Design of Small-Phase Time-Variant Low-Pass Digital Fractional Differentiators and Integrators
References
- K. Oldham and J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier, 1974.
- H. Sun, Y. Zhang, D. Baleanu, W. Chen, and Y. Chen, “A new collection of real world applications of fractional calculus in science and engineering,” Communications in Nonlinear Science and Numerical Simulation, vol. 64, pp. 213–231, 2018.
- I. Petráš, Fractional-order nonlinear systems: modeling, analysis and simulation. Springer Science & Business Media, 2011.
- J. D. Colín-Cervantes et al., “Rational approximations of arbitrary order: A survey,” Fractal and Fractional, vol. 5, no. 4, p. 267, 2021.
- B. T. Krishna, “Studies on fractional order differentiators and integrators: A survey,” Signal processing, vol. 91, no. 3, pp. 386–426, 2011.
- L. Debnath, “Recent applications of fractional calculus to science and engineering,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 54, pp. 3413–3442, 2003.
- S. Manabe, “The non-integer integral and its application to control systems,” Mitsubishi Denki laboratory reports, vol. 2, no. 2, 1961.
- M. Axtell and M. E. Bise, “Fractional calculus application in control systems,” in IEEE conference on aerospace and electronics, 1990: IEEE, pp. 563–566.
- W. Jifeng and L. Yuankai, “Frequency domain analysis and applications for fractional-order control systems,” in Journal of physics: Conference series, 2005, vol. 13, no. 1: IOP Publishing, p. 268.
- A. Charef, “Analogue realisation of fractional order integrator, differentiator and fractional PI!!!x1D706;D!!!x1D707; controller,” IEE Proceedings-Control Theory and Applications, vol. 153, no. 6, pp. 714–720, 2006.
- N. Ferreira, F. Duarte, M. Lima, M. Marcos, and J. Machado, “Application of fractional calculus in the dynamical analysis and control of mechanical manipulators,” Fractional calculus and applied analysis, vol. 11, no. 1, pp. 91–113, 2008.
- J. Tenreiro Machado and A. Galhano, “A new method for approximating fractional derivatives: application in non-linear control,” ENOC-2008, pp. 1–5, 2008.
- J. Mocak, I. Janiga, M. Rievaj, and D. Bustin, “The use of fractional differentiation or integration for signal improvement,” Measurement Science Review, vol. 7, no. 5, pp. 39–42, 2007.
- G. Li, Y. Li, H. Chen, and W. Deng, “Fractional-order controller for course-keeping of underactuated surface vessels based on frequency domain specification and improved particle swarm optimization algorithm,” Applied Sciences, vol. 12, no. 6, p. 3139, 2022.
- M. Shitikova, “Fractional operator viscoelastic models in dynamic problems of mechanics of solids: A review,” Mechanics of solids, pp. 1–33, 2022.
- F. Özköse, M. Yavuz, M. T. Şenel, and R. Habbireeh, “Fractional order modelling of omicron SARSCoV-2 variant containing heart attack effect using real data from the United Kingdom,” Chaos, Solitons & Fractals, vol. 157, p. 111954, 2022.
- X. Zhang, G. Yang, S. Liu, and A. J. Moshayedi, “Fractional-order circuit design with hybrid controlled memristors and FPGA implementation,” AEU-International Journal of Electronics and Communications, vol. 153, p. 154268, 2022.
- P. Bertsias, S. Kapoulea, C. Psychalinos, and A. S. Elwakil, “A collection of interdisciplinary applications of fractional-order circuits,” in Fractional Order Systems: Elsevier, 2022, pp. 35–69.
- M. Joshi, S. Bhosale, and V. A. Vyawahare, “A survey of fractional calculus applications in artificial neural networks,” Artificial Intelligence Review, pp. 1–54, 2023.
- E. Yumuk, M. Güzelkaya, and İ. Eksin, “A robust fractional-order controller design with gain and phase margin specifications based on delayed Bode’s ideal transfer function,” Journal of the Franklin Institute, vol. 359, no. 11, pp. 5341–5353, 2022.
- G. Gembalczyk, P. Gierlak, and S. Duda, “Modeling and control of an underactuated system for dynamic body weight support,” Applied Sciences, vol. 11, no. 3, p. 905, 2021.
