Distributed LQR Design forIdentical Dynamically Decoupled Systems

Francesco Borrelli∗, Tamas Keviczky

Abstract— We consider a set of identical decoupled dynamicalsystems and a control problem where the performance indexcouples the behavior of the systems. The coupling is describedthrough a communication graph where each system is a nodeand the control action at each node is only function of its stateand the states of its neighbors. A distributed control designmethod is presented which requires the solution of a single LQRproblem. The size of the LQR problem is equal to the maximumvertex degree of the communication graph plus one. The designprocedure proposed in this paper illustrates how stabilityof thelarge-scale system is related to the robustness of local controllersand the spectrum of a matrix representing the desired sparsitypattern of the distributed controller design problem.

I. I NTRODUCTION

D ISTRIBUTED control techniques today can be found in abroad spectrum of applications ranging from robotics and

formation flight to civil engineering. Contributions and interestin this field date back to the early results of [1]. Approachesto distributed control design differ from each other in theassumptions they make on: (i) the kind of interaction betweendifferent systems or different components of the same system(dynamics, constraints, objective), (ii ) the model of the system(linear, nonlinear, constrained, continuous-time, discrete-time),(iii ) the model of information exchange between the systems,(iv) the control design technique used.

In this paper we focus onidentical decoupled linear time-invariant systems. Our interest in distributed control for suchsystems arises from the abundance of networks of indepen-dently actuated systems and the necessity of avoiding central-ized design when this becomes computationally prohibitive.Networks of vehicles in formation, production units in a powerplant, cameras at an airport, an array of mechanical actuatorsfor deforming a surface are just a few examples.

In a descriptive way, the problem of distributed control fordecoupled systems can be formulated as follows. A dynamicalsystem is composed of (or can be decomposed into) distinctdynamical subsystems that can be independently actuated. Thesubsystems are dynamically decoupled but have common ob-jectives, which make them interact with each other. Typicallythe interaction is local, i.e., the goal of a subsystem is afunction of only a subset of other subsystems’ states. Theinteraction will be represented by aninteraction graph, where

∗ Corresponding author.F. Borrelli is with the Department of Mechanical Engi-

neering, University of California, Berkeley, 94720-1740,USA,fborrelli@me.berkeley.edu

T. Keviczky is with the Delft Center for Systems and Control,Delft University of Technology, 2628 CD, Delft, The Netherlands,t.keviczky@tudelft.nl

the nodes represent the subsystems and an edge between twonodes denotes a coupling term in the controller associated withthe nodes. Also, typically it is assumed that theexchangeof information has a special structure, i.e., it is assumedthat each subsystem can sense and/or exchange informationwith only a subset of other subsystems. We will assume thatthe interaction graphand the information exchange graphcoincide. A distributed control scheme consists of distinctcontrollers, one for each subsystem, where the inputs to eachsubsystem are computed only based on local information, i.e.,on the states of the subsystem and its neighbors.

Over the past few years, there has been a renewal of interestin systems composed of a large number of interacting andcooperating interconnected units [2]–[20]. Recent advances instability analysis of such systems yielded greater insightintothe relationship between the spectrum of the interconnectiongraph and global stability of the overall system. Research ef-forts in this area are epitomized by the pioneering work of [5],which provided stability analysis tools for an interconnectionof identical linear dynamical systems each using the samecontrol law that operates on the average information obtainedfrom neighbors.

Most recent results, however, pertain to the study of dis-tributed parameter systems where the underlying dynamics arespatially invariant, and where the controls and measurementsare spatially distributed. The fundamental work of [4], [6]inthis field discusses distributed LQR design purely for infinitedimensional spatially invariant systems, where the problemdiagonalizes exactly into a parameterized family of finite di-mensional LQR problems. It is established that the correspond-ing ARE solutions are translation-invariant operators, and theoptimal controller is a spatially invariant system. The authorsshow that quadratically optimal controllers for spatiallyin-variant systems are themselves spatially invariant. Anotherrecent result for spatially invariant systems and unboundeddomains in [7] considers the problem of distributed controllerdesign, when communication among the sites is limited. Inparticular, the controller is assumed to be constrained sothat information is propagated with a delay that depends onthe distance between subsystems. The authors show that theproblem of optimal design can be cast as a convex problemprovided that the propagation speeds in the controller are atleast as fast as those in the plant. Special spatially invariantstructures composed of aninfinitestring of linear systems havealso received much attention. The paper [8] analyzes inverseoptimality of localized distributed controllers in such systems.

Another significant group of recently explored strategiesinvolves relaxations to the LMI versions of these spatially

invariant problems. In [9], authors consider heterogenoussubsystems and a relaxation, which is used to derive sufficientconditions for the existence of controllers which stabilize thesystem and provide a guaranteed level of performance. Thework in [10] is concerned with finding distributed controllersfor a set of arbitrarily connected, finite and possibly het-erogeneous LTI systems. The authors consider nonorientededges in the graph interconnection and formulate convexconditions for the existence of output-feedback controllers,which achieve a certainH∞ performance. The resulting linearmatrix inequalities grow in size with the number of systemsin the interconnection.

The complexity associated with the computation of dis-tributed optimal controllers is well exemplified by the study in[11]. There, a finite-time LQR synthesis problem in discretetime is considered, where the matrix describing the controllaw is constrained to lie in a particular vector space. Thisvector space represents a pre-specified distributed controlstructure between a network of autonomous agents. Thecomputationally intensive optimal solution for the distributedcontrol problem is presented along with a computationallymore tractable sub-optimal one.

One of the most notable recent result in characterizing thecomplexity of optimal distributed controller design subject toconstraints on the controller structure was presented in [12].The authors establish that in general, efficient solution ofsuchproblems require a special property of the sparsity patternor interconnection structure to hold. Specifically, it is shownthat quadratic invariance of the assumed controller structureimplies that the distributed minimum-norm problem may besolved efficiently via convex programming.

