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Exponential Tracking Control of Hydraulic Proportional Directional Valve and Cylinder via Integrator BacksteppingJ. Chen†, W. E. Dixon‡, J. R. Wagner†, D. M. Dawson††Departments of Mechanical and Electrical/Computer EngineeringClemson University, Clemson, SC 29634 ‡Oak Ridge National Laboratory, Oak Ridge, TN 37831
X Directional Valve and Cylinder via Integrator Backstepping
Hydraulic System Model – Cylinder DynamicsExternally applied load
(neglected for simplicity)
(spring / damping term)
Remark: The hydraulic cylinder is assumed to be constructed such that some
volume always remains in the piston and rod sides of the cylinder
P
z
P
S
T
m
s
P
P
R
P
Q
Q
R
P
(solenoid control force)
neglected for simplicity
(spring / damping term)
Control Voltage Directional Valve and Cylinder via Integrator Backstepping
Proportional Solenoid
Hydraulic Valve
Spool Position
LVDT
Bosch NG6 Servo  Solenoid Control Valve1 Directional Valve and Cylinder via Integrator Backstepping
h(2, z)
1
(1+st)
R
Fg
VL
VS
s
+

f(, z)
g(VL)
z
id
ir
+
+
VR
i
Hydraulic System Model – Solenoid Modelreal model
where
Remark: Supply and tank pressures are assumed to satisfy the following inequalities
where is a positive control gain
Obtained by adding and subtracting the desired hydraulic flow force
(Auxillary force tracking error signal)
where,
Desired spool position function
where,
here, the notation denotes the partial derivative of with respect to
(Spool velocity tracking error)
which allows us to write the closed loop dynamics for as follows
where,
Cylinder Dynamics openloop error system can be written as
Pressure Dynamics
Electrical Dynamics
Solenoid Dynamics
Pressure Control
Control StrategyPressure
Position
Force
Flow
Control
Voltage
Desired
Position
Error
Desired
Force
Desired
Flow
Desired
Pressure
Cylinder Control
Solenoid Control
Electrical Control
Controller
0.02 openloop error system can be written as
0.15
0.01
0.1
Cylinder Position (m)
Position Error (m)
0
0.05
0.01
0
0
20
40
60
0
20
40
60
time(s)
time(s)
0.02
0.15
0.01
0.1
Cylinder Position (m)
Position Error (m)
0
0.05
0.01
0
0
20
40
60
0
20
40
60
time(s)
time(s)
Numerical ResultsPD Controller
Nonlinear Controller
900 openloop error system can be written as
900
850
850
Pressure Pr (psi)
Pressure Pp (psi)
800
800
750
750
700
700
0
20
40
60
0
20
40
60
time(s)
time(s)
900
900
850
850
Pressure Pp (psi)
Pressure Pr (psi)
800
800
750
750
700
700
0
20
40
60
0
20
40
60
time(s)
time(s)
Numerical ResultsPD Controller
Nonlinear Controller
PD Controller openloop error system can be written as
10
Solenoid voltage (Volts)
5
0
0
10
20
30
40
50
60
time(s)
Nonlinear Controller
10
5
Solenoid voltage (Volts)
0
0
10
20
30
40
50
60
time(s)
Numerical ResultsControl Valve
Piston side Pressure Transducer
Rod side Pressure Transducer
LVDT
Cylinder