# Distrib uted LQR Design for Dynamically Decoupled Systems Francesco Borrelli am as viczk Abstract consider set of identical decoupled dynamical systems and contr ol pr oblem wher the perf ormance ind PDF document - DocSlides

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The coupling is described thr ough communication graph wher each system is node and the contr ol action at each node is only function of its state and the states of its neighbors distrib uted contr ol design method is pr esented which equir es the s ID: 23613

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## Presentations text content in Distrib uted LQR Design for Dynamically Decoupled Systems Francesco Borrelli am as viczk Abstract consider set of identical decoupled dynamical systems and contr ol pr oblem wher the perf ormance ind

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Distrib uted LQR Design for Dynamically Decoupled Systems Francesco Borrelli am as viczk Abstract consider set of identical decoupled dynamical systems and contr ol pr oblem wher the perf ormance index couples the beha vior of the systems. The coupling is described thr ough communication graph wher each system is node and the contr ol action at each node is only function of its state and the states of its neighbors. distrib uted contr ol design method is pr esented which equir es the solution of single LQR pr oblem. The size of the LQR pr oblem is equal to the maximum ertex degr ee of the communication graph plus one. The design pr ocedur pr oposed in this paper illustrates ho stability of the lar ge-scale system is elated to the ob ustness of local contr ollers and the spectrum of matrix epr esenting the sparsity patter of the distrib uted contr oller design pr oblem. Decentralized control techniques today can be found in broad spectrum of applications ranging from robotics and formation ight to ci vil engineering. Contrib utions and interest in this eld date back to the early results of [1]. Approaches to decentralized control design dif fer from each other in the assumptions the mak on: the kind of interaction between dif ferent systems or dif ferent compo- nents of the same system (dynamics, constraints, objecti e), ii the model of the system (linear nonlinear constrained, continuous-time, discrete-time), iii the model of informa- tion xchange between the systems, iv the control design technique used. In this paper we focus on identical decoupled linear time- in variant systems Our interest in decentralized control for such systems arises from the ab undance of netw orks of independently actuated systems and the necessity of oid- ing centralized design when this becomes computationally prohibiti e. Netw orks of ehicles in formation, production units in po wer plant, cameras at an airport, an array of mechanical actuators for deforming surf ace are just fe xamples. In descripti ay the problem of distrib uted control for decoupled systems can be formulated as follo ws. dynamical system is composed of (or can be decomposed into) distinct dynamical subsystems that can be indepen- dently actuated. The subsystems are dynamically decoupled ut ha common objecti es, which mak them interact with each other ypically the inter action is local, i.e., the goal of subsystem is function of only subset of other subsystems' states. The interaction will be represented by an “interaction Corresponding author Borrelli is with the Dipartimento di Inge gneria, Uni- ersit de gli Studi del Sannio, 82100 Bene ento, Italy francesco.borrelli@unisannio.it viczk is with Control and Dynamical Systems, California Institute of echnology asadena, CA 91125, USA, tamas@cds.caltech.edu graph”, where the nodes represent the subsystems and an edge between tw nodes denotes coupling term in the controller associated with the nodes. Also, typically it is assumed that the xc hang of information has special structure, i.e., it is assumed that each subsystem can sense and/or xchange information with only subset of other subsystems. will assume that the inter action gr aph and the information xc hang gr aph coincide. distrib uted control scheme consists of distinct controllers, one for each subsystem, where the inputs to each subsystem are computed only based on local information, i.e., on the states of the subsystem and its neighbors. Ov er the past fe years, there has been rene al of inter est in systems composed of lar ge number of interacting and cooperating interconnected units [2]–[15]. short re vie of the these approaches can be found in [16]. This manuscript proposes simple distrib uted controller design approach and focuses on class of systems, for which xisting methods are either not ef cient or ould not en be directly applicable. Our method applies to lar ge-scale systems composed of nite number of identical subsystems where the inter connection structur or spar sity pattern is not equir ed to have any special in variance pr operties The philosoph of our approach uilds on the recent orks [9] [11], where at each node, the model of its neighbors are used to predict their beha vior sho that in absence of state and input constraints, and for identical linear system dynamics, such an approach leads to an xtremely po werful result: the synthesis of stabilizing distrib uted control la ws can be obtained by using simple local LQR design, whose size is limited by the maximum erte de gree of the interconnection graph plus one. Furthermore, the design procedure proposed in this paper illustrates ho stability of the erall lar ge-scale system is related to the rob ustness of local controllers and the spectrum of matrix representing the desired sparsity pattern. In addition, the constructed distrib uted controller is stabilizing independent of the tuning parameters in the local LQR cost function. This leads to method for designing distrib uted controllers for nite number of dynamically decoupled systems, where the local tuning parameters can be chosen to obtain desirable global performance. Such result can be immediately used to impro current stability analysis and controller synthesis in the eld of decentralized receding horizon control for dynamically decoupled systems [3], [9], [12], [13], [17]. denote by the eld of real numbers, the eld of comple numbers and the set of real matrices.

