International Journal of Robust and Nonlinear Control, 29(12): 3892 – 3914. May 2019.
One of the most basic problems in control theory is that of controlling a discrete-time linear system subject to uncertain noise with the objective of minimizing the expectation of a quadratic cost. If one assumes the noise to be white, then solving this problem is relatively straightforward. However, white noise is arguably unrealistic: noise is not necessarily independent, and one does not always precisely know its expectation. We first recall the optimal control policy without assuming independence and show that, in this case, computing the optimal control inputs becomes infeasible. In the next step, we assume only the knowledge of lower and upper bounds on the conditional expectation of the noise and prove that this approach leads to tight lower and upper bounds on the optimal control inputs. The analytical expressions that determine these bounds are strikingly similar to the usual expressions for the case of white noise.