Analysis and Compensation for Friction in Electro-hydraulic Servo System

Junlan Cheng ,Xiaoguang Wang Department of Mechanical Engineering

North China Institute of Aerospace Engineering Langfang , China

lanlan0917@126.com

Abstract-Friction in the electro-hydraulic servo system makes tracking and pointing accuracy lower, it is one of the most important influence factors. This paper proposes a nonlinear adaptive friction compensation method based on analysis of the position-control system of valve-control asymmetrical hydraulic cylinder and the LuGre friction model. Dual nonlinear observers were constructed to estimate the friction state, which is unmeasurable in the electro-hydraulic servo system, and adaptive laws of friction parameters were built to compensate dynamic friction. Finally the application of Simulink has proven this method is validity.

Keywords: Dynamic friction; compensation; LuGre model; Simulink

I. INTRODUCTION Friction is one of the major limitations of hydraulic servo

system in performing high precision manipulation tasks at low velocities. It leads to tracking errors, limit cycles, and undesired stick-slip motion. Thus, compensating for the friction has been one of the main research issues in control. LuGre friction model fits the requirements for friction compensation of hydraulic servo systems because it can describe complex friction behavior, such as stick-slip motion, presliding displacement, Dah1 and Stribeck effects and friction lag. This paper proposes a nonlinear adaptive friction compensation method based on analysis the position-control system of valve-control asymmetrical hydraulic cylinder and the LuGre friction model. Dual nonlinear observers were constructed to estimate the unmeasurable friction state and adaptive laws of friction parameters were built to compensate dynamic friction. Finally the application of simulation has proven this method is validity.

II. ANALYSIS OF SERVO VALVE WITH ASYMMETRICAL CYLINDER

This paper studies the zero opening and four-way valve with asymmetrical cylinder. In ideal condition, [1] defined the load flow and load pressure as

++=

−=

)1/()( 221

21

nnQQQnppp

L

L (1)

Where pA is the area of chamber without piston rod ; hA is the area of chamber with piston rod; 12 QQAAn ph == .

When the valve core moving forward, the flow to the left chamber of the cylinder 1Q and from right chamber 2Q are

( )[ ]( )=

−=

21

22

21

11

/2

/2

ρω

ρω

pxcQ

ppxcQ

vd

svd (2)

When the valve core moving negative, the flow to the right chamber of the cylinder 2Q and from left chamber 1Q are

( )[ ]( )=

−=

21

11

21

22

/2

/2

ρω

ρω

pxcQ

ppxcQ

vd

svd (3)

Where dc is flow coefficient of throttle window;ω is area

gradient of throttle window; ρ is oil density; sp is oil

pressure; 1p is oil pressure in left chamber; 2p is oil pressure in

right chamber; vx is valve displacement.

Combined (2) and (3), replaced by valve opening coefficient 1c , 2c , 3c , 4c gives

−−+=

−+−=

uppcusupcusQ

upcusuppcusQ

s

s

21

2421

232

21

1221

111

)()()(

)()()( (4)

When 0≥u , 1)( =us ;when 0<u , 0)( =us

Linearized flow equation of the valve opening of servo valve is

LcasvLcvsvL pKuKKpKXKQ −=−= (5)

Where LQ is load flow; svK valve flow gain; aK is gain of servo amplifier;u is output of controller.

This study is supported by Langfang Science and Technology Bureau.

90978-1-61284-459-6/11/$26.00 ©2011 IEEE

1V , 2V are the volume of in and out chamber(include the volume of cylinder, valve and joint pipes), omit leakage, gives

)( 111

1 QVV

p e +−=β (6)

)( 222

2 QVVpe

−−=β

(7)

Assumed the initial volume of 1V is 10V , 2V is 02V ,when

piston is at equilibrium point, that xAVV p+= 101 ;

xAVV h−= 022 .If omit the flow changes by the leakage of

cylinder and oil compressed, then 11 VQ = , 22 VQ −= ,and inserting them into (4), gives

nAA

VV

ppp

QQ

p

h

s

==−=−

=1

221

1

2

1

2

(8)

That is

( )12

2 ppnp s −= (9)

Applying (1) together with (9), gives

( ) ( )( ) ( )+−=

++=32

2

331

1

1

nppnpnppnp

Ls

Ls (10)

According to (1), inserting (10) into (6), get the load flow continuity equation of the cylinder

L

e

tLtcL p

nVxApCQ

β)1(2 21 +++= (11)

Where tcC is equivalent leakage coefficient; Lp is load

presser; x is piston relative velocity; tV is equivalent total volume.

