A Partial Order Approach to Decentralized Control by Parikshit Shah B.Tech., Indian Institute of Technology, Bombay (2003) S.M., Stanford University (2005) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2011 !c Massachusetts Institute of Technology 2011. All rights reserved. Author............................................. ................ Department of Electrical Engineering and Computer Science April 29, 2011 Certifiedby......................................... ................ Pablo A. Parrilo Professor Thesis Supervisor Acceptedby......................................... ................ Leslie A. Kolodziejski Chairman, Department Committee on Graduate Students 2 A Partial Order Approach to Decentralized Control by Parikshit Shah Submitted to the Department of Electrical Engineering and Computer Science on April 29, 2011, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this thesis we consider the problem of decentralized control of linear systems. We em- ploy the theory of partially ordered sets (posets) to model and analyze a class of decen- tralized control problems. Posets have attractive combinatorial and algebraic properties; the combinatorial structure enables us to model a rich class of communication structures in systems, and the algebraic structure allows us to reparametrize optimal control problems to convex problems. Building on this approach, we develop a state-space solution to the problem of design- ing H2-optimal controllers. Our solution is based on the exploitation of a key separability property of the problem that enables an efficient computation of the optimal controller by solving a small number of uncoupled standard Riccati equations. Our approach gives im- portant insight into the structure of optimal controllers, such as controller degree bounds that depend on the structure of the poset. A novel element in our state-space characteriza- tion of the controller is a pair of transfer functions, that belong to the incidence algebra of the poset, are inverses of each other, and are intimately related to estimation of the state along the different paths in the poset. We then view the control design problem from an architectural viewpoint. We propose a natural architecture for poset-causal controllers. In the process, we establish interesting connections between concepts from order theory such as M¨obius inversion and control- theoretic concepts such as state estimation, innovation, and separability principles. Finally, we prove that the H2-optimal controller in fact posseses the proposed controller structure, thereby proving the optimality of the architecture. Thesis Supervisor: Pablo A. Parrilo Title: Professor 3 4 Acknowledgments My foremost gratitude is to my advisor, Pablo Parrilo. Pablo shows by example what it means to be a truly great researcher. The enthusiasm that he brings to research is boundless and infectious, and it creates an exciting environment for learning and research. Apart from the many technical skills that I learnt from Pablo, I have been the recipient of much patience, kindness, encouragement and support from him, and for all these things I both thank and admire him. I am also deeply grateful to my committeemembers, Sanjoy Mitter and Munther Dahleh for taking the time to understand my research and sharing their insights and ideas with me. I have learnt a great deal from the professors at LIDS and I thank them all for being such excellent teachers. I would especially like to mention Asuman Ozdaglar, Devavrat Shah, Alan Willsky, and my academic advisor Prof. Bertsekas for fruitful discussions both within and beyond the classroom. Perhaps far more important than the knowledge acquired in classes, was that acquired through interactions with my colleagues at LIDS. Through my almost daily discussions and coffee breaks with Mesrob Ohanessian, I have learnt an incredible amount, and I am very grateful for it. I have similarly benefitted greatly from the many intellectual discussions with Venkat Chandrasekaran and collaborations with Christian Ebenbauer. I thank them for their time and friendship. My officemates, Amir Ali Ahmadi, Noah Stein, and Sidhant Misra have provided a great working environment, good cheer and many stimulating dis- cussions, and I am deeply grateful to them for that. I also thank other members of our research group; James Saunderson, Ozan Candogan and Dan Iancu for many discussions about research from which I learnt a lot. During my stay in MIT, I have been fortunate to acquire many lifelong friends. Among them I would like to especially mention Vikrant Vaze, Vaibhav Rathi, Raghavendra Hosur, Anshul Mohnot, Harshad Kasture, Kushal Kedia, Deep Ghosh, Ahmed Kirmani, Aliaa