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doi: 10.3934/naco.2019044

## Linear optimal control of time delay systems via Hermite wavelet

 1 Department of Mathematics, Faculty of Mathematical Science and Statistics, University of Birjand, Birjand, Iran 2 Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

* Corresponding author: Asadollah Mahmoudzadeh Vaziri

Received  July 2018 Revised  July 2019 Published  September 2019

To solve the time delay optimal control problem with quadratic performance index, a direct numerical method based on Hermite wavelet has been proposed in the present study. The idea is to convert the time delay optimal control problem into a quadratic programming problem. To do so, various time functions in the system are expanded as their truncated series and the properties of the operational matrices of integration, delay and product of two Hermite wavelet vectors are used as well. These matrices are utilized to reduce the solution of optimal control with time delay system, to the solution of a quadratic programming with linear constraints. Finally, three examples of time varying and time invariant coefficients are given to compare the results with some of the existing methods.

Citation: Akram Kheirabadi, Asadollah Mahmoudzadeh Vaziri, Sohrab Effati. Linear optimal control of time delay systems via Hermite wavelet. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2019044
##### References:
 [1] A. Ali, M. A. Iqbal and S. T. Mohyud-Din, Hermite wavelets method for boundary value problems, International Journal of Modern Applied Physics, 3 (2013), 38-47. Google Scholar [2] E. Alirezaei, M. Samavat and M. A. Vali, Optimal control of linear time invariant singular delay systems using the orthogonal functions, Applied Mathematical Sciences, 6 (2012), 1877-1891. Google Scholar [3] U. Brandt-Pollmann, R. Winkler, S. Sager, U. Moslener and J. P. Schlder, Numerical solution of optimal control problems with constant control delays, Computational Economics, 31 (2008), 181-206. doi: 10.1007/s10614-007-9113-3. Google Scholar [4] M. Dadkhah and M. H. Farahi, Optimal control of time delay systems via hybrid of block-pulse functions and orthogonal Taylor series, International Journal of Applied and Computational Mathematics, 2 (2016), 137–152. Available from: https://doi.org/10.1093/imamci/dnv044. doi: 10.1007/s40819-015-0051-9. Google Scholar [5] K. B. Datta and B. M. Mohan, Orthogonal functions in systems and control, in Advanced Series in Electrical and Computer Engineering, World Scientific Publishing Co., (1995), 25–89. doi: 10.1142/2476. Google Scholar [6] L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optimal Control Applications and Methods, 30 (2009), 341-365. doi: 10.1002/oca.843. Google Scholar [7] L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time delays, Journal of Industrial & Management Optimization, 10 (2014), 413-441. doi: 10.3934/jimo.2014.10.413. Google Scholar [8] J. S. Gu and W. S. Jiang, The Haar wavelets operational matrix of integration, International Journal of Systems Science, 27 (1996), 623-628. doi: 10.1080/00207729608929258. Google Scholar [9] N. Haddadi, Y. Ordokhani and M. Razzaghi, Optimal control of delay systems by using a hybrid functions approximation, Journal of Optimization Theory and Applications, 153 (2012), 338-356. doi: 10.1007/s10957-011-9932-1. Google Scholar [10] A. Halanay, Optimal controls for systems with time lag, SIAM Journal on Control and Optimization, 6 (1968), 215-234. Google Scholar [11] I. R. Horng and J. H. Chou, Analysis, parameter estimation and optimal control of time-delay systems via Chebyshev series, International Journal of Control, 41 (1985), 1221-1234. doi: 10.1080/0020718508961193. Google Scholar [12] G. L. Kharatishvili, Maximum principle in the theory of optimal processes involving delay, Dokl. Akad. Nauk SSSR, 136 (1961), 39-42. Google Scholar [13] G. L. Kharatishvili, A Maximum Principle in External Problems with Delays, Mathematical Theory on Control, Academic