Research ArticleObserver Based Robust Position Control of a Hydraulic ServoSystem Using Variable Structure Control

E Kolsi-Gdoura M Feki and N Derbel

Research Group CEMLab National Engineering School of Sfax University of Sfax 1073 Sfax Tunisia

Correspondence should be addressed to M Feki moezfekienigrnutn

Received 8 May 2015 Revised 9 July 2015 Accepted 12 July 2015

Academic Editor Rongwei Guo

Copyright copy 2015 E Kolsi-Gdoura et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper deals with the position control of a hydraulic servo system rod Our approach considers the surface design as a case ofvirtual controller design using the backsteppingmethodWe first prove that a linear surface does not yield to a robust controller withrespect to the unmatched uncertainty and perturbation Next to remedy this deficiency a sliding controller based on the second-order slidingmode is proposedwhich outperforms the first controller in terms of chattering attenuation and robustness with respectto parameter uncertainty only Next based on backstepping a nested variable structure design method is proposed which ensuresthe robustness with respect to both unmatched uncertainty and perturbation Finally a robust sliding mode observer is appendedto the closed loop control system to achieve output feedback control The stability and convergence to reference position with zerosteady state error are provenwhen the controller is constructed using the estimated states To illustrate the efficiency of the proposedmethods numerical simulation results are shown

1 Introduction

Actually the hydraulic servo systems are very popular inseveral industrial applications such as robotics aerospaceflight-control actuators heavy machinery aircrafts automo-tive industry and a variety of automated manufacturingsystems This is mainly due to their ability to producehigh power and accurate and fast responses However thesesystems have a high nonlinear behavior due to the pressureflow characteristics [1] and the leakage model inside theservovalves [2]This fact makes the control design for preciseoutput tracking a very challenging task

Owing to their simplicity linear controllers of PID type[3 4] inputoutput linearization controllers [5ndash9] and alsosliding mode controllers (SMC) [10ndash13] have been used tocontrol the hydraulic servo systems However such con-trollers were designed based on the plant physical modeland therefore the plant parameters knowledge is requiredConsequently they were shown to be highly sensitive tomismatched perturbation and uncertainties thus resulting inperformance degradation

To improve the controller performances several strate-gies have been adopted such as using the self-tuned PID

controller [14 15] and nonlinear adaptive controllers [16ndash18] SMC appended with some improvements have also beenused In [19 20] SMC method has been combined with anadaptive controller which can compensate for the systemuncertain nonlinearities for linear uncertain parametersand especially for the nonlinear uncertain parameters toconstruct an asymptotically stable tracking In [21] SMChas been used with the PID controller to achieve con-trol of asymmetrical hydraulic cylinder trajectory trackingTo drive electrohydraulic actuators various robust controltechniques such as H2 and H

infincontrols were applied

[22ndash24] This approach enabled the compensation for theinherent nonlinearities of the actuator and rejects matchedexternal disturbances and attenuates mismatched externaldisturbances To cope with mismatched disturbances authorsused the integral SMC and to remedy the slow response dueto windup phenomenon a realizable reference compensationhas been used to achieve fast position tracking [25 26]Since it has been proposed by Levant [27 28] higher-orderSMC (HOSMC) has been widely used to control electricaldrives [29 30] electropneumatic actuators [31] and electro-hydraulic actuators [32 33]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 724795 11 pageshttpdxdoiorg1011552015724795

2 Mathematical Problems in Engineering

In the present paper we are interested in controlling theposition of the rod in a hydraulic servo system that consistsof a four-way spool valve supplying a double effect linearcylinder with a double-rodded piston The piston is drivinga load modeled by a mass a spring and a sliding viscousfriction Our work aims to design a controller that mayachieve the reference position in presence of mismatchedparameter uncertainty and perturbation in addition to actua-tor saturation To realize this objective we start in the secondsection by formulating the problem and presenting the effectsof using first- and second-order SMCwith a linear surface InSection 3 we present the design of a sliding surface obtainedusing backstepping method and variable structure controllerleading hence to a nonlinear surface that allows achievingthe reference output despite the presence of uncertainties andperturbations Numerical simulation results are presentedto illustrate the efficiency of the proposed control designIn Section 4 we present a sliding observer and prove theconvergence of the observer as well as the exact positiontracking using the nonlinear surface SMC issued from theobserver states Finally the conclusion and some remarks arepresented in Section 5

2 Problem Statement

The electrohydraulic system that we will deal with in thispaper is depicted in Figure 1 and modeled by the dynamicalsystem (1) It has been shown in [34] that for the symmetricalpiston with equal surfaces 1198781 and 1198782 and assuming equalvolume flow passing through (geometrically) identical portswe can describe the system by three variable states wherea differential pressure state substitutes the pressure of eachchamber This decrease in the system dimension ensures theobservability when the system output is the piston positionConsider

1 =4119861119881119905

(119896119906radic119875119889minus sign (119906) 1199091 minus

12057211990911 + 120574 |119906|

minus 1198781199092) (1)

2 =1119898119905

(1198781199091 minus 1198871199092 minus (119896119897 +Δ119896119897) 1199093) (2)

3 = 1199092 +119889 (119905) (3)

where 1199091 = 1198751 minus 1198752 denotes the difference in pressureinside the two chambers of the cylinder 1199092 and 1199093 respec-tively denote the velocity and the position of the rod and|119889(119905)| lt 119889max is a bounded constant or slowly varyingexternal perturbation In fact from Newtonrsquos law a constantforce perturbation leads to a constant acceleration Thus thevelocity perturbation may be interpreted as the result of animpulsive force that acts abruptly on the system119898

119905= 119898+1198980

is the total mass of the rod and the load 119881119905= 1198811 + 1198812 is the

total volume of the cylinder and 119875119889= 119875119904minus 119875119903is the pressure

difference between the supply pressure 119875119904(pressure of the

pump) and the return pressure 119875119903(atmospheric pressure) 119896

119897

is the spring stiffness constant with an uncertainty Δ119896119897and

119887 is the friction coefficient The system parameters used forsimulations are as follows 119861 = 22times109 Pa 119875

119904= 300times105 Pa

119875119903= 105 Pa1198980 = 50 kg 119878 = 15 times 10minus3 m2 119881

119905= 9 times 10minus4 m3

119898 = 20 kg 119887 = 590 kgs 119896119897= 125000Nm 119896 = 362 times

10minus5 m3 sminus1 Aminus1 Paminus12 120572 = 41816 times 10minus12 m3 s minus1 Paminus1 and120574 = 8571 with 120572 and 120574 being intrinsic constants modeling theleakage within the servovalve [2]

Themost difficult aspect in thismodel is the existence of amismatched perturbation as well as a mismatched parameteruncertainty In addition the leakage model includes non-linearity with respect to the control signal 119906 To deal withthis problem we neglect the leakage term in the design butwe consider it in simulation and consider the system as aswitching system between two models The switching aspectmakes the use of the slidingmode approach a good candidateto design a controller 119906 that can drive the rod position to aconstant reference position 1199093ref

21 First-Order SMC TheSMCdesign consists of two phasesIn the first phase the sliding surface is designed such that thesystem is asymptotically stable when it is confined to it and inthe second phase a switching controller is designed to ensurethe existence of the slidingmodeOur idea consists in viewingthe sliding surface design as a special case of backsteppingdesignTherefore at slidingmode 120590(119909) = 0means 1199091 = 119901(119909)and 1199091 can be viewed as a virtual controller to subsystem((2)-(3)) which describes the system behavior on the slidingsurface

Thus should we choose a linear virtual controller

119901 (119909) =1119878(1198961198971199093ref minus11986221199092 minus1198623 (1199093 minus1199093ref)) (4)

we get the sliding surface

120590 (119909) = 1198781199091 +11986221199092 +1198623 (1199093 minus1199093ref) minus 1198961198971199093ref (5)

In sliding mode and if uncertainty and perturbations areneglected the system is a second dimensional linear systemwith the characteristic equation

1199042+1198622 + 119887

119898119905

119904 +1198623 + 119896119897119898119905

= 0 (6)

Using the pole placement method and imposing a stablemultiple pole at 119904 = minus120582 we can determine the controlparameters 1198622 and 1198623

1198622 = 2120582119898119905minus 119887

1198623 = 1205822119898119905minus 119896119897

(7)

Eventually we obtain an exponentially stable system if wecan guarantee the attractivity of the sliding surface 120590(119909) =0 Indeed the attractivity condition 120590(119909)(119909) lt 0 will besatisfied if we choose (119909) = minus119882sign(120590(119909)) where 119882 gt 0is the sliding gain that should be chosen large enough inorder to ensure the attractivity of the surface in presence ofperturbation and uncertainty Using the system model and(5) we can show that 120590(119909)(119909) lt 0 is attained if the slidinggain satisfies

119882 gt10038161003816100381610038161198623119889max

1003816100381610038161003816 +4119861119878119881119905

120572119875119889+

10038161003816100381610038161003816100381610038161003816

1198811199051198622119878119898119905

Δ119896119897

10038161003816100381610038161003816100381610038161003816

(8)

Mathematical Problems in Engineering 3

x2 x3m0

Sk1

m

b

P1 V1 P2 V2

Pr Ps

u

Figure 1 Hydraulic servo system controlled using a servovalve

where 119889max = max119905ge0119889(119905) Consequently by solving (119909) =

minus119882sign(120590(119909)) the control law is expressed as follows

119906 (119909) =

119873(119909)

(4119861119878119896119881119905)radic119875119889minus 1199091

if 119873(119909) le 0

119873(119909)

(4119861119878119896119881119905)radic119875119889+ 1199091

if 119873(119909) lt 0(9)

where

119873(119909) = minus119882 sign (120590 (119909)) minus1198622119898119905

(1198781199091 minus 1198871199092 minus 1198961198971199093)

minus11986231199092 +41198611198782

119881119905

1199092

(10)

Despite the perturbation 119889(119905) and the uncertaintyΔ119896119897 the

variable structure controller defined by (9) with the slidinggain119882 in (8) ensures the existence of sliding mode howeverwe can easily deduce that we do not achieve the referenceoutput because when the system behavior is confined tothe sliding surface (5) the linear virtual controller does notguarantee any robustnesswith respect to the perturbation andthe uncertainty Indeed it can be easily shown that as far asthe sliding motion is preserved the system is asymptoticallystable if the closed loop eigenvalue is chosen such as

120582 gt radic

10038161003816100381610038161003816100381610038161003816

Δ119896119897

119898119905

10038161003816100381610038161003816100381610038161003816

(11)

and in that case the steady state error due to the uncertaintyis given by

119890119896119897=Δ119896119897

119898119905

11205822 + Δ119896

119897119898119905

1199093ref (12)

We similarly can show that the steady state error due to theconstant perturbation is given by

119890119889= minus

2120582119889 (13)

Figure 2 shows the system behavior with 120582 = 50 119889(119905) =01 Δ119896

119897= 25000 and 1199093ref = 20 cm We clearly notice that

there is a steady state error of more than 2 cm at the positionoutput So the first-order sliding mode is not robust to themismatched uncertainty and perturbation

22 Second-Order SMC Since the system is third-ordersingle input thenwemay think of designing the higher-orderSMC up to second order Let 120590

ℎdenote the sliding variable

defined as

120590ℎ= ℎ (1199093 minus1199093ref) + 1199092 (14)

where ℎ gt 0 is a strictly positive scalar We may verify thatthe relative degree of system ((1)ndash(3)) together with (14) withrespect to the sliding variable 120590