- G. Gembalczyk, P. Domogała, and K. Leśniowski, “Modeling of Underactuated Ball and Beam System—A Comparative Study,” in Actuators, 2023, vol. 12, no. 2: MDPI, p. 59.
- Y. Wang, G. Wang, G. Yao, and L. Shen, “Research on the Characteristics of Operating Non-Uniformity of a High-Pressure Common-Rail Diesel Engine Based on Crankshaft Segment Signals,” IEEE Access, vol. 9, pp. 64906–64917, 2021.
- M. Sugi and K. Saito, “Non-integer exponents in electronic circuits: F-matrix representation of the power-law conductivity,” IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences, vol. 75, no. 6, pp. 720–725, 1992.
- I. Podlubny, I. Petráš, B. M. Vinagre, P. O’Leary, and L’. Dorčák, “Analogue realizations of fractional-order controllers,” Nonlinear dynamics, vol. 29, pp. 281–296, 2002.
- B. Krishna and K. Reddy, “Analysis of fractional order low pass and high pass filters,” Journal of Electrical Engineering, vol. 8, no. 1, pp. 4–4, 2008.
- M. Koseoglu, F. N. Deniz, B. B. Alagoz, and H. Alisoy, “An effective analog circuit design of approximate fractional-order derivative models of M-SBL fitting method,” Engineering Science and Technology, an International Journal, vol. 33, p. 101069, 2022.
- Z. M. Shah, M. Y. Kathjoo, F. A. Khanday, K. Biswas, and C. Psychalinos, “A survey of single and multicomponent Fractional-Order Elements (FOEs) and their applications,” Microelectronics Journal, vol. 84, pp. 9–25, 2019.
- M. E. Van Valkenburg, “Introduction to modern network synthesis,”, 1960.
- M. Saków and K. Marchelek, “Design and optimization of regression-type small phase shift FIR filters and FIR-based differentiators with optimal local response in LS-sense,” Mechanical Systems and Signal Processing, vol. 152, p. 107408, 2021.
- R. Karthick, A. Senthilselvi, P. Meenalochini, and S. Senthil Pandi, “Design and analysis of linear phase finite impulse response filter using water strider optimization algorithm in FPGA,” Circuits, Systems, and Signal Processing, vol. 41, no. 9, pp. 5254–5282, 2022.
- F. Arellano-Espitia, M. Delgado-Prieto, A.-D. Gonzalez-Abreu, J. J. Saucedo-Dorantes, and R. A. Osornio-Rios, “Deep-Compact-Clustering based anomaly detection applied to electromechanical industrial systems,” Sensors, vol. 21, no. 17, p. 5830, 2021.
- Y. Zhou et al., “Transient measurement of mechatronic multi-parameters and kinetics performances for linear piezoelectric motors,” Measurement, vol. 203, p. 112032, 2022.
- D. P. Atherton, N. Tan, and A. Yüce, “Methods for computing the time response of fractional-order systems,” IET Control Theory & Applications, vol. 9, no. 6, pp. 817–830, 2015.
- C. Hwang, J.-F. Leu, and S.-Y. Tsay, “A note on time-domain simulation of feedback fractionalorder systems,” IEEE Transactions on Automatic Control, vol. 47, no. 4, pp. 625–631, 2002.
- R. S. Barbosa, J. T. Machado, and I. M. Ferreira, “Tuning of PID controllers based on Bode’s ideal transfer function,” Nonlinear dynamics, vol. 38, pp. 305–321, 2004.
- A. Oustaloup, F. Levron, B. Mathieu, and F. M. Nanot, “Frequency-band complex noninteger differentiator: characterization and synthesis,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 47, no. 1, pp. 25–39, 2000.
- D. Xue, C. Zhao, and Y. Chen, “A modified approximation method of fractional order system,” in 2006 international conference on mechatronics and automation, 2006: IEEE, pp. 1043–1048.
- A. Charef, H. Sun, Y. Tsao, and B. Onaral, “Fractal system as represented by singularity function,” IEEE Transactions on Automatic Control, vol. 37, no. 9, pp. 1465–1470, 1992.
- G. Carlson and C. Halijak, “Approximation of fractional capacitors (1/s)^(1/n) by a regular Newton process,” IEEE Transactions on Circuit Theory, vol. 11, no. 2, pp. 210–213, 1964.
