An algorithm for state constrained stochastic linear-quadratic control

Here we consider a state-constrained stochastic linear quadratic control problem. This problem has linear dynamics and a quadratic cost, and states are required to satisfy a probabilistic constraint. In this paper, the joint probabilistic constraint in the model is converted to a conservative deterministic one using multi-dimensional Chebyshev bound. A maximum volume inscribed ellipsoid problem is solved to obtain this probability bound. We then design an optimal afflne controller for the resulting problem. The convexity of the Chebyshev bound-constrained problem is proved and a practical algorithm is developed. Two numerical examples show that the algorithm is reliable even when the disturbances are large and the problem horizon grows to as long as 20 stages. It is also shown that the approach proposed in this paper can be used to reformulate some classical problems such as tracking problems.