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On Kernel Design for Regularized Volterra Series Identification of Wiener-Hammerstein Systems

Yu Xu, Biqiang Mu, Tianshi Chen

TL;DR

This paper solves the challenge of kernel design for Volterra-series identification by embedding Wiener–Hammerstein structure directly into off-diagonal kernel blocks. It introduces a WH-informed kernel that preserves the $O(N^3)$ complexity of applicable methods while enabling a direct and flexible incorporation of prior knowledge about the linear blocks and the nonlinear structure; in a special separable-case the cost further reduces to $O(N\gamma^2)$. The approach is validated via Monte Carlo simulations across diverse WH datasets, showing improved prediction and system-identification fidelity over existing kernels. Theoretical results, including PSD guarantees for the kernel and low-rank properties of the output-kernel matrix, underpin practical, scalable implementations. Overall, the method offers a principled, data-efficient alternative for WH system identification with potential broad applicability to nonlinear block-oriented models.

Abstract

There have been increasing interests on the Volterra series identification with the kernel-based regularization method. The major difficulties are on the kernel design and efficiency of the corresponding implementation. In this paper, we first assume that the underlying system to be identified is the Wiener-Hammerstein (WH) system with polynomial nonlinearity. We then show how to design kernels with nonzero off-diagonal blocks for Volterra maps by taking into account the prior knowledge of the linear blocks and the structure of WH systems. Moreover, exploring the structure of the designed kernels leads to the same computational complexity as the state-of-the-art result, i.e., $O(N^3)$, where $N$ is the sample size, but with a significant difference that the proposed kernels are designed in a direct and flexible way. In addition, for a special case of the kernel and a class of widely used input signals, further exploring the separable structure of the output kernel matrix can lower the computational complexity from $O(N^3)$ to $O(Nγ^2)$, where $γ$ is the separability rank of the output kernel matrix and can be much smaller than $N$. We finally run Monte Carlo simulations to demonstrate the proposed kernels and the obtained theoretical results.

On Kernel Design for Regularized Volterra Series Identification of Wiener-Hammerstein Systems

TL;DR

This paper solves the challenge of kernel design for Volterra-series identification by embedding Wiener–Hammerstein structure directly into off-diagonal kernel blocks. It introduces a WH-informed kernel that preserves the complexity of applicable methods while enabling a direct and flexible incorporation of prior knowledge about the linear blocks and the nonlinear structure; in a special separable-case the cost further reduces to . The approach is validated via Monte Carlo simulations across diverse WH datasets, showing improved prediction and system-identification fidelity over existing kernels. Theoretical results, including PSD guarantees for the kernel and low-rank properties of the output-kernel matrix, underpin practical, scalable implementations. Overall, the method offers a principled, data-efficient alternative for WH system identification with potential broad applicability to nonlinear block-oriented models.

Abstract

There have been increasing interests on the Volterra series identification with the kernel-based regularization method. The major difficulties are on the kernel design and efficiency of the corresponding implementation. In this paper, we first assume that the underlying system to be identified is the Wiener-Hammerstein (WH) system with polynomial nonlinearity. We then show how to design kernels with nonzero off-diagonal blocks for Volterra maps by taking into account the prior knowledge of the linear blocks and the structure of WH systems. Moreover, exploring the structure of the designed kernels leads to the same computational complexity as the state-of-the-art result, i.e., , where is the sample size, but with a significant difference that the proposed kernels are designed in a direct and flexible way. In addition, for a special case of the kernel and a class of widely used input signals, further exploring the separable structure of the output kernel matrix can lower the computational complexity from to , where is the separability rank of the output kernel matrix and can be much smaller than . We finally run Monte Carlo simulations to demonstrate the proposed kernels and the obtained theoretical results.

Paper Structure

This paper contains 20 sections, 7 theorems, 45 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminary and Problem Statement
  3. The kernel-based regularization method
  4. Wiener-Hammerstein systems
  5. Problem statement
  6. Kernel Design
  7. Computational Aspects
  8. Efficient construction of the output kernel matrix
  9. A special case: separable structure of the output kernel matrix
  10. Numerical Simulation
  11. Test systems and databanks
  12. Simulation setup
  13. Simulation results and discussions
  14. Conclusion
  15. Appendix
  16. ...and 5 more sections

Key Result

Lemma 2.1

Consider the Volterra series eq:VolterraSeriesModel and its matrix-vector format eq:VecForm. Let $\theta^0=[(\theta_0^0)^T,(\theta_1^0)^T,\cdots,(\theta_M^0)^T]^T$ be the true value of $\theta$ in eq:VecForm. For a given kernel matrix $P$, let $MSE(\hat{\theta}^{\text{R}}(P)) = \mathbb{E}(\hat{\thet where $P^{\text{opt}}$ is the optimal kernel matrix, in the sense that $MSE(\hat{\theta}^{\text{R}}

Figures (6)

  • Figure 1: The block diagram of the WH system where $G_1$ and $G_2$ are the linear blocks and $\varphi(\cdot)$ is the static nonlinearity.
  • Figure 3: Boxplot of the prediction fits of the estimators based on DC2 and DCWH in Databank D1.
  • Figure 4: Boxplot of the prediction fits of the estimators based on SED-MPK, DC-bd, DC-decay and DC-ob with $M=2,3$ and SNR$=10$ dB for configuration A in Databank D2.
  • Figure 5: Boxplot of the prediction fits of the estimators based on SED-MPK, DC-bd, DC-decay and DC-ob with $M=2,3$ and SNR$=1,5,10$ dB for configuration B in Databank D2.
  • Figure 6: Boxplot of the prediction (top), impulse response (middle) and static nonlinearity (bottom) fits of the estimators based on RVS, SEMIP+ARD and PEM+HOCV in Databank D3.
  • ...and 1 more figures

Theorems & Definitions (16)

  • Remark 2.1
  • Lemma 2.1: COL12
  • Remark 2.2
  • Remark 2.3
  • Proposition 3.1
  • Remark 3.1
  • Theorem 3.1
  • Corollary 3.1
  • Remark 3.2
  • Theorem 4.1
  • ...and 6 more