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Moving sliding surfaces for high-order variablestructure systemsDong-Won Park & Seung-Bok Choi

Published online: 08 Nov 2010.

To cite this article:Dong-Won Park & Seung-Bok Choi (1999) Moving sliding surfaces for high-order variable

structure systems, International Journal of Control, 72:11, 960-970, DOI: 10.1080/002071799220506To link to this article: http://dx.doi.org/10.1080/002071799220506

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Moving sliding surfaces for high-order variable structure systems

DONG-WON PARK* and SEUNG-BOK CHOI

A movingsliding surface (MSS) was proposedearlier for thesecond-order variable structure control system(VSCS). The

MSS was designed to pass arbitrary initial conditions, and subsequently moved towards a predetermined sliding surfaceby rotating and/or shifting. This methodology led to fast and robust control responses of the second-order VSCS,especially in a reaching phase. However, the moving algorithm of the MSS was too complicated to be employed forthe high-order VSCS. To resolve this problem, a new moving algorithm based on fuzzy theory is proposed in this paper.For the generalization of the MSS, the conditions for rotating or shifting are rstly investigated. Then the fuzzy algor-ithm is formulated by adopting the values of the surface functionand the total discontinuity gain as input variables andthe variation of the surface functionas output variable. The eciency of the proposed movingalgorithmis illustrated byapplying it to the position control problem of an electrohydraulic servomechanism.

1. Introduction

The sliding mode, which can be obtained by anappropriate discontinuous control law, is the principal

operation mode in the variable structure control system(VSCS). In the sliding mode, the trajectory of the statevector belongs to the hyperplane of lower dimensionthan that of the whole state space. Thus, the order ofdierential equations describing sliding motions is alsoreduced. This allows the initial control problem to bedecoupled into two independent lower dimensional sub-problems; the controller is activated only for creatingthe sliding mode, while the required characteristics ofthe motion over the intersection of the hyperplanes areprovided by a suitable choice of their equations.

Consequently, the design of the sliding hyperplanes isvery important as well as the design of the controller.Thus, the design method for the sliding hyperplane hasbeen investigated in several ways (Utkin 1978,Ashchepkov 1983, Dorling and Zinober 1986,Harashima et al. 1986, Chang and Hurmuzlu 1992,Park and Lee 1993, Choi and Park 1994, Ackermannand Utkin 1994, Guldner and Utkin 1996).

The robustness of the VSCS can be improved byshortening the time required to attain the slidingmode, or may be guaranteed during whole intervals ofcontrol action by eliminating the reaching phase.Among many methods to minimize the reachingphase, the simplest one is to employ a large controlinput. However, this will cause higher chattering,which is undesirable in a physical system, and alsoextreme sensitivity to unmodelled dynamics. Most ofthe sliding surfaces proposed so far have been designedwithout consideration of initial conditions, whether

given or arbitrary. This can make the VSCS very sensi-tive to extraneous disturbance and parameter uncer-tainty during the reaching phase.

The time-varying sliding surface in the state-spacewas introduced to maintain the sliding mode duringtracking control (Utkin 1978). Recently, some research-ers introduced a few kinds of time-varying sliding sur-faces in the error-space, but most of those slidingsurfaces use time-varying intercepts of the sliding sur-faces (Chang and Hurmuzlu 1992, Park and Lee 1993,Ackermann and Utkin 1994, Gulder and Utkin 1996).In addition, thesemethods needmodication of the con-trol law and hence require a larger control input thanthat of the VSCS with a conventional sliding surface.Harashima et al.(1986) suggested a time-varying slidingsurface to improve the robustness, whichwas realized byseveral sliding lines with dierent slopes. The robustnesswas improved by the chattering action at each slidingline. However, they did not study whether faster controlresponses could be obtained by proper rotation of thesliding surface.

Choi et al.(1993) suggested a moving sliding surface(MSS) to improve the performance and robustness ofthe second-order VSCS. The MSS was designed topass the initial conditions at rst and subsequentlymovedtowards a predetermined sliding surfaceby rotat-

ing and/or shifting. Employing the MSS, it was possibleto lessen the sensitivity of the system to extraneous dis-turbances by means of shortening the reaching phasewithout increasing undesirable chattering of the controlsignals. Furthermore, the reaching phase can be almosteliminated byincreasingthe dwelling time of the surface,hence guaranteeing system robustness during wholeintervals of control action. It has also been shown thatthe MSS could be applied to both single-input andmulti-input systems.

