Multivariate Estimator for Linear Dynamical Systems with Additive Laplace Measurement and Process Noises

Abstract

Since uncertainties in many physical systems have impulsive properties poorly modeled by Gaussian distributions, heavier-tailed distributions, such as Laplace, may be used to improve model characteristics. From insights obtained through development of the scalar Laplace estimator, an algorithm is determined for the vector-state case. For a discrete-time vector linear system with scalar additive Laplace-distributed process and measurement noises, the a priori and a posteriori conditional probability density functions (pdfs) of the system states given the measurement history are propagated recursively and in closed form. The conditional pdfs are composed of signs and absolute values of affine functions, and a basis composed of signs of affine functions is constructed to simplify their representation. The a posteriori conditional mean and variance are derived analytically from the conditional pdf using characteristic functions. Generalization to independent vector measurement and process noises is straightforward. From the general method for deriving Laplace estimators in $n$-dimensions, a two-dimensional minimum variance estimator is explicitly developed, and a simulation is presented.