For continuous-time descriptor systems with all system matrices incorporated with norm-bounded uncertainties, this thesis addresses robust admissibility and robust H¡Û analysis and the related state feedback design. The results are further extended to systems with time-varying state delay within a known interval. The former part of the thesis extends the current research of considering uncertainty only at the state derivative matrix to the case uncertainty being assumed at all system matrices. While the latter part of the thesis extends the current research in two folds: the state derivative matrix is allowed to be uncertain and the delay is allowed to be time-varying. Since all the results are derived in the LMI-based framework, examples with efficient numerical verifications are included to illustrate the derived results.

In many applications from industry and technology computer simulations are performed using models which can be formulated by systems of differential equations. Often the equations underlie additional algebraic constraints. In this context we speak of descriptor systems. Very important characteristic values of such systems are the $L_\infty$-norms of the corresponding transfer functions. The main goal of this thesis is to extend a numerical method for the computation of the $L_\infty$-norm for standard state space systems to descriptor systems. For this purpose we develop a numerical method to check whether the transfer function of a given descriptor system is proper or improper and additionally use this method to reduce the order of the system to decrease the costs of the $L_\infty$-norm computation. When computing the $L_\infty$-norm it is necessary to compute the eigenvalues of certain skew-Hamiltonian/Hamiltonian matrix pencils composed by the system matrices. We show how we extend these matrix pencils to skew-Hamiltonian/Hamiltonian matrix pencils of larger dimension to get more reliable and accurate results. We also consider discrete-time systems, apply the extension strategy to the arising symplectic matrix pencils and transform these to more convenient structures in order to apply structure-exploiting eigenvalue solvers to them. We also investige a new structure-preserving method for the computation of the eigenvalues of skew-Hamiltonian/Hamiltonian matrix pencils and use this to increase the accuracy of the computed eigenvalues even more. In particular we ensure the reliability of the $L_\infty$-norm algorithm by this new eigenvalue solver. Finally we describe the implementation of the algorithms in Fortran and test them using two real-world examples.

Les systèmes implicites (dits aussi « descripteurs ») peuvent décrire des processus régis à la fois par des équations dynamiques et statiques et permettent de préserver la structure des systèmes physiques. Ils comportent trois types de modes : dynamiques finis, infinis (réponse temporelle impulsive (en cas continu) ou acausale (en cas discret)) et statiques. Dans le cadre du formalisme descripteur, les contributions de cette thèse sont triples : i) revisiter des résultats existants pour les systèmes d’état, ii) étendre certains résultats classiques au cas des systèmes implicites, iii) résoudre rigoureusement des problèmes de commande non standard. Ainsi, le présent mémoire commence par revisiter les résultats concernant la caractérisation LMI stricte de la dissipativité, les caractérisations de l’admissibilité et des performances H2 ou H∞ par LMI étendues et les équations de Sylvester et de Riccati généralisées. Il aborde dans un deuxième temps, le problème de stabilisation simultanée, avec ou sans critère H∞, à travers l’extension de certains résultats récents au cas des systèmes implicites. La solution proposée s’appuie sur la résolution combinée d’une équation algébrique de Riccati généralisée (GARE) et d’un problème de faisabilité sous contrainte LMI stricte. Il traite enfin des problèmes H2 et H∞ non standards : i) en présence de pondérations instables voire impropres, ii) sous contraintes de régulation; dans le cas des systèmes implicites. Ces dernières contributions permettent désormais de traiter rigoureusement, sans approximations ou transformations, de nombreux problèmes H2 ou H∞ formalisant des problèmes pratiques de commande, dont ceux faisant intervenir une pénalisation haute fréquence de la commande ou un modèle interne instable des signaux exogènes. / The descriptor systems have been attracting the attention of many researchers over recent decades due to their capacity to preserve the structure of physical systems and to describe static constraints and impulsive behaviors. Within the descriptor framework, the contributions of this dissertation are threefold: i) review of existing results for state-space systems, ii) generalization of classical results to descriptor systems, iii) exact and analytical solutions to non standard control problems. A realization independent Kalman-Yakubovich-Popov (KYP) lemma and dilated LMI characterizations are deduced for descriptor systems. The solvability and corresponding numerical algorithms of generalized Sylvester equations and generalized algebraic Riccati equations (GARE) associated with descriptor systems are provided. In addition, the simultaneous H∞ control problem is considered through extending recently reported results. A sufficient condition is proposed through a combination of a generalized algebraic Riccati equation and a set of LMIs. Moreover, the nonstandard H2 and H∞ control problems with unstable and/or nonproper weighting functions or subject to regulation constraints are addressed. These contributions allow, without approximation or transformation, dealing with many practical problems defined within H2 or H∞ control methodologies, where the control signals are penalized at high frequency or unstable internal models specified by external signals is involved.

