On feedback stabilizability of decentralized dynamic systems

Automatiea, Vol. 8, pp. 163-173. Pergamon Press, 1972. Printed in Great Britain.

On Feedback Stabilizability of Decentralized Dynamic Systems + Sur l'aptitude ~ la stabilisation par r6action d'un syst~me dynamique deeentralis6 Uber die Stabilisierbarkeit von dezentralisierten dynamischen Systemen durch Riickftihrung 0 cnocorHOCTrt r cTarrtrm3attm,I 06paTaofi CBII3blO)/rmaMrIqecKofi )letlenTpa)lvI30BarIHO~ CHCTeMbI M. A O K I *

Several control agents having different information on the total state of a dynamic system must in general communicate with each other to stabilize the system by.feedback. Summary--The paper formulates and discusses stabilizability of decentralized linear time invariant dynamical systems with coordination and/or communication among control agents. Decentralized control systems are defined to be dynamical systems with several controllers, each operating on the system with partial information on the states of the systems. This restriction amounts to certain structural constraints on the feedback and other system matrices. With the constraints, controllability of the systems does no longer imply stabilizability. Algebraic and geometric approaches are used to obtain stabilizability conditions for decentralized systems.

Although decentralized systems do not necessarily require a large number of variables to be described, most realistic systems of interest would require a large number of variables; for example, planning processes for a national economy. MCFADDEN [2] considered a very special system of this type that arises in modelling certain aspects of economic systems where several national agencies exercise regulatory control power over (possibly) different aspects of economic activities. Another example of large scale decentralized systems is a system of electric networks belonging to several electric power generating companies connected together by tie lines. Electric power station operators and dispatchers have direct controls of power generation and regulation of frequency and voltage over electric networks in their own regions, but have no direct control in regions belonging to other companies. The electric companies buy and sell electric power among regions belonging to different power companies, hence dynamic behavior of electric currents in these networks are influenced by several control agents acting partially independently of each other. Given large scale decentralized systems, a realistic assumption must therefore be made that no control agent, be it a supervisory control agent, also called central planning board or helmsman in the economic literature, or a local control agent, a local manager, possesses the complete descriptions of the systems to be controlled and of the environments in which the systems are to operate. Since each agent possesses a different set of information on the true system state, it is possible for the system to become unstable in the absence of communication among control agents.

1. INTRODUCTION MORE and more, systems originating outside the traditional domain of control problems are attracting the attention of control engineers. Most of these systems, such as those arising from socioeconomic system modelling, require a large number of variables to describe their dynamic behaviors adequately. Such systems will be called large scale systems in this paper. A feature that is really novel in these large scale systems and that is absent in the traditional control systems is the inevitable and essential presence of what WITSENHAUSENcalled non-classical information pattern [1]. Namely, several control agents having non-identical information on the system structure, state vector, parameters and so on, are involved in controlling the same system. Such systems are called decentralized systems in the economic literature [2]. We will adopt this term. * Research reported in this paper is supported in part by NSF Grant 2032 and in part by AFOSR Grant 1328-67C. t Received 11 January 1971 ; revised 19 May 1971 ; revised 16 August 1971. The original version of this paper was presented at the IFAC Symposium on Systems Engineering Approach to Computer Control which was held in Kyoto, Japan during August 1970. It was recommended for publication in revised form by Associate Editor P. Dorato. 163

164

M. AOKI

In the hierarchical system theory [3], control of large scale dynamic systems is approached by decomposition and attendant hierarchical control schemes. Namely, a large scale problem is broken into a number of smaller or simpler subproblems, and two (or multi)-level control structures are imposed such that local optimization is performed by the first level controllers and the global optimization or co-ordination of local controllers is performed by the second, or still higher level, controllers with certain exchange of information between the controllers. The exchange of information that has been studied in the hierarchical system theory is mostly that between the local controllers and the higher level controllers, usually in the form of Lagrange multipliers. Although the hierarchical system theory provides some principles such as interaction prediction principle, coordination principle and so on, that may be useful conceptually in discussing various questions of large systems performance with the second level, or supervisory, control agents, the questions of other forms of information exchange among local controllers has not been studied to any great extent, such as direct information exchange among local agents. Also the implication of the pattern of information exchange on system performance has not been studied in any detail. For example, non-classical information pattern implies that certain structural constraints must be imposed on the control matrix and observation matrix. Stability of the decentralized system depends on several considerations such as the amount of overlaps in observation data made available to controllers and so on. These have not been considered in usual control problems, nor in the multilevel theory in any detail. We examine the stabilizability question of large decentralized dynamic systems with feedbacks in this paper both with and without higher level control agents, leaving the other questions, such as the estimation or control problems, to be treated in separate papers. We give a formulation of the stabilizability problem for a class of decentralized control systems and give necessary and/or sufficient conditions for stabilizing these systems both with and without communications among control agents. These results show that usual control theoretic results do not necessarily apply to this class of systems without some modification. For example, controllability does not imply stabilizability for decentralized systems. Some of the results are relevant to the stabilizability of output or incomplete state feedback systems [4]. The presence of several control agents and the possibility of communication among agents, however, present new aspects to the stabilizability

