Overlapping decentralized dynamic optimal control

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Overlapping decentralized dynamic optimal controlALTUG İFTAR a

a Elektrik ve Elektronik Mühendisligi Bölümü, Anadolu Üniversitesi , Bademlik, Eskişehir,TurkeyPublished online: 15 Mar 2007.

To cite this article: ALTUG İFTAR (1993) Overlapping decentralized dynamic optimal control, International Journal of Control,58:1, 187-209, DOI: 10.1080/00207179308922997

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INT. J. CONTROL, 1993, VOL. 58, No.1, 187-209

Overlapping decentralized dynamic optimal control

ALTUG iFrARt

An optimal dynamic output feedback controller design approach is proposedfor systems which are required to have an overlapping feedback structure. Sucha structure may be either imposed due to some design considerations orselected to match the overlapping structure of the given system. The proposedapproach first involves transforming the given system into a larger dimensionalsystem. Decentralized optimal controllers are then designed for this expandedsystem. These controllers are contracted for implementation on the originalsystem in the final phase. It is shown that if the controller designed for theexpanded system achieves stability and satisfies the necessary conditions ofoptimality for the expanded system, then the contracted controller achievesstability and satisfies the necessary conditions of optimality for the originalsystem. Furthermore, the costs for the original and the expanded systems areequal. The details of overlapping controller design for a particular pattern ofoverlapping are also illustrated and an example design is presented.

1. IntroductionFor today's complex systems it is usually preferable, if not necessary, to

avoid centralized controllers for a number of reasons, for example the cost ofimplementation, complexity of on-line computations, complexity of controllerdesign, and reliability. In such a situation a decentralized feedback structuremay be imposed. Although, in some cases a simple non-overlapping decentralized feedback structure may suffice, some other cases may require an overlapping decentralized feedback structure (iftar 1991 a). A particular overlappingfeedback structure may be selected either due to certain design considerations(such as geographical location, cost of information transfer, or reliability) orsimply to match the overlapping structure of the given system. The underlyingidea in the latter case is the expectation that such a controller may performbetter than a non-overlapping decentralized controller (and perhaps even betterthan a centralized controller when possible complications are also considered).

The first step in designing a decentralized (overlapping or non-overlapping)controller for a large-scale system might be to obtain a decomposition of thegiven system (Siljak 1991). Many large-scale systems, such as large flexiblestructures (Ozguner et al . 1988), interconnected power systems (Siljak 1978),socio-economic systems (Aoki 1976), and freeway traffic regulation systems(Isaksen and Payne 1973), may consist of subsystems which are stronglyconnected through certain dynamics ('the overlapping part'), but weakly connected otherwise. For those systems, disjoint decompositions, which normallyresult in a non-overlapping decentralized feedback structure, may easily fail toproduce useful results. It has been demonstrated, however, that the recently

Received 10 March 1992. Revised 6 July 1992.t Elektrik ve Elektronik Muhendisligi Bolumu, Anadolu Universitesi, Bademlik, Eskisehir,

Turkey.

0020-7179/93 $10.00 © 1993 Taylor & Francis Ltd

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188 A. iftar

introduced overlapping decompositions (Ikeda and Siljak 1980) may produceuseful solutions in such cases (e.g. see fftar and Ozguner 1987 a). In this lattercase, the resulting controllers exhibit an overlapping feedback structure.

The earlier results on the overlapping decompositions were restricted to thedecomposition of the state space only. Decompositions of the input and outputspaces were first considered by Ikeda and Siljak (1986) and by Ohta et al .(1986). Later, fftar and Ozguner (1990) considered controller design with stateand input inclusion, and introduced a special case of inclusion, called extension.This approach was later extended to the output inclusion and estimator designby iftar (1990). The basic advantage of extension is that any feedback controller,designed using an approach based on this principle, is contractible for implementation on the actual system. It is important to satisfy contractibility in order topreserve the desired relations, such as stability, optimality, and good performance, between the expanded and the original systems following the applicationof the appropriate controllers (Iftar 1990, 1991b).

Optimal control within the framework of the inclusion principle was firstconsidered by Ikeda et al. (1981) for the case of state inclusion only. Thisapproach was later extended to the case of state and input inclusion by Ittar andOzguner (1990). To discuss optimal control within this framework, the inclusionof cost functions must also be addressed besides the inclusion of dynamicsystems. Iftar and Ozguner (1990), showed that the inclusion conditions for thecost functions must, in general, depend on the final control laws. It was alsoshown that certain sufficient conditions, which do not depend on the finalcontrol laws, can be found for quadratic cost functions and linear systems, if theinclusion is taken with respect to the optimal control. The optimal control is,however, a centralized one which requires direct access to all the states (it canalso be cast as an open-loop control, but, for well known reasons, we areinterested in closed-loop solutions only).

The problem of finding optimal static output feedback gains for systemswhich are required to have an overlapping decentralized feedback structure wasconsidered by Ittar (1991 c), where it was shown that the optimal overlappingdecentralized static-output feedback controller design problem for the originalsystem can be cast as an optimal non-overlapping decentralized static-outputfeedback controller design problem for the expanded system. Furthermore, oncea stabilizing and optimal solution is found for the latter (expanded) problem, astabilizing and optimal solution for the original problem can be obtained bysimply contracting the optimal gains obtained for the expanded system. It maybe easier to deal with the expanded problem, since it involves a non-overlappingdecentralized control structure. Many methods and software packages areavailable to solve the optimal non-overlapping decentralized control problem(e.g. Davison and Chang 1986, Ittar and Ozguner, 1989, Khorrami et al. 1988).

