 the identification algo-rithms are explained Finally the active disturbance rejectioncontroller (ADRC) is designed based on the mathematicalmodel and the identified parameters in Section 4 After thatthe model controlling effect and real-time controlling effectare presented

2 Modeling of the Back-Pressure Valve

The ET is featured with high nonlinearity The model of EThas been approached which includes the nonlinear returnspring and friction model Yet the BPV model is differentfrom the traditional ET in that it has to consider the influenceof the flow resistance

The BPV is a variable flow resistance device whichprovides needed pressure based on the position of the valveplate It consists of a brushed DC motor with permanentmagnets linked to the valve shaft by means of a set of spurgearbox a nonlinear return spring system a valve plate andtwo redundant potentiometers (Figure 1) [13]

21 MechanismModeling Themotor voltage 119864119886is defined as

the control input and the BPV angle 120579 as the output where 120579is the angle between valve plate and vertical direction of inletflow shown in Figure 2The angle of the valve is between 105∘and 1015∘ due to the physical limit and the effective openingrange is 0ndash91∘

The BPV is driven by a brushed DC motor The motormodel can be described by the following equations [14]

119864119886= 119877119886119894119886+ 119871119886

119889119894119886

119889119905+ 119881119887 (1)

119881119887= 119896119887

120579119898 (2)

119879119898= 119896119905119894119886 (3)

where119881119887is the back electromotive force and the value of119871

119886is

so tiny that it can be ignoredThe details of other variables are

120579m

120579 Returnspring

Position sensor

Reduction gear Valve

Ea

Ra LaM

Figure 1 The structure of back-pressure valve

120579

Figure 2 The angle of the BPV

defined in the Nomenclature part The mechanical equationsof the motion are described by

119869119898

120579119898= minus119896119898

120579119898minus 119879119871+ 119879119898

119869119892

120579 = minus119879sp minus 119879tf + 119899119879119871(4)

where 119899 = 120579119898120579 and 119879sp and 119879tf are described in the

following

119879sp =

119896op (120579 minus 1205790) + 119896pre 1205790lt 120579 lt 120579op

119896cl (120579 minus 1205790) minus 119896pre 120579cl lt 120579 lt 12057900 120579 = 120579

0

119879tf = 119896119891120579 + 119896tf sgn ( 120579)

(5)

When considering previous equations (1)ndash(5) the equa-tion ofmotion of the electromechanical system can bewrittenin form

119869 120579 + 1198992

119896119898

120579 + 119879sp + 119879tf = 119899119896119905119894119886 (6)

where 119869 is the inertial moment in normalized units and 119869 =119869119892+ 1198992

119869119898

If we define1199091= 120579119909

2= 120579 as state variables 119906 isin [minus119864

119886 119864119886]

as the controlled input then the system can be described by

[1

2

] = [

[

0 1

minus

119896sp

119869minus119886

]

]

[1199091

1199092

] + [

[

0

119899119896119905

119869119877119886

]

]

119906 + [0

119887 (1199092)] + [

0

1] 119889

(7)



(1198992

119896119898+ 119896119891+ 1198992




119889119898

119889119905= 119882in minus119882out (8)



119889119901om119889119905

=

119877119892119879om

119881om

119889119898

119889119905 (9)


119876119873= 285119860119870

11988101198632radic1199012Δ119901

119878 (10)


2120579

(11)

Δ119901 =

1199011minus 1199012

1199012

1199012

gt 05

1199012

1199012

1199012

le 05

(12)

where 1198701198810




1= 119901om



119886 the


max ≫ 119876max



radicΔparadicΔp

Q

Qmax

radicΔpcr

Q998400max


Monitor Plant


Host computer




0is 175∘



119864119886= 119877119886119894119886 (13)



105 1015 105

5485

Volta

ge (V

)

05


Lead potentiometer


TPS 1TPS 2


Curr

ent (

A)

Voltage (V)


1

05

005 1 15

15

2 25



119886= 20Ω


be rewritten as

119881119887= 119899119896119887

120579 = 119864119886minus 119877119886119894119886 (14)



=




2

1

0

minus1

minus2

minus3

minus4

minus51211510510 1195985

Angle (rad)



Time (s)


10 20 30 40 50 60 70 80

0

5

10 20 30 40 50 60 70 80

0

2

4

10 20 30 40 50 60 70 800

1

2

minus5

minus2

minus4

Inpu

t vol

tage

(V)

Curr

ent (

A)

Ang

ular

(rad

)

Time (s)











18

16

14

12

1

08

06

04

02


Ang

le (r

ad)

Load torque (Nm)


16

15

14

13

12

11

1

05 1 15 2 25 3 35 4 45 5

times10minus4



ktlowastkblowasti a



119864119886= 119877119886119894119886+ 119896119887

120579119898

119869119898

120579119898= 119896119905119894119886minus 119896119898

120579119898

(16)


