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The Quadrature Gaussian Sum Filter and Smoother for Wiener Systems

Angel L. Cedeño, Rodrigo A. González, Juan C. Agüero

TL;DR

This work tackles state estimation for Wiener Block-Oriented Nonlinear systems by introducing the Quadrature Gaussian Sum Filter (QGSF) and Two-filter Smoother (QGSS), which exploit Gauss-Legendre quadrature to transform the nonlinear measurement model into a Gaussian mixture. The key idea is to represent $p(y_t|\mathbf{x}_t)$ as a finite Gaussian mixture and propagate it through Bayesian filtering and smoothing recursions, yielding closed-form-like update rules for both filtering and smoothing in the Wiener setting. The paper also develops practical reduction techniques to control the growth of mixture components, provides a detailed parallelization viewpoint, and establishes connections to Kalman and multi-model filters. Numerical experiments across first-, second-, and fourth-order Wiener systems demonstrate that QGSF/QGSS achieve higher accuracy than EKF, QKF, UKF, and particle-filter baselines with fewer components and reduced computation, supporting applicability to control, estimation, and system identification, including embedded hardware implementations.

Abstract

Block-Oriented Nonlinear (BONL) models, particularly Wiener models, are widely used for their computational efficiency and practicality in modeling nonlinear behaviors in physical systems. Filtering and smoothing methods for Wiener systems, such as particle filters and Kalman-based techniques, often struggle with computational feasibility or accuracy. This work addresses these challenges by introducing a novel Gaussian Sum Filter for Wiener system state estimation that is built on a Gauss-Legendre quadrature approximation of the likelihood function associated with the output signal. In addition to filtering, a two-filter smoothing strategy is proposed, enabling accurate computation of smoothed state distributions at single and consecutive time instants. Numerical examples demonstrate the superiority of the proposed method in balancing accuracy and computational efficiency compared to traditional approaches, highlighting its benefits in control, state estimation and system identification, for Wiener systems.

The Quadrature Gaussian Sum Filter and Smoother for Wiener Systems

TL;DR

This work tackles state estimation for Wiener Block-Oriented Nonlinear systems by introducing the Quadrature Gaussian Sum Filter (QGSF) and Two-filter Smoother (QGSS), which exploit Gauss-Legendre quadrature to transform the nonlinear measurement model into a Gaussian mixture. The key idea is to represent as a finite Gaussian mixture and propagate it through Bayesian filtering and smoothing recursions, yielding closed-form-like update rules for both filtering and smoothing in the Wiener setting. The paper also develops practical reduction techniques to control the growth of mixture components, provides a detailed parallelization viewpoint, and establishes connections to Kalman and multi-model filters. Numerical experiments across first-, second-, and fourth-order Wiener systems demonstrate that QGSF/QGSS achieve higher accuracy than EKF, QKF, UKF, and particle-filter baselines with fewer components and reduced computation, supporting applicability to control, estimation, and system identification, including embedded hardware implementations.

Abstract

Block-Oriented Nonlinear (BONL) models, particularly Wiener models, are widely used for their computational efficiency and practicality in modeling nonlinear behaviors in physical systems. Filtering and smoothing methods for Wiener systems, such as particle filters and Kalman-based techniques, often struggle with computational feasibility or accuracy. This work addresses these challenges by introducing a novel Gaussian Sum Filter for Wiener system state estimation that is built on a Gauss-Legendre quadrature approximation of the likelihood function associated with the output signal. In addition to filtering, a two-filter smoothing strategy is proposed, enabling accurate computation of smoothed state distributions at single and consecutive time instants. Numerical examples demonstrate the superiority of the proposed method in balancing accuracy and computational efficiency compared to traditional approaches, highlighting its benefits in control, state estimation and system identification, for Wiener systems.
Paper Structure (27 sections, 9 theorems, 73 equations, 9 figures, 1 table)

This paper contains 27 sections, 9 theorems, 73 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. System Setup and Problem Formulation
  3. Bayesian Filtering and Smoothing Recursions and a Quadrature-based Approach
  4. Bayesian Filtering and Smoothing
  5. Gauss-Legendre Quadrature rule for computing $p(y_t|\mathbf{x}_t)$
  6. The Quadrature Gaussian Sum Filter and Smoother for Wiener Systems
  7. Quadrature Gaussian Sum Filtering for Wiener systems
  8. Quadrature Gaussian Sum Smoothing for Wiener systems
  9. Implementation Aspects and Relations with Established Methods
  10. Gaussian sum reduction
  11. State estimator based on Gaussian mixture models
  12. Relation with the Kalman filter and smoother
  13. Relation with Multiple-model Kalman filters
  14. Relation with Gaussian Sum - Quadrature Kalman filter
  15. Simulation examples
  16. ...and 12 more sections

Key Result

Lemma 1

Assume that the domain of $g(\cdot)$ can be partitioned into $M_1+M_2$ sets, of which $M_1$ correspond to intervals in which it is a differentiable and strictly monotonic function, and $M_2$ correspond to quantization sets of the strictly positive Lebesgue measure in which $g(\cdot)$ is constant and where $\delta(\cdot)$ is the Dirac delta function, and

Figures (9)

  • Figure 1: Block diagram of a Wiener system in state-space form.
  • Figure 2: An arbitrary piecewise nonlinear function $g$ admitted by our approach.
  • Figure 3: Parallel interpretation of the QGSF and QGSS. (a) pseudocode for Kalman measurement and time update stages, (b) block diagram of the QGSF indicating that during both the measurement and time update stages, several instances of the respective stages of the Kalman filter are employed concurrently, (c) pseudocode for the Kalman smoother, (d) block diagram of the QGSS, which uses the Kalman smoother equations in a parallelized manner.
  • Figure 4: Filtered PDF $p(x_t|y_{1:t})$ for selected time instances. We considered 10 Gaussian components for the QGSF, 6 sigma points for the QKF, 500 particles for the PF, and $\alpha=0.001$, $\beta=1$. $\kappa=0.001$ for the UKF.
  • Figure 5: State estimation of the system $\hat{x}_{t|t}=\mathbb{E}\left\lbrace x_t|y_{1:t}\right\rbrace$ for 100 Monte Carlo runs. The shaded area depicts the region encompassing all state sequence estimates of the system. The blue line represents the true state, while the red line represents the mean of the 100 Monte Carlo experiments.
  • ...and 4 more figures

Theorems & Definitions (24)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 14 more