Evolved Extended Kalman Filter for first-order dynamical systems with unknown measurements noise covariance
Introduction
The Extended Kalman Filter (EKF) is a generalized version of the well-known state estimator Kalman Filter (KF) dedicated to nonlinear systems. The EKF has been proved to be useful from the Apollo moon landing project in the sixties (see [1], [2]), and continues being relevant to date [3], [4].
The functionality of the EKF is based on the covariance matrices that characterize the White Gaussian Noise (WGN) in the process and in the measurements. In the standard EKF, both covariance matrices are typically assumed constant values, being either known, proposed by an expert after analyzing the process and the measurements, or found by trial-and-error, among other techniques. Most of the time, the covariance matrices are difficult to be known, hence, the search for a replacement of these parameters is required.
The revision of literature shows several approaches to deal with the uncertainty in the covariance matrices. Some works aim to estimate, or identify, a numeric value for the unknown noise covariances by introducing mostly adaptive models or optimization techniques. Afterwards, the found value can be used either for the EKF, or to cancel the effect of noise, either in the process or in the measurement. In the early 70s, Mehra [5] classified the, so far known, adaptive filtering approaches into four categories: Bayesian, Maximum Likelihood, Correlation, and Covariance Matching. Among the new solutions found in the literature, are innovation-based methods [6], [7], [8]; fuzzy logic and/or neural networks [9], [10], [11], [12], [13], [14]; stochastic or probabilistic approaches [15], [16], [17], [18]; and metaheuristics [19], in particular Genetic Algorithms [20], [21], [22], and Differential Evolution [23]. Despite the large amount of research devoted to this particular aspect of Kalman Filters design, dating more than fifty years, the tuning of the filter’s covariance matrices still remains an open problem [24].
This work focuses on the measurement noise covariance of the filter, i.e., it considers a configuration of noisy measurements coming from a deterministic process, as in [25], [26], [27], [28], [29]. A novel methodology that builds replacement functions for the typically unknown measurement noise covariance, which is an important tuning parameter in the EKF, is proposed. In contrast to the works found in the literature, the proposed methodology is built as an extension of the analytic behaviors framework originally developed for the construction of nonlinear controllers [30], [31], [32], [33], [34]. The novelty of this framework resides in the integration of a theoretical formulation with an evolutionary process based on the Genetic Programming paradigm. Focused in our problem, it allows to find dynamic expressions that automatically adjust themselves to minimize the Kalman criterion, and can deal with parametric variations. A remarkable feature with respect to the literature, where the aim is to propose constant estimates of the unknown covariances, here the measurement noise covariance is replaced by an analytic function that depends on the filter’s intrinsic variables and on the system’s noisy measurement. Another important feature of the methodology is that it produces a large set of suitable solutions to the given problem. From these, the designer can choose the most appropriate analytic function replacement to be used online for real-world applications, such as navigation and control [35], [36], [37]. Tuning the EKF with the new replacement generates a new filter called Evolved Extended Kalman Filter (EEKF).
This document is organized as follows. Section 2 states the theoretical preliminaries for the EKF, a brief introduction of the analytic behaviors framework, and the description of the dynamics of the logistic map system employed as a testbed for the general methodology. The general form of the methodology developed for the construction of EEKFs, applied to nonlinear, First Order Dynamical Systems (FO-DS) is presented in Section 3. The application of the methodology for a particular FO-DS, logistic map system, is described in Section 4. The logistic map system is chosen since, despite its simplicity, its solutions are complex, varying from fixed points to chaos [38], [39], [40], depending on its bifurcation parameter. This system is commonly used to model the growth and decay of a population over time, the presence of turbulence in a fluid, the host–parasite problem, the double pendulum, as well as to describe the chaotic dynamics of phenomena within several research fields [38], [41]. Finally, the conclusions are outlined in Section 5.
Section snippets
Preliminaries
For later use, the theoretical basis of EKF, an overview of the analytic behaviors framework, and the description of the logistic map employed to exemplify the proposed approach, are detailed.
Synthesis of analytic behaviors for EEKFs for nonlinear FO-DS
In this section, the proposed methodology for the construction of EEKFs applied to a nonlinear FO-DS is described in a general way. First, the problem statement is defined, and next, each stage of the approach is detailed.
A motivating example: the logistic map system
The logistic map system is a nonlinear, first-order dynamical system selected to exemplify our methodology due the richness of its behaviors, which range from stable fixed points to chaos. This particular system (13) can be rewritten in the form (1)–(2) with and given as where is the logistic map function, and is the logistic map state corrupted by a noise signal . Then, the construction of the EKF (4)–(8) for (17) is specified with
Conclusions
An extension of the analytic behaviors framework has been developed for the optimal computation of a replacement of the unknown measurement noise covariance matrix, , conventionally used to tune the EKF. Traditionally, the estimation of is a challenge, since, in general, the nature of the noise is not known, and there is not an analytic method to compute it. In this approach, the covariance matrix is replaced by an analytic function. This expression is given in terms of the noisy output,
CRediT authorship contribution statement
Leonardo Herrera: Conceptualization, Methodology, Formal analysis, Software, Writing – original draft, Writing – review & editing. M.C. Rodríguez-Liñán: Conceptualization, Formal analysis, Writing - original draft, Writing – review & editing. Eddie Clemente: Conceptualization, Formal analysis, Software, Visualization, Writing – original draft, Writing – review & editing. Marlen Meza-Sánchez: Conceptualization, Methodology, Formal analysis, Software, Writing – original draft, Writing – review &
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
Eddie Clemente thanks TecNM the support given through the project 11391.21-P. L. Monay-Arredondo thanks CONACYT and TecNM for the support given under scholarship No. 406498.
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