ℎis constant and equal to two

Thus we have

ℎ= 1198861 (119909 119905) + Δ1198861 (119909 119905) + 1198862 (119909) 119906 (15)

where 1198861(119909 119905) and 1198862(119909) are known functions and Δ1198861(119909 119905) isan unknown bounded function in terms of 119889(119905) and Δ119896

119897

1198861 (119909 119905) = minus4119861119878119898119905119881119905

(1205721199091 + 1198781199092)

+ (ℎ

119898119905

minus119887

1198982119905

) (1198781199091 minus 1198871199092 minus 1198961198971199093)

minus119896119897

119898119905

1199092

Δ1198861 (119909 119905) = minus(ℎ

119898119905

minus119887

1198982119905

)Δ1198961198971199093 minus

Δ119896119897

119898119905

1199092

minus119889 (119905)119896119897+ Δ119896119897

119898119905

1198862 (119909) =

4119861119878119898119905119881119905

119896radic119875119889minus 1199091 if 119906 ge 0

4119861119878119898119905119881119905

119896radic119875119889+ 1199091 if 119906 lt 0

(16)

The solution should be understood in the Filippov sense[35] and the trajectories of the system are supposed to beextendible infinitely in time for any bounded measurableinput

4 Mathematical Problems in Engineering

0 01 02 03

03

04 050

50

100

150

200

Time (s)0 01 02 03 04 05

minus1

0

1

2

3

4

Velo

city

(ms

)

Time (s)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

0

5

10

15

20

Posit

ion

(cm

)

minus5

0

5

10

15

20

25

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 2 The behavior of the system under sliding mode controller defined by (9) 120582 = 50

By defining 1199111 = 120590ℎ and 1199112 = ℎ achieving sliding mode120590ℎ= 0 is equivalent to the finite stabilization of the system

1 = 1199112

2 = 1198861 (119909 119905) + Δ1198861 (119909 119905) + 1198862 (119909) 119906(17)

which hence ideally yields to the sliding set120590ℎ= 0 and

ℎ= 0

Taking into account the practical considerations the slidingset is defined as follows [27]

Definition 1 Given the sliding variable 120590ℎ(119909 119905) the ldquoreal

second-order sliding setrdquo associated with ((1)ndash(3)) is definedas

119904119879119890= 119909isin

120594

1003816100381610038161003816120590ℎ1003816100381610038161003816

le 119896111987921198901003816100381610038161003816ℎ1003816100381610038161003816 le 1198962119879119890 (18)

where 119879119890is the finite sampling time (fixed at 119879

119890= 20 120583s in the

sequel) and 1198961 and 1198962 are positive constants

Definition 2 Consider the nonempty real second-order slid-ing set 119904

119879119890given in (18) and assume that it is locally an

integral set in the Filippov senseThe corresponding behaviorof system ((1)ndash(3)) satisfying (18) is called ldquoreal second-ordersliding moderdquo with respect to 120590

ℎ(119909 119905) [27]

The variable structure control law 119906 can be chosen asfollows

119906 =1

1198862 (119909)(minus1198861 (119909 119905) minus11987011199111 minus11987021199112 minus1198821 sign (1199111)

minus1198822sign (1199112)) (19)

where all the controller gains 1198701 11987021198821 and1198822 are strictlypositive Applying controller (19) yields to

[1

2] = [

0 1minus1198701 minus1198702

][1199111

1199112]

+[0 0minus1198821 minus1198822

][sign (1199111)sign (1199112)

] minus [01]Δ1198861 (119909 119905)

(20)

Mathematical Problems in Engineering 5

which can be written in the following compact form

= 119860119870119885+119860

119882sign (119911) minus 119861Δ119886 (119909 119905) (21)

where 119860119870= [

0 1minus1198701 minus1198702

] 119860119882= [

0 0minus1198821 minus1198822

] and 119861 = [ 01 ]We can determine the control parameters 1198701 and 1198702

such that the characteristic equation of 119860119870has two equal

eigenvalues at 119904 = minus120582

1198701 = 1205822

1198702 = 2120582

120582 gt 0

(22)

Since 119860119870is a Hurwitz matrix then it satisfies the Lyapunov

function119860119879119870119875+119875119860

119870= minus119876 where 119875 and119876 are some positive

definite matrices To determine the gains 1198821 and 1198822 wedefine the following Lyapunov function candidate for system(20)

119881 (119885) = 119885119879

119875119885 (23)

Its time derivative in the direction of system (20) trajectoriesis given by

= 119879

119875119885+119885119879

119875

= minus119885119879

119876119885+ 2119885119879119875119860119882sign (119885) minus 2119885119879119875119861Δ119886 (119909 119905)

(24)

If we choose 119875221198821= 119875121198822then 119875119860

119882is a negative definite

matrix and we have

lt minus119885119879

119876119885+ 2120582max (119875119860119882) 1198851

+ 2 1198851 120582min (119875) 1198611 |Δ119886 (119909 119905)| lt 0(25)

provided that1198821 and1198822 are also chosen such that

minus120582max (119875119860119882) gt 120582min (119875)1003816100381610038161003816Δ1198861 (119909 119905)

1003816100381610038161003816 (26)

where 120582max(sdot) and 120582min(sdot) are respectively the largest andleast eigenvalues of the matrix

From Figure 3 we can notice that the second-orderSMC achieves a better performance than the first-orderSMC However the robustness with respect to the constantperturbation is not guaranteed Indeed in sliding mode wehave 120590

ℎ= 0 thus 1199092 = minusℎ(1199093 minus 1199093ref ) = minusℎ1198903 and hence on

the sliding set 120590ℎ= 0 we get

1198903 = minus ℎ1198903 +119889 (119905) (27)

which is a first-order nonautonomous system with a steadystate value equal to 119889(119905)ℎ Therefore for the constantperturbation 119889(119905) = 01 the steady state error is as expectedand delineated in Figure 3 1199093 minus 1199093ref = 01 cm Whenincreasing the value of ℎ the variable structure gains 1198821and1198822 should be increased thus accentuating the chatteringphenomenon which was attenuated by the use of the second-order sliding mode

The boundedness of 1199091 and 1199092 is ensured since lt 0 and119881(119885) depends on 1199111 and 1199112 which in turn depend on 1199091 1199092and 1199093

3 Sliding Mode Controller withNonlinear Surface

To overcome the problem of mismatched perturbation anduncertainty we suggest in this section designing a slidingsurface 120590(119909) based on the backstepping method and usingrobust variable structure virtual controller

In Section 21 the sliding surface 120590(119909) = 0 was obtainedfrom the design of the linear controller 1199091 = 119901(119909) forsubsystem ((2)-(3)) To ensure robustness with respect to theparametric uncertainty a variable structure virtual controller1199091 = 119901(119909) is designed by choosing a sliding surface 1205901(1199092 1199093)and requiring its attractivity by imposing

1 (1199092 1199093) = minus1198821 sign (1205901 (1199092 1199093)) (28)

Therefore the virtual controller 1199091 = 119901(119909) is obtained bysolving (28) and the sliding surface 120590(119909) = 1199091 minus 119901(119909)However when sliding on 1205901(1199092 1199093) is achieved we get1205901(1199092 1199093) = 0 which means 1199092 = V(1199093) and thus in slidingmode we have

3 = V (1199093) + 119889 (119905) (29)

Again to ensure asymptotic convergence of 1199093(119905) to 1199093ref asimple linear proportional term V(1199093) = minus119862(1199093 minus 1199093ref ) is notsatisfactory but a variable structure term can guarantee therequired convergence Indeed with

V (1199093) = minus1198823 (1199093 minus1199093ref) minus1198822 sign (1199093 minus1199093ref) (30)

where the gains are chosen such that1198822 gt 119889max and1198823 gt 0we may easily show the convergence of 1199093(119905) to 1199093ref as far asthe sliding surface

1205901 (1199092 1199093) = 1199092 +1198823 (1199093 minus1199093ref)

+1198822 sign (1199093 minus1199093ref)(31)

is attractiveWhen attempting to achieve the attractivity of 1205901(1199092 1199093)

by imposing (28) we face the attempt to differentiate a dis-continuous function Although this is mathematically impos-sible we can overcome the problem from an engineeringpoint of view by either replacing the discontinuous signumfunction with a smoother saturation function or directlyconsidering that the derivative of the signum function is theDirac impulse (120575(119909)) which is zero everywhere except atan isolated single point Indeed in engineering realizationusing a processor and discretization process the isolateddiscontinuities especially when they are few can easily beavoided and would not cause any problems Therefore from(28) we have

1119898119905

(1198781199091 minus 1198871199092 minus 1198961198971199093) +11988231199092 +11988221199092120575 (1199093 minus1199093ref)

= minus1198821 sign (1205901 (1199092 1199093))(32)

and the sliding surface 120590(119909) becomes

120590 (119909) = 1198781199091 + (1198981199051198823 +1198981199051198822120575 (1199093 minus1199093ref) minus 119887) 1199092

minus 1198961198971199093 +1198981199051198821 sign (1205901 (1199092 1199093))

(33)

6 Mathematical Problems in Engineering

0

50

100

150

200

250

minus05

0

05

1

15

2

25

3

0

5

10

15

20

25

minus5

0

5

10

15

20

03 04 0520

201

202

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 3 Behavior of system under second-order sliding mode controller (19) 120582 = 50 and ℎ = 100

Finally by imposing (119909) = minus119882sign(120590(119909)) the attractivitycondition 120590(119909)(119909) lt 0 is satisfied The choice of thesurface 120590(119909) and its imposed derivative lead to the followingcontroller

119906 (119909) =

119873119889(119909)

(4119861119878119896119881119905)radic119875119889minus 1199091

if 119873119889(119909) ge 0

119873119889(119909)

(4119861119878119896119881119905)radic119875119889+ 1199091

if 119873119889(119909) lt 0

(34)

with

119873119889(119909) = minus119882 sign (120590 (119909)) minus119898

1199051198821120575 (1205901 (119909))

+ (4119861119878120572119881119905

+119887119878

119898119905

minus 1198781198824)1199091

+(41198611198782

119881119905minus1198872

119898119905

+ 119896119897+ 1198871198824)1199092

+(1198961198971198824 minus

119887119896119897

119898119905

)1199093

(35)

where the derivative of the Dirac impulse is consideredas zero and the sliding gain 119882 should be chosen suchthat the attractivity occurs in presence of uncertainty andperturbation

119882 gt

10038161003816100381610038161003816100381610038161003816

119887119881119905

119878119898119905

Δ119896119897

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119881119905

1198781198823Δ119896119897

10038161003816100381610038161003816100381610038161003816

+1003816100381610038161003816119896119897119889max

1003816100381610038161003816 (36)

From Figure 4 we can notice clearly that we haveattained our aim to drive the hydraulic servo system tothe reference position Nevertheless due to the nested slid-ing modes the chattering phenomenon was emphasized

Mathematical Problems in Engineering 7

0

50

100

150

200

250

minus05

0

05

1

15

2

25

3

0

5

10

15

20

25

minus5

0

5

10

15

20

25

03 04 05199

20

201

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 4 Behavior of system under sliding mode controller using nonlinear surface defined by (33)

In order to attenuate it and to avoid the mathematicalproblem of taking the derivative of the signum function wehave used a smooth saturation function sign(119909) ≃ tanh(120583119909)for a sufficiently high positive value 120583 gt 0 The resultsdepicted in Figure 5 show that the use of the smoothfunction helped to achieve the reference value in presenceof mismatched uncertainty and perturbation with attenuatedchattering

4 Sliding Mode Observer Design

As we can notice the controller conceived in the foregoingsection uses all three state variables However measuring thedifferential pressureΔ119875 = 1199091 is a costly task and requires hightechnology procedure to avoid additional leakage To remedythis situation we propose in this section designing a slidingmode observer that may estimate the required states that arethen used to construct the controller

