ON STOCHASTIC PERTURBATION THEORY FOR LINFAR SYSTEMS

Peter N. Nikiforuk Madan M. Gupta, Member IEEE Professor and Head Assistant Professor

Division of Control Engineering University of Saskatchewan

Saskatoon, Saskatchewan, Canada

Abstract

This paper i s concerned with the dynamic response of a genera l c lass o f l inear sys tems the parameters of which undergo s tep per turba t - ions. The maximum s ize o f these per turba t ions a re l imi t ed by t h e s t a b i l i t y bounds of the per- turbed system. Using the assumptions t h a t : ( a ) t he system's input vector i s a s t a t i o n a r y process, random or de te rminis t ic ; and (b) the s t ep pe r tu rba t ions a r e random and s t a t i s t i c a l l y independent of the system input vector, a tech- nique i s developed for the exact analysis of the dynamic response of the system. It i s shown t h a t t h e s t a t e and output covariance mat- rices of the perturbed system can be represent- ed by f i r s t - o r d e r matrix di f fe ren t ia l equa t ions .

I t i s ind ica t ed t ha t a quasi- l inear mathemat- ical model of the perturbation process may be obtained for small per turbat ions. This model descr ibes the dynamics of the parameter-pertur- bation-transmission path from the perturbations i n the parameters to t h e dynamic r e sponse i n t h e system covariance matrix. The theory i s developed for continuous linear systems, but may be analogously extended to discrete cases.

I Introduct ion

A basic problem in t he s tudy o f con t ro l sys- tems i s that of evaluat ing a system's dynamic response to pe r tu rba t ions i n i t s p l a n t para- meters. In most cases, as i s well known, an exac t so lu t ion i s d i f f i c u l t t o o b t a i n , b u t f o r small perturbations an approxinate solution may be obtained as has been reported in the litera- t u r e [l],[a. The problem becomes more d i f f i - cult, of course, i f t h e changes i n t h e p l a n t parameters occur in a t rans ien t fash ion , say due t o component failure o r sudden changes i n the plant environment. If the inf luence of the per turba t ions on the system's behavior i s small, c l a s s i c a l s e n s i t i v i t y a n a l y s i s [4 w i l l provide an adequate assessment of their effects . On the o ther hand, when t h e v a r i a t i o n s i n t h e sys- tem parameters s ign i f icent ly a l te r the behavior of the plant, it i s necessary to incorporate a quan t i t a t ive desc r ip t ion of t h e dynamic e f f e c t s of the per turbat ions into the system [a. This i s a problem tha t i s of p rac t i ca l s ign i f i cance , and f o r this reason it i s considered in this paper.

T h i s research was supported by the National Research C o u x i ; of Canada Under Grants A-5625 and A-1080, and the Defence Research Board of Canada under Grant ,!+003-02.

In this paper a c l a s s of l i n e a r systems t h a t i s subjected to random s t ep pe r tu rba t ions i n the plant parameters i s f i r s t defined and a technique for the derivation of an exact analy- sis of i t s dynamic response t o these per turbat- ions i s then presented. It i s assumed t h a t t h e system's input vector i s s t a t i o n a r y i n t h e sta- t i s t i ca l s ense w i th ze ro mean, and t h a t i t ex- ists f o r a l l t h e t i m e , - m < t < m. I t i s a l s o assumed that the unperturbed system i s exci ted a t some time t, t (a, 0) such that it i s i n s t e a d y - s t a t e a t t = 0, the time a t which the p e r t u r b a t i o n s o c m .

For t he ana ly t i ca l development the unpertur- bed and perturbed systems are assumed to be s t a b l e . Hence, the maximum s i z e of the pertur- bat ions in the system parameters are constrain- ed by t he s t ab i l i t y bounds of the perturbed system.

An exact analyt ical technique i s presented for the evaluat ion of the dynamic response i n the system states fol lowing per turbat ions in the parameters. The ana lys i s i s used to eva l - ua te the dynamic response in the s ta te -covar i - ance matrix of the system.

The analysis considered here presents the main r e s u l t s of the inves t iga t ions , the de ta i l descr ip t ion will fo l low in a l a t e r pape r .

