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Adaptive PID Control Strategy Based on RBF Neural Network Identification Ming-guang Zhang Wen-hui Li Man-qiang Liu School of Electrical and Information Engineering ,Lanzhou university of technology Lanzhou 730050, China E-mail:[email protected], [email protected] Abstract-Radial Basis Function (RBF) neural network(NN) is powerful computational tools that have been used extensively in the areas of pattern recognition, systems modeling and identification. This paper proposes an adaptive PIH) control method based on RBF neural network identification. This approach can on-line identify the controlled plant with the RBF neural network identifier and the weights of the adaptive PED controller are adjusted timely based-on the identification of the plant and self-learning capability of RBFNN. Simulation result shows that the proposed controller has the adaptability, strong robustness and satisfactory control performance in the nonlinear and time varying system. I INTRODUCTION The proportional-integral-derivative (PID) controllers are based on the most common control algorithm. Due to their simplicity and robustness, the PID controllers are widely used in the industrial process control [1]. But conventional PID controller with fixed parameters can hardly adapt to time varying of characteristics in wide range. To improve the control performance, several schemes of self-tuning PID controllers were proposed in the past. Junghui Chen, Tien-Chih Huang proposed using neural network to on-line updated PID controllers for nonlinear process control[2]. Ching-hung Lee,et,al designed a PID controller by fuzzy neural network(FNN)[3]. Radial Basis Function (RBF) network is powerful computational tools that have been used extensively in the areas of pattern recognition, systems modeling and identification. RBF network form a special architecture of neural network, which is characterized by several main advantages: the simplicity of its structure, faster learning algorithms, better approximation capabilities. Due to the popularity of RBF neural network, many researchers have been working during the last decade to develop more efficient training algorithms and applications[4,5,6,7,8]. According to advantages of Radial Basis Function (RBF) Neural Network (NN) and PID controller, the adaptive PID control based on.RBFNN is proposed in this paper.We use self-learning ability of RBFNN to automatically tune and modify the robust PID parameters on-line for the parametric interval system. Simulation on various examples show that the PID controller obtained by using these RBF neural network gives satisfactory results. II. RBF NEURAL NETWORK Radial Basis Function (RBF) NN is a kind of Neural network presented by J.Moody and C.Darken at the end of 1980s [9]. Broomhead and Lowe [10] were the first to exploit the use of radial basis functions in the design of neural networks. Radial functions are a special class of functions, their characteristic feature is that response decreases, or increases, monotonically with distance from a center point[l 1,12]. It is a feed-forward three-layered network with single hidden layer. It simulates the network structure of local readjustment in human brain, covering receptive field with each other. So RBF network is a local approximating one. It was proved that RBF network has the ability of approximating any continuous function with any arbitrary accuracy. A. Structure of RBF Neural Network A RBF network is a three-layer feed-forward neural network[13]. The mapping from input to output is nonlinear, but from hidden layer to output layer is linear. Learning rate is quickened greatly and the problem of local minimum is avoided. A typical RBF network configuration is shown in Fig. 1. hidden layer :1. Atial RBF neworkonfigution. In the structure of RBF Neural Network, the first 0-7803-9422-4/051$20.00 ©2005 IEEE 1854

[IEEE 2005 International Conference on Neural Networks and Brain - Beijing, China (13-15 Oct. 2005)] 2005 International Conference on Neural Networks and Brain - Adaptive PID Control

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Page 1: [IEEE 2005 International Conference on Neural Networks and Brain - Beijing, China (13-15 Oct. 2005)] 2005 International Conference on Neural Networks and Brain - Adaptive PID Control

Adaptive PID Control Strategy Based on RBF

Neural Network IdentificationMing-guang Zhang Wen-hui Li Man-qiang Liu

School of Electrical and Information Engineering ,Lanzhou university of technologyLanzhou 730050, China

E-mail:[email protected], [email protected]

Abstract-Radial Basis Function (RBF) neural network(NN) ispowerful computational tools that have been used extensivelyin the areas of pattern recognition, systems modeling andidentification. This paper proposes an adaptive PIH) controlmethod based on RBF neural network identification. Thisapproach can on-line identify the controlled plant with theRBF neural network identifier and the weights of the adaptivePED controller are adjusted timely based-on the identificationof the plant and self-learning capability of RBFNN. Simulationresult shows that the proposed controller has the adaptability,strong robustness and satisfactory control performance in thenonlinear and time varying system.

