Local Dissipativity Analysis of Nonlinear Systems

TL;DR

This paper introduces a convex optimization-based method to determine local dissipativity properties of nonlinear control systems, enabling better analysis and control synthesis through piecewise affine storage functions.

Contribution

It presents a novel approach to synthesize local dissipativity conditions and storage functions for nonlinear systems using convex optimization and matrix inequalities.

Findings

Successfully determines conic bounds and gains of nonlinear systems.

Always finds a feasible IO characterization for sufficiently strictly locally dissipative systems.

Demonstrates effectiveness on various nonlinear system examples.

Abstract

Dissipativity is an input-output (IO) characterization of nonlinear systems that enables compositional robust control through Vidyasagar's Network Dissipativity Theorem (VDNT). However, determining the dissipativity of a system is an involved and, often, model-specific process. We present a general method to determine the local dissipativity properties of smooth, nonlinear, control affine systems. We simultaneously search for the optimal IO characterization of a system and synthesize a continuous piecewise affine (CPA) storage function via a convex optimization problem. To do so, we reformulate the dissipation inequality as a matrix inequality (MI) and develop novel linear matrix inequality (LMI) bounds for a triangulation to impose the dissipativity conditions on the CPA storage function Further, we develop a method to synthesize a combined quadratic and CPA storage function to expand the systems the optimization problem is applicable to. Finally, we establish that our method will always find a feasible IO characterization and storage function given that the system is sufficiently strictly locally dissipative and demonstrate the efficacy of our method in determining the conic bounds and gain of various nonlinear systems.