Optimal Control of Spatially Distributed Systems

Transcription

1 Optimal Control of Spatially Distribute Systems Naer Motee an Ali Jababaie Abstract In this paper, we stuy the structural properties of optimal control of spatially istribute systems. Such systems consist of an infinite collection of possibly heterogeneous linear control systems that are spatially interconnecte via certain istant epenent coupling functions over arbitrary graphs. The key iea of the paper is the introuction of a special class of operators calle spatially ecaying (SD) operators. We stuy the structural properties of infinite-horizon linear quaratic optimal controllers for such systems by analyzing the spatial structure of the solution to the corresponing operator Lyapunov an Riccati equations. We prove that the kernel of the optimal feeback of each subsystem ecays in the spatial omain at a rate proportional to the inverse of the corresponing coupling function of the system. I. INTRODUCTION Analysis an synthesis of istribute coorination an control algorithms for networke ynamic systems has become a vibrant part of control theory research. Several authors have stuie the problem of optimal control of certain classes of spatially istribute systems with symmetries in their spatial structure. In [], Bamieh et al. use spatial Fourier transforms an operator theory to stuy optimal control of linear spatially invariant systems with stanar H 2 (LQ), an H criteria. It was shown that such problems can be tackle by solving a parameterize family of finiteimensional problems in Fourier omain. Furthermore, the authors show that the resulting optimal controllers have an inherent spatial locality similar to the unerlying system. Another interesting relate work in this area is reporte in [2] where the authors use operator theoretic tools, motivate by results of [3] to analyze time-varying systems, an esign optimal controllers for heterogeneous systems which are not shift invariant with respect to spatial or temporal variables. In [4], the authors introuce the notion of quaratic invariance for a constraint set (e.g. sparsity constraints on communication structure of plant an controller). Using this notion, the authors show that the problem of constructing optimal controllers with certain sparsity patterns on the information structure can be cast as a convex optimization problem. This paper is very close in spirit to []. The objective of this paper is to analyze the spatial structure of infinite horizon optimal controllers of spatially istribute systems. Here, we exten the results of [] to heterogeneous systems with arbitrary spatial structure an show that quaratically optimal controllers inherit the same spatial structure as the N. Motee an A. Jababaie are with the Department of Electrical an Systems Engineering an GRASP Laboratory, University of Pennsylvania, 2 South 33r Street, Philaelphia PA 94. {motee,jababai}@seas.upenn.eu This work is porte in parts by the following grants: ONR/YIP N , NSF-ECS-34728, an ARO MURI W9NF original plant. The key point of eparture from [] is that the systems consiere in this work are not spatially invariant an the corresponing operators are not translation invariant either. The spatial structures stuie in [] are Locally Compact Abelian (LCA) groups [] such as (Z, +) an (Z n, ). As a result, the group operation naturally inuces a translation operator for functions efine on the group. However, when the ynamics of iniviual subsystems are not ientical an the spatial structure oes not necessarily enjoy the symmetries of LCA groups, stanar tools such as Fourier analysis cannot be use to analyze the system. To aress this issue, a new class of linear operators, calle spatially ecaying (SD) operators, are introuce that are natural extension of linear translation invariant operators. It is shown that such operators exhibit a localize behavior in spatial omain, i.e., the norm of blocks in the matrix representation of the operator ecay in space. It turns out that the coupling between subsystems in many well-known cooperative