# American Institute of Mathematical Sciences

June  2020, 10(2): 143-156. 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

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 and Optimization, 2020, 10 (2) : 143-156. 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. [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. [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. [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. [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. [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. [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. [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. [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. [10] A. Halanay, Optimal controls for systems with time lag, SIAM Journal on Control and Optimization, 6 (1968), 215-234. [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. [12] G. L. Kharatishvili, Maximum principle in the theory of optimal processes involving delay, Dokl. Akad. Nauk SSSR, 136 (1961), 39-42. [13] G. L. Kharatishvili, A Maximum Principle in External Problems with Delays, Mathematical Theory on Control, Academic Press: New York, 1967. [14] F. Khellat, Optimal control of linear time-delayed systems by linear Legendre multiwavelets, Journal of Optimization Theory and Application, 143 (2009), 107-121. [15] M. Malek-Zavarei and M. Jamshidi, Time-Delay Systems: Analysis, Optimization and Applications, Elsevier Science Inc., 1987. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

show all references

##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [10] A. Halanay, Optimal controls for systems with time lag, SIAM Journal on Control and Optimization, 6 (1968), 215-234. [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. [12] G. L. Kharatishvili, Maximum principle in the theory of optimal processes involving delay, Dokl. Akad. Nauk SSSR, 136 (1961), 39-42. [13] G. L. Kharatishvili, A Maximum Principle in External Problems with Delays, Mathematical Theory on Control, Academic Press: New York, 1967. [14] F. Khellat, Optimal control of linear time-delayed systems by linear Legendre multiwavelets, Journal of Optimization Theory and Application, 143 (2009), 107-121. [15] M. Malek-Zavarei and M. Jamshidi, Time-Delay Systems: Analysis, Optimization and Applications, Elsevier Science Inc., 1987. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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
Estimated value for $J$ for Example 5.3
 Proposed method Wang[31] 2.7930174564 2.7930174564
 Proposed method Wang[31] 2.7930174564 2.7930174564
 [1] Akram Kheirabadi, Asadollah Mahmoudzadeh Vaziri, Sohrab Effati. Solving optimal control problem using Hermite wavelet. Numerical Algebra, Control and Optimization, 2019, 9 (1) : 101-112. doi: 10.3934/naco.2019008 [2] Ömer Oruç, Alaattin Esen, Fatih Bulut. A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 533-542. doi: 10.3934/dcdss.2019035 [3] Masoumeh Hosseininia, Mohammad Hossein Heydari, Carlo Cattani. A wavelet method for nonlinear variable-order time fractional 2D Schrödinger equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2273-2295. doi: 10.3934/dcdss.2020295 [4] Canghua Jiang, Cheng Jin, Ming Yu, Zongqi Xu. Direct optimal control for time-delay systems via a lifted multiple shooting algorithm. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021135 [5] Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 [6] S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3747-3761. doi: 10.3934/dcdss.2020435 [7] Shujuan Lü, Zeting Liu, Zhaosheng Feng. Hermite spectral method for Long-Short wave equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 941-964. doi: 10.3934/dcdsb.2018255 [8] Qinqin Chai, Ryan Loxton, Kok Lay Teo, Chunhua Yang. A unified parameter identification method for nonlinear time-delay systems. Journal of Industrial and Management Optimization, 2013, 9 (2) : 471-486. doi: 10.3934/jimo.2013.9.471 [9] Antonella Falini, Francesca Mazzia, Cristiano Tamborrino. Spline based Hermite quasi-interpolation for univariate time series. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022039 [10] Zhengshan Dong, Jianli Chen, Wenxing Zhu. Homotopy method for matrix rank minimization based on the matrix hard thresholding method. Numerical Algebra, Control and Optimization, 2019, 9 (2) : 211-224. doi: 10.3934/naco.2019015 [11] Abderrazek Karoui. A note on the construction of nonseparable wavelet bases and multiwavelet matrix filters of $L^2(\R^n)$, where $n\geq 2$. Electronic Research Announcements, 2003, 9: 32-39. [12] Iman Malmir. Caputo fractional derivative operational matrices of Legendre and Chebyshev wavelets in fractional delay optimal control. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 395-426. doi: 10.3934/naco.2021013 [13] Meng Zhao, Aijie Cheng, Hong Wang. A preconditioned fast Hermite finite element method for space-fractional diffusion equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3529-3545. doi: 10.3934/dcdsb.2017178 [14] Yuan Shen, Xin Liu. An alternating minimization method for matrix completion problems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1757-1772. doi: 10.3934/dcdss.2020103 [15] Jie Tang, Ziqing Xie, Zhimin Zhang. The long time behavior of a spectral collocation method for delay differential equations of pantograph type. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 797-819. doi: 10.3934/dcdsb.2013.18.797 [16] Baowei Feng, Carlos Alberto Raposo, Carlos Alberto Nonato, Abdelaziz Soufyane. Analysis of exponential stabilization for Rao-Nakra sandwich beam with time-varying weight and time-varying delay: Multiplier method versus observability. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022011 [17] Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015 [18] Marcus Wagner. A direct method for the solution of an optimal control problem arising from image registration. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 487-510. doi: 10.3934/naco.2012.2.487 [19] Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems and Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757 [20] Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235

Impact Factor: