IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 42, NO. 7, JULY 1997
1043
Book Reviews In this section, the IEEE Control Systems Society publishes reviews of books in the control field and related areas. Readers are invited to send comments on these reviews for possible publication in the Technical Notes and Correspondence section of this TRANSACTIONS. The CSS does not necessarily endorse the opinions of the reviewers. If you have used an interesting book for a course or as a personal reference, we encourage you to share your insights with our readers by writing a review for possible publication in the TRANSACTIONS. All material published in the TRANSACTIONS is reviewed prior to publication. Submit a completed review or a proposal for a review to: D. S. Naidu Associate Editor—Book Reviews College of Engineering Idaho State University 833 South Eighth Street Pocatello, ID 83209 USA
Nonlinear Control Systems—Alberto Isidori, 3rd ed. (New York: Springer-Verlag, 1995). Reviewed by David L. Elliott. I. GEOMETRIC CONTROL THEORY In the control context, the word “nonlinear” has referred to both hard nonlinearities (saturation, dead-band, hysteresis, relay action) and to the analytic nonlinearities found in rigid-body mechanics (robotics and aerospace, where a geometric approach is natural), electrical machines, chemical kinetics, theoretical economics, and ecology. Hard nonlinearity in control systems was extensively studied in the Soviet Union and the U.S. roughly from 1950 to 1965, using such methods as phase-plane geometry and describing functions (harmonic balance). A chief concern was the development of methods for predicting the occurrence and size of limit cycles. (It is becoming obvious to those who worked on these problems that some of the experimental “limit cycles” were more likely bounded chaotic motions!) The techniques developed for predicting stability of feedback loops, such as the Yakubovich–Kalman–Popov approach, worked well only for single nonlinearities in such loops. Analytic nonlinearities were studied by series techniques (Volterra, Wiener–Lee, and Lie series). Subsequent to 1961, the popularity of Pontryagin’s Maximal Principle and other optimal control studies, as well as much work on Kalman’s algebraic criterion for controllability, led to the need to understand controllability and the nature of the sets in state space from which a given control target could be reached. It soon became evident that some technical assumptions about the nonlinear plant (smoothness, and even the property of having convergent Taylor series—called real analyticity) would make possible a careful and general mathematical approach. In this direction, in the late 1960’s Hermann1 studied controllability with methods based on the modern approach to vector fields and differential forms; local controllability could be understood as an algebraic property analogous to Kalman’s criterion for linear systems. Meanwhile back in the linear camp, Wonham and Morse and Basile and Marro developed a systematic geometric attack which The reviewer is with the Institute for Systems Research, University of Maryland, College Park, MD 20742 USA (e-mail:
[email protected]). Publisher Item Identifier S 0018-9286(97)03433-8. 1 See the excellent Bibliographical Notes in the book under review for citations to these papers and many of those mentioned below, as well as more history of the subject.
seemed likely to lead to practical design methods for stability by pole placement, noninteracting control, disturbance decoupling, and regulation. This new approach at first seemed likely not to be generalizable to nonlinear problems since it depends on global linearspace structure. Isidori realized that a local approach of this nature might be worthwhile. For several years (1968–1975) it was thought by many of us that research in nonlinear control should emphasize bilinear systems k (x_ = Ax + i=1 ui Bi x). Isidori and his colleagues were very active in this effort; for instance, he and Grasselli showed the existence of “universal inputs” for the observation of bilinear systems. The bilinear systems research introduced Lie groups and Lie algebras (which originated a century ago in the work of Lie) to the control community. In the early 1970’s Brockett, Boothby, the reviewer, and others were promoting the use of Lie algebra methods to study controllability. Sussmann and Jurdjevic in the U.S. and Stefan in Wales developed, independently, an extensive mathematical apparatus for studying controllability using these methods and other powerful tools of differentiable manifold theory. At the same time, the observability of nonlinearity dynamic systems with scalar outputs, and the construction of nonlinear state observers, became of interest. By 1980 controllability was well understood, Hermann and Krener had shown the importance of “observation algebras,” and Hirschorn had shown that Lie algebra methods could be used to find the inverse of a given bilinear system. Now Brockett (and also Willems) suggested looking for something like Kronecker invariants for nonlinear systems. Brockett considered systems equivalent to linear systems by coordinate change and a class of feedback transformations. Necessary and sufficient conditions for such equivalence in the case of more general nonlinear feedback translations and multiplications were given independently by Respondek and Jakubczyk and Hunt et al. The conditions for this “linearization by feedback” proved to be close to those needed for many other problems, such as system inversion and decoupling. Through a concordat between the University of Rome and Washington University in St. Louis, Isidori began teaching in both schools his own careful version of nonlinear control: the geometry and Volterra series methods are brought together and used appropriately for stabilization, regulation, disturbance decoupling, noninteracting control system inversion, and other tasks. The notes for his course were in demand and became the first edition (1985) of the book under