control an networke control problems can be characterize by an SD operator. A linear control system is calle spatially ecaying if the operators in its statespace representation are SD. It is shown that the unique solution of Lyapunov an algebraic Riccati equations (ARE) corresponing to SD system are inee SD themselves. As a result, the corresponing optimal controllers are SD an spatially localize, meaning that in the optimal controller, the gain of subsystems that are farther away from a given subsystem ecays in space an the resulting controller is inherently localize. The machinery evelope in this paper can be use to analyze the spatial structure of a broaer range of optimal control problems such as constraine, finite horizon control or Moel Preictive Control of spatially istribute systems. This problem has been analyze in etail in [6] an [7]. This paper is organize as follows. We introuce the notation an the basic concepts use throughout the paper in Section II. The optimal control problem for spatially istribute linear systems is presente in Section III. The concept of spatially ecaying operators an their properties are introuce in Section IV. The structural properties of quaratically optimal controllers are aresse in Section V. Simulation results are inclue in Section VI. Finally, our concluing remarks are presente in Section VII. II. PRELIMINARIES R enotes the set of real numbers, R + the set of nonnegative real numbers, an C the set of complex numbers. Consier an unirecte connecte graph with a nonempty set G of noes. We refer to G as the spatial omain.. an. enote the Eucliean vector norm an its corresponing inuce matrix norm, respectively. The Banach space l p (G) for p < is efine to be the set of all sequences

2 x = (x i ) in which x i R n i satisfying x i p < enowe with the norm x p := ( x i p ) p. The Banach space l (G) enotes the set of all boune sequences enowe with the norm x := x i. Throughout the paper, we will use the shorthan notation l p for l p (G). The space l 2 is a Hilbert space with inner prouct x, y := x i, y i for all x, y l 2. An operator Q : l p l q for p, q is boune if it has a finite inuce norm, i.e., the following quantity Q p,q := Qx q x p = is boune. The ientity operator is enote by I. The set of all boune linear operators of l p into itself is enote by B(l p ). An operator Q B(l p ) has an algebraic inverse if it has an inverse Q in B(l p ) [8]. The ajoint operator of Q B(l 2 ) is the operator Q in B(l 2 ) such that Qx, y = x, Q y for all x, y l 2. An operator Q is self-ajoint if Q = Q. An operator Q B(l 2 ) is positive efinite, shown as Q, if there exists a number α > such that x, Qx > α x 2 2 for all nonzero x l 2. The set of all functions from A R into R is a vector space F over R. For f, f 2 F, the notation f f 2 will be use to mean the pointwise inequality f (s) < f 2 (s) for all s A. A family of seminorms on F is efine as {. T T R + } in which f T := s T f(s) for all f F. The topology generate by all open. T -balls is calle the topology generate by the family of seminorms an is enote by. T -topology. Continuity of a function in this topology is equivalent to continuity in every seminorm in the family. Although the results of section III is set up in a general framework, in this paper we are intereste in linear operators which have matrix representations. III. OPTIMAL CONTROL OF SPATIALLY DISTRIBUTED SYSTEMS We begin by consiering a continuous-time linear moel for spatially istribute systems over a iscrete spatial omain G escribe by ψ(t) t = (Aψ) (t) + (Bu) (t) () y(t) = (Cψ) (t) + (Du) (t) (2) with the initial conition ψ() = ψ. All signals are assume to be in L 2 ([, ); l 2 ) space: at each time instant t [, ), signals ψ(t), u(t), y(t) are assume to be in l 2. The state-space operators A, B, C, D are assume to be constant functions of time from l 2 to itself. The semigroup generate by A is strongly continuous on l 2. This assumption guarantees existence an uniqueness of classical solutions of the system given by ()-(2) (cf. Chapter 3 of [9]). Example : Consier