This manuscript proposes asimpledistributed controllerde-signapproach and focuses on a class of systems, for which ex-isting methods outlined above are either not efficient or wouldnot even be directly applicable. Our method applies to large-scale systems composed offinite number ofidentical subsys-tems wherethe interconnection structure or sparsity patternis not required to have any special invariance properties. Thephilosophy of our approach builds on the recent works [13]–[15], where at each node, the model of its neighbors areused to predict their behavior. We show that in absence ofstate and input constraints, and for identical linear systemdynamics, such an approach leads to an extremely powerfulresult: the synthesis of stabilizing distributed control laws canbe obtained by using a simple local LQR design, whose size islimited by the maximum vertex degree of the interconnectiongraph plus one. Furthermore, the design procedure proposedin this paper illustrates how stability of the overall large-scalesystem is related to the robustness of local controllers andthe spectrum of a matrix representing the desired sparsitypattern. In addition, the constructed distributed controller isstabilizing independent of the tuning parameters in the localLQR cost function. This leads to a method for designingdistributed controllers for a finite number of dynamicallydecoupled systems, where the local tuning parameters can bechosen to obtain a desirable global performance. Such resultcan be immediately used to improve current stability analysisand controller synthesis in the field of distributed receding

horizon control for dynamically decoupled systems [3], [13],[16]–[18]. We emphasize that the aforementioned advantagesare the results of two main simplifying assumptions comparedto a general structured optimal design: identical and decoupledsubsystem dynamics and suboptimality of the global controller.

The paper is organized as follows. In Section II, we studythe solution properties of the LQR for a set of identical, decou-pled linear systems. Such properties will be used in the paperto construct a stabilizing distributed controller for arbitraryinterconnection structures. Section III summarizes importantproperties of graph Laplacian and adjacency matrices, whichwill be useful in our proofs. Section IV presents the stabilizingdistributed controller design procedure using the local LQRsolution properties. The proposed distributed control designmethod and the effect of the free tuning parameters in thelocal LQR design is illustrated by a simulation example inSection V. Some concluding remarks are made in Section VI.

NOTATION AND PRELIMINARIES

We denote byR the field of real numbers,C the field ofcomplex numbers andRm×n the set ofm × n real matrices.C−− = {s ∈ C : Re(s) < 0}. C− = {s ∈ C : Re(s) ≤ 0}

The transpose of a vectorx and a matrixM will be denotedby x′ andM ′, respectively. A matrixM ∈ Rn×n is symmetricif M = M ′.

Notation 1: Let M ∈ Rm×n, then M [i : j, k : l] denotesa matrix of dimension(j − i + 1) × (l − k + 1) obtained byextracting rowsi to j and columnsk to l from the matrixM ,with m ≥ j ≥ i ≥ 1, n ≥ k ≥ l ≥ 1.

Notation 2: Im denotes the identity matrix of dimensionm,Im ∈ Rm×m.

Notation 3: Let λi(M) denote thei-th eigenvalue ofM ∈Rn×n, i = 1, . . . , n. The spectrum ofM will be denoted byS(M) = {λ1(M), . . . , λn(M)}.

Definition 1: Let A, B ∈ Rn×n be symmetric.1) A is positive definite ifx′Ax > 0 for all nonzerox ∈

Rn, and A is positive semidefinite ifx′Ax ≥ 0 for allnonzerox ∈ Rn. We denote this byA > 0 andA ≥ 0,respectively.

2) A is negative (semi)definite if −A is positive(semi)definite.

3) A < B and A ≤ B meanA − B < 0 and A − B ≤ 0,respectively.

Definition 2: A matrix M ∈ Rn×n is called Hurwitz(or stable) if all its eigenvalues have negative real part,i.e. λi(M) ∈ C−−, i = 1, . . . , n.

Notation 4: Let A ∈ Rm×n, B ∈ Rp×q. Then A ⊗ Bdenotes the Kronecker product ofA andB:

A ⊗ B =

a11B · · · a1nB...

. . ....

am1B · · · amnB

∈ R

mp×nq. (1)

Proposition 1: Consider two matricesA = αIn and B ∈Rn×n. Thenλi(A + B) = α + λi(B), i = 1, . . . , n.

Proof: Take any eigenvalueλi(B) and the correspondingeigenvectorvi ∈ Cn. Then (A + B)vi = Avi + Bvi =

αvi + λivi = (α + λi)vi. 2

Proposition 2: Given A, C ∈ Rm×m and B ∈ Rn×n,consider two matricesA = In ⊗ A and C = B ⊗ C, whereA, C ∈ Rnm×nm. ThenS(A + C) =

⋃n

i=1 S(A + λi(B)C),whereλi(B) is the i-th eigenvalue ofB.

Proof: Let v ∈ Cn be an eigenvector ofB correspondingto λ(B), andu ∈ Cm be an eigenvector ofM = (A+λ(B)C)with λ(M) as the associated eigenvalue. Consider the vectorv ⊗ u ∈ Cnm. Then(A + C)(v ⊗ u) = v ⊗Au + Bv ⊗Cu =v ⊗ Au + λ(B)v ⊗ Cu = v ⊗ (Au + λ(B)Cu). Since(A +λ(B)C)u = λ(M)u, we get(A+ C)(v⊗u) = λ(M)(v⊗u).2

II. LQR PROPERTIES FORDYNAMICALLY DECOUPLED

SYSTEMS

Consider a set ofNL identical, decoupled linear time-invariant dynamical systems, thei-th system being describedby the continuous-time state equation:

xi = Axi + Bui,

xi(0) = xi0.(2)

wherexi(t) ∈ Rn, ui(t) ∈ Rm are states and inputs of thei-thsystem at timet, respectively. Letx(t) ∈ R

nNL and u(t) ∈RmNL be the vectors which collect the states and inputs ofthe NL systems at timet:

˙x = Ax + Bu,

x(0) = x0 , [x10, . . . , xNL0]′,

(3)

with

A = INL⊗ A,

B = INL⊗ B.

(4)

We consider an LQR control problem for the set ofNL

systems where the cost function couples the dynamic behaviorof individual systems:

J(u, x0) =

∫ ∞

0

(NL∑

i=1

(xi(τ)′Qiixi(τ) + ui(τ)′Riiui(τ)) +

+

NL∑

i=1

NL∑

j 6=i

(xi(τ) − xj(τ))′Qij(xi(τ) − xj(τ))

dτ

(5)

with

Rii = R′ii = R > 0, Qii = Q′

ii ≥ 0 ∀i, (6a)

Qij = Q′ij = Qji ≥ 0 ∀i 6= j. (6b)

The cost function (5) contains terms which weigh thei-thsystem states and inputs, as well as the difference betweenthe i-th and thej-th system states and can be rewritten usingthe following compact notation:

J(u(t), x0) =

∫ ∞

0

(

x(τ)′Qx(τ) + u(τ)Ru(τ))

dτ, (7)

where the matricesQ and R have a special structure definednext. Q andR can be decomposed intoN2

L blocks of dimen-sion n × n andm × m respectively:

Q =

Q11 Q12 · · · Q1NL

.... . .