119864119886minus 119877119886119894119886

119896119905119894119886

=119896119887

119896119898

(17)




11 12 13 14 15 16

0

2

4

6

8

minus2

minus4

minus6

minus8

minus10

Load torque (Nm)


t (s)








119896tf (Nm) 02565 037039119877119886(Ω) 2 19306

1205790(rad) 03053 0302

119899 2068 2068



119869 120579 + (1198992

119896119898+ 119896119891) 120579 + 119896tf + 119896sp (120579 minus 1205790) + 119896pre = 119899119896119905119894119886 (18)





2 4 6 8 10 12 1410

20

30

40

50

60

70

80

90

100

Time (s)

Ang

le (∘

)



1 2 3 4 510

20

30

40

50

60

70

80

90

100

110

Time (s)

Ang

le (∘

)


05 15 25 35 45




4 Controller Design


1= 1199092

2= 119891 (119909

1 1199092 119905) + 119908 (119905) + 119887

0119906

119910 = 1199091

(19)






Transientprofile

generator

1

2

e1

minuse2

Feedbackcontrol

z1

z2

z3Extended

stateobserver

u0

1b0 b0

uminus minus

minus Plant y




states 1199091and 119909

2 and 119911






V1(119896 + 1) = V

1(119896) + ℎ lowast V

2(119896)

V2(119896 + 1) = V

2(119896) + ℎ lowastfhan (119890

1 V2 1199030ℎ0)

1198790= 2radic

119860

1199030

(20)



0


(ℎ0= 119896ℎ 119896 isin 119885




hereinafter)


1198901= V1minus 1199111

1198911= fal (119890

1 05 120575)

1198912= fal (119890

1 025 120575)

1199111(119896 + 1) = 119911

1(119896) + ℎ (119911

2(119896) minus 120573

011198901)

1199112(119896 + 1) = 119911

2(119896) + ℎ (119911

3(119896) minus 120573

021198911+ 0119906)

1199113(119896 + 1) = 119911

3(119896) + ℎ (minus120573

031198912)

(21)


1and 119909

2and 119911

3


03



that [20] if 120575 = ℎ 12057301= 1ℎ then 120573

02and 120573

03are functions of

ℎ



1198901= V1minus 1199111

1198902= V2minus 1199112

1199060= 119896 (119890

1 1198902)

(22)


119906 = 1199060minus1199113

1198870

(23)


119903lt 01 s 119860 =


0can be a


0= ℎ


01= 1ℎ then120573


ℎ 12057301asymp 116ℎ

15 12057302asymp 186ℎ

22 So we can set 12057302= 16000

12057303



1199060= 1198701199011198901+ 119870119894int

119905

0

1198901119889119905 + 119870

1198891198902 (24)






Rise time(ms)

Regulationtime (ms)

Overshoot()

Modelsimulation




119894= 15 and 119896

119889= 10




119901= 20 119896

119894= 5 and 119896

119889= 035



5 Conclusion



5 60

10

20

30

40

50

60

70

80

90

100

Ang

le (d

eg)

Time (s)48 52 54 56 58 62


(a) Step response

5 6

0

500

1000

minus500

minus1000

minus150048 52 54 56 58 62

Time (s)


Tota

l dist

urba

nce



10 15 20 25 30

0

1

2

15

05

minus05

minus1

minus15

minus2

Time (s)

Dyn

amic

erro

rs (d

eg)

(a) Dynamic errors

5 10 15

0

100

200


Tota

l dist

urba

nce

Time (s)

minus100

minus200

minus300

minus400

minus500

minus600

minus700

minus800






12 13 1410

20

30

40

50

60

70

80

90

100

Time (s)


Ang

ular

(deg

)

115 125 135 145


20 25 30 35 40

0

1

2

Time (s)

15

05

minus05

minus1

minus15

minus2

Dyn

amic

erro

rs (d

eg)



Appendix

Consider

119889 = 119903ℎ2

1198860= ℎ1199092

119910 = 1199091+ 1198860

1198861= radic119889 (119889 + 8

10038161003816100381610038161199101003816100381610038161003816)

1198862= 1198860+sign (119910) (119886

1minus 119889)

2

119886 = (1198860+ 119910) fsg (119910 119889) + 119886

2(1 minus fsg (119910 119889))

fhan (1199091 1199092 119903 ℎ) = minus119903 (

119886

119889) fsg (119886 119889)


(A1)

sign (119909) =

1 (119909 gt 0)

0 (119909 = 0)

minus1 (119909 lt 0)

(A2)


2 (A3)

fal (119909 120572 120575) = 119909

1205751minus120572fsg (119909 120575)

+1003816100381610038161003816fdb (119909 120575)

