Identification and Control Algorithm for Bilinear Delta Operator Systems

Copyright @IFAC 12th Triennial World Congress, Sydney, Australia, 1993

IDENTIFICATION AND CONTROL ALGORITHM FOR BILINEAR DELTA OPERATOR SYSTEMS M.A.Hersh D'ptlTtmefll of EI,ctrOflic$ aM EI,ctrical E,.,iMerill" U,.iversity of Glas,ow, GItu,ow G12 BLT, UK

Abstract.

A class cL identification algorithms and a controller based on pole assignment are derived for stochastic bilinear delta operator

systems. Stability and convergence results are then derived under reasonable assumptions on the system for the adaptive algorithms obtained by combining the identification and control algorithms. The algorithms presenled here are the first for delta operator representations d stochastic bilinear systems.

Unlike previous works. where continuous-time or shift-operator

1. INTRODUCTION

time systems representations have been used and generally a very firm distinction has been made between continuous and discrete A large body cL control theory has been developed for linear

time systems. in this work bilinear system representations based on

systems. However. although linear systems possess many properties

the delta-operator (Hersh. 1993; Middleton et al.. 1990) are used.

which make them particularly amenable to analysis. many physical

and consequently the resulting algorithms are more general and can

systems have some degree cL non-linearity. Fortunately bilinear

be applied to a wide range et continuous and discrete time systems

systems, which can be considered as the simplest class of non· linear

with little modification.

systems. can be used to represent a wide range of physical,

The layout of this work is as follows: in Section 2 the identification

chemical, biological and social system as well as manufacturing

of bilinear systems is discussed and a class of algorithms is obtained;

processes (Mohler, 1973ab).

in Section 3 a pole assignment type controller is derived; stability

For this reason there has been more inlerest in the control of bilinear

and convergence results are presented for the adaptive control

systems than in other classes of non-linear systems and a number cL

algorithm in Section 4 and conclusions are discussed in Section 5.

algorithms have been developed by the identification (Fnaiech et 01..

The notation' is used to denote an estimate e.g. i(t) denotes an

1987; Yung et aIoo 1988; Palanisamy et al., 1985; Baheti et al.,

estimate of x at time t.

1980) andIor control (Svoronos et al.. 1981; Derese et al.. 1987 ;

polynomial X(1)l) has degree x.

Unless otherwise stated it is asumcd a

Ryan et al., 1987; Ng, 1984, Ohkawa, 1983; Yea et al.. 1986. Sen, 1986) et bilinear systems. The consideration of identification of

1. IDENTIFICATION OF DELTA·OPERATOR BILINEAR

bilinear systems has in general suffered from the drawback of being

SYSTEMS

related to the search for specific algorithms. rather than presenting a In this section the identification of bilinear delta operator systems is

general framework in which the differences from and similarities to

considered. To reduce n()(ational complexity only the case of single-

the linear case are more apparent It is this latter approach which is

input single-output systems is considered, but

taken here control algorithms developed for bilinear systems have

the algorithm5

obtained can easily be extended to multi variable systems. However

been almost exclusively of the model reference (Svoronos et al..

the presence cL what are in effect quadratic terms in the regression

1981; Derese et al., 1987; Ryan. 1987; Ng. 1984) or optimal

vector will have an effect on the relative rat~ of convergence.

control type (Ohkawa. 1983. Yeo et al.. 1986. Sen. 1986). These

In addition (complete) sets of orthogonal functions such as Walsh

algorithms have also been used in a number of applications (Maeda

functions. Chebyshev functions and block pulse functions can be

et al.• 1991; Zakeeniddin et al.• 1987; Ko et al.. 1982; Toledo et

used to represent bilinear systems. In the delta operator case

al.. 1987). The algorithm presented in this work is of the pole

operational matrices giving the effect of the delta and inverse delta

assignment type.

operators on. for instance. the basis vector of Walsh functions are required. These matrices are analogous to the operational matrices

685

of differentiation and integration in the continuous time case and enable the Walsh function basis vectors to be manipulated easily.

generally

are

represented

by state

estimate any terms in (6) and 4>(t) in this equation is replaced by 4>(t).

space

Bilinear systems descriptions, though polynomial representations are also used and in general give a more suitable representation on which to base an

3. CONTR OL DESIGN FOR DELTA OPERA TOR

identification algorithm. The general polynomial representation of a

BILlNEA R SYSTEM S

delta-operator bilinear system is of the form:

AI(8)y(t) t

=BI(8)u(t)

=(n + E)t.