- F. MatsudaK, “H (infinity) optimized waveabsorbing control-Analytical and experimental results,” Journal of Guidance, Control, and Dynamics, vol. 16, no. 6, p. 1146–1153, 1993.
- A. N. Khovanskii, The application of continued fractions and their generalizations to problems in approximation theory. Noordhoff Groningen, 1963.
- K. Bingi, R. Ibrahim, M. N. Karsiti, S. M. Hassam, and V. R. Harindran, “Frequency response based curve fitting approximation of fractional-order PID controllers,” International Journal of Applied Mathematics and Computer Science, vol. 29, no. 2, pp. 311–326, 2019.
- K. Bingi, R. Ibrahim, M. N. Karsiti, S. M. Hassan, and V. R. Harindran, Fractional-order systems and PID controllers. Springer, 2020.
- C. Sanathanan and J. Koerner, “Transfer function synthesis as a ratio of two complex polynomials,” IEEE transactions on automatic control, vol. 8, no. 1, pp. 56–58, 1963.
- F. N. Deniz, B. B. Alagoz, N. Tan, and M. Koseoglu, “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.
- B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Transactions on power delivery, vol. 14, no. 3, pp. 1052–1061, 1999.
- A. Djouambi, A. Charef, and A. Besançon, “Optimal approximation, simulation and analog realization of the fundamental fractional order transfer function,” International Journal of Applied Mathematics and Computer Science, vol. 17, no. 4, pp. 455–462, 2007.
- G. Maione, “Continued fractions approximation of the impulse response of fractional-order dynamic systems,” IET Control Theory & Applications, vol. 2, no. 7, pp. 564–572, 2008.
- G. Maione, “Closed-form rational approximations of fractional, analog and digital differentiators/integrators,” IEEE journal on emerging and selected topics in circuits and systems, vol. 3, no. 3, pp. 322–329, 2013.
- C.-C. Tseng, “Design of fractional order digital FIR differentiators,” IEEE Signal processing letters, vol. 8, no. 3, pp. 77–79, 2001.
- L’. Dorèák, J. Terpák, I. Petráš, and F. Dorèáková, “Electronic realization of the fractional-order systems,” Acta Montanistica Slovaca, vol. 12, no. 3, pp. 231–237, 2007.
- Y. Chen and B. M. Vinagre, “A new IIR-type digital fractional order differentiator,” Signal processing, vol. 83, no. 11, pp. 2359–2365, 2003.
- B. M. Vinagre, Y. Q. Chen, and I. Petráš, “Two direct Tustin discretization methods for fractional-order differentiator/integrator,” Journal of the franklin institute, vol. 340, no. 5, pp. 349–362, 2003.
- M. Aoun, R. Malti, F. Levron, and A. Oustaloup, “Numerical simulations of fractional systems: an overview of existing methods and improvements,” Nonlinear Dynamics, vol. 38, pp. 117–131, 2004.
- G. Maione, “A digital, noninteger order, differentiator using laguerre orthogonal sequences,” International Journal of Intelligent Control and Systems, vol. 11, no. 2, pp. 77–81, 2006.
- A. Djouambi, A. Charef, and A. V. Besançon, “Approximation and synthesis of non integer order systems,” IFAC Proceedings Volumes, vol. 39, no. 11, pp. 265–268, 2006.
- J. T. Machado, “Analysis and design of fractionalorder digital control systems,” Systems Analysis Modelling Simulation, vol. 27, no. 2-3, pp. 107–122, 1997.
- B. Krishna and K. Reddy, “Design of fractional order digital differentiators and integrators using indirect discretization,” Fractional Calculus and applied analysis, vol. 11, no. 2, pp. 143p-151p, 2008.
- B. Krishna and K. Reddy, “Design of digital differentiators and integrators of order,” World Journal of Modelling and Simulation, vol. 4, no. 3, pp. 182–187, 2008.
- M. A. Al-Alaoui, “Novel approach to designing digital differentiators,” Electronics Letters, vol. 28, no. 15, pp. 1376–1378, 1992.
- M. A. Al-Alaoui, “Novel digital integrator and differentiator,” Electronics letters, vol. 29, no. 4, pp.376–378, 1993.
- J. Le Bihan, “Novel class of digital integrators and differentiators,” Electronics Letters, vol. 29, no. 11, pp. 971–973, 1993.
