Self Tuning Control of Nonlinear Systems Based on Artificial Neural ...

Self Tuning Control of Nonlinear Systems Based on Artificial Neural Network Models* h.1. Iictilnl Cdzz, Iiarl E2ectiiccil and Cornputer. Syracuse U ni v e T’Si t y,

Abstract

ofthe system dynamics, are extensively stiidied in [ 3 ] , and successfrilly applied for rigid robotic riinnipulator control i n ., If tlie structure of t.he nonlinearities of the system dynamics is not known a priori, a pnmirtelric m o d e l of tlie systein \voultf not be possilile. I n this c a w , there is not yet. a wcll-definec1 at1;tptive control algor i t l i r n that would give satisfactory performance. A n alternative approacli t o the problem would be to rise a high gain robust coiitroller wliich would eventually increase system’s halidwidth. In this paper ive introduce a novel a p p r o a c h for the adaptive/ learniiig cont,rol of n class of nonliiiear systriiis y i t l i riiiknown clyiinrriics. \I’r iiinko i i w of inulti-layer art.ificial n ~ ? i i n networks l i\s (lie rtiotlcl approxiniators and incorp0rat.e tlierii i i i ail algorithm for on-line identificat.iori and coiitrol-. System cliarnct.erizat.ioii ancl idciitilicat i o n 11lny a i i important role in system theory. A dyiiaiiiic systeiii call be characterized i n several foriris one of whicli is the input-out piit iiiappiiig. I11 ~riatliciiiatic~d tcrrris, a system can be described b y an opctator P frolii a normed vector space Id ( i n p u t space), to ;tnot.licr normed vector space Y (i.e. 7 : U -+ 3.’). If the operator P can be approximated Iiy P ovcr it coriipnct. snb-spnce h: c N , i i i the seiise t.liat,

I?].

T h r s paper addresses t h e tracking c o n t m l o f rionlinear s y s t e m s with u n k n o w n d y n a m i c s A self-tuning control architecture i s proposed based o n t h e n o n l i n e a r c o m p u t a t i o n a l properties of artificial neural n e i works. A local convergence anolysis o f the weight update equations is given and t h e n ihe algorithm IS s i m ulated a n d successfully tested for drscrete i i m e nonlinear systems.

1

Introduction

Control of nonlinear systeriis w i t h u i i k i i o w i i or partially known dynamics has been a challenging research area in the past two decades. In the czie of linear systems, globally convergent adaptive control techniques have been established for tracking control problems. In this paper we will concentrate on the self-tuning control of discrete time systems, however the results given here can be easily extended to coiitinous time systems. Various parameter estimation algorithms are shown to be convergent [l, 21, if the system under study can be represented in a forin,

where y ( t ) is the system’s output, 4 ( t ) is a regression vector wliich tlcpeiids on tlic s y s t c i i i ’ s I)rcvioris iiiptit output values, and 00 is a set of pararneters, w h i c h is usually referred to as the parameter vector. If a nonlinear s y s t e m can L e w r i t t c i i in t,lle T o r i 1 1 of ( 1 ) (i.e. linear i n the pnrarrietcr space), then oiie of t h e well known parameter estiniatioii techniques can be used in constructing a self-tuniiig corit roller. i\daptive control of a certaiii class of iionliriear systerrls. based 011 1iiiearizat.ioii mid explicit I);rrniiic.tcrizntion ‘This work is partially sponsored by C+.\XE Center of Syracuse University and CZ’esting1ioris.e Educatioil Foundation.

980 24ACSSC-1219010980 $1.00

@I

1990 MAPLE PRESS

W h a t makes A N N s important tools in signal analysis is their capability of approximating arbitraw nonlinear functionals defined on a corilpact subsct of R iiorined vector space, in the seiise dcsctibed by ( 2 ) . There is an increasing amount of literature on mathematical proofs of this claim [5, 6, 7, 8). Here we will quote a realizability theorem given i n [5]. T11@or@l11 k[5]

Let r(.) be a nonconstant, bounded and monotone increasing continuous function. Let I‘\ be a compact subset of R” and f(z1,. . . , z,&) be a real valued continuous function defined on I 0, there exists a n integer “ni2’ and real constants wj”, ( j = 1,. . . ,na), arid w k ( i = I , . . , , n; j = I , . . . , m ) such t h a t ,

