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.

Chapter Contents:

• 5.1 Positive real systems
• 5.2 Absolute stability and circle criterion
• 5.3 Input-to-state stability and small gain theorem
• 5.4 Differential stability

Inspec keywords: linear systems; asymptotic stability; nonlinear control systems; adaptive control; control system analysis; Lyapunov methods

Other keywords: ISS; asymptotic stability; stability analysis; circle criterion; advanced stability theory; memoryless nonlinear components; Lyapunov direct method; linear systems; adaptive control system analysis; input-to-state stability; Lyapunov function; Kalman-Yakubovich lemma; nonlinear system; strict positive real systems; adaptive control laws

Subjects: Stability in control theory; Nonlinear control systems; Control system analysis and synthesis methods; Self-adjusting control systems