Synchronization Motion Control for Quad-Cylinder Lift Systems With Acceleration Coupling

7/30/2019 Synchronization Motion Control for Quad-Cylinder Lift Systems With Acceleration Coupling

1/6

978-1-4673-0311-8/12/$31.00 2012 IEEE

Synchronization Motion Control for Quad-cylinderLift Systems with Acceleration Coupling

Haibin Dou1 Shaoping Wang1,2 Wenkui Zhao1

1. School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China2. Science and Technology on Aircraft Control Laboratory, Beijing 100191, China

[email protected] [email protected]

AbstractHigh-performance robust synchronization motion

control for quad-cylinder electro-hydraulic lift systems with

parametric uncertainties and uncertain nonlinearities is

considered. Adaptive Fuzzy logic systems are introduced as an

approximator to counteract the effects of coupling among

acceleration terms. Robust adaptive control by backstepping is

used to handle parametric uncertainties and uncertain

nonlinearities in quad-cylinder synchronization lift systems. In

order to address the problem of explosion of terms caused by

using backstepping techniques, dynamic surface control

techniques are introduced. Simulation results on a quad-cylinder

synchronization lift system verified the effectiveness of the

proposed approach.

Keywordsdecoupled control, fuzzy logic systems, adaptive

control, robust control

I.INTRODUCTION

Traditional synchronizing lifing system is widely used inheavy-duty application [1], in which the synchronizingadjustment is carried out without considering the leveling time.However in some special applications such as missile liftingequipment, it is necessary to lift the heavy load in short time.

So the synchronous adjustment with high speed of multiplehydraulic actuators under nonlinear factors and variant loadmust be studied. Due to the standard design, this paper focuseson the synchronizing quad-cylinder lift system adjustment inthis paper.

Due to uneven loading and the inherent differences inmultiple hydraulic circuits and components, the lift distanceamong the linear actuators will be different under open-loopoperation. If close-loop operation is used in synchronizingcontrol, the interactions among input channels will be moreserious than open-loop operation. The interactions amonginput channels arise from geometric constraint relationsbetween the lift distance among the linear actuators andposture of load. It brings difficult challenge in controller

design and control performance. So a decoupled approach isneeded to address the problem.

Synchronization lift systems are multi-input/multi-output(MIMO) nonlinear systems, the control problem is verycomplicated due to the couplings among various controlchannels. So there are only a few literatures on synchronizationlift systems. Sun and Chiu [2] proposed a 2-step designapproach to address the motion synchronization problem for amulti-cylinder electro-hydraulic (EH) system. But couplingsare not considered. In recent years, a great deal of progress has

been made in decoupling theories. Chen and Khalil [3], Sastryand Isidori [4] not only realized decoupling among inputchannels based on feedback linearization, but also combinedwith other robust approaches to handle parametricuncertainties in the input coupling matrix. However, to removethe couplings of system inputs, the decoupling matrix isrequired to be invertible. Therefore, the possible singularitywhen calculating the inverse of the decoupling matrix shouldbe avoided. Ge et al. [5] pointed out that a projection algorithm

can be used to avoid the possible singularity problem.However, a priori knowledge for the feasible parameter set isneeded and no systematic procedure is available forconstructing such a set for a general plant[6].

As an alternative, intelligent approaches were developed todeal with coupling among various inputs by neural networks(NNs)[5], or fuzzy-logic-system (FLS) [7]. They are mostlyused as an approximator to counteract the effects of variousinputs. Especially by use of the Lyapunov-synthesis approach,adaptive intelligent methods can provide stabilizingcontrollers. Ge et al. [5] proposed an adaptive neural controlmethod for a class of uncertain MIMO nonlinear systems.Adaptive neural technique is used to approximate the unknownnonlinearities which contain coupling in the input matrices and

