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# Distributed LQR Design for Identical Dynamically Decoupled Systems Francesco Borrelli Tamas Keviczky Abstract We consider a set of identical decoupled dynamical systems and a control problem where PDF document - DocSlides

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### Presentations text content in Distributed LQR Design for Identical Dynamically Decoupled Systems Francesco Borrelli Tamas Keviczky Abstract We consider a set of identical decoupled dynamical systems and a control problem where

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Distributed LQR Design for Identical Dynamically Decoupled Systems Francesco Borrelli , Tam´as Keviczky Abstract — We consider a set of identical decoupled dynamical systems and a control problem where the performance index couples the behavior of the systems. The coupling is describ ed through a communication graph where each system is a node and the control action at each node is only function of its sta te and the states of its neighbors. A distributed control desig method is presented which requires the solution of a single L QR problem. The size of the LQR problem is equal to the maximum vertex degree of the communication graph plus one. The desig procedure proposed in this paper illustrates how stability of the large-scale system is related to the robustness of local con trollers and the spectrum of a matrix representing the desired sparsi ty pattern of the distributed controller design problem. I. I NTRODUCTION ISTRIBUTED control techniques today can be found in a broad spectrum of applications ranging from robotics and formation ﬂight to civil engineering. Contributions and in terest in this ﬁeld date back to the early results of [1]. Approaches to distributed control design differ from each other in the assumptions they make on: ( ) the kind of interaction between different systems or different components of the same syste (dynamics, constraints, objective), ( ii ) the model of the system (linear, nonlinear, constrained, continuous-time, discr ete-time), iii ) the model of information exchange between the systems, iv ) the control design technique used. In this paper we focus on identical decoupled linear time- invariant systems . Our interest in distributed control for such systems arises from the abundance of networks of indepen- dently actuated systems and the necessity of avoiding centr al- ized design when this becomes computationally prohibitive Networks of vehicles in formation, production units in a pow er plant, cameras at an airport, an array of mechanical actuato rs for deforming a surface are just a few examples. In a descriptive way, the problem of distributed control for decoupled systems can be formulated as follows. A dynamical system is composed of (or can be decomposed into) distinct dynamical subsystems that can be independently actuated. T he subsystems are dynamically decoupled but have common ob- jectives, which make them interact with each other. Typical ly the interaction is local, i.e., the goal of a subsystem is a function of only a subset of other subsystems’ states. The interaction will be represented by an interaction 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 Netherl ands, t.keviczky@tudelft.nl the nodes represent the subsystems and an edge between two nodes denotes a coupling term in the controller associated w ith the nodes. Also, typically it is assumed that the exchange of information has a special structure, i.e., it is assumed that each subsystem can sense and/or exchange information with only a subset of other subsystems. We will assume that the interaction graph and the information exchange graph coincide. A distributed control scheme consists of distinc controllers, one for each subsystem, where the inputs to eac subsystem 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 interest in systems composed of a large number of interacting and cooperating interconnected units [2]–[20]. Recent advanc es in stability analysis of such systems yielded greater insight into the relationship between the spectrum of the interconnecti on graph and global stability of the overall system. Research e f- forts in this area are epitomized by the pioneering work of [5 ], which provided stability analysis tools for an interconnec tion of identical linear dynamical systems each using the same control law that operates on the average information obtain ed from neighbors. Most recent results, however, pertain to the study of dis- tributed parameter systems where the underlying dynamics a re spatially invariant, and where the controls and measuremen ts are spatially distributed. The fundamental work of [4], [6] in this ﬁeld discusses distributed LQR design purely for inﬁni te dimensional spatially invariant systems, where the proble diagonalizes exactly into a parameterized family of ﬁnite d i- mensional LQR problems. It is established that the correspo nd- ing ARE solutions are translation-invariant operators, an d the optimal controller is a spatially invariant system. The aut hors show that quadratically optimal controllers for spatially in- variant systems are themselves spatially invariant. Anoth er recent result for spatially invariant systems and unbounde domains in [7] considers the problem of distributed control ler design, when communication among the sites is limited. In particular, the controller is assumed to be constrained so that information is propagated with a delay that depends on the distance between subsystems. The authors show that the problem of optimal design can be cast as a convex problem provided that the propagation speeds in the controller are a least as fast as those in the plant. Special spatially invari ant structures composed of an inﬁnite string of linear systems have also received much attention. The paper [8] analyzes invers optimality of localized distributed controllers in such sy stems. Another signiﬁcant group of recently explored strategies involves relaxations to the LMI versions of these spatially

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invariant problems. In [9], authors consider heterogenous subsystems and a relaxation, which is used to derive sufﬁcie nt conditions for the existence of controllers which stabiliz e the system and provide a guaranteed level of performance. The work in [10] is concerned with ﬁnding distributed controlle rs for a set of arbitrarily connected, ﬁnite and possibly het- erogeneous LTI systems. The authors consider nonoriented edges in the graph interconnection and formulate convex conditions for the existence of output-feedback controlle rs, which achieve a certain performance. The resulting linear matrix inequalities grow in size with the number of systems in the interconnection. The complexity associated with the computation of dis- tributed optimal controllers is well exempliﬁed by the stud y in [11]. There, a ﬁnite-time LQR synthesis problem in discrete time is considered, where the matrix describing the control law is constrained to lie in a particular vector space. This vector space represents a pre-speciﬁed distributed contro structure between a network of autonomous agents. The computationally intensive optimal solution for the distri buted control problem is presented along with a computationally more tractable sub-optimal one. One of the most notable recent result in characterizing the complexity of optimal distributed controller design subje ct to constraints on the controller structure was presented in [1 2]. The authors establish that in general, efﬁcient solution of such problems require a special property of the sparsity pattern or interconnection structure to hold. Speciﬁcally, it is sh own that quadratic invariance of the assumed controller struct ure implies that the distributed minimum-norm problem may be solved efﬁciently via convex programming. This manuscript proposes a simple distributed controller de- sign approach and focuses on a class of systems, for which ex- isting methods outlined above are either not efﬁcient or wou ld not even be directly applicable. Our method applies to large scale systems composed of ﬁnite number of identical subsys- tems where the interconnection structure or sparsity pattern is not required to have any special invariance properties . The philosophy of our approach builds on the recent works [13] [15], where at each node, the model of its neighbors are used to predict their behavior. We show that in absence of state and input constraints, and for identical linear syste dynamics, such an approach leads to an extremely powerful result: the synthesis of stabilizing distributed control l aws can be obtained by using a simple local LQR design, whose size is limited by the maximum vertex degree of the interconnection graph plus one. Furthermore, the design procedure proposed in this paper illustrates how stability of the overall large -scale system is related to the robustness of local controllers and the spectrum of a matrix representing the desired sparsity pattern. In addition, the constructed distributed control ler is stabilizing independent of the tuning parameters in the loc al LQR cost function. This leads to a method for designing distributed controllers for a ﬁnite number of dynamically decoupled systems, where the local tuning parameters can be chosen to obtain a desirable global performance. Such resul can be immediately used to improve current stability analys is and controller synthesis in the ﬁeld of distributed recedin horizon control for dynamically decoupled systems [3], [13 ], [16]–[18]. We emphasize that the aforementioned advantage are the results of two main simplifying assumptions compare to a general structured optimal design: identical and decou pled subsystem dynamics and suboptimality of the global control ler. The paper is organized as follows. In Section II, we study the solution properties of the LQR for a set of identical, dec ou- pled linear systems. Such properties will be used in the pape to construct a stabilizing distributed controller for arbi trary interconnection structures. Section III summarizes impor tant properties of graph Laplacian and adjacency matrices, whic will be useful in our proofs. Section IV presents the stabili zing distributed controller design procedure using the local LQ solution properties. The proposed distributed control des ign method and the effect of the free tuning parameters in the local LQR design is illustrated by a simulation example in Section V. Some concluding remarks are made in Section VI. OTATION AND RELIMINARIES We denote by the ﬁeld of real numbers, the ﬁeld of complex numbers and the set of real matrices. Re Re The transpose of a vector and a matrix will be denoted by and , respectively. A matrix is symmetric if Notation 1: Let , then j, k denotes a matrix of dimension + 1) + 1) obtained by extracting rows to and columns to from the matrix with Notation 2: denotes the identity matrix of dimension Notation 3: Let denote the -th eigenvalue of = 1 , . . ., n . The spectrum of will be denoted by ) = , . . . , Deﬁnition 1: Let A, B be symmetric. 1) is positive deﬁnite if Ax > for all nonzero , and is positive semideﬁnite if Ax for all nonzero . We denote this by A > and respectively. 2) is negative (semi)deﬁnite if is positive (semi)deﬁnite. 3) A < B and mean B < and respectively. Deﬁnition 2: A matrix is called Hurwitz (or stable) if all its eigenvalues have negative real part, i.e. , i = 1 , . . ., n Notation 4: Let . Then denotes the Kronecker product of and 11 mn mp nq (1) Proposition 1: Consider two matrices αI and . Then ) = , i = 1 , . . . , n Proof: Take any eigenvalue and the corresponding eigenvector . Then Av Bv