Press: New York, 1967.Google Scholar [14] F. Khellat, Optimal control of linear time-delayed systems by linear Legendre multiwavelets, Journal of Optimization Theory and Application, 143 (2009), 107-121. Google Scholar [15] M. Malek-Zavarei and M. Jamshidi, Time-Delay Systems: Analysis, Optimization and Applications, Elsevier Science Inc., 1987. Google Scholar [16] I. Malmir, Optimal control of linear time-varying systems with state and input delays by Chebyshev wavelets, Statistics, Optimization and Information Computing, 5 (2017), 302-324. doi: 10.19139/soic.v5i4.341. Google Scholar [17] H. R. Marzban and S. M. Hoseini, A composite Chebyshev finite difference method for nonlinear optimal control problems, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 1347-1361. doi: 10.1016/j.cnsns.2012.10.012. Google Scholar [18] H. R. Marzban and S. M. Hoseini, Solution of linear optimal control problems with time delay using a composite Chebyshev finite difference method, Optimal Control Applications and Methods, 34 (2013), 253-274. doi: 10.1002/oca.2019. Google Scholar [19] H. R. Marzban, Optimal control of linear multi-delay systems based on a multi-interval decomposition scheme, Optimal Control Applications and Methods, 37 (2016), 190-211. doi: 10.1002/oca.2163. Google Scholar [20] B. M. Mohan and S. Kumar Kar, Optimal control of multi-delay systems via block-pulse functions, 5th International Conference on Industrial and Information Systems, India, 2010. doi: 10.1109/ICIINFS.2010.5578634. Google Scholar [21] B. M. Mohan and S. Kumar Kar, Orthogonal functions approach to optimal control of delay systems with reverse time terms, Journal of the Franklin Institute, 347 (2010), 1723-1739. doi: 10.1016/j.jfranklin.2010.08.005. Google Scholar [22] B. M. Mohan and S. Kumar Kar, Optimal control of multi-delay systems via shifted Legendre polynomials, International Conference on Energy, Automation, and Signal (ICEAS), India, 2011. doi: 10.1109/ICEAS.2011.6147161. Google Scholar [23] S. H. Nasehi, M. Samavat and M. A. Vali, Analysis and parameter identification of time-delay systems using the Chebyshev wavelets, Journal of Informatics and Mathematical Sciences, 4 (2012), 51-64. Google Scholar [24] A. Nazemi and M. M. Shabani, Numerical solution of the time-delayed optimal control problems with hybrid functions, IMA Journal of Mathematical Control and Information, 32 (2015), 623-638. doi: 10.1093/imamci/dnu012. Google Scholar [25] K. R. Palanisamy and G. P. Rao, Optimal control of linear systems with delays in state and control via walsh function, IEE Proceedings D (Control Theory and Applications), 130 (1983), 300-312. doi: 10.1049/ip-d.1983.0051. Google Scholar [26] M. Razzaghi and M. Razzaghi, Fourier series approach for the solution of linear two-point boundary value problems with time-varying coefficients, International Journal of Systems Science, 21 (1990), 1783-1794. doi: 10.1080/00207729008910498. Google Scholar [27] H. R. Sharif, M. A. Vali, M. Samavat and A. A. Gharavisi, A new algorithm for optimal control of time-delay systems, Applied Mathematical Sciences, 5 (2011), 595-606. Google Scholar [28] D. Shih, F. Kung and C. Chao, Laguerre series approach to the analysis of a linear control system incorporating observes, International Journal of Control, 43 (1986), 123-128. doi: 10.1080/00207178608933452. Google Scholar [29] O. Stryk and R. Bulirsch, Direct and indirect methods for trajectory optimization, Annals of Operations Research, 37 (1992), 357-373. doi: 10.1007/BF02071065. Google Scholar [30] X. T. Wang, Numerical solutions of optimal control for linear time-varying systems with delays via hybrid functions, Journal of the Franklin Institute, 344 (2007), 941–953. doi: 10.1016/j.jfranklin.2007.03.001. Google Scholar [31] X. T. Wang, Numerical solutions of optimal control for time delay systems by hybrid of block-pulse functions and Legendre polynomials, Applied Mathematics and Computation, 184 (2007), 849-856. doi: 10.1016/j.amc.2006.06.075. Google Scholar