1003816100381610038161003816 |119909|120572 sign (119909)

(A4)


2 (A5)

Nomenclature


















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(1198992

119896119898+ 119896119891+ 1198992




119889119898

119889119905= 119882in minus119882out (8)



119889119901om119889119905

=

119877119892119879om

119881om

119889119898

119889119905 (9)


119876119873= 285119860119870

11988101198632radic1199012Δ119901

119878 (10)


2120579

(11)

Δ119901 =

1199011minus 1199012

1199012

1199012

gt 05

1199012

1199012

1199012

le 05

(12)

where 1198701198810




1= 119901om



119886 the


max ≫ 119876max



radicΔparadicΔp

Q

Qmax

radicΔpcr

Q998400max


Monitor Plant


Host computer




0is 175∘



119864119886= 119877119886119894119886 (13)



105 1015 105

5485

Volta

ge (V

)

05


Lead potentiometer


TPS 1TPS 2


Curr

ent (

A)

Voltage (V)


1

05

005 1 15

15

2 25



119886= 20Ω


be rewritten as

119881119887= 119899119896119887

120579 = 119864119886minus 119877119886119894119886 (14)



=




2

1

0

minus1

minus2

minus3

minus4

minus51211510510 1195985

Angle (rad)



Time (s)


10 20 30 40 50 60 70 80

0

5

10 20 30 40 50 60 70 80

0

2

4

10 20 30 40 50 60 70 800

1

2

minus5

minus2

minus4

Inpu

t vol

tage

(V)

Curr

ent (

A)

Ang

ular

(rad

)

Time (s)











18

16

14

12

1

08

06

04

02


Ang

le (r

ad)

Load torque (Nm)


16

15

14

13

12

11

1

05 1 15 2 25 3 35 4 45 5

times10minus4



ktlowastkblowasti a



119864119886= 119877119886119894119886+ 119896119887

120579119898

119869119898

120579119898= 119896119905119894119886minus 119896119898

120579119898

(16)


119864119886minus 119877119886119894119886

119896119905119894119886

=119896119887

119896119898

(17)




11 12 13 14 15 16

0

2

4

6

8

minus2

minus4

minus6

minus8

minus10

Load torque (Nm)


t (s)








119896tf (Nm) 02565 037039119877119886(Ω) 2 19306

1205790(rad) 03053 0302

119899 2068 2068



119869 120579 + (1198992

119896119898+ 119896119891) 120579 + 119896tf + 119896sp (120579 minus 1205790) + 119896pre = 119899119896119905119894119886 (18)





2 4 6 8 10 12 1410

20

30

40

50

60

70

80

90

100

Time (s)

Ang

le (∘

)



1 2 3 4 510

20

30

40

50

60

70

80

90

100

110

Time (s)

Ang

le (∘

)


05 15 25 35 45




4 Controller Design


1= 1199092

2= 119891 (119909

1 1199092 119905) + 119908 (119905) + 119887

0119906

119910 = 1199091

(19)






Transientprofile

generator

1

2

e1

minuse2

Feedbackcontrol

z1

z2

z3Extended

stateobserver

u0

1b0 b0

uminus minus

minus Plant y




states 1199091and 119909

2 and 119911






V1(119896 + 1) = V

1(119896) + ℎ lowast V

2(119896)

V2(119896 + 1) = V

2(119896) + ℎ lowastfhan (119890

1 V2 1199030ℎ0)

1198790= 2radic

119860

1199030

(20)



0


(ℎ0= 119896ℎ 119896 isin 119885




hereinafter)


1198901= V1minus 1199111

1198911= fal (119890

1 05 120575)

1198912= fal (119890

1 025 120575)

1199111(119896 + 1) = 119911

1(119896) + ℎ (119911

2(119896) minus 120573

011198901)

1199112(119896 + 1) = 119911

2(119896) + ℎ (119911

3(119896) minus 120573

021198911+ 0119906)

1199113(119896 + 1) = 119911

3(119896) + ℎ (minus120573

031198912)

(21)


1and 119909

2and 119911

3


03



that [20] if 120575 = ℎ 12057301= 1ℎ then 120573

02and 120573

03are functions of

ℎ



1198901= V1minus 1199111

1198902= V2minus 1199112

1199060= 119896 (119890

1 1198902)

(22)


119906 = 1199060minus1199113

1198870

(23)


119903lt 01 s 119860 =


0can be a


0= ℎ


01= 1ℎ then120573


ℎ 12057301asymp 116ℎ

15 12057302asymp 186ℎ

22 So we can set 12057302= 16000

12057303



1199060= 1198701199011198901+ 119870119894int

119905

0

1198901119889119905 + 119870

1198891198902 (24)






Rise time(ms)

Regulationtime (ms)