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αv = ( Proposition 2: Given A, C and consider two matrices and , where A, nm nm . Then ) = =1 where is the -th eigenvalue of Proof: Let be an eigenvector of corresponding to , and be an eigenvector of = ( with as the associated eigenvalue. Consider the vector nm . Then )( ) = Au Bv Cu Au Cu Au Cu . Since , we get )( ) = )( II. LQR P ROPERTIES FOR YNAMICALLY ECOUPLED YSTEMS Consider a set of identical, decoupled linear time- invariant dynamical systems, the -th system being described by the continuous-time state equation: Ax Bu (0) = (2) where are states and inputs of the -th system at time , respectively. Let nN and mN be the vectors which collect the states and inputs of the systems at time u, (0) = 10 , . . ., x (3) with A, B. (4) We consider an LQR control problem for the set of systems where the cost function couples the dynamic behavio of individual systems: ( u, ) = =1 ii ) + ii )) + =1 )) ij )) d (5) with ii ii R > , Q ii ii i, (6a) ij ij ji j. (6b) The cost function (5) contains terms which weigh the -th system states and inputs, as well as the difference between the -th and the -th system states and can be rewritten using the following compact notation: ( ) = ) + dτ, (7) where the matrices and have a special structure deﬁned next. and can be decomposed into blocks of dimen- sion and respectively: 11 12 . . . . . . . . . . . . R (8) with ii ii =1 , k ik , i = 1 , . . ., N ij ij , i, j = 1 , . . ., N , i j. R. (9) Remark 1: The cost function structure (5) can be used to describe several practical applications including form ation ﬂight, paper machine control and monitoring networks of cameras [19], [21]. Let and be the optimal controller and the value function corresponding to the following LQR problem: min ( u, subj. to (0) = (10) Throughout the paper we will assume that a stabilizing solu- tion to the LQR problem (10) with ﬁnite performance index exists and is unique (see [22], p. 52 and references therein) Assumption 1: System A, is stabilizable and system A, is observable, where is any matrix such that We will also assume local stabilizability and observabilit y: Assumption 2: System A, B is stabilizable and systems A, C are observable, where is any matrix such that It is well known that P, (11) where is the symmetric positive deﬁnite solution to the following ARE: = 0 (12) We decompose and into blocks of dimension and , respectively. Denote by ij and ij the i, j block of the matrix and , respectively. In the following theorems we show that ij and ij satisfy certain properties which will be critical for the design of stabilizing distrib uted controllers in Section IV. These properties stem from the special structure of the LQR problem (10). Next, the matrix is deﬁned as BR Theorem 1: Let and be the optimal controller and the value function solution to the LQR problem (10). Let ij [( 1) im, 1) jn and ij [( 1)

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in, 1) jn with = 1 , . . ., N = 1 , . . ., N Then, =1 =1 ij =1 ij =1 ij =1 ij ii = 0 (13) Proof: Consider the ARE (12) associated with the LQR problem (10). Without loss of generality, the diagonal bloc ks ii of can be written as ii ii =1 ,j ij (14) for a certain ii . In general, ii would be a function of =1 ,j ij . We will show next that for the LQR prob- lem (10), ii and that it is not a function of =1 ,j ij The equations for the diagonal blocks ii of in the ARE equation (12) are ii ii =1 ik ik ii = 0 (15) for = 1 , . . ., N . Note also that ij ji . Substituting (14) in the above equation leads to ii ii =1 ,k ik =1 ,k ik {z =1 ,k ik ki {z ii ii =1 ,k ik =1 ,l il {z =1 ,k ik ii ii =1 ,k ik {z ii = 0 (16) The equations for the off-diagonal blocks ij , i of in the ARE equation (12) are ij ij =1 ik kj ij = 0 (17) for i, j = 1 , . . ., N , i . Substituting (14) in the above equation leads to ij ij ii ij =1 ik ij ij jj ij =1 jk =1 i,k ik kj ij = 0 (18) Summing up equation (18) for all corresponds to a block-wise row sum of off-diagonal terms in the -th block row of the ARE equation (12). This summation leads to =1 ik =1 ik {z ii =1 ik {z (19a) =1 =1 ik il {z (19b) =1 ik =1 kl =1 =1 i,l il lk {z (19c) =1 ik kk =1 ik = 0 (19d) Notice that (19c) =1 =1 ik kl =1 =1 i,l ik kl =1 ,k ik ki (20) where we used symmetry, switching of sum operators and renaming the indices. Adding equation (19) to equation (16) and using properties of deﬁned in (9), equally numbered terms will cancel each other out leading to ii ii ii ii =1 ik ii kk ii = 0 (21) Summing up equation (21) over all = 1 , . . ., N we obtain =1 ii ii ii ii ii = 0 (22) which proves the theorem. Next we particularize Theorem 1 to the case of identical weights and prove an additional property of LQR for decoupled systems. Theorem 2: Assume the weighting matrices (9) of the LQR problem (10) are chosen as ii = 1 , . . ., N ij i, j = 1 , . . ., N , i j. (23) Let be the value function of the LQR problem (10) with weights (23), and the blocks of the matrix be denoted by ij [( 1) in, 1) jn with i, j = 1 , . . ., N

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Then, (I) =1 ij for all = 1 , . . ., N , where is the symmetric positive deﬁnite solution of the ARE associated with a single node local problem: PA PBR = 0 (24) (II) =1 ij for all = 1 , . . ., N , where (III) ij lm j, is a symmetric negative semideﬁnite matrix. Proof: The assumption in (23) requires that the weight used for absolute states and the weight used for neighboring state differences are equal for all nodes and for all neighbors of a node, respectively. Such an assumption and the fact that and are block-diagonal with identical blocks, imply that the ARE in (12) is a set of identical equations where the matrices ij are all identical and symmetric for all . We denote by , the generic block ij for . The matrices ii =1 ij deﬁned in (14) are all identical and therefore equation (13) in Theorem 1 becomes: ii ii ii ii = 0 (25) which proves property (I) with ii . Property (II) follows from property (I) and from equation (11) which implies that ij ij . Next we prove property (III). The ARE equations (17) for the block ij with become 2) = 0 (26) which can be rewritten as follows in virtue of property (I): XP XP )+( = 0 (27) where is the symmetric positive deﬁnite solution of the ARE (24) associated with a single node local problem. Rewrite equation (27) as XP ) + ( )( XP ) + = 0 (28) Since X > and , equation (28) can be seen as an ARE associated with an LQR problem for the stable system XP, B with weights and . Let the matrix be its positive semideﬁnite solution. Then, the following matrix 1) 1) . . . . . . P 1) (29) is a symmetric positive deﬁnite matrix since Px =1 (30) and it is the unique symmetric positive deﬁnite matrix solut ion to the ARE (12). This proves the theorem. Under the hypothesis of Theorem 2, because of symmetry and equal weights on the neighboring state differences and equal weights on absolute states, the LQR optimal controller will have the following structure: . . . . . . K (31) with and functions of and Remark 2: We conjecture that Theorem 2 is true under much milder assumptions on the tuning parameters (even with different weights on neighboring state differences) as lon g as the cost function is deﬁned as in (9). The authors are current ly studying this conjecture. The following corollaries of Theorem 2 follow from the stability and the robustness of the LQR controller for system XP in (28). Corollary 1: XP is a Hurwitz matrix. From the gain margin properties [23] we have: Corollary 2: XP αN is a Hurwitz matrix for all α > , with Remark 3: XP is a Hurwitz matrix, thus the system in Corollary 2 is stable for = 0 as well. ( XP BK with being the LQR gain for system A, B with weights , R .) The following condition deﬁnes a class of systems and LQR weighting matrices which will be used in later sections to extend the set of stabilizing distributed controller struc tures. Condition 1: XP αN is a Hurwitz matrix for all [0 , with Essentially, Condition 1 characterizes systems for which t he LQR gain stability margin described in Corollary 2 is extend ed to any positive . The fact that XP is already stable, does not necessarily guarantee this property. Checking the validity of Condition 1 for a given tuning of and may be performed as a stability test for a simple afﬁne parameter-dependent model = ( αA {z x, (32) where XP and This test can be posed as an LMI problem (Proposition 5.9 in [24]) searching for quadratic parameter-dependent Lyapun ov functions. In the following section we introduce some basic concepts of graph theory before presenting the distributed control d esign problem. III. L APLACIAN PECTRUM OF RAPHS This section is a concise review of the relationship between the eigenvalues of a Laplacian matrix and the topology of the associated graph. We refer the reader to [25], [26] for a comprehensive treatment of the topic. We list a collection of properties associated with undirected graph Laplacians and adjacency matrices, which will be used in subsequent sectio ns of the paper.