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Re Re Notation 1: Let then denotes matrix of dimension 1) 1) obtained by xtracting ro ws to and columns to from the matrix with Notation 2: denotes the identity matrix of dimension Notation 3: Let denote the -th eigen alue of The spectrum of will be denoted by Denition 1: matrix is called Hurwitz (or stable) if all its eigen alues ha ne ati real part, i.e. Notation 4: Let Then denotes the Kroneck er product of and 11 mn mp nq (1) Pr oposition 1: Consider tw matrices and Then Pr oof: ak an eigen alue and the corresponding eigen ector Then Av Pr oposition 2: Gi en A; and consider tw matrices and where A; nm nm Then =1 where is the -th eigen alue of Pr oof: Let be an eigen ector of corre- sponding to and be an eigen ector of with as the associated eigen alue. Consider the ector nm Then )( Au Au Au Since we get )( )( Consider set of identical, decoupled linear time- in ariant dynamical systems, the -th system being described by the continuous-time state equation: Ax (0) (2) where are states and inputs of the -th system at time respecti ely Let nN and mN be the ectors which collect the states and inputs of the systems at time u; (0) 10 (3) with A; (4) consider an LQR control problem for the set of sys- tems where the cost function couples the dynamic beha vior of indi vidual systems: u; =1 ii ii )+ =1 )) ij d (5) with ii ii ii ii i; (6a) ij ij (6b) The cost function (5) contains terms which weigh the -th system states and inputs, as well as the dif ference between the -th and the -th system states and can be re written using the follo wing compact notation: d (7) where the matrices and ha special structure dened ne xt. and can be decomposed into blocks of dimension and respecti ely: 11 12 (8) with ii =1 ik ij ij i; (9) Remark 1: The cost function structure (5) can be used to describe se eral practical applications including formation ight, paper machine control and monitoring netw orks of cameras [14], [18]. Let and be the optimal controller and the alue function corresponding to the follo wing LQR problem: min subj. to (0) (10) Throughout the paper we will assume that stabilizing solution to the LQR problem (10) with nite performance inde xists and is unique (see [19], p. 52 and references therein): Assumption 1: System A; is stabilizable and system A; is observ able, where is an matrix such that will also assume local stabilizablity and observ ability: Assumption 2: System A; is stabilizable and systems A; are observ able, where is an matrix such that It is well kno wn that where is the symmetric positi denite solution to the follo wing ARE: (11)

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decompose and into blocks of dimension and respecti ely Denote by ij and ij the i; block of the matrix and respecti ely In the follo wing theorems we sho 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). Ne xt, the matrix is dened as Theor em 1: Let and be the optimal controller and the alue function solution to the LQR problem (10). Let ij [( 1) im; 1) and ij [( 1) in; 1) with Then, 1) =1 ij for all where is the symmetric positi denite solution of the ARE associated with single node local problem: (12) 2) =1 ij for all where Pr oof: The proof can be found in [16]. Theor em 2: Assume the weighting matrices (9) of the LQR problem (10) are chosen as ii ij i; (13) Let be the alue function of the LQR problem (10) with weights (13), and the blocks of the matrix be denoted by ij [( 1) in; 1) with i; Then ij is symmetric ne ati semidenite matrix for all Pr oof: The assumption in (13) requires that the weight used for absolute states and the weight used for neighboring state dif ferences are equal for all nodes and for all neighbors of node, respecti ely Such an assumption and the act that and are block-diagonal with identical blocks, imply that the ARE in (11) is set of identical equations where the matrices ij are all identical and sym- metric for all denote by the generic block ij for The ARE equations for the block ij with become 2) (14) which can be re written as follo ws in virtue of Theorem (15) where is the symmetric positi denite solution of the ARE (12) associated with single node local problem. Re write equation (15) as )( +( (16) Since and equation (16) can be seen as an ARE associated with an LQR problem for the stable system with weights and Let the matrix be its positi semidenite solution. Then, the follo wing matrix (17) with is symmetric positi denite matrix and it is the unique symmetric positi denite matrix solution to the ARE (11). This pro es the theorem. Under the ypothesis of Theorem 2, because of symmetry and equal weights on the neighboring state dif ferences and equal weights on absolute states, the LQR optimal controller will ha the follo wing structure: (18) with and functions of and The follo wing corollaries of Theorem follo from the stability and the rob ustness of the LQR controller for system in (16). Cor ollary 1: is Hurwitz matrix. From the ain mar gin properties [20] we ha e: Cor ollary 2: is Hurwitz matrix for all with Remark 2: is Hurwitz matrix, thus the system in Corollary is stable for with being the LQR ain for system A; with weights ). The follo wing condition denes class of systems and LQR weighting matrices which will be used in later sections to xtend the set of stabilizing distrib uted controller struc- tures. Condition 1: is Hurwitz matrix for all [0 with Essentially Condition characterizes systems for which the LQR ain stability mar gin described in Corollary is xtended to an positi Checking the alidity of Condition for gi en tuning of and may be performed as stability test for simple af ne parameter -dependent model {z x; (19) where and This test can be posed as an LMI problem (Proposition 5.9 in [21]) searching for quadratic parameter -dependent yapuno functions. In the follo wing section we introduce some basic concepts of graph theory before presenting the distrib uted control design problem.

