The Generation of Liapunov Functions

Abstract

This chapter presents a variety of ways of generating Liapunov functions to determine a region of asymptotic stability. It discusses methods by which the second method of Liapunov may be used to give an estimate of the region of stability. The second method of Liapunov may be broadly classified as one of the modern approaches to the solution of stability problems in automatic control. The second method is the theoretical justification for the usual procedure of linearization about an operating point. The chapter reviews the study of regions of asymptotic stability for autonomous, nonlinear, and control systems. It discusses the actual generation of Liapunov functions for autonomous systems that includes five major divisions. The chapter outlines the procedure developed by Szego and this is shown to extend and complement the Aizerman approach. It explains the variable-gradient method of generating Liapunov functions.