problem not covered by the literature on incomplete state feedback systems. These are presented mostly in Sec. 3.3.

2. CLASSIFICATION OF I N F O R M A T I O N PATTERN

Consider a linear time invariant dynamic system with k local controllers (control agents) described by k

2=Ax+

~ Biu ,

(1)

i=l

where x is the n-dimensional state vector, u~ is the r~-dimensional control vector i = l . . . . . k. The vector u~ in (I) represents the control by the i-th controller, i = 1. . . . , k. Write B as [B1, B 2. . . . . Bk] where B i is an (n x r3 submatrix with k

ri=r. i=1

Each local controller is assumed to have its own observation scheme* y~(t) = H~x(t),

i= 1, . . . , k

(2)

where y~ is the m~-dimensional observation vector, k mi=ff/. i=1

We use the notation ~ to indicate the direct sum of matrices. For example,

We consider both static controllers and dynamic controllers. Static controllers generate control signals proportional to the current observation records, for example, u(t) = N ( t ) y ( t ) + v(t)

(3)

where u(t)r = (ul(t) T. . . . .

Uk(t) r)

y ( t ) T = ( y t ( t ) T. . . . .

yk(t) T)

v ( W = (Vl(t) ~ . . . . .

vk(t)~).

and where

* A more general situation where the number of observation posts is different from the number of local controllers is not discussed here.

On feedback stabilizability of decentralized dynamic systems The variable v~ is used to represent either the signal from the outside or other control signal that is also generated by the i-th controller as a (measurable) function of his own observation record additionally. Dynamic controllers have dynamic structures given as (4)

= Ls + M y

165

To avoid trivial cases, we assume that the system (1) and (2) are jointly controllable and observable, i.e. (A, B) is a completely controllable pair and (A, H ) is completely observable pair, where H T = [H~ . . . . . /-/[]. For later use, we introduce the notation for the controllable subspace R i and the observable subspace Si of the controller i by

and generate control signals by

RIA_[AIB,I SiA_[AT[Hr] ,

(3A)

u(t) = Ks(t) + v(t)

i=1 .....

k

where where s is the controller state vector and where

n-1

[A[Bi]A E AJ~i, j=o

: = C , . . . . . sD, k

dim s = I,

dim s i = li,

= ~ li. i=!

The variable v in (3A) has similar meanings as that in (3). By imposing various structural constraints on matrices K, L, M and N different information pattern imposed on the decentralized systems can be modelled. In the interest of brevity of exposition we examine only dynamic controllers of the type L=L14

•. • -i-Lk

(4A)

where L i is an (li x l~) matrix. This assumption states that the controller dynamics are decoupled. We allow various structures for K, L, M and N. If static controllers do not communicate among themselves, the matrix N in (3) is taken to be N = NI -I- . . .

4 Nk

...

4 Mkk

(5)

where Mii is an (I~ x m i ) matrix or that K i j ~ O for some (i,j) submatrix pair i ¢ j , i.e. K is not given as K=Kll+...

q-Kkk

and where ~ ( ' ) indicates the range space. Then the complete controllability and observability assumption is expressible as RI + • • • + Rk = R" = $ 1 + . . . +Sk. In addition to the variety of information patterns and the interactions of local controllers for the decentralized control problems mentioned earlier, the observation schemes could be used to classify the decentralized systems. We may distinguish these possibilities: (i) Independent observation case where Si, i = 1. . . . . k are all independent [5]. This case can be further subdivided. For example, if H = H l l & H 2 2 q - . . . - l - H k k where H , is an ( m i x n i ) matrix, then the i-th controller observes only x; through n u where x r = ( x r . . . . . x~), d i m x l = n i, k /'/---~ Z n i " i--I

No other components of x is observed. A special case of this, for example, is y i = x i , i = 1, . . . , k.

where Ni is an (ri x m~) matrix. In dynamic controllers, communication among them may be modelled either by assuming that M i j ¢ 0 for some ( i , j ) submatrix pair, i ~ j , i.e. M is not given as M=M11-b

~i=~(Bi)

(6)

where K , is an (ri x/i) matrix. The latter may be interpreted to model a situation where agents exchange not the raw observation data but some summary of his observations since the controller state vector may be used to represent his state of knowledge of the total system. Any combination of the above is also possible.

