Fundamentals of fractional-order LTI circuits and systems: number of poles, stability, time and frequency responses


This paper investigates some basic concepts of fractional-order linear time invariant systems related to their physical and non-physical transfer functions, poles, stability, time domain, frequency domain, and their relationships for different fractional-order differential equations. The analytical formula that calculates the number of poles in physical and non-physical s-plane for different orders is achieved and verified using many practical examples. The stability contour versus the number of poles in the physical s-plane for different fractional-order systems is discussed in addition to the effect of the non-physical poles on the steady state responses. Moreover, time domain responses based on Mittag-Leffler functions for both physical and non-physical transfer functions are discussed for different cases, which confirm the stability analysis. Many fractional-order linear time invariant systems based on fractional-order differential equations have been discussed numerically in both time and frequency domains to validate the previous fundamentals. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.


Semary M.S., Radwan A.G., Hassan H.N.


control; filters; Fractional-order systems; linear system; physical-plane; poles; stability analysis; time invariant

Document Type



International Journal of Circuit Theory and Applications, Vol. 44, PP. 2114 to 2133, Doi: 10.1002/cta.2215

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