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Identification of non-causal systems with arbitrary switching modes

Yanxin Zhang, Chengpu Yu, Filippo Fabiani

Abstract

We consider the identification of non-causal systems with arbitrary switching modes (NCS-ASM), a class of models essential for describing typical power load management and department store inventory dynamics. The simultaneous identification of causal-and-anticausal subsystems, along with the presence of possibly random switching sequences, however, make the overall identification problem particularly challenging. To this end, we develop an expectation-maximization (EM) based system identification technique, where the E-step proposes a modified Kalman filter (KF) to estimate the states and switching sequences of causal-and-anticausal subsystems, while the M-step consists in a switching least-squares algorithm to estimate the parameters of individual subsystems. We establish the main convergence features of the proposed identification procedure, also providing bounds on the parameter estimation errors under mild conditions. Finally, the effectiveness of our identification method is validated through two numerical simulations.

Identification of non-causal systems with arbitrary switching modes

Abstract

We consider the identification of non-causal systems with arbitrary switching modes (NCS-ASM), a class of models essential for describing typical power load management and department store inventory dynamics. The simultaneous identification of causal-and-anticausal subsystems, along with the presence of possibly random switching sequences, however, make the overall identification problem particularly challenging. To this end, we develop an expectation-maximization (EM) based system identification technique, where the E-step proposes a modified Kalman filter (KF) to estimate the states and switching sequences of causal-and-anticausal subsystems, while the M-step consists in a switching least-squares algorithm to estimate the parameters of individual subsystems. We establish the main convergence features of the proposed identification procedure, also providing bounds on the parameter estimation errors under mild conditions. Finally, the effectiveness of our identification method is validated through two numerical simulations.
Paper Structure (16 sections, 4 theorems, 53 equations, 4 figures, 3 tables, 1 algorithm)

This paper contains 16 sections, 4 theorems, 53 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature review
  3. Summary of contribution
  4. Paper organization
  5. Mathematical formulation
  6. System model description
  7. Problem statement
  8. The EM method for identifying NCS-ASM
  9. Implementation details of the EM algorithm
  10. The E-step
  11. The M-step
  12. Numerical examples
  13. Example 1: Academic NCS-ASM
  14. Example 2: The Department Store Inventory Price Index
  15. Conclusion
  16. ...and 1 more sections

Key Result

Lemma 1

Given a dataset $\bm{y}$, let $\{\theta^k\}_{k\in\mathbb{Z}}$ be the sequence generated by Algorithm alg:EM. Then, the likelihood function in eq:mdf, evaluated along $\{\theta^k\}_{k\in\mathbb{Z}}$, is non-decreasing, thereby yielding $\ln \mathbb P_{\theta^{k+1}}[\bm{y}]\geq\ln \mathbb P_{\theta^k}

Figures (4)

  • Figure 1: The true (blue cross) and estimated (red circle) mode sequences over a certain time window of length $100$.
  • Figure 2: Dynamical evolution of the true state variables $\bm{x}_c$ and $\bm{x}_a$ (solid blue line), and of the estimated ones $\hat{\bm x}_c$, and $\hat{\bm x}_a$ (dashed red lines).
  • Figure 3: Match rates obtained by the proposed algorithm for different noise levels.
  • Figure 4: The estimated prices and the true prices with different numbers of the subsystem. (a) $\bm{s}_c=\bm{s}_a$ and $m_c=m_a=1$; (b) $\bm{s}_c\neq \bm{s}_a$ and $m_c=m_a=1$; (c) $\bm{s}_c\neq \bm{s}_a$ and $m_c=m_a=2$

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Definition 1
  • Lemma 3
  • Lemma 4
  • Remark 3