The majority of practical estimation and control solutions are based on system models with additive Gaussian noises. Gaussian distribution, being light tailed, does not capture significant fluctuations that occur in many engineering applications, such as atmospheric noises and air turbulence, underwater acoustic noises and image processing. It was shown that those noises are better described by heavy tailed distributions that exhibit significant impulsive characteristics. The heavy tailed distributions are characterized by probability density functions with tails that are not exponentially bounded. Named after the French mathematician, Augustin Louis Cauchy, the heavy-tailed Cauchy distribution has been shown to much better represent this type of fluctuations. The challenge of using this distribution is that it does not have a moment generating function. Specifically, its first moment is not well defined and its second and higher moments are infinite.
In the recent years, progress was reported in the area of estimation of linear systems with additive Cauchy noises. As part of its solution, the estimator computes explicitly the conditional probability density function (pdf) of the state given the measurement history, or its characteristic function. Those were used to derive a stochastic optimal-predictive controller for the system. Although demonstrating good performance characteristics, the proposed controller was shown to entail high numerical complexity. This motivated the alternative approach presented in the current study.
In this work a stochastic controller, inspired by the sliding mode control methodology, is proposed for linear systems with additive Cauchy distributed noises. The design goal is to maximize the prior probability of the system state or its linear combination to be within a given bound around the regulation point. The control law utilizes the time propagated pdf of the system state given measurements that is computed by the Cauchy estimator. The single-state controller was derived using two equivalent implementations: one that relies directly on the above mentioned prior pdf while the second uses the characteristic function of that pdf. The latter was addressed mainly because in the multi-state case only the characteristic function can be determined by the respective Cauchy estimator. The controller performance was 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, and is much easier to implement.
