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

TL;DR

This paper reviews the evolution of dissipativity and reciprocity in systems theory, highlighting recent advances in understanding passive and non-controllable systems with applications in network synthesis and control.

Contribution

It presents recent developments in dissipativity and reciprocity for systems lacking controllability and traditional input-output structure, with practical examples in electrical and mechanical networks.

Findings

Advances in dissipativity theory for non-controllable systems

Application of reciprocity concepts to passive networks

Illustrative examples in electrical and mechanical systems

Abstract

The dissipativity concept sits at the intersection of physics, systems theory, and control engineering, as a natural generalisation 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 cyber-physical systems), robustness, and the absolute stability problem, and network synthesis (of electrical, mechanical, and multi-physics systems). 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.