Figure 1: X Generic Systern Identiftcation Scheiric nonlinearly, t h a t is,

where f(.) are assunied t o be a srnoot 11 noriiiiicnr fun c !,io11. 0 t.11c r i I 111) or t a I 1 t. assu I I 1 1 ) t, io I I s t I i at 11 :IV to be itlade arc titat syste~ii’sstates a r c ol)scrval)lp so that the eguntiori e r r o r itlethod c.an be usrd iu the ident ificntioii sclienie arid t.liat. t.lic or(1cr o f t I i f ’ systcni is krtou.ii. 1 1 1 the followirtg atixlq‘sis, ‘ivc will denote the dat.a vector (the arguments of f(.)) by X ( k ) = (y(k),y(k - l),. . . ,y(k - n l ) ,lL(k),Il(k l ) , . . .,ll(k - [)IT. Now, Ixwxi on ‘ I ‘ \ i r ( > r ~ t i 1i , w r c a n st,nt.(* tlint, I I t r ~ r ~ ~ exists a sct of real weiglits clvnotcd hy, IC” for t I i r hidden l a y e r and IV2 for the o i i f p i i t l a y e r , for a three layer A N N tliat would apprositnat,c f(.) 11y j ( . ) i n the sensc of ( 2 ) . ‘Ilicn an itl(,iit,ificntiori iiiotlcl hnsrcl on a t h e e layer A N N can be reprcseritetl by f>

I

satisfies maxxEK f ( z 1 , . . . , z n ) - j ( c 1 , .. . , z n ) l

5

That is Il/(S)- f(S)llm 5 E. In other words, for an arbitrary E > 0, there exists a three layer artificial neural network whose output functions for the hidden layer are 7 ( . ) (e.g., sigmoid), whose output functions for input ancl output layers y e linear and which has an input output fuiictioii f ( z 1 , . . . , z,) such t h a t ,

E.

+

This theorem i s in fact a natural extension of Iiolmogorov’s theorem [9] on the superposition of simple one variable functions, aiid its proof is given [SI.

y(k

+ 1) = j ( S ( k ) ,I t @ ) )

= I V ( S ( k ) ,It@))

(5)

where we dctiot.ed the output of tlie neural rictwork by A’(.),and the estimates of the desired weights 11” and 11’’ by li’(b) = { l V ’ ( k ) , Ibz(k)}.For each time step, tlie error term e ( b + I) wliicl~is SJIO\VI: i n Figiirc ( l ) , is defined as e ( k + l ) = g ( k + l ) - i j ( k + l ) and tlie cost, function t h a t is to be minimized over the weight space is giyeri by,

With this theoretical justification for functional approximation, we can now set u p a mathematical basis for system identification and control utilizing these artificial neural network models.

inear System Identificairng Neural Network

.1

J =7’

Before proposing an adaptive control law, ire first devise a system identificalioll sclieme ucing the real time operational data of tlie system unctcr study. X generic system identificatioll sclienic for a SISO (singlr inpiit single o u t p u t ) system is shown in Figure (1). Here we will assume that the plant o u t p u t g ( k + 1) depends on the previous input samples ant! I I output satnplrs

e

98 1

1+7‘-1

hence the number of the input nodes of the network is taken to be n . T h e network 11% 711 Iiitlden nodes and one output I a y r node. T h e weight, cstiriiatcs httirccri tlie input layer aiid the hidden layrr arc tlcnotcd .1, k’, an 72 x 171 matrix witli elemelits C I ! ~ wit11 i = 1, , . . n and j = 1 , . . . , TIL., and tlie \vriglits Iictivrcii the hidden and output layer are denoted by \?’, art m x 1 vector with elements 6 : with j = 1,. . . ,771. T h e nonlinear functions of the hidden layer nodes ( l f e - r ) ) i and deare in sigmoid form (i.e.y(z) = 2 noted by - y j ( . ) with j = l l , . . l 7n or H I a vect,or f u n c tion form I’(y1,. . . ,-yj) : 72“‘ -+ ‘Em. Then the output of the neural network, with an input vector X ( k ) = ( 2 1 , . . . z , , ) ~ at time k can he written as,

arid for

fll

T h e procedure commonly followed f o r the niinimization of the cost function J , is to adjust the parameters at every sampling time based on ttie error at that, instant. T h e instantaneous cost function J k + l is then defined as,

wl

Jk+l

1

1 2

= ; j e z ( k + 1) = -(y(k

+ I) - Q(k + I ) ) ?

