Kernel methods for the approximation of some key quantities of nonlinear systems

Jake Bouvrie; Boumediene Hamzi

doi:10.3934/jcd.2017001

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June & December 2017, 4(1&2): 1-19. doi: 10.3934/jcd.2017001

Kernel methods for the approximation of some key quantities of nonlinear systems

Jake Bouvrie ^1, and Boumediene Hamzi ^2,,

1.	Laboratory for Computational and Statistical Learning, Massachusetts Institute of Technology, Cambridge, MA, USA
2.	Department of Mathematics, AlFaisal University, Riyadh, KSA

* Corresponding author:Boumediene Hamzi

Published April 2017

Fund Project: BH thanks the European Commission and the Scientific and the Technological Research Council of Turkey (Tubitak) for financial support received through a Marie Curie Fellowship, and JB gratefully acknowledges support under NSF contracts NSF-IIS-08-03293 and NSF-CCF-08-08847 to M. Maggioni.
Parts of this work were done while the authors were at the Department of Mathematics of Duke University and then while the second author was with the Departments of Mathematics of Imperial College London, Yildiz Technical University and Ko¸c University for Marie Curie Fellowships and at the Fields Institute.

We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems -with a reasonable expectation of success -once the nonlinear system has been mapped into a high or infinite dimensional feature space. In particular, we embed a nonlinear system in a reproducing kernel Hilbert space where linear theory can be used to develop computable, non-parametric estimators approximating controllability and observability energy functions for nonlinear systems. In all cases the relevant quantities are estimated from simulated or observed data. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system.

Keywords: Nonlinear systems, machine learning, kernel methods, controllability, energy, observability energy, stationary solution of the Fokker-Planck equation.

Mathematics Subject Classification: Primary: 37M99, 37H99, 65P99, 62-07, 60-08; Secondary: 93A99, 65C50, 65C60.

Citation: Jake Bouvrie, Boumediene Hamzi. Kernel methods for the approximation of some key quantities of nonlinear systems. Journal of Computational Dynamics, 2017, 4 (1&2) : 1-19. doi: 10.3934/jcd.2017001

References:

