Scenario Optimization for MPC

Abstract

In many control problems, disturbances are a fundamental ingredient and in stochastic Model Predictive Control (MPC) they are accounted for by considering an average cost and probabilistic constraints, where a violation of the constraints is accepted provided that the probability of this to happen is kept below a given threshold. This results in a so-called chance-constrained optimization, which however is known for being very hard to deal with. In this chapter, we describe a scheme to approximately solve stochastic MPC using the scenario approach to stochastic optimization. In the scenario approach the probabilistic constraints are replaced by a finite number of constraints, each one corresponding to a realization of the disturbance. Considering a finite sample of realizations makes the problem computationally tractable while the link to the original chance-constrained problem is established by a rigorous theory. With this approach, along with computational tractability, one gains the important advantage that no assumptions on the disturbance, such as boundedness, independence or Gaussianity, are required.