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**Nonlinear dynamical control systems.**
*(English)*
Zbl 0701.93001

New York etc.: Springer-Verlag. xiii, 467 p. DM 98.00 (1990).

This monograph gives a systematic introduction to the differential geometric approach to nonlinear control systems governed by ordinary differential equations. Apparently grown out of lectures for graduate students, it provides a nicely written and very convincing account of the achievements of the differential geometric theory, as well as a very useful starting point from which further research in this field may originate.

The text opens in Chapter 1 with the exposition of four typical nonlinear control systems which serve as illustrative examples for the theory throughout the book. Coming from different applied fields, namely robotics, aeronautics, economics and biology, these examples give a good motivation for the mathematical theory following in the later sections of the book.

In Chapter 2, the necessary concepts for the differential geometric treatment of nonlinear control systems are introduced: manifolds, tangent bundles, vectorfields, Lie algebras and Lie brackets, distributions on manifolds, and Frobenius’ theorem are discussed in some detail.

Chapter 3 discusses the notions of controllability and observability. The classical linear Kalman theory is generalized by deriving nonlinear accessibility rank conditions, and the relevance of the notion of invariant distributions for the study of local decompositions is stressed. Chapter 4 deals with input-output representations of nonlinear control systems. The Wiener-Volterra series and the Fliess functional representations are introduced, followed by the nonlinear generalization of the external differential representation. Moreover, conditions are derived for output invariance under particular inputs.

In the subsequent Chapter 5, state-space and feedback transformations are used to transform nonlinear systems into simpler, especially linear, form. Chapter 6 then brings the full solution of the local feedback linearization problem via static state feedback. After the derivation of the geometric conditions for local feedback linearizability, the computational aspects are touched.

Chapter 7 is devoted to the study of the concept of controlled invariance for nonlinear systems. The crucial role played by the notion of controlled invariant distributions in synthesis problems is discussed in Section 7.2, where a detailed account of the disturbance decoupling problem is given, as well as in Chapter 9, where geometric aspects of the input-output decoupling problem are analyzed. Chapter 8 discusses the input-output decoupling problem via static and dynamic state feedback.

Chapter 10 deals with local stability and stabilization of nonlinear systems. Besides Lyapunov-type techniques, also center manifold theory is used. In Chapter 11 the notion of a controlled invariant submanifold is introduced, and its applications to stabilization, interconnected systems and inverse systems are discussed.

Chapter 12 brings a treatment of the particular class of the mechanical control problems, that is, of systems whose dynamics are governed by the Euler-Lagrangian or Hamiltonian equations of motion. Controllability, observability, local decompositions and stabilization of Hamiltonian control systems are treated in detail.

In contrast to the Chapter 8–11, where the authors have confined their study to affine control actions, in the final Chapters 13 and 14 some of the previously derived main results are generalized to general smooth nonlinear dynamics.

All chapters are supplemented by the relevant literature and, especially welcome to a reader entering the field, a set of carefully selected and illustrative exercises.

In conclusion, this monograph is undoubtedly an important and welcome contribution to the existing literature on nonlinear control systems. The interested student or researcher will enjoy the clarity in which the material is presented.

The text opens in Chapter 1 with the exposition of four typical nonlinear control systems which serve as illustrative examples for the theory throughout the book. Coming from different applied fields, namely robotics, aeronautics, economics and biology, these examples give a good motivation for the mathematical theory following in the later sections of the book.

In Chapter 2, the necessary concepts for the differential geometric treatment of nonlinear control systems are introduced: manifolds, tangent bundles, vectorfields, Lie algebras and Lie brackets, distributions on manifolds, and Frobenius’ theorem are discussed in some detail.

Chapter 3 discusses the notions of controllability and observability. The classical linear Kalman theory is generalized by deriving nonlinear accessibility rank conditions, and the relevance of the notion of invariant distributions for the study of local decompositions is stressed. Chapter 4 deals with input-output representations of nonlinear control systems. The Wiener-Volterra series and the Fliess functional representations are introduced, followed by the nonlinear generalization of the external differential representation. Moreover, conditions are derived for output invariance under particular inputs.

In the subsequent Chapter 5, state-space and feedback transformations are used to transform nonlinear systems into simpler, especially linear, form. Chapter 6 then brings the full solution of the local feedback linearization problem via static state feedback. After the derivation of the geometric conditions for local feedback linearizability, the computational aspects are touched.

Chapter 7 is devoted to the study of the concept of controlled invariance for nonlinear systems. The crucial role played by the notion of controlled invariant distributions in synthesis problems is discussed in Section 7.2, where a detailed account of the disturbance decoupling problem is given, as well as in Chapter 9, where geometric aspects of the input-output decoupling problem are analyzed. Chapter 8 discusses the input-output decoupling problem via static and dynamic state feedback.

Chapter 10 deals with local stability and stabilization of nonlinear systems. Besides Lyapunov-type techniques, also center manifold theory is used. In Chapter 11 the notion of a controlled invariant submanifold is introduced, and its applications to stabilization, interconnected systems and inverse systems are discussed.

Chapter 12 brings a treatment of the particular class of the mechanical control problems, that is, of systems whose dynamics are governed by the Euler-Lagrangian or Hamiltonian equations of motion. Controllability, observability, local decompositions and stabilization of Hamiltonian control systems are treated in detail.

In contrast to the Chapter 8–11, where the authors have confined their study to affine control actions, in the final Chapters 13 and 14 some of the previously derived main results are generalized to general smooth nonlinear dynamics.

All chapters are supplemented by the relevant literature and, especially welcome to a reader entering the field, a set of carefully selected and illustrative exercises.

In conclusion, this monograph is undoubtedly an important and welcome contribution to the existing literature on nonlinear control systems. The interested student or researcher will enjoy the clarity in which the material is presented.

Reviewer: Jürgen Sprekels (Berlin)

### MSC:

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

93C10 | Nonlinear systems in control theory |

93B29 | Differential-geometric methods in systems theory (MSC2000) |

93C15 | Control/observation systems governed by ordinary differential equations |