Construction of Piecewise Linear Lyapunov Functions and Its Fast Method

Abstract

In this paper, the issue of constructing piecewise linear Lyapunov functions (PWLLF) for stability analysis of nonlinear systems satisfying generalized sector conditions is addressed. The existence condition of PWLLF is represented by a linear programming problem (LP in brief). If the optimal value of the LP is negative, then PWLLF is constructed by using the optimal solution of it. When the optimal value of the LP is nonnegative, the candidate of PWLLF is modified with additional freedom and a new LP is formulated corresponding to the new PWLLF candidiate. It is shown that the optimal value of the resulting new LP is always less than or equal to that of the old LP. The main purpose of this paper is to propose a fast method to solve LPs which appear repeatedly in constructing PWLLF. The method is based on a kind of sensitivity analysis approach of LP and utilizes the special structure of LPs that appears in constructing PWLLF.