نوع مقاله : مقاله پژوهشی

نویسندگان

گروه مهندسی کامپیوتر، دانشکده فنی و مهندسی، واحد قوچان، دانشگاه آزاد اسلامی، قوچان، ایران.

چکیده

حل مسائل کنترل بهینه‌یسینگولار به‌روش کلاسیک دارای پیچیدگی­‌بهینه‌سازیی است که برای ساده‌تر‌شدن حل این گونه مسائل با  تقریب توابع موجود در مسئله با پایه‌ی ‌بهینه‌سازی متعامد به‌جای حل دستگاه معادله‌ی دینامیکی یک سری مسئله‌ی استاتیکی حل می‌شود. این مقاله با استفاده از خصوصیات عملگر‌بهینه‌سازی ماتریسیویولت لژاندر و سری فوریه الگوریتمی ارائه شده است. در‌این الگوریتم متغیر‌بهینه‌سازی حالت، متغیر‌بهینه‌سازی مشتق حالت و بردار کنترل توسط پایه‌ی ‌بهینه‌سازی متعامد یکه‌ی ویولت لژاندرو سری فوریه با ضرایب مجهول بسط داده شده است. برای محاسبه‌یبردار کنترل بهینه و مسیر بهینهی سیستم‌‌بهینه‌سازی سینگولارخطی با تابع هزینهی درجه دو معرفی شده است که با استفاده از خصوصیات توابع متعامد معرفی‌شده ارتباط بین ضرایب  و  پیدا می‌شود. با استفاده از روش پیشنهادی، دینامیک‌‌بهینه‌سازی سیستم به معادلات جبری تبدیل شده و مسئله‌ی بهینه‌سازیدینامیکی از فضای دینامیکی به فضای استاتیکی نگاشت داده شده است که باعث‌بهینه‌سازی مسئلهی استاتیکی با تابع هزینه‌یدرجه دوم و قید‌بهینه‌سازی خطی میشود. ابتدا برای حل مسئله با استفاده از این الگوریتم با پایه‌ی متعامد یکهی ویولت لژاندر استفاده شده است و سپس با پایهی متعامد سری فوریه، حل مسئله تکرار می شود.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

Optimal control design for linear - invariant linear singular systems with time using orthogonal functions

نویسندگان [English]

  • Hamid Tabatabaee
  • Mahdi Memari

Department of Computer Engineering, Faculty of Engineering, Quchan branch, Islamic Azad University, Quchan, Iran.

چکیده [English]

The problem of solving optimal control of Singular problems in the classic method has a complexity that is solved by approximation of the equations in the problem with orthogonal bases instead of solving the dynamic equation system of a set of static problems. In return for a more relaxed solution, it will face some errors in the computation .however, it has an appropriate precision. Legendre and Fourier series are presented using the specifications of the Fourier transform of Legendre and Fourier series . In this algorithm, the state variables, and the state - derivative variables and the control vector are extended by the orthogonal basis of Legendre and Fourier series with unknown coefficients. in order to compute optimal control vector and optimal path of linear Singular systems with quadratic cost function , we are introduced by using the properties of orthogonal functions introduced by the coefficients and .using the proposed method , the system dynamics are converted into algebraic equations and the problem of dynamic optimization of dynamic space has been mapped to static space optimization problem with quadratic cost function and linear constraints . First, it is used to solve the problem using an orthogonal basis of raw material and then the problem solving with orthogonal basis of Fourier series is repeated. Finally, the application and effectiveness of the proposed method are presented.

کلیدواژه‌ها [English]

