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Exploring the Links between the Fundamental Lemma and Kernel Regression
Published 8 Mar 2024 in eess.SY, cs.LG, cs.SY, and math.OC | (2403.05368v2)
Abstract: Generalizations and variations of the fundamental lemma by Willems et al. are an active topic of recent research. In this note, we explore and formalize the links between kernel regression and some known nonlinear extensions of the fundamental lemma. Applying a transformation to the usual linear equation in Hankel matrices, we arrive at an alternative implicit kernel representation of the system trajectories while keeping the requirements on persistency of excitation. We show that this representation is equivalent to the solution of a specific kernel regression problem. We explore the possible structures of the underlying kernel as well as the system classes to which they correspond.
- J. C. Willems, P. Rapisarda, I. Markovsky, and B. L. De Moor, “A note on persistency of excitation,” Syst. Control Lett., vol. 54, no. 4, pp. 325–329, 2005.
- I. Markovsky and F. Dörfler, “Behavioral systems theory in data-driven analysis, signal processing, and control,” Ann. Rev. Control, vol. 52, pp. 42–64, 2021.
- P. Verheijen, V. Breschi, and M. Lazar, “Handbook of linear data-driven predictive control: Theory, implementation and design,” Ann. Rev. Control, vol. 56, p. 100914, 2023.
- I. Markovsky, “Data-Driven Simulation of Generalized Bilinear Systems via Linear Time-Invariant Embedding,” IEEE Trans. Autom. Control, vol. 68, no. 2, pp. 1101–1106, 2023.
- J. Berberich and F. Allgöwer, “A trajectory-based framework for data-driven system analysis and control,” in Eur. Control Conf. (ECC), 2020, pp. 1365–1370.
- M. Alsalti, J. Berberich, V. G. Lopez, F. Allgöwer, and M. A. Müller, “Data-Based System Analysis and Control of Flat Nonlinear Systems,” in 60th IEEE Conf. Decis. Control (CDC), 2021, pp. 1484–1489.
- M. Alsalti, V. G. Lopez, J. Berberich, F. Allgöwer, and M. A. Müller, “Data-Based Control of Feedback Linearizable Systems,” IEEE Trans. Autom. Control, vol. 68, no. 11, pp. 7014–7021, 2023.
- J. G. Rueda-Escobedo and J. Schiffer, “Data-Driven Internal Model Control of Second-Order Discrete Volterra Systems,” in 59th IEEE Conf. Decis. Control (CDC), 2020, pp. 4572–4579.
- Z. Yuan and J. Cortés, “Data-Driven Optimal Control of Bilinear Systems,” IEEE Control Syst. Lett., vol. 6, pp. 2479–2484, 2022.
- D. Reinhardt and S. Gros, “Data-Driven Predictive Control for Nonlinear Systems Using Feature Selection,” in 62nd IEEE Conf. Decis. Control (CDC), 2023, pp. 2576–2583.
- M. Lazar, “Basis functions nonlinear data-enabled predictive control: Consistent and computationally efficient formulations,” 2023, preprint, arXiv:2311.05360.
- Y. Lian and C. N. Jones, “Nonlinear Data-Enabled Prediction and Control,” in Proc. 3rd Conf. Learn. Dynam. Control, ser. Proceedings of Machine Learning Research, vol. 144. PMLR, 2021, pp. 523–534.
- L. Huang, J. Lygeros, and F. Dörfler, “Robust and kernelized data-enabled predictive control for nonlinear systems,” IEEE Trans. Control Syst. Techn., pp. 1–14, 2023.
- N. Aronszajn, “Theory of Reproducing Kernels,” Trans. Amer. Math. Soc., vol. 68, no. 3, pp. 337–404, 1950.
- J. Umlauft and S. Hirche, “Feedback Linearization Based on Gaussian Processes With Event-Triggered Online Learning,” IEEE Trans. Autom. Control, vol. 65, no. 10, pp. 4154–4169, 2020.
- M. Maiworm, D. Limon, and R. Findeisen, “Online learning-based model predictive control with Gaussian process models and stability guarantees,” Intern. J. Robust Nonlin. Control, vol. 31, no. 18, pp. 8785–8812, 2021.
- G. Kimeldorf and G. Wahba, “A correspondence between Bayesian estimation on stochastic processes and smoothing by splines,” Ann. Math. Stat., vol. 41, no. 2, pp. 495–502, 1970.
- B. Schölkopf, R. Herbrich, and A. Smola, “A generalized representer theorem,” in Intern. conf. comput. learn. theory. Springer, 2001, pp. 416–426.
- H. Mohammadi, R. Le Riche, N. Durrande, E. Touboul, and X. Bay, “An analytic comparison of regularization methods for Gaussian Processes,” 2017, preprint, arXiv:1602.00853v3.
- G. Fasshauer, “Positive Definite Kernels: Past, Present and Future,” Dolomite Res. Notes Approx., vol. 4, pp. 1–48, 2011.
- Z. Hu, C. D. Persis, and P. Tesi, “Learning controllers from data via kernel-based interpolation,” 2023, preprint, arXiv:2304.09577.
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