State analysis of time-varying singular nonlinear systems using Legendre wavelets
DOI:
https://doi.org/10.4067/S0716-09172015000100006Keywords:
Legendre wavelets, Time-varying, Singular nonlinear systems, Convergence analysis, Operational matrix.Abstract
In this paper, the Legendre wavelet method for State analysis of time-varying singular nonlinear systems is studied. The properties of Legendre wavelets and its operational matrices are first presented and then are used to convert into algebraic equations. Also the convergence and error analysis for the proposed technique have been discussed. Illustrative examples have been given to demonstrate the validity and applicability of the technique. The efficiency of the proposed method has been compared with Haar wavelet method and it is observed that the Legendre wavelet method is more convenient than the Haar wavelet method in terms of applicability, efficiency, accuracy, error, and computational effort.References
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[23] S. Yousefi, M.Razzagi, Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations, Math. Comput. Simul, 70, pp. 1-8, (2005).
[24] S. A .Yousefi, Legendre scaling function for solving generalized EmdenFowler equation, Int. J. Inf. Sys. Sci. 3, pp. 243-250, (2007).
[25] S. A .Yousefi, Legendre wavelets method for solving differential equations of Lane-emden type, Appl. Math. Comput., 181, pp. 1417-1422, (2006).
[26] S.G.Venkatesh, S.K. Ayyaswamy, S. Raja Balachandar, The Legendre wavelet method for solving initial value problems of Bratu-type, Comput. Math. Appl., 63, pp. 1287-1295, (2012).
[27] S. G. Venkatesh, S. K. Ayyaswamy, S. Raja Balachandar, K. Kannan, Convergence Analysis of Legendre wavelets method for solving Fredholm integral equations, Appl. Math. Sci., 6, pp. 2289-2296, (2012).
[28] S. G. Venkatesh, S.K. Ayyaswamy, S. Raja Balachandar, Legendre approximation solution for a class of higher-order Volterra integrodifferential equations, Ain Shams Eng J, 3, pp. 417-422, (2012).
[29] S. G. Venkatesh, S.K. Ayyaswamy, S. Raja Balachandar, Legendre wavelets based approximation method for Cauchy problems, Appl. Math. Sci. 6, pp. 6281-6286, (2012).
[30] S. G. Venkatesh, S. K. Ayyaswamy, S. Raja Balachandar, Legendre Wavelets based approximation method for solving advection problems, Ain Shams Eng J, 4, pp. 925-932, (2013).
[31] S. G. Venkatesh, S. K. Ayyaswamy, S. Raja Balachandar, K. Kannan, Wavelet Solution for Class of Nonlinear Integro-diferential Equations, Indian Journal of Science and Technology, pp. 4670-4677, (2013).
[32] S. G. Venkatesh, S. K. Ayyaswamy, S. Raja Balachandar, K.Kannan, Legendre Wavelet Method for the Singular System of Transistor Circuits, International Journal of Applied Engineering Research, 9, pp. 213-221, (2014).
[33] M. Razzagi, S. Yousefi, The Legendre Wavelet operational matrix of integration, Int. J. Syst. Sci., 32 (4), pp. 495-502, (2001).
[34] P. A. Regalia , S. K. Mitra, Kronecker product, Unitary matrices and signal Processing applications, SIAM , 31, pp. 586-613, (1989).
[35] C. H. Hsiao, W. J. Wang, State analysis of time varying singular nonlinear systems via Haar wavelets, Math. Comput. Simul, 51, pp. 91- 100, (1999).
[2] S. L. Campbell, Bilinear nonlinear descriptor control systems, CRSC Tech. Rept. 102386-01, Dept. of Math., N.C. State Univ., Raleigh, NC 27695, (1987).
[3] N. Declaris, A. Rindos, Semistate analysis of neural networks in Apysia Californica, Proc. 27th MSCS, pp. 686-689, (1984).
[4] N. Wiener, Cybernetics, MIT Press, Cambridge. MA, (1948).
[5] F. L. Lewis, B. G. Mertzios, W. Marszalek, Analysis of singular bilinear
systems using Walsh functions, IEE Proc. Pt. D, pp. 89-92, (1991).
[6] C. F. Chen, C. H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proc. Pt. D, Vol. 144, pp. 87-94, (1997).
[7] C. H. Hsiao, State analysis of linear time delayed systems via Haar wavelets, Math. Comput. Simul., 44, pp. 457-470, (1997).
[8] C. H. Hsiao, W.J. Wang, State analysis and optimal control of linear time- varying systems via Haar wavelets, Optim contr appl met. 19, pp. 423-433, (1998).
[9] S. Sekar, K. Prabakaran, Numerical Solution of higher order Linear Singular System Using Single term Haar Wavelet series method, Int. J. of Mathematical Sciences and Applications, 1, pp. 737-746, (2011).
[10] V. Murugesh, K. Batri, State Analysis of Time-Varying Singular Bilinear Systems by RK-Butcher Algorithms, International Journal of Computers, Communications and Control, 3, pp. 103-109, (2008).
[11] B. Sepehriand, M. Razzaghi, Solution of time varying singular nonlinear systems by single term Walsh series, Math. Prob. Eng. 3, pp. 129-136, (2003).
