Nonsingular and Stable Adaptive Control of Discrete-Time Bilinear Systems

Copvright © (FAC i\onlinear Control Systems Design. Capri. Ital)' 1989

ADAPTIVE CO"TROL OF :\O:-JU;-;EA.R SYSTBIS

NONSINGULAR AND STABLE ADAPTIVE CONTROL OF DISCRETE-TIME BILINEAR SYSTEMS C. Wen and D.

J.

Hill

Department uf Electrical and Computer Engineering, L'niz'l'rsity of .\'1'U'cl15tle, ,\'SW 2308, Australia

Abstract. A certainty equivalence adaptive control algorithm for bilinear systems is studied. Achieving a well-defined control law is seen to be a nontrivial aspect of the algorithm. A scheme for resolving this problem and a stability result are presented. Keywords.

Adaptiv~

control, discrete-time systemms, bilinear systems, stabi lity .

INTRODUCTION considered by the above· mentioned authors . Our scheme automatically can avoid the Singular case and ensure the stability of the closed loop. Singularity is avoided by use of an estimator gain resetting procedure whereby the gain avoids four points at each step (if they exist in a certain range).

The ~heory of adaptive control for linear systems 1S now well- established. Comprehensive reviews of the literature are given by Go~dwin and Sin (1984), Anderson et. al. (1986), Astrom (1987) lien and Hill and Middleton, et. al., (1988). (1988) have interpreted these techniques of adaptive linear control as a valid method to control nonlinear systems. However the conditions for stability are somewhat restrictive and it has been noted elsewhere that performance achieved may be poor (Goodwin and Sin, 1984). This leads to consideration of adaptive control based on estimation of nonlinear models. Bilinear systems are a very useful class of nonlinear models (see Mohl er, 1973; Bruni, et. al. 1974 ). Indeed, many real systems can be represented by bilinear models; for example, chemical and biological processes and electrical machines are typically modelled this way. Also it is natural to expect that bilinear systems give a better approximation to general nonlinear systems than linear systems (Krener, 1975). Bilinear systems have attracted the attention of many researchers. However, there is only a limited range of controller design techniques for general bilinear systems (nonadaptive case).

The structure of the paper is as follows . Section 2 introduces the input- output model and associated assumpt ions. In Sect ion 3, the parameter est imator is cons idered. The des ign of a nonsingular controller is given in Section 4 following a one-step ahead predictive control approach . The stability result is given in Section 5. After an example in Section 6, Section 7 gives some conclusions.

SYSTEM MODEL Consider an input- output model of a bilinear system given by Fnaiech and Ljung (1987) yet) = -a 1y(t- l ) - . . . -any(t-n) + b u(t-1) 1 + . . . + bnu(t-n)+ 13 11 u(t-l)y(t- l) + . . . + 13 1n u(t-l)y(t- n) + 13 u(t-2)y(t- 1) 21 + + 13 2n u(t-2)y(t- n) + + 13 n1 u( t-n)y(t-l) + . + 13 nn u(t- n)y(t- n) + d(t) (2.1) where yet) i s the output of the system, u(t) is the input of the system, and d(t) is a bounded disturbance, i.e. there exists constant A such that Id (t) 1 ~ A for all t. lie make the following assumptions in order to achieve a stability theory.

The situation is much harder in design of adaptive controllers. Only a few results have been reported. Goodwin, et. al. (1980) considered stability of an algorithm for a first order bilinear system and applied this theory to two industrial processes. The analysis took account of the so-called singular problem, i.e. where division by zero gives an ill-defined control signal. Svoronos et. al. (1981) gave a self-tuning algorith m in a stochastic setting and a part ial anal~s is of the scheme. Ohkawa and Yorezawa (1983) presented a model reference adaptive controller for a class of bilinear systems. The ir convergence analysis only ensures the convergence of parameter estimation error. A tracking error limit of zero is not guaranteed as the signal s in the control loop may go unbounded. Yeo and lIilliams (1986) proposed a predictive control algorithm and its convergence analysis. However, the singular case is not considered.

