Online identification of nonlinear spatiotemporal systems using kernel learning approach

IEEE Transactions on Neural Networks

Volumn 22, Issue 9, 2011, Pages 1381-1394

  Ning, Hanwen ^a   Jing, Xingjian ^a   Cheng, Li ^a  

^a HONG KONG POLYTECHNIC UNIVERSITY   (Hong Kong)

Author keywords

Lattice dynamics; least squares support vector machines; nonlinear system identification; partially linear systems; spatiotemporal systems

Indexed keywords

ENGINEERING PRACTICES; ENGINEERING SYSTEMS; ERROR MINIMIZATION; IDENTIFICATION PROBLEM; KERNEL LEARNING; LEAST SQUARE SUPPORT VECTOR MACHINES; LEAST SQUARES SUPPORT VECTOR MACHINES; MULTI-INPUT MULTI-OUTPUT; NON-LINEAR SYSTEM IDENTIFICATION; NONLINEAR DISTRIBUTED PARAMETER SYSTEM; NONLINEAR MECHANISMS; ON-LINE ESTIMATION; ON-LINE IDENTIFICATION; OPTIMAL ANALYSIS; PARTIALLY LINEAR SYSTEMS; PHYSICAL CHARACTERISTICS; PHYSICAL MODEL; SPATIO-TEMPORAL RELATIONSHIPS; SPATIO-TEMPORAL SYSTEM; STRUCTURAL INFORMATION; SYSTEM DYNAMICS; UNDERLYING SYSTEMS;

ALGORITHMS; CRYSTAL LATTICES; DATA REDUCTION; DISTRIBUTED PARAMETER CONTROL SYSTEMS; DYNAMICAL SYSTEMS; DYNAMICS; IDENTIFICATION (CONTROL SYSTEMS); LINEAR SYSTEMS; MODELS; NONLINEAR ANALYSIS; SOLIDS; SUPPORT VECTOR MACHINES;

ONLINE SYSTEMS;

ALGORITHM; ARTICLE; ARTIFICIAL INTELLIGENCE; COMPUTER SIMULATION; HUMAN; NONLINEAR SYSTEM; ONLINE SYSTEM;

ALGORITHMS; ARTIFICIAL INTELLIGENCE; COMPUTER SIMULATION; HUMANS; NONLINEAR DYNAMICS; ONLINE SYSTEMS;

EID: 80052408376     PISSN: 10459227     EISSN: None     Source Type: Journal    
DOI: 10.1109/TNN.2011.2161331     Document Type: Article

References (49)