interconnections in the systems. Li et al. [8] proposed a robustadaptive fuzzy tracking control for a class of MIMO systems.The coupled interconnections in systems are counteracted byusing Takagi-Sugeno (T-S) fuzzy systems. But the couplingsamong input channels are not considered. Lee [9] proposed arobust adaptive fuzzy control for a class of MIMO nonlinearsystems which is different from Li. He claimed that theproposed approach can address the couplings among inputchannels. Unlike the previous decoupling approaches, atransformable decoupling method is proposed in this paper.The couplings among acceleration states are usually commonin real engineering problems, especially in robot control. If weget the inverse coefficient matrix of acceleration states, we cantransform the models which are couplings among acceleration

states to those are couplings among input channels. Butadditional efforts have to be made to avoid the possiblesingularity problem. Moreover, it complicates the design ofcontrollers. So the couplings among acceleration states aredirectly considered by using fuzzy logic systems in this paper.In order to reduce the number of required states in FLS, adynamic surface method is utilized.

978-1-4673-0311-8/12/$31.00 2012 IEEE 752


2/6

z

2x

P

X

Y

4B

3B

2B

1B

2f

2m

o

1x

1f1m

3x

3f

3m

4x

4f

4m

Figure 1. Diagram of Acting Forces in a Quad-Cylinder Hydraulic Lift

System

In this paper, a robust adaptive decoupled control schemewhich is performed by incorporating fuzzy logic systems andDSC techniques is extended to the quad-cylindersynchronization lift systems in the presence of parametricuncertainties and uncertain nonlinearities. The four cylindersare controlled by four individual servovalves. The hydrauliccircuit diagram of the system is shown in Figure 1. Robustadaptive control by backstepping is used to handle parametric

uncertainties and uncertain nonlinearities in quad-cylindersynchronization lift systems. Fuzzy logic systems areintroduced as an approximator to counteract the effects ofcoupling among acceleration states. DSC techniques are usedto address the problem of explosion of terms. Simulationresults on a quad-cylinder synchronization lift system verifiedthe effectiveness of the proposed approach.

The rest of this paper is organized as follows. System

modeling is presented in Section followed by the discussionof the decoupled control design. A robust adaptive controllerby backstepping is discussed in detail in Sectionfollowed bythe discussion of simulation results. Conclusions are presented

in Section.

II. SYSTEM MODELINGA. Coupling Characteristics Modeling

z

xo

y

'

1A

'

2A

'

3A

P

'P

'

1B

'

2B

'

3B

'

4B

1L

2L

3L

4L

1x

2x

3x

4

Figure 2. Diagram of geometric constraint relations in Quad-CylinderHydraulic Lift System

The coordinate of load is shown in Figure 2. When thesynchronization errors are zero, i.e., the pitch angle =0 andthe roll angle =0, the load is in the x-o-y plane. When thesynchronization errors are not zero, the load is in the planeformed by Ai. The point P orP represents the center ofgravity of the load. Li (i=1,2,3,4) is the moment arm forfi inFigure 2 with respect to rotation of the load. xi (i=1,2,3,4)represents the displacement of individual cylinder. The

following practical assumption is made: assuming that themoment arm Li of individual cylinder dose not change duringthe motion of the load, the points on the load that contact thecylinders do not vary with the rotation of the load. Underabove assumption, the deformation of the load caused bysynchronization errors can not be ignored. It can be consideredas uncertain nonlinearities in the design of controller. By usingknowledge of geometric, we can get the relations between thelift distance among the linear actuators and posture of load asfollow:

1

2 4 1 4 1 3 2 3

2

4 4 3 3

3

2 1 1 2

4

/ / / /

/ / / / tan

/ / / / tan

p

xL L L L L L L L L L L L x

xL L L L L L L L

xL L L L L L L L

x

=

(1)

Wherexp represents the position of the center of gravity of theload (xp=0 when the load is on the ground). AndL=(L1+L2)(L3+L4).For simplicity, the equation (1) can be expressed as

1

2

1

3

4

tan

tan

p

xx

x

x

x

=

L(2)

The differentiation of equation (2) can be expressed as

1

2 2

1

3 2

4

1

sec

sec

p

xx

x

x

x

=

L

(3)

The differentiation of equation (3) can be expressed as

1

2

1

3

4

p

xx

x

x

x

= +

2L N L

(4)

Where

2 2

2 2

0

2sec ( ) tan( )

2sec ( ) tan( )

=

N

2

1

cos

cos

=

L

The equations (1)-(3) describe the couplings among thelinear actuators.

