Mathematical Modeling of Erection System 1. Model of the Electro-Hydraulic System
TELKOMNIKA ISSN: 1693-6930
Mathematical Modeling and Fuzzy Adaptive PID Control of Erection … Feng Jiangtao
255 the traditional PID control and fuzzy control has been applied to solve many engineering
problems [12]. This paper is organized as follows. Mathematical model of the electro-hydraulic system
is derived. Kinetic analysis of erection mechanism is accomplished. The relationships of erection force, erection angle and the length of hydraulic cylinder are obtained. Fuzzy adaptive
PID controller is designed in combination of fuzzy logic and PID control method. Models of erection system and fuzzy adaptive controller are established in Simulink. Experimental studies
are completed on laboratory equipment. The aim of this research is to investigate the effect of fuzzy adaptive PID controller on erection mechanism.
2. Mathematical Modeling of Erection System 2.1. Model of the Electro-Hydraulic System
Erection mechanism is mainly composed of hydraulic system and mechanical system. Mathematical models of each system are established separately. Hydraulic system includes
hydraulic pump, relief valve, hydraulic cylinder and electro-hydraulic proportional valve, as is shown in Figure 1.
Figure 1. Hydraulic Principle of Erection System The pump model can be built by the following equation.
p p
p v
Q D w
1
Where D
p
is pump displacement. w
p
is rotational speed of the motor.
v
is volumetric efficiency. The model of the electro-hydraulic proportional directional valve is expressed by the
following formulas.
1 1
2
s d
x
p p
q C A
2
2 2
2
d y
p q
C A
3
Where q
1
and q
2
are fluid flows from and to cylinder. p
s
is the hydraulic supply pressure. A
x
and A
y
are the spool valve areas. C
d
is the discharge coefficient. ρ is the fluid density.
The hydraulic cylinder model is built using the cavity node method. The dynamic equation of the hydraulic cylinder is
described by the second Newton’s Law.
01 1
1 1
1 1
2 ec
ic ic
e
V A y dp
dy q
A C
C p
C p dt
dt
˄ ˅
4
5
M
1 4
3 2
ISSN: 1693-6930
TELKOMNIKA Vol. 15, No. 1, March 2017 : 254
– 263 256
02 2
2 2
2 2
1 ec
ic ic
e
V A y dp
dy q
A C
C p
C p dt
dt
˄ ˅
5
2 1
1 2
2 2
d y p A
p A m
F dt
6 Where A
1
is cross-sectional area of the piston and A
2
is the annular area of piston rod chamber. y is the piston displacement. V
01
and V
02
are the initial volumes of the two cylinder chambers. A
1
y and A
2
y represent the flow rates as a function of volume change due to the piston motion. β
e
is the fluid bulk modulus. p
1
and p
2
are the pressures in the forward and return cylinder chambers.C
ic
and C
ec
are the internal and external leakage coefficients. m is mass of the load. F is the external force.
2.2. Kinetic Analysis of Erection Mechanism Mechanical part of erection mechanism is shown in Figure 2. The load revolves around
the point P
2
driven by erection cylinder.
Figure 2. Kinetic Model of Erection Mechanism We can obtain the following equation through Euler dynamic equations in coordinate
oxy.
2 4
2 5
p
J t
F t P P
G P P
7 Where J is the moment of inertia of the load to point P
2
. t is the erection angle. F
p
t is the thrust force of erection cylinder. G is gravity of the load.
Geometry relationships are described by the following equations.
2 3
2 4
2 2
3 2
3
sin =
sin P P
t P P OP
P OP OP
OP
8
2 5 2
= cos
G
P P P P t
9 The relationship of erection angle and cylinder length can be acquired by using cosine
theorem in ΔOP
2
P
3
.
2 2
2 2
3 2
3 2
3 2
cos 2
P P OP
OP t
P P OP
10
y x
F
p
t G
O
P
3
P
2
P
G
P
4
P
5
θt
α β
TELKOMNIKA ISSN: 1693-6930
Mathematical Modeling and Fuzzy Adaptive PID Control of Erection … Feng Jiangtao
257