A scenario solution to state-feedback controller design for discrete-time linear systems subject to probabilistic constraints

This paper deals with discrete-time linear systems affected by an additive stochastic disturbance and subject to input and state constraints. We consider the problem of designing a linear state-feedback control law so as to minimize a cost function while guaranteeing the satisfaction in probability of the constraints by the steady state solution. The problem can be formulated as a chance-constrained optimization program with constraint satisfaction depending on the stationary state distribution, which in turn depends on the controller gain. A data-driven solution is proposed where constraints are imposed on a finite number of disturbance realizations of finite length. By leveraging the new wait-and-judge paradigm of the scenario approach to address the intrinsic nonconvexity of the resulting optimization problem and by explicitly accounting for the introduced approximation error when using finite length disturbance realizations to approximate the steady state, chance-constrained feasibility of the proposed solution is proven.