doi: 10.3934/jcd.2017001

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

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

Received  June 2016 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.

Citation: Bouvrie Jake, Boumediene Hamzi. Kernel methods for the approximation of some key quantities of nonlinear systems. Journal of Computational Dynamics, 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, 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, merican 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. doi: 10.1109/ALLERTON.2010.5706920.

[6]

F. Bouvrie, B. Hamzi, Empirical estimators for stochastically forced nonlinear systems: Observability, controllability and the invariant measure, Proc. American Control Conference (ACC), 2012, (2012). doi: 10.1109/ACC.2012.6315175.

[7]

B. 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, 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, 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.

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G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223.

[13]

E. De Vito, L. Rosasco, 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). doi: 10.1007/3-540-29021-4.

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G. E. Dullerud, F. Paganini, A Course in Robust Control Theory: A Convex Approach, Springer, (2000). doi: 10.1007/978-1-4757-3290-0.

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G. Froyland, K. Judd, A. I. Mees, K. Murao, 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.

[17]

K. Fujimoto, 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.

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P. Giesl, 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, 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, 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, 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, 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, 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, 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, P. S. Krishnaprasad, Proc. of the Math. Theory of Networks and Systems (MTNS), Springer, (2000).

[31]

H. Risken, The Fokker-Planck Equation, Springer, (1984). doi: 10.1007/978-3-642-96807-5.

[32]

L. Rosasco, M. Belkin, 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, A. J. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, MIT Press, (2001). doi: 10.1007/978-3-642-96807-5.

[36]

S. Smale, 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, 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, merican 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. doi: 10.1109/ALLERTON.2010.5706920.

[6]

F. Bouvrie, B. Hamzi, Empirical estimators for stochastically forced nonlinear systems: Observability, controllability and the invariant measure, Proc. American Control Conference (ACC), 2012, (2012). doi: 10.1109/ACC.2012.6315175.

[7]

B. 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, 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, 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, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223.

[13]

E. De Vito, L. Rosasco, 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). doi: 10.1007/3-540-29021-4.

[14]

G. E. Dullerud, 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, 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.

[17]

K. Fujimoto, 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, 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, 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, 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, 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, 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, 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, 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, P. S. Krishnaprasad, Proc. of the Math. Theory of Networks and Systems (MTNS), Springer, (2000).

[31]

H. Risken, The Fokker-Planck Equation, Springer, (1984). doi: 10.1007/978-3-642-96807-5.

[32]

L. Rosasco, M. Belkin, 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, A. J. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, MIT Press, (2001). doi: 10.1007/978-3-642-96807-5.

[36]

S. Smale, 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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