There are many superiorities of the VSCS with the

MSS such as fast tracking performance, short reaching

International Journal of ControlISSN 00207179 print/ISSN 13665820 online 1999 Taylor & Francis Ltdhttp://www.tandf.co.uk/JNLS/con.htm

http://www.taylorandfrancis.com/JNLS/con.htm

INT. J. CONTROL,1999, VOL.72, NO.11, 960970

Received 10 June 1997. Revised 1 December 1998. Department of Mechanical Engineering, Inha University,

Incheon 402-751, South Korea.

*Author for correspondence.


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phase and so on. However, the movement of the MSSproposed by Choiet al.(1993) depends on exact calcula-tions of the surface equations to make the surface func-tion have a value in a certain range. Hence, thegeneralization of the MSS to higher-order systems isrestricted by the complexity of the equations. To resolvethis problem, a simplied moving algorithm based onfuzzy theory is proposed in this paper. In other words,

this work is an extension of the MSS for the second-order VSCS to the MSS for high-order variable struc-ture systems. This algorithm uses the values of the sur-face function and the discontinuity gain as the fuzzyinput variables, and the variation of the surfacefunctionas the fuzzy output variable. The translationof the vari-ation of the surface function to the variation of theintercept or the desired eigenvalue is also studied. Inthe second-order case, we shift the sliding surface ifthe representative point (RP) is in the rst or third quad-rants, and rotate otherwise. Since the conditions for thedetermination of manner of motion are indispensablefor the generalization of the MSS, they are thoroughlyinvestigated in this paper. As a result, specic conditionsfor third-order and fourth-order systems are derived.The electrohydraulic servomechanism, which is mod-elled as the third-order system, is adopted to demon-strate the eciency of the proposed methodology.

2. Review of variable structure control system

2.1. Controller design

Consider an nth-order single-input non-linear con-trollable canonical form system given by

x1= x2x2= x3

.

.

.

xn=p

i=1

fi(x, t)+q

i=1

daigi(x, t)+b(x, t)u(t)+d(t)

(1)

where x(t) 2 R n is the state vector and u(t)is the controlinput. The daigi(x, t) represents the plant uncertaintyand d(t) denotes the external disturbance. These areunknown but possibly bounded. It is well known thata multi-input multi-output (MIMO) non-linear systemcan be I/O (input/output) linearized, and can bedecoupled if it has some (vector) relative degree at apointx =x 0(Isidori 1985). Therefore the MIMO non-linear system can be expressed in the form of equation(1) (Elmali and Olgac 1992). The control problem is toget x(t)= [x1(t), . . ., xn(t)]

T to track a desired trajectoryxd(t)= [xd1(t), . . ., xdn(t)]

T which belongs to the class ofC

1 functions on[t0,1). In other words, the variable

structure control law should force the tracking error to

be asymptotically zero for arbitrary initial states. Thus,we dene the tracking error as

e(t)= [e1(t), . . ., en(t)]T

= [x1(t) xd1(t), . . ., xn(t) xdn(t)]T (2)

We dene the sliding surface asfe 2 Rn j s(e)= 0g,

and we denote

s(e)= c e =n

i=1

ciei, cn= 1 (3)

The coecients of the sliding surface for a high-ordersystem can be determined by the set of desired eigen-values f 1, 2, . . .,n1g (Chern and Wu 1991). Letp( )be the desired characteristic polynomial, i.e.

p( )=n1

i=1

( i)=n1 +dn2

n2 + +d1 +d0

(4)

Then, to make the VSCS have the desired eigenvaluesduring the sliding mode, we choose the coecient vectorof the sliding surface c as follows

c = [d0 d1 . . . dn2 1] (5)

If the directions of motion along every trajectory ofeither side of the sliding surface are toward this surface,the RP remains on the surface, i.e. sliding motionoccurs. Thus, the condition for the existence of slidingmotion in the vicinity of the sliding surface s(e)= 0 is

s(e)s(e)


