Stochastic Estimation for Vector Linear Systems with Additive Cauchy Noise

The Gaussian paradigm has dominated the foundation of estimation and control algorithms. The assumed Gaussian probability density functions (pdf) have very light tails where almost all of its uncertainty evolves near its mean value. However, in many realistic applications the system can experience large impulsive noises far more often than the Gaussian would admit.  By using the heavy-tailed Cauchy pdf, rather than the Gaussian pdf, a new class of estimators is determined. In particular, for the Cauchy vector–state estimation problem, the characteristic function of the unnormalized conditional probability density function (ucpdf) is generated algebraically and recursively through measurement updates and state propagation.   Once the characteristic function of the ucpdf is obtained, the conditional mean and variance are easily computed from its first and second derivatives.  Two-state and three-state dynamic system examples demonstrates the vector-state Cauchy estimator's performance and the two-state estimator is compared to a conditional mean Gaussian estimator (Kalman filter) in Cauchy and Gaussian simulations.