Overshoot()

Modelsimulation




119894= 15 and 119896

119889= 10




119901= 20 119896

119894= 5 and 119896

119889= 035



5 Conclusion



5 60

10

20

30

40

50

60

70

80

90

100

Ang

le (d

eg)

Time (s)48 52 54 56 58 62


(a) Step response

5 6

0

500

1000

minus500

minus1000

minus150048 52 54 56 58 62

Time (s)


Tota

l dist

urba

nce



10 15 20 25 30

0

1

2

15

05

minus05

minus1

minus15

minus2

Time (s)

Dyn

amic

erro

rs (d

eg)

(a) Dynamic errors

5 10 15

0

100

200


Tota

l dist

urba

nce

Time (s)

minus100

minus200

minus300

minus400

minus500

minus600

minus700

minus800






12 13 1410

20

30

40

50

60

70

80

90

100

Time (s)


Ang

ular

(deg

)

115 125 135 145


20 25 30 35 40

0

1

2

Time (s)

15

05

minus05

minus1

minus15

minus2

Dyn

amic

erro

rs (d

eg)



Appendix

Consider

119889 = 119903ℎ2

1198860= ℎ1199092

119910 = 1199091+ 1198860

1198861= radic119889 (119889 + 8

10038161003816100381610038161199101003816100381610038161003816)

1198862= 1198860+sign (119910) (119886

1minus 119889)

2

119886 = (1198860+ 119910) fsg (119910 119889) + 119886

2(1 minus fsg (119910 119889))

fhan (1199091 1199092 119903 ℎ) = minus119903 (

119886

119889) fsg (119886 119889)


(A1)

sign (119909) =

1 (119909 gt 0)

0 (119909 = 0)

minus1 (119909 lt 0)

(A2)


2 (A3)

fal (119909 120572 120575) = 119909

1205751minus120572fsg (119909 120575)

+1003816100381610038161003816fdb (119909 120575)

1003816100381610038161003816 |119909|120572 sign (119909)

(A4)


2 (A5)

Nomenclature


















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










105 1015 105

5485

Volta

ge (V

)

05


Lead potentiometer


TPS 1TPS 2


Curr

ent (

A)

Voltage (V)


1

05

005 1 15

15

2 25



119886= 20Ω


be rewritten as

119881119887= 119899119896119887

120579 = 119864119886minus 119877119886119894119886 (14)



=




2

1

0

minus1

minus2

minus3

minus4

minus51211510510 1195985

Angle (rad)



Time (s)


10 20 30 40 50 60 70 80

0

5

10 20 30 40 50 60 70 80

0

2

4

10 20 30 40 50 60 70 800

1

2

minus5

minus2

minus4

Inpu

t vol

tage

(V)

Curr

ent (

A)

Ang

ular

(rad

)

Time (s)











18

16

14

12

1

08

06

04

02


Ang

le (r

ad)

Load torque (Nm)


16

15

14

13

12

11

1

05 1 15 2 25 3 35 4 45 5

times10minus4



ktlowastkblowasti a



119864119886= 119877119886119894119886+ 119896119887

120579119898

119869119898

120579119898= 119896119905119894119886minus 119896119898

120579119898

(16)


119864119886minus 119877119886119894119886

119896119905119894119886

=119896119887

119896119898

(17)




11 12 13 14 15 16

0

2

4

6

8

minus2

minus4

minus6

minus8

minus10

Load torque (Nm)


t (s)








119896tf (Nm) 02565 037039119877119886(Ω) 2 19306

1205790(rad) 03053 0302

119899 2068 2068



119869 120579 + (1198992

119896119898+ 119896119891) 120579 + 119896tf + 119896sp (120579 minus 1205790) + 119896pre = 119899119896119905119894119886 (18)





2 4 6 8 10 12 1410

20

30

40

50

60

70

80

90

100

Time (s)

Ang

le (∘

)



1 2 3 4 510

20

30

40

50

60

70

80

90

100

110

Time (s)

Ang

le (∘

)


05 15 25 35 45




4 Controller Design


1= 1199092

2= 119891 (119909

1 1199092 119905) + 119908 (119905) + 119887

0119906

119910 = 1199091

(19)






Transientprofile

generator

1

2

e1

minuse2

Feedbackcontrol

z1

z2

z3Extended

stateobserver

u0

1b0 b0

uminus minus

minus Plant y




states 1199091and 119909

2 and 119911






V1(119896 + 1) = V

1(119896) + ℎ lowast V

2(119896)

V2(119896 + 1) = V

2(119896) + ℎ lowastfhan (119890

1 V2 1199030ℎ0)

1198790= 2radic

119860

1199030

(20)



0


(ℎ0= 119896ℎ 119896 isin 119885




hereinafter)