(ii) Dependent observation case where k

N

i=1

{0}.

It can be shown that if Sic~Sj ~ {0}, then controller i can derive certain information on xj as well as information on the past controls employed by the j-th controller. This topic will be covered in a separate paper. In terms of R i, the case of dependent Ri, i = 1. . . . . k, may be interpreted as overlapping control effects on some system states and would be a more realistic assumption than assuming Ri, i = l , . . . , k are independent. The latter implies that controllers have divided their "sphere of influence" in a noninteractive manner and would not be a realistic assumption for large decentralized systems. Hence the former assumption is used in this paper.

166

M. AOK!

To summarize, the above classification scheme is constructed to reflect intuitively a type of control situation where a given controller has a direct control over a subsystem through its own control matrix, where each of the controllers is supplied with either independent or overhapping information on the state of the whole system and the controller dynamics is influenced solely by the information on the system state vector that the i-th controller possesses (Mij=O, Kii=O for all i#j) or by data received by other controllers also influences the state of the controller (Mij# 0 and/or Kij# 0 for some i¢j). Other modelling possibilities exist and will be introduced as needs arise. K, L and M are the primary design variables. The matrices B and H may be given a priori for some systems.

3. DECENTRALIZED SYSTEM DYNAMICS AND THEIR STABILIZABILITY QUESTION With controller dynamics, the combined plantcontroller state vector is governed from (1),~(6) by

[~]=[MAH~][Xs]+[B]v.

(7)

In case the controller has no dynamics, (7) reduces to

:¢= ( A + BC)x +By

(7A)

where k C = NH =

~

i=1

NIH i

where

system (7A) with

A=

0

1

0

~

1

1 0 B=

0 and C l__

C= 0

c3:0

In this example, the controller 1 makes use of information carried by the first component of the state vector only and the controller 2 makes use of information contained in the second component only. Note that (A, B) is a controllable pair. Its characteristic equation is given as (1 +c~-X),

0,

0

o,

l-k,

1

0,

c2,

2-2

O=IA+BC-XlI=

= (1 + c I - 2)(22- 32 + 2 - c3) Therefore, there is at least one unstable root by any choice of C of the given structural form, hence the system is not stabilizable.

3.1 Systems with dynamic controllers We may assume without loss of generality that L is a stable matrix, i.e. the controller dynamics is stable. The characteristic equation for (7) is 0= A-)'/, MH,

LBxK/

=IL-XZlI(A-Xl)-BK(L-XI)-IMHI

N = [N1 . . . . . Nk] and n T = [HL

....

where from (4A)

( L - X I ) - ' = ( L t - - X I ) -1 + ( L 2 --21) -~ -[- . . . 4-(Lk--XI) -a.

When no structural constraints are imposed on C it is well known that the controllability of a pair (A, B) is equivalent to the pole assignability [6, 7]. That is, there exists a matrix C such that A +BC has a set of prespecified eigenvalues subject only to the condition that the complex eigenvalues appear in complex conjugate pairs. This implies the stabilizability of (A, B) is equivalent to controllability of "unstable modes" of A. These results no longer hold for the decentralized systems under discussion since the matrix C is constrained to be of particular structure. A simple example illustrates this. Note that this example is similar to the one in (7). Consider the

We can resolve stabilizability question immediately for two special cases. These are covered by the next two propositions.

Proposition 1. Suppose that A is block lower triangular Axl

0

A2~

A=

Akx

Akk

On feedback stabilizability of decentralized dynamic systems that

Since the approximation can be improved arbitrarily by choosing a's large, the proposition is true.

