Decentralized model reference adaptive control of interconnected dynamic systems using variable structure design

Proce.dln(ls of the 29th Conference on Declrlon and Control

Honolulu, Hews11 Decembar 1990

WP-15-2

₌

5 2 0

Decentralized Model Reference Adaptive Control of Interconnected Dynamic Systems using Variable Structure Design.

A-Cheng Wu', Li-Chen F U ~ , ~ , and Chen-Fa Hsul Department of Electrical Engineering' & Information Engineering2

National Taiwan University Taipei, Taiwan, R.O.C. ABSTRACT

This paper deal with decentralized model reference adaptive control of large-scale interconnected dynamic systems. Using a variable structure design concept, we show that the s t e a d y s t a t e tracking errors can be made as small as desired despite the strength of the interconnections among the subsystems. Furthermore, the steady-state tracking errors will converge to zero with exponential decay rate provided the interconnections are absent.

I. INTRODUCTION

Motivated by the success in designing model reference adaptive control schemes [l] & [3], some authors explored the field of decentralized adaptive control 151, 161 & 171. Owing to direct a plications

bf

model referknce adaptive controllers [l] & $to each unknown subsystems as if they conditions have been established to limit the size of interconnection terms and, hence, retain the stability of the overall system. As has been pointed out in [6], the standard M-matrix condition seems to be a little far from being practical

so

that

a

modified model reference adaptive control scheme is then developed. There, the unknown interconnection strength is taken care of through the use of some structural constraints on those interconnection terms. However, due to the unsatisfactory performance of traditional model reference adaptive control schemes, the decentralized adaptive controllers stated above give limited transient responses and convergence property. In this paper, using a variable structure design concept motivated by Fu [2], a new decentralized model reference adaptive control scheme is developed. We show that the steady state tracking error can be made as small as desired despite the existence of output interconnections among subsystems which may be nonlinear and time-va.rying with any possible strengths. It is also noted here that our problem formulation based on only I/O measurements which ordinarily meets the practical environments in industrial applications is different from those in [5], 161 &

11. PR.OBLEM STATEMENT

Consider

a

class of large-scale interconnected dynamic were decoupled

1

rom each other, the standard M-matrix

[71.

systems described as follows

N

si: yi = Gi(s)ui

+jC1

Hij(s) Fij(Yj,t),

if

j

i O (1) where ui(t), yi(t) E IR are the input and the output respectively of the subsystem Si, and N ={1,2,,..,N}. The function Fij(yj,t) E IR is the interconnection input to the subsystem Si arising from the output of the subsystems Sj whereas Gi(s) is the transfer function from ui to yi and Hij(s) is the transfer function from the interaction input We assume that for each i there exist nonnegative but Fij(Yj,t) to Yi.

unknown numbers aij, j

#

i, j = 1, 2

,...,

N, such that

I

Fij( Yj? t )

I

S

aij

I

Y j

I

(2) Furthermore, we assume that the dynamics of the subsystem Si are also unknown, that is, the transfer functions Gi(s) and Hij(s), j # i, j = 1, 2

,...,

N, are not specified. An implication shows that the interactions to

each subsystem may not always appear at its control channel. Although transfer functions Hij(s), j # i, j = 1, 2,

...,

N, may differ from Gi(s), it is reasonable to assume that they have the same denominator.

The control objective is t o force yi(t) t o track the output of a given reference model for all i E N. A model for

S is formed by assigning t o each subsystem Si a local model

which possesses the desired characteristics. With appropriate, elementary assumptions, it is clear that in the decoupled case the exact model matching through an input and output feedback techniques is obvious[l]. However, because of the use of switching control input and the interconnections among subsystem, the ideal setting

of

parameters may not be such that the isolated

subs

stems match the corresponding models. Instead, diierent, possibly higher gain settings may have to be chosen by the switchin and adaptive mechanism in order to cancel the effects of the interconnection terms and

to

give faster and better tracking performance.

Mi : ymi = Mi@) ri (3)

In order to make this control design problem to be more tractable, the following conditions are also assumed:

Af)

The plant is strictly proper with relative degree one, that is, npi(s) and dpi(s) are both monic, coprime polynomials of degree (ni-1) and ni respectively, and npi(s) is Hurwitz.

A2) The sign of the high frequency gain, kpi, is known and without loss of generality we assume that kPi

>

0.

A3) The reference model is st:ictly positive- real (SPR), and for simplicity we assume nmi(s)=l and d,,(s) is a monic first order Hurwitz polynomial. The high frequency gain is chosen such that kmi>O.

