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

Fractional Calculus and Applied Analysis https://doi.org/10.1007/s13540-022-00081-9 ORIGINAL PAPER Optimized fractional-order Butterworth filter design in complex F-plane Shibendu Mahata1 ·Norbert Herencsar2 ·David Kubanek2 · I. Cem Goknar3 Received: 26 March 2022 / Revised: 2 August 2022 / Accepted: 2 August 2022 © The Author(s) 2022 Abstract 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. BDavid Kubanek [email protected] Shibendu Mahata [email protected] Norbert Herencsar [email protected] I. Cem Goknar [email protected] 1Department of Electrical Engineering, Dr. B. C. Roy Engineering College, Durgapur, West Bengal 713206, India 2Department of Telecommunications, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technicka 12, 616 00 Brno, Czech Republic 3Department of Electrical and Electronics Engineering, Isik University, University Str. 2, 34980 Sile, Istanbul, Turkey 123 S. Mahata et al. Keywords Fractional-order system (primary) ·Analog filter approximation · F-domain ·Fractional-order capacitor ·Constrained optimization ·Fractional-order Butterworth filter ·Stability Mathematics Subject Classification 26A33 (primary)93B50 ·93C99 ·93E11 1 Introduction The Riemann–Liouville definition for the fractional derivative of arbitrary order υfor a function f(t) is given by aDυ tf(t)=1 (n−υ) dn dtnt a (t−τ)n−υ−1f(τ)dτ, (1) where n∈Z+;n−1<υ≤n;aand tare the limits of the operation; and (·)denotes the gamma function [13,20,23]. For zero initial conditions, the Laplace transform of (1)isgivenbysυF(s). Applications of fractional calculus are pervading in various engineering disciplines [15], including fractional-order (FO) circuit theory and filter design [19]. Due to the presence of additional design parameters or ‘tuning knobs’ (viz., υ), FO filters exhibit amplitude, delay, and transient characteristics, which are not possible to achieve using the classical ones [2,11]. Generalization of integer-order filters to the FO-domain allows the exact order (typically, a real number) to be realized; thereby, providing precise roll-off and strict matching of design specifications [9]. Implementations of analog FO filters involve the fractance devices (e.g., FO capacitor/inductor) [4], whose immittance relationship is I(s)=ksβ,(2) where in the case of FO capacitor, it is appropriate to consider I(s)as an admittance and kis the pseudo-capacitance expressed in Farad/second1−β. In the case of FO inductor, I(s) can be considered as an impedance and kis the pseudo-inductance expressed in Henry/second1−β. In both these cases, it holds β∈(0,1). The frequency-domain expression for sβis given by (jω)β=ωβ∠(βπ/2), where the analog frequency variable ωis expressed in radians per second (rad/s). Various approaches have been reported to emulate the frequency response of sβ,[5]. The squared magnitude function of the normalized FO Butterworth filter (FOBF) [3]isgivenby HB(s)HB(−s)|s=jω=|HB(jω)|2=1 1+ω2M,(3) where M=N+α,N∈Z+and α∈(0,1). In the literature, two different approaches are reported to approximate the magnitude-frequency characteristic of the FOBF. The first technique involves mod123 Optimized fractional-order Butterworth filter… eling in the complex s-plane using the FO [3,10,25] or the integer-order transfer functions (see [16] and the references cited therein). Alternatively, FOBFs were also designed in the complex W-plane (W=s1/q,where α=r/q;r,q∈Z+)using the pole-placement algorithms [1,18]. A recent work has demonstrated the effectiveness of the F-domain-based approach in the reduced order modeling of FO commensurate system [17]. While realizing FO filters with optimization techniques exist in the literature [10,16,25], this paper is the first attempt to design the analog FOBF directly in the complex F-plane optimally. A constrained optimization strategy is formulated to guarantee design stability. Relative to the existing W-plane based techniques [1,18], the proposed method provides the following