This talk will present a novel approach for the analysis and design of systems that switch and oscillate. While such nonlinear behaviors abound in control engineering of electrical, mechanical, and biological circuits, it is often considered that they largely fall outside the scope of control theory. In contrast, the proposed approach closely mimics linear-quadratic dissipativity theory, a very foundation of modern control theory. In its classical formulation, dissipativity theory formulates system properties as dissipation inequalities to be satisfied by the storage, an abstraction of the system internal energy. Linear systems admit quadratic storages. When the storage is positive definite, it serves as a Lyapunov function for stability analysis of equilibria. Our generalization rests on two distinct ingredients. First, we apply dissipativity theory differentially: instead of studying the nonlinear system via the nonlinear theory, we apply the linear theory to a family of linearized systems. Second, we relax the positivity constraint of the quadratic storage to a fixed inertia constraint. We allow for one negative eigenvalue in the analysis of switches and two negative eigenvalues in the analysis of clocks. The talk will illustrate the theory in classical models of switches and clocks and discuss the potential of dissipativity theory for the analysis and design of interconnected systems away from equilibrium.
Rodolphe Sepulchre is Professor of Engineering at Cambridge University and a fellow of Sidney Sussex College. His research interests are in dynamics, control and optimization of nonlinear problems. He is a fellow of IEEE and SIAM. He has been Editor-in-Chief of Systems and Control Letters (2009-2018) and an Associate Editor for SIAM Journal of Control and Optimization, the Journal of Nonlinear Science, and Mathematics for Control, Signals, and Systems. In 2008, he was awarded the IEEE Control Systems Society Antonio Ruberti Young Researcher Prize. He has been IEEE CSS distinguished lecturer between 2010 and 2015. A focus of his current research is the ERC advanced grant Switchlets, motivated by neurophysiological questions and aiming at a multiresolution control theory of excitable systems. He co-authored the monographs Constructive Nonlinear Control (Springer-Verlag, 1997) and ?Optimization on Matrix Manifolds? (Princeton University Press, 2008).