Open-loop feedback control under multiple disturbance function hypotheses

We consider the control of a linear time-invariant system that is subject to a systematic, non-zero mean disturbance process drawn from a finite set of hypothesized disturbance functions. In addition to the systematic disturbance, we assume that the system is also subject to zero mean white Gaussian process and sensor noise. Using a quadratic objective function, our setting is a generalization of the well-known framework of Linear Quadratic Gaussian (LQG) control. Decomposing the state estimation problem along the lines of the hypothesized set of systematic disturbances, we derive the optimal filtering equations and corresponding Bayesian update rule for assessing the posterior probability that any given hypothesized disturbance is the one perturbing the system. We also derive the stage-by-stage optimal open-loop feedback control action, based on the current history of measurements and control actions. Finally, we present three numerical examples that illustrate the ability of OLFC to anticipate future perturbations to the system after the initial uncertainty over the hypothesized disturbance set is resolved.