توصيفگر ها :
كنترل بهينه كسري , ماتريس عملياتي مشتق , توابع چبيشف از مرتبه كسري , تابع جريمه
چكيده فارسي :
چكيده:
0، براساس چندجمل هاي هاي چبيشف نوع اول، براي < α ≤ 1 ،α در اين پايان نامه، توابع چبيشف از مرتبه كسري
دستيابي به جواب تقريبي مسائل كنترل بهينه با محدوديت هاي نامساوي، تعريف مي شوند و ويژگي هاي آن ها از نظر
تعامد و كامل بودن، مورد بررسي قرار مي گيرند. چندجمل هاي هاي چبيشف نوع اول روي بازه [ 1, 1 −] نسبت به تابع
متعامدند. همچنين، چندجمله هاي چبيشف انتقال يافته روي بازه [ 0, 1 ] نسبت به تابع w(x) = √ 1
1 − x وزن 2
،α متعامدند. روش حل مسأله، يك روش مستقيم مبتني بر توابع چبيشف كسري از مرتبه w(t) = √ 1
t − t وزن 2
متعامدند و يك پايه كامل w(t) = √ 1
t2−α − t 0 است. اين توابع روي بازه [ 0, 1 ] نسبت به تابع وزن 2 < α ≤ 1
تشكيل مي دهند. بدين منظور، ابتدا با استفاده از روش تابع جريمه، با اضافه كردن توابع L2w [ براي فضاي هيلبرت [ 0, 1
كمكي مثبت، محدوديت هاي نامساوي روي متغيرهاي حالت و كنترل به محدوديت هاي تساوي تبديل مي شوند. سپس با
ساخته مي شود. α استفاده از مفهوم مشتق كسري كاپوتو، ماتريس عملياتي مشتق متناظر با توابع چبيشف از مرتبه كسري
با استفاده از ماتريس مشتق ساخته شده، ديناميك سيستم مسأله كنترل بهينه به يك دستگاه معادلات جبري غير خطي
تبديل مي شود. بدين ترتيب، مسأله كنترل بهينه كسري اوليه به يك مسأله بهينه سازي پارامتريك تبديل مي شود كه مي توان
آن را با استفاده از روش هاي بهينه سازي موجود حل نمود. خاطر نشان مي شود كه حل مسأله بهينه سازي پارامتريك به
دست آمده بسيار ساده تر از حل مسأله كنترل بهينه اصلي است. براي نشان دادن دقت و كارايي روش ارائه شده، مسائل
كنترل بهينه گوناگوني را مورد بررسي و مطالعه قرار مي دهيم
واژگان كليدي: كنترل بهينه كسري، ماتريس عملياتي مشتق، توابع چبيشف از مرتبه كسري، تابع جريمه.
چكيده انگليسي :
In the real world, fractional calculus has been used to describe the behavior of many real-life phenomena such as
hydrologic, viscoelastic modelling, disease control and prevention , the temperature and motor control, growths of
populations modelling , fluid mechanics, bioengineering. It has been shown that materials with memory and hereditary
effects, and dynamical processes, including gas diffusion and heat conduction, parameter identification of nonlinear
complex physical system have more accurate models by fractional-order models than integer-order models.
The objective of this thesis is to present an efficient numerical approximation method for solving nonlinear fractional
optimal control problems with constraints on the state and control variables. The classical Chebyshev polynomials
are orthogonal with respect to the weight function w(x) =
√ 1
1 − x2
on the interval [-1,1] . The shifted classical
Chebyshev polynomials are orthogonal with respect to the weight function w(t) =
√ 1
t − t2
on the interval [0,1] . It
should be mentioned that except some simple cases, the exact solution of fractional optimal control problems are unavailable.
Therefore, to design an efficient numerical method should be developed. The foundation of the developed
method is based on the fractional- order Chebyshev functions. The shifted fractional-order Chebyshev functions form
a complete and orthogonal basis for the Hilbert space L2
w [0, 1]. It is worth noting that the shifted fractional-order
Chebyshev functions are orthogonal with respect to the weight function w(t) =
√ 1
t2−α − t2
, 0 < α ≤ 1 on
the interval [0,1] . The properties of fractional Chebyshev function are discussed. The fractional derivative operator
associated with the proposed fractional basis is obtained and used to convert the original fractional optimal control
problem into a parameter optimization problem. The zeros of the fractional Chebyshev functions are chosen as the
collocation points. The penalty function technique is implemented to convert the inequality constraints on the state
and control functions to equality constraints. The method of Lagrange multipliers method is applied for solving the
obtained parameter optimization problem. For this purpose, positive slack functions are added to inequality conditions
and then the operational matrix for the fractional derivative in the Caputo sense, reduces the problems to those of
solving a system of algebraic equations. It is shown that the solutions converge as the number of approximating terms
increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach one. The
applicability and validity of the method are shown by numerical results of some examples, moreover a comparison
with the existing results shows the preference of this method. It should be noticed that this matrix gives the derivative
exactly in both fractional and integer cases. As a matter of fact, the functions of the problem are approximated by the
fractional order Chebyshev functions with unknown coefficients in the cost function and conditions. Therefore, the
main optimal control problem reduces to an unconstrained optimization problem. Then optimality conditions yield a
system of linear or nonlinear algebraic equations which is solved by the proposed collocation method. As shown, the
method is convergent and has an appropriate accuracy and stability. Illustrative examples show that this method has
good results for linear and nonlinear fractional optimal control problems with constraints