B. Dynamic modelingFigure 2 shows the forces acting on the hydraulic cylinders.

From Newtons second law and the conservation of angular

momentum, the following equations can be obtained torepresent the motion of the load along gravity direction androtations about the axesB2B4 andB1B3:

1

2

3 3 4 4

3

1 2 1 2

4

1 1 1 10

0

0

p

f MgM x

fJ L L L L

fJ L L L L

f

=

(5)

Where M is the mass of the load and g is the gravitationalconstant. fi represents the reaction force acting on cylinder i

753


3/6

(i=1,2,3,4) andJ (J) represents the rotation moment of inertiaof the load about the axisB2B4 (or the axisB1B3).Consider the forces fi (i=1,2,3,4) acting on the cylinders, the

equations of motion for the cylinders can be represented by:

, 1, 2,3, 4i i Li i i i i fi i im x P A B x f F m g f i= + =

(6)

WherePLi=P1i-P2idenotes the load pressure of cylinderi,P1i ispressure inside the forward chamber of cylinder i, P2i ispressure inside the return chamber of cylinder i,Ai represents

the effective piston area. mi is the piston mass of cylinderi.Ffiis the modeled friction and Bi is the viscous friction

coefficient.if

is the uncertain nonlinearities due to external

disturbances, the unmodeled friction forces, interaction forcesdue to deformation of the load and other hard-to-model terms.

The fluid that flows into each cylinder is controlled by aservo-valve. The pressure dynamic in each cylinder can berepresented by the following equation[10]:

, 1,2,3,44

tiLi i i tmi Li Li

e

VP A x C P Q i

b= - - + = (7)

Where e is the bulk modulus of the working fluid, Vtirepresents the total volume from the output port of the valve tothe respective cylinderi chamber and Ctmi is the coefficient ofthe internal leakage of the cylinder i. QLi represents the loadflow. QLi is related to the spool valve displacement of theservo-valvexvi, by [10]

( ), 1,2,3,4s vi LiLi di i vi

P sign x PQ C x ik

r

-= = (8)

Where Cdi is the discharge coefficient, i is the spool valvearea gradient, and ps is the supply pressure of the fluid. Theeffects of servo-valve dynamics have been include by someresearchers, but this requires an additional sensor to obtain thespool position. So the spool valve displacement xvi is directlyproportional to the spool position in this paper, thus thefollowing equation is given by

vi vi i

x K u= (9)

WhereKvi is a positive constant and ui is input voltage.In the Equation (5), it is clear that there are 3 DOF, but

more than three forces. There is one redundant actuation. Theredundant actuation represents that there exists an infinitenumber of forces fi corresponding to a given posture of theload. It complicates the quad-cylinder synchronization motioncontrol problem. Suitable optimization schemes should beintroduced to address the complexity due to the redundancyissue.

C. Actuator Redundancy IssueA well-known mathematical technique traditionally used to

solve underdetermined linear problems is the pseudo-inverse

technique. Compared with other optimization schemes thepseudo-inverse formulation can efficiently reduce thecomputational burden and greatly reduce the conceptualcomplexity of the solution.For simplicity, the equation (5) can be expressed as

q

Mx = Lf G (10)

Where , ,T

q px = x , 1 2 3 4[ , , , ]Tf f f f=f .

The pseudo-inverse of L, L+ yields the minimum normsolution and a solution to the homogeneous system of equation

(10) by:+

+ = ++

q p hf = L (Mx G) + (I - L L)z f f (11)

Where z is a 41 arbitrary vector. The minimum norm

solutionfp can be used to decrease the magnitude of the forcesfi to some extent. And the solutionfh can be designed to satisfya secondary system performance goal. It is desired that fh iszero in this paper. So we make z=0. The equation (11) can beexpressed as

++

qf = L (Mx G) (12)

Where L+=LT(LLT)-1.

From equation (4)qx can be written as

1

2

q 1x = L (L x -N) (13)

Where x=[x1,x2,x3,x4]T.