In the present paper, a dynamic optimal overlapping controller design withinthe framework of the extension principle is investigated. The extension principleand contractible controller design within the framework of this principle arepresented in § 2. The overlapping decentralized dynamic optimal control problem is introduced in § 3. Two optimal control problems, one related to theoriginal system and the other related to the expanded system, are defined andthe relationship between their solutions is discussed in the same section. Thedetails of overlapping controller design for a particular pattern of overlapping

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Overlapping decentralized dynamic optimal control 189

are presented in § 4. A design example is presented in § 5 to illustrate thedeveloped design methodology. Some concluding remarks are included in § 6.

Throughout the paper, IRk denotes k-dimensional real vector space, IRm x n

denotes the space of m x n dimensional real matrices, h denotes the k x kdimensional identity matrix, Omxn denotes the m x n dimensional zero matix,and ( . )T denotes the transpose of ( . ).

2. Controller design with extensionIn this section, we will present the concept of extension as introduced by

iftar (1990). Overlapping controller design within the framework of extensionwill also be discussed. The presented approach first involves transforming thegiven system into a larger-dimensional system. Decentralized controllers arethen designed for this expanded system. These controllers are contracted forimplementation on the original system in the final phase. The theoremspresented in this section were first proved by iftar (1990); hence, the proofs willnot be repeated here.

2.1. ExtensionConsider the following linear time-invariant (LTI) systems:

{X = Ax + Bu

~:

y = Cx

and

(1)

(2)

(3)

{l' = ~x + Bu

~:-Y = CX

Here x E IRn , u E IR m and y E 1R 1 are, respectively, state, input and outputvectors of the system ~. Similarly, x E lR ii , U E IRm and y E IR I are state, inputand output vectors of the system ~. The outputs y and yare assumed to bemeasurable. It is also assumed that n'" n, m'" m, and l » l. In the following,the system ~ is referred to as the original system and ~ is referred to as theexpanded system. The state, input and output spaces IR n

, IR m and 1R 1 of ~ arecalled, respectiyely, original state, input and output spaces; similarly, the spaceslR ii , IRm and IR I of ~ are called expanded state, input and output spaces.

If the given original system exhibits an overlapping structure, then theexpanded system is obtained as a decentralized interconnected system where theoverlapping parts of the original system appear as disjoint. If this is not the case,but the expanded system is to be used simply to facilitate overlapping controllerdesign, then it is obtained in such a way that the controller for the expandedsystem can be completely decentralized. In either case, ~ can be described as:

{t = Ax + 2,1=1 Bjuj

~:Yj = Cjx, i = 1,2, ... , v

where Bj and t: are obtained from the appropriate columns and rows of BandC respectively. Details of obtaining such a decentralized system for a particularpattern of overlapping are presented in § 4.

We now introduce the concept of extension.

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190 A. iftar

Definition 1: The system f is an extension of the system ~, and ~ is adisextension of f, if there exist full-rank transformations T: IR n -> lR ii ,

N: IKl m-> IKl m and S: 1Kl/ -> IKl /, such that for any initial state Xo E IKl n of thesystem ~ and any input il(t) E IKl m, 0", t < ao of the system f, the choiceXo = Txo and u(t) = Nil(t), Vt "" 0 implies that

and

x(t; xo, il) = Tx(t; xo, u), Vt "" 0

y(t; x) = Sy(t; x), v t "" 0

(4 a)

(4b)

Remark 1: The extension defined above is a generalization of the extensionfirst defined by iftar and Ozguner (1990) to the case where the output space isexpanded as well as the state and the input spaces. It is a special case ofinclusion defined by Ikeda and Siljak (1986), and, in fact, is a generalization ofunrestriction first discussed by Ikeda et al. (1984) to the case of state, input andoutput inclusion. However, it is different from the unrestriction defined byIkeda and Siljak (1986), where unrestriction was defined for an arbitrary inputu(t) in the original input space, and the input in the expanded space wasobtained by a transformation: il(t) = Nu(t). Here, on the other hand, theextension is defined for an arbitrary input il(t) in the expanded input space, andthe input in the original space is obtained by the transformation u(t) = Nil(t).Therefore, for the unrestriction the allowable set of inputs for f at any time isonly an m-dimensional subset of IKl m, but for the extension it is IKl m. 0

The necessary and sufficient conditions for the extension are provided by thefollowing theorem.

Theorem 1: The system f is an extension of the system ~ if and only if thereexist full-rank matrices T E lR iix n , N E IR m x m and S E IKl l x l such that

TA = AT

TBN = jj

and

SC = CTThe matrices of ~ and f can be related as:

A = TAT# + E A

jj = TBN + E B

and

(5 a)

(5 b)

(5 c)

(6 a)

(6 b)

C = SCT# + E c (6c)

where T E lR iix n , N E IKl m x iii and S E IR Ix/ are the matrices satisfying(5 a)-(5 c), T# E IR n x ii_ is a matrix satisfying T#T = In' and E A E lR iix ii ,

E B E lR iix m and E c E IR Ixii are constant complementary matrices. In order for fto be an extension of ~, these complementary matrices must satisfy certainconditions provided by the following theorem.