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A graph is deﬁned as = ( (33) where is the set of nodes (or vertices) , . . ., N and A⊆VV the set of edges i, j with ∈V , j ∈V . The degree of a graph vertex is the number of edges which start from . Let max denote the maximum vertex degree of the graph We denote by the (0 1) adjacency matrix of the graph . Let i,j be its i, j element, then i,i = 0 , . . ., N i,j = 0 if i, j ∈A and i,j = 1 if i, j ∈A i, j = 1 , . . ., N, i . We will focus on undirected graphs, for which the adjacency matrix is symmetric. Let )) = )) , . . . , )) be the spec- trum of the adjacency matrix associated with an undirected graph arranged in nondecreasing semi-order. Property 1: )) max This property together with Proposition 1 implies Property 2: ∈S max We deﬁne the Laplacian matrix of a graph in the following way ) = (34) where is the diagonal matrix of vertex degrees (also called the valence matrix). Eigenvalues of Laplacian matrices have been widely studied by graph theorists. Their properties are strongly related to the structural properti es of their associated graphs. Every Laplacian matrix is a singul ar matrix. By Gerˇsgorin’s theorem [27], the real part of each nonzero eigenvalue of is strictly positive. For undirected graphs, is a symmetric, positive semideﬁnite matrix, which has only real eigenvalues. Let )) = )) , . . . , )) be the spectrum of the Laplacian matrix associated with an undirected graph arranged in nondecreasing semi-order. Then, Property 3: 1) )) = 0 with corresponding eigenvector of all ones, and )) = 0 iff is connected. In fact, the multiplicity of as an eigenvalue of is equal to the number of connected components of 2) The modulus of )) , i = 1 , . . ., N is less then The second smallest Laplacian eigenvalue )) of graphs is probably the most important information containe d in the spectrum of a graph. This eigenvalue, called the algebra ic connectivity of the graph, is related to several important g raph invariants, and it has been extensively investigated. Let be the Laplacian of a graph with vertices and with maximal vertex degree max . Then properties of )) include Property 4: 1) )) min , v ∈V} 2) )) 3) )) )(1 cos 4) )) 2(cos cos2 2 cos (1 cos max where is the vertex connectivity of the graph (the size of a smallest set of vertices whose removal renders disconnected) and is the edge connectivity of the graph (the size of a smallest set of edges whose removal renders disconnected) [28]. Further relationships between the graph topology and Lapla cian eigenvalue locations are discussed in [26] for undirec ted graphs. Spectral characterization of Laplacian matrices f or directed graphs can be found in [27]. IV. D ISTRIBUTED ONTROL ESIGN We consider a set of linear, identical and decoupled dynamical systems, described by the continuous-time time- invariant state equation (2), rewritten below Ax Bu (0) = where are states and inputs of the -th system at time , respectively. Let Nn and Nm be the vectors which collect the states and inputs of the systems at time , then u, (0) = 10 , . . . , x (35) with A, B. Remark 4: Systems (35) and (3) differ only in the number of subsystems. We will use system (3) with subsystems when referring to local problems, and system (35) with subsystems when referring to the global problem. According ly, tilded matrices will refer to local problems and hatted matr ices will refer to the global problem. We use a graph to represent the coupling in the control objective and the communication in the following way. We associate the -th system with the -th node of a graph . If an edge i, j connecting the -th and -th node is present, then 1) the -th system has full information about the state of the -th system and, 2) the -th system control law minimizes a weighted distance between the -th and the -th system states. The class of n,m matrices is deﬁned as follows: Deﬁnition 3: n,m ) = nN mN ij if i, j ∈A , M ij [( 1) in, 1) jm , i, j , . . ., N The distributed optimal control problem is deﬁned as follows: min ( u, ) = ) + dτ, (36a) subj. to u, (36b) ) = (36c) ∈K m,n (36d) ∈K n,n R, (36e) (0) = (36f) with and . We also refer to problem (36) without (36d) as a centralized optimal control

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problem . In general, computing the solution to problem (36) is an NP-hard problem. Next, we propose a suboptimal control design leading to a controller with the following properties: 1) ∈K m,n (37a) 2) is Hurwitz. (37b) 3) Simple tuning of absolute and relative state errors, and control effort within K. (37c) Such controller will be referred to as distributed suboptimal controller . The following theorem will be used to propose a distributed suboptimal control design procedure. Theorem 3: Consider the LQR problem (10) with max ) + 1 and weights chosen as in (23) and its solu- tion (29), (31). Let be a symmetric matrix with the following property: ∈S \{ (38) and construct the feedback controller: (39) Then, the closed loop system cl + ( (40) is asymptotically stable. Proof: Consider the eigenvalues of the closed-loop system cl cl ) = XP ) + By Proposition 2: XP ) + =1 XP (41) We will prove that XP is a Hurwitz matrix = 1 , . . ., N , and thus prove the theorem. If ) = 0 then XP is Hurwitz based on Remark 3. If = 0 , then from Corollary 2 and from condition (38), we conclude that XP is Hurwitz. Theorem 3 has several main consequences: 1) If ∈K , then in (39) is an asymptotically stable distributed controller. 2) We can use one local LQR controller to compose dis- tributed stabilizing controllers for a collection of ident ical dynamically decoupled subsystems. 3) The ﬁrst two consequences imply that we can not only ﬁnd a stabilizing distributed controller with a desired sparsity pattern (which is in general a formidable task by itself), but it is enough to solve a low-dimensional problem (characterized by max )) compared to the full problem size (36). This attractive feature of our approach relies on the speciﬁc problem structure deﬁned in Section II and IV. 4) The eigenvalues of the closed-loop large-scale system cl can be computed through smaller eigenvalue computations as =1 XP 5) The result is independent of the local LQR tuning. Thus and in (23) can be used in order to inﬂuence the compromise between minimization of absolute and relative terms, and the control effort in the global perfor- mance. For the special class of systems deﬁned by Condition 1, the hypothesis of Theorem 3 can be relaxed as follows: Theorem 4: Consider the LQR problem (10) with weights chosen as in (23) and its solution (29), (31). Assume that Condition 1 holds. Let be a symmetric matrix with the following property: ∈S (42) Then, the closed loop system (40) is asymptotically stable when is constructed as in (39). Proof: Notice that if Condition 1 holds, then XP is Hurwitz for all (from Corollary 1 and Corollary 2). By Proposition 2 XP ) + =1 XP which together with condition (42) proves the theorem. In the next sections we show how to choose in Theo- rem 3 and Theorem 4 in order to construct distributed subop- timal controllers. The matrix will ( ) reﬂect the structure of the graph , ( ii ) satisfy (38) or (42) and ( iii ) be computed by using the graph adjacency matrix or the Laplacian matrix. In order to make the exposition simpler, we will start with a simple special graph in Section IV-A and generalize the resu lts in Section IV-B to any graph structure. A. Finite Strings Let represent a string interconnection of systems with max ) = 2 and the following Laplacian: ) = 1 0 1 2 1 2 1 1 (43) Construct a distributed controller as follows. Solve the LQ problem (10), (23) with = 2 + 1 = 3 and desired . Consider the decomposition (29) and (31) of local controller and cost function, respectively. Take and construct the global controller gain matrix as 0 0 0 0 aK bK bK aK bK bK aK bK bK aK (44)

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The closed-loop system under the distributed control law can be written as cl with cl + ( K. (45) Rewrite as: aI From Theorem 2, is equal to: ) + and = ( , N = 3 (46) The following corollary of Theorem 3 deﬁnes the range of and which leads to a stable distributed controller. Corollary 3: Consider controller (44). If + 2 b < then the closed loop system (40) is asymptotically stable. If Condition 1 holds then the closed loop system (40) is asymptotically stable for all + 2 Proof: From Property 1, max )) max ) = 2 Since , by using Proposition 1 applied to we derive: min ) = b max )) b, N = 3 (47) Condition (38) of Theorem 3 holds if + 2 b < , which proves the ﬁrst part of the Theorem. Assume that Condition 1 holds and consider equation (47). Then, condition (42) of Theorem 4 holds if + 2 Example 1: With = 0 the distributed controller (44) becomes bK bK bK bK bK bK (48) The closed loop system (40) is asymptotically stable if b < . If Condition 1 holds, then the closed loop system (40) is asymptotically stable if Example 2: Assume that Condition 1 holds. Since the graph Laplacian is a symmetric positive semideﬁnite matrix, aL satisﬁes condition (42) of Theorem 4. Thus any distributed controller ) + aL having the following structure (2 (49) stabilizes system (40) for all B. Arbitrary Graph Structures We consider a generic graph for nodes with an associated Laplacian and maximum vertex degree max Let 0 = < . . . be the eigenvalues of the the Laplacian . In the following Corollaries 4, 5 and 6 we present three ways of choosing in (39) which lead to distributed suboptimal controllers. Corollary 4: Compute in (39) as: aL (50) If a > (51) then the closed loop system (40) is asymptotically stable wh en is constructed as in (39). In addition, if Condition 1 holds, then the closed loop system (40) is asymptotically stable fo all Proof: Apply condition (38) in Theorem 3 to the case aL . Notice that for all = 3 , . . ., N and = 0 by Property 3 of the Laplacian matrix. Therefore condition (38) becomes aL )) . This implies a and thus a > . The ﬁrst part of the Theorem is proven. If then Therefore the application of Theorem 4 to the matrix aL proves the second part of the theorem. Remark 5: By using the third formula in Property 4, con- dition (51) can be linked to the edge connectivity as follows a > cos (52) Remark 6: Corollary 4 links the stability of the distributed controller to the size of the second smallest eigenvalues of the graph Laplacian. It is well known that graphs with large (with respect to the maximal degree) have some properties which make them very useful in several applicati ons such as computer science [29]–[32]. Interestingly enough, this property is shown here to be crucial also for the design of distributed controllers. We refer the reader to [26] for a mo re detailed discussion on the importance of the second smalles eigenvalue of a Laplacian. Corollary 5: Compute in (39) as: aI , b (53) If bd max , then the closed loop system (40) is asymptotically stable when is constructed as in (39). In addition, if Condition 1 holds, then the closed loop system ( 40) is asymptotically stable if bd max Proof: Notice that min ) = b max )) bd max . The proof is a direct consequence of Theorems 3 and 4 and Property 1 of the adjacency matrix. Consider a weighted adjacency matrix deﬁned as follows. Denote by i,j its i, j element, then i,j = 0 , if and i, j ∈A and i,j ij if i, j ∈A

