ABSTRACT
This chapter discusses some mathematical tools which will be used to analyze the stability of the closed-loop systems under the proposed controllers. First, a simple Lyapunov-like lemma is developed to study the performance of the exact model knowledge controllers. To illustrate the control technique, a tracking controller is designed for a first-order nonlinear system. The Lyapunov-like lemma is then used to show that the tracking error is driven to zero exponentially fast. The chapter presents several definitions and lemmas which can be used to study the performance of adaptive control algorithms. The exact model knowledge controller which was formulated for the first-order scalar system is then redesigned as an adaptive controller via the use of two dynamic parameter update laws. The chapter illustrates how the stability tools can be used to show that the tracking error goes to zero asymptotically fast.
