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From Static to Dynamic Couplings in Consensus and Synchronization Among Identical and Non-Identical Systems

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From Static to Dynamic Couplings in Consensus and Synchronization Among Identical and Non-Identical Systems From Static to Dynamic Couplings in Consensus and Synchronization among Identical and Non-Identical Systems Von der Fakult¨at Konstruktions-, Produktions-, und Fahrzeugtechnik der Universit¨at Stuttgart zur Erlangung der Wurde¨ eines Doktors der Ingenieurwissenschaften (Dr.–Ing.) genehmigte Abhandlung Vorgelegt von Peter Wieland aus Stuttgart–Bad Cannstatt Hauptberichter: Prof. Dr.–Ing. Frank Allg¨ower Mitberichter: Prof. Dr. Alberto Isidori Prof. Dr. Rodolphe Sepulchre Tag der mundlichen¨ Prufung:¨ 06.09.2010 Institut fur¨ Systemtheorie und Regelungstechnik Universit¨at Stuttgart 2010 F¨ur Melanie, Lea und Anna Acknowledgments The results presented in this thesis are the outcome of the research I performed during my time as a research assistant at the Institute for Systems Theory and Automatic Control (IST), University of Stuttgart in the years 2005 to 2010. Throughout this exciting journey I had the chance to meet and interact with many interesting and inspiring people. The present thesis was positively influenced by many of them and I wish to convey my sincere gratefulness to them. First, I want to express my gratitude towards my supervisor, Prof. Frank Allg¨ower. He endowed me with lots of freedom in my research and offered me the unique opportunity to be part of a very open-minded, internationally renowned research group. I also want to thank Prof. Rodolphe Sepulchre, with whom I had the chance to have a very fruitful interaction during the last year of my thesis, for sharing his knowledge and insight with me in a very motivating way. Moreover, I am deeply indebted to Prof. Alberto Isidori, Prof. Rodolphe Sepulchre, and Prof. Peter Eberhard for their interest in my research and for accepting to be members of my doctoral exam committee. In addition, I thank Prof. Christian Ebenbauer for accepting to stand in for Prof. Isidori, who could unfortunately not be present at the oral exam. My time as a research assistant became a rich experience by interacting with my col- leagues at the IST. I shared the interest in research on interconnected dynamical systems with Rainer Blind, Jørgen K. Johnsen, Prof. Jung-Su Kim, Ulrich M¨unz, and Gerd S. Schmidt. They contributed with many vivid discussions and useful hints to the results presented in this thesis. Furthermore, I want to thank Tobias Raff, with whom I shared office most of the time, for letting me have part in his experience as a researcher and taking care of me as a novice. In addition to those already mentioned, I wish to thank my colleagues Christoph B¨ohm, Christian Breindl, Jan Hasenauer, Christoph Maier, Marcus Reble, and Simone Schuler as well as Hongkeun Kim from Seoul National University, Korea and Tilman Utz from TU Wien, Austria for their helpful comments on this thesis. Besides those mentioned explicitly, I am indebted to all of my current and former colleagues at the IST for their generous support, uncounted discussions, and lots of fun in academic as well as non-academic interactions. It was a great pleasure being part of this group. Last but not least, I want to thank my parents for their support in all these years, I want to thank Melanie for her love, patience, and encouragements, and I want to apologize to Lea for all the time she was not allowed to disturb me and I was unavailable for playing. Stuttgart, September 2010 Peter Wieland v Contents Symbols and Acronyms xi Abstract xiii Deutsche Kurzfassung xv 1 Introduction 1 1.1 Dimensions of Complexity in Consensus and Synchronization . ... 2 1.2 CouplingMechanisms.............................. 5 1.3 OutlineandContributions ........................... 7 2 Graph Theory for Consensus Problems 9 2.1 Basic Definitions and Link to Diffusive Couplings . 10 2.1.1 GraphConnectedness. 12 2.1.2 TheGraphLaplacian.......................... 14 2.2 Algebraic Graph Properties in Consensus and Synchronization Problems . 15 2.2.1 Connected Components and Algebraic Connectivity for Undirected Graphs.................................. 16 2.2.2 Independent Strongly Connected Components for Directed Graphs. 17 2.3 Summary .................................... 20 3 Consensus with Static Diffusive Couplings 21 3.1 Definition of Consensus and Problem Setup . 23 3.2 Consensus for Fixed Communication Topologies . 25 3.2.1 Integrator Consensus for Fixed Communication Topologies . .. 25 3.2.2 Consensus among Identical Linear Systems with Fixed Communi- cationTopologies ............................ 27 3.3 Consensus for Switching and Time-Varying Communication Topologies . 39 3.4 ConsensuswithLeader ............................. 42 3.4.1 LeaderwithIdenticalDynamics . 43 3.4.2 LeaderwithDifferentDynamics . 