contributions: 1. Unlike [1], the symmetric distribution of coefficients in the denominator polynomial for the integer-order Butterworth filter is also exhibited by the proposed FOBF, 2. Higher-order FOBFs are practically realized in [1] by cascading Nth-order Butterworth filter with an αth-order one. In contrast, the proposed approach can directly generate the optimal FOBF model for any value of M, 3. Unlike [18], the all-pole structure of the Mth-order proposed model always requires N+2 terms, which is the same as that of the (N+1)th order Butterworth filter, and 4. Although the W-plane-based techniques [1,18] generate a stable design, these methods do not provide optimal pole placement in the W-plane. Consequently, the retro-fitted model in the s-plane is also not an optimal one. In contrast, the proposed constrained optimization approach guarantees both stability as well as optimal placement of poles in the unit circle of the complex F-plane for the normalized FOBF approximant. The corresponding s-domain-based proposed model achieves lower error as compared to the cited literature in approximating the theoretical magnitude response. In the rest of the paper, Sect. 2presents the proposed scheme; stability and modeling performances are analyzed in Sect. 3. Experimental results are presented in Sect. 4. Finally, conclusions are drawn in Sect. 5. 2 Proposed technique It is well-known that for a function F=sβ, where β∈(0,1), the region of instability in the s-plane (|θS|<0.5π) is mapped into |θF|<0.5βπ in the complex F-plane [21]. Therefore, the region of stability in the F-plane is given by |θF|>0.5βπ. Since s=F1/β ,then the squared magnitude function in the F-plane for the FOBF (normalized) can be obtained by substituting ω=F1/β /jin (3), as represented by HB(F1/β )HB(−F1/β ) =1 1+F1/β j2M=1 1+(−F2/β )M=1 1+(−1)MF2M/β .(4) 123 S. Mahata et al. Selecting β=M/(N+1)and substituting in (4) leads to the magnitude function as given below. HB(F1/β )HB(−F1/β )=1 1+(−1)MF2(N+1).(5) Thus, the F-domain-based Mth-order Butterworth filter (normalized) is modeled as an all-pole polynomial function in F, as given by H(F)=1 D(F)=1 FN+1+u1FN+u2FN−1+···+u2F2+u1F+1,(6) where uk(k= 1, 2, …) denotes the coefficients of D(F). Recall that the classical Pthorder (P= 1, 2, …) Butterworth filter has an analogous representation in the s-domain [24], as given by HP(s)=1 sP+a1sP−1+a2sP−2+···+a2s2+a1s+1.(7) The objective function for the constrained optimization problem to achieve a stable FOBF model in the F-plane is formulated as f= L  i=120 log10 HB(F1/β ,ω i)HB(−F1/β ,ω i)−20 log10 H2(F,ω i,X)  ,(8) Subject to: θD(F)>θ C, where the frequency variable ωis varied with logarithmic spacing for L=100 sample points between 10−2−102rad/s; θC=0.5βπ denotes the critical phase (in rad); θD(F)represents the minimum absolute phase of the roots of D(F);and the decision variables vector is given by X=[u1u2···uN/2],N:Even [u1u2···u(N+1)/2].N:Odd.(9) Thus, the dimension (d) of the problem is N/2 for even values of Nand (N+1)/2 if Nis an odd number. To maintain conformity with the classical Butterworth filter, the lower bound of Xis set as 0 and the upper bound (Ub) is chosen as the values of the coefficients for the Pth-order Butterworth filter, where P=N+1,as given by Ub =[a1a2···aN/2],N:Even [a1a2···a(N+1)/2].N:Odd.(10) Finally, the s-domain transfer function of the proposed FOBF (normalized) can be obtained as H(s)=1 s(N+1)β +u1sNβ+u2s(N−1)β +···+u2s2β+u1sβ+1.