FP5 = 530 Procwdlngs ot 24th Conference on Decision and C o n t r o l

FI. Lauderdale, FL * December 1985

ADAPTIVE COIVTROL OF DECENTRALIZED SYSTEMS: KNOWY SUBSYSTEMS, UNKNOWN INTERCONNECTIONS *

Donald T . Gavel Dragoslav D. Siljak Lawrence Livermore National Laborat 0 7 7 . The B. & M. Swig Professor

University of Santa Clara Santa Clara. California 95053

University of California Livermore, CA 94550

ABSTRACT

One of the common assumptions in control of large intercon- nected systems is that models of subsystems are to a large extent known to a designer, and an essential modeling uncertainty re- sides in the interconnections. Current decentralized adaptive control schemes take no advantage of this fact. In this paper an algorithm is presented in which adaptation of local feedback gains is in the a direction which compensates for the unknown interconnections, while exploiting the knowledge about the sub- systems. As a result, we broaden considerably the class of inter- connected systems for which decentralized adaptation is feasible

I. INTRODUCTION

In modeling of large systems, it is standard to divide the system into a number of interconnected subsystems. A main advantage of this approach is that the modeling process is used to reflect our knolvledge about the isolated parts of the system and tha t , as a consequence. the essential Uncertainty resides in t,he interconnections. Conirol design is greatly simplified since the large problem has been broken into several smaller ones. Decentralized controllers have been shown to be robust i o a wide 'ange of nonlinear and time-varying elements in the subsystem mterac,t ions I .2

In spite of this progress in decentralized design, theze still remain sit,ualions where fixed decent-alized control laws may not provide t h r desired degree of robustness. For this reason? the adapti\se coni rol methods for single-input ~ single-output (SISO.) sysiems ' : 3 , have been extended i o decentralized sydems j4-6!. These schemes are designed to stabilize an interconnection of uncertain or completely unkno\vn subsystems. The ignorance of subsystem dynamics appears to he an artifice. however, which \\as used to make the centralized adaptive algorithms work in a decentralized setting, rat,her than to reflect a genuine modeling uncertainty about the subsystems. Furthermore, this artifice may unnecessarily prevent extensions of decentralized adaptive control to the broader class of int,erconnected multi-input, multi- output, (MIMO) subsystems.

The purpose of this paper is to explore the possibility of adaptively controlling known MIMO subsystems so as to s ta- hilize the overall system, or track a reference model: in spite of t h e uncertain interconnect,ions between the subsystems. The adaptive schemes presented here, like those in 14-6.; are based on nod el-reference adapiive control. In this approach. parame- t.era of t,he cont,roller are varied to try to match each closed-loop subsystem to a reference model's behavior. We note that in the decentralized model reference scheme. the overall model. con- sisting of isolated submodels, differs in structure from t h e inier- connected large-scale system, hence no variation of parameters will allo\l exact model matching. Instead we rely on high gain feedback to cancel the effect of interconnect,ion distur1)anc.c~ and force the system to closely follow the reference signals. ~-

a Work performed under the auspices of the 1l.S. Depart- ment of Energy by the Lawrence Livermote National Labora- t,ory under contract number W-7405-ENG-48. and .upported in part by the DOE Office of Basic Energy Sciencc:~. Engineering

ECS-831532i Research Program, and the Kational Scienc,e Foundation Grant

CH2245-9/85/0000-1858 $1.00 0 1985 IEEE 1858

11. PROBLEM STATEMENT

We consider the problem of controlling N interconnect,ed subsystems

A, Si : & = A,z, + B,u, t A I 3 x 3 . i 5 N (2.1)

3=1

where z, ( t ) E. R"1 is the state and u, ( t ) E RJ" is the input to S, at time t E R : and N = (1, 2 . . . , A i } . A,, B, , and A,, are constant matrices of appropriate dimension. The overall system of equations (2.1) can be represented in block matrix form as

S : S = A D X T A ~ X T B U ,

where z = (5:: zif, . . . : z!.)~, u = (UT. u r : . . . : u ; , ) ~ : A D = diag(A1: A z , . . . : A N } : B = diag(B1, Bz, . . . ~ B:v}, and Ac =

(2.2)

' A , ? I .

