لایب سان

vBackground During the past years, distributed coordination for networked dynamical systems has been an active area of research. This is due to the broad applications in such areas as sensor networks, smart grids, rendezvous tasks, swarming and flocking, and congestion control in communication networks. One of the critical and central problems in distributed coordination is to design a network consensus algorithm (or protocol) such that a group of agents 1 can reach consensus on the shared information. By reaching consensus, we mean that a network of agents agree on certain variables of interest, such as position, opinion or decision depending on the context. To realize consensus, each agent exchanges information with its nearest neighbors locally. In essence, such a protocol or algorithm propagates information in a diffusive way by taking weighted sums of state or output deviations as inputs. The distributed consensus protocol has been proven to work efficiently to drive all agents to reach agreement and is adopted in this book as well. In general, three factors play important roles in determining the evolution of net- worked dynamical agents, in particular whether consensus among the agents can be realized. They are (1) self-dynamics of each agent, (2) inner coupling describing how input influences state evolution, and (3) the interaction topology among the whole agents. [Refer to the distributed coordination scheme in Fig. 1.1.] It is clear that for single-integrators no self-dynamics exists. In addition, the inner coupling is an Iden- tity map. Hence, the interaction topology itself characterizes the evolution of agents. Following such an idea, the consensus for a network of single-integrators has been investigated from various perspectives. Of particular interest is the consensus over switching interaction topology, see [1–12] for example. This is motivated by the fact that in many scenarios networked systems can possess a dynamic interaction topol- ogy due to node and link failures/creations, packet-loss, asynchronous consensus, state-dependence, formation reconfiguration, evolution, and flocking.

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Outline and Overview of the Chapters . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 Digraph and Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 Dynamically Changing Digraph . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.3 Nonnegative Matrix and M-Matrix Theory . . . . . . . . . . . . . . 12 1.3.4 Kronecker Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.5 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Consensus of Double-Integrators over Switching Network Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Continuous-Time Systems with Piecewise Fixed Network Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.1 Agent Model Without/with Communication Delay . . . . . . . . 18 2.1.2 Stationary Consensus without Communication Delay . . . . . 19 2.1.3 Stationary Consensus with Communication Delay . . . . . . . . 24 2.1.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2 Sampled-Data Systems with Piecewise Continuous Network Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.1 Sampled-Data Agent Model with Nonuniform and Time-Varying Communication Delays . . . . . . . . . . . . . . . 36 2.2.2 Graph Augmentation and Joint Connectivity . . . . . . . . . . . . . 37 2.2.3 Consensus via Product of Stochastic Matrices . . . . . . . . . . . . 41 2.2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.2.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3 Asynchronous Update Systems with Piecewise Continuous Network Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3.1 Asynchronous Update Agent Model . . . . . . . . . . . . . . . . . . . . 48 ix x Contents 2.3.2 Consensus for Asynchronous Update . . . . . . . . . . . . . . . . . . . 51 2.3.3 Consensus Analysis via Augmented Synchronous Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3 Generic Linear Systems over Undirected Network: Controllability and Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1 Piecewise Continuous and Balanced Network Topology . . . . . . . . . . 63 3.1.1 Network Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.1.2 Weakly Connected Network Topology (WCNT) . . . . . . . . . . 65 3.1.3 WCNT with Communication Constraints . . . . . . . . . . . . . . . . 69 3.1.4 Repeatedly Jointly WCNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.1.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.1.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.2 Piecewise Fixed Network Topology: Necessary and Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.2.1 Problem Reformulation and Preliminary Results . . . . . . . . . . 84 3.2.2 A Subspace Method for Consensus Analysis . . . . . . . . . . . . . 87 3.2.3 Extension to Unstable System Dynamics . . . . . . . . . . . . . . . . 93 3.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.2.5 Appendix: Technical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.3 Continuously Time-varying Network Topology via Uniform Complete Observability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.3.1 Agent Model and Network Configuration . . . . . . . . . . . . . . . . 102 3.3.2 Uniform Complete Observability . . . . . . . . . . . . . . . . . . . . . . . 104 3.3.3 A Compact Set of Matrix-Valued Functions and Exponential Consensus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.3.4 Further Discussions and Conclusion . . . . . . . . . . . . . . . . . . . . 118 3.3.5 Appendix: Proof of Lemma 3.56 . . . . . . . . . . . . . . . . . . . . . . . 120 4 Generic Linear Systems over Directed Switching Network Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.1 Discrete-Time Systems with Invertible Input Matrix . . . . . . . . . . . . . 123 4.1.1 Agent Model and Network Configuration . . . . . . . . . . . . . . . . 124 4.1.2 Theoretical Results and Convergence Analysis . . . . . . . . . . . 124 4.1.3 Further Analysis and Extensions . . . . . . . . . . . . . . . . . . . . . . . 130 4.1.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.2 Continuous-Time Systems with Invertible Input Matrix . . . . . . . . . . 135 4.2.1 Agent Model and Network Configuration . . . . . . . . . . . . . . . . 136 4.2.2 Theoretical Results and Convergence Analysis . . . . . . . . . . . 136 4.2.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Contents xi 4.3 Non-invertible Input Matrix: an Algebraic and Geometric Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.3.1 System Model and Preliminary Results . . . . . . . . . . . . . . . . . . 148 4.3.2 Convergence Analysis over Any Subinterval Without Connectivity Requirement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 4.3.3 Synchronization Under Joint Connectivity Condition . . . . . . 156 4.3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.3.5 Proof of the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 4.3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4.3.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5 Extensions to Containment Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.1 Collective Behavior of Generic Linear Agents with Arbitrary Network Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.1.1 Continuous-Time Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.1.2 Discrete-Time Counterpart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 5.1.3 Application to Consensus and Containment Control . . . . . . . 192 5.1.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 5.1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.2 Output Containment Control with Fixed and Switching Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.2.1 Agent Model and Problem Definition . . . . . . . . . . . . . . . . . . . 197 5.2.2 A Necessary Condition from Internal Model Principle . . . . . 197 5.2.3 Output Containment over Fixed Network Topology . . . . . . . 200 5.2.4 Output Containment over Dynamic Network . . . . . . . . . . . . . 201 5.2.5 Optimal Model-Free Output Containment . . . . . . . . . . . . . . . 207 5.2.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 5.2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 5.3 Containment Control: a Novel State Transition Matrix Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 5.3.1 Agent Model and Problem Statement . . . . . . . . . . . . . . . . . . . 217 5.3.2 The Consensus Pattern in Multi-Leader MASs . . . . . . . . . . . 219 5.3.3 Containment of Positive Linear Systems . . . . . . . . . . . . . . . . . 225 5.3.4 Second-Order Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 5.3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

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