[12] M. Garg, L. Dewan, A Generalized approach for state Analysis and parameter estimation of bilinear systems using Haar connection coeffi- cients, World Academy of Science, Engineering and Technology 5, pp. 1094-1099, (2011).
[13] P. L. Liu, Further results on the stability analysis of singular systems with time-varying delay: A delay decomposition approach, International Journal of Analysis, ID 721407, pp. 1-11, (2013).
[14] H. Sandberg, A. Rantzer, Balanced Truncation of Linear Time-Varying Systems, IEEE Transactions on automatic control, 49, pp. 217-229, (2004).
[15] L. Zhou, C. Yang, Q. Zhang, Y. Lai, Finite-time stability analysis for linear time-varying singular impulsive systems, Scientific Research and Essays, 6, pp. 3344-3350, (2011).
[16] Mirmomeni, M., M. Shafiee, State analysis of time-invariant singular systems via Haar wavelet, 13th Iranian Conference on Electrical Engineering (ICEE 05), Zanjan University, Zanjan, Iran, 2005b.
[17] O. Nilsson, A. Rantzer, A novel approach to balanced truncation of nonlinear systems, European Control Conference, (2009).
[18] K. D. Mease, S. Bharadwaj, S. Iravanchy, Timescale analysis for nonlinear dynamical Systems, Journal of guidance, control, and dynamics, 26, pp. 318-330, (2003).
[19] M. Behroozifar, S. A. Yousefi, A. Ranjbar N, Numerical solution of optimal control of time-varying singular systems via operational matrices, International Journal of Engineering, 27, pp. 523-532, (2014).
[20] K. Balachandran, K. Murugesan, Analysis of Electronic circuits using single term Walsh series approach , Int. J. Electron. 69, pp. 327-332, (1990).
[21] K. Balachandran, K. Murugesan, Analysis of transistor circuits using the single term Walsh series technique ,International Journal of Electronics, 71, pp. 397-401, (1991).
[22] M. Razzagi, S. Yousefi, Legendre wavelets method for the solution of nonlinear problems in the calculus of variations, Math. Comput. Modell., 34, pp. 45-54, (2001).
[23] S. Yousefi, M.Razzagi, Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations, Math. Comput. Simul, 70, pp. 1-8, (2005).
[24] S. A .Yousefi, Legendre scaling function for solving generalized EmdenFowler equation, Int. J. Inf. Sys. Sci. 3, pp. 243-250, (2007).
[25] S. A .Yousefi, Legendre wavelets method for solving differential equations of Lane-emden type, Appl. Math. Comput., 181, pp. 1417-1422, (2006).
[26] S.G.Venkatesh, S.K. Ayyaswamy, S. Raja Balachandar, The Legendre wavelet method for solving initial value problems of Bratu-type, Comput. Math. Appl., 63, pp. 1287-1295, (2012).
[27] S. G. Venkatesh, S. K. Ayyaswamy, S. Raja Balachandar, K. Kannan, Convergence Analysis of Legendre wavelets method for solving Fredholm integral equations, Appl. Math. Sci., 6, pp. 2289-2296, (2012).
[28] S. G. Venkatesh, S.K. Ayyaswamy, S. Raja Balachandar, Legendre approximation solution for a class of higher-order Volterra integrodifferential equations, Ain Shams Eng J, 3, pp. 417-422, (2012).
[29] S. G. Venkatesh, S.K. Ayyaswamy, S. Raja Balachandar, Legendre wavelets based approximation method for Cauchy problems, Appl. Math. Sci. 6, pp. 6281-6286, (2012).
[30] S. G. Venkatesh, S. K. Ayyaswamy, S. Raja Balachandar, Legendre Wavelets based approximation method for solving advection problems, Ain Shams Eng J, 4, pp. 925-932, (2013).
[31] S. G. Venkatesh, S. K. Ayyaswamy, S. Raja Balachandar, K. Kannan, Wavelet Solution for Class of Nonlinear Integro-diferential Equations, Indian Journal of Science and Technology, pp. 4670-4677, (2013).
[32] S. G. Venkatesh, S. K. Ayyaswamy, S. Raja Balachandar, K.Kannan, Legendre Wavelet Method for the Singular System of Transistor Circuits, International Journal of Applied Engineering Research, 9, pp. 213-221, (2014).
[33] M. Razzagi, S. Yousefi, The Legendre Wavelet operational matrix of integration, Int. J. Syst. Sci., 32 (4), pp. 495-502, (2001).
[34] P. A. Regalia , S. K. Mitra, Kronecker product, Unitary matrices and signal Processing applications, SIAM , 31, pp. 586-613, (1989).
[35] C. H. Hsiao, W. J. Wang, State analysis of time varying singular nonlinear systems via Haar wavelets, Math. Comput. Simul, 51, pp. 91- 100, (1999).
How to Cite
[1]
S. Raja Balachandar, S. G. Venkatesh, S. K. Ayyaswamy, and S. Balachandran, “State analysis of time-varying singular nonlinear systems using Legendre wavelets”, Proyecciones (Antofagasta, On line), vol. 34, no. 1, pp. 69-83, 1.
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