the general one-imensional heat equation for a bi-infinite bar [] ψ(x, t) = t x ( c(x) ψ(x, t) x ) + b(x)u(x, t) where x is the spatial inepenent variable, t is the temporal inepenent variable, ψ(x, t) is the temperature of the bar, an u(x, t) is a istribute heat source. The thermal conuctivity c is only a function of x an is ifferentiable with respect to x. The bounary conitions are assume to be ψ(, t) = ψ(, t) =. By inserting finite ifference approximation for the spatial partial erivatives, the following continuous-time, iscrete-space moel can be obtaine: ( ) t ψ(x k, t) = c ψ(xk, t) ψ(x k, t) (x k ) + δ ( ) ψ(xk, t) 2ψ(x k, t) + ψ(x k+, t) c(x k ) + b(x k )u(x k, t) δ 2 where c (x) = xc(x). The iscretization is performe with equal spacing δ = x k x k of the points x k such that there is an integer number of points in space. Hence, after iscretization the spatial omain is G = Z. This moel can be represente as linear system () in which the infinitetuples ψ(t) = (ψ(x k, t)) an u(t) = (u(x k, t)) are the state an control input variables of the infiniteimensional system an the block elements of the state-space operators A an B are efine as follows c (x k )δ+c(x k ) δ, i = k 2 [A] ki = c (x k )δ+2c(x k ) δ, i = k 2 c(x k ) δ, i = k + 2, otherwise an [B] ki = { b(xk ), i = k, otherwise for all k, i G. One can show that A is an unboune operator on l 2. However, the semigroup generate by A is strongly continuous on l 2. A. Exponential Stability Consier the following autonomous system over G ψ(t) = (Aψ) (t) (3) t with initial conition ψ() = ψ. Suppose that A generates a strongly continuous C -semingroup on l 2, enote by T (t). The system (3) is exponentially stable if for some M, α >. T (t) 2,2 Me αt for t Theorem [9]: Let A be the infinitesimal generator of the C -semigroup T (t) on l 2 an Q a positive efinite operator. Then T (t) is exponentially stable if an only if the Lyapunov equation Aφ, Pφ + Pφ, Aφ + φ, Qφ = (4) for all φ D(A), has a positive efinite solution P B(l 2 ). B. LQR control of infinite imensional systems While the main results of this paper are proven for LQ optimal controllers, similar results can be proven for H an H 2 problems. In general, the solutions to these problems can be formulate in terms of two operator AREs. Such problems have been aresse in the literature for general classes of istribute parameter systems [9], []. An elegant analysis for the spatially invariant case can be foun in []. Similar to the finite-imensional case, optimal solutions to infinite-imensional LQR can be formulate in terms of

3 an operator Riccati equation. Consier the quaratic cost functional given by J = ψ(t), Qψ(t) + u(t), Ru(t) t. () The system ()-(2) with cost () is sai to be optimizable if for every initial conition ψ() = ψ l 2, there exists an input function u L 2 ([, ); l 2 ) such that the value of () is finite [9]. Note that if (A, B) is exponentially stabilizable, then the system ()-(2) is optimizable. Theorem 2 [9]: Let operators Q an R be in B(l 2 ). If the system ()-(2) with cost functional () is optimizable an (A, Q /2 ) is exponentially etectable, then there exists a unique nonnegative, self-ajoint operator P B(l 2 ) satisfying the ARE ϕ, PAφ + PAϕ, φ + ϕ, Qφ B Pϕ, R B Pφ = (6) for all ϕ, φ D(A) such that A BR B P generates an exponentially stable C -semigroup. Moreover, the optimal control ũ L 2 ([, ); l 2 ) is given by the feeback law where ψ is the solution of ũ(t) = R B P ψ(t) t ψ(t) = (A BR B P) ψ(t) (7) with initial conition ψ. Solving equations (4) an (6) can be a teious task in general. However, the complexity of the problem will reuce significantly if the unerlying system is spatially invariant with respect to G (cf. Section III.B of []). The main objective of this paper is to analyze the spatial structure of the solutions of operator equations (4) an (6) rather than solving them explicitly. IV. SPATIALLY DECAYING OPERATORS The key ifficulty in extening the results of [] is that the notion of spatial invariance was critical in being able to use Fourier methos which greatly simplifie