A Hamiltonian formulation and a direct numerical scheme for Fractional Optimal Control Problems
A Hamiltonian formulation and a direct numerical scheme for Fractional Optimal Control Problems
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DOI：
10.1177/1077546307077467
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发表时间：
2007-09-01
影响因子：
2.8
通讯作者：
Baleanu, Dumitru
中科院分区：
文献类型：
--
作者：
Agrawal, Om P.;Baleanu, Dumitru
本文研究分数阶最优控制问题的一种直接数值方法。在本文中，我们制定的FOCP的Riemann-Liouville分数阶导数（RLFD）。它表明，右RLFD自动出现在制定，即使当系统的动态描述只使用左RLFD。对于数值计算，FD近似使用Grunwald-Letnikov定义。这导致一组代数方程，可以使用数值技术求解。两个例子，一个时不变的和其他时变的，被认为是证明制定的有效性。结果表明，当导数的阶数接近整数时，这些公式可导出整数阶系统的解。该方法需要将整个时域划分为几个子域。此外，随着子域的大小减小，解收敛到唯一解。然而，收敛是缓慢的。一个计划，提高收敛速度将被认为是在未来的文件。其他问题将在未来考虑包括配方使用其他类型的衍生物，非线性和随机分数最优控制，存在性和唯一性的解决方案，以及误差分析。
This paper deals with a direct numerical technique for Fractional Optimal Control Problems (FOCPs). In this paper, we formulate the FOCPs in terms of Riemann-Liouville Fractional Derivatives (RLFDs). It is demonstrated that right RLFDs automatically arise in the formulation even when the dynamics of the system is described using left RLFDs only. For numerical computation, the FDs are approximated using the Grunwald-Letnikov definition. This leads to a set of algebraic equations that can be solved using numerical techniques. Two examples, one time-invariant and the other time-variant, are considered to demonstrate the effectiveness of the formulation. Results show that as the order of the derivative approaches an integer value, these formulations lead to solutions for integer order system. The approach requires dividing of the entire time domain into several sub-domains. Further, as the sizes of the sub-domains are reduced, the solutions converge to unique solutions. However, the convergence is slow. A scheme that improves the convergence rate will be considered in a future paper. Other issues to be considered in the future include formulations using other types of derivatives, nonlinear and stochastic fractional optimal controls, existence and uniqueness of the solutions, and the error analysis.