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Theorem 2: Let the matrices of the systems ~ and i: be related as in (6 a)-(6 c).Then the system i: is an extension of the system ~ if and only if

and

EcT = 0

(7 a)

(7 b)

(7 c)

(8)

2.2. Contractibility

Next, let us consider a LTI controller for the system ~:

{z = Fz + Gyr:v = Hz + Ky

where z E IR P is the state and v E IR m is the output of the controller I'; theoutput v of r is applied to the input of the system ~ in order to control it; i.e.

u = v (9)

(10)

Also, consider decentralized LTI controllers:

r- .. {t i = Piz j + GiYii- __......., i = 1, 2, ... , v

Vj = Hjz i + K;jij

for the decentralized system i: described by (3). Here Zi E IR P' is the state andVi E IR m, is the output of the ith decentralized controller r; and iii; is thedimension of the ith ineut, £li, of the system i:. The output Vi is applied to theith input of the system ~ for control purposes:

Ui = Vj, i = 1, 2, ... , v (11)

(12)

The overall expanded controller, which consists of decentralized controllers ri ,

i = 1, 2, ... , v, can be compactly described as:

_{~= pz + Gyr: __V = Hz + Ky

where Z~ [zI zi ... Z~]T, V~ [vI vi ... V~]T, P~ blockdiag(P1, P2 , ••• ,- -6 . - - - -6 . - - -F.), G = blockdJagCGI> G2 , ••• , G.), H = blockdlag(H1 , H 2, .. . , H.), andK ~ blockdiag(K 1, K 2, ••• , K.). It is assumed that p, the dimension of r,does not exceed P~ "Li=IPi, the dimension of t; this assumption may bejustified since :E is a part of i', and thus should not require a larger dimensionalcontroller (Ikeda and Siljak 1986).

We now introduce the concept of contractibility of controllers.

Definition 2: The controller (12) is contractible to the controller (8), if thereexist full-rank transformations T: IR n ~ IR", N: IRm~ IR m and M: IRP~ IR P suchthat for any initial state Xo E IR n of the system !, for any input £l(t) E IR m,0,;;; t < 00 of the system f, and for any initial state Zo E IR P of the controller r,the choice

X'o = Txo

u(t) = N£l(t), "It"" 0

(13 a)

(13 b)

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192

and

implies that

and

A. iftar

Zo = Mzo

z(t; Zo, y, u) = Mz(t; ZO, y, G), "It;;" 0

vet; z, y) = No(t; Z, y), "It;;" 0

(13 c)

(14 a)

(14 b)

Remark 2: Note that, since the controller is designed in the expanded spaces,G(t) must be arbitrary (otherwise, there will be restrictions on the controllersthat could be designed-in Ikeda and Siljak (1986) the contractibility wasdefined for arbitrary u(t) E IR m and G(t) was obtained through a transformationof the form G(t) = Nu(t); as a result of this choice, however, restrictions on thecontrollers for the expanded system were needed); thus G(t) is allowed to bearbitrary in the above definition. The outputs vet) of rand o(t) of f' must besuch that, when the loops are closed (i.e. when u(t) = vet) and G(t) = o(t)), thecondition (13 b) is satisfied; requirement (14 b) ensures this condition. 0

It is important to satisfy contractibility in order to preserve the desiredrelations between the expanded and the original systems following the application of the appropriate controllers. The following theorem shows that one suchrelation, namely stability, is preserved when contractibility is satisfied.

Theorem 3: Suppose that i: is an extension of ~ and that the controller r iscontractible to the controller r. Moreover, suppose that the controller r is appliedto the system ~ and that the controller f' is applied to the system i:. Then,stability (respectively asymptotic stability) of the expanded closed-loop system(obtained by applying f' to i:) implies the stability (respectively asymptoticstability) of the original closed-loop system (obtained by applying r to ~).

Other relations can also be preserved using extension and contractibility. Inthe next section we will discuss optimality. Discussions on preserving other typesof performance criteria can be found in Iftar (1990).

The necessary and sufficient conditions for the contractibility of controllersare given by the following theorem.

Theorem 4: The controller (12) is contractible to controller (8), if and only ifthere exist full-rank matrices T E IR n x n , N E IR m x m , and M E IR Px p such that

The above conditions reduce to simpler ones if the expanded system is anextension of the original system.

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Corollary 1: Given that the system f is an extension of the system ~, thecontroller (12) for the system f is contractible to the controller (8) for the system~, if there exists a full-rank matrix M E IR Px jl such that

FM= MF (16 a)

GC= MGSC (16 b)

HM=NN (16 c)

and

KC= NKSC (16 d)

where N E IR mxm and S E IR [xl are the matrices satisfying (5 b) and (5 c).

Since the controller is to be designed in the expanded spaces and thencontracted for implementation, it is important that any control law designed inthe expanded spaces be contractible. In fact, if f is an extension of ~ then sucha property holds.

Corollary 2: If the system i: is an extension of the system ~, then any controllerof the form (12) for the system i: is contractible to a controller of the form (8) for~ery~~wM: .

F= F (17 a)

G= GS (17 b)

H= NN (17 c)

and

K= NKS (17 d)

where N E IR mxm and S E IR /xl are the matrices satisfying (5 b) and (5 c).

3. Optimal controlIn this section we will consider two optimal control problems:

(a) an overlapping decentralized optimal control problem for the originalsystem; and

(b) a non-overlapping decentralized optimal control problem for theexpanded system.

We will also discuss the relations between the optimal controllers for the originaland expanded systems. Specifically, we will show that a candidate solution forthe former problem can be obtained by simply contracting a candidate solutionfor the latter problem.

