Control Lyapunov Functions

Abstract

The present chapter is centered around the CLF paradigm that underlies the principal feedback design methods such as Bellman dynamic programming in optimal control and nonlinear H∞ approach in the robust synthesis. CLFs are first illustrated with quadratic forms, which result in a simple LMI criterion of the asymptotic stability of linear systems. Similar LMI-based conditions are then derived for homogeneous systems by using generalized polynomial forms. After that, the CLF approach is applied to describe the well-known Lyapunov minmax and speed gradient methods, and multiple Lyapunov functions are additionally constructed for VSS based on the Lyapunov gradient PDE. Finally, appropriate solutions of Hamilton–Jacobi PDEs are involved to specify CLFs for achieving a prescribed closed-loop performance of uncertain VSS with resets.