Singular perturbation methods in control: analysis and design.

*(English)*Zbl 0646.93001
London etc.: Academic Press (Harcourt Brace Jovanovich, Publishers). XII, 371 p.; £43.00 (1986).

This is a basic book for applied mathematicians and control engineers who are interested in modeling, analysis and design of controlled dynamical systems. It presents both practical motivations and mathematical backgrounds of the so-called singular perturbation method, having its roots in fluid mechanics and developed recently for the purposes of electronics and aeronautics. In the standard singular perturbation-model of an ordinary differential equation (control system) some of the derivatives are multiplied by a scalar parameter, that is \(\dot x=f(x,y,u,t)\), \(\epsilon\dot y=g(x,y,u,t).\)

Chapter 1 (Time Scale Modeling) exhibits some of the basic concepts of singular perturbation asymptotics and time-scale modeling by way of illustrative examples. Chapter 2 presents the main results concerning linear time-invariant control systems. Chapter 3 deals with the linear feedback control. The respective modifications of the techniques for linear stochastic filtering and control are given in Chapter 4. Chapter 5 considers linear time-varying systems. Chapter 6 discusses some singularly perturbed optimal control problems. In Chapter 7 some stability question are considered.

The book is well written, with accurate mathematics and many examples and exercises, including real life processes. The extended list of references covers practically all existing literature.

Chapter 1 (Time Scale Modeling) exhibits some of the basic concepts of singular perturbation asymptotics and time-scale modeling by way of illustrative examples. Chapter 2 presents the main results concerning linear time-invariant control systems. Chapter 3 deals with the linear feedback control. The respective modifications of the techniques for linear stochastic filtering and control are given in Chapter 4. Chapter 5 considers linear time-varying systems. Chapter 6 discusses some singularly perturbed optimal control problems. In Chapter 7 some stability question are considered.

The book is well written, with accurate mathematics and many examples and exercises, including real life processes. The extended list of references covers practically all existing literature.

Reviewer: A.Dontchev

##### MSC:

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

34E15 | Singular perturbations, general theory for ordinary differential equations |

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

34D15 | Singular perturbations of ordinary differential equations |

93B50 | Synthesis problems |

93C05 | Linear systems in control theory |

93C95 | Application models in control theory |

93E11 | Filtering in stochastic control theory |

93E20 | Optimal stochastic control |