分数微积分的两种系统建模方法

介绍了Riemann-Liouville和Caputo分数微积分的定义及其部分性质。给出了分数阶线性定常微分方程的一般定义形式,并指出它是整数阶线性定常微分方程的推广。给出分数阶线性定常系统的传递函数和状态方程描述,并与整数阶线性定常系统的传递函数和状态方程作一比较,指出它们的异同点。运用拉普拉斯变换推导出其两种求解方法:直接求解法和状态空间法。最后给出一个实例说明这两种方法的有效性。

Two System Modeling Methods Using Fractional Calculus

An introduction of the definitions of Riemann-Liouville and Caputo fractional calculus is given as well as some of their properties. The general form of fractional linear time-invariant (LTI) equations is proposed, and it is pointed out that they are the generalizations of integer LTI equations. The transfer function and the state-space representation are given for fractional LTI systems, and a comparison is made with integer LTI systems, and their differences and similarities are also pointed out. Two solving methods are deduced using Laplace transform: the direct solving method and the state-space method. Finally an example is given to show the effectiveness of the two methods aforementioned.