A general formulation and solution scheme for fractional optimal control problems
A general formulation and solution scheme for fractional optimal control problems
复制标题
DOI：
10.1007/s11071-004-3764-6
复制
发表时间：
2004-12-01
影响因子：
5.6
通讯作者：
Agrawal, OP
中科院分区：
文献类型：
--
作者：
Agrawal, OP
许多动态系统的精确建模导致了一组分数阶微分方程（FDES）。本文给出了这类系统的分数阶最优控制问题的一般形式和求解方案。分数阶导数在Riemann-Liouville意义下描述。FOCP的性能指标被认为是状态变量和控制变量的函数，动态约束由一组FDE表示。利用变分法、拉格朗日乘子和分式积分公式，得到了FOCP的欧拉-拉格朗日方程。所提出的公式和所得到的方程与经典最优控制理论中出现的方程非常相似。因此，本公式实质上扩展了经典控制理论的分数阶动态系统。该公式用于推导二次线性分式控制问题的控制方程。一种类似于变分虚功的方法，结合拉格朗日技术，给出了所得方程的近似数值解。两个分数阶系统，一个时不变的和一个时变的数值解，证明了该方法的可行性。结果表明：（1）解随着逼近项的增加而收敛;（2）当分数阶导数的阶数接近1时，解接近经典解。所提出的公式是简单的，可以扩展到其他FOCP。人们希望，这个配方的简单性将引发一个新的兴趣在分数系统的最优控制领域。
Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann-Liouville sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems.