Linear Vector Norms as Lyapunov Functions

Abstract

This chapter continues on the results proposed in [Pol95, KAS92] for computing vector 1- or ∞-norm based Lyapunov functions for linear discretetime systems by presenting an algorithmwith guaranteed finite termination for the construction of Lyapunov functions of that class. The algorithm utilizes the solution of a finite sequence of feasibility LPs with very few constraints or, equivalently, very simple algebraic tests in contrast to solving the original problem where a bilinear matrix equation with rank and norm constraint needs to be solved. Such Lyapunov functions are utilized for a priori imposing closed-loop stability and feasibility for all time when using a receding horizon control policy. A simple algorithm for this purpose for a class of piecewise affine systems is presented.