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This section is concise re vie of the relationship between the eigen alues of Laplacian matrix and the topology of the associated graph. refer the reader to [22], [23] for comprehensi treatment of the topic. list collection of properties associated with undirected graph Laplacians and adjacenc matrices, which will be used in subsequent sections of the paper graph is dened as (20) where is the set of nodes (or ertices) and the set of edges i; with The de gree of graph erte is the number of edges which start from Let max denote the maximum erte de gree of the graph denote by the (0 1) adjacenc matrix of the graph Let i;j be its i; element, then i;i i;j if i; and i;j if i; i; will focus on undir ected graphs, for which the adja- cenc matrix is symmetric. Let )) be the spectrum of the adjacenc matrix associated with an undirected graph arranged in nondecreasing semi-order Pr operty 1: max This property together with Proposition implies Pr operty 2: max dene the Laplacian matrix of graph in the follo wing ay (21) where is the diagonal matrix of erte de grees (also called the alence matrix). Eigen alues of Laplacian matrices ha been widely studied by graph theorists. Their properties are strongly related to the structural properties of their associated graphs. Ev ery Laplacian matrix is singular matrix. By Ger sgorin' theorem [24], the real part of each nonzero eigen alue of is strictly positi e. or undirected graphs, is symmetric, positi semidenite matrix, which has only real eigen alues. Let )) be the spectrum of the Laplacian matrix associated with an undirected graph arranged in nondecreasing semi-order Then, Pr operty 3: 1) with corresponding eigen ector of all ones, and if is connected. In act, the multiplicity of as an eigen alue of is equal to the number of connected components of 2) The modulus of is less then The second smallest Laplacian eigen alue of graphs is probably the most important information contained in the spectrum of graph. This eigen alue, called the algebraic connecti vity of the graph, is related to se eral important graph in ariants, and it has been xtensi ely in estig ated. Let be the Laplacian of graph with ertices and with maximal erte de gree max Then properties of include Pr operty 4: )) )(1 cos where is the edge connecti vity of the graph [25]. Further relationships between the graph topology and Laplacian eigen alue locations are discussed in [23] for undirected graphs. Spectral characterization of Laplacian matrices for directed graphs can be found in [24]. consider set of linear identical and decoupled dynamical systems, described by the continuous-time time- in ariant state equation (2), re written belo Ax (0) where are states and inputs of the -th system at time respecti ely Let and be the ectors which collect the states and inputs of the systems at time then u; (0) 10 (22) with A; Remark 3: Systems (22) and (3) dif fer only in the number of subsystems. will use system (3) with subsystems when referring to local problems, and system (22) with subsystems when referring to the global problem. Accord- ingly tilded matrices will refer to local problems and hatted matrices will refer to the global problem. use graph topology to represent the coupling in the control objecti and the communication in the follo wing ay associate the -th system with the -th node of graph If an edge i; 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 la minimizes weighted distance between the -th and the -th system states. The class of n;m matrices is dened as follo ws: Denition 2: n;m nN mN ij if i; ij [( 1) in; 1) i; The distrib uted optimal contr ol pr oblem is dened as fol- lo ws: min u; d (23a) subj. to u; (23b) m;n (23c) n;n m;m (23d) with (0) and also refer to problem (23) without (23c) as centr alized

