Stochastic Estimation for Two-State Linear Dynamic Systems With Additive Cauchy Noises

An efficient recursive state estimator is developed for two-state linear systems driven by Cauchy distributed process and measurement noises. For a general vector-state system, the estimator is based on recursively propagating the characteristic function of the conditional probability density function (cpdf), where the number of terms in the sum that expresses this characteristic function grows with each measurement update. Both the conditional mean and the conditional error variance are functions of the measurement history. For systems with two states, the proposed estimator reduces substantially the number of terms needed to express the characteristic function of the cpdf by taking advantage of relationships not yet developed in the general vector-state case. Further, by using a fixed sliding window of the most recent measurements, the improved efficiency of the proposed two-state estimator allows an accurate approximation for real-time computation. In this way, the computational complexity of each measurement update eventually becomes constant, and an arbitrary number of measurements can be processed. The numerical performance of the Cauchy estimator in both Cauchy and Gaussian simulations was demonstrated and compared to the Kalman Filter.