In many applications from industry and technology computer simulations are performed using models which can be formulated by systems of differential equations. Often the equations underlie additional algebraic constraints. In this context we speak of descriptor systems. Very important characteristic values of such systems are the $L_\infty$-norms of the corresponding transfer functions. The main goal of this thesis is to extend a numerical method for the computation of the $L_\infty$-norm for standard state space systems to descriptor systems. For this purpose we develop a numerical method to check whether the transfer function of a given descriptor system is proper or improper and additionally use this method to reduce the order of the system to decrease the costs of the $L_\infty$-norm computation. When computing the $L_\infty$-norm it is necessary to compute the eigenvalues of certain skew-Hamiltonian/Hamiltonian matrix pencils composed by the system matrices. We show how we extend these matrix pencils to skew-Hamiltonian/Hamiltonian matrix pencils of larger dimension to get more reliable and accurate results. We also consider discrete-time systems, apply the extension strategy to the arising symplectic matrix pencils and transform these to more convenient structures in order to apply structure-exploiting eigenvalue solvers to them. We also investige a new structure-preserving method for the computation of the eigenvalues of skew-Hamiltonian/Hamiltonian matrix pencils and use this to increase the accuracy of the computed eigenvalues even more. In particular we ensure the reliability of the $L_\infty$-norm algorithm by this new eigenvalue solver. Finally we describe the implementation of the algorithms in Fortran and test them using two real-world examples.

Dans ce mémoire nous avons développé des méthodes de réduction des systèmes hamiltoniens à port en dimension finie et infinie qui préservent leur structure. Dans la première partie, nous avons défini une représentation des systèmes hamiltoniens à port avec contraintes sous la forme d'équations différentielles algébriques (DEA) de type de système descripteur. De cette forme nous avons déduit une réalisation équilibrée du système hamiltonien à port exprimée sous forme de système descripteur contenant les mêmes systèmes d'équations de contrainte. Dans la deuxième partie, nous avons défini une classe de problèmes de commande LQG tels que le contrôleur dynamique LQG est passif et admet une réalisation hamiltonien à port. Deux méthodes de synthèse de commande passive LQG sont proposées et une de ces méthodes LQG nous a permis de définir une réalisation équilibrée LQG. Puis nous avons appliqué la méthode de contrainte de l'effort pour réduire le système hamiltonien à port et obtenir une commande LQG passive d'ordre réduit. Ce contrôleur LQG admettant une réalisation hamiltonienne, la structure hamiltonienne est préservée pour le système en boucle fermée par interconnexion de systèmes hamiltoniens à port. Dans la troisième partie, nous avons généralisé les résultats précédents aux systèmes hamiltoniens à ports linéaires de dimension infinie. Pour cela nous avons considéré une classe de systèmes hamiltoniens à ports de dimension infinie dont l'opérateur d'entrée est borné et un problème de commande LQG passif. Sous des conditions de nucléarité de l'opérateur de Hankel lié au problème LQG, nous définissons une réalisation équilibrée LQG passive du système et une approximation en dimension finie. Le contrôleur LQG passif d'ordre réduit obtenu par cette approximation admet une réalisation hamiltonienne à port et par conséquent la structure hamiltonienne et la passivité sont préservées en boucle fermée / In this thesis we have developed different structure preserving reduction methods for finite and infinite dimensional port Hamiltonian systems by using a balanced model reduction approach. In the first part we have defined a descriptor representation of port Hamiltonian systems with constraints. The balanced realization of the descriptor system has been used for reducing the port Hamiltonian descriptor system and conserving explicitly the constraint equations. In the second part, conditions have been derived on the weighting matrices of the LQG control problem such that the dynamical LQG controller is passive and has a port Hamiltonian realization. Two passive LQG control design methods have been suggested and one of them allows us to define a LQG balanced realization. Based on this realization, the effort constraint method has been used to reduce the LQG balanced port Hamiltonian system and obtain a reduced order passive LQG controller. In this way the closed-loop system is derived from the interconnection of 2 port Hamiltonian systems, hence the Hamiltonian structure has been preserved. In the third part, the proceeding results have been extended to a class of infinite dimensional port Hamiltonian system with bounded input operator. A passive LQG control design method for infinite dimensional port Hamiltonian system has been derived as by Control by Interconnection (CbI). Based on the balanced realization associated with this passive LQG control design, a finite dimensional approximation has been achieved and a reduced order passive LQG controller has been derived. As a consequence, the system in closed-loop with this reduced order LQG controller again admits a port Hamiltonian structure and satisfies the passivity