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optimal contr ol pr oblem In general, computing the solution to problem (23) is an NP-hard problem. Ne xt, we propose suboptimal control design leading to controller with the follo wing properties: 1) m;n (24a) 2) is Hurwitz. (24b) 3) Simple tuning of absolute and relati state errors and control ef fort within (24c) Such controller will be referred to as distrib uted suboptimal contr oller The follo wing theorem will be used to propose distrib uted suboptimal control design procedure. Theor em 3: Consider the LQR problem (10) with max and weights chosen as in (13) and its solu- tion (17), (18). Let be symmetric matrix with the follo wing property: (25) and construct the feedback controller: (26) Then, the closed loop system cl (27) is asymptotically stable. Pr oof: Consider the eigen alues of the closed-loop system cl cl )) By Proposition 2: )) =1 (28) will pro that is Hurwitz matrix and thus pro the theorem. If then is Hurwitz based on Remark 2. If then from Corollary and from condition (25), we conclude that is Hurwitz. Theorem has se eral main consequences: 1) If then in (26) is an asymptotically stable distrib uted controller 2) can use one local LQR controller to compose distrib uted stabilizing controllers for collection of identical dynamically decoupled subsystems. 3) The rst tw consequences imply that we can not only nd stabilizing distrib uted controller with desired sparsity pattern (which is in general formidable task by itself), ut it is enough to solv lo w-dimensional problem (characterized by max )) compared to the full problem size (23). This attracti feature of our approach relies on the specic problem structure dened in Section II and IV. 4) The eigen alues of the closed-loop lar ge-scale system cl can be computed through smaller eigen alue computations as =1 5) The result is independent from the local LQR tuning. Thus and in (13) can be used in order to inuence the compromise between minimization of absolute and relati terms, and the control ef fort in the global performance. or the special class of systems dened by Condition 1, the ypothesis of Theorem can be relax ed as follo ws: Theor em 4: Consider the LQR problem (10) with max and weights chosen as in (13) and its solu- tion (17), (18). Assume that Condition holds. Let be symmetric matrix with the follo wing property: (29) Then, the closed loop system (27) is asymptotically stable when is constructed as in (26). Pr oof: Notice that if Condition holds, then is Hurwitz for all (from Corol- lary and Corollary 2). By Proposition )) =1 which together with condition (29) pro es the theorem. In the ne xt sections we sho ho to choose in Theorem and Theorem in order to construct distrib uted suboptimal controllers. The matrix will reect the structure of the graph ii satisfy (25) or (29) and iii be computed by using the graph adjacenc matrix or the Laplacian matrix. Ne xt we pr esent the distrib uted contr ol design for eneric gr aph structur Illustr ative xamples for simple nite string and for nite squar mesh can be found in [16], [26]. A. Arbitr ary Gr aph Structur es consider generic graph for nodes with an as- sociated Laplacian and maximum erte de gree max Let be the eigen alues of the the Laplacian In the ne xt Corollaries 3, and we present three ays of choosing in (26) which lead to distrib uted suboptimal controllers. Cor ollary 3: Compute in (26) as aL If (30) then the closed loop system (27) is asymptotically stable when is constructed as in (26). In addition, if Condition holds, then the closed loop system (27) is asymptotically stable for all Pr oof: The proof is direct consequence of Theorems and 4, and Property of the Laplacian matrix. Remark 4: By using Property 4, condition (30) can be link ed to the edge connecti vity as follo ws )1 cos (31)

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Remark 5: Corollary links the stability of the distrib uted controller to the size of the second smallest eigen alues of the graph Laplacian. It is well kno wn that graphs with lar ge (with respect to the maximal de gree) ha some properties which mak them ery useful in se eral applications such as computer science. Interestingly enough, this property is sho wn here to be crucial also for the design of distrib uted controllers. refer the reader to [23] for more detailed discussion on the importance of the second lar gest eigen alue of Laplacian. Cor ollary 4: Compute in (26) as aI If bd max then the closed loop system (27) is asymptotically stable when is constructed as in (26). In addition, if Condition holds, then the closed loop system (27) is asymptotically stable if bd max Pr oof: Notice that min b max )) bd max The proof is direct consequence of Theorems and and Property of the adjacenc matrix. Consider weighted adjacenc matrix dened as follo