Distributed Computation of a Robust Estimator Based on Cauchy Noises
Nathaniel Snyder, Moshe Idan, Jason Lee Speyer · 2021 60th IEEE Conference on Decision and Control (CDC) · 2021
A real-time, recursive, multivariate estimation algorithm for time-invariant and time-varying linear systems with modelled Cauchy noises is developed. When previously compared to the Kalman Filter, the Multivariate Cauchy Estimator was shown to be robust against impulsive disturbances in the process or measurement functions, but proved computationally intractable for real-time estimation applications. Two significant insights allow for a reformulation of the Multivariate Cauchy Estimator to possess a streamlined recursive and computationally reduced characteristic function of the conditional probability density function of the system state-vector given the measurement sequence. This characteristic function is represented by a sum of terms, expanding with each measurement. First, we show that a cell-enumeration matrix can be computed for each hyperplane arrangement embedded within each term of the characteristic function of the Cauchy Estimator. We then show that functions used to formulate the terms of this characteristic function can be expressed as a vector of parameters operating on basis functions constructed from this enumeration matrix. This vector is obtained by solving an under-determined system of equations. We demonstrate that our reformulation allows all terms with equal hyperplane arrangements to be reduced into a unique set. Secondly, we take advantage of advances in parallel processing to exploit the inherent parallelism found in the characteristic function of the Cauchy Estimator. A three state time-invariant system example is used to illustrate the performance of the Cauchy Estimator against the Kalman Filter when subjected to Gaussian and Cauchy noises. We report computational savings of over 99% when compared to the previous formulation. Furthermore, we discuss the real-time architecture of the Cauchy Estimator and report the execution speeds for a three-state system implemented on a single NVIDIA GeForce GTX 1060 graphics processing unit (GPU).