Nonlinear and adaptive control of complex dynamical systems.
(Nelinejnoe i adaptivnoe upravlenie slozhnymi dinamicheskimi sistemami.)

*(Russian)*Zbl 0962.93001
Analiz i Sintez Nelinejnykh Sistem. Sankt-Petersburg: Nauka. 549 p. (2000).

This interesting textbook belongs to a series dedicated to nonlinear systems analysis and synthesis. As other books of the series, it occurred as a result of the merging of what used to be the best of the Russian-Soviet school of nonlinear systems with the newest achievements of West-European and American scientists and engineers working in the field.

The book is structured from applications (and motivations) and basic theory to analysis and synthesis procedures and back to applications. This conception explains the structure of the book. It starts with a section on problem statements in a geometric form (oscillation synchronization, motion control, partial stabilization) and an overview (Section 2) of the methods of nonlinear analysis and synthesis (e.g., stability and partial stability, canonical forms, backstepping, passivity and passivation). Further the picture is detailed. The next three sections are like a “Zoom” on nonlinear synthesis. Lyapunov based control and gradient methods are exposed starting from the merging of Soviet and West European and American schools. The control of Lagrangean and Hamiltonian systems presented here will find its applications in the last section dedicated to the control of mechanical systems.

Another basic direction of the book deals with the geometric approach to nonlinear control. The tackled problems are: equilibrium stabilization by linearization (exact and approximate); attractors and stabilization of sets; programmed state control. A natural development of these topics led the authors to the nonlinear control of multi-channel systems. This area includes output control and matched control. The typical application is to space motion control viewed as connected to the geometry of manifolds.

The book contains still two other topics: robust and adaptive control and system decomposition based on several time scales. Adaptive control is the field where one of the authors (Fradkov) has obtained many interesting results (especially on adaptive stabilization). For this reason this topic is very well represented. System decomposition is analyzed in connection with singular perturbations and is applied mainly to adaptive control. Even decentralized control is performed here adaptively. The book ends with the control of mechanical systems (already mentioned above).

The book being a valuable synthesis of two schools of nonlinear control is extremely useful for people from one of these schools who are willing to get a better knowledge about the other one. At the same time it is both a textbook and a handbook.

The book is structured from applications (and motivations) and basic theory to analysis and synthesis procedures and back to applications. This conception explains the structure of the book. It starts with a section on problem statements in a geometric form (oscillation synchronization, motion control, partial stabilization) and an overview (Section 2) of the methods of nonlinear analysis and synthesis (e.g., stability and partial stability, canonical forms, backstepping, passivity and passivation). Further the picture is detailed. The next three sections are like a “Zoom” on nonlinear synthesis. Lyapunov based control and gradient methods are exposed starting from the merging of Soviet and West European and American schools. The control of Lagrangean and Hamiltonian systems presented here will find its applications in the last section dedicated to the control of mechanical systems.

Another basic direction of the book deals with the geometric approach to nonlinear control. The tackled problems are: equilibrium stabilization by linearization (exact and approximate); attractors and stabilization of sets; programmed state control. A natural development of these topics led the authors to the nonlinear control of multi-channel systems. This area includes output control and matched control. The typical application is to space motion control viewed as connected to the geometry of manifolds.

The book contains still two other topics: robust and adaptive control and system decomposition based on several time scales. Adaptive control is the field where one of the authors (Fradkov) has obtained many interesting results (especially on adaptive stabilization). For this reason this topic is very well represented. System decomposition is analyzed in connection with singular perturbations and is applied mainly to adaptive control. Even decentralized control is performed here adaptively. The book ends with the control of mechanical systems (already mentioned above).

The book being a valuable synthesis of two schools of nonlinear control is extremely useful for people from one of these schools who are willing to get a better knowledge about the other one. At the same time it is both a textbook and a handbook.

Reviewer: Vladimir Răsvan (Craiova)

##### MSC:

93-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory |

93C10 | Nonlinear systems in control theory |

93C40 | Adaptive control/observation systems |

93C70 | Time-scale analysis and singular perturbations in control/observation systems |