......

QNL1 . . . . . . QNLNL

, R =

R 0 · · · 0

.... . .

......

0 . . . . . . R

(8)with

Qii = Qii +

NL∑

k=1, k 6=i

Qik, i = 1, . . . , NL.

Qij = −Qij , i, j = 1, . . . , NL, i 6= j.

R = INL⊗ R.

(9)

Remark 1:The cost function structure (5) can be usedto describe several practical applications including formationflight, paper machine control and monitoring networks ofcameras [19], [21].

Let K and x′0P x0 be the optimal controller and the value

function corresponding to the following LQR problem:

minu

J(u, x0)

subj. to ˙x = Ax + Bu

x(0) = x0

(10)

Throughout the paper we will assume that a stabilizing solu-tion to the LQR problem (10) with finite performance indexexists and is unique (see [22], p. 52 and references therein):

Assumption 1:SystemA, B is stabilizable and systemA, Cis observable, whereC is any matrix such thatC′C = Q.We will also assume local stabilizability and observability:

Assumption 2:System A, B is stabilizable and systemsA, C are observable, whereC is any matrix such thatC′C =Q.

It is well known that

K = −R−1B′P , (11)

where P is the symmetric positive definite solution to thefollowing ARE:

A′P + P A − P BR−1B′P + Q = 0 (12)

We decomposeK andP into N2L blocks of dimensionm×n

and n × n, respectively. Denote byKij and Pij the (i, j)block of the matrixK and P , respectively. In the followingtheorems we show thatKij and Pij satisfy certain propertieswhich will be critical for the design of stabilizing distributedcontrollers in Section IV. These properties stem from thespecial structure of the LQR problem (10). Next, the matrixX is defined asX = BR−1B′.

Theorem 1:Let K and x′0P x0 be the optimal controller

and the value function solution to the LQR problem (10). LetKij = K[(i−1)m : im, (j−1)n : jn] andPij = P [(i−1)n :

in, (j − 1)n : jn] with i = 1, . . . , NL, j = 1, . . . , NL.Then,

NL∑

i=1

A′

NL∑

j=1

Pij +

NL∑

j=1

PijA −

NL∑

j=1

PijX

NL∑

j=1

Pij + Qii

= 0,

(13)

Proof: Consider the ARE (12) associated with the LQRproblem (10). Without loss of generality, the diagonal blocksPii of P can be written as

Pii = Fii −

NL∑

j=1,j 6=i

Pij (14)

for a certain Fii. In general,Fii would be a function of∑NL

j=1,j 6=i Pij . We will show next that for the LQR prob-

lem (10),Fii = P and that it is not a function of∑NL

j=1,j 6=i Pij .The equations for the diagonal blocksPii of P in the ARE

equation (12) are

A′Pii + PiiA −

NL∑

k=1

(

PikXPik

)

+ Qii = 0 (15)

for i = 1, . . . , NL. Note also thatPij = Pji. Substituting (14)in the above equation leads to

A′Fii + FiiA − A

′

NL∑

k=1,k 6=i

Pik

−

NL∑

k=1,k 6=i

Pik

A

︸ ︷︷ ︸

1

−

NL∑

k=1,k 6=i

(

PikXPki

)

︸ ︷︷ ︸

2

−FiiXFii

−

NL∑

k=1,k 6=i

Pik

X

NL∑

l=1,l6=i

Pil

︸ ︷︷ ︸

3

+

NL∑

k=1,k 6=i

PikXFii + FiiX

NL∑

k=1,k 6=i

Pik

︸ ︷︷ ︸

4

+Qii = 0.

(16)

The equations for the off-diagonal blocksPij , i 6= j of Pin the ARE equation (12) are

A′Pij + PijA −

NL∑

k=1

(

PikXPkj

)

+ Qij = 0 (17)

for i, j = 1, . . . , NL, i 6= j. Substituting (14) in the aboveequation leads to

A′Pij + PijA − FiiXPij +

NL∑

k=1k 6=i

Pik

XPij − PijXFjj

+ PijX

NL∑

k=1k 6=j

Pjk

−

NL∑

k=1k 6=i,k 6=j

(

PikXPkj

)

+ Qij = 0.

(18)

Summing up equation (18) for allj 6= i corresponds to ablock-wise row sum of off-diagonal terms in thei-th blockrow of the ARE equation (12). This summation leads to

A′

NL∑

k=1k 6=i

Pik

+

NL∑

k=1k 6=i

Pik

A

︸ ︷︷ ︸

1

− FiiX

NL∑

k=1k 6=i

Pik

︸ ︷︷ ︸

4

(19a)

+

NL∑

l=1l6=i

NL∑

k=1k 6=i

Pik

XPil

︸ ︷︷ ︸

3

(19b)

+

NL∑

k=1k 6=i

PikX

NL∑

l=1l6=k

Pkl

−

NL∑

k=1k 6=i

NL∑

l=1l6=i,l6=k

(

PilXPlk

)

︸ ︷︷ ︸

2

(19c)

−

NL∑

k=1k 6=i

PikXFkk +

NL∑

k=1k 6=i

Qik = 0. (19d)

Notice that

(19c)=NL∑

k=1k 6=i

NL∑

l=1l 6=k

PikXPkl −

NL∑

k=1k 6=i

NL∑

l=1l 6=i,l 6=k

PikXPkl

=

NL∑

k=1,k 6=i

PikXPki,

(20)

where we used symmetry, switching of sum operators andrenaming the indices. Adding equation (19) to equation (16)and using properties ofQ defined in (9), equally numberedterms will cancel each other out leading to

A′Fii+FiiA−FiiXFii+

NL∑

k=1k 6=i

(

PikX(

Fii − Fkk

))

+Qii = 0.

(21)Summing up equation (21) over alli = 1, . . . , NL we obtain

NL∑

i=1

(

A′Fii + FiiA − FiiXFii + Qii

)

= 0, (22)

which proves the theorem.2Next we particularize Theorem 1 to the case of identical

weights and prove an additional property of LQR fordecoupled systems.

