Instrumentation & Measurement Magazine 24-2 - 111

Table 1 - Elements of an EHA circuit circuit,
shown in Fig. 2
Element
number

Circuit

Description

1

Main

Actuator

2

Main

Mass of piston and rod

3, 4

Main

Check valve

5, 6

Main

Relief valve

7

Main

Bidirectional pump

8

Main

Servomotor

9

Auxiliary

Auxiliary pump

10

Auxiliary

Relief valve

11

Auxiliary

Tank

V1

Main

Three-position four-way
directional valve

V2

Auxiliary

One-directional flow control
valve

	

	

Q1  Qac  Qa  Aa x p 

Voa  Aa xp

Qbc  Q2 
Qb 
 Ab x p 

e

p b	(7)

Equations (1) through (11) describe the behavior of the system, and are then used as the actual model of the system.

Controller Design
In this section, first the Oustaloup recursive method for
approximation of fractional-order transfer functions is introduced. Then, the modified Nelder-Mead algorithm is used for
parameter tuning of the FOPID controller.

Approximation of Fractional-order Transfer
Functions
Fractional-order derivator and integrators in the Laplace
domain are shown as Sa and S−a, respectively. Thus, a fractional-order integration of input U(s) is obtained from [6]:

Qb   Qa	(9)

When the actuator is retracting (Quadrants III and IV), the
oil flows from either side of the actuator to the tank through
valves V1 and V2 and the relief valve. In Quadrant IV, pL > 0 and
Qac=0. The flow through the orifice at the flow-control valve is:




 U  s 	(12)


where ωu is called the unit frequency gain. Based on (12), the
transmittance (s) is defined as [6]:


 S 
	(13)
D  s  
  
 u
S
The  transfer function can then be limited to a predetermined frequency range [ωA,ωB], by cutting off high and low
S
frequencies. Thus, w is replaced by:

	

u

u

	

and:

April 2021	

 S
Y s  

 u

S

u

1
 C0

1

mrod 
xp  Aa pL  fx p  FL 	(8)
 

where xp is the actuator acceleration, mrod represents the piston
rod mass, and f is the viscous damping coefficient. It should be
noted that the load force FL may vary as the cylinder moves, being either resistant or assistant.
When the actuator is extending as in Quadrant I (pL > 0 and
Qac=0) and Quadrant II (pL < 0 and Qbc=0), the charge pump alternately supplies an oil flow to the cap and rod sides of the
actuator through the flow-control valve, V1. Thus, (9) describes the relationship between the two flows:
	

Qac 
CV  pa  p0  / sg 	(11)

	

	

where Qac and Qbc are the compensation flows coming from the
charge pump through the directional and flow-control valves
to the actuator sides, a and b, respectively; Qa is the flow into
the actuator and Qb is the flow out of the actuator, respectively;
xp is the displacement of the piston rod;  x p is the velocity of the
piston; βe is the effective bulk modulus; and Voa and Vob each
represent the inner pipe and cylinder chamber volumes at the
associate sides of the actuator.
Newton's second law for the actuator piston gives:
	

where CV is valve flow coefficient, sg is the specific gravity of the fluid and p0 is the cracking pressure of the relief
valve. In Quadrant III, pL < 0 and Qbc=0, and Qac can also be
obtained as:

p a	(6)

Vob  Ab xp

e

Qbc 
CV  pb  p0  / sg 	(10)

	

S

b 	(14)
S

h

	

    0.5
 u  b h 

b u 	(15)


C
0
u h


	

b   A	(16)

b  B

where ωb and ωh denote the low-transitional and high-transitional frequencies, respectively.
Substituting (14) into (13) gives:

	


D  s   u

 h








S
 1

b


S
 1
h






 	(17)




IEEE Instrumentation & Measurement Magazine	111



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