a design of nonlinear pid controllers with a neural-net based system estimador

8/3/2019 A Design of Nonlinear PID Controllers With a Neural-net Based System Estimador

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A Design of Nonlinear PID Controllerswith a Neural-Net Based System Estimator

Yoshih i r o Ohnish iDepartment of Electrical Engineeringand Information ScienceKure National Co llege of Technology2-2-1I Aga-Minimi, Kure, Hiroshima,

737-8506, JapanEmail: [email protected]

Abstract-" control schemes based on the dassieal controltheory, have been widely used for various process control systemsfor a long time. However, since such processes have nonlinearproperties, it is difficult to determine 'optimal' PID parameters.In this paper, a system identification scheme by using a neuralnetwork is proposed. Furthermore, a PID control scheme basedon the e stimates is considered. According to the newly proposedscheme, it is possible to employ to nonlinear systems. Finally, thebehavior of the newly proposed control scheme is investigated ona numerical simulation example.

I. INTRODUCTIONPID control schemes based on the classical control theory,

have been widely used for various industrial control systemsfo r a long time[l],[2].This is mainly because PID controllershave simple control structures, and simple maintain and tune.However, since such processes have nonlinear properties anduncertainties caused by modeling errors and process fluctu-ation, it is difficult to determine 'optimal' PID parameters.Som e PID control schemes have been proposed based onthe self-tuning control algorithm for the uncertain systems.However, as many self-tuning PID con trol algorithm calculatetheir PID parameters based on the estimates by least squaresmethod, it is difficult to employ for the systems which canbe used the least squares method. In th e real industrial controlsystems, the linear mathematical m odel is used to calculate thePID parameters. The parameters of mathematical model aremade variable. and the se are assigned by using the referencetable generated by the priori information. This procedure canclear on the system properties, and can employ the linearcontrol theory for the nonlinear systems. However, it is verydifficult to generate the reference table. Th e generation of thistable is required cat-and-try for each processes.

On the other hand, the effectiveness of neural networksis discussed for nonlinear systems[3]. Som e control schemesby using the neural network have been proposed up to now[4]. Th e conventional neural-net based co ntrol schemes can be

Tom YamamotoDepartment of Technologyand Information Education

Graduate School of EducationHiroshima University

1-1-1 Kagamiyama. Higashihiroshima,Hiroshima, 739-8524, Japan

Email: [email protected]

classified into the two groups. The one is that control inputis directly calculated by the neural network. As the controlinput is given by the output of the neural network, it is easyto employ for the controlled object. However, it is difficult toexpress the structure of the controller as th e transfer function.The other is that control parameters are calculated by theneural network, and this control input can be calculated bythese control parameters. Although this scheme makes easyto grasp the physical meanings of control parameters, theproperties of controlled object can not be directly understood.Furthermore, these schemes require the information of systemJacobian to update weighting factors of neural networks. It isdifficult to obtain the system Jacobian of such the nonlinearsystems.

In this paper, the neural network is utilized for the purposeof system identification. That is, th e system parameters areestimated by using the neural network. Based on these esti-mates, PID parameters an: determined in an on-line manner.According to the newly proposed scheme, PID parameters areadequately adjusted corresponding to the nonlinear properties.This paper is organized as follows. The neural-net basedsystem estimator is first proposed. Next, the design schemeof PID controller is explained, whose PID parameters areadjusted based on the relation to the generalized minimumvariance control(GMVC). Finally, the effectiveness of thenewly proposed scheme is numerically evaluated on a sim-ulation example.

11. N E U R A L - N E TIASED S Y S T E M ESTIM A TO RConsider a single-input and single-output system described

as the following nonlinear discrete-time model:

0-7803-7906-3/03/$17 oO02003 IEEE. 1938
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i n p u t h i d d e n o u t p u t

u(t-k-1) ----)...u(t-k-m)-

YO-1) +..Fig. 1. Schematic figure of a neural network.

the input to the neuron j and the output from the neuron j ,respectively. A nd f( . ) s the sigmoidal function used by

1. (4)( z ) =__ -Since it is assumed that the absolute values of the systemparameters are under 1.0, the sigmoidal function describedns eqn.(4) can be adopted. The output range of the sigmoidalfunction (4) should be tuned in proportion to the absolute valuein the case w here the absolute value in order 1.0.

The weighting factors included in the neural network areupdated based on the back-propagation method whose cost

21+ e c a 2

function E, is given by:( 5 )1where U an d y denote the control input signal vector and the

corresponding output signal vector, which are given byEn = 2 { ~ ( t ) i(t)}'.