The hydraulic driving force on hydraulic cylinder's piston is

21 pApAF hpg −= (12)

Differential to (12) gives

21 pApAF hpg −= (13)

The friction include the max-static friction sF ,Coulomb friction

cF and viscous friction bF .Static friction displays the

resistance when load is static and has the movement tendency, can omit when the load moving. Viscous friction is proportional to velocity of load xBF mb = . Because the Coulomb friction has the character of essential nonlinear, it brings the noticeable influence to the performance of servo systems. When servo systems moving, the load force on the actuator shaft usually contain Coulomb friction, viscous

friction and inertia force. Thus, the force balance equation effected on piston is

ftcmtLpg FxmFxBxmpAF +=++== (14)

Where gF is hydraulic driving force on piston; tm the

equivalent mass of piston, oil and load ; mB is damping

coefficient of piston and load; fF is total friction on piston.

Integrated (5), (11), (14) gives dynamical equilibrium equation of hydraulic servo systems:

fg

cee

t

ce

p

ce

pasvt FF

KV

xKA

uKAKK

xm −−−=β4

2

(15)

Where ctcce KCK += .Assume that

cepasv KAKKa /= , cep KAb /2= , ceet KVc β4/= ,(15) becomes

fgt FFcxbauxm −−−= (16)

Combined (4), (6), (7) and (13) gives

uppAvVA

VA

xF php

eg ),,()( 212

2

1

2

++−= β (17)

In above equation:

−+

+−=

])([

])([),,(

21

22

421

11

2

21

22

321

11

1

21

ppVcAp

VcA

pVcA

ppVcA

ppAv

shp

e

hs

pe

p

β

β

0

0

<

≥

u

u

Equation (17) relates the control signal on servo valve and the driving force on hydraulic cylinder piston. This obtained the mathematic relationship between control signal on servo valve and the force that system request.

III. FRICTION MODELING LuGre dynamic friction model includes Coulomb, viscous,

static and Stribeck frictional effects. It includes all static and dynamic friction characteristics. In addition, it can be accepted by controller, so it is very fit for friction compensation.

Reference [2] give LuGre model as follows

( ) zxgx

xdtdz −=

(18)

xdtdzzFf 210 σσσ ++= (19)

( ) ( ) ( )2/0

sxxcsc eFFFxg −−+=σ (20)

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Where sx is the Stribeck velocity; 0σ is a stiffness

coefficient of deformation; 1σ is a damping coefficient; 2σ is a coefficient of viscous friction.

IV. DESIGN OF ADAPTIVE CONTROLLER FOR NONLINEAR FRICTION

In order to design system’s non-linear friction adaptive rate, [3] established the system error equation as follows

)()()(1 txtxte d−= (21)

)()()( 112 tketete += (22)

Where )(txd is expected location value, which is a smooth and a boundary signal that have the second derivative. k is positive feedback gain. So tracking )(tx turn to minimizing

2e , insert (18), (19) into (16) gives

)(10 xg

xxzFcxbauxm gt σβσ +−−−−=

(23)

Where 21 σσβ += , dirivate to (22) and inserting it to (23) gives

)()(

1

102

ekxmFcx

zxgx

zxbauem

dtg

t

−−−−

+−−=

β

σσ (24)

Because nonlinear state z in LuGre model is unknown, here uses two nonlinear observers to estimate z

( ) 00

0 ˆˆ

τ+−= zxgx

xdtzd

(25)

( ) 11

1 ˆˆ τ+−= z

xgx

xdtzd (26)

Where 0z , 1z are estimated value of z; 0τ , 1τ are dynamic item of observer need to design. Assume

00 ˆ~ zzz −= , 11 ˆ~ zzz −= , then the error of estimation may calculated as

00

0 ~)(

~τ+−= z

xgx

xdtzd

(27)

11

1 ~)(

~τ+−= z

xgx

xdtzd (28)

Since the real value of 0σ , 1σ , β are unknown, so use

their estimated value 0σ , 1σ , β to replace. Use the following control rate

)](ˆ

ˆ)(

ˆˆˆ[111002

ekxmFcx

zxgx

zxbhea

u

dtg −++

+−++−=

β

σσ (29)

Where h is positive design constant, insert (19) into (24) gives

1111

00022

ˆ)(

~~)(

ˆ~~~

zxgx

zxgx

zzxheemt

σσ

σσβ

+

+−−−−= (30)

Use the following adaptive rate to each unknown parameter and the dynamic friction state observer

2000 ˆˆ ezγσ −= , 2111 ˆ)(

ˆ ezxgx

γσ = , 2ˆ exβγβ −= ,

20 e−=τ , 21 )(e

xgx

=τ (31)

Where 0γ , 1γ , βγ is positive design constant. Following proof hydraulic system (16) is closed loop stability at adaptive rate (29).