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u(t)= k +q

i=1

j ~gi(x, t)j sgn(s)p

i=1

fi(x, t)

q

i=1

gi(x, t)n1

i=1

ciei+1+ xdn [b(x, t)] (9)

where

gi(x, t)= aigi(x, t), ai=(ldai+ hdai)/2~gi(x, t)= ~aigi(x, t), ~ai= hdai ai

k>max(jldj, jhdj)

(10)

2.2. Moving sliding surface for the second-order VSCS

As mentioned in the Introduction, the performanceof the VSCS associated with the conventional slidingsurface (3) is very sensitive to the surface coecient ci,the discontinuity gain kand the initial conditions. To

observe this, we consider the nominal control system of(1) and (9) in the second-order case. Then the trackingproblem reduces to the following equations

e1= e2

e2=ce2 k sgn(s(e))(11)

We see from the second equation of (11) that if the RPlies in the stable zone (the second and fourth quadrants)the absolute value ofe2becomes small for a xed gain kas we choose a larger value forc. A large value ofcwill

yield a longer reaching time of the RPto the surface. Onthe other hand, if we choose a small value for c, it willtake a long time to track because of the slow conver-gence speed on the surface. Therefore, we can nd anoptimal (xed) sliding surface for an initial condition(Choi and Park 1994). Further, if we choose an appro-priate time-varying slope according to the trajectory ofthe RP, we can obtain faster and more robust controlresponses of the system than those of the system havingan optimal xed sliding surface.

Toachieve this, the moving sliding surface (MSS) forthe second-order VSCS was proposed as follows (Choi

et al.1993)

sm(e, t)= c(t)e1(t)+e2(t) a(t) (12)

The MSS was initially designed to pass arbitrary initialerror states, and subsequently moved towards a prede-termined sliding surface. The movement can be executedby rotating and/or shifting. The rotating is associatedwith the time-varying slope(c(t)) of the surface, whichbelongs to a step function(see Apostol 1974) or piece-wise constant function, whereas the shifting is accom-plished by employing the time-varying intercept(a(t))

of the surface which also belongs to a step function.

Figure 1 illustrates the two patterns of movement ofthe sliding surface: rotating and shifting.

For the second-order system, the conditions forrotating or shifting can be determined from intuition.If the initial conditions are located in the unstablezone, the surface is shifted until the RP moves to thestable zero, and subsequently the surface is rotated

throughout. If the initial RP is in the unstable zone,

962 D.-W. Park and S.-B. Choi

Figure 1. Illustrattion of two typesof movingslidingsurface:(a) rotating sliding surface; (b) shifting sliding surface.


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the RP cannot converge to the origin without enteringthe stable zone. This moving algorithm depends onexact computations of the surface equation, whichmake the surface function remain in a certain vicinityuntil the MSS reaches the predetermined surface.However, the equations for high-order systems are toosophisticated to be solved repeatedly during the periodof rotation or shift. Therefore, a new moving algorithm

is to be established for the generalization of the MSS.

3. Moving sliding surface for a high-order VSCS

The MSS (12) can be generalized for the controlsystem (1) as follows

sm(e, t)=n

i=1

ci(t)ei(t) a(t), cn= 1 (13)

whereci(t)and a(t)arestep functions. It is noted thatsince ci(t)and a(t)are piecewise constant, the controller

(9) associated with (13) satises the sliding mode con-dition (6) at almost all times (Choi et al.1993). To sim-plify the rotating procedure, the sliding surface isassumed to have equal time-varying desired eigenvalues.Then,ci(t)becomes

ci(t)=n 1i 1

( d(t))ni

=(n 1) !( d(t))

ni

(i 1)!(n i) !,

i= 1, . . ., n 1 (14)

Now, the sliding surface (13) can be rotated by changingthe desired eigenvalue d(t), and can be shifted by

adapting the intercept a(t). In the second-order case,we rotate the sliding surface when the RP is in the sec-ond or fourthquadrant. In the high-order case, it is verydicult to determine whether rotating or shiftingshouldbe executed. Thus, the conditions for rotating or shiftingare, rstly, to be investigated. Then, a new fuzzy algor-ithm is proposed to determine appropriated(t)or a(t).Using the proposed algorithm, we can rotate or shift thesliding surface from the initial surface to the predeter-mined surface, maintaining the RP in a certain vicinityof the surface.

