Demonstrating Passivity and Dissipativity using Computational Methods

Passivity and dissipativity are energy based properties of dynamical systems that may be used for the analysis and synthesis of linear and nonlinear systems. The two properties provide valuable stability results as well as compositional results for the analysis of interconnected systems. Using both the stability and compositionality results, large scale systems can be determined to be stable by analyzing the components in terms of energy dissipation and then sequentially analyzing the system interconnections. One of the drawbacks of this approach is that demonstrating that a system is passive or dissipative typically requires finding an energy storage function, which is analogous to a Lyapunov function. As with Lyapunov stability, the search for a storage function to show dissipativity is in general an open-ended search. This paper surveys computational methods for finding energy storage functions. This includes linear matrix inequality (LMI) methods for linear systems and sum of squares (SOS) methods for polynomial nonlinear systems. When these methods are applicable, the search for storage functions can be automated to greatly simplify analysis and synthesis of linear and nonlinear systems. New material is provided on the application of these methods to find passivity indices for dynamical systems. Additional material is provided on using SOS methods to demonstrate dissipativity for switched systems. Examples are provided to illustrate how these methods may be used in practice.