Optimal asynchronous estimation of 2D Gaussian–Markov processes

Abstract

In this article, we consider the problem of trajectory estimation of a continuous-time two-dimensional (2D) Gaussian–Markov processes based on noisy measurements executed in non-uniformly distributed time moments. In such a case, a discrete-time prediction has to be performed in each cycle of estimation (by means of a Kalman filter). This task can, however, be computationally expensive. To solve this problem, we derive explicit formulae for predicting the 2D process based on explicit forms of the matrix exponential. The effects of the resulting estimator are confronted with those of the classical Kalman filter. Simulated experiments illustrate the effectiveness of the proposed approach.

Keywords:

• Gaussian processes
• estimation and detection
• filtering

Notes

1. That is given a probability space (Ω, ℱ, P) and a measurable space (ℝ, ℬ(ℝ)), a function f : Ω → ℝ is measurable, if for any Borel set B ∈ ℬ(ℝ), the inverse image f ⁻¹(B) ∈ ℱ.