Polyhedral Lyapunov Functions with Fixed Complexity

TL;DR

This paper introduces a linear programming-based algorithm to find fixed-complexity polyhedral Lyapunov functions, enabling efficient stability analysis and control synthesis for linear systems without increasing computational costs.

Contribution

The paper presents a novel fixed-size polyhedral Lyapunov function algorithm that limits complexity while ensuring stability, improving computational efficiency over traditional methods.

Findings

Algorithm successfully finds polyhedral Lyapunov functions with fixed complexity.

Method handles additional linear constraints and objectives.

Enables control synthesis using fixed-complexity Lyapunov functions.

Abstract

Polyhedral Lyapunov functions can approximate any norm arbitrarily well. Because of this, they are used to study the stability of linear time varying and linear parameter varying systems without being conservative. However, the computational cost associated with using them grows unbounded as the size of their representation increases. Finding them is also a hard computational problem. Here we present an algorithm that attempts to find polyhedral functions while keeping the size of the representation fixed, to limit computational costs. We do this by measuring the gap from contraction for a given polyhedral set. The solution is then used to find perturbations on the polyhedral set that reduce the contraction gap. The process is repeated until a valid polyhedral Lyapunov function is obtained. The approach is rooted in linear programming. This leads to a flexible method capable of handling additional linear constraints and objectives, and enables the use of the algorithm for control synthesis.