On the construction of piecewise linear Lyapunov functions

Abstract

The issue of constructing piecewise linear Lyapunov functions (PLLFs) for the stability analysis of polytopic uncertain linear systems is considered. PLLF candidates are parametrized by hyperplanes which intersect the given region, and stability conditions are formulated as linear programming (LP) problems in terms of the parameters inserted by the hyperplanes. If the optimal value of the LP problem is negative, then the PLLF is constructed by using the optimal solution. When the optimal value of the LP problem is non-negative, the candidate PLLF is modified by adding new dividing hyperplanes to increase the freedom, and a new LP problem is formulated corresponding to the new PLLF candidate. The main result of this paper is to propose a method generating additional dividing hyperplanes which strictly decrease the optimal values of the LP problems which appear in constructing PLLFs. An example is used to illustrate the obtained results.