Assumption A1

o

13 1n Assumption A2

In this paper, we will consider a class of gen eral discrete bilinear systems suggested by Fnaiech and Ljung (1987) (without the stochastic basis). The model can cover all the systems

lie know Y such that sign (13 ) is known. 11

229

113

11 1

~

Y

>0

and

230

C. \"'en and D.J. Hill

i311 (t) Assumption A3 There exists finite constants monotone

f: ffi +

~ ffi

=

and

S.t.

+

for all t Remarks 2.1 (a) Assumption Al makes the problem considerably simpler. We see later that the coefficients /3 1 i all appear in the coeff icient of the control signal u(t). This assumption can be relaxed at the expense of a more compl icated estimat ion a lgori thm (choice of gain). (b) Assumption A2 is consistent with a common assu mp tion in adaptive linear control. (c) Assumpt ion A3 corresponds to the restr ict ion of a minimum phase plant in auaptive linear control. Techniques for analysing the validity of this property are emerging in the nonlin ear control literature (Byrnes and Isidori, 1984; Glad, 1987; Hill , 1989).

,if sign (13

11

)i3 (t) 2 Y 11

1

sign(/311)Y:=YI,otherwise

(3.5)

A(t - I) is chosen to be in the interval (0,2) and will be restricted further later on. The features to no te in the estimator are a fixed deadzone and projection on parameter 13 , 11 Some useful properties of the esti mator are given by the following result. Firstly, we introduc~ the parameter estimation error 8 (t) := 8 (t) - 8 , 0

Lemma 3.1

a)

The sequence

b)

lim t-

118 (t) 1I

converges.

g2(tl 2 +1I(t-l)1I

o

(3.6)

Proof: a)

119 (t) 11 2 -

119(t)

-

119(t- 1) 11 2

PARAMETER ESTIMATION System model (2.1) can be rewr i t ten in the so-called regression form (Goodwin and Sin, 1984)

~

2A(t-I)tI>Ct-

J)Tii~t-llg(tl

1+1I(t-1)1I (3.1)

+ ACt-

J)2g(t~2

1+1I(t-I)1I

·2A(t-l)(e(t)- d(t))g(t)+ A(t- J)2g(tl 2

where

1+ 1I( t-l) 1I

~

(y(t-I), ... , y(t- n), u(t-l), ... u(t - n) , u(t- I)y(t-l), u(t- 2)y(t-I), ... u(t- 2)y(t- n), u(t - n)y(t- n))

2

- A(t-l)(2 - A(t-l1lg2(t)

(3.7)

I + 11( t- 1) 11 (e(t) - d(t)) get) 2 g2(t)

since

Now A(t-l) E (0,2) for all t. i s nonlncreaslng, but bounded below.

(- ai' . . . , - an' b l ... , b n ,/3 II ,/321 ... /3 2n ... /3 nn ) Then we can use the following modified gradient algorithm to estimate 8

So

limIl8 (t) 1I

So

118 (t) 11

exists.

t-

0

(b) From (3.7), we can conc lude

9(t- l) + A(t- J)(t-I)g~t) I+II(t- I)II

9(t)

(3.2) g2(tl

~

0

o

.here Remark 3.1

-

.

-

As in similar versions of this result for linear systems (Goodwin and Sin, 1984), clearly this result is independent of the control law used to compu te u.

... bn (t-l),/311(t- l) , ···/3nn(t-I)] 9(t)T

get)

[-al(t), ... -a (t) , b (t), 1 n. . . bn(t) ,i311 (t) ,/321 (t), ... 13 nn (t)]

.-

r:

J

A

e(t) + A

e(t) /3

11

(t)

.-

yet) . (t-l)

if if if

O\E- STEP AHEAD CO\TROLLER DESIG\

e(t)

> A

I

A

le (t)

e(t)

< -A (3.3)

T • 8 (t- l)

i s chosen by the projection step

(3 .4)

Th e adaptive controller combines the estimator .ith a (non -adaptiv e) controller via the certainty equivalence idea (Goodwin and Sin, 1984). A convenient controller for discrete- time bilinear systems is the one-step- ahead control l er based on a predictor (Goodwin and Sin, 1984). The predictor in our case is just the system model (2.1). The control u(t) i s computed to make y(t+l) = Y* (t+l) where y* is the desired set-point . In this section, we consider the

;\Ionsingular and Slable Adapl ive Control of Discrele-lime Bili near SYSlems

detail s of this CUlitIllllt'l il,dllding lIIodifications to ensure thC' adapt in' systelll lIIodel has we ll- defined solut iOlls . I n the Systl'lII model (2. I). Asslllllption A1 imp l ies that the locfficic'ilt of the control u(t) is given by

Then if

c (t )

.-

c(t )

#

I, I .;. !.l11 y ( t )

be computed alld Cl t) (- IJ dctcI'llI ill cd in the case g(t) # O. To enSU I l' e(\) # () I.·hen g(t) = O. I,e II~C it modified set- point sequence.

