Applications of Lyapunov Theory

Abstract

In this chapter we will give the reader an idea of the many different ways that Lyapunov theory can be utilized in applications. We start with some very classical examples of stabilization of nonlinear systems through their Jacobian linearization, and the circle and Popov criteria (which we have visited already in Chapter 4) for linear feedback systems with a nonlinearity in the feedback loop. A very considerable amount of effort has been spent in the control community in applying Lyapunov techniques to adaptive identification and control. We give the reader an introduction to adaptive identification techniques. Adaptive control is far too detailed an undertaking for this book, and we refer to Sastry and Bodson {[}259] for a more detailed treatment of this topic. Here, we study multiple time scale systems and singular perturbation and averaging from the Lyapunov standpoint here. A more geometric view of singular perturbation is given in the next chapter.