Our proposed variable structure adaptive decentralized control law is given by

where

is a vector of available sjgnals,

-

ui

=o;

wi

+

spi

(7)

T T. T

oi = ( coj, C t i ,

dei,

dTi )T

wi = ( ri, wpi, yi, w f l )

I

is a vector of adaptation gains, SPi is an auxiliary signal t o be specified, and eoi=yi-ymi is the i-th output error. The signals wpi and W f i in (3.5) are states of the precompensator and the postcompensator defined as follows :

wpi=Aiwpi+bAiui, wfi=Aiwfi+bAiyi (8) Under the assumptions AI)--A3), it is well known [l] that there exist unique Oi = ( Coi, cpi, dei, d r i ) such that the decoupled closed-loop subsystem's model match the corresponding reference model. Since, 0; is known, we use their estimate Oi(t) in the control law (7) and update them according to the following adaptive law:

*

*

*T * * T T

Bi(t)=-ri(eOiwi+ uOi) (9) where l'i is

a

constant, symmetric, positive definite matrix of dimension 2nix2ni and u is a scalar constant chosen to perform

a

similar It u-modification 'I originally proposed in

[3]. Define the parameter errors @i(t) = Oi(t)

-

OT,

so that the 1-th isolated closed-loop subsystem is described by

T

Sci : xci = Acixci

+

bci( @ i wi - d*,i ymi

+

Spi )

+jCl

dcij Fij(yj, t ) N

i f j Y i = Cci Xci

A i

J

bci = [ bT 7 h i i 9 0 IT, cci = [ ci , O 1 0

]

T T

dcij = [ d i j , 0, 0 ]

.

Then, it is easy to derive the error model of the i-th subsystem as: [4]

s .

_{el: e,} . -

_-

_{Aci ei}

₊

_{bci (}

_@T

_{wi - d:i ymi}

₊

_{Spi )} N

+jFl

Dcij Fij (Yj, t)

i f j

eoi = Cci ei (11)

Spi=-sgn(eoi)[Miillwi(lSMiz 1ymi

I]

(12) If we choose Spi, a switching compensation signal defined as :

where positive numbers Mil

I

(1

0;

)I

and Mi2

>

I

d*,i

1.

We can conclude that the state error e as well as controller parameter 6' of the system Se are globally ultimately bounded. The above results can be refer to [4] for a more detailed insight.

V. SIMULATION

Consider the following possibly nonlinear time varying interconnected dynamic systems:

YI = Gi(s) UI

+

Hiz(s p(t)yz

YZ = HZl(S)P(t)Yl

+

c!

Z(S)UZ

where transfer functions

s + l 2 s + 4 G l W = s"+2s-1'H'2= s 2 + 2 s - 1

G2(s) =

pq

7 HZl(S) =

Sz-3

s

-3

s

$1

Function p(t) is the proportional gain of the interconnections. The results of simulations are shown in two cases

Case 1) p(t) = 1 for all t >_ 0.

Case 2) p(t) = IOsin(100t) for all t

>

0, and it results in fast time-varying interconnections.

2

1 .I 6 6 h

Fig. 1 The output error of subsystem 1 using current scheme-case 1

:I---

i 4 6 B ib

Fig. 2 The output error of subsystem 1 using current In Fig. 1-2, it is obvious to observe that in the above two cases the variable structure decentralized adaptive scheme force the output error to drop into zero in finite time.

scheme-case 2

VI. CONCLUSION

In this paper, a variable structure decentralized model reference adaptive control scheme for a class of large-scale interconnected dynamic systems has been developed.

Inaccessibility to states of each subsystem is assumed here. The controller is constructed based on the concept of variable structure design which, in turn, to provided much better transient performance than those obtained so far. Computer simulations are performed in two cases, namely, coupled case and strongly fast time-varying coupled case. References :

[l] Narendra, K.S. and Valavani. L.S.. "Stable Adaptive Controller Design-Direct 'Control", IEEE Trans. Autom. Contr., Vol. AC-23, Aug Fu, L.C., "A Robust Model Reference Adaptive Control using Variable Structure Adaptation for a Class of Plants", Proc. of American Control Conference, 1989.

Ioannou, P.A. and Tsakalis. K.. "A Robust 1978, pp. 570-583.

Adaptive Controller", IEEE Trans. Autom. Contr. Vol AC-31, NO. 11, 1986, pp1033-1043. Wu, A.C., M.S. Dissertation. National Taiwan University, Taipei, Taiwan, R:O.C., 1990. Ioannou, P.A., "Decentralized Adaptive Control of Interconnected Systems" IEEC Trans. Autom. Contr. Vol. AC-31, NO. 4, 1986. Gavel, D.T. and Siljak, D.D., "Decentralized Adaptive Control : Structural Conditions for Stability" IEEE Trans. Autom. Contr. Vol AC-34, " 0 . 4 , April 1989.

Hmamed, A. and Radouane, "Decentralized Nonlinear Adaptive Feedback Stabilization of Large-scale Interconnected Systems", IEE Proc. VOl-130, P t . D,NO. 2, March, 1983.