Model predictive control of constrained discrete-time nonlinear systems: Stability and robustness

In this work we present stability and robustness results for constrained discrete-time nonlinear systems stabilized using model predictive control (MPC). The systems are not required to have stabilizable linearizations and the attractors are allowed to be general sets. The stage and terminal costs are allowed to be positive semidefinite. Properties on the terminal cost are not imposed to assert stability, but can be used to reduce the horizon lengths required for stability. The key assumptions are that the MPC value function can be bounded above by a function of a measure of the state and that this state measure be detectable through the stage cost.; These assumptions allow us to assert stability for MPC closed-loop systems for sufficiently large horizon lengths. In general, the results are semiglobal and practical in the horizon length; however, if the functions characterizing the key assumptions satisfy certain linearity conditions locally, respectively globally, the results are semiglobal in the horizon length, respectively global.; Motivated by a discussion of the lack of robustness in general of MPC closed-loop systems, we introduce an assumption which, if satisfied, allows us to assert that the closed-loop system is robust to sufficiently small disturbances. The assumption imposes a property on the optimal control sequence, requiring it to be robustly feasible for a modified optimization problem that has a sequence of nested state constraint sets. By imposing this sequence of constraint sets, robust stability can be asserted. As in the case of stability, these results are semiglobal and practical in the horizon but can be made semiglobal or global if linearity conditions are satisfied. We apply the robustness result to show that MPC-stabilized certainty equivalence output feedback closed-loop systems are robust when interconnected with sufficiently fast observers.; Finally, we present results on the use of model predictive control for obstacle avoidance tasks. These tasks are problematic from a robustness standpoint if the state feedback controller is not designed properly. Modifications to standard MPC algorithms are presented so that MPC can be used for the task and provide some level of robustness.