1198901= V1minus 1199111

1198911= fal (119890

1 05 120575)

1198912= fal (119890

1 025 120575)

1199111(119896 + 1) = 119911

1(119896) + ℎ (119911

2(119896) minus 120573

011198901)

1199112(119896 + 1) = 119911

2(119896) + ℎ (119911

3(119896) minus 120573

021198911+ 0119906)

1199113(119896 + 1) = 119911

3(119896) + ℎ (minus120573

031198912)

(21)


1and 119909

2and 119911

3


03



that [20] if 120575 = ℎ 12057301= 1ℎ then 120573

02and 120573

03are functions of

ℎ



1198901= V1minus 1199111

1198902= V2minus 1199112

1199060= 119896 (119890

1 1198902)

(22)


119906 = 1199060minus1199113

1198870

(23)


119903lt 01 s 119860 =


0can be a


0= ℎ


01= 1ℎ then120573


ℎ 12057301asymp 116ℎ

15 12057302asymp 186ℎ

22 So we can set 12057302= 16000

12057303



1199060= 1198701199011198901+ 119870119894int

119905

0

1198901119889119905 + 119870

1198891198902 (24)






Rise time(ms)

Regulationtime (ms)

Overshoot()

Modelsimulation




119894= 15 and 119896

119889= 10




119901= 20 119896

119894= 5 and 119896

119889= 035



5 Conclusion



5 60

10

20

30

40

50

60

70

80

90

100

Ang

le (d

eg)

Time (s)48 52 54 56 58 62


(a) Step response

5 6

0

500

1000

minus500

minus1000

minus150048 52 54 56 58 62

Time (s)


Tota

l dist

urba

nce



10 15 20 25 30

0

1

2

15

05

minus05

minus1

minus15

minus2

Time (s)

Dyn

amic

erro

rs (d

eg)

(a) Dynamic errors

5 10 15

0

100

200


Tota

l dist

urba

nce

Time (s)

minus100

minus200

minus300

minus400

minus500

minus600

minus700

minus800






12 13 1410

20

30

40

50

60

70

80

90

100

Time (s)


Ang

ular

(deg

)

115 125 135 145


20 25 30 35 40

0

1

2

Time (s)

15

05

minus05

minus1

minus15

minus2

Dyn

amic

erro

rs (d

eg)



Appendix

Consider

119889 = 119903ℎ2

1198860= ℎ1199092

119910 = 1199091+ 1198860

1198861= radic119889 (119889 + 8

10038161003816100381610038161199101003816100381610038161003816)

1198862= 1198860+sign (119910) (119886

1minus 119889)

2

119886 = (1198860+ 119910) fsg (119910 119889) + 119886

2(1 minus fsg (119910 119889))

fhan (1199091 1199092 119903 ℎ) = minus119903 (

119886

119889) fsg (119886 119889)


(A1)

sign (119909) =

1 (119909 gt 0)

0 (119909 = 0)

minus1 (119909 lt 0)

(A2)


2 (A3)

fal (119909 120572 120575) = 119909

1205751minus120572fsg (119909 120575)

+1003816100381610038161003816fdb (119909 120575)

1003816100381610038161003816 |119909|120572 sign (119909)

(A4)


2 (A5)

Nomenclature


















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










18

16

14

12

1

08

06

04

02


Ang

le (r

ad)

Load torque (Nm)


16

15

14

13

12

11

1

05 1 15 2 25 3 35 4 45 5

times10minus4



ktlowastkblowasti a



119864119886= 119877119886119894119886+ 119896119887

120579119898

119869119898

120579119898= 119896119905119894119886minus 119896119898

120579119898

(16)


119864119886minus 119877119886119894119886

119896119905119894119886

=119896119887

119896119898

(17)




11 12 13 14 15 16

0

2

4

6

8

minus2

minus4

minus6

minus8

minus10

Load torque (Nm)


t (s)








119896tf (Nm) 02565 037039119877119886(Ω) 2 19306

1205790(rad) 03053 0302

119899 2068 2068



119869 120579 + (1198992

119896119898+ 119896119891) 120579 + 119896tf + 119896sp (120579 minus 1205790) + 119896pre = 119899119896119905119894119886 (18)





2 4 6 8 10 12 1410

20

30

40

50

60

70

80

90

100

Time (s)

Ang

le (∘

)



1 2 3 4 510

20

30

40

50

60

70

80

90

100

110

Time (s)

Ang

le (∘

)


05 15 25 35 45




4 Controller Design


1= 1199092

2= 119891 (119909

1 1199092 119905) + 119908 (119905) + 119887

0119906

119910 = 1199091

(19)






Transientprofile

generator

1

2

e1

minuse2

Feedbackcontrol

z1

z2

z3Extended

stateobserver

u0

1b0 b0

uminus minus

minus Plant y




states 1199091and 119909

2 and 119911






V1(119896 + 1) = V

1(119896) + ℎ lowast V

2(119896)

