Deterministic and Stochastic Analysis of Fractional-Order Legendre Filter with Uncertain Parameters

Fractional order filters are increasingly used due to their flexibility and continuous stepped stopband attenuation rate. The current work presents a deterministic design plan for an optimal fractional-order Legendre low-pass filter along with a stochastic investigation of its parametric uncertainty. First, the filter’s order was determined using the provided parameters, then the flower pollination algorithm was used to tune the transfer function parameters. This method uses the phase delay and magnitude response functions to quantify the desired output. Circuit diagrams, LT spice simulations, and a case study were used to validate the method. In addition, the effects of various components on stability and the performance metrics were further examined. Next, each of the described fractional system parameters (R1, R2, the ratio R4R3, Cα, and Cβ) was modeled as an uncertain term in a distinct cases, referred to as Cases I–V, respectively, and their combined effect was investigated as Case VI. These uncertain parameters were implemented using both random variables and stochastic processes. The system response was assessed using the Monte Carlo simulation method, and the mean, standard deviation, probability density function, and lower and upper bounds were plotted. Additionally, the key statistics of the cutoff frequency were tabulated in all cases. Many findings are addressed by the provided system solutions; briefly, the results revealed that the impact of uncertainty cases on system response, in descending order, was Case VI, Case III, Case V, Case II, Case I, and Case IV. Furthermore, the system demonstrated instability in Cases III and VI, which drew the designers’ attention to these two cases.