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=Σ_i=0^j-1x_i+1TQx_i+1+u _i^TRu_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∈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