A sliding mode stochastic controller for scalar linear systems with additive cauchy noises

In this work a stochastic controller, motivated by the sliding mode control methodology, is proposed for linear, single-state system with additive Cauchy distributed noises. The control law utilizes the time propagated probability density function (pdf) of the system state given measurements that has been derived in recent studies addressing the Cauchy estimation problem. The controller is derived using two equivalent implementations: one that relies directly on the above mention pdf while the second uses the characteristic function of that pdf. The motivation for the proposed approach is mainly the high numerical complexity of the currently available methods for such systems. The controller performance is evaluated numerically and compared to an alternative approach presented recently and to a Gaussian approximation to the problem. A fundamental difference between the Cauchy and the Gaussian controllers is their response to noise outliers. While all controllers respond to process noises, even to the outliers, the Cauchy controllers drive the state faster towards zero after those events. On the other hand, the Cauchy controllers do not respond to measurement noise outliers, while the Gaussian does. The newly proposed Cauchy controller exhibits similar performance to the previously proposed one, while requiring lower computational effort.