An Optimal Design of Fractional Order Butterworth Filter With Optimized Magnitude and Phase Response

International Journal of Electronics and Communication Engineering

© 2024 by SSRG - IJECE Journal

Volume 11 Issue 10

Year of Publication : 2024

Authors : Sanjay Ambadas Patil, Uday Pandit Khot

10.14445/23488549/IJECE-V11I10P113

How to Cite?

Sanjay Ambadas Patil, Uday Pandit Khot, "An Optimal Design of Fractional Order Butterworth Filter With Optimized Magnitude and Phase Response," SSRG International Journal of Electronics and Communication Engineering, vol. 11,  no. 10, pp. 161-175, 2024. Crossref, https://doi.org/10.14445/23488549/IJECE-V11I10P113

Abstract:

Frequency responses are frequently referred to in the stability analysis of fractional order control systems. Frequency response-based methods have been introduced in the literature to reduce complex fractional order systems. However, the magnitude and phase response improvement is not handled by these techniques. The optimization methods are utilized to improve the approximation of fractional order filters in the desired frequency range. The article discusses the ideal Fractional-Order Butterworth Filter (FOBF) configuration by utilizing integer-order rational approximations to achieve a precise magnitude and phase response. Additionally, this study utilizes optimal FOBF magnitude and phase characteristics to minimize errors and expands the approximation bandwidth to cover multiple decades in both pass and stop bands. Optimized fractional order Butterworth filter designs have been implemented in biomedical signal processing, audio engineering, telecommunications, and control systems, enhancing performance in noise reduction and signal fidelity across these applications. It is essential to precisely determine these transfer functions regarding frequency and time responses. Therefore, there is room for further enhancements in these approximation methods to reduce errors and enhance the accuracy of real-world implementations. In pursuit of this goal, the study introduced a powerful metaheuristic optimization technique called Enhanced Colliding Bodies Optimization (ECBO), which not only showcases better precision in modelling but also exhibits a higher level of stability when compared to existing methods. This process guides all entities towards an optimal solution in each successive round, boosting the likelihood of finding a superior solution and thoroughly examining the full range of potential solutions. This technique better fits the BF filter function to a fractional-order continuous filter. The digital integrator is optimized in the frequency domain using the Coyote Optimization Algorithm (COA) because of its effectiveness, ease of use, and strength in tackling various complex optimization challenges. This optimizes the magnitude and phase reaction of the low-pass BF filter depending on the Minimum Square Error (MSE). Expanding the scope of the ECBO’s search range allows the enhanced filter to closely replicate the frequency behaviour of fractional order continuous filter functions.

Keywords:

Fractional Order Butterworth Filter, Magnitude Response, Phase Response, Enhanced Colliding Bodies Optimization, Fractional-Order Continuous Filter, and Coyote Optimization.

References:

[1] Tapaswini Sahu, Kumar Biswal, and Madhab Chandra Tripathy, “Time and Frequency Domain Analysis of a Fractional-Order All-Pass Filter,” Advances in Intelligent Computing and Communication, Lecture Notes in Networks and Systems, vol. 430, pp. 263-272, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[2] Vlad Mihaly et al., “Approximating the Fractional-Order Element for the Robust Control Framework,” 2022 American Control Conference, Atlanta, GA, USA, pp. 1151-1157, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[3] Stavroula Kapoulea, Costas Psychalinos, and Ahmed S. Elwakil, “Power Law Filters: A New Class of Fractional-Order Filters without a Fractional-Order Laplacian Operator,” AEU - International Journal of Electronics and Communications, vol. 129, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[4] Suryadi, and Abdul Halim, “Evaluation of Approximation and Reduction Method for Fractional Order Transfer Function,” 2021 17th International Conference on Quality in Research (QIR): International Symposium on Electrical and Computer Engineering, Depok, Indonesia, pp. 5-10, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[5] Imen Deghboudj, and Samir Ladaci, “Automatic Voltage Regulator Performance Enhancement Using a Fractional Order Model Predictive Controller,” Bulletin of Electrical Engineering and Informatics, vol. 10, no. 5, pp. 2424-2432, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[6] Ivo Petráš, “Fractional-Order Control: New Control Techniques,” Fractional Order Systems, vol. 1, pp. 71-106, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[7] Nitisha Shrivastava, and Pragya Varshney, “Efficacy of NPC Technique in Approximation of Fractional Order Differentiators and Integrators,” 2021 2nd International Conference for Emerging Technology, Belagavi, India, pp. 1-6, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[8] Shibendu Mahata et al., “Optimal F-Domain Stabilization Technique for Reduction of Commensurate Fractional-Order SISO Systems,” Fractional Calculus and Applied Analysis, vol. 25, pp. 803-821, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[9] Yiheng Weis et al., “Multiple Fixed Pole-Based Rational Approximation for Fractional Order Systems,” Journal of Dynamic Systems, Measurement, and Control, vol. 143, no. 6, pp. 1-10, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[10] Sahaj Saxena, and Shivanagouda Biradars, “Fractional-Order IMC Controller for High-Order System Using Reduced-Order Modelling Via Big-Bang, Big-Crunch Optimisation,” International Journal of Systems Science, vol. 53, no. 1, pp. 168-181, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[11] Anand Gudnavar, and N. Manjanaik, “A Survey on Energy-Efficient MAC Protocols for Wireless Sensor Networks,” Smart Sensors Measurements and Instrumentation, Lecture Notes in Electrical Engineering, vol. 750, pp. 177-188, 2021. [CrossRef] [Google Scholar] [Publisher Link]
[12] Evisa Memlikai et al., “Design of Fractional-Order Lead Compensator for a Car Suspension System Based on Curve-Fitting Approximation,” Fractal and Fractional, vol. 5, no. 2, pp. 1-11, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[13] P.M. Balasubramaniam et al., “Design of Automotive Accident-Avoidance System at Speed Limit Zone Using GPS,” Innovations in Computer Science and Engineering, Lecture Notes in Networks and Systems, vol. 385, pp. 271-279, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[14] Stavroula Kapoulea, Costas Psychalinos, and Ahmed S. Elwakil, “Double Exponent Fractional-Order Filters: Approximation Methods and Realization,” Circuits, Systems, and Signal Processing, vol. 40, pp. 993-1004, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[15] Harjasdeep Singh, and Durgesh Srivastava, “Sentiment Analysis: Quantitative Evaluation of Machine Learning Algorithms,” 2023 5th International Conference on Smart Systems and Inventive Technology, Tirunelveli, India, pp. 946-951, 2023. [CrossRef] [Google Scholar] [Publisher Link]
[16] Shibendu Mahata, Rajib Kar, and Durbadal Manda, “Optimal Rational Approximation of Bandpass Butterworth Filter with Symmetric Fractional-Order Roll-Off,” AEU - International Journal of Electronics and Communications, vol. 117, 2020.
[CrossRef] [Google Scholar] [Publisher Link]
[17] M. Shanmukhi et al., “Big Data: Query Processing,” Journal of Advanced Research in Dynamical and Control Systems, vol. 10, no. 7, pp. 244-250, 2018.
[Google Scholar]
[18] Stavroula Kapoulea, Costas Psychalinos, and Ahmed S. Elwakil, “Passive Approximations of Double-Exponent Fractional-Order Impedance Functions,” International Journal of Circuit Theory and Applications, vol. 49, no. 5, pp. 1274-1284, 2021. [CrossRef] [Google Scholar] [Publisher Link]
[19] S. Karthik et al., “Crypto AI: Digital Nostalgic Art Generation Using GAN and Creation of NFT Using Blockchain,” International Research Journal of Modernization in Engineering Technology and Science, vol. 4, no. 7, pp. 2408-2413, 2022.
[Google Scholar] [Publisher Link]
[20] Furkan Nur Deniz et al., “Revisiting Four Approximation Methods for Fractional Order Transfer Function Implementations: Stability Preservation, Time and Frequency Response Matching Analyses,” Annual Reviews in Control, vol. 49, pp. 239-257, 2020.
[CrossRef] [Google Scholar] [Publisher Link]
[21] Om Prakash Goswami et al., “Optimal Design of Digital Low-pass Filters Using Multiverse Optimization,” Journal of Electrical Engineering, vol. 75, no. 4, pp.253-257, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[22] Bala Bhaskar Duddeti et al., “Fomcon Toolbox-based Direct Approximation of Fractional Order Systems Using Gaze Cues LearningBased Grey Wolf Optimizer,” Fractal and Fractional, vol. 8, no. 8, pp. 1-20, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[23] Amr M. AbdelAty et al., “Improving Small-signal Stability of Inverter-based Microgrid Using Fractional-Order Control,” International Journal of Electrical Power & Energy Systems, vol. 156, pp. 1-15, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[24] Shibendu Mahata, Rajib Kar, and Durbadal Mandal, “Optimal Design of Fractional-Order Butterworth Filter with Improved Accuracy and Stability Margin,” Fractional-Order Modeling of Dynamic Systems with Applications in Optimization, Signal Processing and Control, Academic Press, pp. 293-321, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[25] Julia Nako et al., “Design of Higher-order Fractional Filters with Fully Controllable Frequency Characteristics,” IEEE Access, vol. 11, pp. 43205-43215, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[26] Andrew Amgad Amin, Mennatullah Mahmoud Albarawy, and Amr Mabrouk, "Survey on Designing Fractional-order Filters: Metaherustic Approach," Fayoum University Journal of Engineering, vol. 6, no. 2, pp. 1-12, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[27] Andrew Amgad et al., “Design of Fractional-order Filters From the Fractional Chebyshev Polynomials: A Meta-Heuristic Approach,” Authorea Preprints, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[28] S. Swain, M.C. Tripathy, and S. Behera, “Realization of Optimized Fractional-Order Symmetric-Slope Bandpass Filter Using SwitchedCapacitors,” Sādhanā, vol. 48, no. 2, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[29] Jiyao Yang et al., “Optimal Design of Digital FIR Filters Based on Back Propagation Neural Network,” IEICE Electronics Express, vol. 20, no. 1, pp. 1-6, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[30] Yotaka TUNGTRAGUL, Atirarj SUKSAWAD, and Worawat SA-NGIAMVIBOOL, “The Optimal Design of Finite Impulse Response High Pass Filter Using Bee Colony Algorithm,” PRZEGLAD ELEKTROTECHNICZNY, vol. 2023, no. 12, 2023.[CrossRef] [Google Scholar] [Publisher Link]