the analysis. Simply put, if we replace space with time, we get a more familiar analogue of this problem: Fourier methos can not be use irectly for analysis of linear time-varying systems. In the following, we will generalize the notion of regions of analyticity of transforms to a larger class of linear operators. Without loss of generality, in the following efinitions it is assume that all operators are self-ajoint. Definition : A istance function on a iscrete topology with a set of noes G is efine as a single-value, nonnegative, real function is(k, i) efine for all k, i, j G which has the following properties: (i) is(k, i) = iff k = i. (ii) is(k, i) = is(i, k). (iii) is(k, i) is(k, j) + is(j, i). Definition 2: A nonecreasing continuous function χ : R + [, ) is calle a coupling characteristic function if χ() = an χ(s + t) χ(s) χ(t) for all s, t R +. The constant coupling characteristic function with unit value everywhere is enote by. In orer to be able to characterize rates of ecay we efine a one-parameter family of coupling characteristic functions as follows. Definition 3: A one-parameter family of coupling characteristic functions C is efine to be the set of all characteristic functions χ α, χ β for α, β R + such that (i) χ =. (ii) χ α χ β = χ α+β. (iii) For α < β, relation χ α χ β hols. (iv) χ α is a continuous function of α in. T -topology. Using this efinition, we can now formally efine a spatially ecaying (SD) operator. Definition 4: Suppose that a istance function is(.,.) an a one-parameter family of parameterize coupling characteristic functions C are given. A linear operator Q is SD with respect to C if there exists τ > such that the auxiliary operator Q, efine block-wise as [ Q] ki = [Q] ki χ α (is(k, i)) is boune on l p for all α < τ. The number τ is referre to as the ecay margin. In general, etermining the bouneness of the auxiliary operator epens on the choice of p. The result of lemma gives us a simple sufficient conition for an operator to be SD in terms of all l p -norms. Uner the assumptions of efinition 4, we also assume that the following conition hols χ α (is(k, i)) < for all < α < τ. Lemma : A linear operator Q is SD with respect to the one-parameter family of coupling characteristic functions C on all l p if there exists τ > such that the following hols for all α < τ. Proof: See [2] for a proof. [Q] ki χ α (is(k, i)) < (8) Examples of SD operators appear naturally in many applications. Intuitively, we may interpret the norm of each block element [Q] ki as the coupling strength between subsystems k an i. Given the one-parameter family of coupling characteristic functions C, fix a value for α [, τ). For an infinite graph, if we fix a noe k an move on the graph away from noe k, the coupling strength ecays proportional to the inverse of the coupling characteristic function χˆα with α < ˆα < τ so that relation (8) hols. The notion of an SD operator will be key in proving spatial locality of optimal controllers. Throughout the rest of the paper, we say an operator is SD if it satisfies conition (8). A. Examples of Spatially Decaying Operators The following class of operators which are use extensively in cooperative an istribute control are interesting special classes of SD operators. ) Spatially Truncate Operators: These are operators with finite range couplings. Examples of such operators arise in motion coorination of autonomous agents such as the

4 Laplacian operator. Given the coupling range T >, the following class of linear operators are SD with respect to every coupling characteristic functions { Qki if is(k, i) T [Q] ki = (9) if is(k, i) > T where Q ki R n n. For such operators an every given noe k G, we have that [Q] ki χ α (is(k, i)) [Q] ki χ α (T ) <. () i k The relation is the neighborhoo relation efine as i k if an only if is(k, i) T. Inequality () shows that Q is SD with respect to every C an the ecay margin is τ =. 