一种分数阶微积分算子的有理函数逼近阶数最小化方法

Minimum method of rational function orders for approximation fractional differential and integral operators

针对分数阶微积分算子的实现问题,基于对数幅频特性,导出分数阶积分算子1/sγ(0<γ<1)的一种有理函数逼近公式,该式与Manabe提出的公式类似,但比它更便于分析和应用,讨论了该式应用范围的拓展。为了改善相位逼近精度,提出有理函数构建频率区间概念,它包含逼近频率区间。在满足逼近精度和逼近频率区间条件下,提出使有理函数阶数最小化的两点措施:(1)充分利用对数幅频特性渐近线与准确曲线之差,适当加宽分数阶积分算子与有理函数二者对数幅频特性之间的误差带;(2)根据逼近频率区间,合理选择函数构建频率区间。计算实例表明上述工作的有效性。

Aiming at the problem of implementation of fractional differential and integral operators, an ra- tional function approximation formula for 1/s^γ （0 〈γ 〈 1 ）is derived based on logarithmic frequency characteristic. The formula is similar to the Manabe formula, but is more convinient for analysis and applica- tion. Its extension of application scope was discussed. In order to improve the accuracy of phase approximation, a rational function constructing the frequency interval is proposed. It contained the approximation frequency interval. To meet the conditions of approximation accuracy and frequency interval approximation, two measures to minimize rational function orders was presented ： firtly, make full use of the error between the asymptote and the actual value of the logarithm amplitude- frequency characteristic, and appropriately broaden the error strip of the logarithm amplitude- frequency characteristic of the fractional integral operator vs the rational function;secondly, select the rational function formation frequency area reasonably based on the approximation of the frequency interval. Computation examples show that above work is valid.