Stochastic Model Predictive Control
Stochastic MPC handles random disturbances with chance constraints, enforcing limits with a specified probability rather than for the worst case.
Probabilistic constraints
When disturbances are random and possibly unbounded, worst-case robust MPC is either impossible or crushingly conservative. Stochastic MPC instead uses chance constraints: a state or input limit must hold with at least a prescribed probability, say 95 percent. This admits rare, controlled violations in exchange for far better average performance, which matches how many real limits are actually specified.
Reformulating chance constraints
Chance constraints are not directly tractable, so they are reformulated. For Gaussian disturbances a linear chance constraint becomes a tightened deterministic constraint whose margin is the standard deviation times a quantile of the normal distribution. For general distributions, sample-based scenario methods replace the probabilistic constraint with a finite set of sampled disturbance realizations, giving guarantees that depend on the sample count.
Cost and stability
The objective is typically the expected cost over the disturbance distribution. Recursive feasibility requires care, because a chance constraint feasible now might be infeasible after an unlikely disturbance; remedies include soft constraints with recovery, or conditioning the guarantee on an initial feasible region. Stability is usually established in an expected or mean-square sense.
- Enforce limits with a set probability, not worst case
- Gaussian case yields tightened deterministic constraints
- Scenario methods sample disturbances for general distributions
- Guarantees are probabilistic or in expectation
The design trade-off is the violation probability: a smaller allowed probability tightens constraints toward the robust case, a larger one relaxes them toward nominal. Choosing it is a genuine engineering decision reflecting how costly a rare violation is.
For a design-stage plant driven by broadband stochastic disturbances, stochastic MPC would keep constraints satisfied with high probability while extracting more performance than robust MPC, all in simulation. It sits between nominal and robust predictive control on the conservatism scale.