یک روش ساده و دقیق برای حل مسائل کنترل بهینه با استفاده از فرمول تفاضلات متناهی مرتبه‌ی دوم

نوع مقاله : مقاله پژوهشی

نویسندگان

1 گروه آموزشی مهندسی برق، دانشکده فنی و مهندسی، دانشگاه بجنورد، بجنورد، ایران

2 دانشکده علوم پایه مهندسی، دانشگاه صنعتی سهند تبریز، تبریز، ایران

3 دانشکده ریاضی، آمار و علوم کامپیوتر، دانشگاه سمنان، سمنان، ایران

چکیده

در این مقاله، یک روش تقریبی ساده و با مرتبه‌ی دقت بالا برای حل مسائل کنترل بهینه ارائه می‌شود. در این روش، ابتدا با استفاده از شرایط لازم بهینگی مبتنی بر اصل ماکزیمم‌یابی پونتریاگین، مسأله‌ی کنترل بهینه‌ی داده شده به یک مسأله‌ی مقدار مرزی دو نقطه‌­ای تبدیل می­‌شود. سپس با اعمال یک فرمول­ تفاضلات متناهی مرتبه‌ی دوم، مسأله‌ی مقدار مرزی حاصل را گسسته نموده و یک دستگاه معادلات جبری را فرمول‌­بندی می‌کنیم. آنالیز همگرایی روش پیشنهادی نیز مورد بحث قرار می‌گیرد و روابط ماتریسی برای سهولت پیاده‌سازی ارائه می‌شود. نتایج عددی به‌دست آمده در این تحقیق با برخی روش‌های دیگر مقایسه می‌شود، که این نتایج مبین دقت، سرعت و کارایی بالای روش پیشنهادی برای حل مسائل کنترل بهینه‌ی خطی و غیرخطی است.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

An accurate and straightforward method for solving optimal control problems using the second-order finite difference formula

نویسندگان [English]

  • Amin Jajarmi 1
  • Mojtaba Hajipour 2
  • Leila Torkzadeh 3
1 Department of Electrical Engineering, University of Bojnord, Bojnord, Iran
2 Department of Mathematics, Faculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran
3 Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran
چکیده [English]

In this paper, a simple and highly accurate approximate method for solving optimal control problems (OCPs) is presented. In this method, initially, by using the necessary optimality conditions based on the Pontryagin's maximum principle, the given OCP is transformed into a two-point boundary value problem (BVP). Then, by applying a second order finite difference formula, the resulting BVP is discretized, and a system of algebraic equations is formulated. Convergence analysis of the proposed method is also discussed, and matrix formulations are provided for ease of implementation. The numerical results obtained in this study are compared with some other methods, demonstrating the high accuracy, speed, and efficiency of the proposed method for solving both linear and nonlinear OCPs.

کلیدواژه‌ها [English]

  • Optimal control problem
  • Pontryagin’s maximum principle
  • Finite difference method

مراجع

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فایل ها

سابقه مقاله

  • تاریخ دریافت: 30 فروردین 1402
  • تاریخ بازنگری: 25 بهمن 1402
  • تاریخ پذیرش: 11 اسفند 1402

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