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Fractional Order Sallen–Key and KHN Filters: Stability and Poles Allocation

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Abstract

This paper presents the analysis for allocating the system poles and hence controlling the system stability for KHN and Sallen–Key fractional order filters. The stability analysis and stability contours for two different fractional order transfer functions with two different fractional order elements are presented. The effect of the transfer function parameters on the singularities of the system is demonstrated where the number of poles becomes dependent on the transfer function parameters as well as the fractional orders. Numerical, circuit simulation, and experimental work are used in the design to test the proposed stability contours.

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References

  1. A. Acharya, S. Das, I. Pan, S. Das, Extending the concept of analog butterworth filter for fractional order systems. Signal Process. 94, 409–420 (2014)

    Article  Google Scholar 

  2. R.P. Agarwal, D. O’Regan, Ordinary and Partial Differential Equations: With Special Functions, Fourier Series, and Boundary Value Problems (Springer, New York, 2009)

    Google Scholar 

  3. R. Alikhani, F. Bahrami, Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations. Commun. Nonlinear Sci. Numer. Simul. 18(8), 2007–2017 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. H. Bateman, A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. 2 (McGraw-Hill, New York, 1953)

    Google Scholar 

  5. B. Bhattacharyya, W.B. Mikhael, A. Antoniou, Design of rc-active networks using generalized-immittance converters. J. Frankl. Inst. 297(1), 45–58 (1974). doi:10.1016/0016-0032(74). 90137–9

    Article  Google Scholar 

  6. R. Caponetto, Fractional Order Systems: Modeling and Control Applications, vol. 72 (World Scientific Publishing Company, Singapore, 2010)

    Google Scholar 

  7. S. Das, Application of generalized fractional calculus in electrical circuit analysis, in Functional Fractional Calculus for System Identification and Controls, (Springer, Berlin, 2008), pp. 157–180

  8. A.S. Elwakil, Fractional-order circuits and systems: an emerging interdisciplinary research area. IEEE Circuits Syst. Mag. 10(4), 40–50 (2010)

    Article  Google Scholar 

  9. P. Fanghella, Fractional-order control of a micrometric linear axis. J. Control Sci. Eng. (2013)

  10. R.H. Fox, J.W. Milnor et al., Singularities of 2-spheres in 4-space and cobordism of knots. Osaka J. Math. 3(2), 257–267 (1966)

    MATH  MathSciNet  Google Scholar 

  11. T.J. Freeborn, B. Maundy, A.S. Elwakil, Field programmable analogue array implementation of fractional step filters. IET Circuits Devices Syst. 4(6), 514–524 (2010)

    Article  Google Scholar 

  12. T.J. Freeborn, B. Maundy, A.S. Elwakil, Fractional step analog filter design, in Analog/RF and Mixed-Signal Circuit Systematic Design, ed. by M. Fakhfakh, E. Tlelo-Cuautle, R. Castro-Lopez. Lecture Notes in Electrical Engineering, vol. 233 (Springer, Berlin, 2013), pp. 243–267

  13. L. Galeone, R. Garrappa, Explicit methods for fractional differential equations and their stability properties. J. Comput. Appl. Math. 228(2), 548–560 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. L.T. Grujić, Non-lyapunov stability analysis of large-scale systems on time-varying sets. Int. J. Control 21(3), 401–415 (1975). doi:10.1080/00207177508921999

    Article  MATH  Google Scholar 

  15. L.T. Grujić, Practical stability with settling time on composite systems. Automatika (Yug), Ljubljana pp. 1–11 (1975)

  16. H.J. Haubold, A.M. Mathai, R.K. Saxena, Mittag-leffler functions and their applications. J. Appl. Math. 235(5), 1311–1316 (2011)

    MATH  MathSciNet  Google Scholar 

  17. A. Hegazi, E. Ahmed, A. Matouk, On Chaos control and synchronization of the commensurate fractional order liu system. Commun. Nonlinear Sci. Numer. Simul. 18(5), 1193–1202 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  18. M.S. Hirano, Y. Miura, Y.F. Saito, K. Saito, Simulation of fractal immittance by analog circuits: an approach to the optimized circuits. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 82(8), 1627–1635 (1999)

    Google Scholar 

  19. T. Kaczorek, New stability tests of positive standard and fractional linear systems. Circuits Syst. 2(4), 261–268 (2011)

    Article  MathSciNet  Google Scholar 

  20. M.P. Lazarevi, Finite time stability analysis of pd fractional control of robotic time-delay systems. Mech. Res. Commun. 33(2), 269–279 (2006)

    Article  Google Scholar 

  21. M.P. Lazarevi, A.M. Spasi, Finite-time stability analysis of fractional order time-delay systems: Gronwalls approach. Math. Comput. Modelling 49(34), 475–481 (2009)

    Article  MathSciNet  Google Scholar 

  22. K. Li, Delay-dependent stability analysis for impulsive BAM neural networks with time-varying delays. Comput. Math. Appl. 56(8), 2088–2099 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. H. Li, Y. Luo, Y. Chen, A fractional order proportional and derivative (FOPD) motion controller: tuning rule and experiments. IEEE Trans. Control Syst. Technol. 18(2), 516–520 (2010)

    Article  Google Scholar 

  24. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993)

