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LSSVM approximate solution for affine nonlinear systems with partially unknown functions
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 
References:
[1] 
A. AkyyuzDascioglu and H. CerdikYaslan, The solution of highorder nonlinear ordinary differential equations by Chebyshev Series, Applied Mathematics and Computation, 217 (2011), 56585666. doi: 10.1016/j.amc.2010.12.044. 
[2] 
S. J. An, W. Q. Liu and S. Venkatesh, Fast Exact crossvalidation of least squares support vector machines, Pattern Recognition, 40 (2007), 21542162. 
[3] 
T. Falck, K. Pelckmans, J. A. K. Suykens and B. De Moor, Identification of WienerHammerstein Systems using LSSVMs, 15th IFAC Symposium on System Identification, SaintMalo, 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, SpringerVerlag, 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), 9871000. doi: 10.1109/72.712178. 
[8] 
H. Lee and I. S. Kang, Neural algorithm for solving differential equations, Journal of Computational Physics, 91 (1990), 110131. doi: 10.1016/00219991(90)90007N. 
[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), 12211233. 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), 13561367. 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), 11651173. doi: 10.1016/j.na.2003.11.009. 
[12] 
J. I. Ramos, Linearization techniques for singular initialvalue problems of ordinary differential equations, Applied Mathematics and Computation, 161 (2005), 525542. doi: 10.1016/j.amc.2003.12.047. 
[13] 
P. Ramuhalli, L. Udpa and S. S. Udpa, Finiteelement neural networks for solving differential equations, IEEE Transactions on Neural Networks, 16 (2005), 13811392. 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), 2335. doi: 10.1016/S08936080(00)000770. 
[16] 
I. G. Tsoulos, D. Gavrilis and E. Glavas, Solving differential equations with constructed neural networks, Neurocomputing, 72 (2009), 23852391. doi: 10.1016/j.neucom.2008.12.004. 
[17] 
V. Vapnik, The Nature of Statistical Learning Theory,, 1st edition, (). 
[18] 
A. M. Wazwaz, A new method for solving initial value problems in secondorder ordinary differential equations, Applied Mathematics and Computation, 128 (2002), 4557. doi: 10.1016/S00963003(01)000212. 
show all references
References:
[1] 
A. AkyyuzDascioglu and H. CerdikYaslan, The solution of highorder nonlinear ordinary differential equations by Chebyshev Series, Applied Mathematics and Computation, 217 (2011), 56585666. doi: 10.1016/j.amc.2010.12.044. 
[2] 
S. J. An, W. Q. Liu and S. Venkatesh, Fast Exact crossvalidation of least squares support vector machines, Pattern Recognition, 40 (2007), 21542162. 
[3] 
T. Falck, K. Pelckmans, J. A. K. Suykens and B. De Moor, Identification of WienerHammerstein Systems using LSSVMs, 15th IFAC Symposium on System Identification, SaintMalo, 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, SpringerVerlag, 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), 9871000. doi: 10.1109/72.712178. 
[8] 
H. Lee and I. S. Kang, Neural algorithm for solving differential equations, Journal of Computational Physics, 91 (1990), 110131. doi: 10.1016/00219991(90)90007N. 
[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), 12211233. 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), 13561367. 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), 11651173. doi: 10.1016/j.na.2003.11.009. 
[12] 
J. I. Ramos, Linearization techniques for singular initialvalue problems of ordinary differential equations, Applied Mathematics and Computation, 161 (2005), 525542. doi: 10.1016/j.amc.2003.12.047. 
[13] 
P. Ramuhalli, L. Udpa and S. S. Udpa, Finiteelement neural networks for solving differential equations, IEEE Transactions on Neural Networks, 16 (2005), 13811392. 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), 2335. doi: 10.1016/S08936080(00)000770. 
[16] 
I. G. Tsoulos, D. Gavrilis and E. Glavas, Solving differential equations with constructed neural networks, Neurocomputing, 72 (2009), 23852391. doi: 10.1016/j.neucom.2008.12.004. 
[17] 
V. Vapnik, The Nature of Statistical Learning Theory,, 1st edition, (). 
[18] 
A. M. Wazwaz, A new method for solving initial value problems in secondorder ordinary differential equations, Applied Mathematics and Computation, 128 (2002), 4557. doi: 10.1016/S00963003(01)000212. 
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