یک الگوریتم برای کنترل بهينه يك كلاس از سيستم‌هاي خطي متغير با زمان تأخیری با رویکرد کاهش زمان محاسبات و افزایش سرعت در مسائل مهندسی
محورهای موضوعی : مهندسی برق و کامپیوترمهدی یوسفی طبری 1 , زهرا رحمانی 2 , علی وحیدیان کامیاد 3 , سیدجلیل ساداتی 4
1 - دانشکده مهندسی برق، دانشگاه صنعتی نوشیروانی بابل
2 - دانشکده مهندسی برق، دانشگاه صنعتی نوشیروانی بابل
3 - دانشکده علوم ریاضی، دانشگاه فردوسی مشهد
4 - دانشکده مهندسی برق، دانشگاه صنعتی نوشیروانی بابل
کلید واژه: کنترل بهینه, سیستم‌های تأخیری, اصل حداکثر پونتریاگین,
چکیده مقاله :
سیستم‌های تأخیر زمانی در چند دهه اخیر بسیار مورد توجه قرار گرفته‌اند و بسیاری از آنها در سیستم‌ها و شاخه‌های مختلف علوم مانند مهندسی، شیمی، فیزیک و مدل‌های بیماری ظاهر می‌شوند. وجود تأخیر، تحلیل و کنترل چنین سیستم‌هایی را بسیار پیچیده‌تر می‌کند. استفاده از اصل حداکثر پونتریاگین برای مسائل کنترل بهینه با تأخیر زمانی منجر به یک مسأله مقدار مرزی می‌شود که شامل هر دو شرایط تأخیر و تقدم است. در این مقاله، یک مسأله کنترل بهینه با تأخیر زمانی را در نظر می‌گیریم. در بخش اول ابتدا با استفاده از اصل حداکثر پونتریاگین برای مسائل کنترل بهینه با تأخیر زمانی، شرایط بهینه لازم را برای این مسأله به‌دست می‌آوریم و سپس الگوریتمی جدید برای حل عددی این مسأله ارائه می‌گردد که بر پایه یک تقریب برای مشتقات و درون‌یابی خطی برای جملات تأخیر است. نهایتاً معادلات حاصل به یک مسأله برنامه‌ریزی خطی تبدیل می‌شوند که می‌توان آنها را به‌صورت عددی حل نمود. کارايي روش پيشنهادی با شبیه‌سازی عددی مورد ارزیابی قرار می‌گیرد.
Time-delay systems have been very much considered in the last few decades. Many of these time-delay systems appear in different systems and branches of science such as engineering, chemistry, physics, disease models. The presence of delay makes the analysis and control of such systems much more complicated. In fact, the application of Pontryagin’s maximum principle to the optimal control problems with time-delay results in boundary value problem involving both delay and advance terms. In this paper, we consider a time-delay optimal control problems. The first section, using the Pontryagin's maximum principle for optimal control problems with time delay, the necessary optimality conditions for this problem, are obtained. Then a new algorithm is proposed to solve this problem numerically. This algorithm is based on an approximation for derivatives and linear interpolation for delayed arguments. Finally, the resulting equations becomes a linear programming problem that can be solved numerically. The efficiency of the proposed method is evaluated by solving several numerical examples.
[1] M. R. Hestenes, Calculus of Variations and Optimal Control Theory, Wiley, 1966.
[2] A. E. Bryson, "Optimal control-1950 to 1985," IEEE Control Systems Magazine, vol. 16, no. 3, pp. 26-33, Jun. 1996.
[3] M. Jamshidi and C. Wang, "A computational algorithm for large-scale nonlinear time-delay systems," IEEE Trans. on Systems, Man, Cybernetics, vol. 14, no. 1, pp. 2-9, Jan.-Feb. 1984.
[4] M. Malek-Zavarei and M. Jamshidi, Time-Delay Systems: Analysis, Optimization and Applications, Elsevier Science Inc., 1987.
[5] H. J. Sussmann and J. C. Willems, "300 years of optimal control: from the brachystochrone to the maximum principle," IEEE Control Systems Magazine, vol. 17, no. 3, pp. 32-44, 1997.
[6] G. Kharatishdi, "The maximum principle in the theory of optimal processes with time lags," Dokl. Akad. Nauk SSSR, vol. 136, no. 1, pp. 39-43, 1961.
[7] D. Eller, J. Aggarwal, and H. Banks, "Optimal control of linear time-delay systems," IEEE Trans. on Automatic Control, vol. 14, no. 6, pp. 678-687, Dec. 1969.
[8] K. Palanisamy and R. G. Prasada, "Optimal control of linear systems with delays in state and control via Walsh functions," IEE Proc. D-Control Theory and Applications, vol. 130, no. 6, pp. 300-312, Nov. 1983.
[9] K. Inoue, H. Akashi, K. Ogino, and Y. Sawaragi, "Sensitivity approaches to optimization of linear systems with time delay," Automatica, vol. 7, no. 6, pp. 671-679, Nov. 1971.
[10] J. Banas and A. Vacroux, "Optimal piecewise constant control of continuous time systems with time-varying delay," Automatica, vol. 6, no. 6, pp. 809-811, Nov. 1970.
