Abstract
This paper introduces new biquadratic approximation algorithms to approximate fractional-order Laplacian operators of order \(\alpha \,;\,s^{ \pm \alpha } , 0 < \alpha \le 1\), by finite-order rational transfer functions. The significance of this approach lies in developing an algorithm that depends only on \(\alpha\), which enables one to synthesize both fractional-order inductors and capacitors. The fundamental biquadratic transfer function enjoys a flat phase characteristics at its corner frequency. The bandwidth of approximation can be extended by cascading several biquadratic modules, each centered at calculated corner frequency. The proposed approach is straightforward and outperforms similar approximation algorithms. The equal orders of the biquadratic structure allow one to synthesize both fractional-order inductors and fractional-order capacitors using passive circuits. The same approach can easily be extended to design active filters, or fractional-order proportional-integral-derivative controllers. The main ideas of the proposed method are demonstrated using several numerical examples.
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References
Krishna, B. T. (2011). Studies on fractional order differentiators and integrators: A survey. Signal Process, 91(3), 386–426.
Pudlubny, I. (1999). Fractional differential equations. New York: Academic Press.
Oldham, K. B., & Spanier, J. (1974). Fractional calculus. New York: Academic Press.
Heaviside, O. (1950). Electromagnetic theory (pp. 128–129). New York: Dover Publications.
Samko, S. G., Kilbas, A. A., & Marichev, O. I. (1993). Fractional integrals and derivatives. Theory and Applications. Amsterdam: Gordon and Breach.
El-Khazali, R. (2013). Fractional-order PI λ D μ controller design. Computers & Mathematics with Applications, 5(66), 639–646.
Vinagre, B. M., Podlubny, I., Hernandez, A., & Feliu, V. (2000). Some approximation of fractional-order operators used in control theory and applications. Fractional Calculus and Applied Analysis, 3(3), 241–248.
Freeborn, T.J., Maundy, B., & Elwakil, A. (2010). Second-Order Approximation of the Fractional Laplacian Operator for Equal-Ripple Response, 53rd IEEE International Midwest Symposium (MWSCAS), 2010.
Chen, Y. Q., Vinagre, B. M., & Podlubny, I. (2004). Continued fraction expansion approaches to discretizing fractional order derivative: An expository review. Nonlinear Dynamics, 38, 155–170.
Charef, A., Sun, H., Tsao, Y., & Onaral, B. (1992). Fractal system as represented by singularity function. IEEE Transactions on Automatic Control, 37, 1465–1470.
Oustaloup, A., Levron, F., & Mathieu, B. (2000). Frequency-band complex noninteger differentiator: Characterization and synthesis. IEEE Transactions on Circuits and Systems I, 47, 25–39.
Gao, Zhe., & Liao, Xiaozhong. (2012). Improved Oustaloup approximation of fractional-order operators using adaptive chaotic particle swarm optimization. Journal of Systems Engineering and Electronics, 23(1), 145–153.
El-Khazali, R., Biquadratic Approximation of Fractional-Order Laplacian Operators. IEEE 56th International Midwest Symposium on Circuits and Systems (MWSCAS) (pp. 69–72), August 4–7, 2013, Ohio, USA.
Monje, C. A., Chen, Y., Vinagre, B. M., Xue, D., & Feliu-Batlle, V. (2010). Fractional-order systems and control. London: Springer.
Miller, K. S., & Ross, B. (1993). An introduction to the fractional calculus and fractional differential equations. New York: Wiley.
Freeborn, T., Maundy, B., & Elwakil, A. S. (2010). Field programmable analog array implementation of fractional step filters. IET Circuits, Devices & Systems, 4, 514–524.
Ahmad, W., El-Khazali, R., & Elwakil, A. S. (2001). Fractional-order Wien-bridge oscillator. IET Electronic Letters, 37(18), 1110–1112.
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El-Khazali, R. On the biquadratic approximation of fractional-order Laplacian operators. Analog Integr Circ Sig Process 82, 503–517 (2015). https://doi.org/10.1007/s10470-014-0432-8
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DOI: https://doi.org/10.1007/s10470-014-0432-8