Theorem 2:Assume the weighting matrices (9) of the LQRproblem (10) are chosen as

Qii = Q1 ∀i = 1, . . . , NL

Qij = Q2 ∀i, j = 1, . . . , NL, i 6= j.(23)

Let x′0P x0 be the value function of the LQR problem (10)

with weights (23), and the blocks of the matrixP be denotedby Pij = P [(i−1)n : in, (j−1)n : jn] with i, j = 1, . . . , NL.

Then,

(I)∑NL

j=1 Pij = P for all i = 1, . . . , NL, where P isthe symmetric positive definite solution of the AREassociated with a single node local problem:

A′P + PA − PBR−1B′P + Q1 = 0. (24)

(II)∑NL

j=1 Kij = K for all i = 1, . . . , NL, where K =

−R−1B′P .(III) Pij = Plm , P2 ∀i 6= j, ∀ l 6= m is a symmetric

negative semidefinite matrix.

Proof: The assumption in (23) requires that the weightQ1 usedfor absolute states and the weightQ2 used for neighboringstate differences areequal for all nodes and for all neighborsof a node, respectively. Such an assumption and the fact thatA and B are block-diagonal with identical blocks, imply thatthe ARE in (12) is a set ofNL identical equations where thematricesPij are all identical and symmetric for alli 6= j. Wedenote byP2, the generic blockPij for i 6= j. The matricesFii =

∑NL

j=1 Pij defined in (14) are all identical and thereforeequation (13) in Theorem 1 becomes:

NL

(

A′Fii + FiiA − FiiXFii + Q1

)

= 0, (25)

which proves property (I) withFii = P . Property (II) followsfrom property (I) and from equation (11) which implies thatKij = −R−1B′Pij . Next we prove property (III).The ARE equations (17) for the blockPij with i 6= j become

A′P2 + P2A− P2XP1− P1XP2−(NL−2)P2XP2−Q2 = 0,(26)

which can be rewritten as follows in virtue of property (I):

(A−XP )′P2 + P2(A−XP )+(NL)P2XP2−Q2 = 0. (27)

where P is the symmetric positive definite solution of theARE (24) associated with a single node local problem.

Rewrite equation (27) as

(A − XP )′(−NLP2) + (−NLP2)(A − XP )−

− (−NLP2)′X(−NLP2) + NLQ2 = 0.

(28)

Since X > 0 and Q2 ≥ 0, equation (28) can be seenas an ARE associated with an LQR problem for the stablesystem (A − XP, B) with weights NLQ2 and R. Let thematrix−NLP2 be its positive semidefinite solution. Then, thefollowing matrix

P =

P − (NL − 1)P2 P2 · · · P2

P2 P − (NL − 1)P2 · · · P2

.... . .

. . ....

P2 . . . . . . P − (NL − 1)P2

(29)

is a symmetric positive definite matrix since

x′P x = x′iPxi +

NL∑

i=1

NL∑

j 6=i

(xi − xj)′P2(xi − xj) (30)

and it is the unique symmetric positive definite matrix solutionto the ARE (12). This proves the theorem.2

Under the hypothesis of Theorem 2, because of symmetryand equal weightsQ2 on the neighboring state differencesand equal weightsQ1 on absolute states, the LQR optimalcontroller will have the following structure:

K =

K1 K2 · · · K2

K2 K1 · · · K2

.... . .

. . ....

K2 . . . . . . K1

, (31)

with K1 andK2 functions ofNL, A, B, Q1, Q2 andR.Remark 2:We conjecture that Theorem 2 is true under

much milder assumptions on the tuning parameters (even withdifferent weights on neighboring state differences) as long asthe cost function is defined as in (9). The authors are currentlystudying this conjecture.

The following corollaries of Theorem 2 follow fromthe stability and the robustness of the LQR controller−R−1B′(−NLP2) for systemA − XP in (28).

Corollary 1: A − XP + NLXP2 is a Hurwitz matrix.From the gain margin properties [23] we have:

Corollary 2: A−XP + αNLXP2 is a Hurwitz matrix forall α > 1

2 , with α ∈ R.Remark 3:A−XP is a Hurwitz matrix, thus the system in

Corollary 2 is stable forα = 0 as well. (A−XP = A+ BKwith K being the LQR gain for system(A, B) with weights(Q1, R).)

The following condition defines a class of systems and LQRweighting matrices which will be used in later sections toextend the set of stabilizing distributed controller structures.

Condition 1: A−XP + αNLXP2 is a Hurwitz matrix forall α ∈ [0, 1

2 ], with α ∈ R.Essentially, Condition 1 characterizes systems for which the

LQR gain stability margin described in Corollary 2 is extendedto any positiveα. The fact thatA−XP is already stable, doesnot necessarily guarantee this property.

Checking the validity of Condition 1 for a given tuning ofP and P2 may be performed as a stability test for a simpleaffine parameter-dependent model

x = (A0 + αA1)︸ ︷︷ ︸

A(α)

x, (32)

where A0 = A − XP , A1 = NLXP2 and 0 ≤ α ≤ 12 .

This test can be posed as an LMI problem (Proposition 5.9 in[24]) searching for quadratic parameter-dependent Lyapunovfunctions.

In the following section we introduce some basic conceptsof graph theory before presenting the distributed control designproblem.

III. L APLACIAN SPECTRUM OFGRAPHS

This section is a concise review of the relationship betweenthe eigenvalues of a Laplacian matrix and the topology ofthe associated graph. We refer the reader to [25], [26] fora comprehensive treatment of the topic. We list a collectionof properties associated with undirected graph Laplaciansandadjacency matrices, which will be used in subsequent sectionsof the paper.

A graphG is defined as

G = (V ,A) (33)

whereV is the set of nodes (or vertices)V = {1, . . . , N} andA ⊆ V × V the set of edges(i, j) with i ∈ V , j ∈ V . Thedegreedj of a graph vertexj is the number of edges whichstart fromj. Let dmax(G) denote the maximum vertex degreeof the graphG.

We denote byA(G) the(0, 1) adjacency matrix of the graphG. Let Ai,j ∈ R be its i, j element, thenAi,i = 0, ∀ i =1, . . . , N , Ai,j = 0 if (i, j) /∈ A and Ai,j = 1 if (i, j) ∈ A,∀ i, j = 1, . . . , N, i 6= j. We will focus onundirectedgraphs,for which the adjacency matrixA(G) is symmetric.

