Extended H2 - H∞ controller synthesis for linear time invariant descriptor systems. (Commande H2 - H∞ non standard des systèmes implicites)

Descriptor systems constitute an important class of systems of both theoretical and practical interests and have been attracting the attention of many researchers over recent decades. This dissertation is concerned with non-standard H2 and H∞ control for linear time-invariant descriptor systems. Within the descriptor framework, the contributions of this dissertation are threefold: i) review of existing results for state-space systems by the use of the descriptor representation, ii) generalizations of some classical results to descriptor systems, iii) exact and analytical solutions to nonstandard control problems with unstable and nonproper weighting functions or subject to regulation constraints. The first part of this dissertation is concerned with a development of useful tools for analysis and control synthesis for descriptor systems. A realization independent Kalman-Yakubovich-Popov lemma is deduced in terms of strict linear matrix inequalities (LMIs) for descriptor systems. This new condition removes the equality constraints found in the reported results and outperforms the existing methods in the viewpoint of numerical computation. The issue of dilated LMI characterizations, which have been extensively studied for conventional state-space systems, is also investigated in this dissertation. Based on reciprocal application of the projection lemma, dilated LMI conditions with regard to admissibility, H2 and dissipative properties are derived for both continuous-time and discrete-time settings. The proposed formulations review the existing results reported in the literature, and also complete some missing conditions. The known dilated LMIs for state-space systems can be regarded as special cases of the proposed results. As known, the Sylvester equations and Riccati equations play an important role in control theory, and some control issues are directly concerned with these equations. This dissertation deals with these two topics for descriptor systems as well. The solvability of a generalized Sylvester equation is transformed into a linear programming problem which can be solved efficiently using available techniques. Moreover, a generalized algebraic Riccati equation (GARE) is considered and a numerical algorithm relying on a generalized eigenvalue problem (GEP) is given for solving it. Moreover, the strong H∞ stabilization and simultaneous H∞ control problems for continuous-time descriptor systems are considered. As a generalization of the existing