Substituting equation (13) into equation (12), we can getvectorf:

1

2

++ +1f = L ML (L x -N) L G (14)

Substituting equation (14) into equation (6), the coupleddynamic models can be expressed as

2 1 1 f= + +0(M + N )x AP Bx N G F f (15)

Where N1=L+ML2-1N, N2=L+ML2-1L1,G1=L+G+diag[mig]TEquation (15) describes a nonlinear four-input four-output

system, where the four cylinders are coupled through thematrix M0+N2. In order to facilitate the process of designingthe controller, equation (15) is divided into four vectors.Define state variables of the ith control channel xi=[xi1,xi2,xi3].Combined with equations (7)-(9), the ith subsystem can beshown in a state-space form as

1 2

2 3 2 1 1 1 2

3 3 3 2

1[ ] ( , , )

( )

4( ( ) ), 1,2,3,4

i i

i i i i i ic i i fi i i i

i i

eii qi vi i s i i tmi i i i

ti

x x

x Ax B x f N G F d x x tm a

x K K u P sign u x C x Ax i

V

=

= + ++

= =

(16)Where ai is the diagonal element of the matrix N2,

2 2 2( )ic j j m m n nf a x a x a x= + + is the coupling from the other

three control channels.xj2 (xm2,xn2) is the second state variableof the other three control channels.N1i is the ith component ofthe vectorN1. G1i is the ith component of the vector G1. ui iscontrol input of the ith control channel.Kqi is defined as

di iqi

CK

k

r= (17)

III. DECOUPLING ISSUEThe coupled term fic can be considered as a nonlinearfunction with parametric uncertainties. So an adaptive fuzzy

logic system is introduced as an approximator to counteractthe effects offic in this section. A fuzzy logic system iscomposed of three main components, fuzzy rule base,fuzzification and defuzzification operators. The fuzzy rulebase is made up of the following inference rules:

1 1 2 2: IF is and is and and is ,

then is ( 1, 2, , )

i i i i

n n

i

R x F x F x F

y B i N

=

754


4/6

Where x=[x1, , xn]TRn andyRare the input and output

of the fuzzy systems, respectively, Fji and Bi are fuzzy sets

corresponding to xj and y. N is the number of rules in thefuzzy rule base. Through singleton function, center averagedefuzzification and product inference, the fuzzy logic systemcan be expressed as

1 1

1 1

( )( )

( )

ij

ij

nN

i jFi j

nN

jFi j

y xy x

x

= =

= =

=

(18)

Where max ( )iii y R B

y y

= , ( )ij

jFx and ( )iB y are fuzzy

membership functions ofxj andy.Define the fuzzy basis functions as

1

1 1

( )

( )

ij

ij

n

jFj

i nN

jFi j

x

x

=

= =

=

(19)

Denote:

1 2 1 2[ , , , ] [ , , , ]T

N Ny y y = =

1 2( ) [ ( ), ( ), , ( )]T

N = x x x x

The fuzzy logic system (18) can be rewritten as

( ) ( )Ty =x x (20)

Lemma1. Let h(x) be a continuous function defined on acompact set . Then for any constant >0, there exists a fuzzylogic system (20) such as

sup ( ) ( )T

x

h x

x (21)

It has been proven that the FLS (20) can approximate anysmooth function over a compact set to arbitrarily anyaccuracy. So the coupled termfic can be expressed as

( ) * ( ) *,Ticf = + x x x (22)

Where * is the optimal estimation normalization factor forfic,* is the optimal estimation error and *>0.

The inputs of (22) are the derivative ofxj2

, xm2

and xn2

. Inorder to avoid additional sensors, a dynamic surface methodis introduced. Introduce a new variablesk(k=j,m,n), and letxk2,(k=j,m,n) pass through a first-order low-pass filter withconstant k(k=j,m,n) as follows:

2 2, (0) (0)

k k k k k k s s x s x + = = (23)

By defining the output error of this filter as k= xk2-sk, it

yields /k k ks = , and

2k

k k

k

x

= + (24)

The signalks will be used as the inputs of (22) to counteract

the effects offic.Define the error of the estimation normalization factor:

*= (25)Where is the actual estimation factor vector.

Define *|z| as the upper boundary of the compensation forestimation error*, that is

* * z (26)

Where z is the tracking error, it will be given in the design ofrobust adaptive controller.

Define the estimation error of*

* = (27)The adaptive laws for adjusting the parameter vectors

and will be designed to counteract the effects of couplingtermfic.

IV. ROBUST ADAPTIVE CONTROLLERDESIGNIn this section, a robust adaptive controller combined with

FLS and DSC will be presented for the ith control channel.

The controllers for the other three control channels can bedesigned in the same way. The goal is to make the four linearhydraulic actuators track the same control command as closelyas possible to realize the synchronization motion of quad-cylinder lift systems.

A. Controller designIn general, the subsystem (16) is subjected to parametric

uncertainties due to the variations ofai,Bi,Ffi, ei,Kqi,Kvi andCtmi. For simplicity, in this paper, we only consider theparametric uncertainties due to Bi, ei, Kqi, Kvi and di. Otherparametric uncertainties can be dealt with in the same way ifnecessary. In order to use parameter adaptation to improveperformance, it is necessary to linearly parametrize the state-

space equation (16) in terms of a set of unknown parameters.To achieve this, define the unknown parameter set i=[i1, i2,i3]

T as i1=Bi/(mi+ai), i2= di, i3=4eiKqiKvi/Vti. The state-space equation (16) can be linearly parametrized in terms ofias

1 2

2 3 1 2 1 1 2

3 3 3 2

3

( )

/ ( )

i i

i i i i i ic i i fi i i

i i i tmi i i i

i i s i i

x x

x A x x f N G F d

x C x A x

u P sign x

=

= + + + +

=

=

(28)

Where / ( )i i i iA A m a= + , / ( )ic ic i if f m a= + ,

1 1 / ( )i i i iN N m a= + , 1 1 / ( )i i i iG G m a= + ,

/ ( )fi fi i iF F m a= + , 2i i id d = ,

/ 4tmi tmi ti ei qi viC C V K K = , / 4i i ti ei qi viA AV K K=

The following practical assumption is made: assuming thatthe extent of parametric uncertainties and uncertainnonlinearities are known, i.e.,

min max

1 2 1 2

{ : }

( , , ) ( , , )

ii i i i i

i i i id i id x x t x x t

< 0. So it isassumed that imax>0 and imin>0.Step1:xi2 can be thought as the virtual control input to the first

equation of (28). Then a control function 1 can beconstructed for the virtual control inputxi2such that the outputtracking errorz1=xi1-x1d converges to zero or a small valuewith a guaranteed transient performance. x1d is the desiredtrajectory to be tracked byxi1.

Differentiating z1 as

1 2 1i dz x x= (30)

The resulting control function 1is given by

1 1 1 1dz x = + (31)

755


5/6

Let2 2 1i

z x = denote the input discrepancy. For the

positive-semidefinite (p.s.d) function V1 defined by2

1 1 1(1/ 2)V z= , substituting the control function (31) into

the derivative ofV1, it can be shown that2

1 1 1 2 1 1 1 1 1 2( )i idV z x x z z z = = + (32)

Where 1 is any positive scalar, 1 is a positive weightingfactor.

Step2: In step 1, as seen from (32), ifz2=0, output trackingwould be achieved. Therefore, Step 2 is to synthesize a virtualcontrol function 2 such that z2 converges to zero or a smallvalve with a guaranteed transient performance as follows.From (28) and (32)

2 2 1 3 1 2 1 1 2 1i i i i i ic i i fi i iz x A x x f N G F d = = + + + +

(33)The resulting control function 2 and adaptive function 2

are given by

2 2 2 2a c s = + +

2 1 2 1 1 2 1 1 (1/ )( )a i i i i i fi iA x N G F z = + + +

2 2(1 / )( ( ) )T

c i kA z = x

2 2 1 2 2s s s = + 2

2 1 2 1 2 2 1 2 2 2 2(1/ ) ,s i s sA k z k C = +

2 2 2 2 2, 0z = > (34)

Where C2 a positive-definite (p.d.) constant diagonal matrix,2 is any positive scalar,2 is a positive weighting factor, and

2 2[ ,1,0]

ix = (35)

In (34), 2a functions as an adaptive control law used toachieve improved model compensation through onlineparameter adaptation combined with discontinuous projection,2c as a adaptive fuzzy logic control law used to counteract theeffects of coupling termfic, and 2s as a robust control law, inwhich 2s is any function satisfying the following conditions:

2 2 2 2 2( )T

i s i iz A d + 2 2 2 0sz (36)

Where 2 is a positive design parameter which can be

arbitrarily small,i

denotes the estimation error.

