An algorithm for constructing nonsmooth Lyapunov functions for continuous nonlinear time invariant systems

Abstract

This paper presents an algorithm based on the Generalized Lyapunov Theorem (GLT) for constructing nonsmooth Lyapunov Function (LF) for nonlinear time invariant continuous dynamical systems which can be differentiable almost every-where. A new method is firstly defined that a neighborhood of the equilibrium point (origin) is partitioned into several regions by means of the coordinate hyperplans (axes) and system state equations (nullclines); hence, the number of regions is a function of number of system states. Then, this method selects a LF in each region by original nonlinear model of system, based on the several proposed analytical Notes. These Notes select LF’s and solve continuity problem of them on the boundaries of regions in more cases. The existing methods that use piecewise model of system in each region for constructing piecewise LF are approximate and computational, but, the defined method is completely exact and analytic. The different steps of this method are proposed by means of a non-iterative algorithm for constructing a nonsmooth continuous Generalized Lyapunov Function (GLF) in whole neighborhood of the origin. The ability of this algorithm is demonstrated via a few examples for constructing LF and analyzing system stability.