1. Exponential Tracking Control of Hydraulic Proportional Directional Valve and Cylinder via Integrator Backstepping J. Chen†, W. E. Dixon‡, J. R. Wagner†, D. M. Dawson† †Departments of Mechanical and Electrical/Computer Engineering Clemson University, Clemson, SC ‡Oak Ridge National Laboratory, Oak Ridge, TN 37831

2. Presentation Outline Literature Review Hydraulic System Model
Differentiable Approximation for Fluid Dynamics
Control Objective
Error System Development
Nonlinear Controller Design
Stability Analysis
Numerical Results
Summary

3. Literature Review
Gamble et al. (1994) presented a comparison of sliding mode control with state feedback and PID control for proportional solenoid valves.
Vossoughi et al. (1995) created, and experimentally verified, a globally linearizing feedback control law for electro-hydraulic valves.
Bobrow et al. (1996) developed an adaptive hydraulic servo-valve controller which uses full-state feedback for simultaneous parameter identification and tracking control.
Alleyne (1996) developed a Lyapunov-based control algorithm for the control of an electro-hydraulic actuator. A gradient parameter adaptation scheme was included for compensating parametric model uncertainties.
Zheng et al. (1998) proposed a nonlinear adaptive learning algorithm for a proportional valve to accommodate valve dead zones, valve flow saturation, and cylinder seal friction.
Bu et al. (1999) designed a discontinuous projection based adaptive robust controller for single-rod hydraulic actuators with time-varying unknown inertia.

4. Hydraulic System Model – Cylinder DynamicsX
Cylinder dynamics can be written as
The hydraulic flow force F is defined as
Externally applied load
(neglected for simplicity)
(spring / damping term)

5. Hydraulic System Model – Pressures and Flows
The pressures dynamics of the piston and rod side can be written as
Fluid flow of the piston and rod side can be written as
Remark: The hydraulic cylinder is assumed to be constructed such that some
volume always remains in the piston and rod sides of the cylinder

6. Hydraulic System Model – Spool Dynamicsz T S m s P P P R Q Q
The spool dynamics can be related to as follows
The spool dynamics can now be rewritten as P R
(solenoid control force)
neglected for simplicity
(spring / damping term)

7. Bosch NG6 Servo - Solenoid Control Valve
Hydraulic Valve
Proportional Solenoid
LVDT
Control Voltage
Spool Position

8. Hydraulic System Model – Solenoid Model1
(1+st) s R VR i VS Fg z  VL
h(2, z)
g(VL)
f(, z) id ir + -

9. Differentiable Approximation for Fluid Dynamics
Differentiable approximation for the fluid dynamics
real model
where
Remark: Supply and tank pressures are assumed to satisfy the following inequalities

10. Control Objective
The control objective is to force the piston position of a hydraulic cylinder to track a time varying reference trajectory.
It is assumed that all system parameters are known (Exact Model Knowledge) and all signals are measurable (Full State Feedback)
Define the piston tracking error as
Define the filtered tracking error as
where is a positive control gain
It can be shown that if , then

11. After taking the time derivative of , the primary open-loop error system can be written as
Based on the previous equation and the subsequent stability analysis, the desired hydraulic flow force is designed as follows
After substituting the control design into the open-loop error system, the closed-loop error system for can be obtained as
Obtained by adding and subtracting the desired hydraulic flow force
(Auxillary force tracking error signal)

12. Error System Development
After taking the time derivative of , using the pressure dynamics, the second open-loop error system is obtained as
where,
The auxillary control input is now designed as
Based on this design, the closed loop error system for becomes
Desired spool position function

13. Error System Development (cont.)
After taking the time derivative of , the open-loop error system for can be obtained
Then the open-loop error system for can be rewritten as
where,
here, the notation denotes the partial derivative of with respect to
The desired spool velocity is now designed as
(Spool velocity tracking error)

14. Error System Development (cont.)
After substituting into the open-loop error system for ,we obtain
After taking the time derivative of , the open-loop dynamics for can be determined as follows
Based on the subsequent stability analysis, the control input is designed as follows
which allows us to write the closed loop dynamics for as follows

15. Stability Analysis where,
A non-negative function is defined as follows
Then V(t) can be lower and upper bounded as follows
where,
After taking the time derivative of V(t) and substituting the closed loop error system, we obtain

16. Control Strategy After utilizing the above inequalities, we can show
Electrical Dynamics
Force
Solenoid Dynamics
Pressure Dynamics
Flow
Pressure
Cylinder Dynamics
Position
Control
Voltage
Desired
Position
Error
Pressure Control
Electrical Control
Desired
Force
Solenoid Control
Desired
Flow
Desired
Pressure
Cylinder Control
Controller

17. Numerical Results
A PD and nonlinear controller shall track a sinusoidal position for the cylinder piston subject to nonlinear load conditions within 3% of the specified position
The desired trajectory is
Comparison of Commanded solenoid voltages for PD and nonlinear controllers
Comparison of Fluid pressures for PD and nonlinear controllers
Comparison of Positions and tracking errors for PD and Nonlinear controllers

18. Numerical Results PD Controller Nonlinear Controller 20 40 60 0.05 0.120 40 60
0.05
0.1
0.15
time(s)
Cylinder Position (m) 20 40 60
-0.01
0.01
0.02
time(s)
Position Error (m)
Nonlinear Controller 20 40 60
0.05
0.1
0.15
time(s)
Cylinder Position (m) 20 40 60
-0.01
0.01
0.02
time(s)
Position Error (m)

19. Numerical Results PD Controller Nonlinear Controller 20 40 60 700 75020 40 60
700
750
800
850
900
time(s)
Pressure Pp (psi) 20 40 60
700
750
800
850
900
time(s)
Pressure Pr (psi)
Nonlinear Controller 20 40 60
700
750
800
850
900
time(s)
Pressure Pp (psi) 20 40 60
700
750
800
850
900
time(s)
Pressure Pr (psi)

20. Numerical Results PD Controller Nonlinear Controller 10 20 30 40 50 6010 20 30 40 50 60 5
PD Controller
time(s)
Solenoid voltage (Volts) 10 20 30 40 50 60 5
Nonlinear Controller
time(s)
Solenoid voltage (Volts)

21. Experimental Test Stand
Control Valve
Piston side Pressure Transducer
Rod side Pressure Transducer
LVDT
Cylinder

22. Summary Present the mathematical models for hydraulic system
Develop a differentiable approximation for the fluid flow dynamics
Proposal a model based nonlinear controller for hydraulic system.
Validate the controller with numerical and experimental results
Investigate both PD and nonlinear controller