Instrumentation & Measurement Magazine 24-2 - 112

In the Oustaloup recursive method, to approximate the
fractional order transfer function of (17), a recursive distribution of real poles and zeros is used that leads to:
	

D  s   lim Dm  S 	(18)
M 

and:

	


Dm  S    u

 h







M



k  M

1
1

S
ώk
	(19)
S

k

where ώk and ωk are defined as:

	

 
ώk  b  u 
 
 h



 k  M  0.5
2

2 M 1











 k  M 0.5
2

2 M 1









6.	If the better of contracted points is such that f(xc)  ≤  f(xn+1),
the algorithm replaces x n+1 by x c and makes a new
simplex.
7.	If f(x c)  >  f (x n+1), the algorithm replaces all points of
simplex except x1 by xi  =  x1 + σ(xi − x1) for i = {2,  · · ·,  n + 1}
where σ = 0.5 is shrinkage coefficient.
The Nelder-Mead algorithm terminates when at least one
of the small simplex convergence test, function value convergence test, or no convergence test becomes true.
The method used in this work has been developed by Butt
et al. [7] based on the Guin variation of the Nelder-Mead algorithm. In this method, based on the bound on the designed
variables, a n-dimensional search space is formed. The algorithm then identifies the region of interest S to search for
optimal parameters as follows:

	(20)

	(21)

 V1 
 
V
S   2 	(23)

 
V
 2n 

Thus, if we select M between 2 and 4 for the EHA system [5],
Dm(S) becomes an integer-order transfer function approximating the fractional-order differentiator D(S). Also, it should be
noted that there are 2M+1 numbers of zeros and poles in Dm(S).
To find an approximation of the non-integer integrators,
the fractional order α takes negative values [5]. Finally, the
FOPID controller can be formulated as follows:

where V1 , V1 ,  , V2n are vertices of n-dimensional search space
with the order of 1 × n. This algorithm runs for a preset maximum of R restarts. For the first run (r = 1), the initial simplex is
obtained from:
0 0 0  0


 1 0 0  0 
T
	

IS X Tc .J1,n1   0  2 0  0 	(24)


   0

0 0 0   
n


	

	

 u 

 h 

k  b 

	

G  s 
K p  KiS    KdS  	(22)

The five parameters [Kp, Ki, Kd, μ, λ] of the FOPID controller in
(22) are optimized in the following subsection.

Modified Nelder-Mead (MNM) Algorithm
The Nelder-Mead downhill simplex algorithm [9] is a nonlinear
optimization algorithm which is gradient-free and unconstrained.
The Guin variation of this method [10] incorporates implicit
constraints in the algorithm. This algorithm uses reflection, expansion, contraction, and shrinkage to find local minimizer following
five steps. It uses n+1 vertices x1, x2, ..., xn+1, as initial simplex where
n is the number of parameters to be optimized for minimization of
objective function. The algorithmic steps are as follows:
1.	The algorithm takes the centroid of all points x0 except
xn+1 where f(x1)  ≤  f(x1)  ≤  ...  ≤  f(xn+1).
2.	It calculates the reflected point xr = x0 + α(x0 − xn+1) where α
= 1 is reflection coefficient.
3.	If the reflected point satisfies f(xr)  ≤  f(xn+1), it calculates the
expanded point xe =  x0 + γ(x0 − xn+1) where γ = 2 is expansion.
4.	If the expanded point is such that f(xe)  ≤  f(xr), the algorithm replaces x n+1 by x e and makes a new simplex.
Otherwise, it replaces xn+1 by xr and makes a new simplex.
5.	If f(xe) > f(xn+1), the algorithm computes the contracted
points xc = x0 + ρ(xn+1 − x0) where ρ = 0.5 is contraction
coefficient.
112	





where Xc is the centroid of the search space and J is the matrix
of ones with the order shown in subscript. Furthermore, αi is
obtained from:

	


xi  max  xi  min
     i 
1
 


i  
    	(25)
 xi  max  xi  min      i 2, , n



Thus, by running the Guin augmented variant of the
Nelder-Mead algorithm in each run, the local minimizer X0 is
found and used for constructing the initial simplex for the succeeding run.
If the centroid is at the origin, X0 is projected to 2n points P
as follows:
	





T


P 0.5S  0.5 X T0 .J1,n1 	(26)

If the centroid of the search space does not lie at the origin,
the search space is moved to make the centroid lie at origin:
	

Xˆ c 
X c   a1

a2  an 	(27)

where Xˆ c is the translated centroid, and ai variables are the distances from the origin.
Thus, variables Sˆ and Xˆ 0 are also the translated versions
by the matrix  a1 a2  an . Consequently, the translated

IEEE Instrumentation & Measurement Magazine	

April 2021



Instrumentation & Measurement Magazine 24-2

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