Let S(A(G)) = {λ1(A(G)), . . . , λN (A(G))} be the spec-trum of the adjacency matrixA associated with an undirectedgraphG arranged in nondecreasing semi-order.

Property 1: λN (A(G)) ≤ dmax(G).This property together with Proposition 1 implies

Property 2: γi ≥ 0, ∀γi ∈ S(dmaxIN − A).We define the Laplacian matrix of a graphG in the following

wayL(G) = D(G) − A(G), (34)

where D(G) is the diagonal matrix of vertex degreesdi

(also called the valence matrix). Eigenvalues of Laplacianmatrices have been widely studied by graph theorists. Theirproperties are strongly related to the structural properties oftheir associated graphs. Every Laplacian matrix is a singularmatrix. By Gersgorin’s theorem [27], the real part of eachnonzero eigenvalue ofL(G) is strictly positive.

For undirected graphs,L(G) is a symmetric, positivesemidefinite matrix, which has only real eigenvalues. LetS(L(G)) = {λ1(L(G)), . . . , λN (L(G))} be the spectrum ofthe Laplacian matrixL associated with an undirected graphGarranged in nondecreasing semi-order. Then,

Property 3:

1) λ1(L(G)) = 0 with corresponding eigenvector of allones, andλ2(L(G)) 6= 0 iff G is connected. In fact, themultiplicity of 0 as an eigenvalue ofL(G) is equal to thenumber of connected components ofG.

2) The modulus ofλi(L(G)), i = 1, . . . , N is less thenN .The second smallest Laplacian eigenvalueλ2(L(G)) of

graphs is probably the most important information contained inthe spectrum of a graph. This eigenvalue, called the algebraicconnectivity of the graph, is related to several important graphinvariants, and it has been extensively investigated.

Let L(G) be the Laplacian of a graphG with N verticesand with maximal vertex degreedmax(G). Then properties ofλ2(L(G)) include

Property 4:1) λ2(L(G)) ≤ N

N−1 min{d(v), v ∈ V},2) λ2(L(G)) ≤ ν(G) ≤ η(G),3) λ2(L(G)) ≥ 2η(G)(1 − cos π

N),

4) λ2(L(G)) ≥ 2(cos πN

− cos 2 πN

)η(G)−2 cos π

N(1 − cos π

N)dmax(G),

where ν(G) is the vertex connectivity of the graphG (thesize of a smallest set of vertices whose removal rendersGdisconnected) andη(G) is the edge connectivity of the graph

G (the size of a smallest set of edges whose removal rendersG disconnected) [28].

Further relationships between the graph topology and Lapla-cian eigenvalue locations are discussed in [26] for undirectedgraphs. Spectral characterization of Laplacian matrices fordirected graphs can be found in [27].

IV. D ISTRIBUTED CONTROL DESIGN

We consider a set ofN linear, identical and decoupleddynamical systems, described by the continuous-time time-invariant state equation (2), rewritten below

xi = Axi + Bui,

xi(0) = xi0.

wherexi(t) ∈ Rn, ui(t) ∈ R

m are states and inputs of thei-th system at timet, respectively. Letx(t) ∈ RNn andu(t) ∈RNm be the vectors which collect the states and inputs of theN systems at timet, then

˙x = Ax + Bu,

x(0) = x0 , [x10, . . . , xN0]′,

(35)

with

A = IN ⊗ A,

B = IN ⊗ B.

Remark 4:Systems (35) and (3) differ only in the numberof subsystems. We will use system (3) withNL subsystemswhen referring to local problems, and system (35) withNsubsystems when referring to the global problem. Accordingly,tilded matrices will refer to local problems and hatted matriceswill refer to the global problem.

We use a graph to represent the coupling in the controlobjective and the communication in the following way. Weassociate thei-th system with thei-th node of a graphG =(V ,A). If an edge(i, j) connecting thei-th andj-th node ispresent, then 1) thei-th system has full information about thestate of thej-th system and, 2) thei-th system control lawminimizes a weighted distance between thei-th and thej-thsystem states.

The class ofKNn,m(G) matrices is defined as follows:

Definition 3: KNn,m(G) = {M ∈ RnN×mN |Mij =

0 if (i, j) /∈ A, Mij = M [(i−1)n : in, (j−1)m : jm], i, j =1, . . . , N}Thedistributed optimal control problemis defined as follows:

minK

J(u, x0) =

∫ ∞

0

(

x(τ)′Qx(τ) + u(τ)′Ru(τ))

dτ,

(36a)

subj. to ˙x = Ax + Bu, (36b)

u(t) = Kx(t), (36c)

K ∈ KNm,n(G), (36d)

Q ∈ KNn,n(G), R = IN ⊗ R, (36e)

x(0) = x0 (36f)

with Q = Q′ ≥ 0 and R = R′ > 0. We also refer toproblem (36) without (36d) as acentralized optimal control

problem. In general, computing the solution to problem (36) isan NP-hard problem. Next, we propose a suboptimal controldesign leading to a controllerK with the following properties:

1) K ∈ KNm,n(G) (37a)

2) A + BK is Hurwitz. (37b)

3) Simple tuning of absolute and relative state errors,

and control effort withinK. (37c)

Such controller will be referred to asdistributed suboptimalcontroller. The following theorem will be used to propose adistributed suboptimal control design procedure.

Theorem 3:Consider the LQR problem (10) withNL =dmax(G) + 1 and weights chosen as in (23) and its solu-tion (29), (31).

Let M ∈ RN×N be a symmetric matrix with the followingproperty:

λi(M) >NL

2, ∀λi(M) ∈ S(M)\{0}. (38)

and construct the feedback controller:

K = −IN ⊗ R−1B′P + M ⊗ R−1B′P2. (39)

Then, the closed loop system

Acl = IN ⊗ A + (IN ⊗ B)K (40)

is asymptotically stable.Proof: Consider the eigenvalues of the closed-loop system

Acl:

S(Acl) = S(

IN ⊗ (A − XP ) + M ⊗ (XP2))

By Proposition 2:

S(

IN ⊗ (A − XP ) + M ⊗ (XP2))

=

=

N⋃

i=1

S(

A − XP + λi(M)XP2

)

.(41)

We will prove that(A−XP+λi(M)XP2) is a Hurwitz matrix∀i = 1, . . . , N , and thus prove the theorem. Ifλi(M) = 0thenA−XP +λi(M)XP2 is Hurwitz based on Remark 3. Ifλi(M) 6= 0, then from Corollary 2 and from condition (38),we conclude thatA − XP + λi(M)XP2 is Hurwitz. 2

Theorem 3 has several main consequences:

1) If M ∈ KN1,1(G), then K in (39) is an asymptotically

stabledistributedcontroller.2) We can useone local LQR controller to compose dis-

tributed stabilizing controllers for a collection of identicaldynamically decoupled subsystems.