Let us consider the cost function:

J = f(xTQX + uTRu + zTVz + iTWi + 2xTUz + 2uTYi)dt (18)

where Q, R, V, W, U and Yare appropriately dimensioned constant matrices

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194

such that

A. iftar

~J (19)

are respectively symmetric positive semi-definite and symmetric positive definitematrices, x and u are respectively the state and the control vectors of the system~, z is the state vector of the controller I', and i: is the time derivative of z ,Also consider:

J = f'(.~TQi + ilTRil + lTVl + 'FW'i" + 2iTUl + 2ilTY'i" )dt (20)

where Q, R, V, W, U and Yare appropriately dimensioned constant matricessuch that

is a symmetric positive semi-definite matrix,

(21)

R = blockdiag(R1, R 2 , ••• , R.),

W= blockdiag(W b W2 , ••• , W.),Y = blockdiag(Yb Y2 , •.. , Y.),

W· E IRP'xp,I ,

i = 1,2, , v

i = 1,2, , v

i = 1, 2, , v

(22 a)

(22 b)

(22 c)

where iiij is the dimension of the ith control input of the system (3) and Pi isthe dimension of the state of the ith decentralized controller r., described in(10), and

c: f::. [kR· - _~1- Y

j

YJ.-...' , i = 1,2, ... , VW j

(23)

are symmetric positive definite matrices, i and il are respectively the state andthe control vectors of the system ~, l is the state vector of the controller r, andt is the time derivative of l.

We now assume that ~ is an extension of ~ and introduce the following sets:

- f::. - - - - 1- -x-'?J'= {F = blockdiag I Fj , F2, ... , F.) F, E IRP' P', i = 1,2, ... , v}

(24 a)

- f::. - . - - - - -.x /<g = {G = blockdiag'(Gj , G2, ... , G.)IG i E IRP' " i = 1,2, ... , v}

(24 b)

'ge~ (H = blockdiag(H1, H2 , •.. , H.)IH i E 1Rii'i,xp" i = 1,2, ... , v}

(24 c)

and

'X ~ {K = blockdiag(K1, K2 , ••• , K.)IKj E lRii'iix 7" i = 1,2, ... , v}

(24 d)

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where 1; is the dimension of the ith output y; of system (3). We also introduce:

gji~f#i (25 a)

'§~ ~S = {G = OSlo E~} (25 b)

';£ ~ N'fie = {H = N ilIii E 'fie} (25 c)and

X~ NXS == {K == NKSIK EX} (25 d)

where N E IR mxm and S E IR /xl are the matrices satisfying (5 b) and (5 c). Weare now ready to define the following two problems, where the dimensions, PI>Pz, ... , Pv, of the decentralized controllers f'j, i == 1, 2, ... , v, and the initialstates Xo E IR" of ~ and Zo E IR P of f' are given a priori and the initial states of fand of r are related to those of Land r by X'0 == Tx 0 and z0 == Zo respectively.

Original problemFind FE gji, G E '!l, H E ';£ and K E ':if such that the closed-loop system:

(26)

is asymptotically stable and

J(F, G, H, K, xo, zo) ,,;; Jet, G, it, K, xo, zo), V(t, G, ti, K) E Q (27)

where

Q ~ {(F, G, H, K)IF E s, G E '!l, H E x, K E x;Lc is asymptotically stable} (28)

and J(F, G, H, K, xo, zo) denotes the value of the cost function (18) for theclosed-loop system z, with initial state [x6 zJ]T.

Expanded problemFind FE f#i, a E~, ii E 'fie, and K E Xsuch that the closed-loop system:

(29)

is asymptotically stable and

I(F, 0, ii, K, xO, zo) ,,;; J(F, G, ii, K, X'o, Zo), V(F, G, Ii, K) E Q

(30)

where-f':, ----- -- -- ---Q == {(F, G, H, K)IF E gji, G E '!l, H E ';£, K EX,

f c is asymptotically stable} (31)

and left, 0, it, K,xo, zo) denotes the value of the cost function (20) for theclosed-loop system f c with initial state [x6 z61 T

.

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196 A. lftar

(32)

Remark 3: There exist solutions to the original and the expanded problems ifand only if the original and the expanded systems are stabilizable under theimposed dynamic feedback structures described by (25 a)-(25 d) and(24 a)-(24 d), respectively. There exist a set of non-negative integers PI, Pz,... , Pv, such that the original system, L, is stabilizable under the imposedfeedback structure with controller dimensions PI"" PI' Pz"" Pz, ... , o,> Pv, ifand only if the system L does not have any unstable decentralized fixed modes(i.e. decentralized fixed modes with non-negative real parts) with respect to the(static) feedback structure described by (25 d) (Wang and Davison 1973).Similarly, there exists a set of non-negative integers PI> Pz, ... , pv, such thatthe expanded system, :E, is stabilizable under the imposed feedback structurewith controller dimensions PI"" PI, pz:;;' Pz, ... , Pv"" Pv, if and only if thesystem :E does not have any unstable decentralized fixed modes with respect tothe (static) feedback structure described by (24 d). Furthermore, by Corollary 2and Theorem 3, if the expanded system :E is stabilizable under the imposedfeedback structure described by (24 a)-(24 d), then the original system L isstabilizable under the imposed feedback structure described by (25 a)-(25 d). 0

To derive the necessary conditions of optimality for the expanded problem,let us introduce the following (artificial) system:

'" 11 = X~ + ±~ifriL: i=l~ A A

Yi = CiX, i = 1,2, ... , v

where

and

~ ~ [iT zT zi Z~]T

fr i ~ [iiT tilT, i = 1,2, ... , v

.;:r t:, [-T -T]T . - 1 2.J'j= y,. Zj ,1- , , .. o,v

'" t:, -A = blockdiag (A, Opxp),"'t:, . - TB, = blockdiagt B}, ~d, i = 1,2, ... , v