(8)

then by using ( 5 ) , we get 1

Jk+l

= ,(U(k

+ 1) - N ( X ( k ) ,It.(k)))2

Hence tlie update equation on

w(k

LP can be written

+ 1) = W ( k )- CuVJk+]

(9)

as, (10)

where a E R is a n adjustment constant and VJk+l is the g r a d i e n t of J k + l with respect to L^v. T h e first step.in computing the update equation is to obtain the gradient VJk+’. VJk+1

= e(k

+ 1)V*e(k + 1)

(111

982

\i-’ we get,,

T h e steps followed in the design of the controller arcvery similar to those in its linear corint,crpart, [I]. 'I'hc overall arc1iitrct.urr of tlic coiit rol1t.r i s s l i o w i i i i t I'iKU F C (3). 'lhe control prchlenn c a n br st.iit,ctlas fullows:

where vet(.) operator givcs a column vector obtained by stacking the columns of i t s n i a t n x argurrierii anti Ol(k) and 82(k) are resultant paranicter vectors of sizes rra. arid inn, rerpectively. 'Clieu eqnatnons ( 2 I ) and (22) can be rewritten in a combined form as,

O(k

+ 1)

= (I - cYG(k))O(k) = Z(k)O(k)

(25)

where I is an m(n + 1) x ni(n + 1) identity matrix, and G ( k )is an m(n+ 1) x rn(n+ 1) partitioned matrix of the form, I

Update

euiat1ons

whose entries C,D,G anti F are given as [I I],

c = D = .E = F =

VVT

(27) (28)

T!TlTA .1iTF?jT

T~T~T~TT~T~

where TI = I , 8 C ' ( k ) X T ( k ) ,

TZ =

13 8 I,,

Figure 2 : Overall archikctiire for self-tuning troller.

(2'3) (30) 1'3

Given a SISO systcin of t,lie foriii (32), find a control law such tliat the output y ( k 1) follows a desired sequence y'(k + I ) wit.li a small error, asyrriptot,ically. L he niain assrii~iptionsa r e t.lint tlic nonlinear systern given by ( 3 2 ) can be represciitetf b y multilayer A N N s in the S L ' I I Sdescribed ~ i r i Sccf i o i i 2 n ~ i dtllc systcin is i n t ~ r i ~ f I b kil'e . first, c l v f i r i c tlic o i i f , l ) i i t , t,rackiiig error as, ( ( k -t 1 ) = ( ( k 1) .- y(k I). Then using t.lie certainty equivalence principii- arid assuming that the syskrn given by (32) c a n be represented by A N N niodels we l i a v e ,

+

=

(W2)*@Ir,, T4 = (LVz)T@In, T 5= III,@S(k)ST(k)

r .

and @ denotes the Kronecker product. Now we can state a theorem giving the sufficient cond'tiond for convergence of tlie linearized recursion eqiiae,ion given by (25).

+

Theorem 2: [11] The recursion for tlie parameter errors given in (25) converges exponexitially for a srua!l adjustment. constant a,if there exists a T and a positive real /3 such

+

((k 41) = y " ( k + 1) - i j ( T i / ( k ) )i-il(.Y(k), l i / ( k ) ) u ( k ) (33) where Ib'(k) are t,lie adnpt,ive weiglits of t,hc A N N a i i t l S ( k ) is tlie state vector as explained i n Srct.ion 2. I n order to make the tracking error zero u ( k ) has to he choscn t.o satisfy,

that,

( G T ( i )+ G"(i)) 2 /'3Pm(n+1)1 f = k , . . . , E:k + T - 1 ) . (31) holds €or all k. Tire proof of the theorem is given in

[Ill. The above theorem gives the sufIicicrit condit,ioiis for convergence. JIotvevec the theorein has to be further extended for the conditions on the i n p u t signal vector X ( k ) for persistent excitation.

y'(k 4 1) = i j ( S ( k ) , i V ( k ) )

-+-

L ( A - ( k ) 1Lyk))tljA.) , (31)

#

and under the assumption f l l i ~ t j. ( , Y ( k ) , Ti'(!,) (invertibility), cont.ro1 input. is cIiosc,n :IS

BaCjer! on the ident ilication model l i l t rodliceti in S c WF will iiow introduce a self t:!ning controller for a class o f noirlincar s j s f m i S of (110 i o r i i ~ ,

y(k

tion 2 ,

+ 1 ) = g ( S ( b ) )+ I t ( . ) -

1 /!(.)

y(k -I-t ) = g ( y ( k ) , . . , y ( k - n ) )

4-h ( y ( k ) , . . y(k - n ) ) r t ( k )

COII-

(32)

983

o

References

and A(.) respectively, under the update laws given by (19) and (20). This leads to, y(k 1) = y’(k 1) which means tracking of the desired sequence.

+

+

[I]

G .C . Goodwin predicttoll

a n d I