New Approach to Realize Fractional Power in $z$-Domain at Low Frequency

Optimizing Continued Fraction Expansion Based IIR Realization of Fractional Order Differ-Integrators with Genetic Algorithm

“…without confining the study in semidifferintegrators only. The concept proposed in this paper can also be extended for higher order operators like [17], [18], [2] in future research.…”

“…Also, as the order of generating function increases, the region of performance in the frequency domain gets shrinked and also order of the IIR filter will be high. So we have restricted this up to second order realization only unlike [17]- [18] and optimized within a chosen structure like Al-Alaoui [19] to obtain an optimum generating function. The generalized AlAlaoui type generating function (6) is ideal for applications where the requirement is to maximize accuracy without going for a higher order realization.…”

“…The mapping relation or formula for conversion from continuous time to discrete time operator ( s z ↔ ) is known as the generating function. Among the family of expansion methods, CFE based digital realization has been extensively studied with various types of generating functions like Tustin [10], Simpson [11], Al-Alaoui [12], mixed TustinSimpson [13], mixed Euler-Tustin-Simpson [14], impulse response based [15] and other higher order generating functions [16]- [18]. This paper focuses on the CFE based realization of the fractional order differ-integrators with an optimization based approach for the mixed type generating functions.…”

Optimization of Integer Order Integrators for Deriving Improved Models of Their Fractional Counterparts

“…(3) In the third step, this expanded series is truncated up to some arbitrary order ( ), which defines the order of the new fractional operator ( ± ). Direct discretization [17] works with the help of different expansion techniques such as Taylor Series Expansion (TSE) [18], Power Series Expansion (PSE) [19,20], and Continued Fraction Expansion (CFE) [21,22], whereas for indirect discretization [23][24][25][26] different rational approximations have been developed by many mathematicians, namely, Carlson and Halijak [27], Steiglitz [28], Khovanskii [29], Roy [30], Oustaloup et al [31], and Maione [32], which proved like boon for the development of fractional integrators/differentiators.…”

Improved Digital Rational Approximation of the Operator $$S^{\alpha }$$ S α Using Second-Order s-to-z Transform and Signal Modeling

“…The obtained approximations have been designed using PSE-truncation approach and compared with Al-Alaoui (CFE)-based approximations. These higher order s-to-z transforms have been modified in [32] to improve the approximations in the low frequencies. However, simulation results showed that the Schneider operator and Al-Alaoui-SKG (PSE-truncation) based approximations possess complex conjugate poles and zeros which may be not desirable since it is well known that, for a better fit to continuous frequency response of s α , it would be of high interest to obtain discrete approximation with poles and zeros distributed in alternating fashion on the real axis in the unit circle in the z plane [25,31].…”