As in the linear case, a variety of different approaches can be used to design control algorithms for bilinear delta-operator systems and the

+ 01 (8)u(t)02 (O)y(t) + CI(8)e(t)

0.$ E < I

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choice of particular algorithm depends largely on the application under consideration. Again , a5 in the linear case. since the system is

where Al(O), BI(O), CI(O), 01(8) and 02(0) are polynomials of appropriate degree in the delta-operator, yet), u(t) and e(t) are the

linear in the parameters. the certainty equivalence principle can be used and unknown system parameters replaced by estimates in a

system output, input and disturbance respectively. Analogously to the properties of zero-mean and bounded variance in the continuous

controller design based on assumed system knowledge . Therefore the algorithms considered in this section will be derived for the non-

time case, it is assumed that e(t) is a random disturbance consisting of zero-mean white noise. However linear transformation, division

adaptive case of known system parameters.

by on and manipulation can be used to give

yet)

=8T 4>(t) + e(t)

4>(t)T = [8- 1y, ... 8-ny.8Jl}·nu ..... 8-nu.8P"ne •.... 8-ne.8m·-nz,....8-nz]

Adaptive algorithms

can be obtained by combination of these algorithms with the identification methods considered in the previous section.

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The circumstances under which the use of feedback is appropriate are similar to those in the linear case and therefore will not be

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discussed here. An issue of relevance in the bilinear. but not the linear cases, owi ng to the different system structures is whether the

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controller itself should be designed linear or bilinear in the control and output feedback. A linear control structure is simpler. whereas a bilinear controller is in general likely to give beller performance, since it uses the system structure and is designed to control the system. rather than just its linear components.

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'The algorithms

developed in this work will all be of the feedback type and bilinear. rather than linear.

Assuming that the system is proper, (2) only contains terms in negative powers of 0 and can be therefore used as a basis for system

The majority of algorithms developed for bilinear systems have been s of the model reference or oprional control type. These algorithm

identification. Therefore any identification algorithm which can be applied to linear systems can be used to identify bilinear systems.

can easily be generalised to the case of delta operator bilinear system5. However. analogously to the linear case these bilinear

For example one fairly large class of identification algorithms for

algorithms require satisfaction of a minimum -phase condition and are therefore not applicable to all systems. This restriction is

bilinear systems can be expressed as

generally more severe for shift operator than continuous time or delta operator systems. In this section pole assignment type

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controllers will be developed.

It is a5sumed that the system is

represented by equation (2) and that a bilinear feedback controller where ~(t) is defined analogously to 4>(t) with e(t) replaced by e(t)

of the form

and the gain R(t) is either a positive definite matrix or non-negative scalar function of the data As in the linear case matrix gajn leads to

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extended least squares type algorithms, whereas scalar gajn gives extended stochastic gradient type algorithms. The algorithms with matrix gain will in general also have beller convergence rates thall

where F(o-I). G(8- 1) and 11(0- 1) are polynomials of degree f. g and h respectively and F(8- 1) has constant term one.

the scalar ones. but the relationship between the convergence rates and the correspondence between this relationship and that in the

A simple controller which gives the closed loop system

linear case remain to be investigated. Again. as in the linear case the deterministic algorithms are obtained from the stochastic ones by

(T I + T 2u)y =Fe

putting e(t) equal to zero. so that it is no longer necessary to

686

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where TI(O-I) and T2(0-1) are stable polynomials which determin

Strong consistency of the system parameter estimates results when

the closed loop system properties. is obtained by choosing F. T I

an apropriate persistent eltcitation condition holds in the scalar case

and T2 to be divisible by Bo i.c.

as well.

Boundedness properties are presented in the following

lemma for the vector d(t) of coefficients of the controller (10)

polynomials and stability and convergence results for the algorithm are given in theorems I and 2 respectively.

and choosing the controller polynomials F. G and H to satisfy

Lemmat (11) Under assumptions of system properness. positivity. boundedness Cancellation of the factor Bo however requires stability of this

and stability of B(OI). mutual coprimeness of A(o'). B(o') and

polynomial. but this assumption is not as restrictive as it would be in

0(0') and the assumptions of lemma 2. the pole assignment

the shift-operator case. since unstable zeros are less frequently

regulator has the following properties with probability one

introduced into minimum phase continuous time systems by discrete time representations based on delta-operators than into those based on shift operators. Combining the two equations (11) gives (12) C2) which has a well defined solution if A(O-I) and 0(0- 1) are mutually

Ilod(t~1

C3) 3

t

< K(w) < ""

> 0 and To

s. t.llod(t~1

<

E

for t > To

coprime. However. since all three controller polynomials are required for The proof follows from using lemma 2 to show that equations (11)

controller implementation. it is necessary to solve the paired

have a unique solution and then using similar techniques to in the

diophantine equations (11). Thus solution of a single diophantine

proof of associated results for linear systems.

equation in the linear case is replaced by solution of two paired diophantine equations in the bilinear case. Conditions under which

Theorem 1

this set of paired equations has a unique solution with FW I) of minimal degree are given in Lemma I in the appendices ..

Under the assumptions of lemma 1 and stability of 0 0 (0-') and and bounded mean square summability of e(t). the closed loop system is bounded mean square stable and the prediction error tends to the

4. STABILITY AND CONVERGENCE RESULTS

system disturbance in mean square. In this section the behaviour of the adaptive algorithm obtained by Proof of the result is similar to that of analogous results for the

substituting the estimates generated by (6)-(7) in the diophantine

linear case in (Goodwin et al .. 1980: Fuchs. 1980; and Hersh et al ..

equations (11) with controller (8) will be considered.