Theorem: Given the hydraulic system (16) and the friction force in the system indicated by (18), (19); two nonlinear friction observer is expressed by (25), (26).Choose adaptive rate (29) and parameter adaptive rate (31), then (16) can global asymptotically tracking the position signal.

Proof: Define the estimation error is

000 ˆ~ σσσ −= , 111 ˆ~ σσσ −= , βββ ˆ~ −= .Choose Lyapunov function[4]

22

11

20

0

211

200

22

~2

1~21~

21

~21~

21

21)(

βγ

σγ

σγ

σσ

β

+++

++= zzemtV t

(32)

Derivate to the above (32) and insert (27), (28), (30), (31) into it gives

22

211

200

22

~)(

~)(

)(

he

zxgx

zxgx

hetV

−≤

−−−= σσ (33)

As h, 0σ , 1σ is all positive,

and )(xg >0,so )(tV <0.According to Lyapunov stability theorem, system (16) is global asymptotically stable. This means 2e is boundary. According to Barbalat theorem, while

∞→t get 02 →e ,according to(22) get 01 →e ,this

guaranteed x restrain to dx asymptotically. The theorem is proven.

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V. SIMULATION Because the friction effect low speed tracking accuracy and

pointing accuracy (i.e. static error) of position control servo systems mainly, so inspect the changes of the two indexes after compensation[5].

The main parameters of the study system are as follows, mt=52kg, e=7×108Pa,Ksv=0.02625m3/(sA),Ka=0.02A/V,Kc

e=7.5×10-12m3/(sPa), 2-4100.31 mAp ×= , 2σ =24.5, 33101.6 mVt

−×= , sx =0.015, 0σ =120000, 1σ =759.3, h=100, k=50.

Fig.1 and Fig.2 are the response curve when input unit step signal; Fig.3 and Fig.4 are the response curve when input 0.01 mm/s low speed ramp signal.

Figure 1. Step response of non-compensation

Figure 2. Step response of compensated system

Figure3. Ramp response of non-compensation

Figure4. Ramp response of compensated system

See from the simulation result, before the friction

compensation, the step response adjusting time is 0.18s approximately, over approximately is 13.5%, but after compensating, the adjusting time is 0.04 s approximately, over approximately is 8%,both of the rising time is 0.02 s approximately. Without the friction compensation, the curve of low speed ramp response is rather stable than before compensation.

VI. CONCLUSIONS Use the most perfect dynamic friction model LuGre at

present to construct two nonlinear observers to estimate the internal behavior of friction to increase the friction compensating precision. Carry on the dynamic compensating the friction through set adaptive rate of friction model’s parameter, and had proven that the system is global stability with the Lyapunov theorem, the simulation result indicated this method may increase system's tracking accuracy and the dynamic performance, so it is a new strategy to compensate friction effectively.

REFERENCES [1] W.Bernzen, “Nonlinear Control of Hydraulic Cylinders-Theoretical and

Experimental Results”, Department of Measurement and Control, 1995, 40(3), pp.60-65.

[2] C.Canudas de Wit, “A New Model for Control of Systems with friction”, IEEE Transction on Automatic Control, 1995, 40(3), pp.419-425.

[3] Zhang B,Dong Y L,Zhao K D, “Study on the friction nonlinear control of force control system”, Proceeding of the 2007 IEEE International Conference on Mechanical & Automation. Harbin: 2007:3695-3699

[4] P. Lischinsky, C. Canudas de Wit, G. Morel:Friction Compensation for an Industrial Hydraulic Robot, IEEE Transactions on Automatic Control, 1999,(2)

[5] Ying Li, Dynamic System Modeling and Simulation with Simulink(The second edition), Xi’an University of Electronic Science and Technology Press, 2009 (In Chinese)

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