43 3.4.3 Connecting the Leader to an Existing Network . 47 3.5 Summary .................................... 48 4 Observer-Based Output Consensus with Relative Output Sensing 51 4.1 ProblemStatement ............................... 52 4.1.1 Dynamic Diffusive Couplings and Output Consensus Trajectories . 52 4.1.2 Limitations of Luenberger Observers in Output Consensus with Rel- ativeOutputSensing .......................... 54 vii Contents 4.2 Consensus with Full Order Unknown-Input Observers . ..... 55 4.3 Consensus with Reduced Order Unknown-Input Observers . ...... 57 4.3.1 Maximal Functional Unknown-Input Observers . 57 4.3.2 Diffusive Couplings with Reduced State Information . 60 4.3.3 Example................................. 62 4.4 Summary .................................... 64 5 The Internal Model Principle for Consensus and Synchronization 67 5.1 TheLinearCase ................................ 68 5.1.1 ProblemSetup ............................. 68 5.1.2 The Internal Model Principle for Synchronization among Linear Systems ................................. 72 5.2 TheNonlinearCase............................... 82 5.2.1 ProblemSetup ............................. 82 5.2.2 The Internal Model Principle for Synchronization among Nonlinear Systems ................................. 85 5.3 Summary .................................... 95 6 Synchronization in Heterogeneous Networks with Dynamic Couplings 97 6.1 Synchronization in Networks of Non-Identical Linear Systems . ....... 98 6.1.1 ProblemStatement ........................... 98 6.1.2 An Internal Model is Necessary and Sufficient for Linear Output Synchronization ............................ 99 6.1.3 Discussion................................102 6.1.4 Example.................................104 6.2 Synchronization of Nonlinear Oscillators . 106 6.2.1 ProblemSetup .............................106 6.2.2 Entrainment of Oscillators by Small Periodic Forcing . 109 6.2.3 Phase Synchronization through Entrainment by a Consensus Input 111 6.2.4 Discussion................................113 6.2.5 Example.................................115 6.3 Summary ....................................117 7 Conclusions 119 7.1 SummaryandDiscussion. .. .. .119 7.2 Outlook .....................................121 A Intermediate Results and Proofs for Chapter 2 125 A.1 Perron-FrobeniusTheorem . 125 A.2 IntermediateResults ..............................126 A.3 Proofs ......................................127 A.3.1 ProofofTheorem2.12 . .127 A.3.2 ProofofTheorem2.13 . .128 B Asymptotic Properties of Ordinary Differential Equations 131 B.1 Linear Systems with Exponentially Decaying Inputs . 131 viii Contents B.2 LimitSets....................................133 B.2.1 AutonomousSystems. .133 B.2.2 Asymptotically Autonomous Systems . 136 C Geometric Concepts from Linear Control Theory 139 C.1 FactorSpaces ..................................139 C.2 Observers, Observability, and Detectability . 139 C.2.1 Observers and Conditionally Invariant Subspaces . 139 C.2.2 UnknownInputObservers . .141 Bibliography 145 ix Symbols and Acronyms Symbols R set of real numbers R+ set of positive real numbers R+ set of non-negative real numbers R set of non-positive real numbers − C set of complex numbers j the imaginary unit (j = √ 1) jR set of imaginary numbers − C+ the closed right-half complex plane C the open left-half complex plane − C the closed left-half complex plane − N the natural numbers N the natural numbers from 1 to k (N = 1,...,k N) k k { } ⊂ Z the integers S1 the circle Re(z) the real part of a complex number z C ∈ Im(z) the imaginary part of a complex number z C z the complex conjugate of z C ∈ ∈ I the n n identity matrix n × 1n the n-dimensional vector of ones the Kronecker product of two matrices (cf. Horn and Johnson, 1991) ⊗ n n σ(A) the spectrum (i.e., set of eigenvalues) of a square matrix A R × n n ∈ or A C × ∈ n n n n λ (A) the kth eigenvalue of a square matrix A R × or A C × with k ∈ ∈ the convention that Re(λk(A)) Re(λk+1(A)), k Nn 1 ≤ n n ∈ − n n ρ(A) the spectral radius of a square matrix A R × or A C × H ∈ ∈n m A the complex conjugate transpose of some matrix A C × + ∈ n m A the Moore-Penrose generalized inverse of some matrix A R × n m ∈ + or C × (cf. Horn and Johnson, 1985); if rank(A) = n, A = T T 1 + A (AA )− is a right-inverse of A; if rank(A) = m, A = T 1 T (A A)− A is a left-inverse of A ker(A) the kernel of a matrix or linear map A im(A) the image of a matrix or linear map A span(v ,...,v ) the subspace of Rn spanned by the vectors v Rn, k N 1 N k ∈ ∈ N xi Symbols and Acronyms diag(x) foravector x Rn the n n diagonal matrix with the elements of ∈ × x on the diagonal / the factor space of equivalence classes x + , x for a linear X Y vector space and a subspace (see AppendixY ∈ X C) X Y ⊂ X A/ the map induced by the linear map A : on the factor space Y / with A , defined as x + X(Ax →)+ X (see Appendix C) X Y Y⊂Y
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