[1]	N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404. doi: 10.1090/S0002-9947-1950-0051437-7.
[2]	G. Biau, B. Cadre and B. Pelletier, Exact rates in density support estimation, J. Multivariate Anal., 99 (2008), 2185-2207. doi: 10.1016/j.jmva.2008.02.021.
[3]	V. I. Bogachev, Gaussian Measures, American Mathematical Society, 1998.
[4]	J. Bouvrie and B. Hamzi, Balanced reduction of nonlinear control systems in reproducing kernel Hilbert space, in Proc. 48th Annual Allerton Conference on Communication, Control, and Computing, (2010), 294-301, http://arxiv.org/abs/1011.2952. doi: 10.1109/ALLERTON.2010.5706920.
[5]	J. Bouvrie and B. Hamzi, Kernel methods for the approximation of nonlinear systems, in SIAM Journal on Control and Optimization, (2017), to appear, https://arxiv.org/abs/1108.2903.
[6]	J. Bouvrie and B. Hamzi, Empirical estimators for stochastically forced nonlinear systems: Observability, controllability and the invariant measure, in Proc. American Control Conference (ACC), 2012, (2012). doi: 10.1109/ACC.2012.6315175.
[7]	R. Brockett, Stochastic Control, Lecture Notes, Harvard University Press, 2009.
[8]	R. L. Butchart, An explicit solution to the Fokker-Planck equation for an ordinary differential equation, Int. J. Control, 1 (1965), 201-208. doi: 10.1080/00207176508905472.
[9]	A. Caponnetto and E. De Vito, Optimal rates for the regularized least-squares algorithm, Found. Comput. Math., 7 (2007), 331-368. doi: 10.1007/s10208-006-0196-8.
[10]	F. Cucker and S. Smale, On the mathematical foundations of learning, Bull. AMS, 39 (2002), 1-49. doi: 10.1090/S0273-0979-01-00923-5.
[11]	G. Da Prato, An Introduction to Infinite Dimensional Analysis, Springer, 2006. doi: 10.1007/3-540-29021-4.
[12]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223.
[13]	E. De Vito, L. Rosasco and A. Toigo, Spectral Regularization for Support Estimation, in J. Shawe-Taylor et al., eds., Advances in Neural Information Processing Systems (NIPS), 24, Vancouver, Curran Associates, Inc., 2010.
[14]	G. E. Dullerud and F. Paganini, A Course in Robust Control Theory: A Convex Approach, Springer, 2000. doi: 10.1007/978-1-4757-3290-0.
[15]	G. Froyland, K. Judd, A. I. Mees, K. Murao and D. Watson, Constructing invariant measures from data, Int. J. Bifurcat. Chaos, 5 (1995), 1181-1192.
[16]	G. Froyland, Extracting dynamical behaviour via Markov models, In Alistair Mees, ed., Nonlinear Dynamics and Statistics: Proceedings, Newton Institute, Cambridge, (2001), 281–321, Birkhauser.
[17]	K. Fujimoto and D. Tsubakino, Computation of nonlinear balanced realization and model reduction based on Taylor series expansion, Systems and Control Letters, 57 (2008), 283-289. doi: 10.1016/j.sysconle.2007.08.015.
[18]	A. T. Fuller, Analysis of nonlinear stochastic systems by means of the Fokker-Planck equation, Int. J. Control, 9 (1969), 603-655. doi: 10.1007/3-540-29021-4.
[19]	P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to dynamical systems, SIAM J. Num. Anal., 45 (2007), 1723-1749. doi: 10.1137/060658813.
[20]	P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Springer, 2007.
[21]	W. S. Gray and E. I. Verriest, Algebraically defined gramians for nonlinear systems, Proc. of the 45th IEEE CDC , (2006). doi: 10.1109/CDC.2006.376840.
[22]	J. Guinez, R. Quintero and A. D. Rueda, Calculating steady states for a Fokker-Planck equation, Acta Math. Hungar., 91 (2001), 311-323. doi: 10.1023/A:1010615818034.
[23]	C. Hartmann and C. Schuette, Balancing of partially-observed stochastic differential equations, Proc. of the 47th IEEE CDC, (2008), 4867-4872. doi: 10.1007/3-540-29021-4.
[24]	D. Kilminster, D. Allingham and A. Mees, Estimating invariant probability densities for dynamical systems: Nonparametric approach to time series analysis, Ann. Ⅰ. Stat. Math., 39 (2002), 1-49. doi: 10.1023/A:1016134209348.
[25]	A. J. Krener, The Important State Coordinates of a Nonlinear System, In Advances in control theory and applications, C. Bonivento, A. Isidori, L. Marconi, C. Rossi, editors, 353 (2007), 161-170, Springer. doi: 10.1007/978-3-540-70701-1_8.
[26]	A. J. Krener, Reduced order modeling of nonlinear control systems, In Analysis and Design of Nonlinear Control Systems, A. Astolfi and L. Marconi, editors, (2008), 41-62, Springer. doi: 10.1007/978-3-540-74358-3_4.
[27]	S. Lall, J. Marsden and S. Glavaski, A subspace approach to balanced truncation for model reduction of nonlinear control systems, nt. J. on Robust and Nonl. Contr., 12 (2002), 519-535. doi: 10.1002/rnc.657.
[28]	D. Liberzon and R. W. Brockett, Nonlinear feedback systems perturbed by noise: Steady-state probability distributions and optimal control, IEEE T. Automat. Control, 45 (2000), 1116-1130. doi: 10.1109/9.863596.
[29]	B. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE T. Automat. Control, 26 (1981), 17-32. doi: 10.1109/TAC.1981.1102568.
[30]	A. J. Newman and P. S. Krishnaprasad, Computing balanced realizations for nonlinear systems, Proc. of the Math. Theory of Networks and Systems (MTNS), (2000).
[31]	H. Risken, The Fokker-Planck Equation, Springer, 1984. doi: 10.1007/978-3-642-96807-5.
[32]	L. Rosasco, M. Belkin and E. De BVito, On learning with integral operators, J. Mach. Learn. Res., 11 (2010), 905-934.
[33]	C. W. Rowley, Model reduction for fluids using balanced proper orthogonal decomposition, Int. J. Bifurcat. Chaos, 11 (2010), 905-934. doi: 10.1142/S0218127405012429.
[34]	J. M. A Scherpen, Balancing for nonlinear systems, Systems & Control Letters, 21 (1993), 143-153. doi: 10.1016/0167-6911(93)90117-O.
[35]	B. Schölkopf and A. J. Smola, Learning with Kernels: Support Vector Machines,Regularization, Optimization, and Beyond, MIT Press, 2001.
[36]	S. Smale and D. X. Zhou, Learning theory estimates via integral operators and their approximations, Constr. Approx., 26 (2007), 153-172. doi: 10.1007/s00365-006-0659-y.
[37]	G. Wahba, Spline Models for Observational Data, SIAM CBMS-NSF Regional Conference Series in Applied Mathematics, Philadelphia, PA, 1990. doi: 10.1137/1.9781611970128.
[38]	H. Wendland, Scattered Data Approximation, Cambridge Monogr. Appl. Comput. Math., Cambridge University Press, Cambridge, UK, 2005.
[39]	M. Zakai, A Lyapunov criterion for the existence of stationary probability distributions for systems perturbed by noise, SIAM J. Control, 1 (1969), 390-397. doi: 10.1137/0307028.