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i, j = 1 , . . ., N, i . Assume ij ji . Deﬁne max as max = max ij Corollary 6: Compute in (39) as: aI (54) If a > w max , then the closed loop system (40) is asymptotically stable when is constructed as in (39). In addition, if Condition 1 holds, then the closed loop system ( 40) is asymptotically stable if max Proof: max and by Perron-Frobenius The- orem max max . Notice that min ) = max )) max , then the proof is a direct con- sequence of Theorems 3 and 4. The results of Corollaries 4-6 are summarized in Table I. Choice of Stability Condition Stability Condition If Condition 1 Holds aL a > aI bd max bd max aI a > w max max TABLE I UMMARY OF STABILITY CONDITIONS IN OROLLARIES 4-6 FOR THE CLOSED LOOP SYSTEM (39)-(40). Corollaries 4-6 present three choices of distributed contr ol design with increasing degrees of freedom. In fact, and ij are additional parameters which, together with and , can be used to tune the closed-loop system behavior. We recall here that from Theorem 3, the eigenvalues of the closed-loop large-scale system are related to the eigenval ues of through the simple relation (41). Thus as long as the stability conditions deﬁned in Table I are satisﬁed, the ove rall system architecture can be modiﬁed arbitrarily by adding or removing subsystems and interconnection links. This leads to a very powerful modular approach for designing distributed control systems. The difference between the performance of the distributed control law and the performance of a central ized optimal solution can be also computed in a simple way. This is discussed in the next section. We remark that detailed perfo r- mance analysis of the proposed distributed control design, or the design of distributed control laws with performance gua r- antees are not discussed within this manuscript and represe nt future research topics. C. Measure of Suboptimality Consider system (35) and let and be an LQR controller and the corresponding ARE solution associated w ith the weights Q > and R > . Therefore, ) = minimizes the following cost function for any ( ) = ) + dτ, (55) i.e., ( min ( ) = ) = (56) The next results aim at comparing the optimal centralized controller and the distributed suboptimal controller presented in the previous section. The comparison will be made assuming (55) reﬂects the desired performance index and computing the loss of performance introduced by the distributed control law (39) with chosen as in Corollaries 4- 6. Proposition 3: Consider the distributed controller in (39) and the closed loop system (40). If (40) is stable, then ) = (57) where is the positive deﬁnite solution of the following Lyapunov equation: + ( + ( + ( ) = 0 (58) Proof: By assumption, system (40) is stable and thus the positive deﬁnite solution to the Lyapunov equation (58) can be written as: d (59) Consider . From direct computation: ) = ) d (60) By substituting with we obtain (57). Proposition 3 shows that the cost associated with the dis- tributed controller according to the performance index (55) can be computed by solving a Lyapunov equation for the closed loop system with weight equal to Since is optimal, for all and thus is a positive semideﬁnite matrix which is equal to zero if . Any norm of is a measure of the suboptimality of the distributed control ler The “best” linear state-feedback distributed controller m,n could be computed by solving: ( ) = min (61a) subj. to u, (61b) ) = (61c) ∈K m,n (61d) (0) = (61e) As discussed in the previous sections, computing the soluti on to problem (61), is a difﬁcult problem in general without further assumptions. Its efﬁcient solution is the topic of c urrent research.

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V. E XAMPLE Consider a 10 10 mesh interconnection of = 100 identical, dynamically decoupled and independently actua ted linear systems moving in a plane with double integrator dynamics in both spatial dimensions: x,i y,i = 1 , . . . , 100 (62) The interconnection structure is depicted in Figure 1. The control objective is to move each subsystem to a desired abso lute position corresponding to its location in the pre-spec iﬁed rectangular grid, which has equal separation distances de ned between each orthogonal neighbor. Fig. 1. The 10 10 ﬁnite mesh grid interconnection structure of the simulation example. The maximum vertex degree of such an interconnection graph is 4, as the nodes located at the “corners” of the rectangular grid have 2 neighbors, those along the “edges” o the rectangle have 3 neighbors and the ones in the “middle have 4 each. A stabilizing distributed controller is designed for the me sh interconnection of systems, by solving the following, simp le LQR problem involving only = 5 nodes ( max + 1). min ( u, subj. to (0) = (63) where A, B, 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 , B 0 0 1 0 0 0 0 1 (64) The cost function deﬁned in (5) uses the following weights for absolute and relative state information ii ij j. (65) The weight on the control effort was kept constant in the following examples ( = 1 for each control input in both spatial dimensions). The subsystem model and chosen weight satisfy Condition 1 (we tested the stability of (32) by solvi ng an LMI problem searching for quadratic parameter-dependen Lyapunov functions), thus we can use less conservative stab il- ity ranges in Table I. Alternatively, one can choose the more conservative stability ranges in Table I which can be always satisﬁed. The solution of the above local LQR problem will yield a controller matrix of the following form in this speciﬁc example: (66) The distributed controller for the grid of = 100 inter- connected systems has the same structure as the underlying graph and is constructed in the following simple way based on results presented in Section IV-B: (67) where denotes the adjacency matrix of the mesh grid interconnection of systems. The distributed controller in (67) can be rewritten as (4 , where +4 . This corresponds to = 4 in Table I with stability condition max , which is satisﬁed in this example since max = 4 Two simulation results, starting from the same initial cond i- tions, are presented with different choices for the paramet ers and in the local LQR problem tuning (65). The ﬁrst simulation weighs the absolute state information in the loc al LQR solution more heavily than the second simulation and uses (1 1) = 1 and (1 1) = 1 values for position states in both spatial dimensions. Velocity states were not weight ed in these simulation examples. Snapshots of the simulation a re shown in Figure 2, indicating that the subsystems converge to their desired absolute positions along fairly straight l ines starting from their initial conditions. The second simulat ion illustrates the behavior of the large-scale distributed co ntrol system, when much more emphasis is put on the relative state information in the local LQR cost function, by selecti ng (1 1) = 0 001 and (1 1) = 1000 . As the snapshots in Figure 3 demonstrate, the behavior of the overall inter- connected system has changed fundamentally. Although the distributed controller is still stabilizing, the dynamic b ehavior has changed drastically and shows wave-like oscillations d ue to 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 numerical exercise that the stabilizing effect of the proposed distri buted control design is independent of the local LQR weighting pa- rameter selection, thus these parameters are freely availa ble for tuning and achieving different global performance objecti ves. VI. C ONCLUDING EMARKS We have introduced a stabilizing distributed control desig method for large-scale interconnections of identical, dyn am- ically decoupled and independently actuated linear system s. The procedure requires the solution of a single local LQR

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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 on absolute 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 on relative position references. problem, whose size depends only on the maximum vertex degree of the interconnection graph. Special properties of the local LQR problem were derived, which enable the con- struction of stabilizing distributed controllers indepen dently of the choice of weighting matrices. The relationship betwe en stability of the overall large-scale system, the robustnes s of local controllers and the spectrum of a sparsity pattern mat rix has been highlighted. The proposed distributed controller will be stabilizing even if subsystems and communication links are added to or removed from the overall system, as long as this does not change the maximum vertex degree or violate the conditions given in Table I. This feature provides a very powerful modularity to our approach. VII. A CKNOWLEDGEMENTS We thank the anonymous reviewers for the suggestions on how to improve the paper and for pointing out an error in the original version of the manuscript. Tam´as Keviczky’s rese arch was supported in part by the Boeing Corporation. EFERENCES [1] S. Wang and E. J. Davison, “On the stabilization of decent ralized control systems, IEEE Trans. Automatic Control , vol. 18, no. 5, pp. 473–478, 1973. [2] R. D’Andrea and G. E. Dullerud, “Distributed control des ign for spa- tially interconnected systems, IEEE Trans. Automatic Control , vol. 48, no. 9, pp. 1478–1495, Sept. 2003. [3] E. Camponogara, D. Jia, B. Krogh, and S. Talukdar, “Distr ibuted model predictive control, IEEE Control Systems Magazine , vol. 22, no. 1, pp. 44–52, Feb. 2002. [4] B. M. Mirkin and P.-O. Gutman, “Output-feedback coordin ated decen- tralized adaptive tracking: the case of MIMO subsystems wit h delayed interconnections, Int. J. Adapt. Control and Signal Processing , vol. 19, no. 8, pp. 639–660, 2005. [5] J. A. Fax and R. M. Murray, “Information ﬂow and cooperati ve control of vehicle formations, IEEE Trans. Automatic Control , vol. 49, no. 9, pp. 1465–1476, 2004. [6] B. Bamieh, F. Paganini, and M. A. Dahleh, “Distributed co ntrol of spatially invariant systems, IEEE Trans. Automatic Control , vol. 47, no. 7, pp. 1091–1107, July 2002. [7] B. Bamieh and P. G. Voulgaris, “A convex characterizatio n of distributed control problems in spatially invariant systems with commu nication constraints, Systems & Control Letters , vol. 54, pp. 575–583, 2005. [8] M. R. Jovanovi´c, “On the optimality of localized distri buted controllers, in Proc. American Contr. Conf. , June 2005, pp. 4583–4588. [9] G. E. Dullerud and R. D’Andrea, “Distributed control of h eterogeneous systems, IEEE Trans. Automatic Control , vol. 49, no. 12, pp. 2113 2128, Dec. 2004. [10] C. Langbort, R. S. Chandra, and R. D’Andrea, “Distribut ed control design for systems interconnected over an arbitrary graph, IEEE Trans. Automatic Control , vol. 49, no. 9, pp. 1502–1519, Sept. 2004. [11] V. Gupta, B. Hassibi, and R. M. Murray, “A sub-optimal al gorithm to synthesize control laws for a network of dynamic agents, Int. J. Control vol. 78, no. 16, pp. 1302–1313, Nov. 2005. [12] M. Rotkowitz and S. Lall, “A characterization of convex problems in decentralized control, IEEE Trans. Automatic Control , vol. 51, no. 2, pp. 274–286, Feb. 2006. [13] T. Keviczky, F. Borrelli, and G. J. Balas, “A study on dec entralized receding horizon control for decoupled systems,” in Proc. American Contr. Conf. , 2004, pp. 4921–4926. [14] ——, “Stability analysis of decentralized RHC for decou pled systems, in 44th IEEE Conf. on Decision and Control, and European Contro Conf. , Seville, Spain, Dec. 2005. [15] ——, “Decentralized receding horizon control for large scale dynami- cally decoupled systems, Automatica , vol. 42, no. 12, pp. 2105–2115, Dec. 2006. [16] W. B. Dunbar and R. M. Murray, “Receding horizon control of multi- vehicle formations: A distributed implementation,” in Proc. 43rd IEEE Conf. on Decision and Control , 2004, pp. 1995–2002. [17] V. Gupta, B. Hassibi, and R. M. Murray, “On the synthesis of control laws for a network of autonomous agents,” in Proc. American Contr. Conf. , 2004, pp. 4927–4932. [18] A. Richards and J. How, “A decentralized algorithm for r obust con- strained model predictive control,” in Proc. American Contr. Conf. , 2004, pp. 4261–4266. [19] F. Borrelli, T. Keviczky, G. J. Balas, G. Stewart, K. Fre gene, and D. Godbole, “Hybrid decentralized control of large scale sy stems, in Hybrid Systems: Computation and Control , ser. Lecture Notes in Computer Science, vol. 3414. Springer Verlag, Mar. 2005, pp . 168–183. [20] D. Stipanovic, G. Inalhan, R. Teo, and C. J. Tomlin, “Dec entralized overlapping control of a formation of unmanned aerial vehic les, Auto- matica , vol. 40, no. 8, pp. 1285–1296, 2004. [21] T. Keviczky, F. Borrelli, G. J. Balas, K. Fregene, and D. Godbole, “Decentralized receding horizon control and coordination of autonomous vehicle formations, IEEE Trans. Control Systems Technology , Jan. 2007, to appear. [22] B. D. O. Andreson and J. B. Moore, Optimal Control: Linear Quadratic Methods . Englewood Cliffs, N.J.: Prentice Hall, 1990. [23] M. G. Safonov and M. Athans, “Gain and phase margin for mu ltiloop LQG regulators, IEEE Trans. Automatic Control , vol. AC-22, no. 2, pp. 173–179, Apr. 1977. [24] C. Scherer and S. Weiland, Linear Matrix Inequalities in Control , 2000, version 3.0. [25] R. Merris, “Laplacian matrices of graphs: A survey, Linear Algebra and Its Applications , vol. 197, pp. 143–176, 1994.

