Abstract

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.