Optimized fractional-order Butterworth filter design in complex F-plane

This paper introduces a new technique to optimally design the fractional-order Butterworth low-pass filter in the complex F-plane. Design stability is assured by incorporating the critical phase angle as an inequality constraint. The poles of the proposed approximants reside on the unit circle in the stable region of the F-plane. The improved accuracy of the suggested scheme as compared to the recently published literature is demonstrated. A mixed-integer genetic algorithm which considers the parallel combinations of resistors and capacitors for the Valsa network is used to optimize the frequency responses of the fractional-order capacitor emulators as part of the experimental verification using the Sallen–Key filter topology. The total harmonic distortion and spurious-free dynamic range of the practical 1.5th-order Butterwoth filter are measured as 0.13% and 62.18 dBc, respectively; the maximum and mean absolute relative magnitude errors are 0.03929 and 0.02051, respectively.Publisher's VersionQ1WOS:00085474580000

On the Design of Power Law Filters and Their Inverse Counterparts

This paper presents the optimal modeling of Power Law Filters (PLFs) with the low-pass (LP), high-pass (HP), band-pass (BP), and band-stop (BS) responses by means of rational approximants. The optimization is performed for three different objective functions and second-order filter mother functions. The formulated design constraints help avoid placement of the zeros and poles on the right-half s-plane, thus, yielding stable PLF and inverse PLF (IPLF) models. The performances of the approximants exhibiting the fractional-step magnitude and phase responses are evaluated using various statistical indices. At the cost of higher computational complexity, the proposed approach achieved improved accuracy with guaranteed stability when compared to the published literature. The four types of optimal PLFs and IPLFs with an exponent alpha of 0.5 are implemented using the follow-the-leader feedback topology employing AD844AN current feedback operational amplifiers. The experimental results demonstrate that the Total Harmonic Distortion achieved for all the practical PLF and IPLF circuits was equal or lower than 0.21%, whereas the Spurious-Free Dynamic Range also exceeded 57.23 and 54.72 dBc, respectively

Optimal approximation of analog PID controllers of complex fractional-order

Complex fractional-order (CFO) transfer functions, being more generalized versions of their real-order counterparts, lend greater flexibility to system modeling. Due to the absence of commercial complex-order fractance elements, the implementation of CFO models is challenging. To alleviate this issue, a constrained optimization approach that meets the targeted frequency responses is proposed for the rational approximation of CFO systems. The technique generates stable, {minimum-phase, and real-valued coefficients based approximants}, which are not always feasible for the curve-fitting approach reported in the literature. {Stability and performance studies of the CFO proportional-integral-derivative (CFOPID) controllers for the Podlubny's, the internal model control, and the El-Khazali's forms are considered to demonstrate the feasibility of the proposed technique}. Simulation results highlight that, for a practically reasonable order, all the designs achieve good agreement with the theoretical characteristics. {Performance comparisons with the CFOPID controller approximants determined by the Oustaloup's CFO differentiator based substitution method justify the proposed approach

Further Generalization and Approximation of Fractional-Order Filters and Their Inverse Functions of the Second-Order Limiting Form

This paper proposes a further generalization of the fractional-order filters whose limiting form is that of the second-order filter. This new filter class can also be regarded as a superset of the recently reported power-law filters. An optimal approach incorporating constraints that restricts the real part of the roots of the numerator and denominator polynomials of the proposed rational approximant to negative values is formulated. Consequently, stable inverse filter characteristics can also be achieved using the suggested method. Accuracy of the proposed low-pass, high-pass, band-pass, and band-stop filters for various combinations of design parameters is evaluated using the absolute relative magnitude/phase error metrics. Current feedback operational amplifier-based circuit simulations validate the efficacy of the four types of designed filters and their inverse functions. Experimental results for the frequency and time-domain performances of the proposed fractional-order band-pass filter and its inverse counterpart are also presented

A Fractional-Order Transitional Butterworth-Butterworth Filter and Its Experimental Validation

This paper introduces the generalization of the classical Transitional Butterworth-Butterworth Filter (TBBF) to the Fractional-Order (FO) domain. Stable rational approximants of the FO-TBBF are optimally realized. Several design examples demonstrate the robustness and modeling efficacy of the proposed method. Practical circuit implementation using the current feedback operational amplifier employed as an active element is presented. Experimental results endorse good agreement (R2= 0.999968) with the theoretical magnitude-frequency characteristic

Optimal Modelling of (1 + α) Order Butterworth Filter under the CFE Framework

This paper presents the optimal rational approximation of (1+α) order Butterworth filter, where α ∊ (0,1) under the continued fraction expansion framework, by employing a new cost function. Two simple techniques based on the constrained optimization and the optimal pole-zero placements are proposed to model the magnitude-frequency response of the fractional-order lowpass Butterworth filter (FOLBF). The third-order FOLBF approximants achieve good agreement to the ideal characteristic for six decades of design bandwidth. Circuit realization using the current feedback operational amplifier is presented, and the modelling efficacy is validated in the OrCAD PSPICE platform

Infinite Impulse Response Approximations to the Non-integer Order Integrator Using Cuckoo Search Algorithm

Part 6: Modelling and OptimizationInternational audienceA popular metaheuristic global optimization technique called Cuckoo Search Algorithm (CSA) is employed to design non-integer order integrators (NOIs) in terms of the Infinite Impulse Response (IIR) templates in this paper. Extensive comparisons on the basis of design quality robustness, error convergence, and optimization time of the CSA-based NOIs are carried out with the Particle Swarm Optimization (PSO) based designs. Results demonstrate the efficient performance of CSA in exploring the multimodal, non-linear, and non-uniform error surface for this optimization problem. The CSA-based designs also outperform the recent literature by 9.67 decibel (dB) and 19.26 dB in terms of mean absolute relative magnitude error (MARME) and maximum absolute magnitude error (MAME) metrics, respectively