ws. Denote by i;j its i; element, then i;j if and i; and i;j ij if i; i; Assume ij Dene max as max max ij Cor ollary 5: Compute in (26) as aI If max then the closed loop system (27) is asymptotically stable when is constructed as in (26). In addition, if Condition holds, then the closed loop system (27) is asymptotically stable if max Pr oof: max and by Perron-Frobenius The- orem max max Notice that min max )) max then the proof is direct consequence of Theorems and 4. The results of Corollaries 3-5 are summarized in able I. Choice of S.C. S.C. if Cond. Holds aL aI bd max bd max aI max max ABLE Corollaries 3-5 present three choices of distrib uted control design with increasing de grees of freedom. In act, and ij are additional parameters which, together with and can be used to tune the closed-loop system beha vior recall here that from Theorem 3, the eigen alues of the closed-loop lar ge-scale system are related to the eigen alues of through the simple relation (28). Thus as long as the stability conditions dened in able are satised, the erall system architecture can be modied arbitrarily by adding or remo ving subsystems and interconnection links. This leads to ery po werful modular approach for designing distrib uted control systems. [1] S. ang and E. J. Da vison, “On the stabilization of decentralized control systems, IEEE ans. utomatic Contr ol ol. 18, no. 5, pp. 473–478, 1973. [2] R. D'Andrea and G. E. Dullerud, “Distrib uted control design for spatially interconnected systems, IEEE ans. utomatic Contr ol ol. 48, no. 9, pp. 1478–1495, Sept. 2003. [3] E. Camponog ara, D. Jia, B. H. Krogh, and S. alukdar “Distrib uted model predicti control, IEEE Contr ol Systems Ma gazine Feb 2002. [4] J. A. ax and R. M. Murray “Graph laplacians and stabilization of ehicle formations, in Pr oc. 2002 IF orld Congr ess, Bar celona, Spain June 2002. [5] M. R. Jo ano vi c, “On the optimality of localized distrib uted con- trollers, in Pr oc. 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[11] ——, “Decentralized receding horizon control for lar ge scale dynam- ically decoupled systems, utomatica to appear 2006. [12] B. Dunbar and R. M. Murray “Receding horizon control of multi- ehicle formations: distrib uted implementation, in Pr oc. 43r IEEE Conf on Decision and Contr ol 2004. [13] A. Richards and J. Ho decentralized algorithm for rob ust constrained model predicti control, in Pr oc. American Contr Conf 2004. [14] Borrelli, viczk G. J. Balas, G. Ste art, K. Fre gene, and D. Godbole, “Hybrid decentralized control of lar ge scale systems, in Hybrid Systems: Computation and Contr ol ser Lecture Notes in Computer Science, ol. 3414. Springer erlag, Mar 2005, pp. 168 183. [15] D. Stipano vic, G. Inalhan, R. eo, and C. J. omlin, “Decentralized erlapping control of formation of unmanned aerial ehicles, utomatica ol. 40, no. 8, pp. 1285–1296, 2004. [16] Borrelli and viczk “Distrib uted lqr design for identical dynamically decoupled systems, Uni ersit del Sannio, Bene ento, Italy ech. Rep. TR390, Semptember 2006. [Online]. ailable: http://www .grace.ing.unisannio.it/publication/390 [17] Gupta, B. Hassibi, and R. M. Murray “On the synthesis of control la ws for netw ork of autonomous agents, in Pr oc. American Contr Conf 2004. [18] viczk Borrelli, G. J. Balas, K. Fre gene, and D. Godbole, “De- centralized receding horizon control and coordination of autonomous ehicle teams, IEEE ansaction Contr ol System ec hnolo gy to appear 2006. [19] B. D. O. Andreson and J. B. Moore, Optimal Contr ol: Linear Quadr atic Methods Engle ood Clif fs, N.J.: Prentice Hall, 1990. [20] M. G. Safono and M. Athans, “Gain and phase mar gin for multiloop LQG re gulators, IEEE ans. utomatic Contr ol ol. C-22, no. 2, pp. 173–179, Apr 1977. [21] C. Scherer and S. eiland, Linear Matrix Inequalities in Contr ol 2000, ersion 3.0. [22] R. Merris, “Laplacian matrices of graphs: surv Linear Alg ebr and Its Applications ol. 197, pp. 143–176, 1994. [23] B. Mohar “The laplacian spectrum of graphs, Gr aph Theory Com- binatorics, and Applications ol. 2, pp. 871–898, 1991. [24] R. Ag ae and Chebotare “On the spectra of nonsymmetric laplacian matrices, Linear Alg ebr and Its Applications ol. 399, no. 5, pp. 157–168, 2005. [25] B. Bollob as, Modern Gr aph Theory Springer 2002. [26] Online, “www .aem.umn.edu/˜k viczk y/Simulations.html, 2005.