Maximum Conditional Probability Stochastic Controller for Linear Systems with Additive Cauchy Noises

Motivated by the sliding mode control approach, a stochastic controller design methodology is developed for discrete-time, vector-state linear systems with additive Cauchy-distributed noises, scalar control inputs, and scalar measurements. The control law exploits the recently derived characteristic function of the conditional probability density function of the system state given the measurements. This result is used to derive the characteristic function of the conditional probability density function of the sliding variable, utilized in the design of the stochastic controller. The incentive for the proposed approach is mainly the high numerical complexity of the currently available method for such systems, that is based on the optimal predictive control paradigm. The performance of the proposed controller is evaluated numerically and compared to the alternative Cauchy controller and a controller based on the Gaussian assumption. A fundamental difference between controllers based on the Cauchy and Gaussian assumptions is the superior response of Cauchy controllers to noise outliers. The newly proposed Cauchy controller exhibits similar performance to the optimal predictive controller, while requiring significantly lower computational effort.