Instrumentation & Measurement Magazine 24-2 - 113

projected pints Pˆ is computed using the translated variables
from:





T


Pˆ 0.5Sˆ  0.5 Xˆ T0 .J1,n1 	(28)

	

The projected points are then translated back to the original
search space using:





T

T
P
Pˆ   a1 a2  an  .J1,n1 	(29)

	

For all points in P, the objective function is evaluated and
one of the points is chosen as P0 which is then added to the linearly growing memory vector M.
For r = 2, the point that leads to the minimum objective
function is P0. For r = 3, ..., R, the point that leads to the minimum objective function and is not an element of M is P0. The
initial simplex for all the next runs is calculated from:

	


IS

 P .J 
T
0

1,n1

0 0

 1 0
T
  0 2



0 0


0 
0 
0 
 
0 

0

0
0 	(30)

0
 n 

where for each of i dimensions (i = 1, ..., n):

xi  max  xi  min
    P0  exceeds  X c
 


i  
    	(31)
 xi  max  xi  min      X  equal  or  exceeds  P  
c
0



	

This multimodal optimization algorithm stops either
when one of the found minima is less than or equal to the
desired minimum, or when all projected points already exist in memory vector in which case it selects the optimal
local minimum found as the global minimum. Otherwise,
the algorithm runs to finish the predetermined maximum

number of restarts and then points with the minimum objective function.
For our FOPID controller, the five parameter Λ = [Kp, Ki, λ, Kd, μ]
from (22) are tuned based on the objective function of an error
and jerk as shown in (32) for each simulation of the program:
	

 errori2
 jerki2
 wJ
	(32)
n
n


OF  w E

where wE and wJ are the error and jerk term weights, and n is
the number of samples.

Experimental Results
In this section, first, the parameters of the FOPID controller are
tuned using the modified Nelder-Mead (MNM) algorithm. Based
on the tuned parameters, the controller is used to control the behavior of the EHA. Then, the tuned FOPID controller is used for
controlling the experimental setup and the results are discussed.
The parameters of the excavator system [11] are listed in
Table 2. These parameters are used in the simulations and the
Simulink model of the system.

Parameter Tuning of the FOPID Controller
In order to tune the parameters using the MNM algorithm, a
Simulink model is used with fS = 1 kHz sampling frequency for
a duration of 45 seconds. The objective function is as described
by (32). Fig. 3 shows a block diagram of the simulation. This
simulation runs at each point of the MNM algorithm to evaluate the objective function.
Table 3 shows the parameters that are used in the simulations in addition to the parameters of Table 2.
Fig. 4 shows the results of the optimization simulations
for three cases: Case 1: wE = 1, wJ = 0.1111; Case 2: wE = 1.5, wJ =
0.1111, and Case 3: wE = 1, wJ = 0.2111. Fig. 5 describes the same
results, but zoomed around the minima. These figures show
the objective function with respect to the fractional order exponents of the integrator and derivate for the 20 runs.

Table 2 - Parameters of the Excavator System
Parameter

Symbol
3

Value
Nominal
−6

Displacement of the bidirectional pump (m /rev)

Vd

Time constant of the servo motor (1/s)

τm

3

Servomotor gain (rev/(sV))

Km

5.8

2

8×10

Range
2.3-4.0
5.6-6.0

−6

Piston-side area (m )

Aa

Area ratio of the actuator

α

Effective bulk modulus (Pa)

βe

689×10

(356-1030) ×106

Hydraulic compliance (m3/Pa)

C

3.46×10−12

(2.20-7.03) ×10−12

Piston and rod mass (kg)

mrod

10

9-11

Viscous damping coefficient (Ns/m)

f

900

600-1200

Load mass (kg)

mL

-

0-367

Length of effort arm (m)

l1

0.3

-

Length of the load arm (m)

l2

1.2

-

April 2021	

3167×10
0.75

-

6

IEEE Instrumentation & Measurement Magazine	113



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