The' update rule of Wkj can be derived by the followingprocedure. First, differentiating E, with Wkj yields= [u(t- k - 1),u(t- - 2 ) , . . u ( t - k - m ) ]

Y = [ y ( t - l ) , y ( t - Z ) , . . . ,Y( t -n ) l . (2)Moreover, k denotes the time-delay, and G denotes a nonlinearfunction.

In this paper, a parameter estimation scheme by using theneural network is proposed. First, the following identificationmodel is introduced:

Furthermore, 6 k is given by(7 )

On the other hand, the update rule of Wji can be derivedaEn am) aok6k :=-- = e @ - .anetk aOk anet,

as follows. Differentiating E, with Wji yieldsB ( t ) = -i iy(t - 1 ) - &y ( t - 2) + bou(t - k, - 1 )(8)aE, aE, &etjaWji &etj aW,,_ - -6,O,,&u(t- ", - 2) + . . .

where+bm,u(t- k , - m' - 1) (3 )

C6kWkj03(1- Oj). (9)where, k , denotes the minimum value of estimated time-delay. 3 ' p &etjIn this paper, the neural network as shown in Fig.1 isconstructed for the parameter estimation. No te that every unit

6 . .- _- =Thus, update rules of Wkj an d Wji are given by

included in the output layer of the neural network correspondsto the estimated parameter. Wkj(t+ 1 ) =Wkj(t)+ j 6 k O k + C?Awk,(t) (lo)

In Fig.1, 0 t the hidden and the output layers mean the Wji(t+ 1 ) = Wj,(t)+ q6jOj +aAWj,(t) (11)neuron shown in Fig.2.12ig.2, wji, netj an d 0, mean the weighing factor

between the neuron i and the neuron j , the summation ofFurthermore, r j an d OL denote the learning rate and the

mOmentUm rate, reSpeCtiVeb'.The system identification can be realized via the above

procedure. Note that according to the proposed scheme, apriori information abou t the system Jacobean is not necessary.This gives us a feature that it is possible to deal with unknowntime-delay system s. In m ost industrial processes, it is relativelydifficult to identify the time-delay exactly. Therefore, this isan advantage in designing process control systems.

According to the newly proposed scheme, variable systemparameters can be obtained corresponding to nonlinear proper-ties. Therefore, by regarding the controlled object a s the linear

neti9i,Fig. 2. Mathematical model of a neuron.1939


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mod el, the controller can b e designed the similar way to linearcontrol scheme. In the following section, PID controller isdesigned, where PID parameters are adjusted based on therelation to the GM VC . where

Next, (12) is rewritten as:C ( z - l ) y ( t )+Au( t )- C(z- ' )w(t)= 0 (20)

111. PID C O N T R O L C(z-1) = + q t - 1 + c2.5-2The following velocity-type PID controller i s employed for

the controlled object given by eqn.(l):Au(t)= k,-e(t), - k,(A + - A 2)y(t), (12)TI TS

where e ( t ) denotes the control error signal given bye ( t ) := r ( t )- (t ) (13)

and k , , TI and TD are the proportional gain, the reset time andthe derivative time, respectively. And , Ts denotes the samplinginterval.

These PID parameters are strongly depend on the controlperformance. The design method is considered based on therelation to the GMVC by following procedure.

First, consider the following cost function to derive aGM VC control Iaw[5]:

J = E[@(t+ k , + I ) ] (14)where

+(t + k, + 1 ) := P ( z - ' ) y ( t + k, + 1 ) + XAu(t)- R ( z - ' ) w ( t ) (15)

In this cost function, X is the weighting factor of the controlinput signal which is th e user-specified parameter. P(2- l ) isthe user-specified polynomial defined as:

P(2-1) = 1 +p*r-' + p 2 t - 2 (16)Furthermore, the polynomial R(z-') is designed by the rela-tionship with PID control law.

