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Problem statement: More advanced control techniques have been developed in recent decades following the great progress of calculation s means. Without considering the constraints on system variables, the response of controlled system moves away from the desired response. Hence the control strategy must provide the ability to integr ate these constraints in the design phase of the controller. Approach: This study presents a design of robust dynamic con troller which was based on the control strategy MPC for an uncertain discrete system described by a multimodel by solving an optimization problem. The Model Predictive Control (MPC) strategy uses a dynamic model of the process in order to predict its future behavior. Th is control strategy, that we propose, makes it poss ible to integrate these constraints in the design phase of the controller. The design of the robust dynamic controller must maintain the stability and performa nce of the system in the presence of uncertainties. The principle of this method was to solve, at each calculation step, a convex optimization problem tha t calculates the matrices characterizing the dynamic controller. LMI formulation of the constraints, on process variables, was introduced into the design p hase of the dynamic controller robust. The optimization study was also implemented in the MATLAB software and simulation studies had been presented. Results: Simulation results had proved the effectiveness of this study. The robust dynamic controller designed for uncertain systems guarantee s stability in closed loop systems by integrating constraints on system variables. Conclusion: The described approach explain how to integrate constraints, MPC type, during the phase of design o f the robust controller: the simulation concluded o n a benchmark proof the powerful of this approach.

Problem statement: The analysis and control of delayed systems are be coming more and more research topics in progress. This is mainly du e to the fact that the delay is frequently encounte red in technological systems. This can affect their sig nificantly operations. Most control command laws are based on current digital computers and delays a re intrinsic to the process or in the control loop caused by the transmission time control sequences, or computing time. The delay may affect one or more states of the considered system. It may also a ffect the establishment of the command. Several studies have investigated the stability of delay sy stems under the assumption that the delay is a vari able phenomenon; such variation is considered to be bounded or limited to facilitate analysis of the system . In this study we propose a modelling of delayed sys tem by using the multimodels and switched system theory. The analysis of stability is based on the u se of second Lyapunov method. The issued stability conditions are expressed as Bilinear Matrix Inequal ities impossible to resolve. That's why we propose the same original relaxations to come over this dif ficulty, an example of induction machine is given t o illustrate over approach. Approach: We propose to use the control theory developed for switched systems to synthesis a control laws for the stabili sation of delays system. Results: We stabilize the induction machine around many operating points despite the non linearities. Conclusion: The developed method is less conservative and less pess imistic than the used classical methods.

Problem statement: The estimation of states and the unknown inputs of a nonlinear system described by a multimodel are done by a multiobserver. The stabilization of the multiobserver calls upon uses both quadratic and no quadratic functions of Lyapunov. Although the stabilization using the quadratic approach is interesting from the point of view implementation, the step showed its limits for the multimodel. However, the problem paused by the quadratic method lies in the obligation to satisfy several LMI with respect to the same Lyapunov matrix P, these results are shown very conservative. Approach: To reduce the conservatism of the quadratic approach we propose another approach which is exclusively based on Lyapunov piecewise quadratic functions. The conditions obtained by the stabilization of the multiobserver are expressed in term of matrix inequalities with constraints on the matrices rank. Results: The estimation of both states and unknown inputs of a multimodel using the quadratic approach per pieces leads to results less conservative than the quadratic approach. Academic examples illustrate the robustness of the piecewise quadratic approach. Conclusion: In this article we proposed new sufficient conditions of stability of a multiobserver able to the estimation of states and unknown inputs of a nonlinear system describes by a multimodel subjected to the influence of the unknown inputs. The study in was carried out by considering two approaches. The first approach is based on Lyapunov quadratic functions; it is significant to note the great difficulty in finding satisfying results by this approach for the multimodel systems. For this reason we proposed an approach based on piecewise quadratic functions which led to interesting results (proposition 1) and less conservative than the quadratic approach. The conditions suggested in this article concern both the multiobserver stabilization and the estimation of states and the unknown inputs of a multimodel with measurable variables of decision (µξ(k)). The synthesis of a multiobserver with no measurable variables of decision is not approached. This point can constitute an interesting prospect for this study.