Standard LS Parameter Estimators Ensure Finite Convergence Time for Linear Regression Equations Under an Interval Excitation Assumption

https://doi.org/10.58571/CNCA.AMCA.2025.016

Resumen: In this brief note we recall the little-known fact that, for linear regression equations with intervally excited (IE) regressors, standard Least Squares parameter estimators ensure finite convergence time of the estimated parameters. The convergence time being equal to the time length needed to comply with the IE assumption. As is wellknown, IE is necessary and sufficient for the identifiability of the linear regression equation—hence, it is the weakest assumption for the on- or off-line solution of the parameter estimation problem.

¿Cómo citar?

Ortega, R., Romero, J., Aranovskiy, S. & Tao, G. (2025). Standard LS Parameter Estimators Ensure Finite Convergence Time for Linear Regression Equations Under an Interval Excitation Assumption. Memorias del Congreso Nacional de Control Automático 2025, pp. 92-96. https://doi.org/10.58571/CNCA.AMCA.2025.016

• V. Adetola and M, Guay, Finite-time parameter estimation in adaptive control of nonlinear systems, IEEE Trans. Automatic Control. vol. 53, no. 3, 807-811, 2008.
• S. Aranovskiy, R. Ortega, J. Romero and D. Sokolov, A globally exponentially stable speed observer for a class of mechanical systems: experimental and simulation comparison with high-gain and sliding mode designs, Int. J. of Control, vol. 92, no. 7, pp 1620-1633, 2019.
• V. Andrieu, L. Praly and A. Astolfi, Homogeneous approximation, recursive observer design, and output feedback, SIAM J. Control and Optimization, vol. 47, no. 4, pp. 1814-1850, 2008.
• M. Basin, Finite- and fixed-time convergent algorithms: Design and convergence time estimation, IFAC Annual Reviews in Control, vol. 48, 209-221, 2021.
• S. Bhat and D. S. Bernstein, Geometric homogeneity with applications to finite-time stability, Mathematics of Control, Signals and Systems, vol. 17, pp. 101-127, 2005.
• Y. Chen, F. Wang and B. Wang, Fixed-time convergence in continuous-time optimization: A fractional approach, IEEE Control Systems Letters, vol. 7, pp. 631-636, 2023.
• E. Cruz-Zavala, E. Nu˜no and J. A. Moreno, Trajectory-tracking in finite-time for robot manipulators with bounded torques via output-feedback, Int. J. of Control, vol. 96, no. 4, pp. 907-921, 2023.
• Ph. de Larminat, On the stabilizability condition in indirect adaptive control, Automatica, vol. 20, no, 6, pp. 793-795, 1984.
• D. Efimov and A. Polyakov, Finite-time stability tools for control and estimation, Foundations and Trends in Systems and Control, vol. 9, no. 2-3, pp. 171-364, 2021.
• C. Huang, H. Shim, S. Yu and B. D. O. Anderson, Mode consensus algorithms with finite convergence time, Systems and Control Letters, arXiv:2403.00221, 2024.
• M. Kawski, Families of dilations and asymptotic stability, Analysis of Controlled Dynamical Systems, pp. 285-294, 1991.
• J. Krause and P. Khargonekar, Parameter information content of measurable signals in direct adaptive control, IEEE Trans. Automatic Control, vol. 32, no. 9, 1987
• G. Kreisselmeier and G. Rietze-Augst, Richness and excitation on an interval—with application to continuous-time adaptive control, IEEE Trans. Automatic Control, vol. 35, no. 2, pp. 165-171, 1990.
• R. Lozano and X. Zhao, Adaptive pole placement without excitation probing signals, IEEE Trans. Automatic Control, vol. 39, no. 1, pp. 47-58, 1994.
• R. Marino and P. Tomei, On exponentially convergent parameter estimation with lack of persistency of excitation, Systems and Control Letters, vol. 159, 105080, 2022.
• H. Min, S. Shi, J. Gu and N. Dua, Further results on fixed-time control for nonlinear systems with asymmetric output constraints, Automatica, vol. 171, 111933, 2025.
• R. Ortega, An on-line least-squares parameter estimator with finite convergence time, Proc. IEEE, vol. 76, no. 7, 1988.
• R. Ortega, A. Bobtsov and N. Nikolayev, Parameter identification with finite-convergence time alertness preservation, IEEE Control Systems Letters, vol. 6, pp. 205-210, 2022.
• R. Ortega, J. G. Romero and S. Aranovskiy, A new least squares parameter estimator for nonlinear regression equations with relaxed excitation conditions and forgetting factor, Systems and Control Letters, vol. 169, no. 105377, 2022.
• Y. Pan, T. Shi and R. Ortega, Comparative análisis of parameter convergence for several least-squares estimation schemes, IEEE Trans. Automatic Control, vol. 69, no. 5, pp. 3341-3348, 2023.
• C. Rao and H. Toutenburg, H, Linear Models: Least Squares and Alternatives, Springer Series in Statistics (3rd ed.). Berlin, Springer, 2008.
• H. Rios, D. Efimov, J. A. Moreno, W. Perruquetti, and J. G. Rueda-Escobedo, Time-varying parameter identification algorithms: Finite and fixed-time convergence, IEEE Trans. Automatic Control, vol. 62, no. 7, pp. 3671-3678, 2017.
• J.-J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, New Jersey, USA, 1991.
• G. Tao, Adaptive Control Design and Analysis, vol. 37, John Wiley & Sons, New Jersey, 2003.
• L. Wang, R. Ortega, A. Bobtsov, J. G. Romero and B. Yi, Identifiability implies robust, globally exponentially convergent on-line parameter estimation, Int. J. of Control, vol. 97, no. 9, pp. 967-1983, 2024.
• J. Wang, D. Efimov and A. Bobtsov, On robust parameter estimation in finite-time without persistence of excitation, IEEE Trans. Automatic Control, vol. 65, no. 4, pp. 1731-1738, 2020.