(4 . I )

• Y (t).

() I'

-

if b l (t·l) " 0 and or -

(1+11

liW.

is given by ~II (t- I )

-

et t)

([y" (t) - 4, / (t)

or y" (t) - .T (t'I)9(t_I ) . y. (t) ... a, othen.. ise

(.).2)

where

of c(t) = 0 for all t . To sohe this problelll, we need to fUI·thel· 1'C':;Lrin the I'illue of A(t- I) over (0,2) ami definc it lIIodified sC't-po int sequence. Cons ider the res tl' i Cl. i 011 on A( t- 1) . In order to keep c(t) off the Si ll.!!,lIlar slIdace, i t turns out that lie need A(t) \~ illoid ee l·tai n points in (0,2). As g(t.) . y(t) . 9(t- l) are available at time t, clearly I,e Ci,n 11:;(' thl'se data to cOllljJute Aj , A2 , A3 and \1 as 1'01101,';: b1 (t-l) + YIy(I) ') nit-I ) .!!,(1) ( l+II (t- I)Ir-) (4.4a)

A1 if g(t) # 0

(b 1 (\,- 1) ( I +

-

+

.)

(311 (t - l)y(t))(1+II(t-1) 11-) y(\ )y(l-l) )1I(t-1)g(t) (4.4b)

if g(t)

#

0 and

l~y (t) y ( I-1)

1-()

~ ( t-I )u( t-I ) "

0

is a constallt cho:3clI suc h that

a

(U (t)

1

U1 (t- 1)

0

4J

+

(4.5 )

!.l11 (t- 1 ) Clearly, c(t) C,III 1)(' , 'TO 1J('lilusc y(t) is varying fr eely. I II onicr 10 Ct)lllpllt.e lhe contro~ via one-step-ahe'ild contloller i' I.~) I'ith 9 rep l acing 9 , \Ct' h,II'C' t u al'oi d the singular case

. • f[Y ( t)- 4.y (t)+ 41 b (t - I ) l

P ll (t· I )

T ) t ) 8 (t) + c( t ) 11 (t)

From the estilllatC'd lIIudd. tll!' estilllate> of e(t), denoted c(t). ill 1 illll' 1 is

z*

The modified setpoin\

¥

u(t)

23 1

[/ (t) and

#

0 only)

i.

+ a-A. y*(t)+a+A]

*

T , y (t) + a - (t-l)O (t -l) + c(t-1)II(t-1)

0

#

(1 .G) Recall !.l11(t ) # 0 is ell~lIrcd by (~Lj) . Clearly, an a all 2A "i 11 1'0l'l,. Also note t hat if v* is bOlllllied. th en so is z * . The control uti) is given by z * (t+1)

(1 .7)

(rewritten as (4.2)). \ole tltat (4 .6 ) implies the control u(t)_ calculated fr01l1 (4 .7) i s nonzero for all t if c(t) # O. Lemma. 4.1 Under the al,ol'c choice uf A(~-l) and control 11 ( 1- 1) calcnLtt.cd hOIll _(cl.i). c(t) will be lIonzel'O to! ,ill t i f b1 (0) and are do se ll s . t .

/311(0) c(O) ;t 0

b1 (0) + !.l11 (0)

(- ytO) )

#

0

if

(4.8a) y(O) # 0 (4.8b)

and !.l11(0) satisfie s d",,;ullljJtiOIl .\:2.

A3

if g(t) # 0

allli

y(t ) # ()

I

[b (t-1) + !.ll1 (l-1)(1 (1 -

4fBI)

~ )J\l + 1I (t -1 ) 1I 2 )

u(t - 1) g (t ) (·Ud)

if g(t) # 0 alld :-l\-1 ) # qt) . y;, \-l) I- 0 The n lie use lhe ~l'ltt' i'i\' E(0,2) sho~n ill Fig. 1.