V2(119896 + 1) = V

2(119896) + ℎ lowastfhan (119890

1 V2 1199030ℎ0)

1198790= 2radic

119860

1199030

(20)



0


(ℎ0= 119896ℎ 119896 isin 119885




hereinafter)


1198901= V1minus 1199111

1198911= fal (119890

1 05 120575)

1198912= fal (119890

1 025 120575)

1199111(119896 + 1) = 119911

1(119896) + ℎ (119911

2(119896) minus 120573

011198901)

1199112(119896 + 1) = 119911

2(119896) + ℎ (119911

3(119896) minus 120573

021198911+ 0119906)

1199113(119896 + 1) = 119911

3(119896) + ℎ (minus120573

031198912)

(21)


1and 119909

2and 119911

3


03



that [20] if 120575 = ℎ 12057301= 1ℎ then 120573

02and 120573

03are functions of

ℎ



1198901= V1minus 1199111

1198902= V2minus 1199112

1199060= 119896 (119890

1 1198902)

(22)


119906 = 1199060minus1199113

1198870

(23)


119903lt 01 s 119860 =


0can be a


0= ℎ


01= 1ℎ then120573


ℎ 12057301asymp 116ℎ

15 12057302asymp 186ℎ

22 So we can set 12057302= 16000

12057303



1199060= 1198701199011198901+ 119870119894int

119905

0

1198901119889119905 + 119870

1198891198902 (24)






Rise time(ms)

Regulationtime (ms)

Overshoot()

Modelsimulation




119894= 15 and 119896

119889= 10




119901= 20 119896

119894= 5 and 119896

119889= 035



5 Conclusion



5 60

10

20

30

40

50

60

70

80

90

100

Ang

le (d

eg)

Time (s)48 52 54 56 58 62


(a) Step response

5 6

0

500

1000

minus500

minus1000

minus150048 52 54 56 58 62

Time (s)


Tota

l dist

urba

nce



10 15 20 25 30

0

1

2

15

05

minus05

minus1

minus15

minus2

Time (s)

Dyn

amic

erro

rs (d

eg)

(a) Dynamic errors

5 10 15

0

100

200


Tota

l dist

urba

nce

Time (s)

minus100

minus200

minus300

minus400

minus500

minus600

minus700

minus800






12 13 1410

20

30

40

50

60

70

80

90

100

Time (s)


Ang

ular

(deg

)

115 125 135 145


20 25 30 35 40

0

1

2

Time (s)

15

05

minus05

minus1

minus15

minus2

Dyn

amic

erro

rs (d

eg)



Appendix

Consider

119889 = 119903ℎ2

1198860= ℎ1199092

119910 = 1199091+ 1198860

1198861= radic119889 (119889 + 8

10038161003816100381610038161199101003816100381610038161003816)

1198862= 1198860+sign (119910) (119886

1minus 119889)

2

119886 = (1198860+ 119910) fsg (119910 119889) + 119886

2(1 minus fsg (119910 119889))

fhan (1199091 1199092 119903 ℎ) = minus119903 (

119886

119889) fsg (119886 119889)


(A1)

sign (119909) =

1 (119909 gt 0)

0 (119909 = 0)

minus1 (119909 lt 0)

(A2)


2 (A3)

fal (119909 120572 120575) = 119909

1205751minus120572fsg (119909 120575)

+1003816100381610038161003816fdb (119909 120575)

1003816100381610038161003816 |119909|120572 sign (119909)

(A4)


2 (A5)

Nomenclature


















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2 4 6 8 10 12 1410

20

30

40

50

60

70

80

90

100

Time (s)

Ang

le (∘

)



1 2 3 4 510

20

30

40

50

60

70

80

90

100

110

Time (s)

Ang

le (∘

)


05 15 25 35 45




4 Controller Design


1= 1199092

2= 119891 (119909

1 1199092 119905) + 119908 (119905) + 119887

0119906

119910 = 1199091

(19)






Transientprofile

generator

1

2

e1

minuse2

Feedbackcontrol

z1

z2

z3Extended

stateobserver

u0

1b0 b0

uminus minus

minus Plant y




states 1199091and 119909

2 and 119911






V1(119896 + 1) = V

1(119896) + ℎ lowast V

2(119896)

V2(119896 + 1) = V

2(119896) + ℎ lowastfhan (119890

1 V2 1199030ℎ0)

1198790= 2radic

119860

1199030

(20)



0


(ℎ0= 119896ℎ 119896 isin 119885




hereinafter)


1198901= V1minus 1199111

1198911= fal (119890

1 05 120575)

1198912= fal (119890

1 025 120575)

