Necessary and sufficient conditions for absolute stability: the case of second-order systems

We consider the problem of absolute stability of linear feedback systems in which the control is a sector-bounded time-varying nonlinearity. Absolute stability entails not only the characterization of the "most destabilizing" nonlinearity, but also determining the parametric value of the nonlinearity that yields instability of the feedback system. The problem was first formulated in the 1940s, however, finding easily verifiable necessary and sufficient conditions for absolute stability remained an open problem all along. Recently, the problem gained renewed interest in the context of stability of hybrid dynamical systems, since solving the absolute stability problem implies stability analysis of switched linear systems. In this paper, we introduce the concept of generalized first integrals and use it to characterize the "most destabilizing" nonlinearity and to explicitly construct a Lyapunov function that yields an easily verifiable, necessary and sufficient condition for absolute stability of second-order systems.

I. Introduction

Absolute stability is perhaps the most well-known open problem in control theory as easily verifiable necessary and sufficient conditions for absolute stability have remained unattainable since the 1940s. Many important results in the theory of stability and control, such as Popov's criterion, the circle criterion [11, Ch. 5], the positive-real lemma [3, Ch. 2], and the theory of integral quadratic constraints [7] are closely related to the problem. However, these results do not address the fundamental issue of characterizing the “most destabilizing” nonlinearity and, therefore, they yield sufficient (and rather conservative [2]), but not necessary and sufficient, conditions for absolute stability.