LS-SVM approximate solution for affine nonlinear systems with partially unknown functions

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April 2014, 10(2): 621-636. doi: 10.3934/jimo.2014.10.621

LS-SVM approximate solution for affine nonlinear systems with partially unknown functions

Guoshan Zhang ^1, , Shiwei Wang ^1, , Yiming Wang ^1, and Wanquan Liu ^2,

1.	Tianjin Key Laboratory of Process Measurement and Control, School of Electrical Engineering and Automation, Tianjin University, Tianjin, 300072, China, China, China
2.	Department of Computing, Curtin University of Technology, Perth, WA 6102

Received January 2013 Revised June 2013 Published October 2013

Abstract
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By using the Least Squares Support Vector Machines (LS-SVMs), we develop a numerical approach to find an approximate solution for affine nonlinear systems with partially unknown functions. This approach can obtain continuous and differential approximate solutions of the nonlinear differential equations, and can also identify the unknown nonlinear part through a set of measured data points. Technically, we first map the known part of the affine nonlinear systems into high dimensional feature spaces and derive the form of approximate solution. Then the original problem is formulated as an approximation problem via kernel trick with LS-SVMs. Furthermore, the approximation of the known part can be expressed via some linear equations with coefficient matrices as coupling square matrices, and the unknown part can be identified by its relationship to the known part and the approximate solution of affine nonlinear systems. Finally, several examples for different systems are presented to illustrate the validity of the proposed approach.

Keywords: Least Squares Support Vector Machines (LS-SVM), coupling square matrices., approximate solutions, measured data points, affine nonlinear systems.

Mathematics Subject Classification: Primary: 65D15; Secondary: 65C20, 65K1.

Citation: Guoshan Zhang, Shiwei Wang, Yiming Wang, Wanquan Liu. LS-SVM approximate solution for affine nonlinear systems with partially unknown functions. Journal of Industrial and Management Optimization, 2014, 10 (2) : 621-636. doi: 10.3934/jimo.2014.10.621

References:

[1]	A. Akyyuz-Dascioglu and H. Cerdik-Yaslan, The solution of high-order nonlinear ordinary differential equations by Chebyshev Series, Applied Mathematics and Computation, 217 (2011), 5658-5666. doi: 10.1016/j.amc.2010.12.044.
[2]	S. J. An, W. Q. Liu and S. Venkatesh, Fast Exact cross-validation of least squares support vector machines, Pattern Recognition, 40 (2007), 2154-2162.
[3]	T. Falck, K. Pelckmans, J. A. K. Suykens and B. De Moor, Identification of Wiener-Hammerstein Systems using LS-SVMs, 15th IFAC Symposium on System Identification, Saint-Malo, France, 2009.
[4]	Z. Guan and J. F. Lu, Basic of Numerical Analysis(Chinese), 2nd edition, Higher Education Press,Beijing, 2010.
[5]	A. Isidori, Nonlinear Control Systems: An Introduction, 3rd edition, Springer-Verlag, London, 1995.
[6]	D. R. Kincaid and E. W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3rd edition, Brooks/Cole, Pacific Grove, CA, 2002.
[7]	I. E. Lagaris, A. Likas and D. I. Fotiadis, Artificial neural networks for solving ordinary and partial differential equations, IEEE Transactions on Neural Networks, 9 (1998), 987-1000. doi: 10.1109/72.712178.
[8]	H. Lee and I. S. Kang, Neural algorithm for solving differential equations, Journal of Computational Physics, 91 (1990), 110-131. doi: 10.1016/0021-9991(90)90007-N.
[9]	K. S. McFall and J. R. Mahan, Artificial neural network method for solution of boundary value problems with exact satisfaction of arbitrary boundary conditions, IEEE Transactions on Neural Networks, 20 (2009), 1221-1233. doi: 10.1109/TNN.2009.2020735.
[10]	S. Mehrkanoon, T. Falck and J. A. K. Suykens, Approximate solutions to ordinary differential equations using least squares support vector machines, IEEE Trans. on Neural Networks and Learning Systems, 23 (2012), 1356-1367. doi: 10.1109/TNNLS.2012.2202126.
[11]	M. Popescu, On minimum quadratic functional control of affine nonlinear systems, Nonlinear Analysis: Theory, Methods & Applications, 56 (2004), 1165-1173. doi: 10.1016/j.na.2003.11.009.
[12]	J. I. Ramos, Linearization techniques for singular initial-value problems of ordinary differential equations, Applied Mathematics and Computation, 161 (2005), 525-542. doi: 10.1016/j.amc.2003.12.047.
[13]	P. Ramuhalli, L. Udpa and S. S. Udpa, Finite-element neural networks for solving differential equations, IEEE Transactions on Neural Networks, 16 (2005), 1381-1392. doi: 10.1109/TNN.2005.857945.
[14]	J. A. K. Suykens, T. V. Gestel, J. Brabanter,B. D. Moor and J. Vandewalle, Least Squares Support Vector Machines, 1st edition, World Scientific, Singapore, 2002.
[15]	J. A. K. Suykens, J. Vandewalle and B. D. Moor, Optimal control by least squares support vector machines, Neural Networks, 14 (2001), 23-35. doi: 10.1016/S0893-6080(00)00077-0.
[16]	I. G. Tsoulos, D. Gavrilis and E. Glavas, Solving differential equations with constructed neural networks, Neurocomputing, 72 (2009), 2385-2391. doi: 10.1016/j.neucom.2008.12.004.
[17]	V. Vapnik, The Nature of Statistical Learning Theory, 1st edition, Springer- Verlag, New York,1995.
[18]	A. M. Wazwaz, A new method for solving initial value problems in second-order ordinary differential equations, Applied Mathematics and Computation, 128 (2002), 45-57. doi: 10.1016/S0096-3003(01)00021-2.

