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**Linear multivariable control. A geometric approach. 3rd ed.**
*(English)*
Zbl 0609.93001

Applications of Mathematics, 10. New York etc.: Springer-Verlag. XVI, 334 p. DM 126.00 (1985).

Since the first edition of this book appeared in 1974, followed by a second one in 1979, the geometric theory of linear dynamical systems has undergone a great development. Unfortunately, for the reader there is scarcely a trace of this in this new edition of Wonham’s book, which does not differ from the previous one, except for minor changes and some additions to the list of references. The content of the book has already been reviewed [see the reviews of the first and second edition, Zbl 0314.93007 and Zbl 0424.93001].

After some preliminaries, the fundamental ”Pole assignment theorem” is proved in Chapter 2. It states that, for finite-dimensional linear systems over \({\mathbb{R}}\), controllability is equivalent to the possibility of moving the poles, by a state feedback, to any symmetric set of complex numbers. In Chapter 3, results on observability and detectability are obtained by duality. The important geometric notion of controlled invariant subspace, or (A,B)-invariant subspace, of a control system \(\dot x(t)=Ax(t)+Bu(t)\), is introduced and discussed in Chapter 4. A subspace V of the state space X is (A,B)-invariant if there exists a state feedback \(F: X\to U\) such that V is \((A+BF)\)-invariant. Controlled invariant subspaces are used to solve a disturbance decoupling problem and an output stabilization problem. Controllability subspaces and controllability indices are treated in Chapter 5. The geometric machinery so far developd is used, in the following Chapters 6-11, to deal with various synthesis problems, such as servo and regulator problems, decoupling problems and construction of minimal order compensators. The last two Chapters contain a standard treatment of linear optimal control.

After some preliminaries, the fundamental ”Pole assignment theorem” is proved in Chapter 2. It states that, for finite-dimensional linear systems over \({\mathbb{R}}\), controllability is equivalent to the possibility of moving the poles, by a state feedback, to any symmetric set of complex numbers. In Chapter 3, results on observability and detectability are obtained by duality. The important geometric notion of controlled invariant subspace, or (A,B)-invariant subspace, of a control system \(\dot x(t)=Ax(t)+Bu(t)\), is introduced and discussed in Chapter 4. A subspace V of the state space X is (A,B)-invariant if there exists a state feedback \(F: X\to U\) such that V is \((A+BF)\)-invariant. Controlled invariant subspaces are used to solve a disturbance decoupling problem and an output stabilization problem. Controllability subspaces and controllability indices are treated in Chapter 5. The geometric machinery so far developd is used, in the following Chapters 6-11, to deal with various synthesis problems, such as servo and regulator problems, decoupling problems and construction of minimal order compensators. The last two Chapters contain a standard treatment of linear optimal control.

Reviewer: G.Conte

### MSC:

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

93B27 | Geometric methods |

93C05 | Linear systems in control theory |

47A15 | Invariant subspaces of linear operators |

93B05 | Controllability |

93B07 | Observability |

93B50 | Synthesis problems |

93B55 | Pole and zero placement problems |

93Dxx | Stability of control systems |

93C35 | Multivariable systems, multidimensional control systems |