Preface

Existing computational tools for control synthesis and verification do not scale well to today’s large-scale networked systems. Recent advances, such as sum-of-squares relaxations for polynomial nonnegativity, have made it possible to numerically search for Lyapunov functions and to certify measures of performance; however, these procedures are applicable only to problems of modest size. In this book we address networks where the subsystems are amenable to standard analytical and computational methods but the interconnection, taken as a whole, is beyond the reach of these methods. To break up the task of certifying network properties into subproblems of manageable size, we make use of dissipativity properties which serve as abstractions of the detailed dynamical models of the subsystems. We combine these abstractions to derive network level stability, performance, and safety guarantees in a compositional fashion. Dissipativity theory, which is fundamental to our approach, is reviewed in Chap. 1 and enriched with sum-of-squares and semidefinite programming techniques, detailed in Appendices A and B respectively. Chapter 2 derives a stability test for interconnected systems from the dissipativity characteristics of the subsystems. This approach is particularly powerful when one exploits the structure of the interconnection and identifies subsystem dissipativity properties favored by the type of interconnection. We exhibit several such interconnections that are of practical importance, as subsequently demonstrated in Chap. 4 with case studies from biological networks, multiagent systems, and Internet congestion control. Before proceeding to the case studies, however, in Chap. 3 we point out an obstacle to analyzing subsystems independently of each other: the dissipativity properties must be referenced to the network equilibrium point which depends on all other subsystems. To remove this obstacle we introduce the stronger notion of equilibrium independent dissipativity, which requires dissipativity with respect to any point that has the potential to become an equilibrium in an interconnection. In Chap. 5 we extend the compositional stability analysis tools to performance and safety certification. Performance is defined as a desired dissipativity property