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Balancing Application Relevant and Sparsity Revealing Excitation in Input Design

Javad Parsa, Cristian R. Rojas, Håkan Hjalmarsson

TL;DR

This paper tackles sparse system identification when input design can inadvertently raise mutual coherence of regressors by proposing a joint design of the input and a linear coordinate transform. The key idea is to minimize the discrepancy between the achieved Fisher information matrix and a desired one while enforcing a low-coherence transformation, via a three-step alternating minimization that leverages proximal mappings and a Toeplitz regressor representation. The resulting LCID method demonstrates substantial improvements in sparse parameter estimation accuracy and model-order selection in a model-reference control setting, achieving a much lower coherence in the transformed coordinates ($\mu_{\bf H} \approx 0.05$) than the original regressor ($\mu_{\boldsymbol\Phi} \approx 0.98$) and outperforming several baselines, especially at low SNR. The approach is applicable to a range of common linear identification models (FIR, Laguerre, Kautz, generalized basis functions) and can be extended to nonlinear sparse systems, offering practical impact for robust sparse identification and controller design.

Abstract

The maximum absolute correlation between regressors, which is called mutual coherence, plays an essential role in sparse estimation. A regressor matrix whose columns are highly correlated may result from optimal input design, since there is no constraint on the mutual coherence, making it difficult to handle sparse estimation. This paper aims to tackle this issue for fixed denominator models, which include Laguerre, Kautz, and generalized orthonormal basis function expansion models, for example. The paper proposes an optimal input design method where the achieved Fisher information matrix is fitted to the desired Fisher matrix, together with a coordinate transformation designed to make the regressors in the transformed coordinates have low mutual coherence. The method can be used together with any sparse estimation method and any desired Fisher matrix. A numerical study shows its potential for alleviating the problem of model order selection when used in conjunction with, for example, classical methods such as the Akaike Information Criterion.

Balancing Application Relevant and Sparsity Revealing Excitation in Input Design

TL;DR

This paper tackles sparse system identification when input design can inadvertently raise mutual coherence of regressors by proposing a joint design of the input and a linear coordinate transform. The key idea is to minimize the discrepancy between the achieved Fisher information matrix and a desired one while enforcing a low-coherence transformation, via a three-step alternating minimization that leverages proximal mappings and a Toeplitz regressor representation. The resulting LCID method demonstrates substantial improvements in sparse parameter estimation accuracy and model-order selection in a model-reference control setting, achieving a much lower coherence in the transformed coordinates () than the original regressor () and outperforming several baselines, especially at low SNR. The approach is applicable to a range of common linear identification models (FIR, Laguerre, Kautz, generalized basis functions) and can be extended to nonlinear sparse systems, offering practical impact for robust sparse identification and controller design.

Abstract

The maximum absolute correlation between regressors, which is called mutual coherence, plays an essential role in sparse estimation. A regressor matrix whose columns are highly correlated may result from optimal input design, since there is no constraint on the mutual coherence, making it difficult to handle sparse estimation. This paper aims to tackle this issue for fixed denominator models, which include Laguerre, Kautz, and generalized orthonormal basis function expansion models, for example. The paper proposes an optimal input design method where the achieved Fisher information matrix is fitted to the desired Fisher matrix, together with a coordinate transformation designed to make the regressors in the transformed coordinates have low mutual coherence. The method can be used together with any sparse estimation method and any desired Fisher matrix. A numerical study shows its potential for alleviating the problem of model order selection when used in conjunction with, for example, classical methods such as the Akaike Information Criterion.
Paper Structure (8 sections, 3 theorems, 58 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 8 sections, 3 theorems, 58 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Recap of sparse estimation and optimal input design
  3. Sparse estimation
  4. Optimal input design
  5. An input design criterion incorporating mutual coherence
  6. Algorithm
  7. NUMERICAL STUDY
  8. Conclusion

Key Result

Theorem 1

Consider the model eq:spsys11, where ${\bf e}\sim{\@fontswitch\mathcal{N}}(0, \sigma^2{\bf I})$. Here, ${\hbox{\boldmath $\theta$}}_0$ denotes a parameter vector with sparsity characterized by the support set $\Lambda_0$, and $\|{\hbox{\boldmath $\theta$}}_0\|_0 = s$. Assume that the mutual coherenc the support set of the solution ${{\hbox{\boldmath $\theta$}}}^{\epsilon}_1$ obtained through $P_1^

Figures (5)

  • Figure 1: Evaluation with assumption: ${\hbox{\boldmath $\Phi$}}\in{\mathbb{R}}^{100\times 40}$, $s = 10$ and $\zeta = 0.1$.
  • Figure 2: Evaluation of performance degradation with assumptions: ${\hbox{\boldmath $\Phi$}}\in{\mathbb{R}}^{100\times 40}$ and $s = 10$.
  • Figure 3: Evaluation of NRMSE with assumptions: ${\hbox{\boldmath $\Phi$}}\in{\mathbb{R}}^{100\times 40}$ and $s = 10$.
  • Figure 4: Evaluation of performance degradation with assumptions: ${\hbox{\boldmath $\Phi$}}\in{\mathbb{R}}^{100\times 30}$ and $s = 10$.
  • Figure 5: Evaluation of NRMSE with assumptions: ${\hbox{\boldmath $\Phi$}}\in{\mathbb{R}}^{100\times 30}$ and $s = 10$.

Theorems & Definitions (5)

  • Theorem 1: Stability of $P_1^{\tau}$ Ben-Haim2010
  • Lemma 2: 10195960
  • Definition 1: PariB14
  • Lemma 3: Moreau decomposition PariB14
  • Remark 4