Section: New Results

Explicit robust constrained control for linear systems : analysis, implementation and design based on optimization

Participants : Sorin Olaru [correspondent] , Ngoc Anh Nguyen [L2S] , Pedro Rodriguez Ayerbe [L2S] , Martin Gulan [STUBA, Bratislava] .

Piecewise affine (PWA) feedback control laws is relevant for the control of constrained systems, hybrid systems; equally for the approximation of nonlinear control. However, they are associated with serious implementation issues. Motivated from the interest in this class of particular controllers, the thesis of Ngoc Anh Nguyen is mostly related to their analysis and design. The first part of this thesis aims to compute the robustness and fragility margins for a given PWA control law and a linear discrete-time system. More precisely, the robustness margin was defined as the set of linear time-varying systems such that the given PWA control law keeps the trajectories inside a given feasible set. On a different perspective, the fragility margin contains all the admissible variations of the control law coefficients such that the positive invariance of the given feasible set is still guaranteed. The second part of this thesis focuses on inverse optimality problem for the class of PWA controllers. Namely, the goal is to construct an optimization problem whose optimal solution is equivalent to the given PWA function. The methodologyis based on convex lifting [33] : an auxiliary 1-dimensional variable which enhances the convexity characterization into recovered optimization problem. Accordingly, if the given PWA function is continuous [35] , the optimal solution to this reconstructed optimization problem will be shown to be unique. Otherwise, if the continuity of this given PWA function is not fulfilled, this function will be shown to be one optimal solution to the recovered problem [36] . In view of applications in linear model predictive control (MPC), it was shown that any continuous PWA control law can be obtained by a linear MPC problem with the prediction horizon at most equal to 2 prediction steps [65] . Aside from the theoretical meaning, this result can also be of help to facilitate implementation of PWA control laws by avoiding storing state space partition [66] . Another utility of convex liftings [34] will be shown in the last part of this thesis to be a control Lyapunov function. Accordingly, this convex lifting will be deployed in the so-called robust control design based on convex liftings for linear system affected by bounded additive disturbances and polytopic uncertainties [60] . Both implicit and explicit controllers can be obtained. This method can also guarantee the recursive feasibility and robust stability. However, this control Lyapunov function is only defined over the maximal $\lambda$-contractive set for a given subunitary $\lambda$ which is known to be smaller than the maximal controllable set. Therefore, an extension of the above method to the N-steps controllable set will be presented. This method is based on a cascade of convex liftings where an auxiliary variable will be used to emulate a Lyapunov function. Namely, this variable will be shown to be non-negative, to strictly decrease for N first steps and to stay at 0 afterwards. Accordingly, robust stability is sought.