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[26] B. Mohar, “The Laplacian spectrum of graphs, Graph Theory, Combi- natorics, and Applications , vol. 2, pp. 871–898, 1991. [27] R. Agaev and P. Chebotarev, “On the spectra of nonsymmet ric Laplacian matrices, Linear Algebra and Its Applications , vol. 399, no. 5, pp. 157 168, 2005. [28] B. Bollob´as, Modern Graph Theory . Springer, 2002. [29] M. Ajtai, J. Koml´os, and E. Szemer´edi, “Sorting in c lo g n parallel steps, Combinatorica , vol. 1, no. 9, 1983. [30] R. M. Tanner, “Explicit concentrators from generalize d n-gons, SIAM J. Alg. Discr. Meth , vol. 5, pp. 287–293, 1984. [31] M. Tompa, “Time space tradeoffs for computing function s, using con- nectivity properties of their circuits, J. Comp. and Sys. Sci , vol. 20, pp. 118–132, 1980. [32] L. Valiant, “Graph theoretic properties in computatio nal complexity, J. Comp. and Sys. Sci , vol. 13, pp. 278–285, 1976. [33] Online, “http://www.cds.caltech.edu/˜tamas/simul ations.html,” 2006.

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Distributed LQR Design for Identical Dynamically Decoupled Systems Francesco Borrelli , Tam´as Keviczky Abstract — We consider a set of identical decoupled dynamical systems and a control problem where the performance index couples the behavior of the systems. The coupling is describ ed through a communication graph where each system is a node and the control action at each node is only function of its sta te and the states of its neighbors. A distributed control desig method is presented which requires the solution of a single L QR problem. The size of the LQR problem is equal to the maximum vertex degree of the communication graph plus one. The desig procedure proposed in this paper illustrates how stability of the large-scale system is related to the robustness of local con trollers and the spectrum of a matrix representing the desired sparsi ty pattern of the distributed controller design problem. I. I NTRODUCTION ISTRIBUTED control techniques today can be found in a broad spectrum of applications ranging from robotics and formation ﬂight to civil engineering. Contributions and in terest in this ﬁeld date back to the early results of [1]. Approaches to distributed control design differ from each other in the assumptions they make on: ( ) the kind of interaction between different systems or different components of the same syste (dynamics, constraints, objective), ( ii ) the model of the system (linear, nonlinear, constrained, continuous-time, discr ete-time), iii ) the model of information exchange between the systems, iv ) the control design technique used. In this paper we focus on identical decoupled linear time- invariant systems . Our interest in distributed control for such systems arises from the abundance of networks of indepen- dently actuated systems and the necessity of avoiding centr al- ized design when this becomes computationally prohibitive Networks of vehicles in formation, production units in a pow er plant, cameras at an airport, an array of mechanical actuato rs for deforming a surface are just a few examples. In a descriptive way, the problem of distributed control for decoupled systems can be formulated as follows. A dynamical system is composed of (or can be decomposed into) distinct dynamical subsystems that can be independently actuated. T he subsystems are dynamically decoupled but have common ob- jectives, which make them interact with each other. Typical ly the interaction is local, i.e., the goal of a subsystem is a function of only a subset of other subsystems’ states. The interaction will be represented by an interaction 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 Netherl ands, t.keviczky@tudelft.nl the nodes represent the subsystems and an edge between two nodes denotes a coupling term in the controller associated w ith the nodes. Also, typically it is assumed that the exchange of information has a special structure, i.e., it is assumed that each subsystem can sense and/or exchange information with only a subset of other subsystems. We will assume that the interaction graph and the information exchange graph coincide. A distributed control scheme consists of distinc controllers, one for each subsystem, where the inputs to eac subsystem 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 interest in systems composed of a large number of interacting and cooperating interconnected units [2]–[20]. Recent advanc es in stability analysis of such systems yielded greater insight into the relationship between the spectrum of the interconnecti on graph and global stability of the overall system. Research e f- forts in this area are epitomized by the pioneering work of [5 ], which provided stability analysis tools for an interconnec tion of identical linear dynamical systems each using the same control law that operates on the average information obtain ed from neighbors. Most recent results, however, pertain to the study of dis- tributed parameter systems where the underlying dynamics a re spatially invariant, and where the controls and measuremen ts are spatially distributed. The fundamental work of [4], [6] in this ﬁeld discusses distributed LQR design purely for inﬁni te dimensional spatially invariant systems, where the proble diagonalizes exactly into a parameterized family of ﬁnite d i- mensional LQR problems. It is established that the correspo nd- ing ARE solutions are translation-invariant operators, an d the optimal controller is a spatially invariant system. The aut hors show that quadratically optimal controllers for spatially in- variant systems are themselves spatially invariant. Anoth er recent result for spatially invariant systems and unbounde domains in [7] considers the problem of distributed control ler design, when communication among the sites is limited. In particular, the controller is assumed to be constrained so that information is propagated with a delay that depends on the distance between subsystems. The authors show that the problem of optimal design can be cast as a convex problem provided that the propagation speeds in the controller are a least as fast as those in the plant. Special spatially invari ant structures composed of an inﬁnite string of linear systems have also received much attention. The paper [8] analyzes invers optimality of localized distributed controllers in such sy stems. Another signiﬁcant group of recently explored strategies involves relaxations to the LMI versions of these spatially

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invariant problems. In [9], authors consider heterogenous subsystems and a relaxation, which is used to derive sufﬁcie nt conditions for the existence of controllers which stabiliz e the system and provide a guaranteed level of performance. The work in [10] is concerned with ﬁnding distributed controlle rs for a set of arbitrarily connected, ﬁnite and possibly het- erogeneous LTI systems. The authors consider nonoriented edges in the graph interconnection and formulate convex conditions for the existence of output-feedback controlle rs, which achieve a certain performance. The resulting linear matrix inequalities grow in size with the number of systems in the interconnection. The complexity associated with the computation of dis- tributed optimal controllers is well exempliﬁed by the stud y in [11]. There, a ﬁnite-time LQR synthesis problem in discrete time is considered, where the matrix describing the control law is constrained to lie in a particular vector space. This vector space represents a pre-speciﬁed distributed contro structure between a network of autonomous agents. The computationally intensive optimal solution for the distri buted control problem is presented along with a computationally more tractable sub-optimal one. One of the most notable recent result in characterizing the complexity of optimal distributed controller design subje ct to constraints on the controller structure was presented in [1 2]. The authors establish that in general, efﬁcient solution of such problems require a special property of the sparsity pattern or interconnection structure to hold. Speciﬁcally, it is sh own that quadratic invariance of the assumed controller struct ure implies that the distributed minimum-norm problem may be solved efﬁciently via convex programming. This manuscript proposes a simple distributed controller de- sign approach and focuses on a class of systems, for which ex- isting methods outlined above are either not efﬁcient or wou ld not even be directly applicable. Our method applies to large scale systems composed of ﬁnite number of identical subsys- tems where the interconnection structure or sparsity pattern is not required to have any special invariance properties . The philosophy of our approach builds on the recent works [13] [15], where at each node, the model of its neighbors are used to predict their behavior. We show that in absence of state and input constraints, and for identical linear syste dynamics, such an approach leads to an extremely powerful result: the synthesis of stabilizing distributed control l aws can be obtained by using a simple local LQR design, whose size is limited by the maximum vertex degree of the interconnection graph plus one. Furthermore, the design procedure proposed in this paper illustrates how stability of the overall large -scale system is related to the robustness of local controllers and the spectrum of a matrix representing the desired sparsity pattern. In addition, the constructed distributed control ler is stabilizing independent of the tuning parameters in the loc al LQR cost function. This leads to a method for designing distributed controllers for a ﬁnite number of dynamically decoupled systems, where the local tuning parameters can be chosen to obtain a desirable global performance. Such resul can be immediately used to improve current stability analys is and controller synthesis in the ﬁeld of distributed recedin horizon control for dynamically decoupled systems [3], [13 ], [16]–[18]. We emphasize that the aforementioned advantage are the results of two main simplifying assumptions compare to a general structured optimal design: identical and decou pled subsystem dynamics and suboptimality of the global control ler. The paper is organized as follows. In Section II, we study the solution properties of the LQR for a set of identical, dec ou- pled linear systems. Such properties will be used in the pape to construct a stabilizing distributed controller for arbi trary interconnection structures. Section III summarizes impor tant properties of graph Laplacian and adjacency matrices, whic will be useful in our proofs. Section IV presents the stabili zing distributed controller design procedure using the local LQ solution properties. The proposed distributed control des ign method and the effect of the free tuning parameters in the local LQR design is illustrated by a simulation example in Section V. Some concluding remarks are made in Section VI. OTATION AND RELIMINARIES We denote by the ﬁeld of real numbers, the ﬁeld of complex numbers and the set of real matrices. Re Re The transpose of a vector and a matrix will be denoted by and , respectively. A matrix is symmetric if Notation 1: Let , then j, k denotes a matrix of dimension + 1) + 1) obtained by extracting rows to and columns to from the matrix with Notation 2: denotes the identity matrix of dimension Notation 3: Let denote the -th eigenvalue of = 1 , . . ., n . The spectrum of will be denoted by ) = , . . . , Deﬁnition 1: Let A, B be symmetric. 1) is positive deﬁnite if Ax > for all nonzero , and is positive semideﬁnite if Ax for all nonzero . We denote this by A > and respectively. 2) is negative (semi)deﬁnite if is positive (semi)deﬁnite. 3) A < B and mean B < and respectively. Deﬁnition 2: A matrix is called Hurwitz (or stable) if all its eigenvalues have negative real part, i.e. , i = 1 , . . ., n Notation 4: Let . Then denotes the Kronecker product of and 11 mn mp nq (1) Proposition 1: Consider two matrices αI and . Then ) = , i = 1 , . . . , n Proof: Take any eigenvalue and the corresponding eigenvector . Then Av Bv