I INTRODUCTION

The proportional-integral-derivative (PID) controllersare based on the most common control algorithm. Due totheir simplicity and robustness, the PID controllers arewidely used in the industrial process control [1]. Butconventional PID controller with fixed parameters canhardly adapt to time varying of characteristics in wide range.To improve the control performance, several schemes ofself-tuning PID controllers were proposed in the past.Junghui Chen, Tien-Chih Huang proposed using neuralnetwork to on-line updated PID controllers for nonlinearprocess control[2]. Ching-hung Lee,et,al designed a PIDcontroller by fuzzy neural network(FNN)[3].

Radial Basis Function (RBF) network is powerfulcomputational tools that have been used extensively in theareas of pattern recognition, systems modeling andidentification. RBF network form a special architecture ofneural network, which is characterized by several mainadvantages: the simplicity of its structure, faster learningalgorithms, better approximation capabilities. Due to thepopularity of RBF neural network, many researchers havebeen working during the last decade to develop moreefficient training algorithms and applications[4,5,6,7,8].

According to advantages of Radial Basis Function(RBF) Neural Network (NN) and PID controller, theadaptive PID control based on.RBFNN is proposed in thispaper.We use self-learning ability of RBFNN toautomatically tune and modify the robust PID parameterson-line for the parametric interval system. Simulation onvarious examples show that the PID controller obtained by

using these RBF neural network gives satisfactory results.

II. RBF NEURAL NETWORK

Radial Basis Function (RBF) NN is a kind of Neuralnetwork presented by J.Moody and C.Darken at the end of1980s [9]. Broomhead and Lowe [10] were the first toexploit the use of radial basis functions in the design ofneural networks. Radial functions are a special class offunctions, their characteristic feature is that responsedecreases, or increases, monotonically with distance from acenter point[l 1,12]. It is a feed-forward three-layerednetwork with single hidden layer. It simulates the networkstructure of local readjustment in human brain, coveringreceptive field with each other. So RBF network is a localapproximating one. It was proved that RBF network has theability of approximating any continuous function with anyarbitrary accuracy.

A. Structure ofRBF Neural Network

A RBF network is a three-layer feed-forward neuralnetwork[13]. The mapping from input to output is nonlinear,but from hidden layer to output layer is linear. Learning rateis quickened greatly and the problem of local minimum isavoided. A typical RBF network configuration is shown inFig. 1.

hidden layer

:1. Atial RBF neworkonfigution.

In the structure of RBF Neural Network, the first

0-7803-9422-4/051$20.00 ©2005 IEEE1854

Page 2: [IEEE 2005 International Conference on Neural Networks and Brain - Beijing, China (13-15 Oct. 2005)] 2005 International Conference on Neural Networks and Brain - Adaptive PID Control

layer X =[[xI,x2,.. x]]T is the input vector of the cji(k)=cji(k-1)+qAcji+a(cji(k-1)-cji(k-2)) (8)network. Neurons at the second layer (hidden layer) isactivated by a radial-basis function. Suppose the radialvector of RBF network is H =[h ...h .hmITwhere hj is multivariate Gaussian function.

2

X-C.jhi= exp(- 2b2I

The center vector of the network at node j is

C = [cjl ,c ...ci,-n].. =jIT,-,And we suppose the radial width vector isB = [b1,b2 )...bm ]T, where bj is the radial parameter andb,>O.

The weight vector of the network is WandW = [WI W2 '..Wj*. Wm]T

The network output y is formed by a linearly weightedsum of the number of basis functions in the hidden layer.ym(k)=wO+wihi+w2A+ +wmhm

m=w0+w

j=l

(1)

where TI is learning rate, Cc is momentum gene. Jacobianmatrix (sensitivity of plant output to controlled input)algorithm is as follows.

ay(k) aym (k) = C.wjj izauy(k) aYm(k) -Zw11 bwhere xl = u(k).

