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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
many of the results based on differential geometric approaches. A significant component of the analysis uses the accessibility filtrations
H0 = spanK fdx; dug Hj = spanK f! 2 Hj 01 j !_ 2 Hj01 g
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]. Differential geometric methods provide a powerful framework for solving several practical nonlinear control problems that are of compelling engineering interest such as model matching and disturbance decoupling; see, e.g., [5] and [7]. However, there are significant classes of problems (involving, e.g., the synthesis of stabilizing feedback, or realization 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 obstacles 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 differential 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
= f (x) + g(x)u 6 = yx_ = h (x )
(1)
(where x 2 n ; u 2 m ; y 2 p , and f; g , and h are meromorphic functions of x) from a linear algebraic standpoint including several appealing control applications such as model matching, output feedback control, and disturbance decoupling. The authors’ methods provide a systematic approach for tackling such problems as system inversion, 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 involving 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, interspersed with worked out examples to show how the methods can be used in practice. The essential contents are as follows. Chapter 1 introduces the basic differential form setting used throughout the book. This helps make the book self contained. The notation can be fairly complicated 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 University, Baton Rouge, LA 70803 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TAC.2007.911476
(2)
over the field K of meromorphic functions in a finite number of variables y; u, and their time derivatives. The simpler framework lends itself 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 elimination 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
=
(1)
x; u; u
;
. . . ; u(s)
(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 y
(k)
='
_ . . . ; y(k01) ; u; u:_ . . . ; u(s)
y; y;
:
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 computational 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 x(t) of the system from an output y (t), the input u(t), and finitely many of their time derivatives y (k) and u(l) . The results of this chapter apply to systems satisfying what the authors term “inherent linearity” which amounts to assuming that the system takes the form
_ = A + ' y = C
_ . . . ; u(s)
y; u; u;
(4)
up to a change in coordinates (3) for which rank((@ )=(@x)) = n, where (C; A) 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 u(t) necessary to obtain the desired output y (t), i.e., (pseudo)inversion. It provides necessary and sufficient conditions for the existence of inversions. Another important feature in this chapter is the inversion algorithm that was first presented in [3]. Chapter 6 is devoted to methods for transforming nonlinear systems into canonical forms, with the understanding 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
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 12, DECEMBER 2007
output injection based on transforming (1) using a relation of the type (r ) x_ 7! (x; _ y; y; _ . . . ; y ). The rest of the book focuses on control applications, while retaining a precise theorem-proof style. Chapters 7–9 address important problems related to linearization and methods of transforming systems so that their outputs only depend on a component of the input. These include input-output linearization (leading to linear relations between the input and output variables under changes of coordinates). In terms of the formalism (2), this leads to the elegant conclusion that the inputstate linearization problem for x_ = f (x) + g (x)u is solvable by static state feedback if and only if i) H = 0 and ii) Hk is closed for any k 1. Here H is such that there exists j ? for which Hj = Hj +r = H for all r 1. The practical value of these results stems from the relative ease with which one can handle linear systems (as opposed to nonlinear systems), as well as the possibility of using SISO techniques when the noninteracting control problem can be solved. Chapter 10 provides methods for using feedbacks to transform systems so that the output is independent of the state components whose dynamics are influenced by disturbances, i.e., disturbance decoupling. Chapter 11 revisits the model matching problem, using a formulation that is similar to [4]. It provides a necessary and sufficient condition for the solvability of the problem. Finally, Chapter 12 is devoted to the important and more realistic situation where the full current state may not be available for measurement, leading to methods for solving control problems based on output feedback or measurement feedback. Here the authors revisit the input-output linearization and input-output decoupling problems and provide corresponding algorithms that can be applied when the state cannot be measured.
1 1
1
III. SUMMARY The book is an updated version of the authors’ work [2], which has been used in graduate courses, summer schools, and tutorial workshops. The section on differential algebraic approaches for more general systems f (x; x; _ u) = 0; h(y; x) = 0 from the first edition has
been removed, but the authors added several new motivating examples from mechanics and other areas. The chapter on output feedback is also new to the second edition. The changes make the book seem more user friendly. However, the reviewer believes that additional sections about the known extensions to discrete time systems, time-varying systems, and systems with time delays could have made the book even more interesting since these extensions do not seem obvious. Another useful feature is the inclusion of problems at the ends of the chapters and a URL where their solutions can be found. The authors are well-regarded researchers with a knack for explaining complicated ideas in a clear and concise way. The book could certainly be considered for use by graduate students and researchers whose background includes the fundamental ideas of mathematical control theory, e.g., the material in [8].
REFERENCES [1] F. Clarke, Y. Ledyaev, R. Stern, and P. Wolenski, Nonsmooth Analysis and Control Theory. New York: Springer-Verlag, 1998, vol. 178, Graduate Texts in Mathematics. [2] G. Conte, C. H. Moog, and A. M. Perdon, Nonlinear Control Systems: An Algebraic Setting. London, U.K.: Springer, 1999, vol. 242, Lecture Notes in Control and Inf. Sci. . [3] M. D. Di Benedetto, J. W. Grizzle, and C. H. Moog, “Rank invariants of nonlinear systems,” SIAM J. Control Optim., vol. 27, no. 3, pp. 658–672, 1989. [4] M. D. Di Benedetto and A. Isidori, “The matching of nonlinear models via dynamic state-feedback,” SIAM J. Control Optim., vol. 24, no. 5, pp. 1063–1075, 1986. [5] A. Isidori, Nonlinear Control Systems. An Introduction, ser. Commun. Control Eng., 2nd ed. Berlin: Springer, 1989. [6] C. Lobry, “Contrôlabilité des systèmes non linéaires,” SIAM J. Control Optim., vol. 8, pp. 573–605, 1970. [7] H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems. New York: Springer, 1990. [8] E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd ed. New York: Springer-Verlag, 1998, Texts in Applied Math. 6. [9] H. Sussmann and V. Jurdjevic, “Controllability of nonlinear systems,” J. Differ. Equ., vol. 12, pp. 95–116, 1972.