I1 The Definit ion of the Svstem

In this section the system i s defined using its input-output s ta te representat ion.

Consider an nth-order linear dynamic system t h a t is completely controllable and observable. The system i s exci ted by a m-dimensional input vector u ( t ) which e x i s t s f o r a l l t h e t i m e t a-, m).

The unperturbed system So i s described by

% ( t ) = A. q, ( t ) + B o u ( t ) , - < t < 0, (1)

a vec tor d i f fe ren t ia l equa t ion

So:

% ( t ) = Co ( t ) + Do U_ ( t ) , < t < 0, (2)

where % ( t ) i s the n-dimensional state vector

The ou tpu t vec to r fg ( t ) i n (2 ) i s the res- ponse of the unperturbed plant a t a given poin t o f in te res t .

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I t i s assumed that the input vec tor g ( t ) to sys tem due t o t he i n i t i a l cond i t ion vec to r x_(O), S, i s a s ta t ionary p rocess , random or determin- and

v

i s t i c w i t h z e r o mean

E[g(tg = Q, f o r a l l t E( a, a), (3 )

where E i s the expectation (ensemble average) operator. The covariance matrix of t he i npu t vector i s

Cov [g(t); :(X)] = E [g(t) g' (X,] = RUU(t-X) (4)

The steady-state solution of the unperturbed system S i s given by 0

+ ( t ) = $o(t) Bo g(t-A) dX, LO < t < 0 (5) $' where @ ( t ) is the state t r a n s i t i o n matrix of the syshm sob

Let the value of the systam parameters under considerat ions be K i 5 p 5 n, f o r t < 0. Assume that all or some of these parameters are subjected to step perturbationsAK., i 5 p 5 n a t time t = 0. The new values of the perturbed system parameters are given by

o i '

K. = K +AKi, i 5 p 5 n, t 2 0. (6) I O i

The maximum s i zes o f the perturbationsAKi

are assumed t o be constrained by t h e s t a b i l i t y of the pertrubed system S. Thus, following the s tep per turba t ions , the perturbed system equa- t i o n s are

where x( t) and g ( t ) a r e t h e state and output vectors of the perturbed system, and the i n i t i a l condition vector i s ~ ( 0 ) . The matrices of the perturbed system S are : A = A. +AA, B = Bo tAB

C = %+AC, and D = Do +AD, whereAA,AB,dC

andAD are the s t ep pe r tu rba t ions i n t he matri- ces as e f fec ted by t h e s t e p p e r t u r b a t i o n s i n the parameters Ki a t t = 0.

Following the s t e p p e r t u r b a t i o n s i n K. , i I n the solution of the perturbed system S is given bY

x ( t ) = d t ) ~ ( 0 ) +$dh)B g( t -1) d l , t 2 0

( t ) =pX) u _ ( t - X ) dX, t 2 0 (12)

i s the s teady-s ta te so lu t ion , where \k(t) = d t ) B .

To evaluate the i n i t i a l condition vector ~ ( 0 ) we apply the equat ion of cont inui ty to the sol- utions of (5) and (10) a t t = 0. Thus,

which f ie lds

- x(0) = $*(X) g( -X) dX

where A *( t ) = @o( t)Bo - $( t ) B

= - Wt) ( 14)

Subst i tut ion of (13) i n t o (11) f ie lds a new express ion for the t rans ien t so lu t ion of the s ta te vec tor g iven by

q ( t ) = d t ) rA*(A) g(-X) dX (15) U

Using the p roper t ies o f the input vec tor as g iven in ( 3 ) , the expected value of the solut- ion i s

I11 The Sta t i s t ica l ProDer t ies

In process optimization and estimation pro blems, it i s convenient t o o b t a i n an estimate o f t he s t a t e s of a system subjected to pertur- bat ions. Thus, from an engineering point of view, t h e statistical propert ies of the per tur- bed systems considered in this paper are impoa- t a n t . I n this sec t ion , we p r e s e n t b r i e f l y t h e theory f o r an exact evaluat ion of the state- covariance matrix of the system fol lowing s tep perturbations in the parameters of the systems

The system and i t s solut ion are descr ibed in Section 11. I n the following, it w i l l be assumed t h a t t h e i n p u t v e c t o r g ( t ) is uncor- r e l a t ed w i th t he i n s t an t of the per turba t ions . To emphasize this the i npu t vec to r g ( t ) will be wr i t t en a s u_( t + Y ), where Y is a random var i - able with uniform dis t r ibut ion over a l l time . m < t < m .

