Nonlinear and adaptive control of complex systems.

*(English)*Zbl 0934.93002
Mathematics and its Applications (Dordrecht). 491. Dordrecht: Kluwer Academic Publishers. xviii, 510 p. (1999).

This book presents a theoretical framework and control methodology for a class of complex dynamical systems characterised by high state space dimension, multiple inputs and outputs, significant nonlinearity, parametric uncertainty, unmodeled dynamics. A unique feature of the authors’ approach is the combination of rigorous concepts and methods of nonlinear control (invariant and attracting submanifolds, Lyapunov functions, exact linearisation, passivation) with approximate decomposition results based on singular perturbations and decentralisation. Some results published previously in the Russian literature and not well known in the West or other countries are exposed. Basic concepts of modern nonlinear control and motivating examples are given.

The book starts with an introductory Chapter 1 where the peculiarities of control problems for complex systems are discussed and motivating examples from different fields of science and technology are given.

Chapter 2 presents some results of nonlinear control theory which assist in reading subsequent chapters. The main notions and concepts of stability theory are introduced, and problems of nonlinear transformation of system coordinates are discussed. On this basis, the authors consider different design techniques and approaches to linearization, stabilization and passivation of nonlinear dynamical systems.

Chapter 3 gives an exposition of the Speed-Gradient method and its applications to nonlinear and adaptive control. Convergence and robustness properties are examined. Problems of regulation, tracking, partial stabilization and control of Hamiltonian systems are considered.

In Chapter 4 the authors introduce the main notions related to the properties of regular hypersurfaces of being an invariant set and nontrivial attractor of a dynamical system. Then, the authors present a methodology of system analysis in the state space and design tools for solving the problems of equilibrium and set stabilization, as well as tracking control, for nonlinear multivariable systems having several controlling inputs.

In Chapter 5 the authors study multi-dimensional problems of output regulation, coordinating control and curve-(surface-)following, having an evident geometric nature similar to the one of the problems considered in Chapter 4. However, unlike the previous parts, the emphasis is here placed on the output space where the majority of the real problems are originally stated.

In Chapter 6 the basic design methods of adaptive, robust adaptive and robust nonlinear control of uncertain plants are presented in the form of universal design tools. Various methodologies (including recursive design, augmented error based design, high-order tuner based design and reduced order reference model design), which allow to overcome structural obstacles caused by violation of the matching condition or by high relative degree, are considered in the chapter. The practical applicability of the introduced design tools is illustrated by the example of output-feedback control of uncertain single-input/single-output linear systems.

Chapter 7 is devoted to decomposition methods in adaptive control based on the separation of slow and fast motions in the system. Convergence and accuracy of decomposition for singularly perturbed and discretized systems are examined. The Speed-Gradient approach to decentralized adaptive control of nonlinear systems is presented.

In Chapter 8 the authors study applied nonlinear control problems for providing the required spatial motion of complex mechanical systems described by the Newton, Euler and Lagrange equations. The presentation begins with investigating the problem of motion of a rigid body, which is the basis for further consideration of multi-body mechanical systems such as multi-link manipulation robots and multi-drive wheeled mechanisms. Also applications to control of oscillatory mechanical systems, based on the material of Chapter 3, are presented.

Finally, in Chapter 9 the relations between control and physics are discussed. New concepts of “feedback resonance”, “excitability index” are introduced with the purpose to better understand behaviour of nonlinear nearly conservative systems under feedback action. The Speed-Gradient method of Chapter 3 is applied both to organize resonant system behaviour and to reformulate the laws of dynamics for a wide class of physical systems. Applications to the escape from a potential well, stabilization of unstable modes, feedback spectroscopy and derivation of the Onzager principle are given. The chapter outlines a new field of research that may be called cybernetical physics.

This book is the Volume 491 of the series Mathematics and Its Applications edited by M. Hazewinkel. It contains 510 pages, 87 figures, and 294 references. This book will be useful for researchers, engineers, university lecturers and postgraduate students specialising in the fields of applied mathematics and engineering, such as automatic control, robotics, control of vibrations.

The book starts with an introductory Chapter 1 where the peculiarities of control problems for complex systems are discussed and motivating examples from different fields of science and technology are given.

Chapter 2 presents some results of nonlinear control theory which assist in reading subsequent chapters. The main notions and concepts of stability theory are introduced, and problems of nonlinear transformation of system coordinates are discussed. On this basis, the authors consider different design techniques and approaches to linearization, stabilization and passivation of nonlinear dynamical systems.

Chapter 3 gives an exposition of the Speed-Gradient method and its applications to nonlinear and adaptive control. Convergence and robustness properties are examined. Problems of regulation, tracking, partial stabilization and control of Hamiltonian systems are considered.

In Chapter 4 the authors introduce the main notions related to the properties of regular hypersurfaces of being an invariant set and nontrivial attractor of a dynamical system. Then, the authors present a methodology of system analysis in the state space and design tools for solving the problems of equilibrium and set stabilization, as well as tracking control, for nonlinear multivariable systems having several controlling inputs.

In Chapter 5 the authors study multi-dimensional problems of output regulation, coordinating control and curve-(surface-)following, having an evident geometric nature similar to the one of the problems considered in Chapter 4. However, unlike the previous parts, the emphasis is here placed on the output space where the majority of the real problems are originally stated.

In Chapter 6 the basic design methods of adaptive, robust adaptive and robust nonlinear control of uncertain plants are presented in the form of universal design tools. Various methodologies (including recursive design, augmented error based design, high-order tuner based design and reduced order reference model design), which allow to overcome structural obstacles caused by violation of the matching condition or by high relative degree, are considered in the chapter. The practical applicability of the introduced design tools is illustrated by the example of output-feedback control of uncertain single-input/single-output linear systems.

Chapter 7 is devoted to decomposition methods in adaptive control based on the separation of slow and fast motions in the system. Convergence and accuracy of decomposition for singularly perturbed and discretized systems are examined. The Speed-Gradient approach to decentralized adaptive control of nonlinear systems is presented.

In Chapter 8 the authors study applied nonlinear control problems for providing the required spatial motion of complex mechanical systems described by the Newton, Euler and Lagrange equations. The presentation begins with investigating the problem of motion of a rigid body, which is the basis for further consideration of multi-body mechanical systems such as multi-link manipulation robots and multi-drive wheeled mechanisms. Also applications to control of oscillatory mechanical systems, based on the material of Chapter 3, are presented.

Finally, in Chapter 9 the relations between control and physics are discussed. New concepts of “feedback resonance”, “excitability index” are introduced with the purpose to better understand behaviour of nonlinear nearly conservative systems under feedback action. The Speed-Gradient method of Chapter 3 is applied both to organize resonant system behaviour and to reformulate the laws of dynamics for a wide class of physical systems. Applications to the escape from a potential well, stabilization of unstable modes, feedback spectroscopy and derivation of the Onzager principle are given. The chapter outlines a new field of research that may be called cybernetical physics.

This book is the Volume 491 of the series Mathematics and Its Applications edited by M. Hazewinkel. It contains 510 pages, 87 figures, and 294 references. This book will be useful for researchers, engineers, university lecturers and postgraduate students specialising in the fields of applied mathematics and engineering, such as automatic control, robotics, control of vibrations.

Reviewer: Fuhua Ling (Cumming)

##### MSC:

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

93C40 | Adaptive control/observation systems |

93C10 | Nonlinear systems in control theory |

93B51 | Design techniques (robust design, computer-aided design, etc.) |

93D15 | Stabilization of systems by feedback |

93D21 | Adaptive or robust stabilization |

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

93C85 | Automated systems (robots, etc.) in control theory |