Dissipativity, reciprocity and passive network synthesis: from the seminal Dissipative Dynamical Systems articles of Jan Willems to the present day

Abstract

The dissipativity concept sits at the intersection of physics, systems theory, and control engineering as a natural generalization of passive systems that dissipate energy. It relates the external behavior of systems to their internal state and connects the subjects of optimal control, algebraic Riccati equations, linear matrix inequalities, complex functions, and spectral factorization. Within control, its applications include the analysis and design of interconnected systems (such as cyberphysical systems), robustness, and the absolute stability problem as well as network synthesis (of electrical, mechanical, and multiphysics systems). Dissipativity emerged as a stand-alone concept following the seminal “Dissipative Dynamical Systems” articles of Jan Willems, which drew on the work of Kalman, Youla, Anderson, Yakubovich, Popov, and others. This article details recent developments in the treatment of dissipativity and the related concept of reciprocity for systems that are not necessarily controllable and need not lend themselves naturally to an input–state–output perspective, as is the case for many physical and passive systems. We illustrate these concepts using simple electric circuit and mechanical network examples. We also draw inspiration from the behavioral theory developed by Jan Willems and collaborators, a natural formalism for analyzing physical systems that need not be controllable and characterized in terms of inputs and outputs.