توصيفگر ها :
سيستم كسري تاخيري , كنترل بهينه كسري , مرتبه قطعه اي ثابت , توابع چبيشف مرتبه كسري , نرم سوبولوف , عملگرهاي انتگرال ريمان-ليوويل و كاپاتو
چكيده فارسي :
امروزه مسائل كنترل بهينه تاخيري كسري با مرتبه متغير با زمان يكي از چالش هاي اساسي در نظريه كنترل
است كه توجه بسياري از پژوهشگران را به خود جلب كرده است. خاطر نشان مي شود آن رده از مسائل كنترل
بهينه كسري كه در آنها ديناميك سيستم يك معادله ديفرانسيل كسري با مرتبه قطعه اي ثابت باشد، مورد بررسي
و مطالعه قرار نگرفته است. در اين رساله، يك روش عددي كارآمد براي حل مسائل كنترل بهينه خطي تاخيري
كسري با مرتبه قطعه اي ثابت يا با تابع تاخير قطعه اي ثابت، ارائه شده است. اساس روش پيشنهادي مبتني
، يك تابع قطعه اي ثابتα است كه در آن α بر توابع تركيبي بلاك-پالس و توابع چبيشف كسري از مرتبه
مي باشد. ماتريس عملياتي انتگرال كسري و ماتريس عملياتي مشتق كسري متناظر با توابع تركيبي ياد شده،
محاسبه شده اند. با استفاده از ماتريس هاي عملياتي مزبور و ماتريس عملياتي تاخير، مساله كنترل بهينه كسري
مورد مطالعه به يك مساله بهينه سازي پارامتري تبديل مي شود كه حل آن به مراتب ساده تر از حل مساله كنترل
بهينه اصلي است . همچنين، همگرايي روش ارائه شده در فضاهاي سوبولوف و هيلبرت اثبات شده است. براي
ارزيابي دقت و كارايي روش پيشنهادي، مثال هاي گوناگوني مورد بررسي و مطالعه قرار گرفته است.
چكيده انگليسي :
In this thesis, we study optimal control of a subclass of variable-order fractional delay systems whose its order is a piecewise constant function. This category of systems has not been discussed in the literature yet. An effective methodology based on a generalization of the fractional-order Chebyshev functions is offered for providing a solution with high level of precision. A detailed consideration regarding the convergence of the new framework is furnished. Moreover, two important estimates connected to the best approximation of the mentioned fractional basis in the Sobolev space and Hilbert space are achieved. Because direct implementation of the Riemann-Liouville integral operator leads to probably some serious drawbacks, such as numerical challenges, instability and unexpected oscillatory behavior of the system under examination, a key integral operator connected to the basis under consideration is attained. The capacity and capability of the suggested numerical scheme are illustrated and verified through our numerical findings. It is recognized that fractional calculus is a well established mathematical instrument for describing the behavior and modelling of many challenging phenomena in which non-locality is an essential issue. Because of the nonlocality and memory properties of the fractional derivative and integral operators, many physical systems and chemical processes can be analyzed more accurately by utilization of the two mentioned operators. Based on this fact, this area of mathematical analysis has progressively grown and many excellent papers have been devoted to study of various fractional order systems involving control of dynamic complex systems, fluid mechanics, oil industry, signal analysis, quantum mechanics, viscoelastic materials, diffusion in porous media and wave propagation, dynamics of earthquakes, biomathematics, biomedical engineering, parameter estimation of systems and economic models. From the theoretical and experimental point of view, we may classify fractional order systems into three categories in accordance with their order as clarified below.
• Constant order fractional systems,
• Variable order fractional models,
• Fractional systems with a piecewise constant order.
It is recognized that the integer-order derivative can be utilized in characterizing the short memory of systems, while the constant order fractional derivative has advantages in characterizing the long memory of systems. Additionally, the variable order fractional derivative can be implemented to describe the variable memory of systems. Some physical meaning of variable-order operators have been described by Ramirez and Coimbra. However, optimal control of fractional delay systems with a piecewise constant order have not been studied in the literature yet. Many real-life phenomena and practical systems can be modeled by various types of delay differential equations. Typical examples are aerospace engineering, transmission lines, chemical processes, climate models, population dynamics, robotics, nuclear reactors, economics, communication networks, biological models, manufacturing processes and power systems. The optimal control problem for linear systems with delays is still open, depending on the delay type, specific systems equation, criterion, etc. It is known that except for some special cases, it is either difficult or impossible to obtain a closed-form solution for delay differential equations. Indeed, time-delay systems is one of the most important subject in optimal control theory. There are two different methods for solving various types of optimal control problems which are known as indirect methods and direct methods.
