1. Introduction

In this chapter, we introduce formal methods and practical tools from dynamics and control. Our intent is to design, analyze, and evaluate robust and adaptive control algorithms for continuous dynamical systems. We begin our material with the presentation of optimal control algorithms to regulate linear-time-invariant systems while enforcing adequate stability margins, and thus providing necessary robustness properties to the overall design. We then introduce output feedback architectures to design optimal controllers for linear systems whose states are not available as measurements. We shall also discuss numerically efficient analysis methods to evaluate the robustness of the synthesized linear controllers. All these topics are covered in Part I of this book, whereby many examples are given to help the reader isolate and focus on the key points and features of the presented methodologies. Part II covers adaptive methods to control both linear and nonlinear systems with uncertainties in their dynamics. The material is self-contained. It includes a brief introduction into Lyapunov’s stability, followed by an exposition of model reference adaptive controllers, with their design and analysis methods illustrated on a series of practical examples from aerospace applications. After we introduce what is now known as the “classical adaptive control,” we focus our attention on extensions and design modifications to improve transient performance and utilize output measurements (as opposed to state feedback connections). Also, in this chapter, we offer design methods to combine robust and adaptive controllers into a unified resilient to uncertainties control architecture. Throughout the book, we will always strive to motivate and rationalize problem formulations, their solutions, and practicality of the methods.