Let us consider the observer model given in (37) andinferred from the step-by-step observer presented in [36 37]

1 =4119861119881119905

(119896119906radic119875119889minus sign (119906) 1 minus

12057211 + 120574 |119906|

minus 1198782)

+1198711 sign (1 minus 1199111) (37)

2 =1119898119905

(1198781199111 minus 1198872 minus 1198961198971199113) + 1198712 sign (2 minus 1199112) (38)

3 = 1199112 +1198713 sign (1199093 minus z3) (39)

where 1198711 1198712 and 1198713 are the observer gain and 1 and 2 aredefined as

1 = 1199111 +119896119897

119878(1199093 minus 1199113) +

119898119905

1198781198712 sign (2 minus 1199112)

2 = 1199112 +1198713 sign (1199093 minus 1199113) (40)

8 Mathematical Problems in Engineering

0

50

100

150

200

250

minus05

0

05

1

15

2

25

0

5

10

15

20

25

minus5

0

5

10

15

20

03 04 0519995

20

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 5 Behavior of system under slidingmode controller using nonlinear surface defined by (33) and a smooth saturation function tanh(sdot)

To prove the efficiency of the observer and the fact thatthe estimated states based controller can also achieve accuratepositioning in presence of perturbation and uncertainty wewill proceed by a step-by-step proof

Step 1 Let 1198903 = 1199093 minus 1199113 and let 1198902 = 1199092 minus 1199112 from (3) and (39)the error dynamics are expressed as

1198903 = 1198902 +119889 (119905) minus 1198713 sign (1198903) (41)Thus if 1198713 is chosen such that

1198713 gt sup119905gt01198902 (119905) + 119889 (119905) (42)

then a sliding mode is established at the observer slidingsurface 1198903 = 0 within a finite timeMoreover at slidingmodewe obtain 1198903 = 0 and thus from (41) we have

0 = 1199092 minus 1199112 +119889 (119905) minus 1198713 sign (1198903) (43)that is

1199092 +119889 (119905) = 1199112 +1198713 sign (1198903) = 2 (44)

Now if the sliding surface 1205901 of the nonlinear surface basedSMC is expressed in terms of the estimated state 2

1205901 (2 1199093) = 2 +1198823 (1199093 minus1199093ref)

+1198822 sign (1199093 minus1199093ref) (45)

then by using (3) (44) and (45) the position dynamics aregiven by

3 = minus1198823 (1199093 minus1199093ref) minus1198822 sign (1199093 minus1199093ref) (46)

Therefore within a finite time we obtain 1199093 = 1199093ref

Step 2 Let 1198901 = 1199091 minus1199111 from (2) and (38) the error dynamicsare expressed as

1198902 =1119898119905

(1198781198901 minus 1198871198902 minus 1198871198713 sign (1198903) minus 1198961198971198903) minusΔ119896119897

119898119905

1199093

minus1198712 sign (2 minus 1199112) (47)

Mathematical Problems in Engineering 9

Thus if 1198712 is chosen such that

1198712 gt1119898119905

1003816100381610038161003816Δ1198961198971199093 + 1198781198901 + 119887119889max1003816100381610038161003816 (48)

then we can deduce that 1198902 + 119889 tend to zero and hence 1199112tend to 1199092 + 119889 That is the observer will estimate the velocitywith the constant perturbation Moreover when the errordynamics are sliding on 1198903 = 0 and 1198902 + 119889 = 0 then 1198902 = 0and we get

1 = 1199091 +119887

1198781198902 minus

Δ119896119897

1198781199093 (49)

That is the observer will estimate the differential pressurewith a constant difference proportional to the perturbationand the uncertainty Now if the controller sliding surface 120590 isexpressed using the observer state variables

120590 (119911) = 1198781 + (1198981199051198823 +1198981199051198822120575 (1199093 minus1199093ref) minus 119887) 2

minus 1198961198971199113 +1198981199051198821 sign (1205901 (2 1199093))

(50)

then at sliding mode of the controller (119911) = 0 the velocitydynamics are given by

2 = minus1198823 (1199092 +119889) minus1198821 sign (1205901 (2 1199093)) (51)

Hence the velocity of the system rod will tend to minus119889

Step 3 Finally the differential pressure error dynamics canbe roughly expressed as

1198901 = 1198911 (119909) minus1198911 (119911) minus 1198711 sign(1198901 +119887

1198781198902 minus

Δ119896119897

1198781199093) (52)

if the system is already sliding on the surfaces 1199093 = 1199093ref and1198902 = minus119889 then by choosing 1198711 gt |1198911(119909)minus1198911(119911)| the differentialpressure is estimated with the constant difference minus(119887119878)119889 +Δ119881119905Figure 6 shows the convergence of the observer state 119911

3to

1199093although they are starting fromdifferent initial conditions

indeed the observer is starting at rod position 10 cm whereasthe system is starting at the origin As expected from theabove analysis 1199112 tends to 1199092 + 119889 and 1199111 tends to 1199091 with aconstant difference of 33 bar The observer gains are chosenas 1198711 = 1000 1198712 = 100 and 1198713 = 1 In Figure 7 the observerand system behaviors are shown with the controller beingcalculated using the estimated states The observer initialposition is minus10 cm

5 Simulation Results Analysis

The controllers designed in this paper used the sliding modetheory which is the most used approach to deal with systemsrunning under uncertainty conditions However we haveseen in the second section that the first-order sliding modecontroller with a linear sliding surface is not robust withrespect to perturbation and mismatched uncertainty In factby using the fact that on the sliding surface the systembehavesin a similar way to a linear second-order system it can be

0

100

200

300

minus10123

0

10

20

30

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 6 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with states feedback

minus1000

100200300

minus20246

minus100

102030

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 7 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with estimated states feed-back

easily shown that as far as the slidingmotion is preserved thesystem is asymptotically stable if the closed loop eigenvaluesare chosen as in (11) and thus the steady state error due to theuncertainty is expressed by (12) Also the steady state errordue to the constant perturbation is given by (13)

Therefore the steady state error gets smaller as the closedloop eigenvalues have a larger amplitude This is illustratedby the simulation results delineated in Figure 2 where 120582 = 50and the total error is 246 cm

To circumvent the problem we have suggested a second-order sliding mode controller Applying this controller we

10 Mathematical Problems in Engineering

can notice that it achieves a better performance than the firstSMC In fact the second-order SMC outperforms the first-order SMC in sense of robustness with respect tomismatchedperturbation but it cannot guarantee the robustness withrespect to the constant perturbation The steady state errordue to the perturbation is equal to 119889(119905)ℎ as demonstratedpreviously When the perturbation 119889(119905) = 01 the erroris equal to 01 cm as delineated in Figure 3 If we increasethe value of ℎ we should also increase the value of thevariable structure gains1198821 and1198822 So we obtain a chatteringphenomenon

To overcome the problem of mismatched perturbationand uncertainty we have suggested a sliding mode controllerwith nonlinear surfaceThe idea is based on the backsteppingmethod and using a robust variable structure virtual con-troller The obtained results achieved robustness with respectto parameter uncertainty and perturbation The zoomedcurve on Figure 4 shows that we have attained our aim todrive the hydraulic servo system to a reference position butwith a chattering phenomenon So to attenuate this problemwe suggest substituting the discontinuous function sign(119909)with a smooth saturation function tanh(120583119909) for sufficientlyhigh positive value 120583 gt 0 Owing to this smooth function wecan achieve the reference value in presence of mismatcheduncertainty and perturbation with attenuated chattering

Finally in Section 4 we designed a slidingmode observerin order to estimate the required states The efficiency of thisobserver is proved mathematically and also by simulationspresented in Figures 6 and 7

Compared to other methods such as that presented in[17] which is based on adaptive approach to achieve positioncontrol we notice that our method provides faster responsesince the adaptive controller proposed therein attempts toestimate the mismatched nonlinear uncertainty Indeed therod makes a displacement of 30 cm within more than 08 swhereas with our method a maximum of 02 s will beneeded to make the same displacement In [21] the authorspresented a controller based on variable structure PID todrive the hydraulic system position Although fast responsehas been achieved which is comparable to our results thegood robustness was obtained since the uncertainties andthe perturbation were matched with the controller In ourcase the robustness against the mismatched character of theperturbation and uncertainty presented the main challengeto take In [26] authors obtained comparable results tothose presented in this paper using integral sliding modecontroller with realizable reference compensation applied toan asymmetric piston

6 Conclusion

In this paper we developed several controllers based onthe sliding mode theory Our aim was to control theposition of a hydraulic servo system piston in presence ofmismatched uncertainty and perturbation We have shownthat a first-order sliding mode controller did not achieveany robustness Next we have developed a second-ordersliding mode controller that has shown robustness withrespect to parametric uncertainty but was not robust to

the perturbation Finally we have suggested a sliding modecontroller based on a nonlinear sliding surface The designis based on the backstepping method where on each stepa variable structure virtual controller design leads to thedesign of the sliding surface This controller emphasized thechattering phenomenon due to the nested sliding modesAs a remedy we suggested substituting the discontinuousfunction with a smooth saturation function Eventually wehave designed a robust sliding observer in order to substitutethe unmeasured states with their estimates A step-by-stepproof has shown that the controller issued from the estimatedstates achieved the position tracking

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H E Merritt Hydraulic Control Systems John Wiley amp SonsNew York NY USA 1967

[2] M Feki and E Richard ldquoIncluding leakage flow in the ser-vovalve static modelrdquo International Journal of Modelling andSimulation vol 25 no 1 pp 4004ndash4009 2005

[3] A Alleyne R Liu and H Wright ldquoOn the limitations offorce tracking control for hydraulic active suspensionsrdquo inProceedings of the American Control Conference (ACC rsquo98) vol1 pp 43ndash47 June 1998

[4] Y Cheng andB L RDeMoor ldquoRobustness analysis and controlsystem design for a hydraulic servo systemrdquo IEEE Transactionson Control Systems Technology vol 2 no 3 pp 183ndash197 1994

[5] M Feki E Richard and F Gomes Almeida ldquoCommande eneffort drsquoun verin hydraulique par linearization entreesortierdquoin Journees Doctorales drsquoAutomatiques (JDA rsquo99) pp 181ndash184Nancy-France 1999

[6] H Hahn A Piepenbrink and K-D Leimbach ldquoInputoutputlinearization control of an electro servo-hydraulic actuatorrdquo inProceedings of the 3rd IEEE Conference on Control Applicationspp 995ndash1000 August 1994

[7] M Jovanovic ldquoNonlinear control of an electrohydraulic velocityservosystemrdquo in Proceedings of the 2002 American ControlConference vol 1 pp 588ndash593 2002

[8] G A Sohl and J E Bobrow ldquoExperiments and simulationson the nonlinear control of a hydraulic servosystemrdquo IEEETransactions on Control Systems Technology vol 7 no 2 pp238ndash247 1999

[9] A G Alleyne and R Liu ldquoSystematic control of a class ofnonlinear systems with application to electrohydraulic cylinderpressure controlrdquo IEEE Transactions on Control Systems Tech-nology vol 8 no 4 pp 623ndash634 2000

[10] M A Avila A G Loukianov and E N Sanchez ldquoElectro-hydraulic actuator trajectory trackingrdquo in Proceedings of theAmerican Control Conference (AAC rsquo04) vol 3 pp 2603ndash2608IEEE Boston Mass USA July 2004

[11] A Bonchis P I Corke D C Rye and Q P Ha ldquoVariable struc-ture methods in hydraulic servo systems controlrdquo Automaticavol 37 no 4 pp 589ndash595 2001

