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

Abstract

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.

Time Dimensional Consistency Aware Analysis of Voltage Mode and Current Mode Active Fractional Circuits

“…For studying the stability of any linear fractional system, its pole location on either the F-plane or Wplane must be determined as the applicability of splane ceases to be valid [13], [36]. These planes can be respectively depicted in Fig.…”

Fractional operator without singular kernel: Applications to linear electrical circuits

“…Taking the inverse Laplace transform to Equation 35, the solution is given by Equation 29 may be rewritten as follows:…”

“…The first operator is the derivative of the convolution of a given function and a power-law kernel, and the second one is the convolution of the local derivative of a given function with power-law function. In other studies, [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] authors developed electrical circuits models using FC; they used the Riemann-Liouville or Liouville-Caputo fractional-order derivative operators. The Liouville-Caputo operator is more suitable for modeling real-world problems since it allows using initial conditions.…”

“…[19][20][21][22][23][24][25] Due to FC is one of the most powerful mathematical tools used in the recent decades to model real-world problems, several existing electrical circuits models have been generalized, and fractional derivatives models have been developed to represent the behavior of fractional linear electrical systems, for the designing of analog and digital filters, as well as to describe the magnetically coupled coils or the behavior of circuits and systems with memristors, meminductors, or memcapacitors. In other studies, [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] authors developed electrical circuits models using FC; they used the Riemann-Liouville or Liouville-Caputo fractional-order derivative operators.…”

The minimax approach for a class of variable order fractional differential equation

“…Therefore, the numerical methods for these equations have undergone fast growth in recent years. These methods are based on an optimization technique, Laplace transforms, spectral techniques, and iterative techniques . Other techniques are based on the operational matrix of fractional calculus of orthogonal polynomials with integer orders (Lucas polynomials, Legendre polynomials, and other) .…”