Mathematical Physics Vol 1

Chapter A. Fractional Calculus: A Survey of Useful Formulas to make ω ℓ ≪ ω C ≪ ω h (e.g. ω C = √ ω ℓ ω h ). Or, if 1 ∈ [ ω ℓ , ω h ] , calculations can be simplified making ω C = 1 ⇒ C = j α N ∏ m = 1 1 + j ω z , m 1 + j ω p , m

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R CRONE is an acronym of Commande Robuste d’Ordre Non-Entier , Flench for Non-Integer Order Robust Control.

A.7.3 Matsuda approximation

Given the frequency behaviour G ( j ω ) of transfer function G ( s ) (which may be fractional), at frequencies ω 0 , ω 1 ,..., ω N (which do not need to be ordered),

= d 0 ( ω 0 ) ;

s − ω k − 1 d k ( ω k )

N

s − ω 0 d 1 ( ω 1 )+

s − ω 1 d 2 ( ω 2 )+

s − ω 2 d 3 ( ω 3 )+ ···

G ( s ) ≈ d 0 ( ω 0 )+

(A.94)

k = 1

where

ω − ω k

− 1 − 1 (

d 0 ( ω )= | G ( j ω ) | and d k ( ω )=

( k = 1 , 2 ,..., N ) .

ω ) − d k

ω − 1 )

d k

− 1 (

Approximation (A.94) only works if all orders involved are real.

A.7.4 General comments on approximations

The following applies to ail approximations. • s α can be approximated as 1 s − α

R

which may be useful one approximation is stable and

causal and the other is not. • can be approximated as s α = s ⌈ α ⌉ s α −⌈ α ⌉ or as s α = s ⌊ α ⌋ s α −⌊ α ⌋ , to limit approxima tion orders to the [ − 1 , 1 ] range. • Discrete approximations of s α can be converted into continuous approximations and continuous ones into discrete ones using the Tustin method or any other such method. Notice that usually continuous-time approximations outperform discrete approximations. • Transfer function b 1 s β 1 + b 2 s β 2 + ··· + b m s β m a 1 s β 1 + a 2 s β 2 + ··· a + ms β m can be approximated finding approximations for s β 1 , s β 2 ,..., s β m and linearly combining them. But it can also be approximated as a whole, save if the CRONE approximation is used. • To find more information on the topic consider in the section consult, for instance, [14-18] and the references included in it.

A.8 An Introduction to Fractional Vector Operators

In this section we provide some topics about the fractional multidimensional operators. A first capital contribution was introduced in 1936 by Riesz. He generalized the Riemann-Liouville integral looking for a solution for some problem in potencial theory in connection with partial differential equations for parabolic and hyperbolic cases. He gave two n-dimensional integral operators which are known as Riesz potencial (see e.g. [19,20,1,10]). The inverse operator of I α given by Riesz, corresponding to the parabolic case, is usually considered