##
**Stabilization of nonlinear uncertain systems.**
*(English)*
Zbl 0906.93001

Communications and Control Engineering Series. London: Springer. xii, 193 p. (1998).

The present monograph addresses stabilization and disturbance attenuation problems for different types of nonlinear systems, in relation with associated (inverse) optimal control problems. The authors promise a “uniform presentation of robust/stochastic/adaptive control” and – within the chosen approach and the scope of the book – in fact manage to keep their promise.

The material – primarily research results of the authors from the last years – is organized in three parts, dealing with deterministic systems, stochastic systems and adaptive control. Here the last part mainly (but not exclusively) covers deterministic systems and is the most specialized one. The book starts with a short summary of the theory of control Lyapunov functions for deterministic systems in Chapter 1, and then consecutively addresses deterministic disturbance attenuation, stochastic stability, stochastic disturbance attenuation, deterministic adaptive tracking, and stochastic adaptive regulation in the Chapters 2-6. These chapters provide a uniform approach (as far as permitted by the different problems addressed), in which the authors in each chapter introduce the appropriate stability concept (e.g., input-to-state, in probability, noise-to-state) and then formulate and solve a corresponding inverse optimal control problem. This, in turn, is done by means of feedback laws obtained by “Sontag-type” formulas from suitable Lyapunov functions which are designed via backstepping techniques. The remaining chapters somewhat deviate from this approach: Chapter 7 is devoted to the asymptotic analysis of the parameter estimates in adaptive control laws using invariant manifold theory, and Chapter 8 addresses extremum searching control (i.e. the adaptive stabilization of a point realizing some extremum in the reference-to-output map) by means of singular perturbations and averaging techniques.

The main concern of the authors is “constructiveness”, meaning that they provide algorithms and formulas rather than abstract existence theorems (one exception from that rule is the equivalence proof between input-to-state stabilizability and the solvability of the inverse optimal gain assignment problem). Of course, this approach bears inherent drawbacks, such as the restriction to specific classes of systems (here typically systems in strict feedback form), and the presence of huge equations in the proofs (see, e.g., Pages 51ff.) sometimes hiding the geometric nature of the underlying problems. On the other hand, the reader is rewarded by algorithms and formulas yielding analytic expressions for the resulting feedback control laws, which are well illustrated by several examples.

The material in the Chapters 1, 2 and 5 of this book is accessible to students, engineers and mathematicians with a background in basic deterministic systems or control theory, or in basic stability theory for ordinary differential equations. Chapters 3, 4 and 6 require basic knowledge on stochastic systems, and Chapters 7 and 8 need some more knowledge on invariant manifolds and singular perturbations (both for deterministic systems), respectively.

Apart from being a valuable source for anyone interested in constructive methods for stabilization, due to the uniform presentation especially in the Chapters 2-6 this book may well serve for an advanced course on nonlinear control.

The material – primarily research results of the authors from the last years – is organized in three parts, dealing with deterministic systems, stochastic systems and adaptive control. Here the last part mainly (but not exclusively) covers deterministic systems and is the most specialized one. The book starts with a short summary of the theory of control Lyapunov functions for deterministic systems in Chapter 1, and then consecutively addresses deterministic disturbance attenuation, stochastic stability, stochastic disturbance attenuation, deterministic adaptive tracking, and stochastic adaptive regulation in the Chapters 2-6. These chapters provide a uniform approach (as far as permitted by the different problems addressed), in which the authors in each chapter introduce the appropriate stability concept (e.g., input-to-state, in probability, noise-to-state) and then formulate and solve a corresponding inverse optimal control problem. This, in turn, is done by means of feedback laws obtained by “Sontag-type” formulas from suitable Lyapunov functions which are designed via backstepping techniques. The remaining chapters somewhat deviate from this approach: Chapter 7 is devoted to the asymptotic analysis of the parameter estimates in adaptive control laws using invariant manifold theory, and Chapter 8 addresses extremum searching control (i.e. the adaptive stabilization of a point realizing some extremum in the reference-to-output map) by means of singular perturbations and averaging techniques.

The main concern of the authors is “constructiveness”, meaning that they provide algorithms and formulas rather than abstract existence theorems (one exception from that rule is the equivalence proof between input-to-state stabilizability and the solvability of the inverse optimal gain assignment problem). Of course, this approach bears inherent drawbacks, such as the restriction to specific classes of systems (here typically systems in strict feedback form), and the presence of huge equations in the proofs (see, e.g., Pages 51ff.) sometimes hiding the geometric nature of the underlying problems. On the other hand, the reader is rewarded by algorithms and formulas yielding analytic expressions for the resulting feedback control laws, which are well illustrated by several examples.

The material in the Chapters 1, 2 and 5 of this book is accessible to students, engineers and mathematicians with a background in basic deterministic systems or control theory, or in basic stability theory for ordinary differential equations. Chapters 3, 4 and 6 require basic knowledge on stochastic systems, and Chapters 7 and 8 need some more knowledge on invariant manifolds and singular perturbations (both for deterministic systems), respectively.

Apart from being a valuable source for anyone interested in constructive methods for stabilization, due to the uniform presentation especially in the Chapters 2-6 this book may well serve for an advanced course on nonlinear control.

Reviewer: Lars Grüne (Frankfurt)

### MSC:

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

93D15 | Stabilization of systems by feedback |

93C10 | Nonlinear systems in control theory |

93D21 | Adaptive or robust stabilization |

93D30 | Lyapunov and storage functions |