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A finite-sample bound for identifying partially observed linear switched systems from a single trajectory

Daniel Racz, Mihaly Petreczky, Balint Daroczy

TL;DR

This work addresses identifying Linear Switched Systems (LSS) from a single observed trajectory by estimating Markov parameters and applying a reduced-basis Ho-Kalman realization. It proves a finite-sample probabilistic bound on the parameter estimation error under quadratic stability, leveraging concentration results for weakly dependent processes. The main result shows an $O(1/\sqrt{N})$ convergence rate for the Frobenius-norm error in the recovered system matrices, with high-probability guarantees that improve as the sample size grows. This provides a practical consistency guarantee for using Ho-Kalman-based identification on LSSs from a single long sequence, informing sample complexity and design of the reduced-basis Hankel approach for control-oriented modeling.

Abstract

We derive a finite-sample probabilistic bound on the parameter estimation error of a system identification algorithm for Linear Switched Systems. The algorithm estimates Markov parameters from a single trajectory and applies a variant of the Ho-Kalman algorithm to recover the system matrices. Our bound guarantees statistical consistency under the assumption that the true system exhibits quadratic stability. The proof leverages the theory of weakly dependent processes. To the best of our knowledge, this is the first finite-sample bound for this algorithm in the single-trajectory setting.

A finite-sample bound for identifying partially observed linear switched systems from a single trajectory

TL;DR

This work addresses identifying Linear Switched Systems (LSS) from a single observed trajectory by estimating Markov parameters and applying a reduced-basis Ho-Kalman realization. It proves a finite-sample probabilistic bound on the parameter estimation error under quadratic stability, leveraging concentration results for weakly dependent processes. The main result shows an convergence rate for the Frobenius-norm error in the recovered system matrices, with high-probability guarantees that improve as the sample size grows. This provides a practical consistency guarantee for using Ho-Kalman-based identification on LSSs from a single long sequence, informing sample complexity and design of the reduced-basis Hankel approach for control-oriented modeling.

Abstract

We derive a finite-sample probabilistic bound on the parameter estimation error of a system identification algorithm for Linear Switched Systems. The algorithm estimates Markov parameters from a single trajectory and applies a variant of the Ho-Kalman algorithm to recover the system matrices. Our bound guarantees statistical consistency under the assumption that the true system exhibits quadratic stability. The proof leverages the theory of weakly dependent processes. To the best of our knowledge, this is the first finite-sample bound for this algorithm in the single-trajectory setting.

Paper Structure

This paper contains 9 sections, 9 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Main result
  4. Conclusions and future work
  5. Remaining proofs
  6. Proof of Lemma \ref{['lem:stab:stat']}
  7. Multi-output case
  8. Numerical example
  9. Effect of observed switching signal on Theorem \ref{['thm:main']}

Key Result

Lemma II.1

Let $B'_q=[B_q \mid I_n \mid 0 ]$, and for all $v \in Q^{*}$, $M'_{qv}=CA_vB'_q$ and $M'_{\epsilon}=[ D, 0 ] \in \mathbb{R}^{1 \times m+1}$, and $m' = m + n + 1$. Under Assumption ass:main the followings hold:

Figures (3)

  • Figure 1: Effect of stability on the parameter estimation error
  • Figure 2: Effect of the variance $4K_{u,inp}^2/12$ of $u$ on the estimation error: smaller values of $K_{u,inp}$ correspond to smaller variance.
  • Figure 3: Effect of the magnitude $K_u$ of the input and noise on the estimation error

Theorems & Definitions (13)

  • Lemma II.1
  • Lemma II.2: MertBastug:TAC
  • Lemma II.3: petreczky2010spaces
  • Lemma III.1
  • proof
  • Theorem III.2
  • Lemma III.3
  • Lemma III.4
  • Theorem III.5: Main result
  • proof
  • ...and 3 more