show all references

References:

[1]	N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404. doi: 10.1090/S0002-9947-1950-0051437-7.
[2]	G. Biau, B. Cadre and B. Pelletier, Exact rates in density support estimation, J. Multivariate Anal., 99 (2008), 2185-2207. doi: 10.1016/j.jmva.2008.02.021.
[3]	V. I. Bogachev, Gaussian Measures, American Mathematical Society, 1998.
[4]	J. Bouvrie and B. Hamzi, Balanced reduction of nonlinear control systems in reproducing kernel Hilbert space, in Proc. 48th Annual Allerton Conference on Communication, Control, and Computing, (2010), 294-301, http://arxiv.org/abs/1011.2952. doi: 10.1109/ALLERTON.2010.5706920.
[5]	J. Bouvrie and B. Hamzi, Kernel methods for the approximation of nonlinear systems, in SIAM Journal on Control and Optimization, (2017), to appear, https://arxiv.org/abs/1108.2903.
[6]	J. Bouvrie and B. Hamzi, Empirical estimators for stochastically forced nonlinear systems: Observability, controllability and the invariant measure, in Proc. American Control Conference (ACC), 2012, (2012). doi: 10.1109/ACC.2012.6315175.
[7]	R. Brockett, Stochastic Control, Lecture Notes, Harvard University Press, 2009.
[8]	R. L. Butchart, An explicit solution to the Fokker-Planck equation for an ordinary differential equation, Int. J. Control, 1 (1965), 201-208. doi: 10.1080/00207176508905472.
[9]	A. Caponnetto and E. De Vito, Optimal rates for the regularized least-squares algorithm, Found. Comput. Math., 7 (2007), 331-368. doi: 10.1007/s10208-006-0196-8.
[10]	F. Cucker and S. Smale, On the mathematical foundations of learning, Bull. AMS, 39 (2002), 1-49. doi: 10.1090/S0273-0979-01-00923-5.
[11]	G. Da Prato, An Introduction to Infinite Dimensional Analysis, Springer, 2006. doi: 10.1007/3-540-29021-4.
[12]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223.
[13]	E. De Vito, L. Rosasco and A. Toigo, Spectral Regularization for Support Estimation, in J. Shawe-Taylor et al., eds., Advances in Neural Information Processing Systems (NIPS), 24, Vancouver, Curran Associates, Inc., 2010.
[14]	G. E. Dullerud and F. Paganini, A Course in Robust Control Theory: A Convex Approach, Springer, 2000. doi: 10.1007/978-1-4757-3290-0.
[15]	G. Froyland, K. Judd, A. I. Mees, K. Murao and D. Watson, Constructing invariant measures from data, Int. J. Bifurcat. Chaos, 5 (1995), 1181-1192.
[16]	G. Froyland, Extracting dynamical behaviour via Markov models, In Alistair Mees, ed., Nonlinear Dynamics and Statistics: Proceedings, Newton Institute, Cambridge, (2001), 281–321, Birkhauser.
[17]	K. Fujimoto and D. Tsubakino, Computation of nonlinear balanced realization and model reduction based on Taylor series expansion, Systems and Control Letters, 57 (2008), 283-289. doi: 10.1016/j.sysconle.2007.08.015.
[18]	A. T. Fuller, Analysis of nonlinear stochastic systems by means of the Fokker-Planck equation, Int. J. Control, 9 (1969), 603-655. doi: 10.1007/3-540-29021-4.
[19]	P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to dynamical systems, SIAM J. Num. Anal., 45 (2007), 1723-1749. doi: 10.1137/060658813.
[20]	P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Springer, 2007.
[21]	W. S. Gray and E. I. Verriest, Algebraically defined gramians for nonlinear systems, Proc. of the 45th IEEE CDC , (2006). doi: 10.1109/CDC.2006.376840.