3.1. The conditions for rotating or shifting

The dynamics of the system in the sliding mode isruled by the sliding hyperplane. Hence, the error-stateequations of the sliding mode are

e1=e2

e2=e3

.

.

.

en=

c1

e2

c2e

3 c

n1e

n

(15)

The system (15) has a zero-eigenvalue, which does notappear in the sliding mode. Thus, the dimension of thecharacteristic equation reduces to n 1 (Dorling andZinober 1986). Therefore, the characteristic equationof the system in the sliding mode is

p( )=n1

i=1

( di)= n1 +cn1

n2 + +c2+c1

(16)Let us assume that the desired eigenvalues of the slidingsurface are the same, i.e. di d for i= 1, . . ., n 1.The characteristics of the systemin the sliding mode canbe determined by the desired eigenvalue d. However,we can obtain better control response by changing thevalue of d, i.e. by rotating the sliding hyperplane. Weshould maintain the RPin a certain vicinity of the slid-ing hyperplane to get fast and robust control responsesduring the rotation. To nd an appropriate value ofdfor a certain time t, we rewrite the characteristic equa-

tion of an nth-order system in the sliding mode (16),using the polynomial ofdas follows

f( d, t) qn1n1d + +q1d+ q0 (17)

where

qni=(1)i1(n 1)!(i 1)!(n i) !

ei(t) (18)

If equation (17) has a negative real root for the statevariable of the system, there is a sliding hyperplanewhich guarantees the stability of the VSCS and crosses

the RP. We rotate the sliding hyperplane from the initialRP to the predetermined sliding hyperplane, maintain-ing the RPin a certain vicinity of the sliding hyperplane.If equation (17) does not have any negative real roots,the sliding surface should be shifted to the predeter-mined sliding surface until (17) has a negative realroot, so as not to obstruct the stability of the systemduring the reaching phase.

If the sliding mode controller is designed to satisfythe sliding mode condition strictly, the sliding hyper-plane should not rotate or shift in the reverse direction.Hence, the initial sliding hyperplane moves to the pre-

determined sliding hyperplane by a nite amount and ina nite time of movement. It is obvious that a nite timeof nite movement of the stable sliding hyperplane doesnot aect the stability of the variable structure controlsystem (Choi et al.1993, Filippov 1964). Inother words,the stepwise variation ofdguarantees the stability ofthe variable structure control system ifdis negative.

In the case that the polynomialf( d)is odd-ordered,there is a negative real root if the signs of the coecientof the highest-order termand the intercept are the same.On the other hand, when the polynomial f( d)is even-

ordered, there is a negative real root if the signs of the

Moving sliding surfaces 963


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coecient of the highest-order term and the interceptare dierent. Since the sign of q0in (17) alternates asthe order increases, we can obtain the following con-dition

e1en


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Now, the linguistic input variables are dened todescribe jsmjandg(x, t)as follows

~s = fVS, SM,ME,L A,V Lg

~g= fVS, SM,ME,L A,V Lg (29)

where VS=very small, SM= small, ME=medium,L A =largeand VL=very large, respectively. The lin-guistic output variable is also dened to describe theamount of rotating or shifting as follows

~s =fVS, SM,ME,L A,V Lg (30)

All of the input and output fuzzy variables are chosen tobe positive. This is possible because we can know the

sign of the fuzzy variables from the direction of rotatingor shifting and it is useful to reduce the calculation timeand requiredmemory. An example of the fuzzy variablesis illustrated in gure 3, which will be used in the appli-cation of the electrohydraulic servomechanism. Theinputoutput relation of the fuzzy algorithm withfuzzy variables (29) and (30) is given by

~s, ~g! ~s (31)

Table 1 presents an example of formulated fuzzy control

rulesR11, R12, . . ., R55, i.e.

R11: IF~sis VSand ~g is VS,THEN ~sis VS

R12: IF~sis VSand ~g isSM,THEN ~sisSM

.

.

.

R55: IF~sis V Land ~g is V L,THEN ~sis VS

(32)

This fuzzy algorithmcan be inferred fromthe centre-of-gravity method (Jamshidi et al.1993).