2) Exponentially Decaying Operators: Consier the oneparameter family of coupling characteristic functions C E efine by χ ζ (s) = ( + ζ) s () where ζ R +. Operator Q is sai to be exponentially SD if conition (8) hols with respect to C E efine by () for all ζ [, τ) where τ > is the ecay margin. An important example of exponentially SD operators is the class of translation invariant operators with G = Z. It can be shown that, uner some mil assumptions, a translation invariant operator in B(l 2 ) is exponentially SD with is(k, i) = k i as a natural notion of istance [2]. The ecay margin of Q is equal to r, the istance of the nearest pole of the Fourier transform of Q to the unit circle in C. 3) Algebraically Decaying Operators: Consier the parameterize family of characteristic functions C A efine as χ ν (s) = ( + λs) ν (2) in which λ > an ν R +. Operator Q is sai to be algebraically SD if conition (8) hols with respect to C A efine by (2) for all ν [, τ) where τ > is the ecay margin. Such functions are often use as pair-wise potentials among agents in flocking an cooperative control problems [3]. Another example of such coupling functions arises in wireless networks. The coupling between noes, which is consiere as the power of the communication signal between agents, ecays with the inverse fourth power law, i.e., is(k,i) 4. B. Properties of SD Operators We efine the operator norm Q τ = [Q] ki χ α (is(k, i)) α [,τ) an the norme vector space S τ (C ) = {Q : Q τ < }. It can be shown that the operator norm satisfies the following properties [2], for all Q, P S τ (C ) an c C, (i) Q τ an Q τ = iff Q =. (ii) c Q τ = c Q τ. (iii) Q + P τ Q τ + P τ. (iv) QP τ Q τ P τ. Property (iv) is calle the submultiplicative property. Theorem 3: Given a one-parameter family of coupling characteristic functions C an τ >, the operator space S τ (C ) forms a Banach Algebra with respect to the operator norm. τ uner the operator composition operation. Proof: See [2] for a proof. The above theorem is a key ingreient in proving that optimal controllers of SD systems are SD. We have shown that operator space S τ (C ) is close uner aition, multiplication, an limit properties (cf. [2], Theorem ). Furthermore, if an SD operator has an algebraic inverse on B(l 2 ), the inverse operator Q is also SD [4]. It is straightforwar to check that the serial, parallel, an well-pose feeback interconnection of two SD systems are also SD. In the next section, using the closure properties of SD operators, it is shown that the solution of ifferential Lyapunov an Riccati equations converge to an SD operator. V. STRUCTURE OF QUADRATICALLY OPTIMAL CONTROLLERS As iscusse in section III, our aim is not to solve the Lyapunov equation (4) an ARE (6) explicitly but to stuy the spatial structure of the solution of these algebraic equations by means of tools evelope in the previous sections. In the following, it is shown that the solution of equations (4) an (6) have an inherent spatial locality an the characteristics of the coupling function will etermine the egree of localization. Theorem : Assume that operators A, Q S τ (C ) an Q is positive efinite. If A is the infinitesimal generator of an exponentially stable C -semigroup T (t) on l 2, then the unique positive efinite solution of operator Lyapunov equation (4) satisfies P S τ (C ). Proof: See [2] for a proof. In the next theorem without loss of generality, we will assume that R = I. Otherwise, by only assuming that R has an algebraic inverse on B(l 2 ), it can be shown that R is SD [4]. Accoring to the closure uner multiplication property of SD operators, if P an B are SD, then the optimal feeback operator K = R B P will be SD. Theorem 6: Let A, B, Q S τ (C ) an Q. Moreover, assume that conitions of Theorem 2 hol. Then the unique positive efinite solution of operator ARE (6) satisfies P S τ (C ). t Proof: Consier the Differential Riccati Equation ϕ, P(t)φ = ϕ, P(t)Aφ + P(t)Aϕ, φ + ϕ, Qφ B P(t)ϕ, B P(t)φ with P() =. We enote the unique solution of this ifferential Riccati equation in the class of strongly continuous, self-ajoint operators in B(l 2 ) by the one-parameter family of operator-value function P(t) for t. The nonnegative operator P, the unique solution of ARE, is the strong limit of P(t) on l 2 as t (see theorem of [9]). Therefore, we have that lim P(t) P 2,2 =. (3) t

