Nonlinear systems.
3rd ed.

*(English)*Zbl 1003.34002
Upper Saddle River, NJ: Prentice Hall. xv, 750 p. (2002).

This is the third edition of a well-written book whose author has been awarded the IFAC Control Engineering Textbook Prize for the second edition [Nonlinear systems (Macmillan Publishing Company, New York, NY) (1992; Zbl 0969.34001)] in 2002. The new edition has been reorganized, partly rewritten, new material has been added, and some material has been dropped. These modifications has been made in order to improve the readability of the book making it accessible to first year graduate students (in fact, the first four chapters can be successfully used now for advanced level undergraduate courses), to restructure the material making the design of courses on nonlinear systems and control easier, to update the text including the material useful in nonlinear control design in recent years, and to update the exercises with more than 170 new exercises added.

In the third edition, all the material is divided logically into four parts. The first part, entitled Basic Analysis, contains material from Chapters 1 to 4. Chapters 5 to 7 compose the second part, Analysis of Feedback Systems. These two parts constitute the background for further streaming into either Part 3, Advanced Analysis (Chapters 8 to 11), or Part 4, Nonlinear Feedback Control (Chapters 12 to 14).

In what follows, we discuss main changes from the second edition. In order to re-arrange the exposition into the four blocks, a lot of material has been moved (for instance, the section on mathematical preliminaries moved from Chapter 2 of the second edition to an appendix, two sections on boundedness, ultimate boundedness, and input-to-state stability moved from the chapter on stability of perturbed systems to Chapter 4 in the new edition, the section on stability of periodic solutions has moved from Chapter 7 in the second edition to Chapter 8, etc.) Several parts of the text have been rewritten in an easier style (for example, section 2.6 and material on nonautomous systems in Chapter 8), while some chapters have seen major revision (for instance, Chapter 6 on passivity or Chapter 13 on feedback linearization). New material has been also added: section 2.7 on bifurcations has been included, the small-gain theorem has been added to Chapter 5, a new theorem has been included in section 4.7 on converse Lyapunov theorems, etc. In Chapter 6, an expanded treatment of the passivity compared to the second edition is given. Furthermore, the results on frequency-domain analysis of the feedback connection of a linear time-invariant system with a static nonlinearity in Chapter 7 are more general than those in the second edition. Several results in Chapters 9 to 11 have been stated as nonlocal results by combining local exponential stability with global uniform asymptotic stability. On the other hand, some material has been dropped (the use of simultaneous Lyapunov functions for studying absolute stability in Chapter 7, the PoincarĂ© map and the section on adaptive control in Chapter 8, etc.).

Finally, we mention that the author maintains the lists of corrections to all three editions (the last one is very short at the moment, just half a page) on the textbook’s web site, where the list of chapter-by-chapter changes can be also found. Although this nice book was written mainly for the graduate students in electrical and mechanical engineering, it will be beneficial for everyone with research interests in nonlinear systems, stability, and control.

In the third edition, all the material is divided logically into four parts. The first part, entitled Basic Analysis, contains material from Chapters 1 to 4. Chapters 5 to 7 compose the second part, Analysis of Feedback Systems. These two parts constitute the background for further streaming into either Part 3, Advanced Analysis (Chapters 8 to 11), or Part 4, Nonlinear Feedback Control (Chapters 12 to 14).

In what follows, we discuss main changes from the second edition. In order to re-arrange the exposition into the four blocks, a lot of material has been moved (for instance, the section on mathematical preliminaries moved from Chapter 2 of the second edition to an appendix, two sections on boundedness, ultimate boundedness, and input-to-state stability moved from the chapter on stability of perturbed systems to Chapter 4 in the new edition, the section on stability of periodic solutions has moved from Chapter 7 in the second edition to Chapter 8, etc.) Several parts of the text have been rewritten in an easier style (for example, section 2.6 and material on nonautomous systems in Chapter 8), while some chapters have seen major revision (for instance, Chapter 6 on passivity or Chapter 13 on feedback linearization). New material has been also added: section 2.7 on bifurcations has been included, the small-gain theorem has been added to Chapter 5, a new theorem has been included in section 4.7 on converse Lyapunov theorems, etc. In Chapter 6, an expanded treatment of the passivity compared to the second edition is given. Furthermore, the results on frequency-domain analysis of the feedback connection of a linear time-invariant system with a static nonlinearity in Chapter 7 are more general than those in the second edition. Several results in Chapters 9 to 11 have been stated as nonlocal results by combining local exponential stability with global uniform asymptotic stability. On the other hand, some material has been dropped (the use of simultaneous Lyapunov functions for studying absolute stability in Chapter 7, the PoincarĂ© map and the section on adaptive control in Chapter 8, etc.).

Finally, we mention that the author maintains the lists of corrections to all three editions (the last one is very short at the moment, just half a page) on the textbook’s web site, where the list of chapter-by-chapter changes can be also found. Although this nice book was written mainly for the graduate students in electrical and mechanical engineering, it will be beneficial for everyone with research interests in nonlinear systems, stability, and control.

Reviewer: Yuri V.Rogovchenko (Famagusta)

##### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

34Cxx | Qualitative theory for ordinary differential equations |

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

93-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory |

34H05 | Control problems involving ordinary differential equations |

93C10 | Nonlinear systems in control theory |