(pr

citoice of

A(t- 1)

The motivation for thi ..; "clll'I:lc' h('comes clearer in later analvsb. \0\(' \ hill \he ~(' h(:'m(' ('II ;;lI res that A , A2' A3' and AI ill 'C I'cll-d('i'illed rc'al numbers. 1 Once choice of A( \ - I ) i:) det ermined, 9( t) can

Proof: c(O) Citll I~(' l!tadl' lIollz~ro by iljipropriate choice of inili,)l hl(O ) alld !.l11 (0) satisfying assumption ,13. ~'e assume c (t - 1 ) is nonzero and prol'e c (t) at (4.3) is nonz('ro . lie halt' lI(t-l) cOlllputed from (4.7) and non ze ro from (-1.6 ) . The proof proceeds by consider,H ion of ,cases . 1) If g(t) = O. tlt l'lI 9 ( t) 8 ( t- 1) and

Iy(t)

* z (t) # 0

*

z (t) I ~ A. + A]. SO c(t) as -

J

b 1 (t- I ) q,

!.l11 ( t - 1 )

•

.c. y (t) E [z (t) - A, b j (t-1) + !.l11 (t-1) y(t)

rz

*

(t)

A, z (t)

+

A]

232

2)

C. Wen and D.]. Hill

from the def initiolt of z * (t). If (I(t);t. 0, I,e havl' the following cases: A)

=

P11(t)

Yj

b1 (t-1)

=

+ Y

1

y(t )

+

,,(t-l)

u(t-l)g(t).) 1 + 1I
gi ven by (4 Ad) .

Thll s I;e have

If g(t-1)

then

=

.

For ,,(t-1)e(0,2), ~l' hav e e(t) ;t. 0 if where

"1

\l

1I
=

;t.

j ) )

o

2

=

s (t)

+

0

bj (t- :2) + PI

j ( 1-

then I ~)( . y (t _ ~) )

s( t- 1) ;t. () Thus we

. =

e(t)

b (t -l) + P (t-l) y(t) ll 1 (!+v(tlv(t-l).») u(t-l) g(t) 1 + II
i)

If

+

,,(t-1)

1 + y(t)y(t-l) ;t. 0, lie can choose allY ,,(t- 1)e(0 ,2) except "2 given by (4.4b).

ii) If 1 + y(t)y(t-1) y(t- 1) ;t. 0 and

=

0, then

y(t);t. 0,

y(t) = - )'(t: I) . Thu s . ' . ' 1 c(t) = b (t-1) + P (t-1)(- Y(t-1T) l ll in thi s casl'. J3cfol·e preced ing, consider the sequcnce {s(t)} lihich is defined as

have

:;(t)

y(t-1)

=0

and s(O);t. O. Clcarly c(t) set) in this case. So if ~c keep {s (t)} nOIlZC1-O at each t, then c(t);t. 0 for this case. As s(O) ;t. 0 , so "e assume s(t-1);t. O. .11 t hi s stage , ~e go back a time step alld I·CCOllsidl'r CilSC,; A), IJ ).

.

Y

=

11 Then

ha~

!?(t-1) , )'(t:l). 8 (t-:2) In the preVIOU S step.

been carried out

o

A~ALYSIS

In this section, I;C consider the stability of the adaptive system. Th e solutions (sholm to exist in Section 4) ill·C scen to be bounded under the assumpt ion s made ill S('ct i on :2. Theorem 5.1 if Id' [lppJy the estimato r (3.2) - (3.5) alld the (OlllrD] ""; ( 1.7) to the class of systcms (:2.1 1 (nllde]' .h~nlllpt.ion::; Al-A3), then thl' clo,.;ed loop ";VSr('1I1 is stable in the sense that {u (t) } . {v ( t) } arc bounded if

*

lim Iy (t) - y * (t) I

V(~-I))

.

p~ssible

c(t) ;t. 0 i" PCIIOIIII!'d ,It every step. So the above proccd;1 rc' to clioose I\( t- :2) from

I

+ Y I (-

all

The gain and set- point, modification to ensure ,,(t-1) avoids points "1' "2' "3' "4 and

{y (t)} is bounded alld

bl(t- ~)

s(t)

for

y (t-1) ;t. 0

if

(t-1)

;t. 0

cases. To ensure s(O) ;t. 0, lie need b (0) + 1 • 1 /3 (0) (- Y(O)) ;t. 0 for y(O);t. 0 or any 11 nonzero value for y(O) = O.