B = BI14- . . . -[- Bkk

where (A u, Bu) is a controllable pair in R"', i.e. [ A , I B , ] = R " ' , for i = 1 . . . . . k, and that H = H 1 x 4- • • • Hkk, HU = I,,, m i = n t, i = 1 . . . . . k, where I~, is the identity matrix of order nt. Assume that the controller i knows A u and Bit, i = 1. . . . , k. There always exists L and v r = Iv r, . . . . Vkr], where v t is generated by the controller i as a proportional feedback, for any choice of M and K, that stabilizes (7) with no communication among agents. Proof. Since y~-- x t by assumption, the i-th agent can generate vi as vi = Cixi where Ci is an (ri x nt) matrix such that A , + B u C t is a stable matrix. This is possible since agent i knows A u and Bu, (A u, Bit ) is a controllable pair and since no further constraints are imposed on Ct. Then the dynamic matrix in (7) becomes

M,

,

B C = B l l C ~ - [ - . . . 4-BkkC k

(8)

where A + B C is a stable matrix since all the submatrices on the diagonal are stable matrices ( A . + BitCt) by choice of Ci, i = 1. . . . . k. Let/~t . . . . ,/~. be the eigenvalue o f ( A + B C ) and let r=max]gtl

167

Remark. If M = 0 , i.e. if a static controller is employed, then the system in Proposition 1 can always be made stable. There are cases where dynamic controllers are desired from other considerations such as observers, aggregated system models and so on. Thus the case M # 0 is of interest. This remark applies also to Proposition 2. The dynamic system of Proposition 1 may be considered to be composed of k subsystems in a chain, each with its own controllers and with dynamic coupling on the subsystem j from all the subsystems i, i < j . Proposition 2. Let the system be such that k---n, B, H, K, L and M be all diagonal matrices, i.e. n=r=m=l. If bthl~O for all i = 1. . . . . n where B=diag(b t . . . . . b,), H = d i a g ( h t . . . . . h.) then (7) is stabilizable. Proof. Let vi=diyt, i = 1 . . . . . becomes

where D = diag(d 1. . . . . d,). Then by Gersgorin's theorem, the eigenvalues of the system lies in the closed region

•

I2-(a,t + b, dtht)l <- E

j=t

Choose L to be diagonal diag(el . . . . . all i and define

R=min[at[. l

at), at < 0 for

Now choose e's so that R is so large that the matrix (8) satisfies the condition of weakly coupled systems [8], that is n612621 < 1 g2

and

r/R< 1 where 6ij is the maximum of the modulus of the elements in (i, j ) submatrix, where (1, 2) submatrix is B K and M is the (2, 1) submatrix in (8). Then the eigenvalues of the matrix in (8) are approximately/2~ . . . . . /~., al . . . . , at by the weakcoupling condition where Pt i = 1. . . . . n are given as the solution of JM-~I] =0.

The condition of weak-coupling implies that the term B K L - X M contributes only a small change to #'S.

Then (7)

MH,

i

[A + B C - B K L -

n.

[2-ohl
lat, I+ bik,

i = 1. . . .

,n

(9)

where L = diag(al . . . . , a , ) . By choosing d i and at sufficiently negative 2 can be made to have negative real parts. We next consider the case where H is not diagonal. The stabilizability of the system (7) is reduced to that of A + BDH, MH,

by choosing v = Dy, where D = D 1 4 - . . . 4-Dk. The stabilizability of the system with dynamic controllers (7) can then be implied by that of the system with static controllers A + B C , where C = D H , since stable eigenvalues of A + B D H will not be moved to make them unstable for a sufficiently large diagonal matrix L such that the plant and the controller dynamics become weakly coupled. We turn next to static controllers for this reason.

168

M. AOKI

3.2 Systems with static controllers First, we examine a special case of k = 2 , r = 2 . B=[bl, b2]. Assume that [Albdc=R", and b#[AIb d, or the whole system is not controllable by b~ alone.* Let h~ be the smallest positive integer such that Ah'b~ is linearly dependent on bt, Abx . . . . . A h~-tbl. Then b2 is linearly independent of these vectors. Using the set of basis vectors introduced by LANGENHOP[9], we can put A and B into a canonical form

A=[~

B=[e~,

~ A::],

e~J'

nl+n2=n

where e~ is an (i x 1) matrix with 1 in the last row, and where

0 0. . . . .

i: I.....

nl

/~j=/~j-~2~,

j = 1. . . . .

n2

where Yu is thefith component of Yi, i = 1, 2. Thus, a desired pole assignment, hence stability of system is achieved if the affine spaces a+R(H~x),qfl+R(H~2 ) contain vectors c( and /3' respectively, representing desired (stable) pole assignments, where aT=(ax . . . . .

~,,) andqflr=(fll . . . . .

:

n I X n 1

1

ut=d~'y 1 and --~1,

--a2,

0,1, ...,

" " " ,

: n 2 x n 2.

