Linear Model Predictive Control Stability and Robustness

Most real systems are subjected to constraints, both on the available control effort and the controlled variables. Classical linear feedback is in some cases not enough for such systems. This has motivated the development of a more complicated, nonlinear controller, called model predictive control, MPC. The idea in MPC is to repeatedly solve optimization problems on-line in order to calculate control inputs that minimize some performance measure evaluated over a future horizon.MPC has been very successful in practice, but there are still considerable gaps in the theory. Not even for linear systems does there exist a unifying stability theory, and robust synthesis is even less understood.The thesis is basically concerned with two different aspects of MPC applied to linear systems. The first part is on the design of terminal state constraints and weights for nominal systems with all states avaliable. Adding suitable terminal state weights and constraints to the original performance measure is a way to guarantee stability. However, this is at the cost of possible loss of feasibility in the optimization. The main contribution in this part is an approach to design the constraints so that feasibility is improved, compared to the prevailing method in the literature. In addition, a method to analyze the actual impact of ellipsoidal terminal state constraints is developed.The second part of the thesis is devoted to synthesis of MPC controllers for the more realistic case when there are disturbances acting on the system and there are state estimation errors. This setup gives an optimization problem that is much more complicated than in the nominal case. Typically, when disturbances are incorporated into the performance measure with minimax (worst-case) formulations, NP-hard problems can arise. The thesis contributes to the theory of robust synthesis by proposing a convex relaxation of a minimax based MPC controller. The framework that is developed turns out to be rather flexible, hence allowing various extensions.

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