Distributionally robust discrete LQR optimal cost

This paper presents some results on the robustness of a classical discrete Linear Quadratic Regulator (LQR) Cost Function with uncertainty in the inputs and state variables. For the typical LQR Cost Function J = E-i=0(j-1) x(i+1)(T)Qx(i+1) + u(i)(T)Ru(i) with Q and R positive definite and symmetric, we consider the expectation and variance of J given unknown independent uncertainties supported by a class of probability distribution functions f epsilon F. We find that the assumption on the uncertainty structure allows straightforward optimization of the cost function in a distributionally robust sense. We show the methodology to derive the expectation and variance of the cost and find inputs which yield robust optimizations of the cost.