[12] M Jerouane and F Lamnabhi-Lagarrigue ldquoA new robust slidingmode controller for a hydraulic actuatorrdquo in Proceedings of the

Mathematical Problems in Engineering 11

40th IEEE Conference on Decision and Control (CDC rsquo01) vol 1pp 908ndash913 December 2001

[13] H-M Chen J-C Renn and J-P Su ldquoSlidingmode control withvarying boundary layers for an electro-hydraulic position servosystemrdquo The International Journal of Advanced ManufacturingTechnology vol 26 no 1-2 pp 117ndash123 2005

[14] J M Zheng S D Zhao and S G Wei ldquoApplication of self-tuning fuzzy PID controller for a SRM direct drive volumecontrol hydraulic pressrdquo Control Engineering Practice vol 17no 12 pp 1398ndash1404 2009 Special Section The 2007 IFACSymposium on Advances in Automotive Control

[15] A Aly ldquoPid parameters optimization using genetic algorithmtechnique for electrohydraulic servo control systemrdquo IntelligentControl and Automation vol 2 no 2 pp 69ndash76 2011

[16] B Yao F Bu and G T C Chiu ldquoNonlinear adaptive robustcontrol of electro-hydraulic servo systems with discontinuousprojectionsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control (CDC rsquo98) pp 2265ndash2270 December1998

[17] B Yao F Bu J Reedy and G T-C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[18] K K Ahn and Q T Dinh ldquoSelf-tuning of quantitative feedbacktheory for force control of an electro-hydraulic test machinerdquoControl Engineering Practice vol 17 no 11 pp 1291ndash1306 2009

[19] E Kolsi Gdoura M Feki and N Derbel ldquoSliding mode controlof a hydraulic servo system position using adaptive slidingsurface and adaptive gainrdquo International Journal of ModellingIdentification and Control vol 23 pp 211ndash219 2015

[20] C Guan and S Pan ldquoAdaptive sliding mode control of electro-hydraulic systemwith nonlinear unknown parametersrdquoControlEngineering Practice vol 16 no 11 pp 1275ndash1284 2008

[21] D Liu Z Tang and Z Pei ldquoVariable structure compensationPID control of asymmetrical hydraulic cylinder trajectorytrackingrdquo Mathematical Problems in Engineering vol 2015Article ID 890704 9 pages 2015

[22] A G Loukianov J Rivera Y V Orlov and E Y MoralesTeraoka ldquoRobust trajectory tracking for an electrohydraulicactuatorrdquo IEEE Transactions on Industrial Electronics vol 56no 9 pp 3523ndash3531 2009

[23] M Ye Q Wang and S Jiao ldquoRobust 1198672119867infin

control forthe electrohydraulic steering system of a four-wheel vehiclerdquoMathematical Problems in Engineering vol 2014 Article ID208019 12 pages 2014

[24] W Shen J Jiang H R Karimi and X Su ldquoObserver-basedrobust control for hydraulic velocity control systemrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 6891329 pages 2013

[25] E Kolsi-Gdoura M Feki and N Derbel ldquoSliding mode-based robust position control of an electrohydraulic systemrdquoin Proceedings of the 10th International Multi-Conference onSystems Signals and Devices (SSD rsquo13) pp 1ndash5 March 2013

[26] E Kolsi Gdoura M Feki and N Derbel ldquoControl of ahydraulic servo system using sliding mode with an integral andrealizable reference compensationrdquo Control Engineering andApplied Informatics vol 17 pp 111ndash119 2015

[27] A Levant ldquoSliding order and sliding accuracy in sliding modecontrolrdquo International Journal of Control vol 58 no 6 pp 1247ndash1263 1993

[28] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[29] M Djemai K Busawon K Benmansour and A MaroufldquoHigh-order sliding mode control of a DC motor drive viaa switched controlled multi-cellular converterrdquo InternationalJournal of Systems Science vol 42 no 11 pp 1869ndash1882 2011

[30] M Lavanya R M Brisilla and V Sankaranarayanan ldquoHigherorder sliding mode control of permanent magnet dc motorrdquoin Proceedings of the 12th International Workshop on VariableStructure Systems (VSS rsquo12) pp 226ndash230 January 2012

[31] S Laghrouche M Smaoui F Plestan and X Brun ldquoHigherorder sliding mode control based on optimal approach of anelectropneumatic actuatorrdquo International Journal of Control vol79 no 2 pp 119ndash131 2006

[32] A G Loukianov E Sanchez and C Lizalde ldquoForce trackingneural block control for an electro-hydraulic actuator viasecond-order slidingmoderdquo International Journal of Robust andNonlinear Control vol 18 no 3 pp 319ndash332 2008

[33] J Komsta N van Oijen and P Antoszkiewicz ldquoIntegral slidingmode compensator for load pressure control of die-cushioncylinder driverdquo Control Engineering Practice vol 21 no 5 pp708ndash718 2013

[34] M Feki Synthese de commandes et drsquoobservateurs pour lessystemes non-lineaires application aux systemes hydrauliques[PhD thesis] Universite de Metz Metz France 2001

[35] V V Filippov ldquoOn a scalar equation with discontinuous right-hand side and the uniqueness theoremrdquo Differential Equationsvol 38 no 10 pp 1435ndash1445 2002

[36] T Boukhobza M Djemai and J P Barbot ldquoNonlinear slidingobserver for systems in output and output derivative injectionformrdquo in Proceedings of the IFAC World Congress pp 299ndash305San Antonio Tex USA 1996

[37] H Saadaoui N Manamanni M Djemaı J P Barbot andT Floquet ldquoExact differentiation and sliding mode observersfor switched Lagrangian systemsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 65 no 5 pp 1050ndash1069 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

In the present paper we are interested in controlling theposition of the rod in a hydraulic servo system that consistsof a four-way spool valve supplying a double effect linearcylinder with a double-rodded piston The piston is drivinga load modeled by a mass a spring and a sliding viscousfriction Our work aims to design a controller that mayachieve the reference position in presence of mismatchedparameter uncertainty and perturbation in addition to actua-tor saturation To realize this objective we start in the secondsection by formulating the problem and presenting the effectsof using first- and second-order SMCwith a linear surface InSection 3 we present the design of a sliding surface obtainedusing backstepping method and variable structure controllerleading hence to a nonlinear surface that allows achievingthe reference output despite the presence of uncertainties andperturbations Numerical simulation results are presentedto illustrate the efficiency of the proposed control designIn Section 4 we present a sliding observer and prove theconvergence of the observer as well as the exact positiontracking using the nonlinear surface SMC issued from theobserver states Finally the conclusion and some remarks arepresented in Section 5

2 Problem Statement

The electrohydraulic system that we will deal with in thispaper is depicted in Figure 1 and modeled by the dynamicalsystem (1) It has been shown in [34] that for the symmetricalpiston with equal surfaces 1198781 and 1198782 and assuming equalvolume flow passing through (geometrically) identical portswe can describe the system by three variable states wherea differential pressure state substitutes the pressure of eachchamber This decrease in the system dimension ensures theobservability when the system output is the piston positionConsider

1 =4119861119881119905

(119896119906radic119875119889minus sign (119906) 1199091 minus

12057211990911 + 120574 |119906|

minus 1198781199092) (1)

2 =1119898119905

(1198781199091 minus 1198871199092 minus (119896119897 +Δ119896119897) 1199093) (2)

3 = 1199092 +119889 (119905) (3)

where 1199091 = 1198751 minus 1198752 denotes the difference in pressureinside the two chambers of the cylinder 1199092 and 1199093 respec-tively denote the velocity and the position of the rod and|119889(119905)| lt 119889max is a bounded constant or slowly varyingexternal perturbation In fact from Newtonrsquos law a constantforce perturbation leads to a constant acceleration Thus thevelocity perturbation may be interpreted as the result of animpulsive force that acts abruptly on the system119898

119905= 119898+1198980

is the total mass of the rod and the load 119881119905= 1198811 + 1198812 is the

total volume of the cylinder and 119875119889= 119875119904minus 119875119903is the pressure

difference between the supply pressure 119875119904(pressure of the

pump) and the return pressure 119875119903(atmospheric pressure) 119896

119897

is the spring stiffness constant with an uncertainty Δ119896119897and

119887 is the friction coefficient The system parameters used forsimulations are as follows 119861 = 22times109 Pa 119875

119904= 300times105 Pa

119875119903= 105 Pa1198980 = 50 kg 119878 = 15 times 10minus3 m2 119881

119905= 9 times 10minus4 m3

119898 = 20 kg 119887 = 590 kgs 119896119897= 125000Nm 119896 = 362 times

10minus5 m3 sminus1 Aminus1 Paminus12 120572 = 41816 times 10minus12 m3 s minus1 Paminus1 and120574 = 8571 with 120572 and 120574 being intrinsic constants modeling theleakage within the servovalve [2]

Themost difficult aspect in thismodel is the existence of amismatched perturbation as well as a mismatched parameteruncertainty In addition the leakage model includes non-linearity with respect to the control signal 119906 To deal withthis problem we neglect the leakage term in the design butwe consider it in simulation and consider the system as aswitching system between two models The switching aspectmakes the use of the slidingmode approach a good candidateto design a controller 119906 that can drive the rod position to aconstant reference position 1199093ref

21 First-Order SMC TheSMCdesign consists of two phasesIn the first phase the sliding surface is designed such that thesystem is asymptotically stable when it is confined to it and inthe second phase a switching controller is designed to ensurethe existence of the slidingmodeOur idea consists in viewingthe sliding surface design as a special case of backsteppingdesignTherefore at slidingmode 120590(119909) = 0means 1199091 = 119901(119909)and 1199091 can be viewed as a virtual controller to subsystem((2)-(3)) which describes the system behavior on the slidingsurface

Thus should we choose a linear virtual controller

119901 (119909) =1119878(1198961198971199093ref minus11986221199092 minus1198623 (1199093 minus1199093ref)) (4)

we get the sliding surface

120590 (119909) = 1198781199091 +11986221199092 +1198623 (1199093 minus1199093ref) minus 1198961198971199093ref (5)

In sliding mode and if uncertainty and perturbations areneglected the system is a second dimensional linear systemwith the characteristic equation

1199042+1198622 + 119887

119898119905

119904 +1198623 + 119896119897119898119905

= 0 (6)

Using the pole placement method and imposing a stablemultiple pole at 119904 = minus120582 we can determine the controlparameters 1198622 and 1198623

1198622 = 2120582119898119905minus 119887

1198623 = 1205822119898119905minus 119896119897

(7)

Eventually we obtain an exponentially stable system if wecan guarantee the attractivity of the sliding surface 120590(119909) =0 Indeed the attractivity condition 120590(119909)(119909) lt 0 will besatisfied if we choose (119909) = minus119882sign(120590(119909)) where 119882 gt 0is the sliding gain that should be chosen large enough inorder to ensure the attractivity of the surface in presence ofperturbation and uncertainty Using the system model and(5) we can show that 120590(119909)(119909) lt 0 is attained if the slidinggain satisfies

119882 gt10038161003816100381610038161198623119889max

1003816100381610038161003816 +4119861119878119881119905

120572119875119889+

10038161003816100381610038161003816100381610038161003816

1198811199051198622119878119898119905

Δ119896119897

10038161003816100381610038161003816100381610038161003816

(8)

Mathematical Problems in Engineering 3

x2 x3m0

Sk1

m

b

P1 V1 P2 V2

Pr Ps

u

Figure 1 Hydraulic servo system controlled using a servovalve

where 119889max = max119905ge0119889(119905) Consequently by solving (119909) =

minus119882sign(120590(119909)) the control law is expressed as follows

119906 (119909) =

119873(119909)