3) The first two consequences imply that we can not onlyfind a stabilizing distributed controller with a desiredsparsity pattern (which is in general a formidable taskby itself), but it is enough to solve a low-dimensionalproblem (characterized bydmax(G)) compared to thefull problem size (36). This attractive feature of ourapproach relies on the specific problem structure definedin Section II and IV.

4) The eigenvalues of the closed-loop large-scale systemS(Acl) can be computed throughN smaller eigenvalue

computations as⋃N

i=1 S(

A − XP + λi(M)XP2

)

.5) The result is independent of the local LQR tuning. Thus

Q1, Q2 andR in (23) can be used in order to influencethe compromise between minimization of absolute andrelative terms, and the control effort in the global perfor-mance.

For the special class of systems defined by Condition 1, thehypothesis of Theorem 3 can be relaxed as follows:

Theorem 4:Consider the LQR problem (10) with weightschosen as in (23) and its solution (29), (31). Assume thatCondition 1 holds.

Let M ∈ RN×N be a symmetric matrix with the following

property:λi(M) ≥ 0, ∀λi ∈ S(M). (42)

Then, the closed loop system (40) is asymptotically stablewhenK is constructed as in (39).

Proof: Notice that if Condition 1 holds, thenS(

A − XP + λiXP2

)

is Hurwitz for all λi(M) ≥ 0

(from Corollary 1 and Corollary 2). By Proposition 2

S(

IN ⊗ (A − XP ) + M ⊗ (XP2))

=

=

N⋃

i=1

S(

A − XP + λi(M)XP2

)

,

which together with condition (42) proves the theorem.2

In the next sections we show how to chooseM in Theo-rem 3 and Theorem 4 in order to construct distributed subop-timal controllers. The matrixM will ( i) reflect the structureof the graphG, (ii ) satisfy (38) or (42) and (iii ) be computedby using the graph adjacency matrix or the Laplacian matrix.

In order to make the exposition simpler, we will start with asimple special graph in Section IV-A and generalize the resultsin Section IV-B to any graph structure.

A. Finite Strings

Let G represent a string interconnection ofN systems withdmax(G) = 2 and the following Laplacian:

L(G) =

1 −1 0 · · · 0−1 2 −1 · · · 0...

. . .. . .

. . ....

0 · · · −1 2 −10 · · · 0 −1 1

. (43)

Construct a distributed controller as follows. Solve the LQRproblem (10), (23) withNL = 2 + 1 = 3 and desiredQ1,Q2, R. Consider the decomposition (29) and (31) of localcontroller and cost function, respectively. Takeb ≥ 0 andconstruct the global controller gain matrixK as

K =

K1 0 0 · · · 00 K1 0 · · · 0...

. . .. . .

. . ....

0 · · · 0 K1 00 · · · 0 0 K1

+

aK2 bK2 0 · · · 0bK2 aK2 bK2 · · · 0

.... . .

. . .. . .

...0 · · · bK2 aK2 bK2

0 · · · 0 bK2 aK2

(44)

The closed-loop system under the distributed control lawKcan be written asx = Aclx with

Acl = IN ⊗ A + (IN ⊗ B)K. (45)

RewriteK as:

K = −IN⊗(R−1B

′P1)−aIN⊗(R−1

B′P2)−bA(G)⊗(R−1

B′P2).

From Theorem 2,K is equal to:

K = −IN ⊗ (R−1B′P ) + M ⊗ (R−1B′P2)

and

M = (NL − 1 − a)IN − bA(G), NL = 3. (46)

The following corollary of Theorem 3 defines the range ofaandb which leads to a stable distributed controller.

Corollary 3: Consider controller (44). Ifa + 2b < 12 ,

then the closed loop system (40) is asymptotically stable.If Condition 1 holds then the closed loop system (40) isasymptotically stable for alla + 2b ≤ 2.

Proof: From Property 1,λmax(A(G)) ≤ dmax(G) = 2.Sinceb ≥ 0, by using Proposition 1 applied toM we derive:

λmin(M) = NL − 1 − a − bλmax(A(G)) ≥

≥ NL − 1 − a − 2b, NL = 3.(47)

Condition (38) of Theorem 3 holds ifa + 2b < 12 , which

proves the first part of the Theorem.Assume that Condition 1 holds and consider equation (47).

Then, condition (42) of Theorem 4 holds ifa + 2b ≤ 2. 2

Example 1:With a = 0 the distributed controller (44)becomes

K =

K1 bK2 0 · · · 0bK2 K1 bK2 · · · 0

.... . .

. . .. . .

...0 · · · bK2 K1 bK2

0 · · · 0 bK2 K1

. (48)

The closed loop system (40) is asymptotically stable ifb <14 . If Condition 1 holds, then the closed loop system (40) isasymptotically stable ifb ≤ 1.

Example 2:Assume that Condition 1 holds. Since the graphLaplacianL(G) is a symmetric positive semidefinite matrix,M = aL(G), a ≥ 0 satisfies condition (42) of Theorem 4.Thus any distributed controllerK = −IN ⊗ (R−1B′P ) +aL(G) ⊗ (R−1B′P2) having the following structure

K =

0 K2 0 · · · 0K2 0 K2 · · · 0...

. . ... .

. . ....

0 · · · K2 0 K2

0 · · · 0 K2 0

+ IN ⊗ (K1 − (2− a)K2)

(49)stabilizes system (40) for alla ≥ 0.

B. Arbitrary Graph Structures

We consider a generic graphG for N nodes with anassociated LaplacianL(G) and maximum vertex degreedmax.Let 0 = λ1(G) < λ2(G) . . . ≤ λN (G) be the eigenvalues ofthe the LaplacianL(G). In the following Corollaries 4, 5 and6 we present three ways of choosingM in (39) which lead todistributed suboptimal controllers.