M ' c , shall assume that the syst>ems S is partially unknown. and thus some form of adaptive control is required. Further- more. we assume that all of the modeling uncertainty is in the interconnection matrix. A < , , hut that the subsystem matrices A , and are known.

The control laws are restrict,ed t>o be decentralized, that is each controller operates on its local subsystem exclusively, with no exchange of information bet,ween subsystems. In the adaptive controllers to be analyzed in this paper, this means that not only the feedback control law, but, also the adaptation mechanism, must involve local information only.

As is well known, even in the non-adaptive case, decentral- ized stabilization of S is not in general possible with an arbitrarl- AT. LVe shall assume a certain structure to Ac which makes sta- bilization possible. In the first algorithm presented, we require that the A,, be factorable as

where H,, '1 R f J r ' "i are bounded but otherwise arbitrary ma- trices. This is equivalent to requiring that:

Q ( A , ] ) : R ( B , ) . i . 3 t N (2.4)

where Q(.) denotes the range space of the indicated matrix.

\Ye will he applying the direct adaptive control philosophy in the schemes presented in this paper. In the direct approach. the feedback gain vector is adapted diredy. as opposed to a two step process of system identification followed by control de- sign. Direct adaptive control has been more sucresnful than the indirect method for large-scale systems because the difficult in- termediate step of subsystem identification is avoided. In the direct method, a reference model must be specified:

2, = A , T , - B,rt , z E .U (2 .5)

where % , ( t ) E Rn>. and r , ( t ) E RJ': is a bounded piecewise continuous reference signal. Equivalently:

. - 3 = A D Z - B ~ ! (2 .6)

where 3 = (ZT, 2; . . . . . P;.) . r = (r:. r T , . . . , r:,) . A D = diag{A,}. B = diag{B,}. The matrices A , and B , are computed as follow-s. Since we know the subsystem plant models. we can require that B , = B,. z E ,4. lye then must provide a feedback matrix h', E RJ, ' " i which satisfies

T T

'4, = A , - B,K, . i 6 ,V . ( 2 . 7 )

so that the A,. i 6 A'. are stable. These x, w i l l be known to the designer. The condition ( 2 . 7 ) is called the model malchirlg condition. and i, the model matching gains. It is well k n o \ v i I that choice of f i > (-an be used tl7 produce arbitraril! stabl(, -4 ; .

The state feedback law implied by (2 .7 ) :

u I = ~ R,x, .- r , . (2.8)

stabilizes the isolated subsystems. although it may not stabilize the system as a Mhole due to the rlffect of interconnecting sig- nals. . 4 1 1 ~ I . Zote that this is quite a different situation than in ordinary model reference adaptive control. where it is assumed t h a t there exist feedback gains that will exactly match the plant 1 0 a stable reference model. Decentralized adaptive controllers In-tead must rely on increasing the stability of the local subS!s- t cms (through. possibly. high gains) in order to stabilizr to the overall system.

111. ADAPTIVE SCHEME

The adaptive scheme to be presented is based on increasing optimal feedback gains until o\-erall stability is achieved. The optimal' feedback gain for subsystem i is:

K: = R; BTP, , (3 .1 )

where R, -- R' 'i . R, = RT ; 0. and PI is the positive definite solution to the algebraic LiapuLov equation:

P7.i, - .ATP, = -Q, , (3 .2 )

where Q7 -: QT '., 0. The control to be used for subsystem i is:

l i , - R,x, - a,K:e, - r , , (3 .3)

where e, = I, ~ 5,. Equivalently:

u = - K z -~ aK'e - r . (3 .4)

where e = (e:. e : , . . . e:.) a = diag{a,Ii,,}. K = diag(E1. K?. . . . . Rh'}, K' = diag{Ki, Ki . . . . . K.;.}. and IJj7 represents the p , x p , identity matrix. The a, are parameters to be locally adapted. Combining the control (3 .4 ) with the plant (2 .2) . we get:

i. = ( A D - B K ) x - aBK'e - Br . (3 .5)

T

Lsing (2 .6) an error differen~ial equation can be written

i = .Toe ~ nBK'e - Ace 7 . A c Z . (3 .6)

For preliminar? analysis. conqider the unforced case. r , ( t ) 0 for all 1 . Thii implies ? : i t ) ~- 0 .

