# Rational Approximations of Arbitrary Order: A Survey

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## Abstract

**:**

## 1. Introduction

## 2. Basic Definitions

#### 2.1. Arbitrary-Order Integral and Derivative of Riemann–Liouville

#### 2.2. Arbitrary-Order Integral and Derivative of Grünwald–Letnikov

#### 2.3. Arbitrary-Order Derivative of Caputo

## 3. Rational Approximations in the Frequency Domain

#### 3.1. Oustaloup’s Approximation

#### 3.2. Refined Oustaloup’s Approximation

#### 3.3. Charef’s Approximation Version 1

#### 3.4. Charef’s Approximation Version 2

#### 3.5. Carlson’s Approximation

#### 3.6. Matsuda’s Approximation

#### 3.7. Continued Fraction Expansion Approximation

#### 3.8. Curve-Fitting Approximation

#### 3.9. Modified Stability Boundary Locus (MSBL) Fitting Approximation

## 4. Model Order Reduction

#### 4.1. Pade’s Approximation

#### 4.2. Stochastic Balancing Method

## 5. Results and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Ideal behavior (black line), Oustaloup’s (blue line), refined Oustaloup’s (red line), Charef’s V1 (brown line), ideal behavior (gray line) of (26), Charef’s V2 (maroon line) and Carlson’s (gold line) approximation for $\frac{1}{{s}^{0.5}}$. (

**a**) Magnitude response. (

**b**) Magnitude error. (

**c**) Phase response. (

**d**) Phase error.

**Figure 3.**Ideal behavior (black line), Matsuda’s (blue line), continued fraction expansion (red line), curve fitting (brown line) and MSBL (maroon line) approximation for $\frac{1}{{s}^{0.5}}$. (

**a**) Magnitude response. (

**b**) Magnitude error. (

**c**) Phase response. (

**d**) Phase error.

**Figure 4.**Magnitude and phase responses of reduced order transfer functions by applying Pade’s approximation. (

**a**) Ideal behavior (black line) of $\frac{1}{{s}^{0.5}}$, Oustaloup’s (blue line), refined Oustaloup’s (red line), Charef’s V1 (brown line), ideal behavior (gray line) of $\frac{1}{1+{s}^{0.5}}$, Charef’s V2 (maroon line) and Carlson’s (gold line), (

**b**) ideal behavior (black line) of $\frac{1}{{s}^{0.5}}$, Matsuda’s (blue line), continued fraction expansion (red line), curve fitting (brown line) and MSBL (maroon line), (

**c**) phase responses of (

**a**,

**d**), phase responses of (

**b**).

**Figure 5.**Magnitude and phase diagrams of reduced order transfer functions by applying stochastic balancing method. (

**a**) Ideal behavior (black line) of $\frac{1}{{s}^{0.5}}$, Oustaloup’s (blue line), refined Oustaloup’s (red line), Charef’s V1 (brown line), ideal behavior (gray line) of $\frac{1}{1+{s}^{0.5}}$, Charef’s V2 (maroon line) and Carlson’s (gold line), (

**b**) ideal behavior (black line) of $\frac{1}{{s}^{0.5}}$, Matsuda’s (blue line), continued fraction expansion (red line), curve fitting (brown line) and MSBL (maroon line), (

**c**) phase responses of (

**a**,

**d**), phase responses of (

**b**).

**Table 1.**Maple 18 code of (1).

restart: |

assume(alpha <= 1); additionally(0 < alpha); |

AOI_RL := proc(alpha,f) |

1/GAMMA(alpha)*int(f*(t-tau)${}^{\wedge}$(alpha-1),tau = 0..t,‘AllSolutions’) assuming t > 0; |

end proc: |

**Table 2.**Maple 18 code of (4).

restart: |

assume(p-1 < alpha);additionally(alpha <= p); |

AOD_RL := proc(alpha,f,p) |

diff(1/GAMMA(p-alpha)*int((t-tau)${}^{\wedge}$(p-alpha-1)*f,tau = 0..t),t$p) assuming t >= 0; |

end proc: |

**Table 3.**Maple 18 code of (7).

restart: |

f := subs(t = t − p*h,f): f := unapply(f,t,p,h): |

AOI_GL := proc(alpha,h,tf,a) |

local t,p,Sa,S1 := []: |

for t from 0 by h to tf do |

Sa := 0: |

for p from 0 by 1 to (t-a)/h do |

Sa := Sa + h${}^{\wedge}$alpha*GAMMA(alpha + p)/(p!*GAMMA(alpha))*f(t,p,h): |

od: |

S1 := [op(S1),[t,Sa]]: |

od:return(S1): |

end proc: |

**Table 4.**Maple 18 code of (9).

restart: |

f := subs(t = t − p*h,f): f := unapply(f,t,p,h): |

AOD_GL := proc(alpha,h,tf,a) |

local t,p,Sa,S1 := []: |

for t from 0 by h to tf do |

Sa := 0: |

for p from 0 by 1 to (t-a)/h do |

Sa := Sa + (−1)${}^{\wedge}$p/h${}^{\wedge}$alpha*GAMMA(alpha + 1)/(p!*GAMMA(1 + alpha-p))*f(t,p,h): |