III. DESIGN OF ADAPTIVE PID CONTROLLER BASED ON RBFNN

This paper proposes adaptive PID control based onRBF neural network identification. Its structure is shown inFig.2.The PID parameters are modified on-line using ofresults ofRBFNN identification.

(2)

Where wI is the weight and hj is the output of the jthnode in the hidden layer.

B. Identification Algorithm of Controlled Plant

The identification algorithm of Jacobian information ofcontrolled plant is stated below. The performance indexfunction of controller is defined as

i =I

(y(k)-Ym (k))2 (3)'2In order to minimize the error between the

identification model output ym(k) and that of the real planty(k), gradient descent method is adopted here to modifyweights of the output layer, node center and node radialparameters. The corresponding modifier formulas are asfollows.wj (k) = wj (k -1) + q(y(k) -Ym (k))hj

(4)+ a(wj (k - 1) - wj (k - 2))2X-C.

Ab1 = (y(k) -Ym(k))wjhj b3 (5)

bj(k) =b (k-1)+ Aj +a(b (k -1) -bj(k-2)) (6)2

Ac1= (y(k) - ym (k))w1 (7)

Figure2 structure of adaptive PID control based on RBFNN

The algorithm of the basic PID Controller that iscommonly known in the literature is as follows.

u(k) =kpe(k)+k,Ze(j)+kd(e(k)-e(k -1)) (10)j=1

where kp,ki,kd was respectively proportion coefficient,integral coefficient and differential coefficient, and are theinput/output sampling sequence of the control system, y(k)is the controller output, e(k)is the system error ande(k) = r(k) - y(k) (11)The three inputs of PID are as follows.xl (k) = e(k) - e(k - 1) = Ae(k)X2(k) = e(k) (12)x3(k) = e(k) - 2e(k- 1)+e(k- 2)

So the controller output follows as.u(k) = wlx1 (k) + W2x2 (k) + w3x3 (k) (13)where wi is the weight of RBFNN.

The task is to reduce the system average square errorE(k) to zero by adjusting all the variable weights of theneural network. E(k) is defined as

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Page 3: [IEEE 2005 International Conference on Neural Networks and Brain - Beijing, China (13-15 Oct. 2005)] 2005 International Conference on Neural Networks and Brain - Adaptive PID Control

E(k)= 1e2(k) (14)

The adjustment of kp,ki,kd parameters adopts gradientdescent method.

EEayau -ef x

aki d- au akp au

Akd -y --17j =-7e(k) -x 3(k) (17)akd dy au akd au

where is Jacobian information of controlled plant. It isauobtained by RBFNN identification results.

IV. SIMULATION EXPERIMENTAND RESULTS

The simulation adopts a nonlinear discrete system inthis paper, whose transfer function is described by

y(k) = .(18)

1+y2(k - 1)We adopts proposed algorithm and controller in the

simulation. According to Figure 1, RBF network adopts a3-6-1 structure. u(k), y(k) and y(k-J) are the inputs of theRBFNN identification. The PID parameters kp,ki,kd isadjusted by self-learning of RBF neural network until theyapproach zero. Suppose input signal isr(t) = 1.Osgn(sin(2zt)) and thus the output results ofPID controller based on RBFNN identification are shownin Fig.3 and Fig.4. The simulation results is shown theoutputs of the identification network can match the output ofthe closed-loop controlled plant excellently. The adaptivetuning curve of PID controller parameters are shown inFig.5, which indicates kp,ki,kd parameters of PID approachzero.At the same time Jacobian information of identificationis shown in Fig.6 in simulation. We can see that the systemoutput tract the reference input satisfactorily and theperformance of the proposed controller is better than that ofthe conventional PID controller.

O8f

0.6-0.2 .r

0.4

0.2 .

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

0 0.2 0.4 0os 0.8 1 1-2 1.4 1.6 1-8 2

Fig.3 Square wave response of adaptive PID control based on

RBFNN

-A

---k

----1---- I-20 0.2 0.4 0.6 0. 1 1.2 14 1.6 1.8 2

Fig.4 Output curve of controlled plant and RBFNN

01

o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. ...............................

