Stochastic LQ Control with Probabilistic Constraints

In this thesis we focus on stochastic LQ control problems with probabilistic
constraints. We will briefly review the history of LQ control and the related classical results for constrained and unconstrained cases. Then we formulate the problem studied, where the system has linear dynamics and a quadratic cost, and states are required to satisfy a probabilistic constraint. We sample the most recent techniques for such probability constrained control problems and propose our own approach. We consider two types of probability constraints: all-stage and per-stage. In the first case, there is a joint probabilistic constraint on all the system states over the whole predicting horizon while in the second one the states are restricted by individual probability constraints at each stage. The contribution of this thesis has two parts: first we develop a recursive state feedback control algorithm for a special class of state constrained stochastic LQR, and a disturbance feedback controller for the general case using quadratic programming for the all-stage problems. Second, we design a recursive algorithm for the per-stage problem based on sub-gradient method. We also implement a practical Model Predictive Control algorithm for such problems. The control algorithm is tested on a simple temperature control problem for analysis.