(33 a)

(33 b)

(33 c)

(33 d)

(33 e)

where

~t:, .-c, = blockdiag ICi, ~i)' i = 1,2, ... , v (33 f)

~i ~ [Op- x•. i-l p- I p- Op-.x•. '. IP-']' i = 1,2, ... , v- t "1-1 j i I .l.J"""+ J

Also define

~ ~ blockdiag (8, I p)

and

?: ~ blockdiag tC, I p)

(33 g)

(34 a)

(34 b)

Let us als!1, introduce the following decentralized feedback control law forthe system 1::

A "'~IIi = KYi, i = 1,2, ... , v (35)

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where


iii] , 1 2Fi

' 1= , , ... , v (36)

Note that the closed-loop system cbtained by applying the decentralized (static)feedback law (35) to the system l' is equivalent to the closed-loop system f cintroduced in (29), Therefore, the expanded problem is equivalent to findingstabilizing decentralized feedback gain matrices K; (i = 1,2, . , " v) for thesystem l' which minimizes the cost function:

(37)

which, with Qand Ri as defined in (21) and (23) respectively, is equivalent to(20),

Assuming the stabilizability of the expanded system under the imposedfeedback structure, the optimal cost for the expanded problem is (Iftar andOzguner 1989):

J*(xo, zo) = trace(PYo)

h X.c t,.c .c T .c t, [-T - T]T d p-, th I ti fwere 0 = x 0 x 0, x 0 = x 0 z0 , an IS e so u Ion 0 :

cpTp + PCP + Q+ ?:Tf(T~f(?: = 0

(38)

(39)

(40)

and Qis as defined as (21). A

Necessary conditions for K to be an optimal solution are:

o _.c __----;c-trace[PXo + L1] = 0 (41)s«,

for i = 1, 2, "', v, where X is the left-hand side of (39) and the Lagrangemultiplier matrix i: satisfies:

cpl + lcpT + Yo = 0 (42)

The conditions (41) can be reduced to:

(43)

or:

(44)

equivalently:

(45)

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198 A. lftar

for i = 1,2, ... , v. Hence, assuming that ?:il?:T is invertible, the matrices ofthe optimal controller are given by:

[~iGi

iii] = _ R.c:-l-B~Tp-L--9<T(?<.L-?<T)-1 . 1 2F. I I C, «""', C, ,I = , , .. -, VI

(46)

Therefore, the optimal expanded controller can be determined by simultaneously solving (39), (42) and (45). A number of algorithms have been proposedto solve such equations simultaneously. The descent Anderson-Moore method(Makila and Toivonen 1987), which is based on the Anderson-Moore algorithm(Anderson and Moore 1971), is an effective method. The Levine-Athansalgorithm (Levine and Athans 1970, Levine et al. 1971) provides anothereffective method. The nonlinear programming approach of Davison and Chang(1986) is yet another alternative. Although some of these methods were.originally developed for the centralized optimal output feedback problem, theirextension to the decentralized case is straightforward (iftar and Ozguner 1989).Software packages have also been developed to numerically solve this problem(e.g. Khorrami et al. 1988).

Also, note that the optimal solution depends on the initial conditions.c

through the initial condition matrix X o- If the actual initial conditions of thesystem and/or the controller are not known, one may choose to minimize theexpected value of the cost (37) over the initial conditions. Such a choice allowsthe designer to re-define i 0 in (42) as i 0 ~ 't:[i0 i JJ, where 't:[.] is theexpectation operator. This matrix can be calculated in terms of certain statisticalmoments of the system and controller initial conditions (i ftar and Ozguner1987b). Note that such a modification would not affect the above-discussedsolution methods.

Once the expanded problem is solved, the question remains about how tosolve the original problem. The following theorem shows that a controller T,with satisfies the necessary conditions of optimality for the original problem, cansimply be obtained by contracting the controller T, which satisfies the necessaryconditions of optimality for the expanded problem.

Theorem 5: Let i be an extension of ~ and let T, Nand S be the transformations satisfying the conditions of extension as introduced in Definition 1. Supposethat the controller t, described in (12), stabilizes the system i and that thematrices FE'!}, G E <§, H E ~ and K E '5l of t satisfy the necessary conditions ofoptimality for the expanded problem for the initial state vectors x0 = Tx 0 andzo = zo, for some Xo E IR n and Zo E IR P • Furthermore, suppose thatrank(N[KS H]) = rank ([KS H]), and let the cost weighting matrices R, YandW be chosen such that

where

s~ blockdiag(S, 1p ) , N ~ blockdiag(N, Ip )

f( and N are as defined in (40), and R is as defined in (19).remaining weighting matrices, Q, U and V, be chosen such that

Q = TTQT

U= TTV

(47)

(48)

Also let the

(49 a)

(49 b)

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and

Then the controller I', described in (8), where

F= F

G= GSH= Nil

and

K= NKS

199

(49 c)

(50 a)

(50 b)

(50 c)

(50 d)

stabilizes the system L, the matrices F, G, Hand K of r satisfy the necessaryconditions of optimality for the original problem with initial states x 0 and zo, and

J(F, G, H, K, xo, zo) = f(F, G, it, K, i'o, zo) (51)

(52)

Proof: The stability of the original closed-loop system follows from Corollary 2and Theorem 3.

