Stochastic Estimation and Control for Linear Systems with Cauchy Noise

The light-tailed Gaussian paradigm has dominated the foundation of estimation and control algorithms. However, in many realistic applications the system can experience large impulsive noises far more often than the Gaussian would admit. In this paper the Cauchy probability density function (pdf) is used to develop a new class of estimation and control algorithms. First, the scalar Cauchy estimation problem is addressed which entails the generation of the state pdf conditioned on the measurement history. Next, based on this Cauchy conditional pdf, a model predictive optimal controller is developed. Finally, the vector stated estimator is derived by recursively propagating the characteristic function of the unnormalized conditional pdf through measurement updates and dynamic state propagation. The conditional mean and variance are easily computed from the first and second derivatives of this characteristic function.