Networked Control of Multi-Agent Systems: Consensus and synchronisation, Communication structure design, Self-organisation in networked systems, Event-triggered control

During the last decade, networked control systems have developed as a new area of control theory, which studies a set of physical components and control systems that are connected by a digital communication network. Potential application areas are multi-vehicle systems (e.g. platoons on highways) or power grids. There are two new challenges of this theory. First, the data network provides flexible communication means in which every component in a control loop may send information to any other component. The methodological problem is to elaborate systematic ways for the selection of information structures that are suitable for a given control task. Second, as many networks used for control purposes act simultaneously as communication medium for other tasks, time delays in the delivery of the information or packet loss regularly occur and the control systems have to be made tolerant against such imperfections. This is the first textbook about networked control that gives a thorough introduction to the graph-theoretical and systems-theoretical foundations and combines both to get systematic ways for analysing networked control systems and for choosing the communication structure of the controller. It starts with consensus and synchronisation as the classical problems of the field and continues with the communication structure design, self-organisation and event-based control, which are novel problems for which solutions have been elaborated recently. The common methodological foundation for the solution of these problems is network thinking, which means that the overall system is seen as a compound of coupled subsystems which gets its dynamics from the subsystem properties and from the interconnection structure. The main problem is to bring graph theory and system theory together to investigate which collective behaviour results from the interactions of subsystems with specific properties. The book concentrates on multi-agent systems, which are physical subsystems that do not have any physical couplings, but have to satisfy common control goals. Cooperative controllers have to be found that introduce couplings among the controlled agents. This situation leads to the general problem of the book to find methods for the selection of the information structure of networked control. The main difference between the scope of this textbook and the contents of several monographs on networked control systems lies in the emphasis on the communication structure design. The competing books, which are characterised below, concentrate on the classical problem of making networked controllers tolerant to imperfections of the communication (packet loss, transmission delay, bandwidth limitations etc.).

Table of Contents:
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Application examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv 1 Introduction to networked control systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Networked systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Networked control systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Survey of the control methods tought in these lecture notes . . . . . . . . . . . . . . . . . 16 2 Algebraic graph theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 Matrix representation of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3 Analysis of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3 Consensus in multi-agent systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.1 Consensus problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Consensus in networked systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3 Specific multi-agent systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.4 Discrete-time consensus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.5 Consensus over switching networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4 Synchronisation of multi-agent systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.1 Synchronisation problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.3 Asymptotic synchronisation of identical agents . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.4 Synchronisable agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.5 Example: Synchronising oscillator networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.6 Synchronisation of agents with individual dynamics . . . . . . . . . . . . . . . . . . . . . . 160 4.7 Synchronisation of multi-agent systems by means of dynamical networked controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4.8 Synchronisation of Kuramoto oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 4.9 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 5 Design of the communication structure of networked controllers . . . . . . . . . . . . . 233 5.1 Control aims and design steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 5.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 5.3 Delay measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 5.4 Example: Distance control in vehicle platoons . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 6 Random graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 6.1 Sparse graphs with random edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 6.2 Some further notions of graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 6.3 Erdös-Renyi graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 6.4 Results from Network Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 7 Consensus with random communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 7.1 Stochastic convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 7.2 Stochastic stability of dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 7.3 Random agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 8 Self-organisation in networked systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 8.1 The idea of self-organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 8.2 Self-connecting networked systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 8.3 Small-world architecture of networked systems . . . . . . . . . . . . . . . . . . . . . . . . . . 409 8.4 Self-organised disturbance attenuation in multi-agent systems . . . . . . . . . . . . . . 420 8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 9 Event-triggered control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 9.1 The main idea of event-triggered control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 9.2 Event-triggered stabilisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 9.3 Disturbance attenuation by event-triggered state feedback . . . . . . . . . . . . . . . . . . 462 9.4 Event-triggered control of interconnected systems . . . . . . . . . . . . . . . . . . . . . . . . 480 9.5 Event-triggered synchronisation of multi-agent systems . . . . . . . . . . . . . . . . . . . 488 9.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 Appendix 1: Solutions of the exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 Appendix 2: Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 A2.1 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 A2.2 Inverse and pseudoinverse matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 A2.3 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 A2.4 Specific matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 A2.5 Stability analysis of complex-valued matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 A2.6 Kronecker product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 Contents ix Appendix 3: Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 A3.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 A3.2 Further system properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 A3.3 Comparison principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 Appendix 4: Probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 A4.1 Probability space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 A4.2 Expectation and variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 A4.3 Random sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 Appendix 5: MATLAB functions for graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 A5.1 Matrix functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 A5.2 Graph objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 A5.3 Graph algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 Appendix 6: English–German dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 Index . . . . . . . . . . . . . . . . . . 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