[11] H. R. Marzban and M. Razzaghi, "Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials," J. of the Franklin Institute, vol. 341, no. 3, pp. 279-293, May 2004.
[12] L. Y. Lee, "Numerical solution of time‐delayed optimal control problems with terminal inequality constraints," Optimal Control Applications and Methods, vol. 14, no. 3, pp. 203-210, Jul./Sept. 1993.
[13] ها. چهکندی نژاد، م. فرشاد و ر هاونگی، "ارائه یک روش جدید به منظور تخمین برخط تأخیر زمانی در سیستم‌های SISO-LTI با تأخیر زمانی متغیر با زمان و نامعلوم در ورودی کنترلی،" نشریه مهندسی برق و مهندسی کامپیوتر ایران، الف- مهندسی برق، سال 18، شماره 1، صص. 44-36، بهار 1399.
[14] H. Marzban and S. Hoseini, "Solution of linear optimal control problems with time delay using a composite Chebyshev finite difference method," Optimal Control Applications and Methods, vol. 34, no. 3, pp. 253-274, May/Jun. 2013.
[15] X. T. Wang, "Numerical solutions of optimal control for time delay systems by hybrid of block-pulse functions and Legendre polynomials," Applied Mathematics, vol. 184, no. 2, pp. 849-856, 15 Jan. 2007.
[16] A. Jajarmi and M. Hajipour, "An efficient finite difference method for the time‐delay optimal control problems with time‐varying delay," Asian J. of Control, vol. 19, no. 2, pp. 554-563, Mar. 2017.
[17] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 1998.
[18] O. P. Agrawal and D. Baleanu, "A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems," J. of Vibration and Control, vol. 13, no. 9-10, pp. 1269-1281, 2007.
[19] R. L. Burden, J. D. Faires, and A. Reynolds, Numerical Analysis Prindle, Weber & Schmidt, pp. 28-34, 1985.
[20] H. Banks and J. Burns, "Hereditary control problems: numerical methods based on averaging approximations," SIAM J. on Control Optimization, vol. 16, no. 2, pp. 169-208, 1978.
[21] A. Jajarmi and D. Baleanu, "Suboptimal control of fractional-order dynamic systems with delay argument," J. of Vibration and Control, vol. 24, no. 12, pp. 2430-2446, 2017.
[22] L. Moradi, F. Mohammadi, and D. Baleanu, "A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets," J. of Vibration and Control, vol. 25, no. 2, pp. 310-324, 2019.
[23] S. Sabermahani, Y. Ordokhani, and S. A. Yousefi, "Fractional-order Lagrange polynomials: an application for solving delay fractional optimal control problems," Trans. of the Institute of Measurement and Control, vol. 41, no. 11, pp. 2997-3009, 2019.
[24] H. R. Marzban and F. Malakoutikhah, "Solution of delay fractional optimal control problems using a hybrid of block-pulse functions and orthonormal Taylor polynomials," J. of the Franklin Institute, vol. 356, no. 15, pp. 8182-8215, Oct. 2019.
[25] N. Haddadi, Y. Ordokhani, and M. Razzaghi, "Optimal control of delay systems by using a hybrid functions approximation," J. of Optimization Theory and Applications, vol. 153, no. 2, pp. 338-356, 12 Oct. 2012.
[26] K. Palanisamy, K. Balachandran, and R. Ramasamy, "Optimal control of linear time-varying delay systems via single-term Walsh series," IEE Proc. D-Control Theory and Applications, vol. 135, no. 4, pp. 332-332, Jul. 1988.
[27] S. Dadebo and R. Luus, "Optimal control of time‐delay systems by dynamic programming," Optimal Control Applications and Methods, vol. 13, no. 1, pp. 29-41, Jan./Mar. 1992.
[28] J. R. Ockendon and A. B. Tayler, "The dynamics of a current collection system for an electric locomotive," Proc. of the Royal Society A: Mathematical Physical Sciences, vol. 322, no. 1551, pp. 447-468, 4 May 1971. 1971.
[29] F. Ghomanjani, M. H. Farahi, and A. V. Kamyad, "Numerical solution of some linear optimal control systems with pantograph delays," IMA J. of Mathematical Control and Information, vol. 32, no. 2, pp. 225-243, Jun. 2015.
[30] N. Ghaderi and M. H. Farahi, "The numerical solution of some optimal control systems with constant and pantograph delays via bernstein polynomials," Iranian J. of Mathematical Sciences Informatics, vol. 15, no. 2, pp. 163-181, 2020.
[31] M. Fatehi, M. Vali, and M. Samavat, "State analysis and optimal control of linear time-invariant scale systems using the legendre wavelets," Canadian J. on Automation, Control & Intelligent Systems, vol. 3, no. 1, pp. 1-7, May 2012.
[32] M. Fatehi, M. Samavat, M. Vali, and F. Khaleghi, "State analysis and optimal control of linear time-invariant scaled systems using the Chebyshev wavelets," Contemporary Engineering Sciences, vol. 5, no. 2, pp. 91-105, 2012.