(4119861119878119896119881119905)radic119875119889minus 1199091

if 119873(119909) le 0

119873(119909)

(4119861119878119896119881119905)radic119875119889+ 1199091

if 119873(119909) lt 0(9)

where

119873(119909) = minus119882 sign (120590 (119909)) minus1198622119898119905

(1198781199091 minus 1198871199092 minus 1198961198971199093)

minus11986231199092 +41198611198782

119881119905

1199092

(10)

Despite the perturbation 119889(119905) and the uncertaintyΔ119896119897 the

variable structure controller defined by (9) with the slidinggain119882 in (8) ensures the existence of sliding mode howeverwe can easily deduce that we do not achieve the referenceoutput because when the system behavior is confined tothe sliding surface (5) the linear virtual controller does notguarantee any robustnesswith respect to the perturbation andthe uncertainty Indeed it can be easily shown that as far asthe sliding motion is preserved the system is asymptoticallystable if the closed loop eigenvalue is chosen such as

120582 gt radic

10038161003816100381610038161003816100381610038161003816

Δ119896119897

119898119905

10038161003816100381610038161003816100381610038161003816

(11)

and in that case the steady state error due to the uncertaintyis given by

119890119896119897=Δ119896119897

119898119905

11205822 + Δ119896

119897119898119905

1199093ref (12)

We similarly can show that the steady state error due to theconstant perturbation is given by

119890119889= minus

2120582119889 (13)

Figure 2 shows the system behavior with 120582 = 50 119889(119905) =01 Δ119896

119897= 25000 and 1199093ref = 20 cm We clearly notice that

there is a steady state error of more than 2 cm at the positionoutput So the first-order sliding mode is not robust to themismatched uncertainty and perturbation

22 Second-Order SMC Since the system is third-ordersingle input thenwemay think of designing the higher-orderSMC up to second order Let 120590

ℎdenote the sliding variable

defined as

120590ℎ= ℎ (1199093 minus1199093ref) + 1199092 (14)

where ℎ gt 0 is a strictly positive scalar We may verify thatthe relative degree of system ((1)ndash(3)) together with (14) withrespect to the sliding variable 120590

ℎis constant and equal to two

Thus we have

ℎ= 1198861 (119909 119905) + Δ1198861 (119909 119905) + 1198862 (119909) 119906 (15)

where 1198861(119909 119905) and 1198862(119909) are known functions and Δ1198861(119909 119905) isan unknown bounded function in terms of 119889(119905) and Δ119896

119897

1198861 (119909 119905) = minus4119861119878119898119905119881119905

(1205721199091 + 1198781199092)

+ (ℎ

119898119905

minus119887

1198982119905

) (1198781199091 minus 1198871199092 minus 1198961198971199093)

minus119896119897

119898119905

1199092

Δ1198861 (119909 119905) = minus(ℎ

119898119905

minus119887

1198982119905

)Δ1198961198971199093 minus

Δ119896119897

119898119905

1199092

minus119889 (119905)119896119897+ Δ119896119897

119898119905

1198862 (119909) =

4119861119878119898119905119881119905

119896radic119875119889minus 1199091 if 119906 ge 0

4119861119878119898119905119881119905

119896radic119875119889+ 1199091 if 119906 lt 0

(16)

The solution should be understood in the Filippov sense[35] and the trajectories of the system are supposed to beextendible infinitely in time for any bounded measurableinput

4 Mathematical Problems in Engineering

0 01 02 03

03

04 050

50

100

150

200

Time (s)0 01 02 03 04 05

minus1

0

1

2

3

4

Velo

city

(ms

)

Time (s)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

0

5

10

15

20

Posit

ion

(cm

)

minus5

0

5

10

15

20

25

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 2 The behavior of the system under sliding mode controller defined by (9) 120582 = 50

By defining 1199111 = 120590ℎ and 1199112 = ℎ achieving sliding mode120590ℎ= 0 is equivalent to the finite stabilization of the system

1 = 1199112

2 = 1198861 (119909 119905) + Δ1198861 (119909 119905) + 1198862 (119909) 119906(17)

which hence ideally yields to the sliding set120590ℎ= 0 and

ℎ= 0

Taking into account the practical considerations the slidingset is defined as follows [27]

Definition 1 Given the sliding variable 120590ℎ(119909 119905) the ldquoreal

second-order sliding setrdquo associated with ((1)ndash(3)) is definedas

119904119879119890= 119909isin

120594

1003816100381610038161003816120590ℎ1003816100381610038161003816

le 119896111987921198901003816100381610038161003816ℎ1003816100381610038161003816 le 1198962119879119890 (18)

where 119879119890is the finite sampling time (fixed at 119879

119890= 20 120583s in the

sequel) and 1198961 and 1198962 are positive constants

Definition 2 Consider the nonempty real second-order slid-ing set 119904

119879119890given in (18) and assume that it is locally an

integral set in the Filippov senseThe corresponding behaviorof system ((1)ndash(3)) satisfying (18) is called ldquoreal second-ordersliding moderdquo with respect to 120590

ℎ(119909 119905) [27]

The variable structure control law 119906 can be chosen asfollows

119906 =1

1198862 (119909)(minus1198861 (119909 119905) minus11987011199111 minus11987021199112 minus1198821 sign (1199111)

minus1198822sign (1199112)) (19)

where all the controller gains 1198701 11987021198821 and1198822 are strictlypositive Applying controller (19) yields to

[1

2] = [

0 1minus1198701 minus1198702

][1199111

1199112]

+[0 0minus1198821 minus1198822

][sign (1199111)sign (1199112)

] minus [01]Δ1198861 (119909 119905)

(20)

Mathematical Problems in Engineering 5

which can be written in the following compact form

= 119860119870119885+119860

119882sign (119911) minus 119861Δ119886 (119909 119905) (21)

where 119860119870= [

0 1minus1198701 minus1198702

] 119860119882= [

0 0minus1198821 minus1198822

] and 119861 = [ 01 ]We can determine the control parameters 1198701 and 1198702

such that the characteristic equation of 119860119870has two equal

eigenvalues at 119904 = minus120582

1198701 = 1205822

1198702 = 2120582

120582 gt 0

(22)

Since 119860119870is a Hurwitz matrix then it satisfies the Lyapunov

function119860119879119870119875+119875119860

119870= minus119876 where 119875 and119876 are some positive

definite matrices To determine the gains 1198821 and 1198822 wedefine the following Lyapunov function candidate for system(20)

119881 (119885) = 119885119879

119875119885 (23)

Its time derivative in the direction of system (20) trajectoriesis given by

= 119879

119875119885+119885119879

119875

= minus119885119879

119876119885+ 2119885119879119875119860119882sign (119885) minus 2119885119879119875119861Δ119886 (119909 119905)

(24)

If we choose 119875221198821= 119875121198822then 119875119860

119882is a negative definite

matrix and we have

lt minus119885119879

119876119885+ 2120582max (119875119860119882) 1198851

+ 2 1198851 120582min (119875) 1198611 |Δ119886 (119909 119905)| lt 0(25)

provided that1198821 and1198822 are also chosen such that

minus120582max (119875119860119882) gt 120582min (119875)1003816100381610038161003816Δ1198861 (119909 119905)

1003816100381610038161003816 (26)

where 120582max(sdot) and 120582min(sdot) are respectively the largest andleast eigenvalues of the matrix

From Figure 3 we can notice that the second-orderSMC achieves a better performance than the first-orderSMC However the robustness with respect to the constantperturbation is not guaranteed Indeed in sliding mode wehave 120590

ℎ= 0 thus 1199092 = minusℎ(1199093 minus 1199093ref ) = minusℎ1198903 and hence on

the sliding set 120590ℎ= 0 we get

1198903 = minus ℎ1198903 +119889 (119905) (27)

which is a first-order nonautonomous system with a steadystate value equal to 119889(119905)ℎ Therefore for the constantperturbation 119889(119905) = 01 the steady state error is as expectedand delineated in Figure 3 1199093 minus 1199093ref = 01 cm Whenincreasing the value of ℎ the variable structure gains 1198821and1198822 should be increased thus accentuating the chatteringphenomenon which was attenuated by the use of the second-order sliding mode

The boundedness of 1199091 and 1199092 is ensured since lt 0 and119881(119885) depends on 1199111 and 1199112 which in turn depend on 1199091 1199092and 1199093

3 Sliding Mode Controller withNonlinear Surface

To overcome the problem of mismatched perturbation anduncertainty we suggest in this section designing a slidingsurface 120590(119909) based on the backstepping method and usingrobust variable structure virtual controller

In Section 21 the sliding surface 120590(119909) = 0 was obtainedfrom the design of the linear controller 1199091 = 119901(119909) forsubsystem ((2)-(3)) To ensure robustness with respect to theparametric uncertainty a variable structure virtual controller1199091 = 119901(119909) is designed by choosing a sliding surface 1205901(1199092 1199093)and requiring its attractivity by imposing

1 (1199092 1199093) = minus1198821 sign (1205901 (1199092 1199093)) (28)

Therefore the virtual controller 1199091 = 119901(119909) is obtained bysolving (28) and the sliding surface 120590(119909) = 1199091 minus 119901(119909)However when sliding on 1205901(1199092 1199093) is achieved we get1205901(1199092 1199093) = 0 which means 1199092 = V(1199093) and thus in slidingmode we have

3 = V (1199093) + 119889 (119905) (29)

Again to ensure asymptotic convergence of 1199093(119905) to 1199093ref asimple linear proportional term V(1199093) = minus119862(1199093 minus 1199093ref ) is notsatisfactory but a variable structure term can guarantee therequired convergence Indeed with

V (1199093) = minus1198823 (1199093 minus1199093ref) minus1198822 sign (1199093 minus1199093ref) (30)

where the gains are chosen such that1198822 gt 119889max and1198823 gt 0we may easily show the convergence of 1199093(119905) to 1199093ref as far asthe sliding surface

1205901 (1199092 1199093) = 1199092 +1198823 (1199093 minus1199093ref)

+1198822 sign (1199093 minus1199093ref)(31)

is attractiveWhen attempting to achieve the attractivity of 1205901(1199092 1199093)

by imposing (28) we face the attempt to differentiate a dis-continuous function Although this is mathematically impos-sible we can overcome the problem from an engineeringpoint of view by either replacing the discontinuous signumfunction with a smoother saturation function or directlyconsidering that the derivative of the signum function is theDirac impulse (120575(119909)) which is zero everywhere except atan isolated single point Indeed in engineering realizationusing a processor and discretization process the isolateddiscontinuities especially when they are few can easily beavoided and would not cause any problems Therefore from(28) we have

1119898119905

(1198781199091 minus 1198871199092 minus 1198961198971199093) +11988231199092 +11988221199092120575 (1199093 minus1199093ref)

= minus1198821 sign (1205901 (1199092 1199093))(32)

and the sliding surface 120590(119909) becomes

120590 (119909) = 1198781199091 + (1198981199051198823 +1198981199051198822120575 (1199093 minus1199093ref) minus 119887) 1199092

minus 1198961198971199093 +1198981199051198821 sign (1205901 (1199092 1199093))

(33)

6 Mathematical Problems in Engineering

0

50

100

150

200

250

minus05

0

05

1

15

2

25

3

0

5

10

15

20

25

minus5

0

5

10

15

20

03 04 0520

201

202

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 3 Behavior of system under second-order sliding mode controller (19) 120582 = 50 and ℎ = 100