0018–9286/97$10.00 1997 IEEE
1044
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 42, NO. 7, JULY 1997
review; it was revised soon into a more complete textbook, with both necessary mathematical prerequisites and expositions of new results. The book’s intent is to provide a systematic exposition of its subject, developing nonlinear versions of both input–output and statespace methods with proofs that illuminate their relationships. There are several other, quite different, textbooks on nonlinear control systems: [8] is a first introduction to mathematical control theory, [7] is written for the applications-oriented undergraduate or practicing engineer, while [2], [3], and [9] are broad surveys of stability and control for advanced undergraduate or first-year graduate classes. Recent books that share many of the concerns of Isidori’s book are [6], which pays special attention to Hamiltonian mechanics; [5] which emphasizes robust and adaptive design methods; and for the “back-stepping” approach to nonlinear design, [4]. Older books have discussed the multiple Laplace transform approach to Volterra series (Rugh), the Small Gain theorem in input–output stability (Desoer and Vidyasagar), and Lyapunov stability theory. Recent nonlinear control studies not discussed in the book under review include system factorization methods and “control of chaos.” A necessary disclaimer may be mentioned here: the reviewer and the author have a long acquaintance and have both worked in the same Department of Systems Science and Mathematics at Washington University. Different interests and work styles, and the reviewer’s retirement in 1992, leave us with no “conflicts of interest” in discussing each other’s work. II. ISIDORI’S BOOK This is revised third edition of Nonlinear Control Systems. Like the second edition it begins with the most important facts about manifolds, vector fields, distributions, local decompositions, reachability and, observability. Chapter 2, “Global Decompositions of Control Systems,” gives the core of the differential geometric theory. Chapter 3, “Input–Output Maps and Realization Theory” uses functional expansions (Volterra, Fliess) to relate the input–output and state-space descriptions. This unified approach is an important feature of Isidori’s work in that it makes it possible to teach the most important facts about nonlinear control theory in one semester. The author suggests that the second and third chapters may be skipped on first reading and approached once the reader is motivated by Chapters 4–8, which discuss nonlinear feedback for singleinput/single-output and multi-input/multi-output systems, tools, and applications for state feedback: zero dynamics, controlled invariant distributions, stabilization, decoupling, noninteracting control with stability, tracking, and regulation. The book is self-contained to the extent that it is possible for the reader who already has had some background in nonlinear stability (and a liking for mathematics) to work through most of the book with
the help of Appendixes A and B. (A readable book on Differentiable Manifolds or Differential Geometry, such as [1], will help with the first three chapters and Appendix A.) Appendix B provides the needed theorems on the Center Manifold Theorem, Lyapunov stability, and singular perturbations, without proofs. There have been several important revisions; Chapters 8 and 9 (the last two) are new. Chapter 8 deals with tracking and regulation, culminating in the design of regulators having a degree of structural stability. Chapter 9 gives results of a global or “semiglobal” nature for stabilization and disturbance attenuation, involving the assumption of uniform relative degree. Many of the results of nonlinear control theory are local in nature, failing on singular sets of low dimension. This is analogous to some well-known problems in mechanics, e.g., the nonexistence of global coordinates for otherwise nice objects like spheres, or the occurrence of gimbal lock in gyroscopes. Nevertheless, it is of great interest to see when global results may be obtained, especially for those systems which are equivalent to linear systems by a change of coordinates and feedback transformation. A semiglobal result is one which can be achieved on a closed, bounded region (such as stabilization) but whose criteria and conclusions change quantitatively as the region is enlarged. III. IN CONCLUSION Isidori’s book is essential for anyone preparing for serious reading or basic research in the differential geometric approach to control theory and will not disappoint those mathematically trained. I have observed its use in the hands of two teachers other than the author; the students enjoyed it and made good use of it later. There is no universal solvent for nonlinear control problems, but the methods presented here are powerful. REFERENCES [1] W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. Orlando, FL: Academic, 1986. [2] P. A. Cook, Nonlinear Dynamical Systems, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1994. [3] H. Khalil, Nonlinear System Theory, 2nd ed. New York: Macmillan, 1995. [4] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [5] R. Marino and P. Tomei, Nonlinear Control Design: Geometric, Adaptive, and Robust. London: Prentice Hall, 1995. [6] H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems. Berlin: Springer-Verlag, 1990. [7] J.-J. E. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991. [8] E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems. New York: Springer-Verlag, 1990. [9] M. Vidyasagar, Nonlinear System Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1993.
