توصيفگر ها :
كنترل بهينه , سيستم هاي تاخيري , روش ريتز , چندجمله اي هاي مونتس-لژاندر , مشتق كسري كپوتو
چكيده فارسي :
در مسائل كنترل بهينه كسري تأخيري، به علت وجود تأخير، پيدا كردن يك روش تحليلي دشوار و تا حدودي غيرممكن است، بنابراين ارائه روش‌هاي عددي موثر و كارا براي حل اين نوع مسائل از اهميت خاصي برخوردار است. هدف اين پايان نامه، ارائه روش محاسباتي ريتز بر پايه چندجمله‌اي‌هاي مونتس-لژاندر براي حل مسائل كنترل بهينه كسري تأخيري است. تابع حالت، تابع حالت تاخيري و تابع كنترل را توسط چندجمله‌اي‌هاي مونتس-لژاندر به‌گونه‌اي تقريب مي‌زنيم كه در شرايط اوليه داده شده مسأله صدق كنند. اگر تقريب توابع حالت و كنترل در ديناميك سيستم مسأله كنترل بهينه كسري تأخيري مورد مطالعه جايگذاري شوند، آنگاه ديناميك سيستم كه يك معادله ديفرانسيل كسري تأخيري از مرتبه آلفا است، به يك دستگاه معادلات جبري خطي يا غيرخطي تبديل خواهد شد. بدين ترتيب، مسأله كنترل بهينه كسري تأخيري مورد مطالعه به يك مسأله بهينه‌سازي پارامتري تبديل مي‌شود. براي بررسي دقت، كارايي و كاربرد روش ارائه شده، مثال‌هاي مختلف و گوناگوني مورد بررسي و ارزيابي قرار خواهند گرفت.
چكيده انگليسي :
In this thesis, it is demonstrated that the utilization of the Ritz method based on the Müntz-Legendre polynomials can provide an effective and efficient approach for solving delayed fractional optimal control problems. This method not only offers high accuracy in approximating state and control functions but also, due to its simple and comprehensible structure, can be easily applied to more complex problems.
The Ritz method is one of the most widely used direct methods for solving variational and optimal control problems. Additionally, the high accuracy of the results and the ease of use of the Müntz-Legendre polynomials are two main reasons why they are employed in this study.
Initially, state functions and delayed state functions are approximated based on the Müntz-Legendre polynomials. These approximations are chosen in such a way that they can model the dynamic behavior of the system well. Then, using the given constraints, the control functions are computed. Finally, by substituting the obtained functions approximations into the objective function, an algebraic equation, either linear or nonlinear in terms of unknown coefficients, is derived.
Thus, the delayed fractional optimal control problem under study is transformed into a parametric optimization problem. This transformation allows us to utilize algebraic equation-solving techniques, which are considerably simpler than directly solving fractional delayed differential equations. This parametric optimization problem can be solved using numerical optimization methods, and we can find the optimal coefficients that approximate the state and control functions.
To ensure the accuracy and efficiency of the proposed method, several numerical examples are examined and analyzed. These examples include various systems with complex dynamics and different boundary conditions. The results demonstrate that the proposed method can achieve high accuracy and can be implemented for solving delayed fractional optimal control problems. The time-delay fractional optimal control problems are an extension form of fractional optimal control
problems which at least one of the state or control variables in the dynamic system or performance index has the delay. If
the dynamic system of the problem is governed by the previous information at the specified time, then the The time-delay fractional optimal control problem needs
to be defined. In The time-delay fractional optimal control problems, the assumptions for the traditional point wise modeling are replaced by the realistic distributed
assumptions. In fact, in the time-delay mathematical models, the time trajectory changes do not depend only on t moment
itself. Some certain conditions and their reflections before that moment have effects on the trajectory changes. Time delays
are appeared in every situation and their significant effects and presence cannot be ignored. They appear in transportation,
economics, power systems, engineering, chemical, electronics, biological, manufacturing. Due to the presence
of delay in a The time-delay fractional optimal control problem, finding the analytical solution is impossible and extremely difficult. Therefore, a numerical method
for the solution is required. In this thesis, the Ritz computational method based upon Müntz–Legendre polynomials for solving a class of The time-delay fractional optimal control problems
is presented. The Ritz method is one of the most extensively applied direct methods for solving the variational
problems. In general terms, the Ritz direct method has a greater convergence radius than indirect one. In this
method, the unknown functions are approximated by a linear combination of a finite number of the basis functions. The
high accuracy of the results and ease of use of Müntz–Legendre polynomials are two reasons why they have been applied
in this study.
