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Achieving $\widetilde{O}(1/ε)$ Sample Complexity for Bilinear Systems Identification under Bounded Noises

Hongyu Yi, Chenbei Lu, Jing Yu

Abstract

This paper studies finite-sample set-membership identification for discrete-time bilinear systems under bounded symmetric log-concave disturbances. Compared with existing finite-sample results for linear systems and related analyses under stronger noise assumptions, we consider the more challenging bilinear setting with trajectory-dependent regressors and allow marginally stable dynamics with polynomial mean-square state growth. Under these conditions, we prove that the diameter of the feasible parameter set shrinks with sample complexity $\widetilde{O}(1/ε)$. Simulation supports the theory and illustrates the advantage of the proposed estimator for uncertainty quantification.

Achieving $\widetilde{O}(1/ε)$ Sample Complexity for Bilinear Systems Identification under Bounded Noises

Abstract

This paper studies finite-sample set-membership identification for discrete-time bilinear systems under bounded symmetric log-concave disturbances. Compared with existing finite-sample results for linear systems and related analyses under stronger noise assumptions, we consider the more challenging bilinear setting with trajectory-dependent regressors and allow marginally stable dynamics with polynomial mean-square state growth. Under these conditions, we prove that the diameter of the feasible parameter set shrinks with sample complexity . Simulation supports the theory and illustrates the advantage of the proposed estimator for uncertainty quantification.
Paper Structure (15 sections, 7 theorems, 51 equations, 1 figure)

This paper contains 15 sections, 7 theorems, 51 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Bilinear Dynamical System Model
  4. Set-membership Identification
  5. Assumptions and Definitions
  6. Finite Sample Analysis
  7. Main Results
  8. Simulation
  9. Conclusion
  10. Proof of Lemma \ref{['lem:BMSB']}
  11. $\mathbb{P}(\mathcal{E}_w | \mathcal{E}_u)$
  12. $\mathbb{P}(\mathcal{E}_u)$
  13. Obtain BMSM condition
  14. Proof of Lemma \ref{['lem:E2c']}
  15. Proof of Lemma \ref{['lem:E1E2']}

Key Result

Lemma 1

Under Assumption ass:input and Assumption ass:noise, if the augmented matrix $\tilde{\mathbf A}$ satisfies $\rho(\tilde{\mathbf A})\le 1$, where with $\mathbf F := \mathbf A + \sum_{k=1}^m \mathbb{E}[\mathbf{u}_t[k]]\, \mathbf B_k$ and $\gamma_{k\ell} := \mathbb{E}[\mathbf{u}_t[k]\mathbf{u}_t[\ell]]- \mathbb{E}[\mathbf{u}_t[k]]\,\mathbb{E}[\mathbf{u}_t[\ell]]$, then $\tilde{\mathbf A}$ can be wr

Figures (1)

  • Figure 1: Diameters of Uncertainty Sets Contraction

Theorems & Definitions (14)

  • Definition 1: Diameter of a set
  • Definition 2: Persistent excitation
  • Definition 3: BMSB condition
  • Lemma 1: Polynomial mean-square growth
  • proof
  • Lemma 2: BMSB for $\mathbf{z}_t$
  • Theorem 1: Sample complexity guarantee
  • proof
  • Lemma 3: Bound on $\mathbb{P}(\mathcal{E}_2^c)$
  • proof
  • ...and 4 more