Let z3=xi3-2 be the input discrepancy. Consider theaugmented p.s.d function V2 given by

2 2 2

2 1 2 2

, ,1 2

1 1 1 1

2 2 2 2

T

k

k j m n

V V z =

= + + + + (37)

Substituting the control function (34) into the derivative ofV2,it can be shown that

2 2

2 1 1 1 2 2 1 2 2 1 2 3

2 2 2 2 2

1 2

2 2 2

, ,

1 1( )

( ( ) )

s

T T

i s i i

T

ic k k k

k j m n

V z k z A z z

z A d

z f z

=

= +

+ + + +

+ +

x

(38)

Choose the time constant of filters as2

2 2

11

4

k

k

M

= + (39)

If the adaptive laws are chosen as

2 1 2( )

k z = = x 22 2 2z = =

(40)

The equation (43) becomes2 2

2 1 1 1 2 2 1 2 2 1 2 3

2 2 2 2

2 22 22

, , 2 2 2 2

2 2

1 1 1 2 2 1 2 2 1 2 3

2 2 22 2

2 22, , 2 2

2 2

1 1 1 2 2 1 2 2

+ [( (1 ( )) ) ] 44 4

[ (1 ) ] 4

4

s

k k k k

k

k j m n k

s

k k k

k

k j m n k

s

V z k z A z z

M M x

M

z k z A z z

x M

M

z k z A

=

=

+

+ + +

+

+ +

+

1 2 3 2 24z z +

(41)

Step 3: Similar to (33), thez3 dynamic can be obtained as

3 3 2 3 3 2 2( )

i i i tmi i i iz x C x A x = = (42)

In (42), differentiating 2 is very complex. It is so called theproblem of explosion of terms. Let 2pass through a first-order low-pass filter with constant 2 as follows:

2 2 2 2 2 2, (0) (0)

d d d + = = (43)

By defining the output error of this filter as 2=2-2d, it

yields 2 2 2/d = , and

2 22 2 1 2

2 2

( , , , , , , , )i i j m n iD x x x x x

= + = + (44)

Using2d

instead of2

in (42), equation (42) becomes:

3 3 3 2 2( )i i tmi i i i d z C x A x = (45)

The resulting control function iand adaptive function aregiven by

i ia is = +

3 2 2 2 1 2

3

1( )

ia tmi i i i d i

C x A x A z

= + +

1 2is is isv v = +

1 3 1 3 3 1 3 3 3 3

3,min

1,is s s

i

k z k C

= +

2 3 3 3 3, 0z = + > (46)Where3>0 is a constant, and 3 3 2[0,0, ]ia tmi i i iC x A x =

In (46), is2 is a robust control law satisfying the followingconditions:

3 3 2 3 3( )T

i is iz 3 2 0isz (47)

in which3 is a design parameter.Consider the augmented p.s.d function V2 given by

2 2

3 2 3 3 2

1 1

2 2V V z = + + (48)

Substituting the function (45) and (46) into the derivative ofV3, it can be shown that

2 2 23

3 1 1 1 2 2 1 2 2 2 3 3 1 33,min

2

3 3 3 2 3 2 2 2

4

( ) /

i

s si

T

i is i

V z k z k z

z D

+

+ +

(49)

Choose the time constant 2 of filters as2

2 3 3

11

4

DM

= + (50)

So the equation (49) becomes

756


6/6

2 2 233 1 1 1 2 2 1 2 2 2 3 3 1 3

3,min

2 2 222 2

3 3 2 2

3 3 3 3

2 2 231 1 1 2 2 1 2 2 2 3 3 1 3

3,min

2 222 2

3 3 2 2

3 3

2

1 1 1 2

4

2 (1 )4 4

4

2 (1 )4

i

s s

i

DD

D

is s

i

D

D

V z k z k z

M DM

M

z k z k z

MD

M

z

+

+ + +

+

+

2 2

2 2 3 3 3 2 2 3 34 2z z + +

(51)

Using equation (46) and the last equation of (28), the inputcontrol signals can be achieved.