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αv = ( Proposition 2: Given A, C and consider two matrices and , where A, nm nm . Then ) = =1 where is the -th eigenvalue of Proof: Let be an eigenvector of corresponding to , and be an eigenvector of = ( with as the associated eigenvalue. Consider the vector nm . Then )( ) = Au Bv Cu Au Cu Au Cu . Since , we get )( ) = )( II. LQR P ROPERTIES FOR YNAMICALLY ECOUPLED YSTEMS Consider a set of identical, decoupled linear time- invariant dynamical systems, the -th system being described by the continuous-time state equation: Ax Bu (0) = (2) where are states and inputs of the -th system at time , respectively. Let nN and mN be the vectors which collect the states and inputs of the systems at time u, (0) = 10 , . . ., x (3) with A, B. (4) We consider an LQR control problem for the set of systems where the cost function couples the dynamic behavio of individual systems: ( u, ) = =1 ii ) + ii )) + =1 )) ij )) d (5) with ii ii R > , Q ii ii i, (6a) ij ij ji j. (6b) The cost function (5) contains terms which weigh the -th system states and inputs, as well as the difference between the -th and the -th system states and can be rewritten using the following compact notation: ( ) = ) + dτ, (7) where the matrices and have a special structure deﬁned next. and can be decomposed into blocks of dimen- sion and respectively: 11 12 . . . . . . . . . . . . R (8) with ii ii =1 , k ik , i = 1 , . . ., N ij ij , i, j = 1 , . . ., N , i j. R. (9) Remark 1: The cost function structure (5) can be used to describe several practical applications including form ation ﬂight, paper machine control and monitoring networks of cameras [19], [21]. Let and be the optimal controller and the value function corresponding to the following LQR problem: min ( u, subj. to (0) = (10) Throughout the paper we will assume that a stabilizing solu- tion to the LQR problem (10) with ﬁnite performance index exists and is unique (see [22], p. 52 and references therein) Assumption 1: System A, is stabilizable and system A, is observable, where is any matrix such that We will also assume local stabilizability and observabilit y: Assumption 2: System A, B is stabilizable and systems A, C are observable, where is any matrix such that It is well known that P, (11) where is the symmetric positive deﬁnite solution to the following ARE: = 0 (12) We decompose and into blocks of dimension and , respectively. Denote by ij and ij the i, j block of the matrix and , respectively. In the following theorems we show that ij and ij satisfy certain properties which will be critical for the design of stabilizing distrib uted controllers in Section IV. These properties stem from the special structure of the LQR problem (10). Next, the matrix is deﬁned as BR Theorem 1: Let and be the optimal controller and the value function solution to the LQR problem (10). Let ij [( 1) im, 1) jn and ij [( 1)

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in, 1) jn with = 1 , . . ., N = 1 , . . ., N Then, =1 =1 ij =1 ij =1 ij =1 ij ii = 0 (13) Proof: Consider the ARE (12) associated with the LQR problem (10). Without loss of generality, the diagonal bloc ks ii of can be written as ii ii =1 ,j ij (14) for a certain ii . In general, ii would be a function of =1 ,j ij . We will show next that for the LQR prob- lem (10), ii and that it is not a function of =1 ,j ij The equations for the diagonal blocks ii of in the ARE equation (12) are ii ii =1 ik ik ii = 0 (15) for = 1 , . . ., N . Note also that ij ji . Substituting (14) in the above equation leads to ii ii =1 ,k ik =1 ,k ik {z =1 ,k ik ki {z ii ii =1 ,k ik =1 ,l il {z =1 ,k ik ii ii =1 ,k ik {z ii = 0 (16) The equations for the off-diagonal blocks ij , i of in the ARE equation (12) are ij ij =1 ik kj ij = 0 (17) for i, j = 1 , . . ., N , i . Substituting (14) in the above equation leads to ij ij ii ij =1 ik ij ij jj ij =1 jk =1 i,k ik kj ij = 0 (18) Summing up equation (18) for all corresponds to a block-wise row sum of off-diagonal terms in the -th block row of the ARE equation (12). This summation leads to =1 ik =1 ik {z ii =1 ik {z (19a) =1 =1 ik il {z (19b) =1 ik =1 kl =1 =1 i,l il lk {z (19c) =1 ik kk =1 ik = 0 (19d) Notice that (19c) =1 =1 ik kl =1 =1 i,l ik kl =1 ,k ik ki (20) where we used symmetry, switching of sum operators and renaming the indices. Adding equation (19) to equation (16) and using properties of deﬁned in (9), equally numbered terms will cancel each other out leading to ii ii ii ii =1 ik ii kk ii = 0 (21) Summing up equation (21) over all = 1 , . . ., N we obtain =1 ii ii ii ii ii = 0 (22) which proves the theorem. Next we particularize Theorem 1 to the case of identical weights and prove an additional property of LQR for decoupled systems. Theorem 2: Assume the weighting matrices (9) of the LQR problem (10) are chosen as ii = 1 , . . ., N ij i, j = 1 , . . ., N , i j. (23) Let be the value function of the LQR problem (10) with weights (23), and the blocks of the matrix be denoted by ij [( 1) in, 1) jn with i, j = 1 , . . ., N

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Then, (I) =1 ij for all = 1 , . . ., N , where is the symmetric positive deﬁnite solution of the ARE associated with a single node local problem: PA PBR = 0 (24) (II) =1 ij for all = 1 , . . ., N , where (III) ij lm j, is a symmetric negative semideﬁnite matrix. Proof: The assumption in (23) requires that the weight used for absolute states and the weight used for neighboring state differences are equal for all nodes and for all neighbors of a node, respectively. Such an assumption and the fact that and are block-diagonal with identical blocks, imply that the ARE in (12) is a set of identical equations where the matrices ij are all identical and symmetric for all . We denote by , the generic block ij for . The matrices ii =1 ij deﬁned in (14) are all identical and therefore equation (13) in Theorem 1 becomes: ii ii ii ii = 0 (25) which proves property (I) with ii . Property (II) follows from property (I) and from equation (11) which implies that ij ij . Next we prove property (III). The ARE equations (17) for the block ij with become 2) = 0 (26) which can be rewritten as follows in virtue of property (I): XP XP )+( = 0 (27) where is the symmetric positive deﬁnite solution of the ARE (24) associated with a single node local problem. Rewrite equation (27) as XP ) + ( )( XP ) + = 0 (28) Since X > and , equation (28) can be seen as an ARE associated with an LQR problem for the stable system XP, B with weights and . Let the matrix be its positive semideﬁnite solution. Then, the following matrix 1) 1) . . . . . . P 1) (29) is a symmetric positive deﬁnite matrix since Px =1 (30) and it is the unique symmetric positive deﬁnite matrix solut ion to the ARE (12). This proves the theorem. Under the hypothesis of Theorem 2, because of symmetry and equal weights on the neighboring state differences and equal weights on absolute states, the LQR optimal controller will have the following structure: . . . . . . K (31) with and functions of and Remark 2: We conjecture that Theorem 2 is true under much milder assumptions on the tuning parameters (even with different weights on neighboring state differences) as lon g as the cost function is deﬁned as in (9). The authors are current ly studying this conjecture. The following corollaries of Theorem 2 follow from the stability and the robustness of the LQR controller for system XP in (28). Corollary 1: XP is a Hurwitz matrix. From the gain margin properties [23] we have: Corollary 2: XP αN is a Hurwitz matrix for all α > , with Remark 3: XP is a Hurwitz matrix, thus the system in Corollary 2 is stable for = 0 as well. ( XP BK with being the LQR gain for system A, B with weights , R .) The following condition deﬁnes a class of systems and LQR weighting matrices which will be used in later sections to extend the set of stabilizing distributed controller struc tures. Condition 1: XP αN is a Hurwitz matrix for all [0 , with Essentially, Condition 1 characterizes systems for which t he LQR gain stability margin described in Corollary 2 is extend ed to any positive . The fact that XP is already stable, does not necessarily guarantee this property. Checking the validity of Condition 1 for a given tuning of and may be performed as a stability test for a simple afﬁne parameter-dependent model = ( αA {z x, (32) where XP and This test can be posed as an LMI problem (Proposition 5.9 in [24]) searching for quadratic parameter-dependent Lyapun ov functions. In the following section we introduce some basic concepts of graph theory before presenting the distributed control d esign problem. III. L APLACIAN PECTRUM OF RAPHS This section is a concise review of the relationship between the eigenvalues of a Laplacian matrix and the topology of the associated graph. We refer the reader to [25], [26] for a comprehensive treatment of the topic. We list a collection of properties associated with undirected graph Laplacians and adjacency matrices, which will be used in subsequent sectio ns of the paper.

