%0 Journal Article
%T Online Stochastic Convergence Analysis of the Kalman Filter
%A Matthew B. Rhudy
%A Yu Gu
%J International Journal of Stochastic Analysis
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/240295
%X This paper presents modifications to the stochastic stability lemma which is then used to estimate the convergence rate and persistent error of the linear Kalman filter online without using knowledge of the true state. Unlike previous uses of the stochastic stability lemma for stability proof, this new convergence analysis technique considers time-varying parameters, which can be calculated online in real-time to monitor the performance of the filter. Through simulation of an example problem, the new method was shown to be effective in determining a bound on the estimation error that closely follows the actual estimation error. Different cases of assumed process and measurement noise covariance matrices were considered in order to study their effects on the convergence and persistent error of the Kalman filter. 1. Introduction Since its introduction in 1960, the linear Kalman filter (LKF) [1] has been used widely in industry. When the LKF is implemented in real-time applications, it is often difficult to quantify the performance of the filter without access to some reference ¡°truth.¡± Offline simulations can provide some indication of the filter performance; however accurate mathematical models are not always available. For the LKF, there are two primary sources of error in the estimation: initialization error and stochastic errors due to the process and measurement noise. In the early stages of the filter, the initialization error is dominant, and it takes some amount of time for the estimated state to converge to the true state from this incorrect initial state. After the initial error convergence, the errors due to the noise terms remain, resulting in ¡°persistent¡± errors. Because of these types of error, there is a need to analyze the performance of the LKF online by quantifying the convergence rate and persistent error bounds of the real system. Such a tool could benefit many safety or performance critical systems, such as the aircraft health management system. Existing techniques for online performance analysis of the LKF include outlier detection [2], performance reliability prediction [3], and confidence bounds from the covariance matrix; for example, see [4]. Confidence bounds can also be established through use of the Chebyshev inequality [5], although these bounds tend to be too large for practical use [6]. Some other investigations for confidence bounds on the Kalman filter consider the non-Gaussian case using enhancements to the Chebyshev inequality [6] or the Kantorovich inequality [7]. The work presented herein offers a novel online method
%U http://www.hindawi.com/journals/ijsa/2013/240295/