Systematic Stabilization of Constrained Piecewise Affine Systems

TL;DR

This paper introduces an offline, semi-definite programming-based method for synthesizing controllers and Lyapunov functions for constrained piecewise affine systems, enabling invariant set computation and minimum-norm control.

Contribution

It proposes a novel triangulation-based approach that avoids non-convex optimization, providing efficient controller synthesis and Lyapunov functions for constrained systems.

Findings

Efficient offline controller synthesis via semi-definite programs

Explicit Lipschitz Lyapunov functions for invariant set computation

Enables minimum-norm control through quadratic programming

Abstract

This paper presents an efficient, offline method to simultaneously synthesize controllers and seek closed-loop Lyapunov functions for constrained piecewise affine systems on triangulated subsets of the admissible states. Triangulation refinements explore a rich class of controllers and Lyapunov functions. Since an explicit Lipschitz Lyapunov function is found, an invariant subset of the closed-loop region of attraction is obtained. Moreover, it is a control Lyapunov function, so minimum-norm controllers can be realized through online quadratic programming. It is formulated as a sequence of semi-definite programs. The method avoids computationally burdensome non-convex optimizations and a-priori design choices that are typical of similar existing methods.