Corollary 4: ComputeM in (39) as:

M = aL(G). (50)

If

a >NL

2λ2(G), (51)

then the closed loop system (40) is asymptotically stable whenK is constructed as in (39). In addition, if Condition 1 holds,then the closed loop system (40) is asymptotically stable forall a ≥ 0.

Proof: Apply condition (38) in Theorem 3 to the caseM = aL(G). Notice thatλi(G) ≥ λ2(G) for all i = 3, . . . , Nand λ2(G) 6= 0 by Property 3 of the Laplacian matrix.Therefore condition (38) becomesλ2(aL(G)) > NL

2 . Thisimplies aλ2(G) > NL

2 and thusa > NL

2λ2(G) . The first partof the Theorem is proven. Ifa ≥ 0 then λi(M) ≥ 0 ∀ i.Therefore the application of Theorem 4 to the matrixM =aL(G) proves the second part of the theorem.2

Remark 5:By using the third formula in Property 4, con-dition (51) can be linked to the edge connectivity as follows

a >NL

2η(G)(1 − cos

(πN

)) . (52)

Remark 6:Corollary 4 links the stability of the distributedcontroller to the size of the second smallest eigenvaluesof the graph Laplacian. It is well known that graphs withlarge λ2 (with respect to the maximal degree) have someproperties which make them very useful in several applicationssuch as computer science [29]–[32]. Interestingly enough,thisproperty is shown here to be crucial also for the design ofdistributed controllers. We refer the reader to [26] for a moredetailed discussion on the importance of the second smallesteigenvalue of a Laplacian.

Corollary 5: ComputeM in (39) as:

M = aIN − bA(G), b ≥ 0. (53)

If a − bdmax > NL

2 , then the closed loop system (40) isasymptotically stable whenK is constructed as in (39). Inaddition, if Condition 1 holds, then the closed loop system (40)is asymptotically stable ifa − bdmax ≥ 0.

Proof: Notice that λmin(M) = a − bλmax(A(G)) ≥a− bdmax. The proof is a direct consequence of Theorems 3and 4 and Property 1 of the adjacency matrix.2

Consider a weighted adjacency matrixAw = Aw(G),defined as follows. Denote byAw

i,j ∈ R its i, j element, thenAw

i,j = 0, if i = j and(i, j) /∈ A andAwi,j = wij if (i, j) ∈ A,

∀ i, j = 1, . . . , N, i 6= j. Assumewij = wji > 0. Definewmax as

wmax = maxi

∑

j

wij

Corollary 6: ComputeM in (39) as:

M = aIN − Aw(G). (54)

If a > wmax − NL

2 , then the closed loop system (40) isasymptotically stable whenK is constructed as in (39). Inaddition, if Condition 1 holds, then the closed loop system (40)is asymptotically stable ifa ≥ wmax.

Proof: Aw1 ≤ wmax1 and by Perron-Frobenius The-

orem λmax(Aw) ≤ wmax. Notice that λmin(M) = a −λmax(A(G)) ≥ a − wmax, then the proof is a direct con-sequence of Theorems 3 and 4.2

The results of Corollaries 4-6 are summarized in Table I.

Choice ofM Stability ConditionStability Condition If

Condition 1 Holds

M = aL(G) a >NL

2λ2(G)a ≥ 0

M = aIN − bA(G),b ≥ 0

a − bdmax >NL

2a − bdmax ≥ 0

aIN − Aw(G) a > wmax −NL

2a ≥ wmax

TABLE I

SUMMARY OF STABILITY CONDITIONS IN COROLLARIES4-6 FOR THE

CLOSED LOOP SYSTEM(39)-(40).

Corollaries 4-6 present three choices of distributed controldesign with increasing degrees of freedom. In fact,a, b andwij are additional parameters which, together withQ1, Q2

andR, can be used to tune the closed-loop system behavior.We recall here that from Theorem 3, the eigenvalues of theclosed-loop large-scale system are related to the eigenvaluesof M through the simple relation (41). Thus as long as thestability conditions defined in Table I are satisfied, the overallsystem architecture can be modified arbitrarily by adding orremoving subsystems and interconnection links. This leadstoa very powerful modular approach for designing distributedcontrol systems. The difference between the performance ofthe distributed control law and the performance of a centralizedoptimal solution can be also computed in a simple way. This isdiscussed in the next section. We remark that detailed perfor-mance analysis of the proposed distributed control design,orthe design of distributed control laws with performance guar-antees are not discussed within this manuscript and representfuture research topics.

C. Measure of Suboptimality

Consider system (35) and letK∗ and P ∗ be an LQRcontroller and the corresponding ARE solution associated withthe weightsQ > 0 and R > 0. Therefore,u(t) = −K∗x(t)minimizes the following cost function for anyx0:

J(u(t), x0) =

∫ ∞

0

(

x(τ)′Qx(τ) + u(τ)′Ru(τ))

dτ, (55)

i.e.,

J∗(x0) , minu(t)

J(u(t), x0) = J(−K∗x(t), x0) = x′0P

∗x0

(56)The next results aim at comparing the optimal centralizedcontroller K∗ and the distributed suboptimal controllerKpresented in the previous section. The comparison will bemade assuming (55) reflects the desired performance indexand computing the loss of performance introduced by thedistributed control law (39) withM chosen as in Corollaries 4-6.

Proposition 3: Consider the distributed controllerK in (39)and the closed loop system (40). If (40) is stable, then

J(−Kx(t), x0) = x′0P x0 (57)

where P is the positive definite solution of the followingLyapunov equation:

(IN ⊗ A + (IN ⊗ B)K)′P

+ P (IN ⊗ A + (IN ⊗ B)K)

+ (Q + K ′RK) = 0

(58)

Proof: By assumption, system (40) is stable and thus thepositive definite solution to the Lyapunov equation (58) canbe written as:

P =

∫ ∞

0

(

e(A+BK)τ′

(Q + K ′RK)e(A+BK)τ)

dτ (59)

ConsiderJ(−Kx(t), x0). From direct computation:

J(−Kx(t), x0) =

∫ ∞

0

(

x(τ)′(Q + K ′RK)x(τ))

dτ (60)

By substitutingx(τ) with e(A+BK)τ x0 we obtain (57).2Proposition 3 shows that the cost associated with the dis-

tributed controllerK according to the performance index (55)can be computed by solving a Lyapunov equation for theclosed loop system with weight equal to(Q + K ′RK).

