Constructing numerically stable Kalman filter‐based algorithms for gradient‐based adaptive filtering

Abstract

SUMMARYThis paper addresses the numerical aspects of adaptive filtering (AF) techniques for simultaneous state and parameters estimation arising in the design of dynamic positioning systems in many areas of research. The AF schemes consist of a recursive optimization procedure to identify the uncertain system parameters by minimizing an appropriate defined performance index and the application of the Kalman filter (KF) for dynamic positioning purpose. The use of gradient-based optimization methods in the AF computational schemes yields to a set of the filter sensitivity equations and a set of matrix Riccati-type sensitivity equations. The filter sensitivities evaluation is usually done by the conventional KF, which is known to be numerically unstable, and its derivatives with respect to unknown system parameters. Recently, a novel square-root approach for the gradient-based AF by the method of the maximum likelihood has been proposed. In this paper, we show that various square-root AF schemes can be derived from only two main theoretical results. This elegant and simple computational technique replaces the standard methodology based on direct differentiation of the conventional KF equations (with their inherent numerical instability) by advanced square-root filters (and its derivatives as well). As a result, it improves the robustness of the computations against roundoff errors and leads to accurate variants of the gradient-based AFs. Additionally, such methods are ideal for simultaneous state estimation and parameter identification since all values are computed in parallel. The numerical experiments are given.

A unified square-root approach for the score and Fisher information matrix computation in linear dynamic systems

“…Lemmas 1 and 2 evidently cover any SR filter/smoother algorithm with the conventional orthogonal transformation by the simple substitution J = I . It also means that the results presented in Lemmas 1 and 2 generalize the schemes developed in [18] that are valid only for usual orthogonal transformations, i.e. when J = I .…”

SVD-Based Kalman Filter Derivative Computation

“…, zN } is N -step measurement history and c0 is a constant value where c0 = Nm 2 ln(2π). Taking into account that the matrix DR e,k is diagonal and using the Jacobi's formula, d A −1 = −A −1 (dA) A −1 , from (29) we obtain the expression for the log LF gradient evaluation in terms of the SVD filter variables and their derivatives computed in the newly-developed Algorithm 2 (for each i = 1, . .…”

Parameter identification of a nonlinear radial basis function‐based state‐dependent autoregressive network with autoregressive noise via the filtering technique and the multiinnovation theory

“…26,27 Since the colored noise has adverse effect on system identification, it is necessary to apply some techniques to reduce the influence of the colored noise. 28,29 The basic idea of the data filtering technique is to filter the input and output data by using a filter 30,31 and the filtered input and output data are used for identification. The filtering in identification is different from the filtering in communication and signal processing.…”

“…Compared with the white noise, the colored noise is more common and representative 26,27 . Since the colored noise has adverse effect on system identification, it is necessary to apply some techniques to reduce the influence of the colored noise 28,29 . The basic idea of the data filtering technique is to filter the input and output data by using a filter 30,31 and the filtered input and output data are used for identification.…”