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On Kernel Design for Regularized Non-Causal System Identification

Xiaozhu Fang, Tianshi Chen

TL;DR

The guidelines for kernel design are introduced and the system theoretic framework is extended to design the so-called non-causal simulation-induced (NCSI) kernel, and its structural properties are studied, including stability and semiseparability.

Abstract

Through one decade's development, the kernel-based regularization method (KRM) has become a complement to the classical maximum likelihood/prediction error method and an emerging new system identification paradigm. One recent example is its application in the non-causal system identification, and the key issue lies in the design and analysis of kernels for non-causal systems. In this paper, we develop systematic ways to deal with this issue. In particular, we first introduce the guidelines for kernel design and then extend the system theoretic framework to design the so-called non-causal simulation-induced (NCSI) kernel, and we also study its structural properties, including stability and semiseparability. Finally, we consider some special cases of the NCSI kernel and show their advantage over the existing kernels through numerical simulations.

On Kernel Design for Regularized Non-Causal System Identification

TL;DR

The guidelines for kernel design are introduced and the system theoretic framework is extended to design the so-called non-causal simulation-induced (NCSI) kernel, and its structural properties are studied, including stability and semiseparability.

Abstract

Through one decade's development, the kernel-based regularization method (KRM) has become a complement to the classical maximum likelihood/prediction error method and an emerging new system identification paradigm. One recent example is its application in the non-causal system identification, and the key issue lies in the design and analysis of kernels for non-causal systems. In this paper, we develop systematic ways to deal with this issue. In particular, we first introduce the guidelines for kernel design and then extend the system theoretic framework to design the so-called non-causal simulation-induced (NCSI) kernel, and we also study its structural properties, including stability and semiseparability. Finally, we consider some special cases of the NCSI kernel and show their advantage over the existing kernels through numerical simulations.
Paper Structure (26 sections, 6 theorems, 60 equations, 8 figures, 2 tables)

This paper contains 26 sections, 6 theorems, 60 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background and Problem Statement
  3. Non-Causal System Identification
  4. Kernel-Based Regularization Method
  5. Problem Statement
  6. Non-Causal Kernel Design
  7. Guidelines for Non-Causal Kernel Design
  8. Non-Causal Simulation-Induced Kernel
  9. The Nominal Model
  10. The Uncertainty
  11. The NCSI Kernel
  12. Stability of NCSI Kernel
  13. Semiseparability of NCSI Kernel
  14. Embedding More Prior Knowledge
  15. Over-Damped Dominant Dynamics
  16. ...and 11 more sections

Key Result

Proposition 3.1

Let $\bar{g}^0$ and $\bar{\hat{g}}$ denote any finite dimensional vector obtained by sampling $g^0(t)$ and its estimate $\hat{g}(t)$ at the same but arbitrary sampling time instants in $\mathbb Z$. The optimal kernel is defined as which minimizes the MSE matrix in the sense that $MSE(k(t,s))-MSE(k^{\text{opt}}(t,s))$ is positive semidefinite for any positive semidefinite kernel $k(t,s)$, where $

Figures (8)

  • Figure 1: The block diagram for the multiplicative uncertainty.
  • Figure 2: Interpretation of the NCSI kernel \ref{['eq:si_expre']} as the infinite sum of rank-1 kernels, $g_0(t-k) g_0(s-k)$, weighted by $b^2(k)$, where we choose $g_0(t)=10\exp(-|t|)$, and $b(k)=\exp (-|k|)$.
  • Figure 3: The time cost for computing the Cholesky factor of $K^{\text{NCSI-FO}}$ by using chol in MATLAB (in blue) and by exploiting its high-order semiseparable structure with Algorithm 3 in MC19 (in red).
  • Figure 4: The illustrations of three TC-like non-causal kernels with $\lambda_c= 0.9$, $\lambda_a=0.8$, $c=1$ for the NC-TC kernel \ref{['eq:nctc']} and its block diagonal variant, the NCBD-TC kernel \ref{['eq:bdtc']}, and $\lambda_c= 0.9$, $\lambda_a=0.8$, $c_c=-1$, $c_0=c_a=1$ for the NCSI-TC kernel \ref{['eq:kernel_si_tc']}. One significant difference of the NCSI-TC kernel \ref{['eq:kernel_si_tc']} is its flexibility of choosing negative off-diagonal blocks, which partially accounts for its advantages over the NC-TC kernel \ref{['eq:nctc']} and the NCBD-TC kernel \ref{['eq:bdtc']}.
  • Figure 5: The illustrations of the true non-causal impulse response $g^0$: the single system \ref{['eq:bosys']} in D1, and five system examples in D2-D4.
  • ...and 3 more figures

Theorems & Definitions (16)

  • Proposition 3.1: Optimal Kernel, COL12PDCDL14
  • Lemma 3.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1
  • Remark 3.3
  • Definition 3.1
  • Theorem 3.2
  • Corollary 3.1
  • Remark 3.4
  • ...and 6 more