Model predictive control of linear systems with multiplicative unbounded uncertainty and chance constraints

This paper presents a novel stochastic Model Predictive Control algorithm for linear systems characterized by multiplicative and possibly unbounded model uncertainty. Probabilistic constraints on the states and inputs are considered, and a quadratic cost function is minimized. The stochastic control problem, and in particular the probabilistic constraints, are reformulated in deterministic terms by means of the Cantelli inequality, so that the on-line computational burden of the algorithm is similar to the one of a standard MPC method. The properties of the algorithm, namely the recursive feasibility and the pointwise convergence of the state, are proven by suitably selecting the terminal cost and the constraints on the mean and the variance of the state at the end of the prediction horizon, and by considering as additional optimization variables also the mean and the covariance of the state at the beginning of the prediction horizon. An extension to deal with the case of expectation, rather than probabilistic, constraints is reported. The numerical issues related to the off-line selection of the algorithm’s parameters and its on-line implementation are discussed. Simulation results referred to a system with unbounded uncertainty are shown to compare the performances achievable with probabilistic and expectation constraints.