Multi-step prediction optimal control for a scalar linear system with additive Cauchy noise

A multi-step predictive optimal control scheme is developed for scalar discrete linear dynamic systems driven by Cauchy distributed process and measurement noises. Although the Cauchy densities that model the process and measurement noise have an undefined first moment and an infinite second moment, the probability density function conditioned on linear noisy measurements does have a finite mean and variance. For the control problem a cost criterion should be defined for which the unconditional expectation of this criterion with respect to the Cauchy densities exists. The cost criterion chosen is functionally similar to the Cauchy density. Although a dynamic programming solution for this criterion is not yet transparent, for the multistage problem an optimal controller is determined that at each time stage minimizes the unconditional expected cost to some terminal time. Numerical results are shown for this m-step predictive optimal control scheme. An essential difference between the proposed controller and one designed for Gaussian noises is that large measurement noises do not produce large control responses, although large and impulsive process noises do induce controls needed for regulation.