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 rela-
tionships 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 invari-
ant systems based on fractional-order differential equations have been discussed numerically in both time and frequency domains to validate the previous fundamentals. |