5 From the ifferential Riccati equation, it follows that t [P(t)] ki = [A P(t) + P(t)A P(t)BB P(t) + Q] ki for all k, i G. For a ifferentiable matrix X(t) C n n for t, we have the following inequality X(t + δ) X(t) X(t) = lim t δ τ lim X(t + δ) X(t) δ δ t X(t). (4) Using inequality (4), we have t P(t) τ t [P(t)] ki χ α (is(k, i)) A P(t) + P(t)A P(t)BB P(t) + Q τ. For simplicity in notations, enote π(t) = P(t) τ. Using the triangle inequality an the fact that norm. τ is submultiplicative, we have the following ifferential inequality π(t) 2 A τ π(t) + ( B τ ) 2 π(t) 2 + Q τ () with initial conition π() = an constraint π(t) for all t. All coefficients A τ, B τ, Q τ in the right han sie of the inequality () are finite numbers. If π(t) for t is a solution of the ifferential inequality (), then it is also a solution of the following ifferential inequality π(t) λ (π(t) + ) 2 (6) with π() = an λ = max( A τ, ( B τ ) 2, Q τ ). In other wors, the set of feasible solutions of () is a subset of solutions of (6). From (6), we have t ( which has the set of solutions π(t) + ) λ π(t) + e λt π() +. Using the fact that π(t) for all t an π() =, it follows that π(t) e λt. The above inequality is feasible, i.e., there exists at least one sequence of solutions satisfying π(t) for all t. The above inequality also proves that π(t) < for all t. Thus, we have that P(t) S τ (C ) for all t. Accoring to Theorem in [2], we can use this result an (3) to conclue that P S τ (C ). This completes the proof. VI. SIMULATION We consier a large network of N linear subsystems couple on an arbitrary graph which can be escribe by ψ(t) = (Aψ) (t) + (Bu) (t). t The coupling characteristic function is χ an the system operators are given by [A] ki = χ(is(k,i)) an B = I. The istance function is Eucliean. We will stuy the LQR problem iscusse in Section III with weighting operators R = I an Q being the corresponing unweighte graph Fig.. N= noes are ranomly an uniformly istribute in a region of area 3 3 (units) 2. Each noe is a linear subsystem which is couple to other subsystems through their ynamic an a central cost function by a given coupling characteristic function. Laplacian. The corresponing ARE is given by A P + PA P 2 + Q =. (7) Then the LQR optimal feeback is given by K = P. In the following simulations, it is assume that N = noes are ranomly an uniformly istribute in a region of area 3 3 (units) 2. Each noe is assume to be a linear system which is couple through its ynamic an the LQR cost functional to other subsystems. In the sequel, three ifferent scenarios are consiere for the coupling characteristic function: algebraical ecay, exponential ecay, an nearest neighbor coupling. The results are shown, respectively, in Figures 2, 3, an 4 where the norm of the LQR feeback gains [K] ki corresponing to agents k =, 4, 2, (their locations are marke by bol stars in Figure ) is epicte versus the istance of other subsystems to subsystem k. As seen from these simulations, for every subsystem k the norm of the optimal feeback kernel [K] ki is envelope by the inverse of the coupling characteristic function. Therefore, the spatial ecay rate of the optimal controller can be etermine priory only using the information of the coupling characteristic function. A. Spatial Truncation Let K T be the spatially truncate operator efine by { [K]ki if is(k, i) T [K T ] ki = if is(k, i) > T. By applying the small-gain stability argument, one can obtain a truncation length T s for which K T is stabilizing for all T T s (cf. Section V.B in []). Figure illustrates the performance loss percentage efine as Trace(P T ) Trace(P) Trace(P) versus ifferent values of T T s for ifferent coupling characteristic functions where P T satisfies (A + BK T ) P T + P T (A + BK T ) + Q + K T RK T =. As seen from Figure, the larger values of truncation length T ensue better close-loop performance. In the above simulations, the extension of this surprising locality result to finite-imensional systems is ue to the fact that the matrices in system s state-space representation are

