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A Weighted Least-Squares Method for Non-Asymptotic Identification of Markov Parameters from Multiple Trajectories

Jiabao He, Cristian R. Rojas, Håkan Hjalmarsson

TL;DR

The paper addresses non-asymptotic identification of Markov parameters for partially observed LTI systems from multiple trajectories. It proposes Weighted Least-Squares (WLS) with an optimal weighting matrix to exploit noise structure, deriving finite-sample error bounds that scale as $O(N^{-1/2})$ and are tighter than those for OLS. It also develops two consistent schemes to estimate the weighting matrix (recursive and Ho-Kalman-based) with proven convergence rates, and analyzes how estimation errors propagate to WLS. Numerical experiments on a marginally stable SISO and a stable MIMO system show WLS outperforms OLS in finite samples, and that weighting-matrix estimates converge toward the optimum with increasing data. This work advances data-efficient identification for control tasks and suggests integration with subspace methods for robust realization.

Abstract

Markov parameters play a key role in system identification. There exists many algorithms where these parameters are estimated using least-squares in a first, pre-processing, step, including subspace identification and multi-step least-squares algorithms, such as Weighted Null-Space Fitting. Recently, there has been an increasing interest in non-asymptotic analysis of estimation algorithms. In this contribution we identify the Markov parameters using weighted least-squares and present non-asymptotic analysis for such estimator. To cover both stable and unstable systems, multiple trajectories are collected. We show that with the optimal weighting matrix, weighted least-squares gives a tighter error bound than ordinary least-squares for the case of non-uniformly distributed measurement errors. Moreover, as the optimal weighting matrix depends on the system's true parameters, we introduce two methods to consistently estimate the optimal weighting matrix, where the convergence rate of these estimates is also provided. Numerical experiments demonstrate improvements of weighted least-squares over ordinary least-squares in finite sample settings.

A Weighted Least-Squares Method for Non-Asymptotic Identification of Markov Parameters from Multiple Trajectories

TL;DR

The paper addresses non-asymptotic identification of Markov parameters for partially observed LTI systems from multiple trajectories. It proposes Weighted Least-Squares (WLS) with an optimal weighting matrix to exploit noise structure, deriving finite-sample error bounds that scale as and are tighter than those for OLS. It also develops two consistent schemes to estimate the weighting matrix (recursive and Ho-Kalman-based) with proven convergence rates, and analyzes how estimation errors propagate to WLS. Numerical experiments on a marginally stable SISO and a stable MIMO system show WLS outperforms OLS in finite samples, and that weighting-matrix estimates converge toward the optimum with increasing data. This work advances data-efficient identification for control tasks and suggests integration with subspace methods for robust realization.

Abstract

Markov parameters play a key role in system identification. There exists many algorithms where these parameters are estimated using least-squares in a first, pre-processing, step, including subspace identification and multi-step least-squares algorithms, such as Weighted Null-Space Fitting. Recently, there has been an increasing interest in non-asymptotic analysis of estimation algorithms. In this contribution we identify the Markov parameters using weighted least-squares and present non-asymptotic analysis for such estimator. To cover both stable and unstable systems, multiple trajectories are collected. We show that with the optimal weighting matrix, weighted least-squares gives a tighter error bound than ordinary least-squares for the case of non-uniformly distributed measurement errors. Moreover, as the optimal weighting matrix depends on the system's true parameters, we introduce two methods to consistently estimate the optimal weighting matrix, where the convergence rate of these estimates is also provided. Numerical experiments demonstrate improvements of weighted least-squares over ordinary least-squares in finite sample settings.
Paper Structure (11 sections, 50 equations, 2 figures)

This paper contains 11 sections, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Identifying Markov Parameters
  4. Ordinary Least-Squares
  5. Weighted Least-Squares
  6. Estimating the Weighting Matrix
  7. Recursive Algorithm
  8. Ho-Kalman Algorithm
  9. Convergence Rate
  10. Simulation Examples
  11. Conclusion

Figures (2)

  • Figure 1: Estimation errors of SISO (left) and MIMO (right).
  • Figure 2: Estimation errors of SISO (left) and MIMO (right).