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##### References:
 [1] A. Ali, M. A. Iqbal and S. T. Mohyud-Din, Hermite wavelets method for boundary value problems, International Journal of Modern Applied Physics, 3 (2013), 38-47. Google Scholar [2] E. Alirezaei, M. Samavat and M. A. Vali, Optimal control of linear time invariant singular delay systems using the orthogonal functions, Applied Mathematical Sciences, 6 (2012), 1877-1891. Google Scholar [3] U. Brandt-Pollmann, R. Winkler, S. Sager, U. Moslener and J. P. Schlder, Numerical solution of optimal control problems with constant control delays, Computational Economics, 31 (2008), 181-206. doi: 10.1007/s10614-007-9113-3. Google Scholar [4] M. Dadkhah and M. H. Farahi, Optimal control of time delay systems via hybrid of block-pulse functions and orthogonal Taylor series, International Journal of Applied and Computational Mathematics, 2 (2016), 137–152. Available from: https://doi.org/10.1093/imamci/dnv044. doi: 10.1007/s40819-015-0051-9. Google Scholar [5] K. B. Datta and B. M. Mohan, Orthogonal functions in systems and control, in Advanced Series in Electrical and Computer Engineering, World Scientific Publishing Co., (1995), 25–89. doi: 10.1142/2476. Google Scholar [6] L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optimal Control Applications and Methods, 30 (2009), 341-365. doi: 10.1002/oca.843. Google Scholar [7] L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time delays, Journal of Industrial & Management Optimization, 10 (2014), 413-441. doi: 10.3934/jimo.2014.10.413. Google Scholar [8] J. S. Gu and W. S. Jiang, The Haar wavelets operational matrix of integration, International Journal of Systems Science, 27 (1996), 623-628. doi: 10.1080/00207729608929258. Google Scholar [9] N. Haddadi, Y. Ordokhani and M. Razzaghi, Optimal control of delay systems by using a hybrid functions approximation, Journal of Optimization Theory and Applications, 153 (2012), 338-356. doi: 10.1007/s10957-011-9932-1. Google Scholar [10] A. Halanay, Optimal controls for systems with time lag, SIAM Journal on Control and Optimization, 6 (1968), 215-234. Google Scholar [11] I. R. Horng and J. H. Chou, Analysis, parameter estimation and optimal control of time-delay systems via Chebyshev series, International Journal of Control, 41 (1985), 1221-1234. doi: 10.1080/0020718508961193. Google Scholar [12] G. L. Kharatishvili, Maximum principle in the theory of optimal processes involving delay, Dokl. Akad. Nauk SSSR, 136 (1961), 39-42. Google Scholar [13] G. L. Kharatishvili, A Maximum Principle in External Problems with Delays, Mathematical Theory on Control, Academic Press: New York, 1967.Google Scholar [14] F. Khellat, Optimal control of linear time-delayed systems by linear Legendre multiwavelets, Journal of Optimization Theory and Application, 143 (2009), 107-121. Google Scholar [15] M. Malek-Zavarei and M. Jamshidi, Time-Delay Systems: Analysis, Optimization and Applications, Elsevier Science Inc., 1987. Google Scholar [16] I. Malmir, Optimal control of linear time-varying systems with state and input delays by Chebyshev wavelets, Statistics, Optimization and Information Computing, 5 (2017), 302-324. doi: 10.19139/soic.v5i4.341. Google Scholar [17] H. R. Marzban and S. M. Hoseini, A composite Chebyshev finite difference method for nonlinear optimal control problems, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 1347-1361. doi: 10.1016/j.cnsns.2012.10.012. Google Scholar [18] H. R. Marzban and S. M. Hoseini, Solution of linear optimal control problems with time delay using a composite Chebyshev finite difference method, Optimal Control