To show that F, G, Hand K, given in (50a)-(50d), satisfy the necessaryconditions of optimality, let us introduce the following system:

~ {i = Ax + BuL: ~

y = CX, i = 1,2, ... , v

where

and

x ~ [xT ZTJT

u~ ruT zTry ~ [y T ZTJT

~ !::,A = blockdiag(A, Opxp)

B~ blockdiag(B, lp)

(53 a)

(53 b)

(53 c)

(53 d)

(53 e)

c~ blockdiag(C, lp) (53!)

Also introduce the following feedback control law for the system f:u = Ky (54)

where

~J (55)

Note that the closed-loop system obtained by applying the (static) feedback law(54) to the system f is equivalent to the closed-loop system Lc introduced in(26). Therefore, the ~original problem is equivalent to finding a stabilizingfeedback gain matrix K E '?If, where

(56)

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200 A. iftar

for the system f which minimizes the cost function:

I = f'(.fTQ£ + uTRu)dt (57)

which, with Q and R as defined in (19), is equivalent to (18). Furthermore, notethat any K E Xcan be represented as K = illKS with some KE ~, where

(58)

Furthermore, minimizing I(F, G, H, K, xo, zo) over

is equivalent to minimizing lei, os, N Ii, N [(S, xo, zo) over

The necessary conditions of optimality for the original problem can now bestated as follows:

(i) there exists a symmetric positive semi-definite matrix P E ~ilXil, wheren~ n + p, satisfying:

cpT P + pcp + Q + cT KTRKC = 0 (59)

where cp ~ A + BKC(ii) there exists a symmetric positive semi-definite matrix L E ~ilxil satisfying:

CPL + LcpT + X'o = 0 (60)

where X'o ~ £0£6, where £0 ~ [x6 Z6]T, and

(iii) the following hold for all i E {I, 2, ... , v}:

a ~~race[PXo + LA] = 0 (61)aK i

where A is the left-hand side of (59) with K = illK,S, and the matricesP and L are matrices introduced in (i) and (ii).

...... ......~A '" '"

To show that K = N K S satisfies condition (i) for some P E ~nxn, let

f ~ blockdiag(T, Ip ) (62)

and pre and post-multiply both sides of (39) by fT and i respectively. Then,use (5 a)-(5 c), (50 a)-(50 d), (49 a)-(49 c), and (47) to obtain:

cpT(fTpf) + (fTpf)cp+ Q + CTKTRKC = 0 (63)

Therefore, (59) is satisfied with P = fT pf.Since r stabilizes the original system k, the eigenvalues of cp have negative

real parts; this fact, together with the fact that X'0 is positive semi-definite,guarantees that there exists a positive semi-definite matrix L satisfying (60).

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Furthermore, by pre and post-multiplying both sides of (60) by T and TTr!s'pe~tively.c using (5 a)-(5 c) and (50 a)-(50 d), and noting the fact thatTXoTT= Xo, we obtain

?J(TLTT) + (TLTT)?JT + Xo = 0 (64)

By comp!rin~ (0) with (42), we conclude that L, which satisfies (60), alsosatisfies L = TLTT .

To show that K = Nf(S satisfies condition (iii), let RE Ihl~x~, where#; ~ m+ p, be a symmetric matrix satisfying:

STKTNTRNKS = STKTRKS (65)

Then (61) can be re-written as:

~ trace [LPBNKSC + iLCTSTKTRKSCj = 0 (66)aK j

Using (5b) and (5c), and substituting TTPT for P and L for TLT T, equation(66) becomes:

a ~~~ ~~~~~

----znrace[lPBKC + ilcTKTRKCj = 0 (67)aK j

.lb' comparing (67) with (43), we conclude that (61) is satisfied with R= ~ andK=K.

To show that (51) holds, note that

J(F, G, H, K, xo, zo) = trace (PXo) (68)

and_..... ....... __ ..... _ _ ..... A. A. AT AT"'" A A

J(F, G, H, K, xo, zo) = trace (PTXoT ) = trace(T PTXo) (69)

But, TT PT = P, hence the result follows. 0

4. Overlapping controller design

As pointed out in the introduction of the present paper, a system may berequired to have an overlapping feedback control structure either due to certaincontroller design considerations or to match the overlapping structure of thegiven system. It is possible to find many different examples of real-life situationsin which a controller is required to have a certain pattern of an overlappingdecentralized structure. Similarly, it is possible to find many different examplesof real-life systems composed of overlapping subsystems arranged in variousdifferent patterns. The patterns which are encountered most often in practiceare, however, one of the following:

(a) subsystems/subcontrollers having overlapping parts with neighbouringsubsystems/subcontrollers (as depicted in Fig. 1),

(b) subsystems/subcontrollers having a common overlapping part (as de-picted in Fig. 2 for the special case of four subsystems).

In this section, we will discuss the overlapping controller design under these twopatterns. For simplicity, we will restrict ourselves to the case of two overlappingsubsystems/subcontrollers. Note that, in this special case the above listed twopatterns are equivalent.

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202 A. iftar

E,

E,

E,

Figure 1. Subsystems having overlapping parts with neighbouring subsystems.

E,

commonE, overlapping E,

part

E.

Figure 2. Four subsystems having a common overlapping part.

