Advanced stability theory

Abstract

Lyapunov direct method provides a tool to check the stability of a nonlinear system if a Lyapunov function can be found. For linear systems, a Lyapunov function can always be constructed if the system is asymptotically stable. In many nonlinear systems, a part of the system may be linear, such as linear systems with memoryless nonlinear components and linear systems with adaptive control laws. For such a system, a Lyapunov function for the linear part may be very useful in the construction for the Lyapunov function for the entire nonlinear system. In this chapter, we will introduce one specific class of linear systems, strict positive real systems, for which, an important result, Kalman-Yakubovich lemma, is often used to guarantee a choice of the Lyapunov function for stability analysis of several types of nonlinear systems. The application of Kalman-Yakubovich lemma to analysis of adaptive control systems will be shown in later chapters, while in this chapter, this lemma is used for stability analysis of systems containing memoryless nonlinear components and the related circle criterion. In Section 5.3 of this chapter, input-to-state stability (ISS) is briefly introduced.