od: |

S1 := [op(S1),[t,Sa]]: |

od:return(S1): |

end proc: |

**Table 5.**Maple 18 code of (11).

restart: |

assume(p − 1 < alpha);additionally(alpha <= p); |

AOD_C := proc(alpha,f,p) |

1/GAMMA(p-alpha)*int(diff(f,tau$p)/(t-tau)${}^{\wedge}$(1 + alpha-p),tau = 0..t) assuming t >= 0; |

end proc: |

restart:Digits := 6: |

Oustaloup := proc(alpha,wl,wh,N) |

local p,z,Ha; |

z := wl*(wh/wl)${}^{\wedge}$((2*k − 1 − alpha)/(2*N)): |

p := wl*(wh/wl)${}^{\wedge}$((2*k − 1 + alpha)/(2*N)): |

Ha := wh${}^{\wedge}$alpha*factor(product((s + z(k))/(s + p(k)),k = 1..N)): |

return(Ha): |

end proc: |

restart:Digits := 6: |

Ref_Oustaloup := proc(alpha,wl,wh,N) |

local p,z,Ha,b := 10,d := 9; |

z := wl*(wh/wl)${}^{\wedge}$((2*k − 1 − alpha)/(2*N)): |

p := wl*(wh/wl)${}^{\wedge}$((2*k − 1 + alpha)/(2*N)): |

Ha := factor((d*wh/b)${}^{\wedge}$alpha*(d*s${}^{\wedge}$2 + b*wh*s)/(d*(1 − alpha)*s${}^{\wedge}$2 + b*wh*s + d*alpha)) |

*factor(product((s + z(k))/(s + p(k)),k = 1..N)): |

return(Ha): |

end proc: |

**Table 8.**Code in Maple 18 environment of (21).

restart:Digits := 6: |

Charef_V1 := proc(alpha,wl,wh,epsilon) |

local p0,p,z,a,b,k,Ha,N; |

p0 := wl*10${}^{\wedge}$(epsilon/(20*alpha)): |

a := 10${}^{\wedge}$(epsilon/(10*(1-alpha))): |

b := 10${}^{\wedge}$(epsilon/(10*alpha)): |

N := ceil(log10(wh/p0)/log10(a*b)): |

p := (a*b)${}^{\wedge}$k*p0: z := (a*b)${}^{\wedge}$k*a*p0: |

Ha := factor(product(1 + s/z,k = 0..N-1)/product(1 + s/p,k = 0..N))*eval(1/s${}^{\wedge}$alpha,s = wl): |

return(Ha): |

end proc: |

**Table 9.**Code in Maple 18 environment of (26).

restart:Digits := 6: |

Charef_V2 := proc(alpha,wl,wh,epsilon) |

local p0,p,z,a,b,k,Ha,N; |

p0 := wl*10${}^{\wedge}$(epsilon/(20*alpha)): |

a := 10${}^{\wedge}$(epsilon/(10*(1-alpha))): |

b := 10${}^{\wedge}$(epsilon/(10*alpha)): |

N := ceil(log10(wh/p0)/log10(a*b)): |

p := (a*b)${}^{\wedge}$k*p0: z := (a*b)${}^{\wedge}$k*a*p0: |

Ha := factor(product(1 + s/z,k = 0..N − 1)/product(1 + s/p,k = 0..N)): |

return(Ha): |

end proc: |

**Table 10.**Code in Maple 18 environment of (29).

restart:Digits := 6: |

Carlson := proc(alpha,G,N) |

local k,Ha := 1: |

for k from 1 to N do |

Ha := Ha*((1-alpha)*Ha${}^{\wedge}$(1/alpha) + (1+alpha)*G)/((1 + alpha)*Ha${}^{\wedge}$(1/alpha) + (1-alpha)*G): |

od: |

return(simplify(expand(numer(Ha))/expand(denom(Ha)))): |

end proc: |

restart:Digits := 6: |

logspace := proc(a,b,n) evalf(10${}^{\wedge}\sim $ <seq(evalf(a)..evalf(b),evalf((b − a)/(n − 1),7))>) end proc: |