1. L. Debnath, Nonlinear Water Waves. Boston, MA: Academic, 1994.
2. L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers. Boston, MA: Birkhäuser, 2005.
3. D. L. Russell, "Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions", SIAM Rev., vol. 20, no. 4, pp. 639-739, Oct. 1978.
4. R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Providence, RI: AMS, 1996.
5. L. Hörmander, "On the existence of real analytic solutions of partial differential equations with constant coefficients", Invent. Math., vol. 21, no. 3, pp. 151-182, 1973.
6. K. Deimling, Nonlinear Functional Analysis. Berlin, Germany: Springer-Verlag, 1985.
7. R. A. Adams and J. F. Fournier, Sobolev Spaces. Boston, MA: Academic, 1975.
8. W. F. Ames, Numerical Methods for Partial Differential Equations. Boston, MA: Academic, 1977.
9. C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge, U. K.: Cambridge Univ. Press, 1987.
10. A. Quartevoni and A. Valli, Numerical Approximation of Partial Differential Equations. New York: Springer-Verlag, 1997.
11. S. N. Chow, Lattice Dynamical Systems (Lecture Notes in Mathematics). New York: Springer-Verlag, 2003.
12. T. D. Martin, Introduction to Lattice Dynamics. Cambridge, U. K.: Cambridge Univ. Press, 1983.
13. D. Coca and S. A. Billings, "Identification of finite dimensional models of infinite dimensional dynamical systems", Automatica, vol. 38, no. 11, pp. 1851-1865, Nov. 2002.
14. L. Guo and S. A. Billings, "State-space reconstruction and spatiotemporal prediction of lattice dynamical systems", IEEE Trans. Autom. Control, vol. 52, no. 4, pp. 622-632, Apr. 2007.
15. L. Guo, S. A. Billings, and D. Coca, "Identification of partial differential equation models for a class of multiscale spatio-temporal dynamical systems", Int. J. Control, vol. 83, no. 1, pp. 40-48, 2010.
16. H. Wei, S. A. Billings, Y. Zhao, and L. Guo, "Lattice dynamical wavelet neural networks implemented using particle swarm optimization for spatio-temporal system identification", IEEE Trans. Neural Netw., vol. 20, no. 1, pp. 181-185, Jan. 2009.
17. S. A. Billings, L. Guo, and H. Wei, "Identification of coupled map lattice models for spatio-temporal patterns using wavelets", Int. J. Syst. Sci., vol. 37, no. 14, pp. 1021-1038, Nov. 2006.
18. M. Milanese and C. Novara, "Set membership identification of nonlinear systems", Automatica, vol. 40, no. 6, pp. 957-975, 2004.
19. S. Chen, S. A. Billings, and W. Luo, "Orthogonal least squares methods and their application to non-linear system identification", Int. J. Control, vol. 50, no. 5, pp. 1873-1896, 1989.
20. S. A. Billings, S. Chen, and M. J. Korenberg, "Identification of MIMO non-linear systems using a forward-regression orthogonal estimator", Int. J. Control, vol. 49, no. 6, pp. 2157-2189, 1989.
21. I. Lind and L. Ljung, "Regressor selection with the analysis of variance method", Automatica, vol. 41, no. 4, pp. 693-700, Apr. 2005.
22. B. Scholkopf and A. J. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization and Beyond. Cambridge, MA: MIT Press, 2002.
23. D. Conus and R. C. Dalan, "The non-linear stochastic wave equation in high dimensions", Electron. J. Probab., vol. 13, pp. 629-670, Jan. 2008
24. J. A. K. Suykens, J. Vandewalle, and B. D. Moor, "Optimal control by least squares support vector machines", Neural Netw., vol. 14, no. 1, pp. 23-35, Jan. 2001.
25. V. Vapnik, The Nature of Statistical Learning Theory. New York: Springer-Verlag, 1995.
26. J. Zhang, T. Sato, and S. Iai, "Novel support vector regression for structural system identification", Struct. Control Health Monit., vol. 14, no. 4, pp. 609-626, Jun. 2007.
27. S. Totterman, H. T. Toivonen, and B. Akesson, "Identification of statedependent parameter models with support vector regression", Int. J. Control, vol. 80, no. 9, pp. 1454-1470, 2007.
28. W. C. Chan, C. W. Chan, K. C. Cheung, and C. J. Harris, "On the modelling of nonlinear dynamic systems using support vector neural networks", Eng. Appl. Artif. Intell., vol. 14, no. 2, pp. 105-113, 2001.
29. Y. Liu, H. Wang, J. Yu, and P. Li, "Selective recursive kernel learning for online identification of nonlinear systems with NARX form", J. Process Control, vol. 20, no. 2, pp. 181-194, Feb. 2010.
30. B. J. D. Kruif and T. J. A. D. Vries, "Pruning error minimization in least squares support vector machines", IEEE Trans. Neural Netw., vol. 14, no. 3, pp. 696-702, May 2003.
31. L. Guo, S. Mei, and B. A. Billings, "Neighbourhood detection and identification of spatio-temporal dynamical systems using a coarse-tofine approach", Int. J. Syst. Sci., vol. 38, no. 1, pp. 1-15, Jan. 2007.
32. H. Wei and B. A. Billings, "Model structure selection using an integrated forward orthogonal search algorithm assisted by squared correlation and mutual information", Int. J. Model. Identif. Control, vol. 3, no. 4, pp. 341-356, 2008.
33. M. Espinoza, J. A. K. Suykens, and B. D. Moor, "Kernel based partially linear models and nonlinear identification", IEEE Trans. Autom. Control, vol. 50, no. 10, pp. 1602-1606, Oct. 2005.
34. J. A. K. Suykens, V. T. Gestel, J. D. Brabanter, B. D. Moor, and J. Vandewalle, Least Squares Support Vector Machines. Singapore: World Scientific, 2002.
35. G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins Univ. Press, 1996.
36. R. Reed, "Pruning algorithms-a survey", IEEE Trans. Neural Netw., vol. 4, no. 5, pp. 740-747, Sep. 1993.
37. H. Tang, S. Xue, R. Chen, and T. Sato, "Online weighted LS-SVM for hysteretic structural system identification", Eng. Struct., vol. 28, no. 12, pp. 1728-1735, Oct. 2006.
38. J. A. K. Suykens, L. Lukas, B. D. Moor, and J. Vandewalle, "Weighted least squares support vector machines: Robustness and sparse approximation", Neurocomputing, vol. 48, nos. 1-4, pp. 85-105, Oct. 2002.
39. S. Chen and J. Wigger, "Fast orthogonal least squares algorithm for efficient subset model selection", IEEE Trans. Signal Process., vol. 43, no. 7, pp. 1713-1715, Jul. 1995.
40. Y. Engel, S. Mannor, and R. Meir, "The kernel recursive least-squares algorithm", IEEE Trans. Signal Process., vol. 52, no. 8, pp. 2275-2285, Aug. 2004.
41. W. Liu, P. P. Pokharel, and J. C. Principe, "The kernel least-mean-square algorithm", IEEE Trans. Signal Process., vol. 56, no. 2, pp. 543-554, Feb. 2008.
42. C. Richard, J. C. M. Bermudez, and P. Honeine, "Online prediction of time series data with kernels", IEEE Trans. Signal Process., vol. 57, no. 3, pp. 1058-1067, Mar. 2009.
43. W. Liu, J. C. Principe, and S. Haykin, Kernel Adaptive Filtering. New York: Wiley, 2010.
44. M. Gerdts, G. Greif, and H. J. Pesch, "Numerical optimal control of the wave equation: Optimal boundary control of a string to rest in finite time", Math. Comput. Simul., vol. 79, no. 4, pp. 1020-1032, Dec. 2008.
45. C. Crossmann and H. G. Roos, Numerical Treatment of Partial Differential Equations. Berlin, Germany: Springer-Verlag, 2007.
46. R. A. Carmona and B. Rozovskii, Stochastic Partial Differential Equations: Six Perspectives. Providence, RI: AMS, 1998.
47. H. G. Matthies and A. Keese, "Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations", Comput. Methods Appl. Mech. Eng., vol. 194, nos. 12-16, pp. 1295-1331, Apr. 2005.
48. E. J. Allen, S. J. Novosel, and Z. Zhang, "Finite element and difference approximation of some linear stochastic partial differential equations", Stochast. Stochast. Rep., vol. 64, nos. 1-2, pp. 117-142, 1998.
49. 1 by a finite-difference method", Math. Comput., vol. 69, no. 230, pp. 653-666, Apr. 1999.