(10 1 Define P(t) as the state-covariance matrix of the perturbed s y s t e m

i s the t r ans i en t so lu t ion of the perturbed Using (101, (12) and (15), it may be shown that 5-C-2

where P ( t ) and P a re r e spec t ive ly t he time- varying and time-Znvariant responses i n t h e state-covariance matrix P ( t ) given by

t

and

These may be evaluated using (12) and (15) .

Further, i t may be shown that the s ta te-co- variance matrix satisfies a f i r s t - o r d e r s y m e t - r i c a l matrix d i f fe ren t ia l equa t ion g iven by

$ ( t ) = AP(t) + P(t)At + BF(t) + F ' ( t ) B ' (20)

In the solut ion of (20); the i n i t i a l s ta te-cor- variance matrix i s

The relat ions der ived above are exact for a gen- e ra l c l a s s o f s t a t iona ry i npu t vec to r as defined in the p receding sec t ion .

A similar expression can be derived for the output-covariance matrix M(t). This l eads t o a convenient expression in the de te rmina t ion of the dynamic response of the perturbed system's performance index. For example, i f n ( t ) i s the system e r ro r vec to r t he dynamic response s ( t ) i n t h e ensemble average of the error-squared s igna l , fo l lowing s tep Per turba t ions in the system parameters, i s

s ( t ) = t r a c e [M( t)] (23)

The covariance matrices P( t) and M ( t ) are thus related dynamical ly and nonl inear ly to the s tep per turbat ions.

V Conclusions

A parameter perturbation process for a c lass of l inear systems subjected t o s t a t iona ry i n - pu t s igna ls and random step per turbat ions has been studied

I t has been shown t h a t t h e dynamic response in the state-covariance matrix i s r e l a t e d dynam- i c a l l y and nonl inear ly to the s tep per turba t - ions. However, a quasi-linear mathematical model of the perturbation process can be ob- t a ined fo r small per turbat ions. This quasi- l inear iza t ion permi ts the responses in the s ta te covar iance matrix P ( t ) and output covar- iance matrix M(t) t o be evaluated for small a r b i t r a r y p e r t u r b a t i o n s i n t h e system using the superposit ion integral . This formulation i s use fu l i n t he s tudy C51 of the dynamic behavior of the parameter-perturbation-transmission paths, from the perturbations in the parameters t o t h e dynamic response in the system covarian- ce matrix M(t) .

The ana lys i s g iven for the l inear time-in- ian t sys tem which may be analogously extended t o a wide class of continuous and discrete systems, including nonlinear and time-varying systems.

References

1. J .L . Douce, K.C. Ng, and M.M. Gupta, Vy- namics of the Parameter Perturbation Pro- cess", Proc. IEE (U.K.), Vol. 113, No. 6, June 1966, pp. 1077-1973,

Per turba t ions in L inear Systems Parameters", &oca Second IFAC Symposim on System Sen- s i t i v i t y and mvity, k b r o v n i k (Yugoslavie) August 1968.

3. D.D. Swoder, "Feedback Control of a c l a s s of Linear Systems with Jump Parameters", IEEE Trans. Automatic Control, Vol AC-14, No. 1, February 1969, pp 9-14.

2. M.M. Gupta, IlEpamic S e n s i t i v i t y t o S t e p

4. P.V. Kokotovic e t a l , " S e n s i t i v i t y methods in the experimental design of adapt ive con- t r o l systems", Proc. Third International Federation of Automatic Control, London, 1966, Paper 548.

5. M.M. Gupta, "On the charac te r i s t ics o f the parameter perturbation process", IEEE Proc, of the Seventh Symposium on Adaptive Proc- esses , Los Angeles, December 1968. Also t o be appeared i n IEEE Transactions on Automat- i c Control, October 1969 issue.

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