Atwi, Shashi Borade, Sujay Sanghavi, Amit Bhatia, Mukul Agarwal, Srikanth Jagabathula, and Sachin Katti. I am deeply grateful for their friendship. My time in the Boston area would not be the same without my friends from way back: Gaurav Mehta, Anuj Shroff, Ronak Mehta, Gaurav Tanna, Komal Sane, Kavita Shah, Kaushal Sanghai and Risheen Reejhsinghani. I thank also my aunts and uncles in Boston, Sujata and Himanshu Shah, and Charu and Chandrahas Shah. They have always provided a welcome home-away-from-home, I thank them for their love and support. I would also like to mention two very special people in my life: my sister Kaushambi for her love and support through the years, and for always being there, and my fianc`ee Shonan for her love and patience during the last year of my PhD and for her solid support and belief in me. It is difficult to express in words my gratitude towards them. Finally, none of this would be possible without the unconditional love, support and belief from my parents Shivani and Mayank Shah. They both encouraged me to dream and taught me to follow my dreams. I dedicate this thesis to them. 5 6 Contents 1 Introduction 15 1.1 Decentralized versus Centralized Control . 15 1.2 InformationFlowinSystems . 16 1.2.1 Poset-CausalInformationFlows . 18 1.3 ComputationalConsiderations . 19 1.4 MainContributions .............................. 20 1.5 RelatedWork ................................. 22 1.6 OrganizationofThesis . .. .. .. .. .. .. .. .. 24 2 Background 27 2.1 OrderTheoreticBackground . 27 2.1.1 Partially Ordered Sets and Incidence Algebras . 27 2.1.2 QuosetsandStructuralMatrixAlgebras . 31 2.1.3 GaloisConnections. 32 2.2 ControlTheoreticBackground . 33 2.2.1 FiniteDimensionalSystems . 33 2.2.2 YoulaParametrization . 36 2.2.3 QuadraticInvariance . 38 2.2.4 StateSpaceRealizations . 39 2.2.5 SystemswithTimeDelays . 40 7 2.2.6 SpatiallyInvariantSystems. 40 3 Poset-Causal Systems, Convexity, and the Youla Parameterization 43 3.1 ControlofPosetCausalSystems . 46 3.1.1 Examples of Communication Structures Arising from Posets . 47 3.2 Systems with Same Plant and Controller Communication Constraints. 49 3.3 Systemswithdifferent plant and controller communication constraints . 57 3.3.1 ModelingviaGaloisConnections . 57 3.4 SystemswithTimeDelays . .. .. .. .. .. .. .. 62 3.5 SpatiallyInvariantSystems . 65 3.5.1 RelationtoFunnelCausality . 71 3.5.2 ExamplesofPosetCausalSystems . 75 3.6 Quadratic Invariance and Poset Structure . 78 3.6.1 ExistenceofPosets. .. .. .. .. .. .. .. 80 3.6.2 ExistenceofQuosets . 81 3.7 Conclusion .................................. 83 4 H2 Optimal Control over Posets 85 4.1 Introduction.................................. 85 4.2 Preliminaries ................................. 87 4.2.1 ControlTheoreticPreliminaries . 88 4.3 SolutionStrategy ............................... 95 4.3.1 Reparametrization . 95 4.3.2 SeparabilityofOptimalControlProblem . 98 4.4 MainResults ................................. 99 4.4.1 Problem Decomposition and Computational Procedure . 99 4.4.2 StructureoftheOptimalController . 102 4.4.3 Interpretation of Φ and Γ .......................104 8 4.4.4 RoleofControllerStatesandtheClosedLoop . 109 4.4.5 ASeparationPrinciple . .112 4.5 DiscussionandExamples. 113 4.5.1 TheNestedCase ...........................113 4.5.2 Discussion Regarding Computational Complexity . 115 4.5.3 DiscussionRegardingDegreeBounds . 115 4.5.4 NumericalExample . .116 4.6 ProofsoftheMainResults . 120 4.7 Conclusions..................................131 5 A General Controller Architecture 133 5.1 Introduction..................................133 5.2 Preliminaries .................................135 5.2.1 Notation ...............................138 5.3 IngredientsoftheArchitecture . 138 5.3.1 LocalVariablesandLocalProducts . 139 5.3.2 TheM¨obiusandZetaoperators . .142 5.4 ProposedArchitecture . .. .. .. .. .. .. .. .. 146 5.4.1 LocalStatesandLocalInputs . .146 5.4.2 ControlLaw .............................149 5.4.3 StatePrediction. .151 5.4.4 SeparationPrinciple . .153 5.4.5 ControllerRealization . .155 5.4.6 StructureoftheOptimalController . 156 5.4.7 Φ and Γ revisited ...........................158 5.5 ABlock-DiagramInterpretation . 160 5.6 Conclusions..................................167 9 6 Conclusions 169 6.1 Posets,Decentralization,andComputation . ........169 6.2 FutureDirections ...............................170 10 List of Figures 1-1 Anexampleofadecentralizedsystem. 19 2-1 Aposetontheset {a, b, c}. .......................... 28 2-2 A quoset and the sparsity pattern of its associated structural matrix algebra. Elements with a ’∗’indicatepossiblenonzeroelements. 32 2-3 ApairofposetsrelatedbyaGaloisconnection.