Optimal Steady-State Disturbance Compensation for Constrained Linear Systems: The Gaussian Noise Case

We consider the problem of designing a disturbance compensator for a discrete time linear system, so as to optimize a performance index while satisfying probabilistic state and input constraints in steady-state conditions. The problem is formulated as a chance-constrained program that depends on the compensator parameters through the state and input stationary distributions. In this article, we focus on the Gaussian noise case and provide an analytic expression of the stationary state distribution as a function of the compensator parameters. This expression can be used in the chance-constrained program, which can then be tackled via the scenario approach. Some useful extensions of the setup are also discussed to further broaden the applicability of the approach. Performance of the proposed design methodology is shown on a building energy management problem where cyclostationary disturbances are compensated, thus providing a stochastic periodic control solution.