IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 12, DECEMBER 2007 2395 Book Review Algebraic Methods for Nonlinear Control Systems—G. Conte, C. Moog, and A. Perdon (Springer, 2007 ISBN: 978-1-84628-594-3). Reviewed by Michael Malisoff I. INTRODUCTION Starting in the 1970s, differential geometric methods began to be systematically applied to the analysis of controllability of nonlinear systems and other difficult problems; see, for example, [6] and [9]. Dif- ferential geometric methods provide a powerful framework for solving several practical nonlinear control problems that are of compelling en- gineering interest such as model matching and disturbance decoupling; see, e.g., [5] and [7]. However, there are significant classes of prob- lems (involving, e.g., the synthesis of stabilizing feedback, or realiza- tion problems) that do not lend themselves to differential geometric approaches, often because their models do not satisfy the necessary regularity assumptions. In fact, it is well appreciated that there are both topological obsta- cles and “virtual” obstacles that preclude the construction of globally stabilizing time-invariant feedback stabilizers e.g., those imposed by Brockett’s criterion [8]. One approach to overcoming some of these limitations involves time-varying or discontinuous feedbacks, or dif- ferential inclusions. Nonsmooth analysis is an important tool for the study of differential inclusions and discontinuous feedbacks; see [1] for a good introduction. A different approach to nonlinear control systems involves algebraic methods, which are the subject of the book under review. II. THE BOOK This book is devoted to the study of finite dimensional time invariant systems (1) (where , and , and are meromorphic functions of ) from a linear algebraic standpoint including several appealing control applications such as model matching, output feed- back control, and disturbance decoupling. The authors’ methods pro- vide a systematic approach for tackling such problems as system inver- sion, the synthesis of dynamic feedbacks, and other challenging areas that are beyond the scope of the well-established differential geometric methods. However, the focus of the book is on structural issues not in- volving stability. The book strikes a balance between methods and applications, with the first six chapters devoted to methodology and the last six to control applications. The authors give precise statements of their results, inter- spersed with worked out examples to show how the methods can be used in practice. The essential contents are as follows. Chapter 1 intro- duces the basic differential form setting used throughout the book. This helps make the book self contained. The notation can be fairly com- plicated at times, but the presentation seems clear enough for readers with a background in basic nonlinear control to follow the main ideas without much difficulty. In fact, the level of mathematics needed to understand this work seems like much less than the background one needs to understand The reviewer is with the Department of Mathematics, Louisiana State Uni- versity, Baton Rouge, LA 70803 USA (e-mail: malisoff@lsu.edu). Digital Object Identifier 10.1109/TAC.2007.911476 many of the results based on differential geometric approaches. A sig- nificant component of the analysis uses the accessibility filtrations (2) over the field of meromorphic functions in a finite number of vari- ables , and their time derivatives. The simpler framework lends it- self to several constructive algorithms that appear later in the text e.g., for finding system inversions. Chapter 2 provides methods for deriving state-space realizations from input-output descriptions, as well as the converse state elimi- nation problem (which involves finding input-output descriptions in terms of state space realizations). A useful feature is that the chapter includes results on classical realizations for nonlinear systems, which makes it easier to appreciate the value added by the authors’ algebraic approach. The authors’ realizations are in a generalized sense where the new state coordinates (3) are parameterized in terms of the inputs and their derivatives, leading to a necessary and sufficient condition for the existence of an observable state space system realization for a given input-output realization The chapter closes with several interesting illustrations involving electromechanical systems and virus dynamics for HIV infection. Chapter 3 is devoted to reachability, controllability, and accessibility for systems without outputs. The chapter includes a practical computa- tional method for deciding strong accessibility, as well as an illustration using a hopping robot model. Chapter 4 deals with observability, i.e., recovering the state of the system from an output , the input , and finitely many of their time derivatives and . The re- sults of this chapter apply to systems satisfying what the authors term “inherent linearity” which amounts to assuming that the system takes the form (4) up to a change in coordinates (3) for which , where is an appropriate constant pair in canonical observer form. The constructions of this chapter are complicated, but the chapter includes two worked out examples that make the material easier to follow. The examples also suggest that the structure (4) is not too restrictive. Chapter 5 involves the problem of recovering the control input necessary to obtain the desired output , i.e., (pseudo)inversion. It provides necessary and sufficient conditions for the existence of inver- sions. Another important feature in this chapter is the inversion algo- rithm that was first presented in [3]. Chapter 6 is devoted to methods for transforming nonlinear systems into canonical forms, with the un- derstanding that the transformations can depend on the state as well as the inputs, the outputs, and time derivatives of the inputs and outputs. This leads to an appealing generalization of the well-known Morse canonical form for linear systems, as well as a generalized notion of 0018-9286/$25.00 © 2007 IEEE