Applications and Methods, 34 (2013), 253-274. doi: 10.1002/oca.2019. Google Scholar [19] H. R. Marzban, Optimal control of linear multi-delay systems based on a multi-interval decomposition scheme, Optimal Control Applications and Methods, 37 (2016), 190-211. doi: 10.1002/oca.2163. Google Scholar [20] B. M. Mohan and S. Kumar Kar, Optimal control of multi-delay systems via block-pulse functions, 5th International Conference on Industrial and Information Systems, India, 2010. doi: 10.1109/ICIINFS.2010.5578634. Google Scholar [21] B. M. Mohan and S. Kumar Kar, Orthogonal functions approach to optimal control of delay systems with reverse time terms, Journal of the Franklin Institute, 347 (2010), 1723-1739. doi: 10.1016/j.jfranklin.2010.08.005. Google Scholar [22] B. M. Mohan and S. Kumar Kar, Optimal control of multi-delay systems via shifted Legendre polynomials, International Conference on Energy, Automation, and Signal (ICEAS), India, 2011. doi: 10.1109/ICEAS.2011.6147161. Google Scholar [23] S. H. Nasehi, M. Samavat and M. A. Vali, Analysis and parameter identification of time-delay systems using the Chebyshev wavelets, Journal of Informatics and Mathematical Sciences, 4 (2012), 51-64. Google Scholar [24] A. Nazemi and M. M. Shabani, Numerical solution of the time-delayed optimal control problems with hybrid functions, IMA Journal of Mathematical Control and Information, 32 (2015), 623-638. doi: 10.1093/imamci/dnu012. Google Scholar [25] K. R. Palanisamy and G. P. Rao, Optimal control of linear systems with delays in state and control via walsh function, IEE Proceedings D (Control Theory and Applications), 130 (1983), 300-312. doi: 10.1049/ip-d.1983.0051. Google Scholar [26] M. Razzaghi and M. Razzaghi, Fourier series approach for the solution of linear two-point boundary value problems with time-varying coefficients, International Journal of Systems Science, 21 (1990), 1783-1794. doi: 10.1080/00207729008910498. Google Scholar [27] H. R. Sharif, M. A. Vali, M. Samavat and A. A. Gharavisi, A new algorithm for optimal control of time-delay systems, Applied Mathematical Sciences, 5 (2011), 595-606. Google Scholar [28] D. Shih, F. Kung and C. Chao, Laguerre series approach to the analysis of a linear control system incorporating observes, International Journal of Control, 43 (1986), 123-128. doi: 10.1080/00207178608933452. Google Scholar [29] O. Stryk and R. Bulirsch, Direct and indirect methods for trajectory optimization, Annals of Operations Research, 37 (1992), 357-373. doi: 10.1007/BF02071065. Google Scholar [30] X. T. Wang, Numerical solutions of optimal control for linear time-varying systems with delays via hybrid functions, Journal of the Franklin Institute, 344 (2007), 941–953. doi: 10.1016/j.jfranklin.2007.03.001. Google Scholar [31] X. T. Wang, Numerical solutions of optimal control for time delay systems by hybrid of block-pulse functions and Legendre polynomials, Applied Mathematics and Computation, 184 (2007), 849-856. doi: 10.1016/j.amc.2006.06.075. Google Scholar
Estimated values for $J$ for Example 5.1
 Proposed method Dadkhah[4] Palanisamy [25] Wang [30] 1.64787419 1.64787419 1.6497 0.85124283
 Proposed method Dadkhah[4] Palanisamy [25] Wang [30] 1.64787419 1.64787419 1.6497 0.85124283
Estimated values for $J$ for Example 5.2
 Proposed method Haddadi[9] Khellat[14] Palanisamy[25] 4.6192 4.7404 5.1713 6.0079
 Proposed method Haddadi[9] Khellat[14] Palanisamy[25] 4.6192 4.7404 5.1713 6.0079
Estimated value for $J$ for Example 5.3
 Proposed method Wang[31] 2.7930174564 2.7930174564
 Proposed method Wang[31] 2.7930174564 2.7930174564
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