(11)

if 0~+ R(Hrx) and fl + R(H~'2) contain vectors , ' and fl' representing stable pole configurations, where Alx and A22 are in the phase canonical form with the last row vectors o~r and flT, as defined in connection with (11). This proposition can be generalized to

1

Proposition 3.1. Let (A, B) a completely controllable pair with

-th .....

Control signals are generated by

Fd~H17 [u17d-Le[d~y17 yd-Ld . j x Assume that H t = [ H l l , 0] a n d / / 2 = [ 0 , H22 ] where H . is an (m~ x n3 matrix, i = l, 2, namely no overlaps in observation.

R"={bl . . . . .

Ah'-lbl, be . . . . .

n

bt. . . . .

A~'-lb,}

where

t
Then

A+BC=A+BDH,

u2=dr2y2

--0~nl

0

A12 = . ..,

qfl.2), and

Proposition 3. Let k = 2 , r = 2 with b2¢[Albl]. Then the system is stabilizable by the output feedback control with the observation scheme H = H l l q-H22 and

0

All =

O~ri=ai--~li,

~' and fl' are defined similarly. Thus we have established

0

0, 1,

Thus (A+nC)kk, k = 1, 2 is in a phase canonical form with ~ and fli replaced by

D=df-~dr: 2xm,

Hxl, 0 . . . . .

0

0, H22 . . . . .

0

=EnT,

where the row vectors of B D H are all zero except for the last row vector yr in (BDH)t t and the last row vector yT in (BDH)22 and where y~ is an (n~ x 1) matrix, i - 1, 2 given by

H=

O,

* ° . ) ntt

H' m

y i - H u d i, __

T

i=1, 2.

u~= d~yi * The notation [A[bd is used to indicate the subspace spanned by bl, Abl . . . . etc. as in [A[Bd.

U j----0,

i=l,...,t j=t+l .....

r if t < r

On feedback stabilizability of decentralized dynamic systems if ~i+R(H~) contains vectors (~,)i representing stable pole configurations, where ct i, i =

l , . . . , t

are the vectors that appear in the phase canonical form of the diagonal submatrices of A in the Langenhop canonical form.

169

where the partitions of A and B conform with that of the state vector. In (14), A2a=A3~=0 since R 0 is A-invariant. A32 = 0 because of the A-invariance of R1 = Ro +/~ 1. Similarly A23=0 from the A-invariance of

R2 = Ro +/~2" By direct calculation we establish

Proposition 4.

Remark. To carry out the test of this proposition is not a trivial numerical problem for large systems. As the preceeding discussions indicate, the conditions we have obtained so far by the algebraic approach becomes rapidly very complex except for some special cases and we tend to lose sight of the significance of the conditions. For the sake of simplicity of presentation, we consider the case k = 2 in the rest of the paper. 3.3 Decomposition of state space I Consider a dynamic system with two local controllers as given by (1), k = 2 . We assume A to be nonsingular without loss of generality.* Let

Ri=Ri(~R o,

(A~ 1 A12"~, A22J

(Bl1"~ ~B21J

is a completely controllable pair in RI. (A33, B22) is a completely controllable pair in/~2.

Corollary 1.

(A22, B21) is a completely control-

lable pair in/~1.

Proposition 5. Suppose Yl = (xl, x2)T and Y2 ----"X3 with respect to the decomposition of (13). Then there exists a matrix C

i = 1, 2

where

such that A +BC is stable.

R 0 -----R 1nR 2 .

Proof. From Proposition 5 there exists [Cll ,

Note that Ro, R,, i = 1, 2 are A-invariant. Since R0,/~1 and/~2 are three independent subspaces decompose R" into

Rn = Ro ~)/~1(~-~2"

(A~1 AI2"~ Bll A22J'[-(B21)[C11' C12]"

(12)

The subspaces/~, i = 1, 2 are not uniquely defined. We choose them to be orthogonal to R o. With this convention,/~i, i = I, 2 becomes unique. Decomposing the state vector into the corresponding three subvectors

x(i)T=(xl(i) r, x2(i) r, x3(i) r)

C12 ] which stabilizes

Similarly there exists C23 that stabilizes A33 +B22 C23. Then A + BC is stabilized by C = [ C11, C12] 4- C23.

Corollary. I f y I = x 2 and Y2 =x3, then the system is stabilized by ux = C12x2 and u 2 = C23x 3 if A 11 in (14) is a stable matrix.

(13)

Remark. Even though no data is exchanged,

where

there is an implicit agreement among agents, or coordination by a higher level agent, in deciding to observe the state vector in "non-conflicting" manner.