V. COMPARATIVE SIMULATIONS

Figure 3. angle and without decoupling

Figure 4. angle and with proposed method

A 1Hz sinusoidal trajectory of xd=0.5sin(2pit) is given toeach control channel. If the couplings among accelerationstates are neglected, only adaptation laws and robust controllaws are used, the rotation angle and caused by

synchronization errors are shown in Figure 3, which angle has the steady absolute maximum of 0.2 and angle has thesteady absolute maximum of 2.86. Using the proposedcontroller, rotation angle , are shown in Figure 4. As seen,angle has the steady absolute maximum of 2.16 and angle has the steady absolute maximum of 4.7. The synchronizationperformance is improved greatly.

VI. CONCLUSIONIn this paper, a robust adaptive decoupled controller has

been developed for quad-cylinder synchronization liftsystems. The proposed controller takes into account thecouplings among acceleration states and uses adaptive fuzzylogic systems to counteract the effect of couplings. Theparameter uncertainties and uncertain nonlinearities areeffectively handled via certain robust adaptive control lawsfor a guaranteed robust performance. Due to backsteppingtechnique used in the design of controller, dynamic surfacecontrol technique is utilized to address the problem ofexplosion of terms. The controller achieves highsynchronization accuracy and final tracking accuracy for eachcylinder in the present of parametric uncertainties and

uncertain nonlinearities. The simulation results verify theeffectiveness of the proposed robust adaptive decoupledcontrol strategy.

ACKNOWLEDGMENT

The authors would like to appreciate the support ofProgram111 of China and Natural Science Foundation(51175014).

REFERENCES

[1] Hong Sun, George T-C Chiu. Motion Synchronization for Dual-Cylinder Electrohydraulic Lift Systems. IEEE/ASME Transactions onMechatronics. 2002, 7(2) :171-181.

[2] Hong Sun, George T-C Chiu. Motion Synchronization for Multi-Cylinder Electro-Hydraulic System. Proceeding of 2001 ASME

international Mechanical Engineering Congress and Exposition. 2001,1-11.

[3] F. C. Chen and H. K. Khalil, Adaptive control of a class of nonlineardiscrete-time systems using neural networks, IEEE Trans. Automat.Contr. 1995, 40(5) : 791-801.

[4] S S Sastry, A Isidori. Adaptive control of linearizable systems. IEEETrans. Automat. Contr., 1989, 34: 11231131.

[5] Shuzhi Sam Ge, Cong Wang. Adaptive Neural Control of UncertainMIMO Nonlinear Systems. IEEE Transaction on Neural Networks.2004, 15(3):674-692

[6] P A Ioannou, J Sun. Robust Adaptive Control. Englewood Cliffs, NJ:Prentice-Hall, 1995.

[7] L X Wang. Stable adaptive fuzzy control of nonlinear systems. IEEETransaction on Fuzzy System. 1993, 1(2):146155.

[8] Tie-Shan Li, Shao-Cheng Tong, Gang Feng. A Novel Robust Adaptive-Fuzzy-Tracking Control for a Class of Nonlinear Multi-Input/Multi-Output Systems. IEEE Transaction on Fuzzy System. 2010, 18(1):150-160.

[9] Hyeongcheol Lee. Robust Adaptive Fuzzy Control by Backstepping fora Class of MIMO Nonlinear Systems. IEEE Transaction on FuzzySystem. 2011, 19(2): 256-275.

[10] H E Merrit. Hydraulic Control Systems. New York: Wiley, 1967.

757

Documents

Synchronization Motion Control for Quad-Cylinder Lift Systems With Acceleration Coupling