' Ki i y the optimal state feedback gain for the isolated. model-matched. subsystem only. In the interconnected case. K,' is a suboptimal de(centra1ized feedback gain.

Theorem 3.1 If we use adaptation laws

\\here ?, > 0. then e ( ! ) -+ 0 as t - 3c and the a, ( t ) are uni- formly bounded.

O.lin Proof. Use the candidate vector Liapunov function

7,,(e,. a : ) ~ T P , : E ~ - . (3 .8)

T11r sj-mbol ci, represents the difference between a , ( t ) and a "sufficientl! stabilizing" constant a;. Taking the time deriva- t ive:

i., : ~ eTQ,e, - 2u,eTP,B,K:e,

(3 .9) , = I

-- 2 6 , e ~ P l H , R t ~ ' B T P , e ,

uzing K! = Rl-'BTP, and &, : a, - a;. and completing the square:

.\ ' 2 (3 .10) + R,' l 2 HlJe3 ~

j = 1

[ S . l l )

and compute

5 - e T ~ e - J H T R ~ ? o x - 1 ~ 1 ' 2 ~ ~

5 !-X, ( Q ) - X.hf ( H T R H ) m y { a : . I } ] e 1 r . (3 .12)

t where and X,(.) represent the minimum and maximum eigenvalues of the indicated matrix. respectively. Therefore. I' : 0 if a+ is chosen sufficiently large. This implies ( e . a ) goes to the invariant set, , e i ' = 0. and both e ( t ) and a(t) are uniformly hounded. This completes the proof.

The proposed adaptive s rhr~ne has the advantagc, thal i t i \ w r > - simple. in\-ol\ ing ani> one adaptive parameter per .suh>ys- tem. Also. the wbsystems are allowed to have multiple control inputs. a relaxation of a constraint to single input s>s tems i m - posed h>- earlier decentralized adaptive controllers.

IV. TRACKING

In th? case \shere model tracking is required. ~ ( t ) is not idcnticall>- zero and therefore acts as a disturbance term through : h c interconnections to drive the error model ( 3 . 6 ) .

e :iDe aBK'e T .4ce - (3 .6 )

This disturbance prevents exact model follow-ing but ire can nlak(, the error bounded. \\-e modif>- the adaptation law slightly and obtain the follouing theorem.

Theorem 4.1

If we use the adaptation laws

where 0, > 0. then the solution ( e ( t ) , o ( t ) ) is globally ultimatelv honnded.

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Proof. Let ?! , (e , , a , ) = e,P,ei +*,,&:. Along (3.6) and (4 .1):

i ’? - e , Q , e , T

- 2e~P,l?,H,e, - a;eTP,B,RFIBTPle,

t 2e~lJiB,I;Lz, - a;e%P,B,R,-’BTP,e, (4 .2 )

- 2n,ti,a; - .,ti; Completing squares

&, 5 - eTQ,e, a:-‘eTH,?R,H,e, - u,&f

, Z, T H , T R H , % , + uta;- . (4.3)

which can be satisfied i f

Choosing V = x:., ?’[ we find that V 5 0 if ( e ( f j . n ( t ) ) lies outside a bounded wgion. This implies e ( t ) and a ( t ) are globally ultimately bounded and proves the theorem.

Corollary 4.1

Tracking error. e ( t ) . converges to a residual set. O P = { e t R” : ‘ , e ’ ~ <. [ } : and can be made as small as de.;ired through choice of 0,.

Proof. From (4.3) we can see that

where g’ = min,(a:), 6: = max,(a;): d = max,(u,). We choose g‘ large enough to make the first term small. and u small enough io make the second term small: so (4 .4) is satisfied for any given 6 ‘> 0. Note that the small a will lead to possibly large values for a ( t ) which is intuitively reasonable since we would ex- pect tighter tracking to require higher feedback gains.