Because we dened the MSSas (13),j smjshouldbetranslated to a for the shifting sliding surface and dfor the rotating sliding surface. To obtain the relation-ship between j smj and aor d, we calculate thepartial derivatives of smwith respect to aand das follows

sm

a =1 (33)

smd

= s0m=

n1

i=1

(n 1)!( d(t)jni1

(i 1)!(n i 1)!ei (34)

Fromthese equations, we can obtain the following rela-tionships

a = ws sm (35)

d= wm sm/s0

m (36)

where wsandwm are scalingfactors which depend onthetime-step and the characteristics of the system. Whens

0

m=0,

dcan be obtained by multiplying a pre-

scribed large number. Figure 4 shows a ow chart ofthe proposed control scheme characterizing the fuzzymoving algorithm.

4. Application to the position control of an

electrohydraulic servomechanism

There are many distinct advantages of a hydrauliccontrol system, such as higher speed of response withfast motions, speed reversals, higher torque stiness and

continuous operation (Merrit 1976). However, some dis-


s m

VS SM ME LA VL

0.0

1.0

g

m

0.20

VS SM ME LA VL

0.0

1.0

0.0Ds m

m

(x)3000 6000 90000.0

0.05 0.10 0.15

4.0

VS SM ME LA VL

0.0

1.0

0.0

m

2.0 3.01.0

0.25

Figure 3. Fuzzy variables for the electrohydraulic servo-mechanism.

~g

~s V S SM ME L A V L

VS VS SM ME LA VL SM VS VS SM ME LAME VS VS VS SM SM L A VS VS VS VS VS VL VS VS VS VS VS

Table 1. Fuzzy rule base of MSS for the electrohydraulicservomechanism.


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advantages tend to limit its use; for example, the highcost of hydraulic components for small allowable toler-ance results, the upper temperature limit and non-linearphenomena, the uid compressibility, and the deadbanddue to internal leakage and hysteresis (Larry 1988).These drawbacks make the control of a hydraulic con-trol system dicult in the sense of implementation.

Furthermore, it has been shown by Petersen (1985)that some uncertain dynamics cannot be quadraticallystabilized via a linear control but admit a non-linearcontrol such as sliding mode control.

Hwang and Lan (1994) studied the position controlof the electrohydraulic servomechanism via the slidingmode controller. In that paper they used a large dis-continuity gain to obtain fast control responses. Asmentioned in 1 of this paper, the increment of thediscontinuity gain may cause extreme sensitivity tounmodelled dynamics, undesirable higher chattering

and large error bound in the steady state. In this

study, we obtain fast response by employing the pro-posed MSS with relatively small discontinuity gain.

4.1. Problem formulation

The positioncontrol block-diagram of the electrohy-draulic servomechanism is shown in gure 5 (Hwangand Lan 1994). The relationship between the valve dis-

placement Xv (in) and the load ow Ql (in3 s1) isdescribed in the following servovalve ow equation

Ql=KqXv KcPl (37)

where Kc is the valve ow-pressure coecient(in3 s1 psi1), Pl is the load pressure dierence (psi)andKqis called the valve ow gain, given by

Kq= CdW [Ps sgn (Xv)Pl]/q (38)

Here Cdis the discharge coecient (dimensionless), Wisthe area gradient (in), q is the uid mass density

(lbs2

in4

)andPsis the supply pressure (psi).


Initial Hyperplane

Determination of Moving Manner

Initialization of Surface Gradient

Calculation of g (x, t)

Current Moving Manner

Fuzzy Rule (Calculate Dld)

Update ld(ld+=Dld)

e t e tn1 0( ) ( ) 1 01n

de f l

N

Y

e t e tn1 0( ) ( )

Predetermined Hyperplane

Figure 4. Flow chart of the control scheme with fuzzy algorithm.


9/12

The continuity equation to the motor chamber isgiven by

Ql=Dm

m+CtmPl+(Vt/4be)

Pl (39)whereDmis the volumetric displacement of the motor(in3 rad1),mis the angular position of the motor shaft(rad), Ctm is the total load leakage coecient of themotor (in3 s1 psi1), Vtis the total compressed volume(in3)andbeis the eective bulk modulus of the system(psi).

The torque balance equation for the motor is givenby

PlDm=Jtm+Bmm+Gm+Tl (40)

whereJtis the total inertia of motor and load (in lbs2),

Bm is the viscous damping coecient of the load (in lbs),G is the torsional spring gradient of the load(inlbrad1) and Tlis the arbitrary load torque on themotor (in lb).

