Yavari Mina, Nazemi Alireza
Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrood, Iran.
ISA Trans. 2020 Jun;101:78-90. doi: 10.1016/j.isatra.2020.02.011. Epub 2020 Feb 20.
Fractional calculus is a powerful and effective tool for modeling of nonlinear systems. In this paper, we first introduce a modification of conformable fractional derivative to solve fractional infinite horizon optimal control problems. We point out that the term t in definition of conformable derivative has some disadvantages and thus must be refined. Using an interesting property of relationship between the presented fractional derivative and the usual first-order derivative, fractional dynamic system in the infinite horizon optimal control problem is transformed into a non-fractional one. By a suitable change of variable, the obtained infinite horizon problem is reduced to a finite-horizon one. According to the Pontryagin minimum principle for optimal control problems and by constructing an error function, an unconstrained minimization problem is then defined. In the achieved minimization problem, trial solutions for state, co-state and control functions are utilized where these trial solutions are constructed by using two-layered perceptron neural network. Some numerical results are solved to explain our main results. Two applicable examples as stabilization and chaos control of fractional order systems are also provided.
分数阶微积分是一种用于非线性系统建模的强大而有效的工具。在本文中，我们首先引入一种修正的一致分数阶导数来求解分数阶无限时域最优控制问题。我们指出，一致导数定义中的项t存在一些缺点，因此必须加以改进。利用所提出的分数阶导数与通常一阶导数之间关系的一个有趣性质，将无限时域最优控制问题中的分数阶动态系统转化为非分数阶系统。通过适当的变量变换，将得到的无限时域问题简化为有限时域问题。根据最优控制问题的庞特里亚金最小值原理，并通过构造一个误差函数，进而定义一个无约束最小化问题。在得到的最小化问题中，利用状态、共态和控制函数的试探解，这些试探解是通过使用两层感知器神经网络构造的。求解了一些数值结果以解释我们的主要结果。还提供了分数阶系统稳定化和混沌控制这两个应用实例。