- M. A. Al-Alaoui, “Novel IIR differentiator from the Simpson integration rule,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 41, no. 2, pp. 186–187, 1994.
- M. A. Al-Alaoui, “Filling the gap between the bilinear and the backward-difference transforms: an interactive design approach,” International Journal of Electrical Engineering Education, vol. 34, no. 4, pp. 331–337, 1997.
- M. A. Al-Alaoui, “Novel stable higher order s-to-z transforms,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 48, no. 11, pp. 1326–1329, 2001.
- A.-A. M. Al, “Al-Alaoui operator and the !!!x1D6FC;-approximation for discretization analog systems,” Facta universitatis-series: Electronics and Energetics, vol. 19, no. 1, pp. 143–146, 2006.
- N. Q. Ngo, “A new approach for the design of wideband digital integrator and differentiator,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 53, no. 9, pp. 936–940, 2006.
- M. A. Al-Alaoui, “Novel approach to analog-to digital transforms,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 54, no. 2, pp.338–350, 2007.
- M. Sakow and K. Marchelek, “Design and optimization of regression-type small phase shift FIR filters and FIR-based differentiators with optimal local response in LS-sense,” Mechanical Systems and Signal Processing, vol. 152, p. 107408, 2021.
- D. Cafagna and G. Grassi, “Fractional-order Chua’s circuit: time-domain analysis, bifurcation, chaotic behavior and test for chaos,” International Journal of Bifurcation and Chaos, vol. 18, no. 03, pp. 615–639, 2008.
- A. Dabiri and E. A. Butcher, “Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods,” Applied Mathematical Modelling, vol. 56, pp. 424–448, 2018.
- A. Dabiri and E. A. Butcher, “Stable fractional Chebyshev differentiation matrix for the numerical solution of multi-order fractional differential equations,” Nonlinear Dynamics, vol. 90, pp. 185–201, 2017.
- K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics, vol. 29, pp. 3–22, 2002.
- W. Deng, “Numerical algorithm for the time fractional Fokker–Planck equation,” Journal of Computational Physics, vol. 227, no. 2, pp. 1510–1522, 2007.
- C. Li, J. Xiong, W. Li, Y. Tong, and Y. Zeng, “Robust synchronization for a class of fractional-order dynamical system via linear state variable,” Indian Journal of Physics, vol. 87, pp. 673–678, 2013.
- M. Saków and K. Marchelek, “Model-free and time-constant prediction for closed-loop systems with time delay,” Control Engineering Practice, vol. 81, pp. 1–8, 2018.
- M. Saków, “Novel robust disturbance observer,” ISA transactions, vol. 104, pp. 255–277, 2020.
- Y. Cui, H. Sun, and L. Hou, “Decentralized event-triggered adaptive neural network control for nonstrict-feedback nonlinear interconnected systems with external disturbances against intermittent DoS attacks,” Neurocomputing, vol. 517, pp. 133–147, 2023.
- D. T. Tena, “Design of disturbance observers based on their implementability in industrial control systems,” Universitat Jaume I, 2022.
- D. Tena, I. Peñarrocha-Alós, and R. Sanchis, “Performance, robustness and noise amplification trade-offs in Disturbance Observer Control design,” European Journal of Control, vol. 65, p. 100630, 2022.
- N. Fink, G. Zenan, P. Szemes, and P. Korondi, “Comparison of Direct and Inverse Model-based Disturbance Observer for a Servo Drive System,” Acta Polytechnica Hungarica, vol. 20, no. 4, 2023.
- V. L. Korupu and M. Muthukumarasamy, “A comparative study of various Smith predictor configurations for industrial delay processes,” Chemical Product and Process Modeling, vol. 17, no. 6, pp. 701–732, 2021.
- S. Chen, W. Xue, and Y. Huang, “Analytical design of active disturbance rejection control for nonlinear uncertain systems with delay,” Control Engineering Practice, vol. 84, pp. 323–336, 2019.
- S. Guo, Y. Liu, Y. Zheng, and T. Ersal, “A Delay Compensation Framework for Connected Testbeds,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 52, no. 7, pp. 4163–4176, 2021.
Language: English
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Submitted on: Oct 3, 2023
Accepted on: Feb 8, 2024
Published on: Jun 23, 2024
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