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Linear-quadratic control. An introduction. (English) Zbl 0838.93004

Englewood Cliffs, NJ: Prentice Hall. xiv,205 p. (1995).
The book contains in a compact form an exellent survey of the problems related to the linear-quadratic control problem. In the successive chapters both the classical as well as more recent problems are very skillfully presented and discussed.
After an introduction, the classical linear-quadratic regulator (LQR) problem (with different versions) is described in chapter 2. The solution to the problem is obtained from the Hamilton-Jacobi equation. Resulting from this equation, the sufficient conditions of optimality for a more general case are also described. Chapter 3 contains tracking and disturbance rejection problems based on the LQ approach. In chapter 4 the robustness properties of the LQR problem are discussed. Especially, Kalman’s identity, Kalman’s inequality and their implications are presented. Chapter 5 contains the basic notions of stochastic control. First, stochastic differential equations are described, then the stochastic Hamilton-Jacobi equation and the sufficient conditions of optimality related to it are formuated. Control problems with additive and multiplicative disturbances are also considered.
The linear-quadratic Gaussian (LQG) problem is described in chapter 6. First, the Kalman-Bucy filter is derived and then, the LQG solution and the separation principle are presented. Finally, fixed-order compensators are discussed. Chapter 7 contains a compact description on robust design. Especially, the LQR/LTR (loop transfer recovery) design, the guaranteed-cost design and a brief survey of \(H^\infty\) methods are given. Finally, chapter 8 contains a description of digital control in which the discrete-time LQR problem plays a crucial role.
Each chapter contains examples and computation which use the MATLAB software. Some possibilities of the MATLAB software are also discussed. Each chapter contains a discussion on the cited references and a set of problems for solving.
The book is very well written, and is easy to read. It contains relatively simple and very skillful derivations; it avoids extensive mathematical proofs. However, the reader should have an appropriate mathematical preparation and a knowledge concerning control problems.

MSC:

93-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory
93B35 Sensitivity (robustness)
93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
93B51 Design techniques (robust design, computer-aided design, etc.)
93E20 Optimal stochastic control

Software:

Matlab
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