Substituting (37) into (39) gives

KqXv=Dmm+(Kc+Ctm)Pl+(Vt/4be) Pl (41)

By combining (37)(41), a set of state equations of the

electrohydraulic servomechanism model can be achievedas follows

x1=x2(t)

x2=x3(t)

x3=3

i=1

ai(t)xi(t)3

i=1

dai(t)xi(t)+b(x)u(t) d(t)

(42)

where

x(t)= [x1(t) x2(t) x3(t)]T

= [m(t) m(t) m(t)]

T

a1(t)=4beVt

KceJt

G

a2(t)=G

Jt+

4beVt

D2m

Jt+

4beVt

KceJt

Bm

a3(t)=BmJt

+4be

VtKce

b(x)=4beVt

DmJt

KqKv

d(t)=4beVt

KceJt

Tl+1Jt

Tl

dai(t) :the parameter uncertainty

Kce= Kc+Ktm: the total flow-pressure coecient

(in3 s1 psi1),

Kv : the servovalve gain

Tl=f(m)(43)

From (42) and (43), it can be seen that the dynamics ofthe electrohydraulic servomechanism is highly non-lin-ear(seeb(x)) , time variant(seeai(t) = 1, 2, 3)and sub-

jected to external load disturbances.

4.2. VSCS design

First, let us dene the error state vector e as follows

e = [e1 e2 e3]T

= [x1 xd1 x2 xd2 x3 xd3]T

(44)

Then the conventional sliding surface is dened by thefollowing equation

s =c1e1+c2e2+e3= 0 (45)

wherec1andc2are determined from (4) and (5).Let the uncertainties of the system dai(t), i= 1, 2, 3,

and disturbance d(t)be assumed to satisfy the condition(8); then sliding mode would occur if we use a controllerof the form (9), i.e.

u(t)= k +3

i=1

j ~gi(x, t)j sgn (s)3

i=1

fi(x, t)

3

i=1

gi(x, t) c1e2 c2e3+ xd3 [b(x, t)] (46)

wheregi(x, t)and ~gi(x, t)are dened in (10).We dene the moving sliding surface for this system

in the form of (13) as

sm= c1(t)e1+c2(t)e2+e3 a(t)= 0 (47)


VSCq +

-

K Ku X Q

1/D 1/s

K C V s

4J s + B

b

q q

T

G

P+

+

+

ed v lv q m

m m

c tmt

e

t m

1/Dm

l

l

+

-

+

-+

+

Figure 5. Position control block diagram of the electrohy-draulic servomechanism.


10/12

Here the time-varying coecients c1(t) and c2(t) aregiven in equation (14). The fuzzy variables areillustrated in gure 3, and the linguistic fuzzy rulebase for the moving algorithm of the MSS is shown in

table 1.

4.3. Simulation results and discussion

For the computer simulations, the initial state istaken as x0= [0 0 2]

T. The desired angular posi-tion is selected as d= 1rad. All of the desired eigen-values of the xed (conventional) sliding surface and ofthe predetermined sliding surface for the MSS are cho-sen tobe 25. The nominal values of the electrohydrau-lic servomechanism parameters are listed in table 2. Theparameter uncertainties and external load disturbances

are conned as follows~ai(t)= 0.3jai(t)j, i= 1, 2

max (jldj , jhdj)= 1000(48)

where a1= 0.04, a2= 12000, a3= 170. The electrohy-draulic servomechanismis simulated with a time intervalof 0.001s using the RungeKutta method. The scalingfactors of the fuzzy algorithm wsand wmare selected as1.0 and 3.2, respectively.

Figure 6 illustrates the angular position trajectories

and surface histories of the electrohydraulic control


Parameter Value Unit

Ps 2000 psiKq 0.01 Ps sgn(Xv)Pl in2 s1be 50000 psiVt 10 in

3

Kce 0.001 in3 spsi

Dm 0.5 in3 rad1

Jt 0.5 in2 lbs2

Bm 75 in2 lbs

Kv 20 in2

V1

G 0.01 in2 lbrad1

Table 2. Nominal values of the electrohydraulic servo-mechanism.