Matsuda := proc(alpha,wl,wh,N,k) |

local w,wa,d,c,r,Ha,Dp := []: |

w := convert(logspace(log10(wl),log10(wh),N + 1),list): |

wa := [seq(abs(w[r]${}^{\wedge}$alpha),r = 1..nops(w))]: |

d := matrix(nops(w),nops(w),0): |

for c from 1 to N + 1 do |

d[1,c] := wa[c]: |

for r from 2 to c do |

d[r,c] := (w[c] − w[r − 1])/(d[r − 1,c] − d[r − 1,r − 1]): |

od: |

Dp := [op(Dp),d[c,c]]: |

od: |

Ha := op(nops(Dp),Dp): |

for r from nops(Dp) − 1 by −1 to 1 do |

Ha := Dp[r] + (s-w[r])/Ha: |

od:return(simplify(k*Ha)): |

end proc: |

**Table 12.**Code in Maple 18 environment of (37).

restart:Digits := 6:with(DynamicSystems):with(numtheory): |

CFE := proc(alpha,N) |

local x,Ha:unassign(‘x’,‘s’): |

Ha := cfrac(cfrac((x + 1)${}^{\wedge}$alpha,x,N)):x := s − 1: |

return(expand(numer(eval(Ha)))/expand(denom(eval(Ha)))): |

end proc: |

CF := proc(alpha,wl,wh,N) |

Matlab[setvar](“alpha”,alpha):Matlab[setvar](“wl”,wl): |

Matlab[setvar](“wh”,wh):Matlab[setvar](“N”,N): |

Matlab[evalM](“CF(alpha,wl,wh,N)”); |

end proc: |

Matlab script of (39)–(41) |

function Ha = CF(alpha,wl,wh,N) |

w = logspace(log10(wl),log10(wh)); |

A = (j*w).${}^{\wedge}$alpha; |

Ha = fitfrd(frd(A,w),N); |

[num,den] = ss2tf(Ha.A,Ha.B,Ha.C,Ha.D); |

Ha = minreal(tf(num,den)) |

end |

restart:Digits := 6:with(LinearAlgebra): |

logspace := proc(a,b,n) evalf(10${}^{\wedge}\sim $ <seq(evalf(a)..evalf(b),evalf((b-a)/(n-1),7))>) end proc: |

MSBL := proc(alpha,wl,wh,N) |

local A := Matrix(N),B := Vector[row](N),C,k,r,w,Ha:unassign(‘s’): |

if N = 1 then w[1] := wh: |

else w := convert(logspace(log10(wl),log10(wh),N),list): fi: |

for k from 1 to N do |

for r from 1 to N do |

A[r,k] := (I*w[k])${}^{\wedge}$r-(I*w[k])${}^{\wedge}$(N-r + 2)*cos(Pi/2*alpha)/w[k]${}^{\wedge}$alpha |

–(I*w[k])${}^{\wedge}$(N-r + 1)*sin(Pi/2*alpha)/w[k]${}^{\wedge}$(alpha-1): |

A[r,k] := Re(A[r,k]) + Im(A[r,k]): |

B[k] := –(I*w[k])${}^{\wedge}$(N + 1) + I*w[k]*cos(Pi/2*alpha)/w[k]${}^{\wedge}$alpha |

+sin(Pi/2*alpha)/w[k]${}^{\wedge}$(alpha-1): |

B[k] := Re(B[k]) + Im(B[k]): |

od: |

od:unassign(’k’): |

C := convert(((1/A)${}^{\wedge}$%T.B${}^{\wedge}$%T)${}^{\wedge}$%T,list): |

Ha := sort(sum(1/N + C[k + 1]*s${}^{\wedge}$(N-k),k = 0..N-1),s,descending)/ |

sort(sum(s${}^{\wedge}$N/N + C[N-k]*s${}^{\wedge}$(N-k-1),k = 0..N-1),s,descending): |

return(Ha): |

end proc: |

restart:Digits := 6:with(SignalProcessing): |

Pade := proc(Ha,m,n) |

local A,Hp:unassign(‘s’): |

A := series(Ha,s = 0.200): |

Hp := convert(A,ratpoly,m,n): |

end proc: |

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**MDPI and ACS Style**

Colín-Cervantes, J.D.; Sánchez-López, C.; Ochoa-Montiel, R.; Torres-Muñoz, D.; Hernández-Mejía, C.M.; Sánchez-Gaspariano, L.A.; González-Hernández, H.G.
Rational Approximations of Arbitrary Order: A Survey. *Fractal Fract.* **2021**, *5*, 267.
https://doi.org/10.3390/fractalfract5040267

**AMA Style**

Colín-Cervantes JD, Sánchez-López C, Ochoa-Montiel R, Torres-Muñoz D, Hernández-Mejía CM, Sánchez-Gaspariano LA, González-Hernández HG.
Rational Approximations of Arbitrary Order: A Survey. *Fractal and Fractional*. 2021; 5(4):267.
https://doi.org/10.3390/fractalfract5040267

**Chicago/Turabian Style**

Colín-Cervantes, José Daniel, Carlos Sánchez-López, Rocío Ochoa-Montiel, Delia Torres-Muñoz, Carlos Manuel Hernández-Mejía, Luis Abraham Sánchez-Gaspariano, and Hugo Gustavo González-Hernández.
2021. "Rational Approximations of Arbitrary Order: A Survey" *Fractal and Fractional* 5, no. 4: 267.
https://doi.org/10.3390/fractalfract5040267