Nonlinear Systems
Third Edition
List of Chapter-by-Chapter Changes

Chapter 1: The material on second-order systems is taken out.

Chapter 2: This is a new chapter that gathers the material on second-order systems which was covered in Chapters 1 and 7 of the 2nd Ed. Sections 2.1 to 2.5 belong to the first group and Section 2.6 belongs to the second group. The chapter has also a new section (2.7) on bifurcation. The material in Section 2.6 has been re-written in an easier style that matches the level of an early chapter.

Chapter 3 (2 in 2nd Ed): The section on mathematical preliminaries has been moved to an appendix. The proof of the existence and uniqueness theorem has been moved to an appendix so that students do not have to deal with the contraction mapping principle in such early chapter. The chapter now is much easier to read compared to the 2nd Ed.

Chapter 4 (3 in 2nd Ed): The first three sections on autonomous systems remain basically the same. The material on nonautonomous systems has been rewritten to make it more accessible. First, comparison functions are treated in a separate section (4.4). Second, the section on nonautonomous system (4.5) has been rewritten to make it easier to read. The basic theorem has been restated in three theorems covering different cases and a sketch was added to illustrate the proof. In the section on converse Lyapunov theorems (4.7), a new theorem was added to cover the case of autonomous systems. The theorem is known as Kurzweil's theorem and provides a Lyapunov function that approaches infinity as the trajectory approaches the boundary of the region of attraction. It is a useful theorem that was used recently by several researchers. It is used later in the book to sharpen some of the results on perturbed systems and perturbations. It is also used in the proof of the nonlinear separation principle in Chapter 14. The chapter has two new sections (4.8 & 4.9) which show how Lyapunov's method can be used to show boundedness, ultimate boundedness, and input-to-state stability. In the 2nd Ed., this material was covered in the chapter on stability of perturbed systems. However, these concepts are important in nonlinear control and it is useful to expose the reader (student) to them without necessarily reading the chapter on stability of perturbed systems. Moving this material to Chapter 4 allowed for reorganization of the book where the reader can go through the chapters on nonlinear control without necessarily reading the chapters on advanced analysis.

Chapter 5 (6 in 2nd Ed): With the changes at the end of Chapter 4, it is now possible to teach input-output stability right after Lyapunov stability. The small-gain theorem has been added to the chapter, so that the chapter actually reads as if we are building the tools to state the small-gain theorem. The chapter is the first of three chapters that form a part of the book that deals with analysis of feedback systems.

Chapter 6 (Section 10.3 in 2nd Ed): This chapter is an expanded treatment of passivity compared to the 2nd Ed. It builds towards passivity theorems for feedback systems. It starts with motivated definitions of passivity for memoryless functions and dynamical systems (6.1 & 6.2) including introduction of sector nonlinearities. Then it presents positive real transfer functions (6.3). In the 2nd Ed, PR transfer functions were introduced in the section on absolute stability. In the new organization, they are introduced in their natural setting as passivity of linear systems. Section 6.4 presents the connection between passivity and stability and Section 6.5 gives passivity theorems for feedback systems, including loop transformations.

Chapter 7 (Sections 10.1 & 10.4 in 2nd Ed): This chapter gathers the classical tools which use frequency-domain analysis of the feedback connection of a linear time-invariant system with a static nonlinearity. Section 7.1 covers the circle and Popov criteria and Section 7.2 covers the describing function methods. The derivation of the circle and Popov criteria makes use of the passivity and loop transformation tools of the previous chapter. The results are more general than those of the 2nd ed. in two aspects: (1) transfer functions are allowed to be proper, not just strictly proper; (2) sectors of the form [K_1,K_2] and [K_1, infinity) are covered, while in the 2nd Ed. only the sector [K_1,K_2] was covered. The use of simultaneous Lyapunov functions for studying absolute stability was dropped.

Chapter 8 (4 in 2nd Ed): The chapter stays almost the same except that a new section on stability of periodic solutions was added (it was in Chapter 7 of the 2nd Ed). In this section, the Poincare map was dropped. In the section on invariance-like theorems, an adaptive control example was added, as the section on adaptive control in the 2nd Ed. (13.4) was dropped.

Chapter 9 (5 in 2nd Ed): The chapter stays almost the same, except that the material on ultimate boundedness and input-to-state stability has been moved to Chapter 4. In the section on continuity of solutions (9.4), the result has been made nonlocal by combining local exponential stability with non-local/global uniform asymptotic stability.

Chapter 10 (8 in 2nd Ed): The structure of the chapter has been changed. It is now closer to its structure in the 1st Ed. In particular, the section on periodic perturbation of autonomous systems (10.3) was brought back and most of the proofs are given within the text rather than in the appendix. The perturbation and averaging theorems on the inifinite-time interval have been stated as nonlocal results by combining local exponential stability with non-local/global uniform asymptotic stability.

Chapter 11 (9 in 2nd Ed): The chapter stays the same as in the 2nd Ed., except that the section on singular perturbation on the infinite-time interval (11.3) has been advanced to appear right after the finite time result (11.2), and its theorem has been stated as a nonlocal result by combining local exponential stability with non-local/global uniform asymptotic stability.

Chapter 12 (11 in 2nd Ed): The chapter stays the same with one exception: integral control has been emphasized and treated in a separate section (12.2) by discussing the principle of integral control of a general nonlinear system, irrespective of the stabilizing feedback control technique. New results on gain scheduling are given in two theorems. In the 2nd edition, the results were discussed but without a formal statement.

Chapter 13 (12 in 2nd Ed): This chapter on feedback linearization has seen major changes. In the 2nd Ed., full-state linearization was treated first followed by input-output linearization. Differential geometric conditions and terminology were delegated to a separate section at the end of the chapter. In the new edition, differential geometric terms are introduced right from the beginning. The chapter starts with a motivation section (13.1) which introduces the idea of feedback linearization via examples. Then, rigorous treatment proceeds with input-output linearization (13.2) followed by full state linearization (13.3).

Chapter 14 (13 in 2nd Ed): The chapter retains the sections on sliding mode control, Lyapunov redesign and backstepping from the 2nd ed., while the section on adaptive control is dropped. The sliding mode control section is revised to expand the discussion on chattering and its connection with unmodeled high-frequency dynamics. The chapter has two new sections on passivity-based control and high-gain observers. A nonlinear separation principle is proved for high-gain observers. In both the sliding mode and high-gain observer sections, a detailed treatment of a second-order system is used to introduce the basic ideas of the method. The last 7 exercises of the chapter include cases studies of induction motor, robot manipulator, and the TORA system.