##
**Optimal control. Linear quadratic methods.**
*(English)*
Zbl 0751.49013

Prentice Hall Information and System Sciences Series. Englewood Cliffs, NJ: Prentice Hall. xi, 380 p. (1990).

This book is the result of revising an earlier work by the authors [“Linear optimal control” (1971; Zbl 0321.49001)], motivated by the fact that in the past two decades a variety of directions in optimal control have evolved. The title has been changed to focus on linear- quadratic methods, as opposed to \(H^ \infty\) and \(L_ 1\) methods. Hence the book deals with linear plants and linear controllers to be designed such as to minimize quadratic performance criteria. Therefore material on relay control systems and dual-mode controllers has been omitted, whereas material on second variation theory, frequency shaping, loop recovery and controller reduction has been added. Many sections have been rewritten, especially on robustness and tracking. The aim of the book is to point out the engineering properties of the solution to the problems presented and to connect them to classical results and ideas.

The book consists of three major parts. Part I (chapters 1 to 4) introduces and outlines the basic theory of linear regulator/tracker, emphasizing time-invariant systems. The Hamilton-Jacobi equation is introduced using the principle of optimality. Finite-time as well as infinite-time problems are considered, including regulator design with a described degree of stability.

Part II (chapters 5 and 6) focusses on the engineering properties of the optimal regulator, such as sensitivity and robustness based on sensitivity functions and singular values. Also the effect of sector nonlinearities and time delay is studied. The relationship between quadratic index weight selection and closed-loop eigenvalues is investigated.

Part III (chapters 7 to 11) considers both deterministic and stochastic state estimation (Kalman-Bucy filter) robust controller design using state estimate feedback, loop transmission recovery and frequency shaping techniques. Since this approach may result in controllers of unacceptable high order, controller reduction methods are presented. Some practical aspects of digital implementation such as sampling time selection and anti-aliasing filter are discussed.

Some theoretical background relevant to the material in the book is summarized in the appendices.

This is a fine textbook written be leading authorities in the field of automatic control. It constructs bridges between familiar classical control engineering and modern control theory and is therefore recommended to graduate students as well as to practicing control engineers.

The book consists of three major parts. Part I (chapters 1 to 4) introduces and outlines the basic theory of linear regulator/tracker, emphasizing time-invariant systems. The Hamilton-Jacobi equation is introduced using the principle of optimality. Finite-time as well as infinite-time problems are considered, including regulator design with a described degree of stability.

Part II (chapters 5 and 6) focusses on the engineering properties of the optimal regulator, such as sensitivity and robustness based on sensitivity functions and singular values. Also the effect of sector nonlinearities and time delay is studied. The relationship between quadratic index weight selection and closed-loop eigenvalues is investigated.

Part III (chapters 7 to 11) considers both deterministic and stochastic state estimation (Kalman-Bucy filter) robust controller design using state estimate feedback, loop transmission recovery and frequency shaping techniques. Since this approach may result in controllers of unacceptable high order, controller reduction methods are presented. Some practical aspects of digital implementation such as sampling time selection and anti-aliasing filter are discussed.

Some theoretical background relevant to the material in the book is summarized in the appendices.

This is a fine textbook written be leading authorities in the field of automatic control. It constructs bridges between familiar classical control engineering and modern control theory and is therefore recommended to graduate students as well as to practicing control engineers.

Reviewer: D.Franke (Hamburg)

### MSC:

49N10 | Linear-quadratic optimal control problems |

49N05 | Linear optimal control problems |

49-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control |

49N35 | Optimal feedback synthesis |

93E10 | Estimation and detection in stochastic control theory |

93E24 | Least squares and related methods for stochastic control systems |