A Weighted Least-Squares Method for Non-Asymptotic Identification of Markov Parameters from Multiple Trajectories (2405.04258v1)
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.
- Bernstein, D.S. (2009). Matrix mathematics: theory, facts, and formulas. Princeton University Press.
- The effect of estimating weights in weighted least squares. J. Am. Stat. Assoc., 83(404), 1045–1054.
- Iterative weighted least squares estimators. Ann. Stat., 21(2), 1071–1092.
- Efficient learning of hidden state LTI state space models of unknown order. arXiv preprint arXiv:2202.01625.
- Fattahi, S. (2021). Learning partially observed linear dynamical systems from logarithmic number of samples. In Proc. Learn. Dyn. Control.
- Parametric identification using weighted null-space fitting. IEEE Trans. Autom. Control, 64(7), 2798–2813.
- Effective construction of linear state-variable models from input/output functions. Automatisierungstechnik, 14(1-12), 545–548.
- Finite-time identification of linear systems: Fundamental limits and optimal algorithms. IEEE Trans. Autom. Control.
- Identification of observer/Kalman filter Markov parameters: Theory and experiments. J. Guid. Control Dyn., 16(2), 320–329.
- Lee, H. (2022). Improved rates for prediction and identification of partially observed linear dynamical systems. In Int. Conf. on Algorithmic Learn. Theory.
- Non-asymptotic identification of lti systems from a single trajectory. In Proc. Amer. Control Conf (ACC).
- Revisiting Ho–Kalman-based system identification: Robustness and finite-sample analysis. IEEE Trans. Autom. Control, 67(4), 1914–1928.
- Near optimal finite time identification of arbitrary linear dynamical systems. In Int. Conf. on Machine Learn., 5610–5618.
- Finite-time system identification for partially observed LTI systems of unknown order. arXiv preprint arXiv:1902.01848.
- Learning linear dynamical systems with semi-parametric least squares. In Conf. on Learn. Theory.
- Learning without mixing: Towards a sharp analysis of linear system identification. In Conf. On Learn. Theory.
- Finite sample system identification: Optimal rates and the role of regularization. In Proc. Learn. Dyn. Control, 16–25.
- Finite sample analysis of stochastic system identification. In IEEE 58th Conf. Decis. Control (CDC).
- Non-asymptotic analysis of robust control from coarse-grained identification. arXiv preprint arXiv:1707.04791.
- Subspace Identification for Linear Systems: Theory-Implementation-Applications. Springer.
- Subspace identification with non-steady Kalman filter parameterization. J. Process Control, 24(9), 1337–1345.
- Non-asymptotic identification of linear dynamical systems using multiple trajectories. IEEE Control Syst. Lett., 5(5), 1693–1698.