Robust Model Predictive Control
Robust MPC guarantees constraint satisfaction and stability for every disturbance in a bounded set, not just for the nominal prediction.
The problem with nominal MPC
Standard predictive control optimizes over a single predicted trajectory assuming the model is exact. Real disturbances and model errors push the true state off that trajectory, and constraints that looked satisfied in prediction can be violated in reality. Robust MPC accounts for a bounded uncertainty set explicitly, so guarantees hold for the worst case.
Min-max and its cost
The most direct formulation is min-max: minimize the worst-case cost over all admissible disturbance sequences. This is correct but computationally explosive, because the disturbance can react to the control at every step, producing a game tree. Feedback min-max, which optimizes over control policies rather than open-loop sequences, is less conservative but even more expensive. Practical robust MPC seeks tractable approximations.
Constraint tightening
A common tractable approach tightens the constraints by the reachable spread of the uncertainty, so that satisfying the tightened constraints in nominal prediction guarantees satisfying the true constraints under any disturbance. The tightening is computed from the disturbance set and a stabilizing feedback that bounds error growth. This is the idea developed fully in tube MPC.
- Guarantees hold for all disturbances in a bounded set
- Min-max is exact but computationally intractable
- Constraint tightening trades exactness for tractability
- Requires a bounded, known uncertainty description
Robust MPC needs a characterization of the uncertainty: a bounded additive disturbance set, a polytopic model set, or norm bounds. The tighter and more accurate that description, the less conservative the controller. Overly large uncertainty sets shrink the feasible region.
When constraints are hard safety limits, as in a design-stage machine's operating envelope, robust MPC provides a principled guarantee that they are never crossed under bounded disturbances, validated in simulation. Its stochastic cousin, stochastic MPC, trades hard guarantees for probabilistic ones.