The main aim in this section is to present a coherent basis for establishing fractional operators compatible with the corresponding classic integer order. In particular, the formulation developed in the sequel allows currently used tools like, IR, TF and FR, and includes the standard derivatives obtained when the orders become integers. To look forward FD formulations consistent with the laws of Physics we recall the most important results from the classic calculus. The standard definition of derivative is Df tf t f ==lim h 0 () () l " or Df tf tl() b ==lim h 0 () " ft hf t() () +h . (49) Substituting - h for h interchanges the definitions, meaning that we only have to consider h 0> In this . case, expression (48) uses the present and past values, while (49) uses the present and future values. In the following, we will distinguish the two cases by using the subscripts f (forward-in the sense that we go from past into future, a direct time flow) and b (backward-meaning a reverse time flow). It is straightforward to invert the above equations to obtain 3 Df lim / n=0 and 3 Df lim / " 0 -1 () ()·. b tftnhh h =+ n=0 (51) These relations motivate the following comments: ■ The different time flow shows its influence: the causality (anti-causality) is clearly stated, ■ We have DD ft DD ft ft== and ()== f ff f 11 () DD ft DD ft ft b bb b 11 Dfth a- / n! a f () = lim+ h " 0 Df te h a h " 0 a b () -= jar lim+ / +3 ^-ahn n=0 n=0 -- () (). -- () () We will call D ,fb -1 antiderivative [63]. We generalize the derivatives to the fractional case ft nh - +3 ^-ahn n! () ft nh + () (52) that are at present called Grünwald-Letnikov (forward and backward) derivative, in spite of their first proposal having been done by Liouville [26]. The symbol ^h-a n ^^ ^hh h In applications where -= -= -+= - aa a 1 n P k n 1 k . ss ss s == which gives us two ways to solve the problem. The first reads [26] Df t = #3 a f () x M--a a 1 C() -+M ft d [26]. The second decomposition, ss , gives - aa= - Df tD a () = ff C() M;#3 -+M a x M 1--a () ,d E ft xx () -xx M () . (56) This is called Liouville-Caputo derivative (LC) [8], MMs (57) that constitutes a derivative of the Riemann-Liouville type, that is also called Liouville derivative [30]. In conclusion, from the IR of the differintegrator, 3 difis the Pochhamer representation of the raising factorial: , 46 IEEE CIRCUITS AND SYSTEMS MAGAZINE ferent integral formulations were obtained from where current expressions can be derived, (55), (56), and (57). A fair comparison of the 3 derivatives lead us to conclude that: SECOND QUARTER 2022 aa MM MM --a -1 () ()· = " f tftnhh h (50) where N Z0 ! + is the greatest integer less than or equal - . We can write to a , so thataa1 N1 # . However, we have two alternatives for applying the convolution, avoiding the singularity. Let M Z0 a#! + ft ft h -h () () , (48) the variable t is not a time the constant factor e , in -jar the backward case, can be removed [63]. In such situations, the derivatives can be called left and right respectively and we can show that they verify L [Dfts Fs lr, a ()]( )( ), a Re() , s = !! $ 0 (53) in agreement with the requirement (46). In the rest of this section we consider merely the forward case. It must be highlighted an important fact: equation (52) is valid for any real (or complex) order. The relation (53) suggests another way of expressing the FD. We only have to remember that in (20) we obtained the impulse response of the causal differintegrator. So, the output for a given function, (),ft is given by the convolution Dftftd C() 0 a fb, () = 1 -a #3 --a xx x " 1 () . (54) Relation (54) is an integral formulation of the FD. However, this expression is not as handy as (52) due to the singularity of x a-- 1 at the origin when a 2 0 . Therefore, we adopt it only for negative orders (anti-derivative) and, for the positive (derivative) case we proceed with the regularization of the integral (we will consider the causal case). The regularized Liouville derivative is given by [91] Dftft--= () / 1 f () = # C() a 3 x xx x, (55) --a -a N () () mm -1 ft mGd () m!

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