[XI(~)r, 0, 0]reR0 etc, the matrices A and B are represented as

All A=

A12 A13 1

0

A22

0

0

0

A33

I B11

J

B12

B=

B21

0

0

B22

(14)

* If N(AT)#{O}, then by aggregating the state vector the state vector dimension can be reduced [10].

Communication among agents. So far no explicit communication among control agents has been introduced. If certain information exchange takes place between the agents, then system is stabilizable from the output feedback under certain conditions. Suppose R" is decomposed as R 1@/~2 and agent 1 controls Rt and agent 2 controls /~2. This assumes that agreement exists between the agents. Let

170

M. AOKI

and z2=x 3 in (13), dimz 1=n I and dimzz=n2, and define

Thus, if l is such that m21>_n2 and if

//22 ={o}

W

Gll \B21 )

G21 = 0 ,

0.

°.:.22

then z2(0) is recovered without error given y2(0) • • • y2(lr) and u2(0) •. • u2((l- 1)T). Proceeding similarly, agent 1 upon receiving z2(0) and u2(0), . . . , u2((k- 1)T) can recover zl(0 ), if he remembers all of his past controls if he knows ~II' ¢~12,t~ll and ~12 and if

Then instead of (14) we have

:1

\ o

FI2 Z1

Gll

a22:~g2J "

IHll

(14A)

Hllq~11

w

={o},

Assume

Hll~b~l-I

0, n22](::):n22z2 where H l l is (rex x n l ) and H22 is (m2 ×n2). If the inputs are changed only at discrete time instants kT, k = 0, 1. . . . . then

z((k + 1)T) = ¢bz(kT) + Wu(kT) where ~ = exp FTA(q~ ~ --

t~12~ (~22)

~=f~er(T-OGdt=(~l

~/12~

for some k such that mlk>nt. Note that the transmission of u's from agent 2 to 1 is necessitated by assigning Ro to agent 1. If Ro is assigned to agent 2, the transmission of ut's by agent 1 is required in addition to the transmission of appropriate initial conditions. Remark. To carry out the procedures indicated above, it is assumed that agent 1 or 2 know F12 and G~2 in addition to their own subsystem dynamics and control matrices, or a supervisory agent must be assumed to exist which supplies the coupling terms to agent 1. The case where the subsystem coupling terms are unknown is not treated in this paper. See for example Ref. [14] where the details of information exchange is discussed. Thus, we have

Proposition 6. If vector quantities can be exIn particular

changed among agents, the system is stabilizable by output feedback ul = Clyl, u2 = C2Y2 where

k-1

(z,)

z2(kT)=dpk2z2(O)+ ~ tkkj-1~O22u2(j).

j=0

y l = [ H 1 1 , 0] z2

Thus

and

//22 H22~b22

I y2(0) 1

z2(0)+ 2

y2(lT) where (2 is a known vector to agent 2 if F22 and G22 are known to it, since t~22---expf22and ~b22=

I"|ref(r -°G2zdt.

3o

if (q~11, Hxt) is an observable pair in R"' and (q~22,//22) is an observable pair in R "~.

Proof. Vector quantity exchanges guarantee that each agent knows pertinent system matrices after a finite time. If such k and / exist, then after a finite time, Zl(0) and Z2(0 ) are known to agent 1 and 2 respectively hence the problem reduces to that covered by Proposition 5.*

• It is possibleto statea propositioninvolvingderivatives of yl(0)andy2(0)[11,12].

On feedback stabilizability of decentralized dynamic systems

Decomposition of state space II. Using S t defined on p. 7, i= 1, 2, Proposition 6 shows that if zi(0)s&, i= 1, 2, the output feedback can stabilize the decentralized dynamic system provided agents can exchange vector quantities. We will next show that even when these assumptions do not hold, appropriate communication among the agents can stabilize the decentralized systems by the output feedback when an additional condition is imposed on the system structures. Assume that some subsets of the natural coordinate basis {el . . . . . e~}, where er=(o, 0 . . . . .

0,1,0 .....

0),

spans Ri and S , i= 1, 2. A sufficient condition for this is that each Jordan block of A has a distinct eigenvalue, i.e. A is cyclic [13]. Under this assumption /~I-=R2 and the intersection operation distributes as

and

B=

Bll

B12

B21

0

B31

0

0

B42

0

352

0

=~H21 i= i, 2.

