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A Two-stage Identification Method for Switched Linear Systems

Zheng Wenju, Ye Hao

TL;DR

This paper tackles the identification of switched linear systems by decoupling switching-instants detection from subsystem parameter estimation under a constrained switching assumption. It introduces a two-stage framework where dynamic programming identifies switching instants with a minimal inter-switch interval, followed by sparsity-inducing, convexified parameter estimation on segmented data to recover subsystem parameters. The authors establish unbiasedness under the constrained switching and propose a new sufficient persistent excitation condition tailored to segmentation, supported by theoretical results and practical algorithms using a weighted $\ell_1$-norm relaxation. Experimental results on synthetic data and real high-speed train data demonstrate robust switching-time identification and improved predictive accuracy relative to traditional LS-based methods, especially in noisy settings.

Abstract

In this work, a new two-stage identification method based on dynamic programming and sparsity inducing is proposed for switched linear systems. Our method achieves sparsity inducing in the identification of switched linear systems by the constrained switching mechanism, in contrast to previous optimization-based identification techniques that rely on the rigid data distribution assumption in the parameter space. The proposed mechanism assumes the existence of a minimal interval between adjacent switching instants. First, an efficient iterative dynamic programming approach is used to determine the switching instants and segments using the constrained switching mechanism. Then, each submodel is identified as a combinatorial $\ell_0$ optimization problem, and the true parameter for each submodel is determined by solving the problem. The problem of combinatorial $\ell_0$ optimization is solved by relaxing it into a convex $\ell_1$-norm optimization problem. Furthermore, the unbiasedness of the switched linear system identification is discussed thoroughly with the constrained switching mechanism and a new persistent excitation condition is proposed. Simulation experiments are conducted to indicate that our algorithms exhibit strong robustness against noise.

A Two-stage Identification Method for Switched Linear Systems

TL;DR

This paper tackles the identification of switched linear systems by decoupling switching-instants detection from subsystem parameter estimation under a constrained switching assumption. It introduces a two-stage framework where dynamic programming identifies switching instants with a minimal inter-switch interval, followed by sparsity-inducing, convexified parameter estimation on segmented data to recover subsystem parameters. The authors establish unbiasedness under the constrained switching and propose a new sufficient persistent excitation condition tailored to segmentation, supported by theoretical results and practical algorithms using a weighted -norm relaxation. Experimental results on synthetic data and real high-speed train data demonstrate robust switching-time identification and improved predictive accuracy relative to traditional LS-based methods, especially in noisy settings.

Abstract

In this work, a new two-stage identification method based on dynamic programming and sparsity inducing is proposed for switched linear systems. Our method achieves sparsity inducing in the identification of switched linear systems by the constrained switching mechanism, in contrast to previous optimization-based identification techniques that rely on the rigid data distribution assumption in the parameter space. The proposed mechanism assumes the existence of a minimal interval between adjacent switching instants. First, an efficient iterative dynamic programming approach is used to determine the switching instants and segments using the constrained switching mechanism. Then, each submodel is identified as a combinatorial optimization problem, and the true parameter for each submodel is determined by solving the problem. The problem of combinatorial optimization is solved by relaxing it into a convex -norm optimization problem. Furthermore, the unbiasedness of the switched linear system identification is discussed thoroughly with the constrained switching mechanism and a new persistent excitation condition is proposed. Simulation experiments are conducted to indicate that our algorithms exhibit strong robustness against noise.
Paper Structure (20 sections, 7 theorems, 38 equations, 7 figures, 6 tables, 4 algorithms)

This paper contains 20 sections, 7 theorems, 38 equations, 7 figures, 6 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Prior works on the identification of switched linear systems
  3. Contributions of this paper
  4. Outline of this paper
  5. Preliminaries and problem formulation
  6. Sparse Optimization
  7. Identification Problem
  8. Persistence of excitation problem
  9. Identification based on dynamic programming and sparsity inducing
  10. Constrained switching mechanism and problem formulation
  11. Identification of the switching instants by dynamic programming
  12. Identification for the switched linear system by sparsity inducing
  13. The complete algorithm
  14. A new sufficient persistent excitation condition for switched linear system
  15. Experiments
  16. ...and 5 more sections

Key Result

Theorem 1

(See Theorem 2 in bruckstein2009sparse) If a system of linear equations $Az=b$ with full row rank matrix $A$ has a solution $z$ obeying $\lVert z \rVert_0 < {spark(A)}/ {2}$, this solution is necessarily the sparsest possible.

Figures (7)

  • Figure 1: A noisy example with data samples $\{x_k\}_{k=1}^8$ for the illustration of the mismatch in data partition. The original data partition corresponding to the true switching instants, $\{\lambda_k\}_{k=1}^8$, is shown on the left graph while the estimated partition with mismatch samples, $\{\hat{\lambda}_k\}_{k=1}^8$, is on the right.
  • Figure 2: 300 identification and 200 test data with $SNR=30dB$. The blue solid line, marked with "Origin", represents the true output. The green solid line, marked with "Sparse", denotes the model prediction by the reproducible algorithm in bako2011identification. The red dotted line, marked as "$\ell_0$-norm" represents the model prediction by the improved OMP algorithm, as Algorithm \ref{['algo:zero']} in Appendix \ref{['append_1']} while the purple dot-dashed line, marked as "$\ell_1$-norm" represents the result of Algorithm \ref{['algo:com']} with $\ell_1$-norm.
  • Figure 3: The histogram of Fit for $100$ independent runs by Algorithm \ref{['algo:com']} with $\ell_0$-norm.
  • Figure 4: The histogram of Fit for $100$ independent runs by Algorithm \ref{['algo:com']} with $\ell_1$-norm.
  • Figure 5: 500 identification and 200 test data with $SNR=\infty$.
  • ...and 2 more figures

Theorems & Definitions (18)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Theorem 2
  • Definition 3
  • Remark 1
  • Definition 4
  • Remark 2
  • Remark 3
  • Lemma 1
  • ...and 8 more