Reliable approximations of probability-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 constraint using a multi-dimensional Chebyshev bound. A maximum volume inscribed ellipsoid problem is solved to obtain this probability bound. Using the probability bound, we develop a recursive state feedback control algorithm for a special class of state-constrained stochastic linear-quadratic regulator (LQR). The performance of this approach is explored in a numerical example.