This paper considers a new class of discrete time stochastic uncertain systems in which the uncertainty is described by a constraint on the relative entropy between a nominal noise distribution and the perturbed noise distribution. This uncertainty description is a natural extension to the case of stochastic uncertain systems, of the sum quadratic constraint uncertainty description. This paper solves problems of worst-case robust performance analysis and output feedback minimax optimal controller synthesis in a general nonlinear setting. Specializing these results to the linear case leads to a minimax linear quadratic Gaussian (LQG) optimal controller. This controller is defined by Riccati difference equations and a Kalman filter-like state equation. The paper also shows that the minimax LQG problem will have a solution if and only if a corresponding H-infinity control problem has a solution. A linear example is presented to illustrate the minimax LQG methodology.