Consider the systems ~ given in (1). Suppose that the state, the input andthe output are partitioned as:

and

Xi E IR ni ,

u, E IR mi ,

i = 1, 2, 3

1 = 1, 2, 3

(70 a)

(70 b)

y=[yLyi,yI]T, UiEIR1" 1=1,2,3 (70 c)

where it is assumed that Xz, Uz and yz correspond to the overlapping parts ofthe state, input and output spaces respectively. Suppose that the system matricesin (1) are also partitioned compatibly:

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and

203

Then, an extension of ~ is obtained by choosing:

[I'a

°l[I", a

ns = a II, a T = a In,a II, a ' a In,a a II, a a

[Im l a a

I~]N = ~ 1m , 1m ,

a a

[In la a

I~]T# = ~ iIn, iIn,a a

E. "[!iA 12 -iA12

niA 22 -iA22E B = a

-iA22 iA 22

-iA32 iA 32

and

E," [!iC12 -!c12

!liC22 -iC22

-iC22 iC22

-iC32 !C32

The resulting expanded system matrices are:

[

A ll A12 i a A13lA 21 A 22 ! a A 23

A = -------------------!--------------------A21 a i A 22 A 23A 31 a i A 32 A 33

[

B ll B12 i B 12 B13lB21 B22 i B22 B23B = --------------------r------------------

B21 B22 i B22 B 23

B 31 B32 ! B32 B33

~12JA 2

~12JB2

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204

and

A. iftar

lCU Cl2 i 0 CO]C21 C22 ! 0 C23

C = ·..-----------------1·------------------C21 0 i C22 C23C3l 0 ! C32 C33

If the given original system does not actually exhibit an overlapping dynamicstructure, but the decomposition is made merely to obtain an overlappingdecentralized controller, then the state space of k need not be decomposed. Inthis case we simply take T = In, resulting in A = A. Whether the state space ofthe given system k is decomposed or not, the expanded system f is in the formof the decentralized control system as described in (3) with v = 2. The inputsand ouputs of the expanded system f are related to those of the original systemk by:

(71 a)

and

(71 b)

respectively.Next, suppose that decentralized controllers of the form (10) are designed for

the expanded system (say, to minimize a cost function of the form (20)). Let uspartition the following matrices of these decentralized controllers as:

OJ = [G\ G~l" i = 1,2

tt, = [H\] i = 1,2t Hi'

and

_ [K\I K\2] i = 1,2K·= .I Kil K' '22

(72 a)

(72 b)

(72 c)

where G: E IR p, X!" G~ E IR p, X!" H: E IR m,x p" H~ E IR m, x.p" K: I E IR m , X!"

K~2 E IR m, X {, and Pi is the dimension of the state of the ith decentralized

controller r; (i = 1,2). These decentralized controllers can now be contracted toa controller of the form (8) for implementation on the original system. Thematrices of the contracted controller are given by:

(73 a)

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and


[C 1C~

~~JC= I0 ci

[Hi ;t]H = ~~H~

205

(73 b)

(73 c)

(73 d)

The implementation of this controller, which consists of overlapping decentralized controllers, is illustrated in Fig. 3. Assuming that the decentralizedcontroller r, described by (12), achieves asymptotic stability and that it satisfiesthe necessary conditions of optimality for the expanded problem discussed in theprevious section, by Theorem 5, the above described contracted controllerachieves asymptotic stability and it satisfies the necessary conditions of optimality for the original problem discussed in the previous section.

5. Example designTo illustrate the design methodology presented we consider the

example system, borrowed (with modifications) from iftar and(1987 a):

followingOzguner

(74 a)

(74 b)

where the dotted lines separate the overlapping subsystems. The initial conditions on the system are assumed to be:

x(O) = DJ (75)

Our design goal is to find overlapping decentralized controllers to stabilizethis system and to minimize a certain quadratic cost function. To achieve this,

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206 A. iftar

Figure 3. Implementation of overlapping decentralized controllers.

we first expand this system as explained in the previous section to obtain thefollowing expanded system:

(76 c)

(76 a)

(76 b)o1

oo

_ [1Yl = 0

_ [0yz = 0

t = [~----~~-----I----~------~] x + [~-----~-] ill + [~-----~] il zo 0 1 -1 0 0 1 1 000 13100 01

o 0J-o 0 x

1 0J-o 1 x

with initial conditions:

(77)

For this expanded system we consider decentralized static controllers of thefollowing form:

(78)

to achieve stability for the closed-loop expanded system and to minimize thefollowing cost function:

(79)

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where

and

- [~Q= oo

oI'2oo

ooI'2o !]

l1 0 i O O

Jo 2 i 0 0

R = ·-----------·1------------o 0 i 2 0o 0 ! 0 1

Note that, since the controller dimensions PI =,i52 = 0, the matrices 0, V, lV,and Y do not appear and we have: Q = Qand R = it

By applying the algorithm presented by Khorrami et at. (1988) we find thatthe feedback gains:

- _ [-1.2035K I - -0.9894

- _ [-1.2263K 2 - -0.9739

-0.9739J-1·2263

-0.9894J-1·2035

(80 a)

(80 b)

stabilize the closed-loop expanded system and satisfy the necessary conditions ofoptimality for the expanded problem.

We now contract the above feedback gains to:

[

- 1.2035K = -0'~894

-0·9739-2·4525-0·9739

-0'~894J-1·2035

(81)

to find the contracted overlapping controller

u = Ky (82)

for the original system (74 a)-(74 b). It can be checked that this controllerstabilizes the original system and satisfies the necessary conditions of optimalityfor the original problem with

J = (xTQX + uTRu)dt (83)

where

Q = [t o1o ~J

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208

and

A. Iftar

[

1·67R = 0·00

-0·67

0·001·000·00

-0'67J0·001·67

which respectively satisfy (49 a) and (47). The resulting cost for both the originaland the expanded problems is: J* = J* = 8·8216.

