The research concretized in this memory is located at the intersection of two important fields, the robust control of discrete-time linear systems (LTI, LPV, switched) affected by bounded disturbances and constraints and the ellipsoidal invariant sets theory.The first part of this memory focuses on the analysis of input-to-state stability (ISS) over a bounded perturbation and the computation of the maximal or minimal invariant ellipsoidal (or truncated ellipsoidal) set satisfying the constraints. The second part is considering the synthesis of a control state feedback law ISS stable and robust over bounded disturbances, ensuring the maximal ellipsoidal invariant set satisfying the constraints, then the synthesis of an observer-based control law ISS stable over bounded disturbances,ensuring a certain performance, and finally the design of a Youla parameter guaranteeing the maximal ellipsoidal projection on the initial state subspace. The resulting projection has a volume greater than the one obtained without the Youla parameter resulting an improvement in terms of robustness. A final step is to obtain a compromise between robustness and performance using criteria based on poles placement or on theLyapunov function decreasing rate. The theoretical results are expressed as matrix inequalities and are validated in simulation and and experimentally on a Buck DC-DC converter.
