This chapter presents an introduction to some of the dominant methods that have been developed for nonlinear control design. The objective of this chapter is to introduce the methods, analysis tools, and key issues of nonlinear control. In this chapter, we set the foundation, but do not yet discuss the use of adaptive approximation to improve the performance of nonlinear controller operation in the presence of nonlinear model uncertainty. Chapters 6 and 7 will discuss the methods, objectives, and outcomes of augmenting nonlinear control with approximation capabilities assuming that the reader is familiar with the material in this chapter.

This chapter begins with a discussion of the traditional and still commonly used approaches of small-signal linearization and gain scheduling. These approaches are based on the principle of linearizing the system around a certain operation point, or around multiple operating points, as in gain scheduling. The method of feedback linearization is presented in Section 5.2. This is one of the most commonly used nonlinear control design tools. In Section 7.2, feedback linearization is extended to include adaptive approximation. The method of backstepping is discussed in Section 5.3 and its extension using adaptive approximation is discussed in Section 7.3. A modification to the standard backstepping approach that simplifies the algebraic manipulations and online computations, especially in adaptive approaches, is presented in Section 5.3.3. Section 5.4 presents a set of robust nonlinear control design techniques, which are based on the principle of assuming that the unknown component of the nonlinearities is bounded by a known function. The methods include bounding control, sliding mode control, Lyapunov redesign, nonlinear damping, and adaptive bounding. These techniques rely on the design of a nonlinear controller that is able to handle all nonlinearities within the assumed bound. As a result, they may result in high-gain control algorithms. As we will see, one of the key motivations of adaptive approximation is to reduce the need for such conservative control design. Finally, Section 5.5 briefly presents the adaptive nonlinear control methodology, which is based on the estimation of unknown parameters in nonlinear systems.

Naturally, it is impossible to cover in a single chapter all nonlinear control design and analysis methods. By necessity, many of the technical details have been omitted. An excellent treatment of nonlinear systems and control methods is given in [134]. The intent of the present chapter is to introduce selected nonlinear control methods, highlight some methods that are robust to nonlinear model errors, and to motivate the use of adaptive approximation in certain situations. Throughout this chapter, the main focus is on tracking control problems, even though where convenient we also consider the regulation problem. Also, the presentation focuses on systems where the full state is measured; output feedback methods are not discussed.