0 1 2 3 4 5

-9

-6

-3

0

3

: MSS with fuzzy algorithm (k=1200)

Time (sec)

s

(t)

-800

-600

-400

-200

0

20 0

s

(t)

: fixed sliding surface (k=12 00)s

(t)

0 1 2 3 4 5

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

(t)

q

: fixed sliding surface (k=1200): fixed sliding surface (k=2400)


An

gularposition

Time (sec)

-800

-600

-400

-2000

200

: fixed sliding surface (k=240 0)

Figure 6. Control responses of the electrohydraulic system.

0 1 2 3 4 5

-0.2

0.0

0.2

0.4

0.6


Time (sec)

u

(t)

-0.2

0.0

0.2

0.4

0.6

u

(t)

: fixed sliding surface (k=1200)

u

(t)

0 1 2 3 4 5

0

3000

6000

9000

12000

Time (sec)


0

3000

6000

9000

12000

(x)

g

(x)

g

(x)

g


0

3000

6000

9000

12000


-0.2

0.0

0.2

0.4

0.6

: fixed sliding surface (k=2400)

Figure 7. Control histories of the electrohydraulic system.


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system. In this gure, we can see that the control per-formance is remarkably enhanced by employing theMSS, and that the fuzzy moving algorithm is success-fully executed. For example, the tracking time of thesystem with the MSS (47) is 0.587s whereas that ofthe system with the xed sliding surface (45) is 3.194s(fork= 1200) and 0.619s (fork= 2400).

Figure 7 shows the control histories and the vari-

ations of the discontinuity gain. Clearly, in the steadystate the discontinuity gain of the VSCS using the MSSis the same as that of the VSCS employing the xedsliding surface with k= 1200. Thus, the controlresponse is improved by using the MSS without entail-ing a larger limit cycle or steady-state error. If weincrease k to 2400 for the xed sliding surface, boththe discontinuity gain and the chattering magnitudeare twice as large as those of the VSCS with k= 1200,in the steady state. Furthermore, we can see that thecontrol responses of the VSCS with the xed sliding

surface depends heavily on the discontinuity gain.The control results presented in this study are quiteself-explanatory in that the proposed MSS for the high-order VSCS is very eective for achieving fast androbust responses. Figure 8 presents the trajectories ofthe desired eigenvalues and the coecients of the slidingsurfaces. Hence, c1(t)and c2(t)are dened as2 d(t)and 2d(t), respectively. Figure 9 shows some sampledsliding hyperplanes to illustrate the rotating procedurein the electrohydraulic servomechanism. The slidinghyperplane 1 is the initial sliding hyperplane, 2 is


0 1 2 3 4 5

0

20

40

60

Time (sec)

c2

(t)

0

20 0

40 0

60 0

80 0

c1

(t)

0 1 2 3 4 5

-3 0

-2 5

-2 0

-1 5

-1 0

-5

0

: fixed sliding surface

: MSS with fuzzy algorithm


: MS S with fuzzy algorithm

(t)

l


: MSS with fuzzy algorithm

Time (sec)

Figure 8. Histories of the desired eigenvalues andcoecientsof the sliding surfaces.

Figure 9. Rotation of the sliding hyperplane of the electrohydraulic servomechanism.


12/12

the hyperplane at0.002s,3 is the one at0.037s, and theothers were sampled at intervals of 0.05s.

It is nally remarked that the computer program tosimulate this control system is written in C++ , and exe-cuted byIntel 80486 DX2TM 66MHz CPU with BorlandC++ 3.1TM compiler. The times required to compute thecontrol input and the sliding hyperplane movement are0.17ms and 0.1 ms, respectively. This indicates that the

proposed control scheme can be employed in realisticsystems.

5. Conclusions

A new moving algorithm for the moving sliding sur-face (MSS) has been proposed on the basis of fuzzytheory for application to high-order variable structuresystems. The values of the surface function and discon-tinuity gain are used as input variables, and the vari-ation of the surface function as output variable. Theconditions for rotating or shifting the surface were

investigated. As a result, the specic conditions forthird- and fourth-order systems were derived. In addi-tion, the position control problem of the electro-hydraulic servomechanism was adopted in order todemonstrate the eciency and feasibility of the pro-posed MSS. The control performance of the systemwas enhanced remarkably by employing the MSS with-out entailing an increment of the discontinuity gain inthe steady state. It has also been demonstrated that theMSS is successfully implemented in the high-orderVSCS with the fuzzy moving algorithm and the con-

ditions for rotating or shifting.A study on an adaptive fuzzy or neuro-fuzzy algor-ithm to avoid the trial and error procedure in decidingthe fuzzy variables and fuzzy rule base for the MSS isbeing undertaken.