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A graph is deﬁned as = ( (33) where is the set of nodes (or vertices) , . . ., N and A⊆VV the set of edges i, j with ∈V , j ∈V . The degree of a graph vertex is the number of edges which start from . Let max denote the maximum vertex degree of the graph We denote by the (0 1) adjacency matrix of the graph . Let i,j be its i, j element, then i,i = 0 , . . ., N i,j = 0 if i, j ∈A and i,j = 1 if i, j ∈A i, j = 1 , . . ., N, i . We will focus on undirected graphs, for which the adjacency matrix is symmetric. Let )) = )) , . . . , )) be the spec- trum of the adjacency matrix associated with an undirected graph arranged in nondecreasing semi-order. Property 1: )) max This property together with Proposition 1 implies Property 2: ∈S max We deﬁne the Laplacian matrix of a graph in the following way ) = (34) where is the diagonal matrix of vertex degrees (also called the valence matrix). Eigenvalues of Laplacian matrices have been widely studied by graph theorists. Their properties are strongly related to the structural properti es of their associated graphs. Every Laplacian matrix is a singul ar matrix. By Gerˇsgorin’s theorem [27], the real part of each nonzero eigenvalue of is strictly positive. For undirected graphs, is a symmetric, positive semideﬁnite matrix, which has only real eigenvalues. Let )) = )) , . . . , )) be the spectrum of the Laplacian matrix associated with an undirected graph arranged in nondecreasing semi-order. Then, Property 3: 1) )) = 0 with corresponding eigenvector of all ones, and )) = 0 iff is connected. In fact, the multiplicity of as an eigenvalue of is equal to the number of connected components of 2) The modulus of )) , i = 1 , . . ., N is less then The second smallest Laplacian eigenvalue )) of graphs is probably the most important information containe d in the spectrum of a graph. This eigenvalue, called the algebra ic connectivity of the graph, is related to several important g raph invariants, and it has been extensively investigated. Let be the Laplacian of a graph with vertices and with maximal vertex degree max . Then properties of )) include Property 4: 1) )) min , v ∈V} 2) )) 3) )) )(1 cos 4) )) 2(cos cos2 2 cos (1 cos max where is the vertex connectivity of the graph (the size of a smallest set of vertices whose removal renders disconnected) and is the edge connectivity of the graph (the size of a smallest set of edges whose removal renders disconnected) [28]. Further relationships between the graph topology and Lapla cian eigenvalue locations are discussed in [26] for undirec ted graphs. Spectral characterization of Laplacian matrices f or directed graphs can be found in [27]. IV. D ISTRIBUTED ONTROL ESIGN We consider a set of linear, identical and decoupled dynamical systems, described by the continuous-time time- invariant state equation (2), rewritten below Ax Bu (0) = where are states and inputs of the -th system at time , respectively. Let Nn and Nm be the vectors which collect the states and inputs of the systems at time , then u, (0) = 10 , . . . , x (35) with A, B. Remark 4: Systems (35) and (3) differ only in the number of subsystems. We will use system (3) with subsystems when referring to local problems, and system (35) with subsystems when referring to the global problem. According ly, tilded matrices will refer to local problems and hatted matr ices will refer to the global problem. We use a graph to represent the coupling in the control objective and the communication in the following way. We associate the -th system with the -th node of a graph . If an edge i, j connecting the -th and -th node is present, then 1) the -th system has full information about the state of the -th system and, 2) the -th system control law minimizes a weighted distance between the -th and the -th system states. The class of n,m matrices is deﬁned as follows: Deﬁnition 3: n,m ) = nN mN ij if i, j ∈A , M ij [( 1) in, 1) jm , i, j , . . ., N The distributed optimal control problem is deﬁned as follows: min ( u, ) = ) + dτ, (36a) subj. to u, (36b) ) = (36c) ∈K m,n (36d) ∈K n,n R, (36e) (0) = (36f) with and . We also refer to problem (36) without (36d) as a centralized optimal control

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problem . In general, computing the solution to problem (36) is an NP-hard problem. Next, we propose a suboptimal control design leading to a controller with the following properties: 1) ∈K m,n (37a) 2) is Hurwitz. (37b) 3) Simple tuning of absolute and relative state errors, and control effort within K. (37c) Such controller will be referred to as distributed suboptimal controller . The following theorem will be used to propose a distributed suboptimal control design procedure. Theorem 3: Consider the LQR problem (10) with max ) + 1 and weights chosen as in (23) and its solu- tion (29), (31). Let be a symmetric matrix with the following property: ∈S \{ (38) and construct the feedback controller: (39) Then, the closed loop system cl + ( (40) is asymptotically stable. Proof: Consider the eigenvalues of the closed-loop system cl cl ) = XP ) + By Proposition 2: XP ) + =1 XP (41) We will prove that XP is a Hurwitz matrix = 1 , . . ., N , and thus prove the theorem. If ) = 0 then XP is Hurwitz based on Remark 3. If = 0 , then from Corollary 2 and from condition (38), we conclude that XP is Hurwitz. Theorem 3 has several main consequences: 1) If ∈K , then in (39) is an asymptotically stable distributed controller. 2) We can use one local LQR controller to compose dis- tributed stabilizing controllers for a collection of ident ical dynamically decoupled subsystems. 3) The ﬁrst two consequences imply that we can not only ﬁnd a stabilizing distributed controller with a desired sparsity pattern (which is in general a formidable task by itself), but it is enough to solve a low-dimensional problem (characterized by max )) compared to the full problem size (36). This attractive feature of our approach relies on the speciﬁc problem structure deﬁned in Section II and IV. 4) The eigenvalues of the closed-loop large-scale system cl can be computed through smaller eigenvalue computations as =1 XP 5) The result is independent of the local LQR tuning. Thus and in (23) can be used in order to inﬂuence the compromise between minimization of absolute and relative terms, and the control effort in the global perfor- mance. For the special class of systems deﬁned by Condition 1, the hypothesis of Theorem 3 can be relaxed as follows: Theorem 4: Consider the LQR problem (10) with weights chosen as in (23) and its solution (29), (31). Assume that Condition 1 holds. Let be a symmetric matrix with the following property: ∈S (42) Then, the closed loop system (40) is asymptotically stable when is constructed as in (39). Proof: Notice that if Condition 1 holds, then XP is Hurwitz for all (from Corollary 1 and Corollary 2). By Proposition 2 XP ) + =1 XP which together with condition (42) proves the theorem. In the next sections we show how to choose in Theo- rem 3 and Theorem 4 in order to construct distributed subop- timal controllers. The matrix will ( ) reﬂect the structure of the graph , ( ii ) satisfy (38) or (42) and ( iii ) be computed by using the graph adjacency matrix or the Laplacian matrix. In order to make the exposition simpler, we will start with a simple special graph in Section IV-A and generalize the resu lts in Section IV-B to any graph structure. A. Finite Strings Let represent a string interconnection of systems with max ) = 2 and the following Laplacian: ) = 1 0 1 2 1 2 1 1 (43) Construct a distributed controller as follows. Solve the LQ problem (10), (23) with = 2 + 1 = 3 and desired . Consider the decomposition (29) and (31) of local controller and cost function, respectively. Take and construct the global controller gain matrix as 0 0 0 0 aK bK bK aK bK bK aK bK bK aK (44)