1987) and therefore omitted.

Since the identification algorithm is linear. under assumptions of system properness. positive realness . bounded system order (and

Theorem 2

persistent eltcitation in the matrilt gain case). it can easily be shown to have the following 'standard' properties with probability one:

Under assumptions of system properness. positivity. boundedness and mutual coprimeness of A(O') and 0(0-'). the closed loop

PI)

119("1:)1 <

K(w) < ""

P2)

109("1:)1

< K(w) < ""

P3)

I.[e(kM - (e(kM)2 r (k4) < ""

poles converge to the desired positions in the sense that

RI)

[A(I)- I )F(t.I)- I) . O(t.I)- I) - Co (1)-1 )TI (1) - 1)1 = 0 'v'1)

lim,-+~ ["(t.I)- I)

k=1

for r(k4)

lim,-+~

= trace R(k4) and R(k4) in the case of scalar and matrilt

- Co(I)- I)F(t.I)-I) . C o (I)-I)T 2 (1)-1)1

=0

'v'1)

When. in addition. the conditions of lemma 2 are satisfied the closed

gain respectively

loop system has the further asymptotic property

9(t) - 9

o

687

The proof is based on manipulation of the system and controller

Yeo. Y.K . and Williams. D.C. (1986). IEEE. AC31. 1071-1073.

equations and omitted to save space.

Ng. S.K. (1984). OIEEE lrans. aul. conlrol, AC29. 271-274.

7. APPENDICES

5. CONCLUSIONS

Lemma 2

A number of the issues surrounding the derivation of identification and control algorithms for bilinear systems have been examined. A class of identification algorithms and a control algorithm based on

The paired set of diophantine equations (11) is in general soluble. If

pole assignment have been obtained for delta operator stochastic

the system is such that

bilinear systems and stability and convergence results have been m'+n~2

obtained for the adaptive algorithm obtained by combining the

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dentification and control algorithms. The algorithms presented here are the first identification and control algorithm for delta operator

then the solution is unique if the degrees of F(O-I). G(OI).H(O-I).

representations of bilinear systems. but it should be nOled that they

T I (Wl) and T2WI) are chosen to satisfy the following conditions

can be easily extended to the continuous time and shift operator

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discrete time cases. Further work is underway to derive further classes of adaptive algorithms for delta operator bilinear systems. when the degrees of A(O) and 0(0) are such that

6. REFERENCES

Baheti. R.S .. Mohler. R. and Sparg. H.A . (1980).

IEEE IrallS

m'=n= I

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f = P + k I - n. g = n - 2. h = P + k2

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AC2S.1141-1146. Derese. I. and Noldus. E. (1987). 1nl. j . .!)Is. sci.. D, 237-246.

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Fnaiech. F. and Ljung. L. (1987). Int. j. colllrol. 45. 453-470. when the degree of A(d) satisfies

Fuchs. J.J.J. (1980). lEE Proc.• 1270. 259-264. Goodwin. G.c.. Ramadge. P.J. and Caines. P.E. (1981). SIAM j.

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control and opt.• 19. 829-853. Goodwin. G.c. and Sen. K.S. (1981) IEEE lrans. AC16. 478-487. Hersh. M.A. (1993). Int. j. control. to appeal.

and (17) with either (16) or

Hersh. MA and Zarrop. M.B . (1987). Int. j. cont.. 45. 1789-1802. Ko. K. Y-J .• McJnnis. B.c. and Goodwin. G.c. (1982). AUlomalica.

f = P + k2 - m. g = p + k I' h = m'- 2

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18. 727- 730. when the degree of 0(0) satisfies

Le. H.• Zakeeniddin. M.. Gainshankar. V. and Rink. R. (1987). J.

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Middleton. R.J. and Goodwin. G.c. (1990) . Digital conlrol and

eslimation. Prentice Hall. Englewood Giffs.

The proof is based on writing the Diophantine equations (11) as

Mohler. R.R. (1973) . Bilinear control processes, Academic Press. Mohler. R.R. (1973).IEEEtrans. SSC6. 192-197.

TX=N

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Yung. c.Y. and Chen. C-K. (1988). Int. j. sys. sci.• 19. 275-288. Ohkawa. F. and Yooezawa. Y. (1983). ftu. j. conI.• 37. 1095-1101.

and showing that

Palanisamy. K.R. and Aninachalam. C. (1985). Im. j. colllrol. 41 . 541-547.

rank T = rank(T N}

Ryan. E.P. (1987).lnt. j. conlrol, 45.1035-1041. del la Sen. M. (1986). lEE Proc.. 133D. 165-171.

Details of the proof are omitted to save space.

Svoronos. S .• Stephanopoulos. G. and Aris. R. (1981). Int. j.

control. 34. 651-684. Toledo. B.C. and GaUegos. J.A. (1987). Int. j. SYS. sci.. 18.22092228.

688

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