Analysis and Identification of Time-Invariant Systems, Time-Varying Systems, and Multi-Delay Systems using Orthogonal Hybrid Functions pp 49-86 | Cite as
Function Approximation via Hybrid Functions
Abstract
In this chapter, square integrable time functions of Lebesgue measure are approximated via hybrid functions and such approximations are compared with similar approximations using BPF and Legendre polynomials. For handling discontinuous functions, a modified method of approximation is suggested in hybrid function domain. This modified approach, named HFm approach, seems to be more accurate than the conventional HF domain technique, termed as HFc approach. The mean integral square errors (MISE) for both the approximations are computed and compared. Finally, error estimates for the SHF domain approximation and TF domain approximation are derived. The chapter contains many tables and graphs along with six illustrative examples.
Keywords
Function Approximation Legendre Polynomial Piecewise Linear Function Jump Discontinuity Hybrid FunctionReferences
- 1.Jiang, J.H., Schaufelberger, W.: Block Pulse Functions and their Application in Control System, LNCIS, vol. 179. Springer, Berlin (1992)CrossRefMATHGoogle Scholar
- 2.Deb, A., Sarkar, G., Sen, S.K.: Block pulse functions, the most fun-damental of all piecewise constant basis functions. Int. J. Syst. Sci. 25(2), 351–363 (1994)CrossRefMATHGoogle Scholar
- 3.Deb, A., Sarkar, G., Bhattacharjee, M., Sen, S.K.: A new set of piecewise constant orthogonal functions for the analysis of linear SISO systems with sample-and-hold. J. Franklin Inst. 335B(2), 333–358 (1998)CrossRefMATHGoogle Scholar
- 4.Deb, A., Sarkar, G., Sengupta, A.: Triangular Orthogonal Functions for the Analysis of Continuous Time Systems. Anthem Press, London (2011)MATHGoogle Scholar
- 5.Deb, A., Sarkar, G., Dasgupta, A.: A complementary pair of orthogonal triangular function sets and its application to the analysis of SISO control systems. J. Inst. Eng. (India) 84, 120–129 (2003)Google Scholar
- 6.Deb, A., Dasgupta, A., Sarkar, G.: A complementary pair of orthogonal triangular function sets and its application to the analysis of dynamic systems. J. Franklin Inst. 343(1), 1–26 (2006)MathSciNetCrossRefMATHGoogle Scholar
- 7.Rao, G.P.: Piecewise Constant Orthogonal Functions and their Application in Systems and Control, LNCIS, vol. 55. Springer, Berlin (1983)CrossRefGoogle Scholar
- 8.Deb, A., Sarkar, G., Mandal, P., Biswas, A., Ganguly, A., Biswas, D.: Transfer function identification from impulse response via a new set of orthogonal hybrid function (HF). Appl. Math. Comput. 218(9), 4760–4787 (2012)MathSciNetCrossRefMATHGoogle Scholar
- 9.Deb, A., Sarkar, G., Ganguly, A., Biswas, A.: Approximation, integration and differentiation of time functions using a set of orthogonal hybrid functions (HF) and their application to solution of first order differential equations. Appl. Math. Comput. 218(9), 4731–4759 (2012)MathSciNetCrossRefMATHGoogle Scholar
- 10.Baranowski, J.: Legendre polynomial approximations of time delay systems. XII international Ph.D workshop OWD 2010, 23–26 Oct 2010Google Scholar
- 11.Tohidi, E., Samadi, O.R.N., Farahi, M.H.: Legendre approximation for solving a class of nonlinear optimal control problems. J. Math. Finance 1, 8–13 (2011)CrossRefGoogle Scholar
- 12.Mathews, J.H., Kurtis, D.F.: Numerical Methods using MATLAB, 4th edn. Prentice Hall of India Pvt. Ltd., New Delhi (2005)MATHGoogle Scholar
- 13.Rao, G.P., Srinivasan, T.: Analysis and synthesis of dynamic systems containing time delays via block pulse functions. Proc. IEE 125(9), 1064–1068 (1978)MathSciNetGoogle Scholar
- 14.Deb, A., Sarkar, G., Sen, SK.: Linearly pulse-width modulated block pulse functions and their application to linear SISO feedback control system identification. Proc. IEE, Part D, Control Theory Appl. 142(1), 44–50 (1995)Google Scholar