A precise control in the stopband attenuation characteristics can be achieved by using the fractional-step filters instead of the traditional integer-order filters. In this paper, nine nature-inspired optimization algorithms, such as five advanced variants of differential evolution (DE), three advanced variants of particle swarm optimization (PSO), and an efficient evolutionary strategy method (CMA-ES-RIS) are employed to design the fractional-step low pass Butterworth filter (FLBF). The proposed $(1+\alpha )$and $(2+\alpha )$ order models, where $\alpha \in (0,\text{ }1),$ are optimally approximated as an integer-order transfer function by using the magnitude-frequency information of the ideal FLBF. Comparisons regarding the solution quality and robustness reveal an improved accuracy for the DE variants and CMA-ES-RIS over all the PSO variants. Results from the pair-wise Wilcoxon rank-sum test demonstrate the superiority of enhanced fitness-adaptive differential evolution (EFADE) algorithm as the most efficient optimization tool for solving this problem. Comparisons with the state-of-the-art approaches also confirm the superior modelling accuracy of the proposed FLBFs. The canonical structure circuit realization of the FLBFs using current feedback operational amplifiers is presented. Simulations carried out in OrCAD PSPICE platform suggest proximity in the magnitude responses between the proposed and the theoretical models. The optimal design of stable, minimum-phase $(2+\alpha )$order FLBFs is also presented for the first time without employing the cascading concept involving the integer-order Butterworth polynomials.

通过使用分数阶滤波器代替传统的整数阶滤波器，可以实现对阻带衰减特性的精确控制。本文采用了9种自然启发式优化算法，例如5种差分进化高级变体（DE），3种粒子群优化高级变体（PSO）和有效的进化策略方法（CMA-ES-RIS）设计分数步低通巴特沃斯滤波器（FLBF）。建议$（\mathrm{1个}+\alpha ）$和 $（2+\alpha ）$ 订购模型 $\alpha \in （0，\text{ }\mathrm{1个}），$通过使用理想FLBF的幅频信息，可以将它们最佳地近似为整数阶传递函数。关于解决方案质量和鲁棒性的比较表明，DE变体和CMA-ES-RIS的精度高于所有PSO变体。成对的Wilcoxon秩和检验的结果表明，增强适应性自适应差分进化（EFADE）算法是解决此问题的最有效的优化工具。与最新方法的比较也证实了所提出的FLBF的卓越建模精度。提出了使用电流反馈运算放大器实现FLBF的规范结构电路实现。在OrCAD PSPICE平台上进行的仿真表明，所提出的模型与理论模型之间的幅度响应接近。稳定，最小相位的最佳设计$（2+\alpha ）$在没有采用涉及整数阶Butterworth多项式的级联概念的情况下，也首次提出了阶FLBF。