    MATH  Google Scholar 

  25. G. Mittag-Leffler, Sur la nouvelle fonction \(E_{\alpha } (x)\). CR Acad. Sci. Paris 137, 554–558 (1903)

    Google Scholar 

  26. K.B. Oldham, J. Spanier, The Fractional Calculus, vol. 1047 (Academic Press, New York, 1974)

    Google Scholar 

  27. G. Peng, Synchronization of fractional order chaotic systems. Phys. Lett. A 363(56), 426–432 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, vol. 198. Access Online via Elsevier (1998)

  29. A.G. Radwan, Stability analysis of the fractional-order \(RL_{\beta }C_{\alpha }\) circuit. J. Fract. Calc. Appl. 3(1), 1–15 (2012)

    Google Scholar 

  30. A.G. Radwan, A.S. Elwakil, A.M. Soliman, Fractional-order sinusoidal oscillators: design procedure and practical examples. IEEE Trans. Circuits Syst. I 55(7), 2051–2063 (2008)

    Article  MathSciNet  Google Scholar 

  31. A.G. Radwan, A.S. Elwakil, A.M. Soliman, On the generalization of second-order filters to the fractional-order domain. J. Circuits Syst. Comput. 18(02), 361–386 (2009)

    Article  Google Scholar 

  32. A.G. Radwan, K. Moaddy, S. Momani, Stability and non-standard finite difference method of the generalized chuas circuit. Comput. Math. Appl. 62(3), 961–970 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  33. A.G. Radwan, K.N. Salama, Fractional-order \(RC\) and \(RL\) circuits. Circuits Syst. Signal Process. 31(6), 1901–1915 (2012)

    Article  MathSciNet  Google Scholar 

  34. A.G. Radwan, A. Shamim, K.N. Salama, Theory of fractional order elements based impedance matching networks. IEEE Microwav. Wirel. Compon. Lett. 21(3), 120–122 (2011)

    Article  Google Scholar 

  35. A.G. Radwan, A.M. Soliman, A.S. Elwakil, A. Sedeek, On the stability of linear systems with fractional-order elements. Chaos Solitons Fractals 40(5), 2317–2328 (2009)

    Article  MATH  Google Scholar 

  36. D. Saha, D. Mondal, S. Sen, Effect of initialization on a class of fractional order systems: experimental verification and dependence on nature of past history and system parameters. Circuits Syst. Signal Process. 32(4), 1501–1522 (2013)

    Article  MathSciNet  Google Scholar 

  37. R. Sallen, E. Key, A practical method of designing rc active filters. IRE Trans. Circuit Theory 2(1), 74–85 (1955)

    Article  Google Scholar 

  38. A.S. Sedra, P.O. Brackett, Filter Theory and Design: Active and Passive (Matrix Publishers, Portland, 1978)

    Google Scholar 

  39. A. Soltan, A.G. Radwan, A.M. Soliman, Butterworth passive filter in the fractional-order, in 2011 International Conference on Microelectronics (ICM), (IEEE, 2011) pp. 1–5

  40. A. Soltan, A.G. Radwan, A.M. Soliman, Fractional order filter with two fractional elements of dependant orders. Microelectron. J. 43(11), 818–827 (2012)

    Article  Google Scholar 

  41. A. Soltan, A.G. Radwan, A.M. Soliman, Measurement fractional order Sallen–Key filters. Int. J. Electr. Electron. Sci. Eng. 7(12), 2–6 (2013)

    Google Scholar 

  42. A. Soltan, A. Radwan, A. Soliman, CCII based KHN fractional order filter, in 2013 IEEE 56th International Midwest Symposium on Circuits and Systems (MWSCAS) (2013), pp. 197–200

  43. A. Soltan, A.G. Radwan, A.M. Soliman, CCII based fractional filters of different orders. J. Adv. Res. 5(2), 157–164 (2014)

    Article  Google Scholar 

  44. M.P. Tripathi, V.K. Baranwal, R.K. Pandey, O.P. Singh, A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions. Commun. Nonlinear Sci. Numer. Simul. 18(6), 1327–1340 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  45. X. Zhang, L. Liu, Y. Wu, The uniqueness of positive solution for a singular fractional differential system involving derivatives. Commun. Nonlinear Sci. Numer. Simul. 18(6), 1400–1409 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Electrical and Electronic Engineering, Newcastle University, Newcastle upon Tyne, UK

    Ahmed Soltan

  2. Department of Engineering Mathematics and Physics, Cairo University, Cairo, Egypt

    Ahmed G. Radwan

  3. Nanoelectronics Integrated Systems Center (NISC), Nile University, Giza, Egypt

    Ahmed G. Radwan

  4. Department of Electronics and Communications Engineering, Cairo University, Cairo, Egypt

    Ahmed M. Soliman

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  1. Ahmed Soltan
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  2. Ahmed G. Radwan
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  3. Ahmed M. Soliman
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Correspondence to Ahmed G. Radwan.

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Soltan, A., Radwan, A.G. & Soliman, A.M. Fractional Order Sallen–Key and KHN Filters: Stability and Poles Allocation. Circuits Syst Signal Process 34, 1461–1480 (2015). https://doi.org/10.1007/s00034-014-9925-z

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  • DOI: https://doi.org/10.1007/s00034-014-9925-z

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