Combine the above with the decomposition (12) to obtain (see JACOBSON[5] p. 30)

Rn=Ro~)Tl~)([~lnSl)t~T2~(l~2nS2)

(12A)

(16)

where the partition of A and B conform with that of the state vector. In (16) A2~ . . . . . As~=0 because Ro is A-invariant. Ro~)T1 ~)(Rl nSl) = Ri is A-invariant making Aa2=A43=As2=A53=O. Similarly R2=Ro~)T2~)(I~2nS2) is A-invariant. Hence A24=A25=A34=A35=O. /~1 =R~ is Ar-invariant, so is $1. Thus (/~1c~$1) is Ar-invariant making .432=0. The subspace (/~2nS2) is also Ar-invariant since/~2 =R~ making A,a =0. We have

B ['Hll =(/~,c~S~)~(/~lnS,)

Hla H14 Hts~

n22

H23

Ti = (/~ic~S~-) ,

0

(17)

H25 ]

since x2 is the only subvector not observed by Controller 1 (x2eS-~) and since x, is the subvector not observed by Controller 2 (x,eS~). From Proposition 4, we have

Corollary 2 (Proposition 4). A22, A23"~, 0, A33]

where

171

(B21~ ~B31/

i=1, 2. is a completely controllable pair, and

The corresponding decomposition of the state and control vectors are

o,

X(i)T=[xl(i) r, x2(i) r . . . . . xs(i)T] r

0]eRo

and

[0, x2(i) T, 0 . . . . .

0]gT1

(15)

etc. we have

x(i + 1) = Ax(i) + Bu(i) where

A=

All

A12 Ala

0

A22 A23

A14 Al5 0

0

0

0

A33

0

0

0

0

0

A,4

A45

0

0

0

0

A55

AsO

(B42~

\852!

is a completely controllable pair.

where [xt(i) r, 0 . . . . .

A44, A45~,

From Proposition 5 we know that if xl, x2 and x3 are available to agent 1 and x4 and x~ are available to agent 2, the system is stabilizable. Note(x r, x2) r ~ and x3 m Proposition 5 correspond to (x r, x2r, xar)T and (x r, xr) T in the decomposition (12A). x 2 cannot be recovered by agent 1 from only his own observation data alone. Since S~=g2, agent 2 can recover x2(0), say from his own observation and knowledge of/~1 and must transmit it to agent 1. Also if Roc~St~{0}, the corresponding components of x~(0) must be recovered by agent 2 and transmitted to agent 2.* Similarly xa(0), say, cannot be reconstructed by agent 2 but must be reconstructed by agent 1 and transmitted to agent 2. The detail is analogous to the constructions in connection with Proposition 6 and hence will be omitted.

* A remark similar to that made relative to Proposition 6 also applies here.

172

M. AOKI

In the above, the subspace Ro was assigned to agent 1 arbitrarily necessitating transmission of u2's from agent 2 to agent t. See the sentence immediately prior to R e m a r k on p. 170, R0 can be assigned to agent 2 or Ro can be further subdivided between agent 1 and 2. A cost of communication must be introduced to be able to decide on an " o p t i m a l " division o f R 0. This topic will be treated separately, see Ref. [13].

4. CONCLUSIONS It has been pointed out in the previous section that the subspace R 0, which is the controllable subspace of both agents, requires either communication and coordination o f control actions by both agents or assignment to one o f the agents, or by a central agent. The subspace R o varies with partial feedback. As an example, consider ui=N~y+vi,

i = 1 , 2.

Then (1) becomes for k = 2 = ( A + B I N 1 H t + B 2 N 2 H 2 ) x + B i v l +B2v2.

Define •~ 1 ( N 2 ) = [ A + B 1 N 1 H 1

+B2N2H2IB1]

= [A + B 2 N 2 H 21B1]

and ~2(Nx) = [A + B I N 1 H t +B2N2H2]B2] =[ A + BIN1HI[B2]

,~ I(N2) + RI(N1) = R l + R 2 = R n .