[22]	J. Guinez, R. Quintero and A. D. Rueda, Calculating steady states for a Fokker-Planck equation, Acta Math. Hungar., 91 (2001), 311-323. doi: 10.1023/A:1010615818034.
[23]	C. Hartmann and C. Schuette, Balancing of partially-observed stochastic differential equations, Proc. of the 47th IEEE CDC, (2008), 4867-4872. doi: 10.1007/3-540-29021-4.
[24]	D. Kilminster, D. Allingham and A. Mees, Estimating invariant probability densities for dynamical systems: Nonparametric approach to time series analysis, Ann. Ⅰ. Stat. Math., 39 (2002), 1-49. doi: 10.1023/A:1016134209348.
[25]	A. J. Krener, The Important State Coordinates of a Nonlinear System, In Advances in control theory and applications, C. Bonivento, A. Isidori, L. Marconi, C. Rossi, editors, 353 (2007), 161-170, Springer. doi: 10.1007/978-3-540-70701-1_8.
[26]	A. J. Krener, Reduced order modeling of nonlinear control systems, In Analysis and Design of Nonlinear Control Systems, A. Astolfi and L. Marconi, editors, (2008), 41-62, Springer. doi: 10.1007/978-3-540-74358-3_4.
[27]	S. Lall, J. Marsden and S. Glavaski, A subspace approach to balanced truncation for model reduction of nonlinear control systems, nt. J. on Robust and Nonl. Contr., 12 (2002), 519-535. doi: 10.1002/rnc.657.
[28]	D. Liberzon and R. W. Brockett, Nonlinear feedback systems perturbed by noise: Steady-state probability distributions and optimal control, IEEE T. Automat. Control, 45 (2000), 1116-1130. doi: 10.1109/9.863596.
[29]	B. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE T. Automat. Control, 26 (1981), 17-32. doi: 10.1109/TAC.1981.1102568.
[30]	A. J. Newman and P. S. Krishnaprasad, Computing balanced realizations for nonlinear systems, Proc. of the Math. Theory of Networks and Systems (MTNS), (2000).
[31]	H. Risken, The Fokker-Planck Equation, Springer, 1984. doi: 10.1007/978-3-642-96807-5.
[32]	L. Rosasco, M. Belkin and E. De BVito, On learning with integral operators, J. Mach. Learn. Res., 11 (2010), 905-934.
[33]	C. W. Rowley, Model reduction for fluids using balanced proper orthogonal decomposition, Int. J. Bifurcat. Chaos, 11 (2010), 905-934. doi: 10.1142/S0218127405012429.
[34]	J. M. A Scherpen, Balancing for nonlinear systems, Systems & Control Letters, 21 (1993), 143-153. doi: 10.1016/0167-6911(93)90117-O.
[35]	B. Schölkopf and A. J. Smola, Learning with Kernels: Support Vector Machines,Regularization, Optimization, and Beyond, MIT Press, 2001.
[36]	S. Smale and D. X. Zhou, Learning theory estimates via integral operators and their approximations, Constr. Approx., 26 (2007), 153-172. doi: 10.1007/s00365-006-0659-y.
[37]	G. Wahba, Spline Models for Observational Data, SIAM CBMS-NSF Regional Conference Series in Applied Mathematics, Philadelphia, PA, 1990. doi: 10.1137/1.9781611970128.
[38]	H. Wendland, Scattered Data Approximation, Cambridge Monogr. Appl. Comput. Math., Cambridge University Press, Cambridge, UK, 2005.
[39]	M. Zakai, A Lyapunov criterion for the existence of stationary probability distributions for systems perturbed by noise, SIAM J. Control, 1 (1969), 390-397. doi: 10.1137/0307028.

Figure 1. Comparison between the exact (red), the kernel-based (green) and the empirical (black) steady-state distribution

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