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Numerical solution of time-varying delay systems by Chebyshev wavelets. (English) Zbl 1228.65105
Summary: The solution of time-varying delay systems is obtained by using Chebyshev wavelets. The properties of the Chebyshev wavelets consisting of wavelets and Chebyshev polynomials are presented. The method is based upon expanding various time functions in the system as their truncated Chebyshev wavelets. The operational matrix of delay is introduced. The operational matrices of integration and delay are utilized to reduce the solution of time-varying delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

65L03Functional-differential equations (numerical methods)
34K07Theoretical approximation of solutions of functional-differential equations
65T60Wavelets (numerical methods)
Full Text: DOI
[1] Jamshidi, M.; Wang, C. M.: A computational algorithm for large-scale nonlinear time-delays systems, IEEE trans. Syst. man cybernet. 14, 2-9 (1984) · Zbl 0538.93002
[2] Chen, W. L.; Shih, Y. P.: Shift Walsh matrix and delay differential equations, IEEE trans. Automat. control 23, 265-280 (1978) · Zbl 0388.93029 · doi:10.1109/TAC.1978.1101888
[3] Rao, G. P.; Sivakumar, L.: Analysis and synthesis of dynamic systems containing time-delays via block-pulse functions, Proc. IEE125, 1064-1068 (1978)
[4] Kung, F. C.; Lee, H.: Solution and parameter estimation of in linear time-invariant delay systems using Laguerre polynomial expansion, J. dyn. Syst. meas. Control-trans. ASME 105, 297-301 (1983) · Zbl 0525.93036 · doi:10.1115/1.3140675
[5] Lee, H.; Kung, F. C.: Shifted Legendre series solution and parameter estimation of linear delayed systems, Int. J. Syst. sci. 16, 1249-1256 (1985) · Zbl 0568.93028 · doi:10.1080/00207728508926748
[6] Horng, I. R.; Chou, J. H.: Analysis parameter estimation and optimal control of time-delay systems via Chebyshev series, Int. J. Control 41, 1221-1234 (1985) · Zbl 0562.93034 · doi:10.1080/0020718508961193
[7] Razzaghi, M.; Razzaghi, M.: Fourier series direct method for variational problems, Int. J. Control 48, 887-895 (1988) · Zbl 0651.49012 · doi:10.1080/00207178808906224
[8] Kajani, M. Tavassoli; Vencheh, A. Hadi: Solving linear integro-differential equation with Legendre wavelet, Int. J. Comput. math. 81, 719-726 (2004) · Zbl 1062.65146 · doi:10.1080/00207160310001650044
[9] Maleknejad, K.; Kajani, M. Tavassoli; Mahmoudi, Y.: Numerical solution of linear Fredholm and Volterra integral equation of the second kind by using Legendre wavelets, Kybernetes 32, 1530-1539 (2003) · Zbl 1059.65127 · doi:10.1108/03684920310493413
[10] Razzaghi, M.; Yousefi, S.: Legendre wavelets direct method for variational problems, Math. comput. Simul. 53, 185-192 (2000)
[11] Razzaghi, M.; Yousefi, S.: The Legendre wavelets operational matrix of integration, Int. J. Syst. sci. 32, 495-502 (2001) · Zbl 1006.65151 · doi:10.1080/002077201300080910
[12] Kajani, M. Tavassoli; Vencheh, A. Hadi; Ghasemi, M.: The Chebyshev wavelets operational matrix of integration and product operation matrix, Int. J. Comput. math. 86, No. 7, 1118-1125 (2009) · Zbl 1169.65072 · doi:10.1080/00207160701736236
[13] Ghasemi, M.; Babolian, E.; Kajani, M. Tavassoli: Numerical solution of linear Fredholm integral equations using sine-cosine wavelets, Int. J. Comput. math. 84, 979-987 (2007) · Zbl 1122.65130 · doi:10.1080/00207160701242300
[14] Kajani, M. Tavassoli; Ghasemi, M.; Babolian, E.: Numerical solution of linear integro-differential equation by using sine-cosine wavelets, Appl. math. Comput. 180, 569-574 (2006) · Zbl 1102.65137 · doi:10.1016/j.amc.2005.12.044
[15] Maleknejad, K.; Kajani, M. Tavassoli: Solving second kind integral equations by Galerkin methods with hybrid Legendre and block-pulse functions, Appl. math. Comput. 145, 623-629 (2003) · Zbl 1101.65323 · doi:10.1016/S0096-3003(03)00139-5
[16] Maleknejad, K.; Kajani, M. Tavassoli: Solving integro-differential equation by using hybrid Legendre and block-pulse functions, Int. J. Appl. math. 11, 67-76 (2002) · Zbl 1029.65147
[17] Marzban, H. R.; Razzaghi, M.: Hybrid functions approach for linearly constrained quadratic optimal control problems, Appl. math. Model. 27, 471-485 (2003) · Zbl 1020.49025 · doi:10.1016/S0307-904X(03)00050-7
[18] Ghasemi, M.; Kajani, M. Tavassoli; Babolian, E.: Hybrid Fourier and block-pulse functions applications in the calculus of variations, Int. J. Comput. math. 83, 695-702 (2006) · Zbl 1114.65071 · doi:10.1080/00207160601056016
[19] Datta, K. B.; Mohan, B. M.: Orthogonal functions in systems and control, (1995) · Zbl 0866.93003
[20] Gu, J. S.; Jiang, W. S.: The Haar wavelets operational matrix of integration, Int. J. Syst. sci. 27, 623628 (1996) · Zbl 0875.93116 · doi:10.1080/00207729608929258
[21] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A.: Spectral methods on fluid dynamics, (1988) · Zbl 0658.76001
[22] Lancaster, P.: Theory of matrices, (1969) · Zbl 0186.05301
[23] Chung, H. Y.; Sun, Y. Y.: Analysis of time-delay systems using an alternative method, Int. J. Control 46, 1621-1631 (1987) · Zbl 0626.93025 · doi:10.1080/00207178708934000
[24] Hwang, C.; Chen, M. Y.: Analysis of time-delay systems using the Galerkin method, Int. J. Control 44, 847-866 (1986) · Zbl 0593.93033 · doi:10.1080/00207178608933636