STAUILITY if

.

~ a + A

(5.1)

t-

+ ,,(t-2)

.)

u(l- :2)g(t-I)/( I+II
(A is the bound 011 the dis\Ul'bilnce in model (2.1) and a is defined in sctpoint modification

(4.4c).

Proof:

Thell

If g(t-1) /3

11

5(t) ;t. 0

= O.

(t- 2) = /3

- A,

If

b (t-2) 1

Pll(t )

Then from (3 .2 )

(a) /3

O.

z * (t -1).

from definitioll of y(t-2) = y(t-l)

i.e .• given by (4.4a). P 11 (t)

s(t)

If

(t- 1) g( t)

+

0,

s(t) ;t.

1

/3 11 (t- 2) (. y (t-

,,(t- 1) ;t. "I

is such thot

b (t-1) + Y y(t) + "I 1 1 B)

u (t- n!!,(t-1)/(1+1I
If g(t-l) ;t. O. [lIld \(t-:2) ;t. y(t-1), choose ,,(t- 2)E(0,2) but ,, (t - ~ I ;t. /\,.j ,;hel'e "4 is

Then from (:3 . ~ ) e(t)

~:i~:i;)

+,,(t-2)(1 -

11

z * (t -1 ) + ~ J .

I') (t-" ) 1

Thus if

b (1-2) 1

J

b (t- 2) 1

;t. O.

definition of (b) /311(t-1) s (t )

11I;der this /31(t-l)

+

"it- 2).

=

b (t·2l, 1

y(t·l )e[z * (t -1)

( t- 1) = Y .
s(t)

TIlen

thcn

s(t)

;t. 0

is 1)()lIlltied. is

and so y(t)

Suppose {y(t)} exists a subsequence

1) ~

O. thell s(t) ;t. 0 as Y1;t. O. thl'n

{y * (1)}

Since

{z * (t)}

control law give s

SO Y 1 (-

(4.5) . )

from the

z * (t- I). = P11 (t·l ) . ) ,. 1 ) b 1 ( t - ~) + PI 1 ( t- L) (- Y(t-1T

lim Iy(t ) I = i

00

=

it

[ollol;s that

bOlllldcd

s inec the

~

z· (t) .

is ulIIJoundcd . {t j } such that

Then there

allll

t1· -

Iy(t) I

~

Ht i ) I

for all

Also from definiti on of !;

< t·1

-

(5.2)

r

:-.Jonsingular and Stable Adaptive Control of Discrete-timt' Bilinear S\stcms

-

if

y(t) - y(t i ) > A if Iy ( t i ) - y~t) I ~ A if y( t i) - y(t i ) < - A

ti ) -,It i )-·

g(t i )=

0_ y(t i )- y(ti )+A

I t follows that g(t i ) -1-+ 0 as ti --> "'. we now follow a line of argument used in the linear case (Goodwin and Sin, 1984) to show a contradiction. From (3- 6), we have

o

lim

(5.3)

t1· -

Th ese tl,-O facts imply lim(l t 1· -

+

1Icj>(t

r

1)112)

Since the y(t i ) are bounded ' we have o

o

(5.4)

To follow the proof of the linear case, we need (5.5) Thi s does not hold for all t due to bilinear terms. It is shown in !.'en and Hill (1989) that (5.5) does hold though on the subsequence {t i} . Then the contradiction can be completed to show {y(t)}, {u( t)} are bounded. ~rom (3.6), we have limg(t) = 0 t-

and so limle (t) I

~ A

t-

That is, (5.1) is established after noting (4.5) . o

EXAMPLE Consider the system y(t)

0.6y(t-1) + O.3y(t- 2) + 0.3y(t-1)u(t-1) + 0.2y(t-2)u(t-2) + 0.8u(t- 1) + 0.5u(t-2) + 2sint

I.·here the disturbance term is s inu so idal. The * des ired set point y is a square wave of magnitude 10 and period 40 time steps. The response of the adaptive cont roller is shown in Fig. 2 and the control inpu t is shown in Fig. 3.