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The closed-loop system under the distributed control law can be written as cl with cl + ( K. (45) Rewrite as: aI From Theorem 2, is equal to: ) + and = ( , N = 3 (46) The following corollary of Theorem 3 deﬁnes the range of and which leads to a stable distributed controller. Corollary 3: Consider controller (44). If + 2 b < then the closed loop system (40) is asymptotically stable. If Condition 1 holds then the closed loop system (40) is asymptotically stable for all + 2 Proof: From Property 1, max )) max ) = 2 Since , by using Proposition 1 applied to we derive: min ) = b max )) b, N = 3 (47) Condition (38) of Theorem 3 holds if + 2 b < , which proves the ﬁrst part of the Theorem. Assume that Condition 1 holds and consider equation (47). Then, condition (42) of Theorem 4 holds if + 2 Example 1: With = 0 the distributed controller (44) becomes bK bK bK bK bK bK (48) The closed loop system (40) is asymptotically stable if b < . If Condition 1 holds, then the closed loop system (40) is asymptotically stable if Example 2: Assume that Condition 1 holds. Since the graph Laplacian is a symmetric positive semideﬁnite matrix, aL satisﬁes condition (42) of Theorem 4. Thus any distributed controller ) + aL having the following structure (2 (49) stabilizes system (40) for all B. Arbitrary Graph Structures We consider a generic graph for nodes with an associated Laplacian and maximum vertex degree max Let 0 = < . . . be the eigenvalues of the the Laplacian . In the following Corollaries 4, 5 and 6 we present three ways of choosing in (39) which lead to distributed suboptimal controllers. Corollary 4: Compute in (39) as: aL (50) If a > (51) then the closed loop system (40) is asymptotically stable wh en is constructed as in (39). In addition, if Condition 1 holds, then the closed loop system (40) is asymptotically stable fo all Proof: Apply condition (38) in Theorem 3 to the case aL . Notice that for all = 3 , . . ., N and = 0 by Property 3 of the Laplacian matrix. Therefore condition (38) becomes aL )) . This implies a and thus a > . The ﬁrst part of the Theorem is proven. If then Therefore the application of Theorem 4 to the matrix aL proves the second part of the theorem. Remark 5: By using the third formula in Property 4, con- dition (51) can be linked to the edge connectivity as follows a > cos (52) Remark 6: Corollary 4 links the stability of the distributed controller to the size of the second smallest eigenvalues of the graph Laplacian. It is well known that graphs with large (with respect to the maximal degree) have some properties which make them very useful in several applicati ons such as computer science [29]–[32]. Interestingly enough, this property is shown here to be crucial also for the design of distributed controllers. We refer the reader to [26] for a mo re detailed discussion on the importance of the second smalles eigenvalue of a Laplacian. Corollary 5: Compute in (39) as: aI , b (53) If bd max , then the closed loop system (40) is asymptotically stable when is constructed as in (39). In addition, if Condition 1 holds, then the closed loop system ( 40) is asymptotically stable if bd max Proof: Notice that min ) = b max )) bd max . The proof is a direct consequence of Theorems 3 and 4 and Property 1 of the adjacency matrix. Consider a weighted adjacency matrix deﬁned as follows. Denote by i,j its i, j element, then i,j = 0 , if and i, j ∈A and i,j ij if i, j ∈A

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i, j = 1 , . . ., N, i . Assume ij ji . Deﬁne max as max = max ij Corollary 6: Compute in (39) as: aI (54) If a > w max , then the closed loop system (40) is asymptotically stable when is constructed as in (39). In addition, if Condition 1 holds, then the closed loop system ( 40) is asymptotically stable if max Proof: max and by Perron-Frobenius The- orem max max . Notice that min ) = max )) max , then the proof is a direct con- sequence of Theorems 3 and 4. The results of Corollaries 4-6 are summarized in Table I. Choice of Stability Condition Stability Condition If Condition 1 Holds aL a > aI bd max bd max aI a > w max max TABLE I UMMARY OF STABILITY CONDITIONS IN OROLLARIES 4-6 FOR THE CLOSED LOOP SYSTEM (39)-(40). Corollaries 4-6 present three choices of distributed contr ol design with increasing degrees of freedom. In fact, and ij are additional parameters which, together with and , can be used to tune the closed-loop system behavior. We recall here that from Theorem 3, the eigenvalues of the closed-loop large-scale system are related to the eigenval ues of through the simple relation (41). Thus as long as the stability conditions deﬁned in Table I are satisﬁed, the ove rall system architecture can be modiﬁed arbitrarily by adding or removing subsystems and interconnection links. This leads to a very powerful modular approach for designing distributed control systems. The difference between the performance of the distributed control law and the performance of a central ized optimal solution can be also computed in a simple way. This is discussed in the next section. We remark that detailed perfo r- mance analysis of the proposed distributed control design, or the design of distributed control laws with performance gua r- antees are not discussed within this manuscript and represe nt future research topics. C. Measure of Suboptimality Consider system (35) and let and be an LQR controller and the corresponding ARE solution associated w ith the weights Q > and R > . Therefore, ) = minimizes the following cost function for any ( ) = ) + dτ, (55) i.e., ( min ( ) = ) = (56) The next results aim at comparing the optimal centralized controller and the distributed suboptimal controller presented in the previous section. The comparison will be made assuming (55) reﬂects the desired performance index and computing the loss of performance introduced by the distributed control law (39) with chosen as in Corollaries 4- 6. Proposition 3: Consider the distributed controller in (39) and the closed loop system (40). If (40) is stable, then ) = (57) where is the positive deﬁnite solution of the following Lyapunov equation: + ( + ( + ( ) = 0 (58) Proof: By assumption, system (40) is stable and thus the positive deﬁnite solution to the Lyapunov equation (58) can be written as: d (59) Consider . From direct computation: ) = ) d (60) By substituting with we obtain (57). Proposition 3 shows that the cost associated with the dis- tributed controller according to the performance index (55) can be computed by solving a Lyapunov equation for the closed loop system with weight equal to Since is optimal, for all and thus is a positive semideﬁnite matrix which is equal to zero if . Any norm of is a measure of the suboptimality of the distributed control ler The “best” linear state-feedback distributed controller m,n could be computed by solving: ( ) = min (61a) subj. to u, (61b) ) = (61c) ∈K m,n (61d) (0) = (61e) As discussed in the previous sections, computing the soluti on to problem (61), is a difﬁcult problem in general without further assumptions. Its efﬁcient solution is the topic of c urrent research.

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V. E XAMPLE Consider a 10 10 mesh interconnection of = 100 identical, dynamically decoupled and independently actua ted linear systems moving in a plane with double integrator dynamics in both spatial dimensions: x,i y,i = 1 , . . . , 100 (62) The interconnection structure is depicted in Figure 1. The control objective is to move each subsystem to a desired abso lute position corresponding to its location in the pre-spec iﬁed rectangular grid, which has equal separation distances de ned between each orthogonal neighbor. Fig. 1. The 10 10 ﬁnite mesh grid interconnection structure of the simulation example. The maximum vertex degree of such an interconnection graph is 4, as the nodes located at the “corners” of the rectangular grid have 2 neighbors, those along the “edges” o the rectangle have 3 neighbors and the ones in the “middle have 4 each. A stabilizing distributed controller is designed for the me sh interconnection of systems, by solving the following, simp le LQR problem involving only = 5 nodes ( max + 1). min ( u, subj. to (0) = (63) where A, B, 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 , B 0 0 1 0 0 0 0 1 (64) The cost function deﬁned in (5) uses the following weights for absolute and relative state information ii ij j. (65) The weight on the control effort was kept constant in the following examples ( = 1 for each control input in both spatial dimensions). The subsystem model and chosen weight satisfy Condition 1 (we tested the stability of (32) by solvi ng an LMI problem searching for quadratic parameter-dependen Lyapunov functions), thus we can use less conservative stab il- ity ranges in Table I. Alternatively, one can choose the more conservative stability ranges in Table I which can be always satisﬁed. The solution of the above local LQR problem will yield a controller matrix of the following form in this speciﬁc example: (66) The distributed controller for the grid of = 100 inter- connected systems has the same structure as the underlying graph and is constructed in the following simple way based on results presented in Section IV-B: (67) where denotes the adjacency matrix of the mesh grid interconnection of systems. The distributed controller in (67) can be rewritten as (4 , where +4 . This corresponds to = 4 in Table I with stability condition max , which is satisﬁed in this example since max = 4 Two simulation results, starting from the same initial cond i- tions, are presented with different choices for the paramet ers and in the local LQR problem tuning (65). The ﬁrst simulation weighs the absolute state information in the loc al LQR solution more heavily than the second simulation and uses (1 1) = 1 and (1 1) = 1 values for position states in both spatial dimensions. Velocity states were not weight ed in these simulation examples. Snapshots of the simulation a re shown in Figure 2, indicating that the subsystems converge to their desired absolute positions along fairly straight l ines starting from their initial conditions. The second simulat ion illustrates the behavior of the large-scale distributed co ntrol system, when much more emphasis is put on the relative state information in the local LQR cost function, by selecti ng (1 1) = 0 001 and (1 1) = 1000 . As the snapshots in Figure 3 demonstrate, the behavior of the overall inter- connected system has changed fundamentally. Although the distributed controller is still stabilizing, the dynamic b ehavior has changed drastically and shows wave-like oscillations d ue to 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 numerical exercise that the stabilizing effect of the proposed distri buted control design is independent of the local LQR weighting pa- rameter selection, thus these parameters are freely availa ble for tuning and achieving different global performance objecti ves. VI. C ONCLUDING EMARKS We have introduced a stabilizing distributed control desig method for large-scale interconnections of identical, dyn am- ically decoupled and independently actuated linear system s. The procedure requires the solution of a single local LQR

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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 on absolute 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 on relative position references. problem, whose size depends only on the maximum vertex degree of the interconnection graph. Special properties of the local LQR problem were derived, which enable the con- struction of stabilizing distributed controllers indepen dently of the choice of weighting matrices. The relationship betwe en stability of the overall large-scale system, the robustnes s of local controllers and the spectrum of a sparsity pattern mat rix has been highlighted. The proposed distributed controller will be stabilizing even if subsystems and communication links are added to or removed from the overall system, as long as this does not change the maximum vertex degree or violate the conditions given in Table I. This feature provides a very powerful modularity to our approach. VII. A CKNOWLEDGEMENTS We thank the anonymous reviewers for the suggestions on how to improve the paper and for pointing out an error in the original version of the manuscript. Tam´as Keviczky’s rese arch was supported in part by the Boeing Corporation. EFERENCES [1] S. Wang and E. J. Davison, “On the stabilization of decent ralized control systems, IEEE Trans. Automatic Control , vol. 18, no. 5, pp. 473–478, 1973. [2] R. D’Andrea and G. E. Dullerud, “Distributed control des ign for spa- tially interconnected systems, IEEE Trans. Automatic Control , vol. 48, no. 9, pp. 1478–1495, Sept. 2003. [3] E. Camponogara, D. Jia, B. Krogh, and S. 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