As NI(N2) are varied ~ l ( N 2 ) ( ~ 2 ( N I ) ) c o u l d become larger or smaller than RI(R2) depending on the relations a m o n g various subspaces involving B ' s and H ' s . State vector feedback results can be extended to output feedback systems under certain conditions. Ref. [13] contains these results. In this paper, the stabilizability question of decentralized dynamic systems has been formulated. For example, Proposition 5 gives the stabilizability conditions for the case no observation data is exchanged a m o n g agents, and Proposition 6 gives the parallel results for the case observation data are exchanged. It is to be noted that even in Proposition 5 where no communication a m o n g agents exists an implicit coordination of agents' actions exist by the assumed observation scheme. When no communication a m o n g agents are allowed, no agreements a m o n g agents or coordination of local

agents by a central agent exist, only weaker stabilizability results are obtainable, even with the complete state vector feedback. This is because agents' action may conflict in R o. As mentioned just prior to Proposition 6, a more realistic treatment of large decentralized systems would require system parameter estimation in decentralized settings as well. Finally, cost for communication, more precisely of transmitting finite dimensional vector quantities a m o n g agents, and cost for computation must also be incorporated to render the results o f this paper applicable to more realistic problems. gratefully acknowledge the fruitful discussions with Mr. M. T. Li. Acknowledgement--I

REFERENCES [1] H. S. WIrSENHAUSEN:A counter-example in stochastic optimum control. SIAMJ. Control 6, 131-147 (1968). [2] D. L. MCFADDEN: On the controllability of decentralized marco-economic systems; assignment problems. In: Mathematical Systems Theory and Economics, pp. 221-239. Springer-Verlag (1969). [3] M. D. MESAROVlC, D. MACKO and Y. TAKAHARA; Theory of Hierarchical, Multilevel Systems. Academic Press, New York (1970). [4] F. M. BRASCHand J. B. PEARSON: Pole placement using dynamiccompensators. IEEE Trans. Automatic Control AC-15, 34--43 (1970). [5] N. JACOB,SON: Lectures on Abstract Algebra, Vol. 2. Van Nostrand. [6] W. M. WONI-IAM: On pole assignment in multi-input controllable linear systems. 1EEE Trans. Aut. Control AC-12 6, 660-665 (1967). [7] W. M. WONHAMand A. S. MORSE: Decoupling and pole assignment in linear multivariate systems: A geometric approach, PM-66 (Revised). SIAMJ. Control 8, 1-18 (1970). [8] R. MILNE: The analysis of weakly coupled dynamical systems. Int. J. Control 2, 171-199 (1965). [9] C. E. LANGENrfOP: On the stabilization of linear systems. Proc. Am. Math. Soc. 15, 735-742 (1964). [10] M. AOKI: Aggregation, Chapter 5 of Optimization Methods and Applications for Large-Scale Systems (D. WXSMEREd.). McGraw-Hill (1972). [11] G. BASILEand G. MARRO: Controlled and conditioned invariant subspaces in linear system theory. J. Optimization Theory and AppL 3, 306-315 (1969). [12] G. BASILEand G. MARRO: On the observability of linear time-invariant systems with unknown inputs. J. Opt. Theo. and Appl. 3, 410---415(1969). [13] M. T. LI: Decentralized Dynamic Systems. Ph.D. dissertation, Department of Systems Science, University of California, Los Angeles, California, August (1971). [14] M. AoKI: Control of large decentralized dynamic systems, XII.l.l-I.5. Proc. 1970 IEEE Symposium on Adaptive Processes, Univ. Texas, Austin, 7-9 December 1970.

R6sumg--Le pr6sent article formule et discute l'aptitude b. la stabilisation de syst/~mes dynamiques lin6aires decentralisds, invariables dans le temps, avec une coordination et/ou une communication entre les moyens de commande. Les syst~mes de commande decentralisds sont d6finis comme 6tant des syst6mes dynamiques avec plusieurs r6gulateurs dont chacun agit sur le syst6me avec une information partielle sur les 6tats des syst6mes. Cette restriction 6quivaut certaines limitations de structure de la r6action et des autres matrices du syst6me. Avec les contraintes, l'aptitude

On feedback stabilizability of decentralized dynamic systems des syst6mes ~ 6tre command6s ne signifie plus leur aptitude it 6tre stabilis6s. Des approches alg6briques et g6ometriques sont utilis6es pour obtenir des conditions d'aptitude tt la stabilisation des syst6mes decentralis6s.

Zusammenfassung--Formuliert und diskutiert wird die Stabilisierung dezentralisierter linearer zeitinvarianter dynamischer Systeme mit Koordinierung und/oder Verbindung unter Steuereinwirkung. Dezentralisierte gesteuerte Systeme sind als dynamische Systeme mit verschiedenen Reglern definiert, deren jeder auf das System mit partieller Information tiber die Zustiinde des Systems operiert. Diese Bedingung zielt auf gewisse Strukturbeschr/inkungen bei den Riickftihrungs- und anderen System-Matrizen. Mit den Beschriinkungen schlieSt die Steuerbarkeit der Systeme nicht mehr die Stabilisierbarkeit ein.

173

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