Exponential Tracking Control of Hydraulic Proportional Directional Valve and Cylinder via Integrator...
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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


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Presentation Outline Directional Valve and Cylinder via Integrator Backstepping

  • Literature Review

  • Hydraulic System Model

  • Differentiable Approximation for Fluid Dynamics

  • Control Objective

  • Error System Development

  • Nonlinear Controller Design

  • Stability Analysis

  • Numerical Results

  • Summary


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Literature Review Directional Valve and Cylinder via Integrator Backstepping

  • 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.


Hydraulic system model cylinder dynamics l.jpg

X Directional Valve and Cylinder via Integrator Backstepping

Hydraulic System Model – Cylinder Dynamics

  • Cylinder dynamics can be written as

  • The hydraulic flow force F is defined as

Externally applied load

(neglected for simplicity)

(spring / damping term)


Hydraulic system model pressures and flows l.jpg
Hydraulic System Model – Pressures and Flows Directional Valve and Cylinder via Integrator Backstepping

  • 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


Hydraulic system model spool dynamics l.jpg
Hydraulic System Model – Spool Dynamics Directional Valve and Cylinder via Integrator Backstepping

P

z

P

S

T

m

s

P

P

R

P

Q

Q

R

P

  • The spool dynamics can be related to as follows

  • The spool dynamics can now be rewritten as

(solenoid control force)

neglected for simplicity

(spring / damping term)


Bosch ng6 servo solenoid control valve l.jpg

Control Voltage Directional Valve and Cylinder via Integrator Backstepping

Proportional Solenoid

Hydraulic Valve

Spool Position

LVDT

Bosch NG6 Servo - Solenoid Control Valve


Hydraulic system model solenoid model l.jpg

1 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 Model


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Differentiable Approximation for Fluid Dynamics Directional Valve and Cylinder via Integrator Backstepping

  • Differentiable approximation for the fluid dynamics

real model

where

Remark: Supply and tank pressures are assumed to satisfy the following inequalities


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Control Objective Directional Valve and Cylinder via Integrator Backstepping

  • 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


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  • 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)


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Error System Development open-loop error system can be written as

  • 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


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Error System Development ( open-loop error system can be written ascont.)

  • 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)


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Error System Development ( open-loop error system can be written ascont.)

  • 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


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Stability Analysis open-loop error system can be written as

  • 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


Control strategy l.jpg

Cylinder Dynamics open-loop error system can be written as

Pressure Dynamics

Electrical Dynamics

Solenoid Dynamics

Pressure Control

Control Strategy

  • After utilizing the above inequalities, we can show

Pressure

Position

Force

Flow

Control

Voltage

Desired

Position

Error

Desired

Force

Desired

Flow

Desired

Pressure

Cylinder Control

Solenoid Control

Electrical Control

Controller


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Numerical Results open-loop error system can be written as

  • 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


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0.02 open-loop 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 Results

PD Controller

Nonlinear Controller


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900 open-loop 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 Results

PD Controller

Nonlinear Controller


Numerical results20 l.jpg

PD Controller open-loop 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 Results


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Experimental Test Stand open-loop error system can be written as

Control Valve

Piston side Pressure Transducer

Rod side Pressure Transducer

LVDT

Cylinder


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Summary open-loop error system can be written as

  • 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


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