##
**Control system synthesis: a factorization approach.**
*(English)*
Zbl 0655.93001

One of the classic debates in systems and control theory (of course, never to be definitively resolved) is the use of state space methods as opposed to input-output techniques usually used in the frequency domain. Proponents of the former methodology tend to prefer linear algebraic methods while proponents of the latter tend to use complex analytic and operator-theoretic techniques. The factorization approach discussed in the book under review is, in a certain sense, a compromise between these two schools of thought and the mathematics discussed is therefore an interesting mix of all of the above.

The author begins his book with a discussion (in Chapter 2) on the ring S of proper stable rational functions. This he shows to be a Euclidean domain. He topologizes S via the sup norm and considers the matrix analogue of this ring. Finally a survey of some results on interpolation in the disc algebra wraps up this preliminary chapter.

In Chapter 3 the author considers some famous results of D. C. Youla (and his co-workers J. J. Bongiorno, C. N. Lu and H. A. Jabr). Explicitly he gives a parametrization of all stabilizing compensators for a given plant in terms of the Bezoutian identity associated to a coprime factorization of the system. Moreover, he discusses the important necessary and sufficient condition derived by Youla and his co-workers, guaranteeing the existence of a stable stabilizing controller for a given plant. These results have had an important influence on the modern robustness literature in systems and control theory.

Chapters 4 and 5 are essentially devoted to the generalization of the previous chapter to multiple input-multiple output systems. Once more, many of the important results are due to Youla and his colleagues. The author also gives some of his own work here, done in collaboration with N. Viswanadham, on the simultaneous stabilization problem (using one compensator to simultaneously internally stabilize a finite number of given plants).

Chapter 6 is one of the most important chapters in this book, since here the author discusses the sensitivity minimization problem, originated and developed by G. Zames, and worked on extensively by numerous engineers and mathematicians. This area has become known as \(H^{\infty}\) opimization theory and has motivated much research in control and applied operator theory. The author states the problem and discusses its solution using matrix Nevanlinna-Pick interpolation theory. We should add that the sensitivity minimization problem, which was motivated completely by engineering considerations, turns out to be equivalent to the Nehari problem (which in the scalar case amounts to finding the distance of an \(L^{\infty}\) function to \(H^{\infty})\) and hence is amenable to the techniques developed by the mathematicians Arov-Adamyan-Kreĭn, Sarason-Sz.-Nagy-Foiaş, and Ball-Helton.

Chapter 7 is, in general, connected with the topic of robustness, and in particular the author’s own work on the graph metric which defines a topology relative to which continuity becomes a necessary and sufficient condition for feedback stability to be robust. Chapter 8 discusses some extensions of the previous chapters to more general settings (systems defined over certain rings of functions), and then the book is completed by several appendices on algebra and topology.

As is clear from the above, this book covers a wide range of topics collected for the first time in a unified setting. Therefore, it should prove to be an important and essential reference to researchers in the field. In general the book is well written and the author demonstrates his mastery of a very large amount of material, both mathematical and control-theoretic. However, there are a number of typographical errors and some erroneous statements in the text. This may make it somewhat difficult for nonexperts and graduate students to read the book. This is a pity, precisely because of the importance and relevance of this material to a fast-growing and central area in systems and control theory.

The author begins his book with a discussion (in Chapter 2) on the ring S of proper stable rational functions. This he shows to be a Euclidean domain. He topologizes S via the sup norm and considers the matrix analogue of this ring. Finally a survey of some results on interpolation in the disc algebra wraps up this preliminary chapter.

In Chapter 3 the author considers some famous results of D. C. Youla (and his co-workers J. J. Bongiorno, C. N. Lu and H. A. Jabr). Explicitly he gives a parametrization of all stabilizing compensators for a given plant in terms of the Bezoutian identity associated to a coprime factorization of the system. Moreover, he discusses the important necessary and sufficient condition derived by Youla and his co-workers, guaranteeing the existence of a stable stabilizing controller for a given plant. These results have had an important influence on the modern robustness literature in systems and control theory.

Chapters 4 and 5 are essentially devoted to the generalization of the previous chapter to multiple input-multiple output systems. Once more, many of the important results are due to Youla and his colleagues. The author also gives some of his own work here, done in collaboration with N. Viswanadham, on the simultaneous stabilization problem (using one compensator to simultaneously internally stabilize a finite number of given plants).

Chapter 6 is one of the most important chapters in this book, since here the author discusses the sensitivity minimization problem, originated and developed by G. Zames, and worked on extensively by numerous engineers and mathematicians. This area has become known as \(H^{\infty}\) opimization theory and has motivated much research in control and applied operator theory. The author states the problem and discusses its solution using matrix Nevanlinna-Pick interpolation theory. We should add that the sensitivity minimization problem, which was motivated completely by engineering considerations, turns out to be equivalent to the Nehari problem (which in the scalar case amounts to finding the distance of an \(L^{\infty}\) function to \(H^{\infty})\) and hence is amenable to the techniques developed by the mathematicians Arov-Adamyan-Kreĭn, Sarason-Sz.-Nagy-Foiaş, and Ball-Helton.

Chapter 7 is, in general, connected with the topic of robustness, and in particular the author’s own work on the graph metric which defines a topology relative to which continuity becomes a necessary and sufficient condition for feedback stability to be robust. Chapter 8 discusses some extensions of the previous chapters to more general settings (systems defined over certain rings of functions), and then the book is completed by several appendices on algebra and topology.

As is clear from the above, this book covers a wide range of topics collected for the first time in a unified setting. Therefore, it should prove to be an important and essential reference to researchers in the field. In general the book is well written and the author demonstrates his mastery of a very large amount of material, both mathematical and control-theoretic. However, there are a number of typographical errors and some erroneous statements in the text. This may make it somewhat difficult for nonexperts and graduate students to read the book. This is a pity, precisely because of the importance and relevance of this material to a fast-growing and central area in systems and control theory.

### MSC:

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

93B28 | Operator-theoretic methods |

93B25 | Algebraic methods |

93B35 | Sensitivity (robustness) |

93B50 | Synthesis problems |

93D15 | Stabilization of systems by feedback |

30D55 | \(H^p\)-classes (MSC2000) |

30E05 | Moment problems and interpolation problems in the complex plane |

46E15 | Banach spaces of continuous, differentiable or analytic functions |

54H20 | Topological dynamics (MSC2010) |