CO\CLCSIO\S This paper gives some preliminary result s .on the design and stability analysis of an ada~tlve control algorithm for a large class of blllnear systems. Particular attention is given to

233

ensuring the system so lution s are well - defined. This is guaranteed by an est imator gain sett ing scheme and set-po in t modifi cation . The system is shown to be stab le in the sense of having bounded Signal s. A tracking property is established which allows for bounded di st urban ces. Several improvements to the present result deserve attention. Ass ump t ion Al on bilinear coeffi cients i3 ij can be relaxed. Some modern tec hniqu es should be used to make Assumpt ion A3 and the stability analysis more explicit.

REFERE:iCES Anderson, B.D.O., et al (1986). Stability of adaptive svstems passivitv and avera~ing analvsis. Th e MIT Press, Cambridge , Mass. Astr~m, K.J. (1987). Adaptive f eedback control . Proc. IEEE , Vol. 75, No.2. Byrnes, C.I. and A. I sidori (1984). A frequency domain philosophy for nonlinear systems with applications to stab ili zat ion and to adaptive control. Proc. of 23rd IEEE Conference on Decision and Control, Las Vegas. Bruni , C., G. DiPillo and G. Koch (1974). Bilinear Systems: an appealing class of I nearly 1inear I systems in theory and app lication s, IEEE Trans . Auto. Controi, Vol. AC- 19, No. 4, August 1974, pp . 334-348. Fnaiech, F. and L. Ljung (1987). Recursive identification of bilinear systems. Int. J. Controi, Vol. 45, No. 2, pp. 453-470. Glad, T. (1987). Output dead- beat control for nonlinear systems with one zero at infinity. Systems and Controi Letters, Vol. 9, pp. 249-255. Goodwin, G.C . and K.S. Sin (1984) . Adaptive filtering predi ction and contro l. Prentice-Hall, New Jersey. Goodwin, G.C. , R.S. Long and B.C. McInn is (1980). Adaptive control of bilinear systems. Department of Electrical and Computer Engineering, University of Nel.·castle, Technicai Report EE8017. Hill , D.J. (1989). A generalisation of the small- gain theorem for nonlinear feedback systems. This proceedings. Kren er, A.J. (1975). Bilinear and nonlinear realization s of input-output maps. SIAlI J . Controi , Vol. 13, No. 4, pp. 827-834. Middl eton, R.H., G.C. Goodwin, D.J . Hill and D.Q. Mayne (1988). Design issues in adaptive contro l. IEEE Trans. Auto Controi , Vol. 33. ~o. 1 , pp. 50-58. Mohl er R.R. (1973). Bilinear control processes. Academic Press, :iew York. Ohkawa, F. and Y. Yon ezawa (1983). A model reference adaptive control system for discret e bil inear systems. Int. J. ControL Vol. 37, No. 5, pp. 1095-1101. Svoronos, S, G. Stephanopoulos and R. Ari s (1981). On bilinear estimation and control. Int. J. Controi, Vol. 34, \0. 4. pp. 651 - 684. ~' en C. and D.J. Hill (1987). Adapti\'.eline~r control of nonlinear systems. LnIversIty of Newcastle , Technicai Report EE8724. wen, C. and D.J. Hill (1989). :ionsingular and stab le adaptive control of discrete-time bilinear systems. l"niversity of ~el.· castle . Technical Report .

Yeo, Y.K. and D.C. williams (1986). Adaptive bilinear model predictive contro l. Proc. American Control Conference. pp. 1455- 1461.

C. Wen and D. J. Hill

234

". y

y

, ,~,--------',:~.------~,~c,------~.~o~ .. ------~:5~'.--

Fig. 3 . N

Fig. 1.

Algorithm for setting A(t-l).

'. ~ , I

I I

u

1

I

,I

0-

'40:.

...

~:.

~.

.. ..

I

l ~

:

\ .

:>---"

::::: .

.. "' :;-

Fig. 2 •

The set-point and output of the system.

The input of the system.