  • Optimal control
  • Orthogonal functions
  • Singular matrix
  • Violet
Fridman, E., & Shaked, U. (2002). H∞-control of linear state-delay descriptor systems: an LMI approach. Linear algebra and its applications, 351, 271-302.
Lovass-Nagy, V., Powers, D. L., & Schilling, R. J. (1994). On regularizing descriptor systems by output feedback. IEEE transactions on automatic control39(7), 1507-1509.
Azarfar, A., Shandiz, H. T., & Shafiee, M. (2014). Adaptive feedback control for linear singular systems. Turkish journal of electrical engineering & computer sciences22(1), 132-142.
Swaidan, W., & Hussin, A. (2013). Feedback control method using Haar wavelet operational matrices for solving optimal control problems. Abstract and applied analysis (Vol. 2013). Hindawi.http://dx.doi.org/10.1155/2013/240352
Duan, G. R. (2010). Analysis and design of descriptor linear systems (Vol. 23). Springer Science & Business Media.
Bajic, V. B. (1992). Lyapunov's direct method in the analysis of singular systems and networks. Shades Technical Publ.
Shafiei, M. (2001). Optimal control for descriptor systems: Tracking problem (research note). International journal of engineering, 14(2), 123-130.
Marzban, H. R., & Razzaghi, M. (2004). Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials. Journal of the franklin institute341(3), 279-293.
Razzaghi, M., & Yousefi, S. (2002). Legendre wavelets method for constrained optimal control problems. Mathematical methods in the applied sciences25(7), 529-539.
Razzaghi, M., & Yousefi, S. (2001). The legendre wavelets operational matrix of integration. International journal of systems science32(4), 495-502.
Azarfar, A., Toosan Shandiz, H., & Shafiee, M. (2013). Self-tuning state feedback control of MIMO singular systems with applications to constrained robot systems. The mechanics of the structures and fluid, 2(4), 37-46.
Liu, Y., Kao, Y., Gu, S., & Karimi, H. R. (2015). Soft variable structure controller design for singular systems. Journal of the franklin institute352(4), 1613-1626.
Lin, D., & Lan, W. (2015). Output feedback composite nonlinear feedback control for singular systems with input saturation. Journal of the franklin institute352(1), 384-398.
Wang, Y., Zou, Y., Liu, Y., Shi, X., & Zuo, Z. (2015). Average dwell time approach to finite-time stabilization of switched singular linear systems. Journal of the franklin institute352(7), 2920-2933.
Shu, Y., & Zhu, Y. (2017a). Optimistic value based optimal control for uncertain linear singular systems and application to a dynamic input-output model. ISA transactions71, 235-251.
Shu, Y., & Zhu, Y. (2017b). Stability and optimal control for uncertain continuous-time singular systems. European journal of control34, 16-23.
Zhang, W., Lin, Y., & Xue, L. (2017). Linear quadratic Pareto optimal control problem of stochastic singular systems. Journal of the franklin institute354(2), 1220-1238.
Zheng, G., & Bejarano, F. J. (2017). Observer design for linear singular time-delay systems. Automatica80, 1-9.
Chávez-Fuentes, J. R., Mayta, J. E., Costa, E. F., & Terra, M. H. (2017). Stochastic and exponential stability of discrete-time Markov jump linear singular systems. Systems & control letters107, 92-99.
Zhang, X., & Chen, Y. (2018). Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional order α: the 0< α< 1 case. ISA transactions82, 42-50.
Marir, S., Chadli, M., & Bouagada, D. (2017). New admissibility conditions for singular linear continuous-time fractional-order systems. Journal of the franklin institute354(2), 752-766.
Kodra, K., & Gajic, Z. (2017). Optimal control for a new class of singularly perturbed linear systems. Automatica81, 203-208.
Guo, B., Han, Y., Huang, D., Bian, D., & Zhang, L. (2017). Global smooth solution of a two-dimensional nonlinear singular system of differential equations arising from geostrophics. Journal of differential equations262(7), 3980-4020.
Niamsup, P., & Phat, V. N. (2016). A new result on finite-time control of singular linear time-delay systems. Applied mathematics letters60, 1-7.
Fu, L., & Ma, Y. (2016). Passive control for singular time-delay system with actuator saturation. Applied mathematics and computation289, 181-193.
Hou, H., & Zhang, Q. (2016). Novel sliding surface design for nonlinear singular systems. Neurocomputing177, 497-508.
Zheng, G., Efimov, D., Bejarano, F. J., Perruquetti, W., & Wang, H. (2016). Interval observer for a class of uncertain nonlinear singular systems. Automatica71, 159-168.
Fang, Q. (2016). Retraction notice to “Synchronization and state feedback control of linearly coupled singular systems” [AMC 232 (2014) 381–390]. Applied mathematics and computation, 273, 1276-1276.
Liu, Y., Kao, Y., Karimi, H. R., & Gao, Z. (2016a). Input-to-state stability for discrete-time nonlinear switched singular systems. Information sciences358, 18-28.
Liu, Z. Y., Lin, C., & Chen, B. (2016b). Admissibility analysis for linear singular systems with time-varying delays via neutral system approach. ISA transactions61, 141-146.
Al-Zhour, Z. (2016). The general solutions of singular and non-singular matrix fractional time-varying descriptor systems with constant coefficient matrices in Caputo sense. Alexandria engineering journal55(2), 1675-1681.
Ren, G., & Liu, B. (2019). Near-optimal control for a singularly perturbed linear stochastic singular system with Markovian jumping parameters. European journal of control, 50, 88-95.
Zhu, J. (2018). Singular optimal control by minimizer flows. European journal of control, 42, 32-37.