Minimizing the cost function J yields the following controllaw:

F ( e - ' ) y ( t ) + {E(z - ' )B(z - ' ) +A}Au(t)-R(t- ' )w(t) = 0 (17)

where F(2- I ) an d E(2-I) is obtained by solving the follow-ing D iophantine equation:P(z-') = AA ( z - ' ) E ( t - ' ) + z - ( ~ * + ' ) Ft I ) (18)

where

Then, by considering the steady property of the controlsystem, E ( z - ~ ) B ( z - ~ )n eqn.(l7) can be replaced to thestatic gain E ( l ) B ( l ) .F ( r - ' ) y ( t )+ { E ( l ) B ( l ) X}Au(t)- R ( z - ' ) w ( t ) = 0

(22)U is defined as:

v := E ( l ) B ( l ) X (23)(22) can be rewritten as:

w(t) = 0 (24)( 2 - l ) R ( z - ' )- y ( t ) + k ( t ) ~v UTherefore, by comparing eqn.(20) and eqn.(24), R(2-I) an dC(2-l) can be designed .as:

R ( 2 - l ) = F( 2 - I )

By this procedure, PID parameters can be calculated basedon GMVC. Therefore, by using eqn.(21) and eqn.(26), PIDparameters can be obtained as the following equations.

1k, = --U1 + 2 j 2 )I/

This algorithm of proposed scheme is summarized below.[Proposed PID control algorithm]

1. Obtain th e input-output date for the learning ofneural network.Design th e structure of identification mod el (3).Learn the system property by using the neural net-work.

2.3.

4. Design P(2-I ) and A.5.6.

Calculate ii nd 21, by using the neural network.Solve E(2- l ) and F ( 2 - l ) based on eqns.(lS).

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7.8. Solve k,, TI an d TDbased on eqn.(27)9. Calculate u(t) rom eqn. ( l2) .10.

Solve U based on eqn.(23).

Update t and return to 5. .IV . SIMULATIONXAMPLE

In orde r to investigate the behavior of the propose d controlscheme, a numerical simulation example is illustrated in thissection.[Case 11

Hammerstein model:Th e controlled object is given by the follow ing second order

y ( t ) = 0.9y ( t - 1 )- . 3y ( t - 2 ) + 0 .4 z ( t - 1)+0.2z ( t - 2 )

z ( t ) = u(t)+ 0.5uZ( t ) . (28)Th e following model w as used for system identification:

&(t )= -81y(t - 1 )-&y(t - 2 )+ i0u(t'- 1 )+ 6,u(t - )(29)

Fig.3 shows the simulation result by using the fixed PIDparameters, whose parameters are k , = 0.17, TI = 1.0 andTo = 0.25. These parameters can be obtained using CHRscheme[2].

The number of units included in the inputs layers and thehidden layers of the neural network were set as 4 and 6. Theparameters used in the training were set as

a = 1.0,q = 0,001, Y = 0.001. (30)This learning was carried on 50000 iterations.

...... . . . . . . . .. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .0 10 n I a Y) 3 n a

2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : ...2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .al . . . . . . . . . . . . . . . . . . . . . . . . : ...... . . . . . . . . . . . ., . .

I 1 0 n a .a Y) a n 10I [*=PI

Fig. 4. Cooml result by using the proposed scheme in Case 1

Fig.4 shows the simulation result by using the proposedscheme. and Fig.5 shows the trajectories of the calculationresults of PID parameters.

From these results, it is clear that the proposed controlscheme can adjust the PID parameters corresponding to thenonlinear property.[Case 21Hamm erstein model:

The controlled object is given by the following third order

y ( t ) = 0.6y ( t - 1)- O.ly( t - ) + 1 . 2 ~ ( t )-O.lz(t - 2)

z ( t ) = 1.5u( t ) + 1.5u2( t )+ 0.5u3( t ) . (31)Fig.6 shows the static property of the controlled object

eqn.(31). From Fig.6, it is clear that this con trolled object hasthe nonlinear property.

Fig.7 shows the simulation result by using the fixed PIDparameters, whose parameters are k , = 0.25, TI = 0.2 an d

... i ..... ..... : ..:.. . . . . . . . . . . :. .. .i . . . . . . . j 0 10 20 a .o Y) I*) 70 a-*.. . : . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . .z . . . . . . . . . . . . . . . . . . . . . . . . . . . .e.., : :. . . . .I-...... . . . . . . . ID I0 10 m .o Y) I*) m a*.r'T. . . . .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .e * : ; 00' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .- :,,,0 10 1 m a Y) a 10 at IWWI 0 70 20 a .O Y) fa 70 mt [step]Fig. 3.TD= 0.25).Control result by fixcd PID parameters(kc = 0.17.TI 1.0 nd Fig. 5 . Trajectoriesof PID parameters io case 1

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input signa1Fig. 6 . Stltic property of controlled object.