Finally by imposing (119909) = minus119882sign(120590(119909)) the attractivitycondition 120590(119909)(119909) lt 0 is satisfied The choice of thesurface 120590(119909) and its imposed derivative lead to the followingcontroller

119906 (119909) =

119873119889(119909)

(4119861119878119896119881119905)radic119875119889minus 1199091

if 119873119889(119909) ge 0

119873119889(119909)

(4119861119878119896119881119905)radic119875119889+ 1199091

if 119873119889(119909) lt 0

(34)

with

119873119889(119909) = minus119882 sign (120590 (119909)) minus119898

1199051198821120575 (1205901 (119909))

+ (4119861119878120572119881119905

+119887119878

119898119905

minus 1198781198824)1199091

+(41198611198782

119881119905minus1198872

119898119905

+ 119896119897+ 1198871198824)1199092

+(1198961198971198824 minus

119887119896119897

119898119905

)1199093

(35)

where the derivative of the Dirac impulse is consideredas zero and the sliding gain 119882 should be chosen suchthat the attractivity occurs in presence of uncertainty andperturbation

119882 gt

10038161003816100381610038161003816100381610038161003816

119887119881119905

119878119898119905

Δ119896119897

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119881119905

1198781198823Δ119896119897

10038161003816100381610038161003816100381610038161003816

+1003816100381610038161003816119896119897119889max

1003816100381610038161003816 (36)

From Figure 4 we can notice clearly that we haveattained our aim to drive the hydraulic servo system tothe reference position Nevertheless due to the nested slid-ing modes the chattering phenomenon was emphasized

Mathematical Problems in Engineering 7

0

50

100

150

200

250

minus05

0

05

1

15

2

25

3

0

5

10

15

20

25

minus5

0

5

10

15

20

25

03 04 05199

20

201

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 4 Behavior of system under sliding mode controller using nonlinear surface defined by (33)

In order to attenuate it and to avoid the mathematicalproblem of taking the derivative of the signum function wehave used a smooth saturation function sign(119909) ≃ tanh(120583119909)for a sufficiently high positive value 120583 gt 0 The resultsdepicted in Figure 5 show that the use of the smoothfunction helped to achieve the reference value in presenceof mismatched uncertainty and perturbation with attenuatedchattering

4 Sliding Mode Observer Design

As we can notice the controller conceived in the foregoingsection uses all three state variables However measuring thedifferential pressureΔ119875 = 1199091 is a costly task and requires hightechnology procedure to avoid additional leakage To remedythis situation we propose in this section designing a slidingmode observer that may estimate the required states that arethen used to construct the controller

Let us consider the observer model given in (37) andinferred from the step-by-step observer presented in [36 37]

1 =4119861119881119905

(119896119906radic119875119889minus sign (119906) 1 minus

12057211 + 120574 |119906|

minus 1198782)

+1198711 sign (1 minus 1199111) (37)

2 =1119898119905

(1198781199111 minus 1198872 minus 1198961198971199113) + 1198712 sign (2 minus 1199112) (38)

3 = 1199112 +1198713 sign (1199093 minus z3) (39)

where 1198711 1198712 and 1198713 are the observer gain and 1 and 2 aredefined as

1 = 1199111 +119896119897

119878(1199093 minus 1199113) +

119898119905

1198781198712 sign (2 minus 1199112)

2 = 1199112 +1198713 sign (1199093 minus 1199113) (40)

8 Mathematical Problems in Engineering

0

50

100

150

200

250

minus05

0

05

1

15

2

25

0

5

10

15

20

25

minus5

0

5

10

15

20

03 04 0519995

20

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 5 Behavior of system under slidingmode controller using nonlinear surface defined by (33) and a smooth saturation function tanh(sdot)

To prove the efficiency of the observer and the fact thatthe estimated states based controller can also achieve accuratepositioning in presence of perturbation and uncertainty wewill proceed by a step-by-step proof

Step 1 Let 1198903 = 1199093 minus 1199113 and let 1198902 = 1199092 minus 1199112 from (3) and (39)the error dynamics are expressed as

1198903 = 1198902 +119889 (119905) minus 1198713 sign (1198903) (41)Thus if 1198713 is chosen such that

1198713 gt sup119905gt01198902 (119905) + 119889 (119905) (42)

then a sliding mode is established at the observer slidingsurface 1198903 = 0 within a finite timeMoreover at slidingmodewe obtain 1198903 = 0 and thus from (41) we have

0 = 1199092 minus 1199112 +119889 (119905) minus 1198713 sign (1198903) (43)that is

1199092 +119889 (119905) = 1199112 +1198713 sign (1198903) = 2 (44)

Now if the sliding surface 1205901 of the nonlinear surface basedSMC is expressed in terms of the estimated state 2

1205901 (2 1199093) = 2 +1198823 (1199093 minus1199093ref)

+1198822 sign (1199093 minus1199093ref) (45)

then by using (3) (44) and (45) the position dynamics aregiven by

3 = minus1198823 (1199093 minus1199093ref) minus1198822 sign (1199093 minus1199093ref) (46)

Therefore within a finite time we obtain 1199093 = 1199093ref

Step 2 Let 1198901 = 1199091 minus1199111 from (2) and (38) the error dynamicsare expressed as

1198902 =1119898119905

(1198781198901 minus 1198871198902 minus 1198871198713 sign (1198903) minus 1198961198971198903) minusΔ119896119897

119898119905

1199093

minus1198712 sign (2 minus 1199112) (47)

Mathematical Problems in Engineering 9

Thus if 1198712 is chosen such that

1198712 gt1119898119905

1003816100381610038161003816Δ1198961198971199093 + 1198781198901 + 119887119889max1003816100381610038161003816 (48)

then we can deduce that 1198902 + 119889 tend to zero and hence 1199112tend to 1199092 + 119889 That is the observer will estimate the velocitywith the constant perturbation Moreover when the errordynamics are sliding on 1198903 = 0 and 1198902 + 119889 = 0 then 1198902 = 0and we get

1 = 1199091 +119887

1198781198902 minus

Δ119896119897

1198781199093 (49)

That is the observer will estimate the differential pressurewith a constant difference proportional to the perturbationand the uncertainty Now if the controller sliding surface 120590 isexpressed using the observer state variables

120590 (119911) = 1198781 + (1198981199051198823 +1198981199051198822120575 (1199093 minus1199093ref) minus 119887) 2

minus 1198961198971199113 +1198981199051198821 sign (1205901 (2 1199093))

(50)

then at sliding mode of the controller (119911) = 0 the velocitydynamics are given by

2 = minus1198823 (1199092 +119889) minus1198821 sign (1205901 (2 1199093)) (51)

Hence the velocity of the system rod will tend to minus119889

Step 3 Finally the differential pressure error dynamics canbe roughly expressed as

1198901 = 1198911 (119909) minus1198911 (119911) minus 1198711 sign(1198901 +119887

1198781198902 minus

Δ119896119897

1198781199093) (52)

if the system is already sliding on the surfaces 1199093 = 1199093ref and1198902 = minus119889 then by choosing 1198711 gt |1198911(119909)minus1198911(119911)| the differentialpressure is estimated with the constant difference minus(119887119878)119889 +Δ119881119905Figure 6 shows the convergence of the observer state 119911

3to

1199093although they are starting fromdifferent initial conditions

indeed the observer is starting at rod position 10 cm whereasthe system is starting at the origin As expected from theabove analysis 1199112 tends to 1199092 + 119889 and 1199111 tends to 1199091 with aconstant difference of 33 bar The observer gains are chosenas 1198711 = 1000 1198712 = 100 and 1198713 = 1 In Figure 7 the observerand system behaviors are shown with the controller beingcalculated using the estimated states The observer initialposition is minus10 cm

5 Simulation Results Analysis

The controllers designed in this paper used the sliding modetheory which is the most used approach to deal with systemsrunning under uncertainty conditions However we haveseen in the second section that the first-order sliding modecontroller with a linear sliding surface is not robust withrespect to perturbation and mismatched uncertainty In factby using the fact that on the sliding surface the systembehavesin a similar way to a linear second-order system it can be

0

100

200

300

minus10123

0

10

20

30

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 6 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with states feedback

minus1000

100200300

minus20246

minus100

102030

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 7 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with estimated states feed-back

easily shown that as far as the slidingmotion is preserved thesystem is asymptotically stable if the closed loop eigenvaluesare chosen as in (11) and thus the steady state error due to theuncertainty is expressed by (12) Also the steady state errordue to the constant perturbation is given by (13)

Therefore the steady state error gets smaller as the closedloop eigenvalues have a larger amplitude This is illustratedby the simulation results delineated in Figure 2 where 120582 = 50and the total error is 246 cm

To circumvent the problem we have suggested a second-order sliding mode controller Applying this controller we

10 Mathematical Problems in Engineering

can notice that it achieves a better performance than the firstSMC In fact the second-order SMC outperforms the first-order SMC in sense of robustness with respect tomismatchedperturbation but it cannot guarantee the robustness withrespect to the constant perturbation The steady state errordue to the perturbation is equal to 119889(119905)ℎ as demonstratedpreviously When the perturbation 119889(119905) = 01 the erroris equal to 01 cm as delineated in Figure 3 If we increasethe value of ℎ we should also increase the value of thevariable structure gains1198821 and1198822 So we obtain a chatteringphenomenon

To overcome the problem of mismatched perturbationand uncertainty we have suggested a sliding mode controllerwith nonlinear surfaceThe idea is based on the backsteppingmethod and using a robust variable structure virtual con-troller The obtained results achieved robustness with respectto parameter uncertainty and perturbation The zoomedcurve on Figure 4 shows that we have attained our aim todrive the hydraulic servo system to a reference position butwith a chattering phenomenon So to attenuate this problemwe suggest substituting the discontinuous function sign(119909)with a smooth saturation function tanh(120583119909) for sufficientlyhigh positive value 120583 gt 0 Owing to this smooth function wecan achieve the reference value in presence of mismatcheduncertainty and perturbation with attenuated chattering

Finally in Section 4 we designed a slidingmode observerin order to estimate the required states The efficiency of thisobserver is proved mathematically and also by simulationspresented in Figures 6 and 7

Compared to other methods such as that presented in[17] which is based on adaptive approach to achieve positioncontrol we notice that our method provides faster responsesince the adaptive controller proposed therein attempts toestimate the mismatched nonlinear uncertainty Indeed therod makes a displacement of 30 cm within more than 08 swhereas with our method a maximum of 02 s will beneeded to make the same displacement In [21] the authorspresented a controller based on variable structure PID todrive the hydraulic system position Although fast responsehas been achieved which is comparable to our results thegood robustness was obtained since the uncertainties andthe perturbation were matched with the controller In ourcase the robustness against the mismatched character of theperturbation and uncertainty presented the main challengeto take In [26] authors obtained comparable results tothose presented in this paper using integral sliding modecontroller with realizable reference compensation applied toan asymmetric piston

6 Conclusion

In this paper we developed several controllers based onthe sliding mode theory Our aim was to control theposition of a hydraulic servo system piston in presence ofmismatched uncertainty and perturbation We have shownthat a first-order sliding mode controller did not achieveany robustness Next we have developed a second-ordersliding mode controller that has shown robustness withrespect to parametric uncertainty but was not robust to

the perturbation Finally we have suggested a sliding modecontroller based on a nonlinear sliding surface The designis based on the backstepping method where on each stepa variable structure virtual controller design leads to thedesign of the sliding surface This controller emphasized thechattering phenomenon due to the nested sliding modesAs a remedy we suggested substituting the discontinuousfunction with a smooth saturation function Eventually wehave designed a robust sliding observer in order to substitutethe unmeasured states with their estimates A step-by-stepproof has shown that the controller issued from the estimatedstates achieved the position tracking

