Robust Constraint Satisfaction: Invariant Sets and Predictive Control

Set invariance plays a fundamental role in the design of control systems for constrained systems since the constraints can be satisfied for all time if and only if the initial state is contained inside an invariant set. This thesis is concerned with robust set invariance theory and its application to guaranteeing feasibility in model predictive control. In the first part of this thesis, some of the main ideas in set invariance theory are brought together and placed in a general, nonlinear setting. The key ingredients in computing robust controllable and invariant sets are identified and discussed. Following this, linear systems with parametric uncertainty and state disturbances are considered and algorithms for computing the respective robust controllable and invariant sets are described. In addition to discussing linear systems, an algorithm for computing the robust controllable sets for piecewise affine systems with state disturbances is described. In the second part, the ideas from set invariance are applied to the problem of guaranteeing feasibility and robust constraint satisfaction in Model Predictive Control (MPC). A new sufficient condition is derived for guaranteeing feasibility of a given MPC scheme. The effect of the choice of horizons and constraints on the feasible set of the MPC controller is also investigated. Following this, a necessary and sufficient condition is derived for determining whether a given MPC controller is robustly feasible. The use of a robustness constraint for designing robust MPC controllers is discussed and it is shown how this proposed scheme can be used to guarantee robust constraint satisfaction for linear systems with parametric uncertainty and state disturbances. A new necessary and sufficient condition as well as some new sufficient conditions are derived for guaranteeing that the proposed MPC scheme is robustly feasible. The third part of this thesis is concerned with recovering from constraint violations. An algorithm is presented for designing soft-constrained MPC controllers which guarantee constraint satisfaction, if possible. Finally, a mixed-integer programming approach is described for finding a solution which minimises the number of violations in a set of prioritised constraints.