(11) 123 Optimized fractional-order Butterworth filter… Table 1 Decision variables vector and pole locations of the proposed FOBFs MX Roots of D(F) 1.5 [0.6400]−0.3200 ±0.9474i 2.5 [1.2325]−1,−0.1162 ±0.9932i 2.8 [1.7030]−1,−0.3515 ±0.9362i 3.2 [1.2068 2.2975]−0.1727 ±0.9850i,−0.4307 ±0.9025i 3.6 [2.0929 2.5454]−0.1525 ±0.9883i,−0.8939 ±0.4482i 4.2 [1.8412 2.8654]− 1,−0.4057 ±0.9140i,−0.0149 ±0.9999i 3 Performance analysis using MATLAB simulations The applicability of metaheuristic algorithms for solving constrained optimization problems is well established in the literature [14]. The search process of these algorithms involves a combination of stochastic and deterministic rules. These methods employ multiple agents that explore the hyper-dimensional problem search space in the initial stages, followed by local search in the later stages. This is unlike the ‘fminsearch’ function in MATLAB employed in [25], where the unconstrained optimization technique involves a direct search strategy based on the Nelder–Mead simplex algorithm. The iterative nature of the metaheuristic search process may lead to higher computational time as compared to the traditional optimization algorithms. It is worth noting that metaheuristic algorithms can only achieve a near-global optimal solution. Furthermore, a closed-form solution is also not attained by a metaheuristic technique. However, these methods have established a niche due to their ease of implementation and ability to effectively handle linear or non-linear, convex or non-convex, multimodal, multidimensional, unconstrained, and constrained optimization problems. The proposed optimization technique is implemented in MATLAB using the constrained composite differential evolution (C2oDE) algorithm [27]. C2oDE has demonstrated effective performance in solving the benchmark optimization problems provided in IEEE CEC2006 and CEC2010 test suites. In this work, the basic parameters for C2oDE are set as population size = 100 and termination condition = 10000d function evaluations, while the internal parameters are the same as specified in [27]. The accuracy and stability of the proposed FOBFs are investigated by considering various design orders. Table 1presents the optimal value of Xobtained for design cases such as M= 1.5, 2.5, 2.8, 3.2, 3.6, and 4.2. The following observations can be made about the pole locations and stability of the designed FOBFs based on the results shown in Table 2: 1. The roots of D(F) exhibit a magnitude of 1, which implies that all the poles of the proposed FOBF reside on the unit circle in the F-plane. A comparable situation exists for the pole locations of the classical Butterworth filter (normalized), where the poles are located on the unit circle in the left-half s-plane [24]. 2. The region of stability in the F-plane is larger than in the s-plane for β<1.All the proposed models achieve stability in the F-plane since all the poles of H(F) have a negative real part. Note also from Table 2that θD(F)>θ Cfor all the 123 S. Mahata et al. Table 2 Phase of the roots of D(F), critical phase and minimum absolute phase angle of the roots of the proposed FOBFs M∠Roots of D(F)θ C(◦)θD(F)(◦) 1.5 ±108.66◦67.50 108.66 2.5 ±96.67◦,180◦74.97 96.67 2.8 ±110.58◦,180◦83.98 110.58 3.2 ±99.94◦,±115.51◦72.00 99.94 3.6 ±98.77◦,±153.37◦81.00 98.77 4.2 ±90.85◦,±113.93◦,180◦75.60 90.85 Fig. 1 Locations of poles in the F-plane for the proposed 2.8th-order Butterworth filter cases, which further confirms the attainment of stability criteria in the F-domain for all the proposed FOBFs. 3. For odd values of N, the poles of H(F) exist in complex conjugate form, whereas, for even values of N, a pole also resides at F=−1. As a representative, the locations of poles in the F-plane for the proposed 2.8th-order Butterworth filter are presented in Fig. 1. Table 3shows the proposed models after transformation from F-domain to the s-domain. If Mis an irrational number or it leads to the occurrence of a recurring decimal for β, then βmay be truncated to three digits after the decimal point (viz., H(s)forM= 2.5, 2.8). The absolute relative magnitude error (ARME) metric, as defined below, is used to evaluate the modeling accuracy. ARME = |H(jω)|−|HB(jω)| |HB(jω)| .