References

Apostol, T. M.,1974, Mathematical Analysis (New York:Addison-Wesley).

Ackermann, J, and Utkin, V. I., 1994, Sliding modecontrol based on Ackermanns formula. Proceedings of the33rd IEEE Conferenceon Decision and Control, vol. 4, Kobe,Japan, pp. 36223627.

Ashchepkov, L. T., 1983, Optimization of sliding motions ina discontinuous systems. Automation and Remote Control,44 , 14081415.

Chang, T. H., and Hurmuzlu, Y.,1992, Trajectory trackingin robotic systems usingvariable structure control without a

reaching phase. Proceedings of the 1992 American ControlConference, Vol. 2, Chicago, Illinois, pp. 15051509.

Chern, T.L., and Wu,Y.C.,1991, Design of integral variablestructure controller and application to electrohydraulic vel-ocity servomechanism. IEE ProceedingsD, 138, 439444.

Choi, S. B., and Park, D. W., 1994, Moving sliding surfacesfor fast tracking control of second-order dynamical systems.ASME Journal of Dynamic Systems, Measurement, andControl, 116, 154158.

Choi, S. B., Cheong, C. C., and Park, D. W., 1993, Moving

switching surfaces for robust control of second-order vari-able structure systems. International Journal of Control, 58 ,229247.

Dorling,C.M., and Zinober,A.S. I.,1986, Twoapproachesto hyperplane design in multivariable variable structuresystems.International Journal of Control, 44 , 6582.

Drazenovic, B., 1969, The invariance conditions in variablestructure systems. Automatica, 5, 287295.

Elmali, H., and Olgac, N., 1992, Robust output trackingcontrol of nonlinear MIMO systems via sliding mode tech-nique.Automatica, 28 , 145151.

Filippov, A. F.,1964, Dierential equations with discontinu-ous right-hand side. American Mathematical Society

Transactions, 42

, 199231.Guldner,J., and Utkin, V. I.,1996, Trackingthe gradient ofarticial eld: sliding mode control for mobile robots.International Journal of Control, 63 , 417432.

Harashima, F., Hashimoto, H., and Maruyama, K., 1986,Sliding mode control of manipulator with time-varyingswitching surfaces, Transactions of the Society ofInstrument and Control Engineers, 22 , 335342.

Hwang, C. L., and Lan, C. H., 1994, The position control ofelectrohydraulic servomechanism via a novel variable struc-ture control. Mechatronics, 4, 369391.

Isidori, A.,1985, Nonlinear Control Systems, An Introduction(Berlin: Springer-Verlag).

Jamshidi, M.,Vadiee, N., and Ross, T. J., 1993,Fuzzy Logic

and Control: Software and Hardware Applications(Englewood Clis, NJ: Prentice Hall).Jayasuriya, S., and Choi, S. B.,1987, On the suciency con-

dition for existence of a sliding mode. Proceedings of the1987 American Control Conference, Vol. 1, Minneapolis,Minnesota, pp. 8489.

Larry, S., 1988, Precision control with hydraulics. MachineDesign, 12 , 6467.

Merrit, H. E., 1976, Hydraulic Control System(New York:John Wiley).

Park, K. B., and Lee,J.J.,1993, Variable structure controllerfor robotic manipulators using time-varying sliding surface.Proceedings of the 1993 IEEE Conference on Robotics andAutomation, Vol. 1, Atlanta, Georgia, pp. 8993.

Petersen, I. R., 1985, Quadratic stabilizability of uncertainlinear systems: existence of a nonlinear stabilizing controldoes not imply existence of a linear stabilizing control.IEEE Transactions on Automatic Control, AC-30, 291293.

Utkin, V. I., 1978, Sliding Modes and their Applications inVariable Structure Systems(Moscow: Mir).

970 Moving sliding surfaces

dong-won park & seung-bok choi 15

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