6. Conclusions

An optimal dynamic output feedback controller design approach has beenproposed for systems which are required to have an overlapping feedbackstructure. Such a structure may be either imposed due to some design considerations or selected to match the overlapping structure of the given system. Theproposed approach first involves transforming the given system into a largerdimensional system. If the given original system exhibits an overlapping structure, then the larger dimensional expanded system is obtained such that theoverlapping parts of the original system appear as disjoint. If this is not the case,but the expanded system is to be used simply to facilitate overlapping controllerdesign, then it is obtained in such a way that the controller for the expandedsystem can be completely decentralized. In either case, non-overlapping decentralized optimal controllers are then designed for this expanded system. Thesecontrollers are contracted for implementation on the original system in the finalphase.

It may be easier to deal with the expanded problem, since it involves anon-overlapping decentralized feedback structure. Many methods and softwarepackages are available to solve this type of optimal control problem (e .g.Davison and Chang 1986, iftar and Ozguner 1989, Khorrami et al . 1988). In thispaper, it has been shown that if the controller designed for the expanded systemachieves stability and satisfies the necessary conditions of optimality for theexpanded system, then the contracted controller achieves stability and satisfiesthe necessary conditions of optimality for the original system. Furthermore, thecosts of the original and the expanded systems are equal.

ACKNOWLEDGMENT

The author was with the Department of Electrical Engineering, University ofToronto, Toronto, Canada during the initial phase of this research.

REFERENCES

ANDERSON, B. D.O., and MOORE, J. B., 1971, Linear Optimal Control (Englewood Cliffs:Prentice Hall).

AOKI, M., 1976, On decentralized stabilization and dynamic assignment problems. Journal ofInternational Economics, 6, 143-171.

DAVISON. E. J., and CHANG, T. N., 1986, Decentralized controller design using parameteroptimization methods. Control Theory and Advanced Technology, 2, 131-154.

iFTAR, A., 1990, Decentralized estimation and control with overlapping input, state, andoutput decomposition. Preprints of IFAC World Congress, Vol. 2, Tallinn, Estonia pp.13-18. An extended version is to appear in Automatica; 1991a, Overlapping decentralized control. Proceedings of the IEEE Conference on Decision and Control, Brighton,

Dow

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ded

by [

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ry]

at 1

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U.K. PI'. 341-342; 1991b, Decentralized opimal control with overlapping decompositions. Proceedings of the IEEE International Conference on Systems Engineering,Dayton, Ohio, PI'. 299-302; 1991c, Decentralized optimal control with extension.Proceedings of the First IFAC Symposium on Design Methods of Control Systems,Zurich, S~itzerland, flP' 747-752.

iFTAR, A., and OZGONER, U, 1987a, Local LQG/LTR controller design for decentralizedsystems. IEEE Transactions on Automatic Control, 32, 926-930; 1987b, A linearquadratic optimal solution to the decentralized servomechanism problem with robustnoisy controllers. Proceedings of the IEEE Conference on Decision and Control, LosAngeles, California, PI'. 2314-2319; 1989, An optimal control approach to thedecentralized robust servomechanism problem. IEEE Transactions on AutomaticControl, 34, 1268-1271; 1990, Contractible controller design and optimal control withstate and input inclusion. Automatica, 26, 593-597.

IKEDA, M., and S,UAK, D. D., 1980, Overlapping decompositions, expansions, and contractions of dynamic systems. Large Scale Systems, 1, 29-38; 1986, Overlapping decentralized control with input, state, and output inclusion. Control Theory and AdvancedTechnology, 2, 155-172.

IKEDA, M., S,UAK, D. D., and WHITE, D. E., 1981, Decentralized control with overlappinginformation sets. Journal of Optimization Theory and Applications, 34, 279-310; 1984,An inclusion principle for dynamic systems. IEEE Transactions on Automatic Control,29, 244-249.

ISAKSEN, L., and PAYNE, H. J., 1973, Suboptimal control of linear systems by augmentationwith application to freeway traffic regulation. IEEE Transactions on AutomaticControl, 18,210-219.

KHORRAMI, F., T'EN, S., and OZGONER, 0., 1988, DOLORES: a software package foranalysis and design of optimal decentralized control. Proceedings of the 40th NationalAerospace and Electronics Conference, Dayton, Ohio, PI'. 434-441.

LEVINE, W. S., and ATHANS, M., 1970, On the determination of the optimal constant outputfeedback gains for linear mullivariable systems. IEEE Transactions on AutomaticControl, 15, 44-48.

LEVINE, W. S., JOHNSON, T. L., and ATHANS, M., 1971, Optimal limited state variablefeedback controllers for linear systems. IEEE Transactions on Automatic Control, 16,785-792.

MAKILA, P. M., and TOIVONEN, H. T., 1987, Computational methods for parametric LQproblems-A survey. IEEE Transactions on Automatic Control, 32, 658-671.

OHTA, Y., S,UAK, D. D., and MATSUMOTO, T., 1986, Decentralized control using quasi-blockdiagonal dominance of transfer function matrices. IEEE Transactions on Automatic

. Control, 31, 420-430.OZGONER, 0., KHORAMI, F., and iFTAR, A., 1988, Two controller design approaches for

decentralized systems. In Proceedings of the AIAA Guidance, Navigation, and ControlConference, Minneapolis, Minnesota, PI'. 337-344.

S,UAK, D. D., 1978, Large-scale Dynamic Systems: Stability and Structure (New York:North-Holland); 1991, Decentralized Control of Complex Systems (San Diego, Calif.:Academic Press).

WANG, S. H., and DAVISON, E. J., 1973, On the stabilization of decentralized control systems.IEEE Transactions on Automatic Control, 18, 473-478.

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Overlapping decentralized dynamic optimal control