Since P ∗ is optimal, J(−Kx(t), x0) ≥ J(−K∗x(t), x0)for all x0 and thus∆P = P − P ∗ is a positive semidefinitematrix which is equal to zero ifK = K∗. Any norm of ∆Pis a measure of the suboptimality of the distributed controllerK.

The “best” linear state-feedback distributed controllerK∗d ∈

KNm,n(G) could be computed by solving:

J∗d (x0) = min

Kd

‖∆P‖ (61a)

subj. to ˙x = Ax + Bu, (61b)

u(t) = Kdx(t), (61c)

Kd ∈ KNm,n(G), (61d)

x(0) = x0 (61e)

As discussed in the previous sections, computing the solutionto problem (61), is a difficult problem in general withoutfurther assumptions. Its efficient solution is the topic of currentresearch.

V. EXAMPLE

Consider a10 × 10 mesh interconnection ofN = 100identical, dynamically decoupled and independently actuatedlinear systems moving in a plane with double integratordynamics in both spatial dimensions:

xi = ux,i

yi = uy,i

i = 1, . . . , 100(62)

The interconnection structure is depicted in Figure 1. Thecontrol objective is to move each subsystem to a desired abso-lute position corresponding to its location in the pre-specifiedrectangular grid, which has equal separation distances definedbetween each orthogonal neighbor.

Fig. 1. The 10 × 10 finite mesh grid interconnection structure of thesimulation example.

The maximum vertex degree of such an interconnectiongraph is 4, as the nodes located at the “corners” of therectangular grid have 2 neighbors, those along the “edges” ofthe rectangle have 3 neighbors and the ones in the “middle”have 4 each.

A stabilizing distributed controller is designed for the meshinterconnection of systems, by solving the following, simpleLQR problem involving onlyNL = 5 nodes (dmax + 1).

minu

J(u, x0)

subj. to ˙x = Ax + Bu

x(0) = x0

(63)

where

A = I5 ⊗ A, B = I5 ⊗ B,

A =

0 1 0 00 0 0 00 0 0 10 0 0 0

, B =

0 01 00 00 1

.(64)

The cost function defined in (5) uses the following weightsfor absolute and relative state information

Qii = Qa,

Qij = Qr, ∀i 6= j.(65)

The weight on the control effort was kept constant in thefollowing examples (R = 1 for each control input in bothspatial dimensions). The subsystem model and chosen weightssatisfy Condition 1 (we tested the stability of (32) by solvingan LMI problem searching for quadratic parameter-dependent

Lyapunov functions), thus we can use less conservative stabil-ity ranges in Table I. Alternatively, one can choose the moreconservative stability ranges in Table I which can bealwayssatisfied.

The solution of the above local LQR problem will yielda controller matrix of the following form in this specificexample:

K =

Ka Kr Kr Kr Kr

Kr Ka Kr Kr Kr

Kr Kr Ka Kr Kr

Kr Kr Kr Ka Kr

Kr Kr Kr Kr Ka

. (66)

The distributed controller for the grid ofN = 100 inter-connected systems has the same structure as the underlyinggraph and is constructed in the following simple way basedon results presented in Section IV-B:

K = IN ⊗ Ka + A ⊗ Kr, (67)

where A denotes the adjacency matrix of the mesh gridinterconnection of systems. The distributed controller in(67)can be rewritten asK = IN ⊗ K − (4IN − A) ⊗ Kr, whereK = Ka +4Kr. This corresponds toM = 4IN −A in Table Iwith stability conditiona − dmax ≥ 0, which is satisfied inthis example sincea = dmax = 4.

Two simulation results, starting from the same initial condi-tions, are presented with different choices for the parametersQa and Qr in the local LQR problem tuning (65). The firstsimulation weighs the absolute state information in the localLQR solution more heavily than the second simulation andusesQa(1, 1) = 1 andQr(1, 1) = 1 values for position statesin both spatial dimensions. Velocity states were not weightedin these simulation examples. Snapshots of the simulation areshown in Figure 2, indicating that the subsystems convergeto their desired absolute positions along fairly straight linesstarting from their initial conditions. The second simulationillustrates the behavior of the large-scale distributed controlsystem, when much more emphasis is put on the relativestate information in the local LQR cost function, by selectingQa(1, 1) = 0.001 and Qr(1, 1) = 1000. As the snapshotsin Figure 3 demonstrate, the behavior of the overall inter-connected system has changed fundamentally. Although thedistributed controller is still stabilizing, the dynamic behaviorhas changed drastically and shows wave-like oscillations dueto the high weights on relative state information.

Videos of these simulation examples can be found at [33].The purpose of these examples was to show with a numericalexercise that the stabilizing effect of the proposed distributedcontrol design is independent of the local LQR weighting pa-rameter selection, thus these parameters are freely available fortuning and achieving different global performance objectives.

VI. CONCLUDING REMARKS

We have introduced a stabilizing distributed control designmethod for large-scale interconnections of identical, dynam-ically decoupled and independently actuated linear systems.The procedure requires the solution of a single local LQR

0 sec

(a)

0.84 sec

(b)

1.33 sec

(c)

1.61 sec

(d)

1.96 sec

(e)

7 sec

(f)

Fig. 2. Snapshots of the mesh grid simulation with larger weight onabsolute position references.

0 sec

(a)

0.35 sec

(b)

1.12 sec

(c)

2.17 sec

(d)

4.62 sec

(e)

7 sec

(f)

Fig. 3. Snapshots of the mesh grid simulation with larger weight onrelative position references.

problem, whose size depends only on the maximum vertexdegree of the interconnection graph. Special properties ofthe local LQR problem were derived, which enable the con-struction of stabilizing distributed controllers independentlyof the choice of weighting matrices. The relationship betweenstability of the overall large-scale system, the robustness oflocal controllers and the spectrum of a sparsity pattern matrixhas been highlighted. The proposed distributed controllerwillbe stabilizing even if subsystems and communication linksare added to or removed from the overall system, as long asthis does not change the maximum vertex degree or violatethe conditions given in Table I. This feature provides a verypowerful modularity to our approach.

VII. A CKNOWLEDGEMENTS

We thank the anonymous reviewers for the suggestions onhow to improve the paper and for pointing out an error in theoriginal version of the manuscript. Tamas Keviczky’s researchwas supported in part by the Boeing Corporation.

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