(12) 123 Optimized fractional-order Butterworth filter… Table 3 s-Domain transfer function of the proposed FOBFs MH(s) 1.5 1 s1.5+0.6400s0.75 +1 2.5 1 s2.5+1.2325s1.666 +1.2325s0.833 +1 2.8 1 s2.8+1.7030s1.866 +1.7030s0.933 +1 3.2 1 s3.2+1.2068s2.4+2.2975s1.6+1.2068s0.8+1 3.6 1 s3.6+2.0929s2.7+2.5454s1.8+2.0929s0.9+1 4.2 1 s4.2+1.8412s3.36 +2.8654s2.52 +2.8654s1.68 +1.8412s0.84 +1 Fig. 2 Comparison of magnitude and ARME responses with the FOBFs published in the literature for 2.8th-order Butterworth filter Figure 2(top) shows the magnitude response comparison of the proposed 2.8thorder Butterworth filter with the corresponding model published in [18,25]. The reported model in [18] exhibits peaking near the cut-off frequency, whereas the rolloff behavior of the FOBF in [25] deviates from the theoretical one. These findings are further confirmed from the ARME comparison plots presented in Fig. 2(bottom). The magnitude response of the proposed 3.2th-order Butterworth filter is compared with that of the published literature [18], as shown in Fig. 3. The reported design exhibits peaking in the magnitude response near the normalized cut-off frequency, whereas, the proposed model stays in proximity to the theoretical characteristic throughout the design range. The maximum (max) and mean (evaluated using 1000 sampled frequency points with logarithmic spacing in the interval [10−3,103] rad/s) ARME comparisons with 123 S. Mahata et al. Fig. 3 Comparison of magnitude responses with the FOBF published in [18] for the 3.2th-order Butterworth filter Table 4 Maximum and mean ARME comparisons with the FOBFs reported in the recent literature MFOBF Model Max ARME Mean ARME 1.5 [18] Pole-placement 0.07590 0.02862 Optimal (Present work) 0.03293 0.01402 2.5 [18] Pole-placement 0.30990 0.05870 [25] Pseudo-differential 0.95980 0.25870 Optimal (Present work) 0.18770 0.02644 2.8 [18] Pole-placement 0.71540 0.10870 [25] Pseudo-differential 0.39190 0.13560 Optimal (Present work) 0.04280 0.01105 3.2 [18] Pole-placement 0.43650 0.04131 Optimal (Present work) 0.03361 0.01443 3.6 [18] Pole-placement 0.15120 0.03979 Optimal (Present work) 0.08938 0.02435 4.2 [18] Pole-placement 0.42250 0.06142 Optimal (Present work) 0.29690 0.03460 the recently published literature are presented in Table 4. The proposed FOBFs achieve the least error for all the considered cases, thus demonstrating superior accuracy. The ARME comparison plots with [18]forM= 1.5, 2.5, 3.2, 3.6, and 4.2 are presented in Fig. 4to justify the improved accuracy of the proposed approach graphically. Further investigations are conducted by considering the design of (2+α)-order Butterworth filter with αvarying from 0.01 to 0.99 in steps of 0.01. Results shown in 123 Optimized fractional-order Butterworth filter… Fig. 4 ARME comparison plots between the proposed optimal FOBFs (solid blue) and the FOBFs designed using the W-plane-based pole-placement method in [18] (dashed red) Fig. 5 Variations of u1,θC,θD(F), and comparison with those in [25] for the proposed (2+α)-order Butterworth filters Fig. 5demonstrate that: (i) coefficient u1increases as αis increased and u1approaches 2asMapproaches 3. It is noteworthy that the third-order Butterworth polynomial is given by s3+2s2+2s+1[24], (ii) θD(F)exceeds θCthroughout the design range, which justifies the stability criterion in the F-plane for the proposed FOBFs, and (iii) the proposed designs outperform the results in [25] regarding the mean ARME for M∈[2.10,2.99]. 123 S. Mahata et al. Acknowledgements This article is based upon work from COST Action CA15225, a network supported by COST (European Cooperation in Science and Technology). Declarations Conflict of interest The authors declare that they have no conflict of interest. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Acharya, A., Das, S., Pan, I., Das, S.: Extending the concept of analog Butterworth filter for fractional order systems. Signal Process. 94, 409–420 (2014). https://doi.org/10.1016/j.sigpro.2013.07.012 2. Adhikary, A., Sen, S., Biswas, K.: Practical realization of tunable fractional order parallel resonator and fractional order filters. IEEE Trans. Circ. Syst. I 63(8), 1142–1151 (2016). https://doi.org/10.1109/ TCSI.2016.2568262 3. 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