1199111(119896 + 1) = 119911

1(119896) + ℎ (119911

2(119896) minus 120573

011198901)

1199112(119896 + 1) = 119911

2(119896) + ℎ (119911

3(119896) minus 120573

021198911+ 0119906)

1199113(119896 + 1) = 119911

3(119896) + ℎ (minus120573

031198912)

(21)


1and 119909

2and 119911

3


03



that [20] if 120575 = ℎ 12057301= 1ℎ then 120573

02and 120573

03are functions of

ℎ



1198901= V1minus 1199111

1198902= V2minus 1199112

1199060= 119896 (119890

1 1198902)

(22)


119906 = 1199060minus1199113

1198870

(23)


119903lt 01 s 119860 =


0can be a


0= ℎ


01= 1ℎ then120573


ℎ 12057301asymp 116ℎ

15 12057302asymp 186ℎ

22 So we can set 12057302= 16000

12057303



1199060= 1198701199011198901+ 119870119894int

119905

0

1198901119889119905 + 119870

1198891198902 (24)






Rise time(ms)

Regulationtime (ms)

Overshoot()

Modelsimulation




119894= 15 and 119896

119889= 10




119901= 20 119896

119894= 5 and 119896

119889= 035



5 Conclusion



5 60

10

20

30

40

50

60

70

80

90

100

Ang

le (d

eg)

Time (s)48 52 54 56 58 62


(a) Step response

5 6

0

500

1000

minus500

minus1000

minus150048 52 54 56 58 62

Time (s)


Tota

l dist

urba

nce



10 15 20 25 30

0

1

2

15

05

minus05

minus1

minus15

minus2

Time (s)

Dyn

amic

erro

rs (d

eg)

(a) Dynamic errors

5 10 15

0

100

200


Tota

l dist

urba

nce

Time (s)

minus100

minus200

minus300

minus400

minus500

minus600

minus700

minus800






12 13 1410

20

30

40

50

60

70

80

90

100

Time (s)


Ang

ular

(deg

)

115 125 135 145


20 25 30 35 40

0

1

2

Time (s)

15

05

minus05

minus1

minus15

minus2

Dyn

amic

erro

rs (d

eg)



Appendix

Consider

119889 = 119903ℎ2

1198860= ℎ1199092

119910 = 1199091+ 1198860

1198861= radic119889 (119889 + 8

10038161003816100381610038161199101003816100381610038161003816)

1198862= 1198860+sign (119910) (119886

1minus 119889)

2

119886 = (1198860+ 119910) fsg (119910 119889) + 119886

2(1 minus fsg (119910 119889))

fhan (1199091 1199092 119903 ℎ) = minus119903 (

119886

119889) fsg (119886 119889)


(A1)

sign (119909) =

1 (119909 gt 0)

0 (119909 = 0)

minus1 (119909 lt 0)

(A2)


2 (A3)

fal (119909 120572 120575) = 119909

1205751minus120572fsg (119909 120575)

+1003816100381610038161003816fdb (119909 120575)

1003816100381610038161003816 |119909|120572 sign (119909)

(A4)


2 (A5)

Nomenclature


















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(ℎ0= 119896ℎ 119896 isin 119885




hereinafter)


1198901= V1minus 1199111

1198911= fal (119890

1 05 120575)

1198912= fal (119890

1 025 120575)

1199111(119896 + 1) = 119911

1(119896) + ℎ (119911

2(119896) minus 120573

011198901)

1199112(119896 + 1) = 119911

2(119896) + ℎ (119911

3(119896) minus 120573

021198911+ 0119906)

1199113(119896 + 1) = 119911

3(119896) + ℎ (minus120573

031198912)

(21)


1and 119909

2and 119911

3


03



that [20] if 120575 = ℎ 12057301= 1ℎ then 120573

02and 120573

03are functions of

ℎ



1198901= V1minus 1199111

1198902= V2minus 1199112

1199060= 119896 (119890

1 1198902)

(22)


119906 = 1199060minus1199113

1198870

(23)


119903lt 01 s 119860 =


0can be a


0= ℎ


01= 1ℎ then120573


ℎ 12057301asymp 116ℎ

15 12057302asymp 186ℎ

22 So we can set 12057302= 16000

12057303



1199060= 1198701199011198901+ 119870119894int

119905

0

1198901119889119905 + 119870

1198891198902 (24)






Rise time(ms)

Regulationtime (ms)

Overshoot()

Modelsimulation




119894= 15 and 119896

119889= 10




119901= 20 119896

119894= 5 and 119896

119889= 035



5 Conclusion



5 60

10

20

30

40

50

60

70

80

90

100

Ang

le (d

eg)

Time (s)48 52 54 56 58 62


(a) Step response

5 6

0

500

1000

minus500

minus1000

minus150048 52 54 56 58 62

Time (s)