~006~

0 02 0.4 0A 08 1 1.2 1.4 1. 1.8 2

02

0.150.1t

0.0

00 02 0.4 06 08 1 12 14 186 1.8 2

0.1 _

0.06ko.wF

0.02 ...l 1__. __._L... _. _..._._.l._...

0.4 0.6 08 1 1.2 1A 16 1.8 2tins(s

V. CONCLUSIONS Fig.5 Adaptive tuning curve of PID parameters

Conventional PID controller tuned at typical operatingpoint can hardly work well at different operating condition.A novel adaptive PID control strategy based on Radial BasisFunction (RBF) Neural Network (NN) is presented in thispaper. The proposed controller has advantages of bothself-learning capability of neural network and simplicity ofPID controller. During the practice, adaptive controller hasthe superiority such as strong robustness, simple theory.

oit

4, ; £

o X 0.4 0.6 O's 1 II

Fig.6 Jacobian information of identification

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I

1.4 1.6 1I8 2

:1

::I:IiIB

2

-0.

Page 4: [IEEE 2005 International Conference on Neural Networks and Brain - Beijing, China (13-15 Oct. 2005)] 2005 International Conference on Neural Networks and Brain - Adaptive PID Control

Simulation result shows that the proposed controller has theadaptability, strong robustness and satisfactory controlperformance in the nonlinear and time varying system.

ACKNOWLEDGEMENTS

This work was supported by the National 863Scientific Project Development Fundation P.R. China undergrant no.2002BA90128A.

REFERENCES

[1]K. J. Astrom, T. Hagglund, C. C. Hang, and W. K. Ho, "Automatictuning and adaptation for PID controllers A survey," IFAC J. Contr. Eng.Practice, vol. 1, no. 4, pp. 699-714, 1993.

[2]Junghui Chen, Tien-Chih Huang.Applying neural network to on-lineupdated PID controllers for nonlinear process control. Journal ofProcessControl ,No.14 (2004), P211-230.

[31 Ching-hung Lee,Yi-Hshiung Lee.A Novel robust PID controller designby fuzzy neural network. Proceedings of the American Controlconference.May 8-10,2002, ppl561-1566.

[41 A. Ai-Amoudi, L. Zhang, Application of radial basis function networkfor solar-array modeling and maximum power-point prediction, IEEPro.Gener. Transm. Distrib. 147 (2000) 310-316.

[5] F. Yoshiba, N. Ono, Numerical analyses of the internal conditions of amolten carbonate fuel cell stack: comparison of stack performances forvarious gas flow types, J. Power Sources 71 (1998) 328-336.

[6] J.B. Gomm, D.L. Yu, Selecting radial basis function network centerswith recursive orthogonal least squares training, IEEE Trans. NeuralNetwork 11 (2) (2000) 306-314.

[71 Haralambos Sarimveis, Alex Alexandridis. A fast training algorithm forRBF network based on subtractive clustering. IEEE Trans NeuralNetworks, Volume: 51, April, 2003, pp. 501-505.

[81Rank,Erhard. Application of Bayesian trained RBF network to nonlineartime-series modeling. IEEE Trans Neural Networks, Volume: 83, Issue:7, July, 2003, pp. 1393-1410.

[9] Suni V. T. Elanayar, Yung C. Shin, Radial basis unction neural networkfor approximation and estimation of nonlinear stochastic dynamicsystems, IEEE Transaction on Neural Network, Vol. 5,No. 4, pp.584-603,Apr.1994.

[10]Broomhead DS. Multi-variable functional interpolation and adaptivenetworks. Complex System, 1988;2: 321-55.

[11] Bianchini M, Frasconi P, Gori M. Learning without local minima inradial basis function networks. IEEE Trans Neural Networks,1995;6:749-56.

[12]Park J, Sandberg IW. Universal approximation using radial basisfunctions network. Neural Comput, 1991;3: 246-57.

[13] K. Warwick, An introduction to radial basis functions for systemidentification: a comparison with other neural network methods, in:Proceedings of the 35th Conference on Decision and Control, Kobe,Japan, 1996, pp. 464 469.

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