Robust control of linear descriptor systems (Q2359213)

Linear Descriptor Systems (LDS) are described by an equation of the form \(E \dot{x}=Ax+Bu\), and \(E=I_{n}\) in the classic case of usual state-space systems. Descriptor systems of differential equations have been studied for a long time mathematically [\textit{Weierstrass}, Berl. Monatsber. 1868, 310--338 (1868; JFM 01.0054.04)], and from the control point of view by \textit{G. C. Verghese} et al. [IEEE Trans. Autom. Control 26, 811--831 (1981; Zbl 0541.34040)] and many others. Their algebraic properties are not different from those of all linear systems, as was shown by \textit{M. Fliess} [Syst. Control Lett. 15, No. 5, 391--396 (1990; Zbl 0727.93024)] in the time-varying case (this reference is not quoted in the book). The originality of the book under review is that LDS are studied from the point of view of robust control (Chapter 3 and following). The \(E\) matrix modifies classic linear matrix inequalities (LMI) of \(H_{2}\) or \(H_{\infty }\) control, in such a way that these classic LMI are recovered bu putting \( E=I_{n}.\) Chapter 4 generalizes the Davison-Wonham internal model principle to LDS and adds dissipativity requirements. In Chapter 5, the weighting problem for, e.g., \(H_{\infty }\) control, is considered. \ It is emphasized that for many practical problems, unstable and/or nonproper weights must be used. Nonproper weights can be easily represented by LDS, and this is clearly an advantage of this formalism. \ The corresponding \(H_{2}\) and \(H_{\infty }\) problems are solved in Chapter 6, both of them illustrated through an example. A desensitized LQ controller is considered in Chapter 7, but here the comparison with the literature is far from being exhaustive. The book ends with a list of references and an index.