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On derivative-free extended Kalman filtering and its Matlab-oriented square-root implementations for state estimation in continuous-discrete nonlinear stochastic systems

Maria V. Kulikova, Gennady Yu. Kulikov

TL;DR

A novel continuous-discrete derivative-free EKF methodology by deriving the related moment differential equations (MDEs) and sample point differential equations (SPDEs) and sample point differential equations (SPDEs) and obtaining several numerically stable derivative-free EKF methods.

Abstract

Recent research in nonlinear filtering and signal processing has suggested an efficient derivative-free Extended Kalman filter (EKF) designed for discrete-time stochastic systems. Such approach, however, has failed to address the estimation problem for continuous-discrete models. In this paper, we develop a novel continuous-discrete derivative-free EKF methodology by deriving the related moment differential equations (MDEs) and sample point differential equations (SPDEs). Additionally, we derive their Cholesky-based square-root MDEs and SPDEs and obtain several numerically stable derivative-free EKF methods. Finally, we propose the MATLAB-oriented implementations for all continuous-discrete derivative-free EKF algorithms derived. They are easy to implement because of the built-in fashion of the MATLAB numerical integrators utilized for solving either the MDEs or SPDEs in use, which are the ordinary differential equations (ODEs). More importantly, these are accurate derivative-free EKF implementations because any built-in MATLAB ODE solver includes the discretization error control that bounds the discretization error arisen and makes the implementation methods accurate. Besides, this is done in automatic way and no extra coding is required from users. The new filters are particularly effective for working with stochastic systems with highly nonlinear and/or nondifferentiable drift and observation functions, i.e. when the calculation of Jacobian matrices are either problematical or questionable. The performance of the novel filtering methods is demonstrated on several numerical tests.

On derivative-free extended Kalman filtering and its Matlab-oriented square-root implementations for state estimation in continuous-discrete nonlinear stochastic systems

TL;DR

A novel continuous-discrete derivative-free EKF methodology by deriving the related moment differential equations (MDEs) and sample point differential equations (SPDEs) and sample point differential equations (SPDEs) and obtaining several numerically stable derivative-free EKF methods.

Abstract

Recent research in nonlinear filtering and signal processing has suggested an efficient derivative-free Extended Kalman filter (EKF) designed for discrete-time stochastic systems. Such approach, however, has failed to address the estimation problem for continuous-discrete models. In this paper, we develop a novel continuous-discrete derivative-free EKF methodology by deriving the related moment differential equations (MDEs) and sample point differential equations (SPDEs). Additionally, we derive their Cholesky-based square-root MDEs and SPDEs and obtain several numerically stable derivative-free EKF methods. Finally, we propose the MATLAB-oriented implementations for all continuous-discrete derivative-free EKF algorithms derived. They are easy to implement because of the built-in fashion of the MATLAB numerical integrators utilized for solving either the MDEs or SPDEs in use, which are the ordinary differential equations (ODEs). More importantly, these are accurate derivative-free EKF implementations because any built-in MATLAB ODE solver includes the discretization error control that bounds the discretization error arisen and makes the implementation methods accurate. Besides, this is done in automatic way and no extra coding is required from users. The new filters are particularly effective for working with stochastic systems with highly nonlinear and/or nondifferentiable drift and observation functions, i.e. when the calculation of Jacobian matrices are either problematical or questionable. The performance of the novel filtering methods is demonstrated on several numerical tests.
Paper Structure (6 sections, 2 theorems, 51 equations, 3 figures, 2 tables)

This paper contains 6 sections, 2 theorems, 51 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The derivative-free EKF-based principle: The mean and covariance approximation at the measurement update
  3. Derivation of continuous-discrete derivative-free EKF
  4. Derivation of Cholesky factored-form continuous-discrete derivative-free EKF methods
  5. Numerical experiments
  6. Concluding remarks

Key Result

Lemma 1

Let the MDE in eq2.4b possess the unique solution ${P}(t)$ in a sampling period $[t_{k-1},t_{k}]$ and has factorization $P(t) = P^{1/2}(t)P^{\top/2}(t)$ where $P^{1/2}(t)$ stands for the lower triangular Cholesky factor and $P^{\top/2}(t)$ denotes its transpose. If the square-root factor $P^{1/2}(t) where the matrix $M(t)$ of size $n\times n$ stands for where the factor ${P}^{-\top/2}(t)$ refers

Figures (3)

  • Figure 1: The accuracies (left graph) and efficiencies (right graph) of various continuous-discrete EKF estimators on the test problem in Example \ref{['ex:2']}.
  • Figure 2: The accuracies degradation of various continuous-discrete EKF estimators on the ill-conditioned problem in Example \ref{['ex:2ill']}.
  • Figure 3: The accuracies of various continuous-discrete EKF estimators depending on the value of the stiffness parameter $\lambda$ in Example \ref{['ex:3']}.

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • Example 1: Gas-phase reversible reaction in CSTR
  • Example 2: The CSTR ill-conditioned measurement scheme
  • Example 3: stochastic Van der Pol oscillator