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H E Merritt Hydraulic Control Systems John Wiley amp SonsNew York NY USA 1967

[2] M Feki and E Richard ldquoIncluding leakage flow in the ser-vovalve static modelrdquo International Journal of Modelling andSimulation vol 25 no 1 pp 4004ndash4009 2005

[3] A Alleyne R Liu and H Wright ldquoOn the limitations offorce tracking control for hydraulic active suspensionsrdquo inProceedings of the American Control Conference (ACC rsquo98) vol1 pp 43ndash47 June 1998

[4] Y Cheng andB L RDeMoor ldquoRobustness analysis and controlsystem design for a hydraulic servo systemrdquo IEEE Transactionson Control Systems Technology vol 2 no 3 pp 183ndash197 1994

[5] M Feki E Richard and F Gomes Almeida ldquoCommande eneffort drsquoun verin hydraulique par linearization entreesortierdquoin Journees Doctorales drsquoAutomatiques (JDA rsquo99) pp 181ndash184Nancy-France 1999

[6] H Hahn A Piepenbrink and K-D Leimbach ldquoInputoutputlinearization control of an electro servo-hydraulic actuatorrdquo inProceedings of the 3rd IEEE Conference on Control Applicationspp 995ndash1000 August 1994

[7] M Jovanovic ldquoNonlinear control of an electrohydraulic velocityservosystemrdquo in Proceedings of the 2002 American ControlConference vol 1 pp 588ndash593 2002

[8] G A Sohl and J E Bobrow ldquoExperiments and simulationson the nonlinear control of a hydraulic servosystemrdquo IEEETransactions on Control Systems Technology vol 7 no 2 pp238ndash247 1999

[9] A G Alleyne and R Liu ldquoSystematic control of a class ofnonlinear systems with application to electrohydraulic cylinderpressure controlrdquo IEEE Transactions on Control Systems Tech-nology vol 8 no 4 pp 623ndash634 2000

[10] M A Avila A G Loukianov and E N Sanchez ldquoElectro-hydraulic actuator trajectory trackingrdquo in Proceedings of theAmerican Control Conference (AAC rsquo04) vol 3 pp 2603ndash2608IEEE Boston Mass USA July 2004

[11] A Bonchis P I Corke D C Rye and Q P Ha ldquoVariable struc-ture methods in hydraulic servo systems controlrdquo Automaticavol 37 no 4 pp 589ndash595 2001

[12] M Jerouane and F Lamnabhi-Lagarrigue ldquoA new robust slidingmode controller for a hydraulic actuatorrdquo in Proceedings of the

Mathematical Problems in Engineering 11

40th IEEE Conference on Decision and Control (CDC rsquo01) vol 1pp 908ndash913 December 2001

[13] H-M Chen J-C Renn and J-P Su ldquoSlidingmode control withvarying boundary layers for an electro-hydraulic position servosystemrdquo The International Journal of Advanced ManufacturingTechnology vol 26 no 1-2 pp 117ndash123 2005

[14] J M Zheng S D Zhao and S G Wei ldquoApplication of self-tuning fuzzy PID controller for a SRM direct drive volumecontrol hydraulic pressrdquo Control Engineering Practice vol 17no 12 pp 1398ndash1404 2009 Special Section The 2007 IFACSymposium on Advances in Automotive Control

[15] A Aly ldquoPid parameters optimization using genetic algorithmtechnique for electrohydraulic servo control systemrdquo IntelligentControl and Automation vol 2 no 2 pp 69ndash76 2011

[16] B Yao F Bu and G T C Chiu ldquoNonlinear adaptive robustcontrol of electro-hydraulic servo systems with discontinuousprojectionsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control (CDC rsquo98) pp 2265ndash2270 December1998

[17] B Yao F Bu J Reedy and G T-C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[18] K K Ahn and Q T Dinh ldquoSelf-tuning of quantitative feedbacktheory for force control of an electro-hydraulic test machinerdquoControl Engineering Practice vol 17 no 11 pp 1291ndash1306 2009

[19] E Kolsi Gdoura M Feki and N Derbel ldquoSliding mode controlof a hydraulic servo system position using adaptive slidingsurface and adaptive gainrdquo International Journal of ModellingIdentification and Control vol 23 pp 211ndash219 2015

[20] C Guan and S Pan ldquoAdaptive sliding mode control of electro-hydraulic systemwith nonlinear unknown parametersrdquoControlEngineering Practice vol 16 no 11 pp 1275ndash1284 2008

[21] D Liu Z Tang and Z Pei ldquoVariable structure compensationPID control of asymmetrical hydraulic cylinder trajectorytrackingrdquo Mathematical Problems in Engineering vol 2015Article ID 890704 9 pages 2015

[22] A G Loukianov J Rivera Y V Orlov and E Y MoralesTeraoka ldquoRobust trajectory tracking for an electrohydraulicactuatorrdquo IEEE Transactions on Industrial Electronics vol 56no 9 pp 3523ndash3531 2009

[23] M Ye Q Wang and S Jiao ldquoRobust 1198672119867infin

control forthe electrohydraulic steering system of a four-wheel vehiclerdquoMathematical Problems in Engineering vol 2014 Article ID208019 12 pages 2014

[24] W Shen J Jiang H R Karimi and X Su ldquoObserver-basedrobust control for hydraulic velocity control systemrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 6891329 pages 2013

[25] E Kolsi-Gdoura M Feki and N Derbel ldquoSliding mode-based robust position control of an electrohydraulic systemrdquoin Proceedings of the 10th International Multi-Conference onSystems Signals and Devices (SSD rsquo13) pp 1ndash5 March 2013

[26] E Kolsi Gdoura M Feki and N Derbel ldquoControl of ahydraulic servo system using sliding mode with an integral andrealizable reference compensationrdquo Control Engineering andApplied Informatics vol 17 pp 111ndash119 2015

[27] A Levant ldquoSliding order and sliding accuracy in sliding modecontrolrdquo International Journal of Control vol 58 no 6 pp 1247ndash1263 1993

[28] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[29] M Djemai K Busawon K Benmansour and A MaroufldquoHigh-order sliding mode control of a DC motor drive viaa switched controlled multi-cellular converterrdquo InternationalJournal of Systems Science vol 42 no 11 pp 1869ndash1882 2011

[30] M Lavanya R M Brisilla and V Sankaranarayanan ldquoHigherorder sliding mode control of permanent magnet dc motorrdquoin Proceedings of the 12th International Workshop on VariableStructure Systems (VSS rsquo12) pp 226ndash230 January 2012

[31] S Laghrouche M Smaoui F Plestan and X Brun ldquoHigherorder sliding mode control based on optimal approach of anelectropneumatic actuatorrdquo International Journal of Control vol79 no 2 pp 119ndash131 2006

[32] A G Loukianov E Sanchez and C Lizalde ldquoForce trackingneural block control for an electro-hydraulic actuator viasecond-order slidingmoderdquo International Journal of Robust andNonlinear Control vol 18 no 3 pp 319ndash332 2008

[33] J Komsta N van Oijen and P Antoszkiewicz ldquoIntegral slidingmode compensator for load pressure control of die-cushioncylinder driverdquo Control Engineering Practice vol 21 no 5 pp708ndash718 2013

[34] M Feki Synthese de commandes et drsquoobservateurs pour lessystemes non-lineaires application aux systemes hydrauliques[PhD thesis] Universite de Metz Metz France 2001

[35] V V Filippov ldquoOn a scalar equation with discontinuous right-hand side and the uniqueness theoremrdquo Differential Equationsvol 38 no 10 pp 1435ndash1445 2002

[36] T Boukhobza M Djemai and J P Barbot ldquoNonlinear slidingobserver for systems in output and output derivative injectionformrdquo in Proceedings of the IFAC World Congress pp 299ndash305San Antonio Tex USA 1996

[37] H Saadaoui N Manamanni M Djemaı J P Barbot andT Floquet ldquoExact differentiation and sliding mode observersfor switched Lagrangian systemsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 65 no 5 pp 1050ndash1069 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

x2 x3m0

Sk1

m

b

P1 V1 P2 V2

Pr Ps

u

Figure 1 Hydraulic servo system controlled using a servovalve

where 119889max = max119905ge0119889(119905) Consequently by solving (119909) =

minus119882sign(120590(119909)) the control law is expressed as follows

119906 (119909) =

119873(119909)

(4119861119878119896119881119905)radic119875119889minus 1199091

if 119873(119909) le 0

119873(119909)

(4119861119878119896119881119905)radic119875119889+ 1199091

if 119873(119909) lt 0(9)

where

119873(119909) = minus119882 sign (120590 (119909)) minus1198622119898119905

(1198781199091 minus 1198871199092 minus 1198961198971199093)

minus11986231199092 +41198611198782

119881119905

1199092

(10)

Despite the perturbation 119889(119905) and the uncertaintyΔ119896119897 the

variable structure controller defined by (9) with the slidinggain119882 in (8) ensures the existence of sliding mode howeverwe can easily deduce that we do not achieve the referenceoutput because when the system behavior is confined tothe sliding surface (5) the linear virtual controller does notguarantee any robustnesswith respect to the perturbation andthe uncertainty Indeed it can be easily shown that as far asthe sliding motion is preserved the system is asymptoticallystable if the closed loop eigenvalue is chosen such as

120582 gt radic

10038161003816100381610038161003816100381610038161003816

Δ119896119897

119898119905

10038161003816100381610038161003816100381610038161003816

(11)

and in that case the steady state error due to the uncertaintyis given by

119890119896119897=Δ119896119897

119898119905

11205822 + Δ119896

119897119898119905

1199093ref (12)

We similarly can show that the steady state error due to theconstant perturbation is given by

119890119889= minus

2120582119889 (13)

Figure 2 shows the system behavior with 120582 = 50 119889(119905) =01 Δ119896

119897= 25000 and 1199093ref = 20 cm We clearly notice that

there is a steady state error of more than 2 cm at the positionoutput So the first-order sliding mode is not robust to themismatched uncertainty and perturbation

22 Second-Order SMC Since the system is third-ordersingle input thenwemay think of designing the higher-orderSMC up to second order Let 120590

ℎdenote the sliding variable

defined as

120590ℎ= ℎ (1199093 minus1199093ref) + 1199092 (14)

where ℎ gt 0 is a strictly positive scalar We may verify thatthe relative degree of system ((1)ndash(3)) together with (14) withrespect to the sliding variable 120590

ℎis constant and equal to two

Thus we have

ℎ= 1198861 (119909 119905) + Δ1198861 (119909 119905) + 1198862 (119909) 119906 (15)

where 1198861(119909 119905) and 1198862(119909) are known functions and Δ1198861(119909 119905) isan unknown bounded function in terms of 119889(119905) and Δ119896

119897

1198861 (119909 119905) = minus4119861119878119898119905119881119905

(1205721199091 + 1198781199092)

+ (ℎ

119898119905

minus119887

1198982119905

) (1198781199091 minus 1198871199092 minus 1198961198971199093)

minus119896119897

119898119905

1199092

Δ1198861 (119909 119905) = minus(ℎ

119898119905

minus119887

1198982119905

)Δ1198961198971199093 minus

Δ119896119897

119898119905

1199092

minus119889 (119905)119896119897+ Δ119896119897

119898119905

1198862 (119909) =

4119861119878119898119905119881119905

119896radic119875119889minus 1199091 if 119906 ge 0

4119861119878119898119905119881119905

119896radic119875119889+ 1199091 if 119906 lt 0

(16)

