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**Stochastic optimal control theory with application in self-tuning control.**
*(English)*
Zbl 0667.93090

Lecture Notes in Control and Information Sciences, 117. Berlin etc.: Springer-Verlag. x, 308 p. DM 74.00 (1989).

The book covers three parts: stochastic optimal control theory, self- tuning control, and a case study.

The first part is mainly theoretical and derives stochastic optimal control theory using the polynomial approach. This gives a good supplement to the traditional way of solving the problem. The resulting controller is obtained by solving polynomial equations such as Diophantine and spectral factorization equations. Feedforward from measurable disturbances are incorporated in the solution. A major contribution of the book is the extension to the cases of unstable reference and disturbance generators.

The second part of the book discusses how the stochastic optimal controller can be transformed into self-tuning controllers. The main idea is to make an estimation of the process parameters and then apply the results from the first part. This leads to indirect adaptive controllers. The robustness properties of the resulting self-tuner is discussed.

The third part of the book uses the self-tuning controller from part two on a simulated model of a gas-cooled nuclear reactor. The model is the same as used in a simulator for training of operators. This implies that the case study reflects many of the practical problems encountered when applying adaptive controllers.

In summary the book gives a coherent treatment of linear quadratic Gaussian self-tuning regulators.

The first part is mainly theoretical and derives stochastic optimal control theory using the polynomial approach. This gives a good supplement to the traditional way of solving the problem. The resulting controller is obtained by solving polynomial equations such as Diophantine and spectral factorization equations. Feedforward from measurable disturbances are incorporated in the solution. A major contribution of the book is the extension to the cases of unstable reference and disturbance generators.

The second part of the book discusses how the stochastic optimal controller can be transformed into self-tuning controllers. The main idea is to make an estimation of the process parameters and then apply the results from the first part. This leads to indirect adaptive controllers. The robustness properties of the resulting self-tuner is discussed.

The third part of the book uses the self-tuning controller from part two on a simulated model of a gas-cooled nuclear reactor. The model is the same as used in a simulator for training of operators. This implies that the case study reflects many of the practical problems encountered when applying adaptive controllers.

In summary the book gives a coherent treatment of linear quadratic Gaussian self-tuning regulators.

Reviewer: B.Wittenmark

### MSC:

93E20 | Optimal stochastic control |

93B35 | Sensitivity (robustness) |

65C20 | Probabilistic models, generic numerical methods in probability and statistics |

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

93C40 | Adaptive control/observation systems |

49J55 | Existence of optimal solutions to problems involving randomness |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

11D41 | Higher degree equations; Fermat’s equation |