Solution of Multi-Delay Dynamic Systems by Using Hybrid Functions
Affiliation(s)
Department of Mathematics, College of Basic Sciences, Karaj Branch, Islamic Azad University, Alborz, Iran.
Department of Mathematics, College of Basic Sciences, Karaj Branch, Islamic Azad University, Alborz, Iran.
ABSTRACT
In this paper, we present a method for finding the solution of the linear multi-delay systems (MDS) by using the hybrid of the Block-Pulse functions and the Bernoulli polynomials. In this approach, the MDS is reduced to a system of linear algebraic equations by expanding various time functions for the hybrid functions and using operational matrices. To demonstrate the validity and the applicability of the technique, some examples are presented.
In this paper, we present a method for finding the solution of the linear multi-delay systems (MDS) by using the hybrid of the Block-Pulse functions and the Bernoulli polynomials. In this approach, the MDS is reduced to a system of linear algebraic equations by expanding various time functions for the hybrid functions and using operational matrices. To demonstrate the validity and the applicability of the technique, some examples are presented.
Cite this paper
Maleknejad, K. , Ezzati, R. and Damercheli, T. (2014) Solution of Multi-Delay Dynamic Systems by Using Hybrid Functions. Applied Mathematics, 5, 2016-2027. doi: 10.4236/am.2014.513194.
Maleknejad, K. , Ezzati, R. and Damercheli, T. (2014) Solution of Multi-Delay Dynamic Systems by Using Hybrid Functions. Applied Mathematics, 5, 2016-2027. doi: 10.4236/am.2014.513194.
References
[1] Malek-Zavarei, M. and Jamshidi, M. (1987) Time Delay Systems: Analysis, Optimization and Applications (North-Holland Systems and Control Series). Elsevier Science, New York. [2] Harrison, R.F. (1993) Optimal Control of Vehicle Suspension Dynamics Incorporating Front-to-Rear Excitation Delays: An Approximate Solution. Journal of Sound and Vibration, 168, 339-354.
http://dx.doi.org/10.1006/jsvi.1993.1377 [3] Lee, Y.S. and Maurer, P. (1995) A Compiled Event-Driven Multi Delay Simulator. Technical Report Number DA-30, Department of Computer Science and Engineering, University of South Florida, Tampa. [4] Cai, G. and Huang, J. (2002) Optimal Control Method with Time Delay in Control. Journal of Sound and Vibration, 251, 383-394.
http://dx.doi.org/10.1006/jsvi.2001.3999 [5] Ji, J.C. and Leung, A.Y.T. (2002) Resonances of a Non-Linear s.d.o.f. System with Two Time-Delays in Linear Feedback Control. Journal of Sound and Vibration, 253, 985-1000.
http://dx.doi.org/10.1006/jsvi.2001.3974 [6] Razzaghi, M. and Marzban, H.R. (2000) Direct Method for Variational Problems via Hybrid of Block-Pulse and Chebyshev Functions. Mathematical Problems in Engineering, 6, 85-97.
http://dx.doi.org/10.1155/S1024123X00001265 [7] Marzban, H.R. and Razzaghi, M. (2004) Optimal Control of Linear Delay Systems via Hybrid of Block-Pulse and Legendre Polynomials. Journal of the Franklin Institute, 341, 279-293.
http://dx.doi.org/10.1016/j.jfranklin.2003.12.011 [8] Maleknejad, K., Basirat, B. and Hashemizadeh, E. (2011) Hybrid Legendre Polynomials and Block-Pulse Functions Approach for Nonlinear Volterra-Fredholm Integro-Differential Equations. Computers and Mathematics with Applications, 61, 2821-2828.
http://dx.doi.org/10.1016/j.camwa.2011.03.055 [9] Marzban, H.R. and Razzaghi, M. (2006) Solution of Multi-Delay Systems Using Hybrid of Block-Pulse Functions and Taylor Series. Journal of Sound and Vibration, 292, 954-963.
http://dx.doi.org/10.1016/j.jsv.2005.08.007 [10] Maleknejad, K. and Mahmoudi, Y. (2004) Numerical Solution of Linear Fredholm Integral Equation by Using Hybrid Taylor and Block-Pulse Functions. Applied Mathematics and Computation, 149, 799-806.
http://dx.doi.org/10.1016/S0096-3003(03)00180-2 [11] Haddadi, N., Ordokhani, Y. and Razzaghi, M. (2011) Optimal Control of Delay Systems by Using a Hybrid Functions Approximation. Journal of Optimization Theory and Applications, 153, 338-356.