Tota

l dist

urba

nce



10 15 20 25 30

0

1

2

15

05

minus05

minus1

minus15

minus2

Time (s)

Dyn

amic

erro

rs (d

eg)

(a) Dynamic errors

5 10 15

0

100

200


Tota

l dist

urba

nce

Time (s)

minus100

minus200

minus300

minus400

minus500

minus600

minus700

minus800






12 13 1410

20

30

40

50

60

70

80

90

100

Time (s)


Ang

ular

(deg

)

115 125 135 145


20 25 30 35 40

0

1

2

Time (s)

15

05

minus05

minus1

minus15

minus2

Dyn

amic

erro

rs (d

eg)



Appendix

Consider

119889 = 119903ℎ2

1198860= ℎ1199092

119910 = 1199091+ 1198860

1198861= radic119889 (119889 + 8

10038161003816100381610038161199101003816100381610038161003816)

1198862= 1198860+sign (119910) (119886

1minus 119889)

2

119886 = (1198860+ 119910) fsg (119910 119889) + 119886

2(1 minus fsg (119910 119889))

fhan (1199091 1199092 119903 ℎ) = minus119903 (

119886

119889) fsg (119886 119889)


(A1)

sign (119909) =

1 (119909 gt 0)

0 (119909 = 0)

minus1 (119909 lt 0)

(A2)


2 (A3)

fal (119909 120572 120575) = 119909

1205751minus120572fsg (119909 120575)

+1003816100381610038161003816fdb (119909 120575)

1003816100381610038161003816 |119909|120572 sign (119909)

(A4)


2 (A5)

Nomenclature


















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 60

10

20

30

40

50

60

70

80

90

100

Ang

le (d

eg)

Time (s)48 52 54 56 58 62


(a) Step response

5 6

0

500

1000

minus500

minus1000

minus150048 52 54 56 58 62

Time (s)


Tota

l dist

urba

nce



10 15 20 25 30

0

1

2

15

05

minus05

minus1

minus15

minus2

Time (s)

Dyn

amic

erro

rs (d

eg)

(a) Dynamic errors

5 10 15

0

100

200


Tota

l dist

urba

nce

Time (s)

minus100

minus200

minus300

minus400

minus500

minus600

minus700

minus800






12 13 1410

20

30

40

50

60

70

80

90

100

Time (s)


Ang

ular

(deg

)

115 125 135 145


20 25 30 35 40

0

1

2

Time (s)

15

05

minus05

minus1

minus15

minus2

Dyn

amic

erro

rs (d

eg)



Appendix

Consider

119889 = 119903ℎ2

1198860= ℎ1199092

119910 = 1199091+ 1198860

1198861= radic119889 (119889 + 8

10038161003816100381610038161199101003816100381610038161003816)

1198862= 1198860+sign (119910) (119886

1minus 119889)

2

119886 = (1198860+ 119910) fsg (119910 119889) + 119886

2(1 minus fsg (119910 119889))

fhan (1199091 1199092 119903 ℎ) = minus119903 (

119886

119889) fsg (119886 119889)


(A1)

sign (119909) =

1 (119909 gt 0)

0 (119909 = 0)

minus1 (119909 lt 0)

(A2)


2 (A3)

fal (119909 120572 120575) = 119909

1205751minus120572fsg (119909 120575)

+1003816100381610038161003816fdb (119909 120575)

1003816100381610038161003816 |119909|120572 sign (119909)

(A4)


2 (A5)

Nomenclature


















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










12 13 1410

20

30

40

50

60

70

80

90

100

Time (s)


Ang

ular

(deg

)

115 125 135 145


20 25 30 35 40

0

1

2

Time (s)

15

05

minus05

minus1

minus15

minus2

Dyn

amic

erro

rs (d

eg)



Appendix

Consider

119889 = 119903ℎ2

1198860= ℎ1199092

119910 = 1199091+ 1198860

1198861= radic119889 (119889 + 8

10038161003816100381610038161199101003816100381610038161003816)

1198862= 1198860+sign (119910) (119886

1minus 119889)

2

119886 = (1198860+ 119910) fsg (119910 119889) + 119886

2(1 minus fsg (119910 119889))

fhan (1199091 1199092 119903 ℎ) = minus119903 (

119886

119889) fsg (119886 119889)


(A1)

sign (119909) =

1 (119909 gt 0)

0 (119909 = 0)

minus1 (119909 lt 0)

(A2)


2 (A3)

fal (119909 120572 120575) = 119909

1205751minus120572fsg (119909 120575)

+1003816100381610038161003816fdb (119909 120575)

1003816100381610038161003816 |119909|120572 sign (119909)

(A4)


2 (A5)

Nomenclature


















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Modeling, Parameters...

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