6 3 4 2 Subsystem No. 2 Subsystem No Norm of LQR Feeback Gain Subsystem No Subsystem No. 2 Norm of LQR Feeback Gain Subsystem No Subsystem No Subsystem No. Subsystem No Distance Distance Fig. 2. Norm of LQR feeback gain [K] ki (bar) an function [K] kk (which is algebraically ecaying) when λ =. an ν = 4 χ ν (is(k,i)) (ashe) for subsystems k =, 4, 2,, respectively, from top to bottom. Fig. 4. Norm of LQR feeback gain [K] ki (bar) an [K] ki pulse function (which represents the nearest neighbor coupling) with length T = (ashe) for subsystems k =, 4, 2,, respectively, from top to bottom. Norm of LQR Feeback Gain Subsystem No Subsystem No Subsystem No Subsystem No. 2 2 Distance Performance Loss Percentange % Exponential Decay Nearest Neighbor Coupling Algebraical Decay Truncation Length (T) Fig. 3. Norm of LQR feeback gain [K] ki (bar) an function [K] kk (which is exponentially ecaying) when ζ = e (ashe) χ ζ (is(k,i)) for subsystems k =, 4, 2,, respectively, from top to bottom. Fig.. Performance Loss percentage of LQR controller after spatial truncation for ifferent types of couplings: (i) algebraical ecay (ii) exponential ecay (iii) nearest neighbor coupling. efine such that the norm of blocks in the matrix ecay as a function of istance between subsystems. VII. CONCLUSIONS In this paper we stuie the spatial structure of infinite horizon optimal controllers for spatially istribute systems. By introucing the notion of SD operators we extene the notion of analytic continuity to operators that are not spatially invariant. Furthermore, we prove that SD operators form a Banach algebra. We use this to prove that solutions of Lyapunov an Riccati equations for SD systems are themselves SD. This result was utilize to show that the kernel of optimal LQ feeback is also SD. Although these results were proven for LQ problems, they can be easily extene to general H 2, an H optimal control problems as the key enabling property is the spatial ecay of solution of the corresponing Riccati equations. One major implication of these results is that the optimal control problem for spatially ecaying systems lens itself to istribute solutions without too much loss in performance as even the centralize solutions are inherently localize. REFERENCES [] B. Bamieh, F. Paganini, an M. A. Dahleh, Distribute control of spatially invariant systems, IEEE Trans. Automatic Control, vol. 47, no. 7, pp. 9 7, 22. [2] G. Dulleru an R. D Anrea, Distribute control of heterogeneous systems, IEEE Trans. on Automatic Control, vol. 49, no. 2, pp , 24. [3] G. Dulleru an S. Lall, A new approach for analysis an synthesis of time-varying systems, IEEE Trans. on Automatic Control, vol. 44, no. 8, pp , 999. [4] M. Rotkowitz an S. Lall, A characterization of convex problems in ecentralize control, IEEE Tran. on Automatic Control, vol., no. 2, pp , 26. [] W. Ruin, Fourier Analysis on Groups. Interscience, 962. [6] N. Motee an A. Jababaie, Distribute receing horizon control of spatially invariant systems, in Proc. of the American Control Conference, Minneapolis, MN, 26. [7], Receing horizon control of spatially istribute systems over arbitrary graphs, in Proc. 4th IEEE Conference on Decision an Control, San Diego, CA, 26. [8] P. Lax, Functional Analysis. John Wiley, 22. [9] R. Curtain an H. Zwart, An introuction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, 99. [] S. Kakaç an Y. Yener, Heat conuction, 3r E. Taylor & Francis, 993. [] B. van Keulen, H -Control for Distribute Parameter Systems: A State-Space Approach. Boston, MA: Birkhauser, 993. [2] N. Motee an A. Jababaie, Optimal control of spatially istribute systems, IEEE Tran. on Automatic Control, September 26, accepte. [Online]. Available: motee/ TACMoteeJ6SD.pf [3] F. Cucker an S. Smale, Emergent behaviour in flocks, 2. [4] N. Motee an A. Jababaie, Distribute quaratic programming over arbitrary graphs, IEEE Tran. on Automatic Control, Jan. 27, submitte. [Online]. Available: motee/ TACMoteeJ6QP.pf