V. OUTPUT FEEDBACK

If only t,he outputs of subsystems are available for feedback then some form of dynamic output feedback compensator is re- quired. Under certain conditions, the decentralized observer presented in T I can be used. These conditions are summarized below.

For ease in presenting the argument below: let us choose the reference model of ( 2 . 5 ) so that K , = K: = R,-’BTP, (this gain is known t,o stabilize the isolated subsystem). Therefore the gains would be K , = (a, t 1)K: if t,he local s h t e variables were available for feedback.

In order for output feedback eventually to have the same effect as state feedback we require that

u , ( t ) - K , T ~ ( ~ ) as t - 00 (5 .4)

and we choose Yl and Z, so that

( 5 . 5 )

(3.(j)

The estimator w i l l converge as in ( 5 . 5 ) : subject to the ob- server dynamics ( 5 . 2 ) , system dynamics (2 .1) , and interconnec- tion structure ( 2 . 3 ) if and only if F,: T,: G,, and G, be selected so t,hat

F,T, - T,A, i G,Cl = 0 , (5.7) T,B, = 0 , ( 5 . 8 ) G , = O , (5.9)

as was shown in ,i]. Furthermore, Y, and Z, must exist to sat- isfy (5.6). That is: it is necessary and sufficient thai,

(5.10)

where ,Y(.) denotes the null space of the indicated matrix. Con- structive methods for computing F,: T,: G,, Yi: and 2, (if they exist) a r e given in 17,.

\Vhen thrse conditions can be satisfied, out,put feedhac,k ~hrough the dynamic compensators, ( S . 2 ) : ( 5 . 3 ) . is assymptot- icall? equivalent i,o state feedback, and the control law u , = (0 - l ) , ( J ’ 7 y I , ~ Z l q ) can be used wit,h t.he adaptive schemes pre- sent#ed in t h ~ s paper.

VI. COXCLUSION

\Ye have provided adaptive control schemes for decentral- ized large-scale systems that exploit a-priori knowledge about t,he sub! st ems and adapt to compensate for uncert.aint?- in the interco1111cc.t i o n s . Such schemes utilize the high gain feedback approach nhich requires that interconnection st,ructure t,o h r rc- qtricted t,o a certain class. that, is, interconnecting signals nus st hr i n t h e range space of t h e control input.

The adaptive decentralized regulator in this context has been proven globally asymptotically stable, and the model t,racking algorithm can be made t,o follow a reference signal t o an! degree of accuracy required.

Improvements over earlier schemes include a simplified adap- tation algorithm and the ability to handle MIhlO subsystems, using decentralized sbate estimators for output feedback if nec- essar!-.

REFERENCES

I ] . : 1). D. Siljak. Large-Scale Dynamic Systems: Stability and . S t r u c t u r e . Sorth-Holland. Kew York, 1978.

2 . 6 . Hiisevin: M. E. Sezer, and D. D. Siljak, “Rohlls t Drc~n- tralisrd Control Using Output Feedback,” IEE Prucetdr?,ys, 12.9-U: No. 6 : (1982), 310-314.

3 . ’ K . S. Narendra, Y. H. Lin, and L. S. Valavani, “Stable Adap- tive Controller Design, Part 11: Proof of Stability,’ IEEE Transactions AC-25 (1980), 440-448.

4.1 A . Hmamed and L. Radouane, “Decentralized Nonlinear Adaptive Feedback Stabilization of Large-Scale Intercon- nected Systems,” IEE Proceedings, 130-D, (1983), 5i-62.

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' 5 . . P. loannou. "Design of Decentralized .4daptive controller^." Proceedings o/ the Conference on Decision a n d Control. (1983). 205-210.

'(i. D. T. Gavel and D. D. Siljak. "High Gain Adaptive De- centralized Control." Proceeding.s of t h e Amencan Control Conference, (19%). 565-573.

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