TD= 1.0. These parameters ware designed by cut-and-try. Itis difficult to obtain the good control result by the fixed PIDparameters.

Th e following model was used for system identif ication:Y ( t )= --81y(t - 1)- &y(t - ) + 6,U(t - 1)+ 61u(t- )

Fig3 shows the simulation result by using the leastsquares method, whose parameters are i1 = - 1 .2296 ,& =+0.2618,60 = 0.4334,61 = -0.3953. These parameters areestimatied in the neighborhood of 1.0. It is also difficult toobtain the good control result by using LSM.

On the other hand, the learning of the neural network wascar r ied out by using input-output data shown as Fig.9

T h e n um b er of units included in the inputs layers and thehidden layers of the neural netw ork were set as 4 and 10. The

. . . . . . . . . : . . . . . . . . . . . . . j n . . . . . . . . . 481 k..;.. . . . .Kd...;.....\TiL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .*.I

1

1942

. . . . . . .. . . . . ,

10 io YJ 40 11 ea 70 at [rtepl

Fig. 7.T~ = 1.0).Control result by fixed PI D parameters(& = 0.25, TI = 0.2 an d

,~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .-/l ........................ 1. . . . . . . . . . . . . . . . .. . .. .. . .. .. . .. .. . .. . .. . .. . .. . ..

..................................................................

I, I.iM .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .D

.I.

~~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 10 m 5 a D a IO II

I (step1Fig. 8 . Coritrol result by using LSM.

parameters used in the training were set asa=1.5 ,~=0.001,a=0.001. (33)

This learning was carried on 50000 iterations.Fig.10 shows the simulation result by using the proposed

scheme. and Fig.11 shows the trajectories of the calculationresults of PID parameters.

From these results, it is clear that the proposed controlscheme works well for nonlinear systems.

V. CONCLUSIONIn this paper, an identif ication method by using the neural

network and a design method of PID parameters based on theestimates have been considered. Furthermore, the effectivenessof the proposed method has been numerically evaluated on asimulation example.

The main features of the proposed control scheme aresummarized as follows.

( 2 . . . . . . . . . . . . . . . . . . .lo - [ . . . . . . ..-r---. . . .To& .... : . . . . . . . . ."I. . . ~ ,.. ,~ .. ;... .... .... ..... ....... . . ' I . . I . . . .i ...........

o , 2 . .... ~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .m Im m zm 211 91

,, J J. . . . i . . . . . . ... . ; . . . . . . . . ; . . . . . . .3pr'o,I . . . . . . . . . . . . . . . . . ..

YJ Im ,S OI SIeQl

Fig. 9. Input-outputdate sets used for learning


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[41 K. S. Narendra and K. Parthasarathy, Identification and Canhol of Dy-( I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . namical Systems Using Neural Networks, IEEE Trans. Neural Networks.Vol. I , No.1. pp.4-27(1990).[SI D.W.Clarke and P.I.Gawthrop : Self-tuning conholler ; EE Pmc.,V01.122D;No. 9, pp.929-934 (1975)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 1 0 m Y ) a a Y ) 7 o a o

~~

0.6

5.o2r

. . . . . . ., ..............

. . . . .

0 10 m a a Y) UI m EoI stePl

Fig. IO. Control result by using Ule proposed scheme in case 2

. . . . . . . . . . . . . . . . . . .....................>.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. his scheme can be employed fo r the nonlinear systems.PID parameters can be obtained based on the relation tothe GMVC.PID parameters are adjusted corresponding to the nonlin-ear properties.

We are now in employ ing this scheme to a real system . Th econtrol results will be shown in that day.

A C K N O W L E D G M E N TThis research was partially supported by The Japan Society

for the Promotion of Science, G rant-in-Aid for Young Scien-tists (A), 14750382.

R E F E R E N C E S[I 1 1. G. Ziegler and N. B.Nichols. Optimum Setting for Automatic Con-

hollers, Trans. ASME, Vol.@%,p.759-768 (1942)I21 K. L. Chien, J. A. Hrons and J. B. Reswick, On theAutomatic Control ofGeneralized Passive Systems, Trans. AMSE, Vo1.74, pp.175-I85 (1972)[31 S.Haykin, Neural Networks. Macmillan College Publishing, New York,(1994).

O rG>.,

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

........ :. . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

~ , ~ . . .=0 1 . '0 , s -b

. . . . . . . . . . . . :. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . ....; . .

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a design of nonlinear pid controllers with a neural-net based system estimador

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