On the Factorization of Rational Discrete-Time Spectral Densities

In this paper, we consider an arbitrary matrix-valued, rational spectral density $\Phi(z)$. We show with a constructive proof that $\Phi(z)$ admits a factorization of the form $\Phi(z)=W^\top (z^{-1})W(z)$, where $W(z)$ is stochastically minimal. Moreover, $W(z)$ and its right inverse are analytic in regions that may be selected with the only constraint that they satisfy some symplectic-type conditions. By suitably selecting the analyticity regions, this extremely general result particularizes into a corollary that may be viewed as the discrete-time counterpart of the matrix factorization method devised by Youla in his celebrated work (Youla, 1961).Comment: 34 pages, no figures. Revised version with partial rewriting of Section I and IV, added Section VI with a numerical example and other minor changes. To appear in IEEE Transactions of Automatic Contro

Optimal LQ-Feedback regulation of a nonisothermal plug flow reactor model by spectral factorization

The linear-quadratic (LQ) optimal temperature and reactant concentration regulation problem is studied for a partial differential equation model of a nonisothermal plug flow tubular reactor by using a nonlinear infinite dimensional Hilbert state space description. First the dynamical properties of the linearized model around a constant temperature equilibrium profile along the reactor are studied: it is shown that it is exponentially stable and (approximately) reachable. Next the general concept of LQ-feedback is introduced., It turns out that any LQ-feedback is optimal from the input-output viewpoint and stabilizing. For the plug flow reactor linearized model, a state LQ-feedback operator is computed via the solution of a matrix Riccati differential equation (MRDE) in the space variable. Thanks to the reachability property, the computed LQ-feedback is actually the optimal one. Then the latter is applied to the nonlinear model, and the resulting closed-loop system dynamical performances are analyzed. A criterion is given which guarantees that the constant temperature equilibrium profile is an asymptotically stable equilibrium of the closed-loop system. Moreover, under the same criterion, it is shown that the control law designed previously is optimal along the nonlinear closed-loop model with respect to some cost criterion. The results are illustrated by some numerical simulations

Contributions à la commande prédictive des systèmes de lois de conservation

La Commande prédictive ou Commande Optimale à Horizon Glissant (COHG) devient de plus en plus populaire dans de nombreuses applications pratiques en raison de ses avantages importants tels que la stabilisation et la prise en compte des contraintes. Elle a été bien étudiée pour des systèmes en dimension finie même dans le cas non linéaire. Cependant, son extension aux systèmes en dimension infinie n'a pas retenu beaucoup d'attention de la part des chercheurs. Ce travail de thèse apporte des contributions à l'application de cette approche aux systèmes de lois de conservation. Nous présentons tout d'abord une preuve de stabilité complète de la COHG pour certaines classes de systèmes en dimension infinie. Ce résultat est ensuite utilisé pour les systèmes hyperboliques 2x2 commandés aux frontières et appliqué à un problème de contrôle de canal d'irrigation. Nous proposons aussi l'extension de cette stratégie au cas de réseaux de systèmes hyperboliques 2x2 en cascade avec une application à un ensemble de canaux d'irrigation connectés. Nous étudions également les avantages de la COHG dans le contexte des systèmes non linéaires et semi-linéaires notamment vis-à-vis des chocs. Toutes les analyses théoriques sont validées par simulation afin d'illustrer l'efficacité de l'approche proposée.The predictive control or Receding Horizon Optimal Control (RHOC) is becoming increasingly popular in many practical applications due to its significant advantages such as the stabilization and constraints handling. It has been well studied for finite dimensional systems even in the nonlinear case. However, its extension to infinite dimensional systems has not received much attention from researchers. This thesis proposes contributions on the application of this approach to systems of conservation laws. We present a complete proof of stability of RHOC for some classes of infinite dimensional systems. This result is then used for 2x2 hyperbolic systems with boundary control, and applied to an irrigation canal. We also propose the extension of this strategy to networks of cascaded 2x2 hyperbolic systems with an application to a set of connected irrigation canals. Furthermore, we study the benefits of RHOC in the context of nonlinear and semi-linear systems in particular with respect to the problem of shocks. All theoretical analyzes are validated by simulation in order to illustrate the effectiveness of the proposed approach.SAVOIE-SCD - Bib.électronique (730659901) / SudocGRENOBLE1/INP-Bib.électronique (384210012) / SudocGRENOBLE2/3-Bib.électronique (384219901) / SudocSudocFranceF