Stability

Introduction

A basic knowledge of stability of linear systems has been assumed throughout the previous chapters. Stability was associated with the location of the poles of the system in the left half plane. In chapter 9, we saw that the poles are the eigenvalues of the system matrix A when the system is written in state variable form. In chapter 10, we examined the Nyquist criterion for closed-loop stability of a SISO system; we concluded on the stability of the closed-loop system G(1 + G)^− 1 from the number of encirclements of − 1 by the open-loop transfer function G(s). In this chapter, we examine the salient results of Lyapunov’s theory of stability; it is attractive for mechanical systems because of its exceptional physical meaning and its wide ranging applicability, especially for the analysis of nonlinear systems, and also in controller design. We will conclude this chapter with a class of collocated controls that are especially useful in practice, because of their guaranteed stability, even for nonlinear systems; we will call them energy absorbing controls. The following discussion will be restricted to time-invariant systems (also called autonomous), but most of the results can be extended to time-varying systems. As in the previous chapters, most of the general results are stated without proof and the discussion is focused on vibrating mechanical systems; a deeper discussion can be found in the references.