Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
156 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Finite Time Performance Analysis of MIMO Systems Identification (2310.11790v1)

Published 18 Oct 2023 in eess.SY and cs.SY

Abstract: This paper is concerned with the finite time identification performance of an n dimensional discrete-time Multiple-Input Multiple-Output (MIMO) Linear Time-Invariant system, with p inputs and m outputs. We prove that the widely-used Ho-Kalman algorithm and Multivariable Output Error State Space (MOESP) algorithm are ill-conditioned for MIMO system when n/m or n/p is large. Moreover, by analyzing the Cramer-Rao bound, we derive a fundamental limit for identifying the real and stable (or marginally stable) poles of MIMO system and prove that the sample complexity for any unbiased pole estimation algorithm to reach a certain level of accuracy explodes superpolynomially with respect to n/(pm). Numerical results are provided to illustrate the ill-conditionedness of Ho-Kalman algorithm and MOESP algorithm as well as the fundamental limit on identification.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (37)
  1. Lennart Ljung. System identification: theory for the user prentice-hall, inc. Upper Saddle River, NJ, USA, 1986.
  2. DYNAMIC SYSTEM IDENTIFICATION. Elsevier, 1977.
  3. Lennart Ljung. Consistency of the least-squares identification method. IEEE Transactions on Automatic Control, 21(5):779–781, 1976.
  4. Consistency and relative efficiency of subspace methods. Automatica, 31(12):1865–1875, 1995.
  5. Yang Zheng and Na Li. Non-asymptotic identification of linear dynamical systems using multiple trajectories. IEEE Control Systems Letters, 5(5):1693–1698, 2020.
  6. Finite sample properties of system identification methods. IEEE Transactions on Automatic Control, 47(8):1329–1334, 2002.
  7. M Vidyasagar and Rajeeva L Karandikar. A learning theory approach to system identification and stochastic adaptive control. Journal of Process Control, 18:421–430, 2008.
  8. Finite-sample system identification: An overview and a new correlation method. IEEE Control Systems Letters, 2(1):61–66, 2017.
  9. Learning sparse dynamical systems from a single sample trajectory. In 2019 IEEE 58th Conference on Decision and Control (CDC), pages 2682–2689. IEEE, 2019.
  10. Learning without mixing: Towards a sharp analysis of linear system identification. In Conference On Learning Theory, pages 439–473. PMLR, 2018.
  11. Finite time identification in unstable linear systems. Automatica, 96:342–353, 2018.
  12. Near optimal finite time identification of arbitrary linear dynamical systems. In International Conference on Machine Learning, pages 5610–5618. PMLR, 2019.
  13. On the sample complexity of the linear quadratic regulator. Foundations of Computational Mathematics, 20(4):633–679, 2020.
  14. Active learning for identification of linear dynamical systems. In Conference on Learning Theory, pages 3487–3582. PMLR, 2020.
  15. A tutorial on concentration bounds for system identification. In 2019 IEEE 58th Conference on Decision and Control (CDC), pages 3741–3749. IEEE, 2019.
  16. Sample complexity lower bounds for linear system identification. In 2019 IEEE 58th Conference on Decision and Control (CDC), pages 2676–2681. IEEE, 2019.
  17. Linear systems can be hard to learn. In 2021 60th IEEE Conference on Decision and Control (CDC), pages 2903–2910. IEEE, 2021.
  18. Sample complexity of linear quadratic gaussian (lqg) control for output feedback systems. In Learning for Dynamics and Control, pages 559–570. PMLR, 2021.
  19. Revisiting ho–kalman-based system identification: Robustness and finite-sample analysis. IEEE Transactions on Automatic Control, 67(4):1914–1928, 2021.
  20. Learning linear dynamical systems with semi-parametric least squares. In Conference on Learning Theory, pages 2714–2802. PMLR, 2019.
  21. Finite sample analysis of stochastic system identification. In 2019 IEEE 58th Conference on Decision and Control (CDC), pages 3648–3654. IEEE, 2019.
  22. On the ill-conditioning of subspace identification with inputs. Automatica, 40(4):575–589, 2004.
  23. N4sid and moesp algorithms to highlight the ill-conditioning into subspace identification. International Journal of Automation and Computing, 11(1):30–38, 2014.
  24. Estimation error analysis of system matrices in some subspace identification methods. In 2015 10th Asian Control Conference (ASCC), pages 1–6. IEEE, 2015.
  25. Logarithmic regret bound in partially observable linear dynamical systems. Advances in Neural Information Processing Systems, 33:20876–20888, 2020.
  26. Gradient descent learns linear dynamical systems. Journal of Machine Learning Research, 19:1–44, 2018.
  27. Fundamental identification limit of single-input and single-output linear time-invariant systems. In 2022 13th Asian Control Conference (ASCC), pages 2157–2162. IEEE, 2022.
  28. Fundamental identification limit on diagonal canonical form for siso systems. In 2022 IEEE 17th International Conference on Control & Automation (ICCA), pages 728–733. IEEE, 2022.
  29. Fundamental limit on siso system identification. In 2022 IEEE 61st Conference on Decision and Control (CDC), pages 856–861. IEEE, 2022.
  30. BL Ho and Rudolf E Kálmán. Effective construction of linear state-variable models from input/output functions. at-Automatisierungstechnik, 14(1-12):545–548, 1966.
  31. Subspace model identification part 2. analysis of the elementary output-error state-space model identification algorithm. International journal of control, 56(5):1211–1241, 1992.
  32. Subspace identification with multiple data sets. In Guidance, Navigation, and Control Conference, page 3716, 1995.
  33. Arun K Tangirala. Principles of system identification: theory and practice. Crc Press, 2018.
  34. Steven M Kay. Fundamentals of statistical signal processing: estimation theory. Prentice-Hall, Inc., 1993.
  35. Network energy minimization via sensor selection and topology control. IFAC Proceedings Volumes, 42(20):174–179, 2009.
  36. On the singular values of matrices with displacement structure. SIAM Journal on Matrix Analysis and Applications, 38(4):1227–1248, 2017.
  37. Suk-Geun Hwang. Cauchy’s interlace theorem for eigenvalues of hermitian matrices. The American mathematical monthly, 111(2):157–159, 2004.
Citations (1)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now