http://dx.doi.org/10.1007/s10957-011-9932-1 [12] Mashayekhi, S., Ordokhani, Y. and Razzaghi, M. (2012) Hybrid Functions Approach for Nonlinear Constrained Optimal Control Problems. Communications in Nonlinear Science and Numerical Simulation, 17, 1831-1843.
http://dx.doi.org/10.1016/j.cnsns.2011.09.008 [13] Costabile, F., Dellaccio, F. and Gualtieri, M.I. (2006) A New Approach to Bernoulli Polynomials. Rendiconti di Matematica, Serie VII, 26, 1-12. [14] Arfken, G. (1985) Mathematical Methods for Physicists. 3rd Edition, Academic Press, San Diego. [15] Lancaster, P. (1969) Theory of Matrices. Academic Press, New York.
[1] Malek-Zavarei, M. and Jamshidi, M. (1987) Time Delay Systems: Analysis, Optimization and Applications (North-Holland Systems and Control Series). Elsevier Science, New York. [2] Harrison, R.F. (1993) Optimal Control of Vehicle Suspension Dynamics Incorporating Front-to-Rear Excitation Delays: An Approximate Solution. Journal of Sound and Vibration, 168, 339-354.
http://dx.doi.org/10.1006/jsvi.1993.1377 [3] Lee, Y.S. and Maurer, P. (1995) A Compiled Event-Driven Multi Delay Simulator. Technical Report Number DA-30, Department of Computer Science and Engineering, University of South Florida, Tampa. [4] Cai, G. and Huang, J. (2002) Optimal Control Method with Time Delay in Control. Journal of Sound and Vibration, 251, 383-394.
http://dx.doi.org/10.1006/jsvi.2001.3999 [5] Ji, J.C. and Leung, A.Y.T. (2002) Resonances of a Non-Linear s.d.o.f. System with Two Time-Delays in Linear Feedback Control. Journal of Sound and Vibration, 253, 985-1000.
http://dx.doi.org/10.1006/jsvi.2001.3974 [6] Razzaghi, M. and Marzban, H.R. (2000) Direct Method for Variational Problems via Hybrid of Block-Pulse and Chebyshev Functions. Mathematical Problems in Engineering, 6, 85-97.
http://dx.doi.org/10.1155/S1024123X00001265 [7] Marzban, H.R. and Razzaghi, M. (2004) Optimal Control of Linear Delay Systems via Hybrid of Block-Pulse and Legendre Polynomials. Journal of the Franklin Institute, 341, 279-293.
http://dx.doi.org/10.1016/j.jfranklin.2003.12.011 [8] Maleknejad, K., Basirat, B. and Hashemizadeh, E. (2011) Hybrid Legendre Polynomials and Block-Pulse Functions Approach for Nonlinear Volterra-Fredholm Integro-Differential Equations. Computers and Mathematics with Applications, 61, 2821-2828.
http://dx.doi.org/10.1016/j.camwa.2011.03.055 [9] Marzban, H.R. and Razzaghi, M. (2006) Solution of Multi-Delay Systems Using Hybrid of Block-Pulse Functions and Taylor Series. Journal of Sound and Vibration, 292, 954-963.
http://dx.doi.org/10.1016/j.jsv.2005.08.007 [10] Maleknejad, K. and Mahmoudi, Y. (2004) Numerical Solution of Linear Fredholm Integral Equation by Using Hybrid Taylor and Block-Pulse Functions. Applied Mathematics and Computation, 149, 799-806.
http://dx.doi.org/10.1016/S0096-3003(03)00180-2 [11] Haddadi, N., Ordokhani, Y. and Razzaghi, M. (2011) Optimal Control of Delay Systems by Using a Hybrid Functions Approximation. Journal of Optimization Theory and Applications, 153, 338-356.
http://dx.doi.org/10.1007/s10957-011-9932-1 [12] Mashayekhi, S., Ordokhani, Y. and Razzaghi, M. (2012) Hybrid Functions Approach for Nonlinear Constrained Optimal Control Problems. Communications in Nonlinear Science and Numerical Simulation, 17, 1831-1843.
http://dx.doi.org/10.1016/j.cnsns.2011.09.008 [13] Costabile, F., Dellaccio, F. and Gualtieri, M.I. (2006) A New Approach to Bernoulli Polynomials. Rendiconti di Matematica, Serie VII, 26, 1-12. [14] Arfken, G. (1985) Mathematical Methods for Physicists. 3rd Edition, Academic Press, San Diego. [15] Lancaster, P. (1969) Theory of Matrices. Academic Press, New York.