show all references

References:

[1]	A. Akyyuz-Dascioglu and H. Cerdik-Yaslan, The solution of high-order nonlinear ordinary differential equations by Chebyshev Series, Applied Mathematics and Computation, 217 (2011), 5658-5666. doi: 10.1016/j.amc.2010.12.044.
[2]	S. J. An, W. Q. Liu and S. Venkatesh, Fast Exact cross-validation of least squares support vector machines, Pattern Recognition, 40 (2007), 2154-2162.
[3]	T. Falck, K. Pelckmans, J. A. K. Suykens and B. De Moor, Identification of Wiener-Hammerstein Systems using LS-SVMs, 15th IFAC Symposium on System Identification, Saint-Malo, France, 2009.
[4]	Z. Guan and J. F. Lu, Basic of Numerical Analysis(Chinese), 2nd edition, Higher Education Press,Beijing, 2010.
[5]	A. Isidori, Nonlinear Control Systems: An Introduction, 3rd edition, Springer-Verlag, London, 1995.
[6]	D. R. Kincaid and E. W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3rd edition, Brooks/Cole, Pacific Grove, CA, 2002.
[7]	I. E. Lagaris, A. Likas and D. I. Fotiadis, Artificial neural networks for solving ordinary and partial differential equations, IEEE Transactions on Neural Networks, 9 (1998), 987-1000. doi: 10.1109/72.712178.
[8]	H. Lee and I. S. Kang, Neural algorithm for solving differential equations, Journal of Computational Physics, 91 (1990), 110-131. doi: 10.1016/0021-9991(90)90007-N.
[9]	K. S. McFall and J. R. Mahan, Artificial neural network method for solution of boundary value problems with exact satisfaction of arbitrary boundary conditions, IEEE Transactions on Neural Networks, 20 (2009), 1221-1233. doi: 10.1109/TNN.2009.2020735.
[10]	S. Mehrkanoon, T. Falck and J. A. K. Suykens, Approximate solutions to ordinary differential equations using least squares support vector machines, IEEE Trans. on Neural Networks and Learning Systems, 23 (2012), 1356-1367. doi: 10.1109/TNNLS.2012.2202126.
[11]	M. Popescu, On minimum quadratic functional control of affine nonlinear systems, Nonlinear Analysis: Theory, Methods & Applications, 56 (2004), 1165-1173. doi: 10.1016/j.na.2003.11.009.
[12]	J. I. Ramos, Linearization techniques for singular initial-value problems of ordinary differential equations, Applied Mathematics and Computation, 161 (2005), 525-542. doi: 10.1016/j.amc.2003.12.047.
[13]	P. Ramuhalli, L. Udpa and S. S. Udpa, Finite-element neural networks for solving differential equations, IEEE Transactions on Neural Networks, 16 (2005), 1381-1392. doi: 10.1109/TNN.2005.857945.
[14]	J. A. K. Suykens, T. V. Gestel, J. Brabanter,B. D. Moor and J. Vandewalle, Least Squares Support Vector Machines, 1st edition, World Scientific, Singapore, 2002.
[15]	J. A. K. Suykens, J. Vandewalle and B. D. Moor, Optimal control by least squares support vector machines, Neural Networks, 14 (2001), 23-35. doi: 10.1016/S0893-6080(00)00077-0.
[16]	I. G. Tsoulos, D. Gavrilis and E. Glavas, Solving differential equations with constructed neural networks, Neurocomputing, 72 (2009), 2385-2391. doi: 10.1016/j.neucom.2008.12.004.
[17]	V. Vapnik, The Nature of Statistical Learning Theory, 1st edition, Springer- Verlag, New York,1995.
[18]	A. M. Wazwaz, A new method for solving initial value problems in second-order ordinary differential equations, Applied Mathematics and Computation, 128 (2002), 45-57. doi: 10.1016/S0096-3003(01)00021-2.

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