The solution should be understood in the Filippov sense[35] and the trajectories of the system are supposed to beextendible infinitely in time for any bounded measurableinput

4 Mathematical Problems in Engineering

0 01 02 03

03

04 050

50

100

150

200

Time (s)0 01 02 03 04 05

minus1

0

1

2

3

4

Velo

city

(ms

)

Time (s)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

0

5

10

15

20

Posit

ion

(cm

)

minus5

0

5

10

15

20

25

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 2 The behavior of the system under sliding mode controller defined by (9) 120582 = 50

By defining 1199111 = 120590ℎ and 1199112 = ℎ achieving sliding mode120590ℎ= 0 is equivalent to the finite stabilization of the system

1 = 1199112

2 = 1198861 (119909 119905) + Δ1198861 (119909 119905) + 1198862 (119909) 119906(17)

which hence ideally yields to the sliding set120590ℎ= 0 and

ℎ= 0

Taking into account the practical considerations the slidingset is defined as follows [27]

Definition 1 Given the sliding variable 120590ℎ(119909 119905) the ldquoreal

second-order sliding setrdquo associated with ((1)ndash(3)) is definedas

119904119879119890= 119909isin

120594

1003816100381610038161003816120590ℎ1003816100381610038161003816

le 119896111987921198901003816100381610038161003816ℎ1003816100381610038161003816 le 1198962119879119890 (18)

where 119879119890is the finite sampling time (fixed at 119879

119890= 20 120583s in the

sequel) and 1198961 and 1198962 are positive constants

Definition 2 Consider the nonempty real second-order slid-ing set 119904

119879119890given in (18) and assume that it is locally an

integral set in the Filippov senseThe corresponding behaviorof system ((1)ndash(3)) satisfying (18) is called ldquoreal second-ordersliding moderdquo with respect to 120590

ℎ(119909 119905) [27]

The variable structure control law 119906 can be chosen asfollows

119906 =1

1198862 (119909)(minus1198861 (119909 119905) minus11987011199111 minus11987021199112 minus1198821 sign (1199111)

minus1198822sign (1199112)) (19)

where all the controller gains 1198701 11987021198821 and1198822 are strictlypositive Applying controller (19) yields to

[1

2] = [

0 1minus1198701 minus1198702

][1199111

1199112]

+[0 0minus1198821 minus1198822

][sign (1199111)sign (1199112)

] minus [01]Δ1198861 (119909 119905)

(20)

Mathematical Problems in Engineering 5

which can be written in the following compact form

= 119860119870119885+119860

119882sign (119911) minus 119861Δ119886 (119909 119905) (21)

where 119860119870= [

0 1minus1198701 minus1198702

] 119860119882= [

0 0minus1198821 minus1198822

] and 119861 = [ 01 ]We can determine the control parameters 1198701 and 1198702

such that the characteristic equation of 119860119870has two equal

eigenvalues at 119904 = minus120582

1198701 = 1205822

1198702 = 2120582

120582 gt 0

(22)

Since 119860119870is a Hurwitz matrix then it satisfies the Lyapunov

function119860119879119870119875+119875119860

119870= minus119876 where 119875 and119876 are some positive

definite matrices To determine the gains 1198821 and 1198822 wedefine the following Lyapunov function candidate for system(20)

119881 (119885) = 119885119879

119875119885 (23)

Its time derivative in the direction of system (20) trajectoriesis given by

= 119879

119875119885+119885119879

119875

= minus119885119879

119876119885+ 2119885119879119875119860119882sign (119885) minus 2119885119879119875119861Δ119886 (119909 119905)

(24)

If we choose 119875221198821= 119875121198822then 119875119860

119882is a negative definite

matrix and we have

lt minus119885119879

119876119885+ 2120582max (119875119860119882) 1198851

+ 2 1198851 120582min (119875) 1198611 |Δ119886 (119909 119905)| lt 0(25)

provided that1198821 and1198822 are also chosen such that

minus120582max (119875119860119882) gt 120582min (119875)1003816100381610038161003816Δ1198861 (119909 119905)

1003816100381610038161003816 (26)

where 120582max(sdot) and 120582min(sdot) are respectively the largest andleast eigenvalues of the matrix

From Figure 3 we can notice that the second-orderSMC achieves a better performance than the first-orderSMC However the robustness with respect to the constantperturbation is not guaranteed Indeed in sliding mode wehave 120590

ℎ= 0 thus 1199092 = minusℎ(1199093 minus 1199093ref ) = minusℎ1198903 and hence on

the sliding set 120590ℎ= 0 we get

1198903 = minus ℎ1198903 +119889 (119905) (27)

which is a first-order nonautonomous system with a steadystate value equal to 119889(119905)ℎ Therefore for the constantperturbation 119889(119905) = 01 the steady state error is as expectedand delineated in Figure 3 1199093 minus 1199093ref = 01 cm Whenincreasing the value of ℎ the variable structure gains 1198821and1198822 should be increased thus accentuating the chatteringphenomenon which was attenuated by the use of the second-order sliding mode

The boundedness of 1199091 and 1199092 is ensured since lt 0 and119881(119885) depends on 1199111 and 1199112 which in turn depend on 1199091 1199092and 1199093

3 Sliding Mode Controller withNonlinear Surface

To overcome the problem of mismatched perturbation anduncertainty we suggest in this section designing a slidingsurface 120590(119909) based on the backstepping method and usingrobust variable structure virtual controller

In Section 21 the sliding surface 120590(119909) = 0 was obtainedfrom the design of the linear controller 1199091 = 119901(119909) forsubsystem ((2)-(3)) To ensure robustness with respect to theparametric uncertainty a variable structure virtual controller1199091 = 119901(119909) is designed by choosing a sliding surface 1205901(1199092 1199093)and requiring its attractivity by imposing

1 (1199092 1199093) = minus1198821 sign (1205901 (1199092 1199093)) (28)

Therefore the virtual controller 1199091 = 119901(119909) is obtained bysolving (28) and the sliding surface 120590(119909) = 1199091 minus 119901(119909)However when sliding on 1205901(1199092 1199093) is achieved we get1205901(1199092 1199093) = 0 which means 1199092 = V(1199093) and thus in slidingmode we have

3 = V (1199093) + 119889 (119905) (29)

Again to ensure asymptotic convergence of 1199093(119905) to 1199093ref asimple linear proportional term V(1199093) = minus119862(1199093 minus 1199093ref ) is notsatisfactory but a variable structure term can guarantee therequired convergence Indeed with

V (1199093) = minus1198823 (1199093 minus1199093ref) minus1198822 sign (1199093 minus1199093ref) (30)

where the gains are chosen such that1198822 gt 119889max and1198823 gt 0we may easily show the convergence of 1199093(119905) to 1199093ref as far asthe sliding surface

1205901 (1199092 1199093) = 1199092 +1198823 (1199093 minus1199093ref)

+1198822 sign (1199093 minus1199093ref)(31)

is attractiveWhen attempting to achieve the attractivity of 1205901(1199092 1199093)

by imposing (28) we face the attempt to differentiate a dis-continuous function Although this is mathematically impos-sible we can overcome the problem from an engineeringpoint of view by either replacing the discontinuous signumfunction with a smoother saturation function or directlyconsidering that the derivative of the signum function is theDirac impulse (120575(119909)) which is zero everywhere except atan isolated single point Indeed in engineering realizationusing a processor and discretization process the isolateddiscontinuities especially when they are few can easily beavoided and would not cause any problems Therefore from(28) we have

1119898119905

(1198781199091 minus 1198871199092 minus 1198961198971199093) +11988231199092 +11988221199092120575 (1199093 minus1199093ref)

= minus1198821 sign (1205901 (1199092 1199093))(32)

and the sliding surface 120590(119909) becomes

120590 (119909) = 1198781199091 + (1198981199051198823 +1198981199051198822120575 (1199093 minus1199093ref) minus 119887) 1199092

minus 1198961198971199093 +1198981199051198821 sign (1205901 (1199092 1199093))

(33)

6 Mathematical Problems in Engineering

0

50

100

150

200

250

minus05

0

05

1

15

2

25

3

0

5

10

15

20

25

minus5

0

5

10

15

20

03 04 0520

201

202

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 3 Behavior of system under second-order sliding mode controller (19) 120582 = 50 and ℎ = 100

Finally by imposing (119909) = minus119882sign(120590(119909)) the attractivitycondition 120590(119909)(119909) lt 0 is satisfied The choice of thesurface 120590(119909) and its imposed derivative lead to the followingcontroller

119906 (119909) =

119873119889(119909)

(4119861119878119896119881119905)radic119875119889minus 1199091

if 119873119889(119909) ge 0

119873119889(119909)

(4119861119878119896119881119905)radic119875119889+ 1199091

if 119873119889(119909) lt 0

(34)

with

119873119889(119909) = minus119882 sign (120590 (119909)) minus119898

1199051198821120575 (1205901 (119909))

+ (4119861119878120572119881119905

+119887119878

119898119905

minus 1198781198824)1199091

+(41198611198782

119881119905minus1198872

119898119905

+ 119896119897+ 1198871198824)1199092

+(1198961198971198824 minus

119887119896119897

119898119905

)1199093

(35)

where the derivative of the Dirac impulse is consideredas zero and the sliding gain 119882 should be chosen suchthat the attractivity occurs in presence of uncertainty andperturbation

119882 gt

10038161003816100381610038161003816100381610038161003816

119887119881119905

119878119898119905

Δ119896119897

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119881119905

1198781198823Δ119896119897

10038161003816100381610038161003816100381610038161003816

+1003816100381610038161003816119896119897119889max

1003816100381610038161003816 (36)

From Figure 4 we can notice clearly that we haveattained our aim to drive the hydraulic servo system tothe reference position Nevertheless due to the nested slid-ing modes the chattering phenomenon was emphasized

Mathematical Problems in Engineering 7

0

50

100

150

200

250

minus05

0

05

1

15

2

25

3

0

5

10

15

20

25

minus5

0

5

10

15

20

25

03 04 05199

20

201

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 4 Behavior of system under sliding mode controller using nonlinear surface defined by (33)

In order to attenuate it and to avoid the mathematicalproblem of taking the derivative of the signum function wehave used a smooth saturation function sign(119909) ≃ tanh(120583119909)for a sufficiently high positive value 120583 gt 0 The resultsdepicted in Figure 5 show that the use of the smoothfunction helped to achieve the reference value in presenceof mismatched uncertainty and perturbation with attenuatedchattering

4 Sliding Mode Observer Design

As we can notice the controller conceived in the foregoingsection uses all three state variables However measuring thedifferential pressureΔ119875 = 1199091 is a costly task and requires hightechnology procedure to avoid additional leakage To remedythis situation we propose in this section designing a slidingmode observer that may estimate the required states that arethen used to construct the controller

Let us consider the observer model given in (37) andinferred from the step-by-step observer presented in [36 37]

1 =4119861119881119905

(119896119906radic119875119889minus sign (119906) 1 minus

12057211 + 120574 |119906|

minus 1198782)

+1198711 sign (1 minus 1199111) (37)

2 =1119898119905

(1198781199111 minus 1198872 minus 1198961198971199113) + 1198712 sign (2 minus 1199112) (38)

3 = 1199112 +1198713 sign (1199093 minus z3) (39)

where 1198711 1198712 and 1198713 are the observer gain and 1 and 2 aredefined as

1 = 1199111 +119896119897

119878(1199093 minus 1199113) +

119898119905

1198781198712 sign (2 minus 1199112)

2 = 1199112 +1198713 sign (1199093 minus 1199113) (40)

8 Mathematical Problems in Engineering

0

50

100

150

200

250

minus05

0

05

1

15

2

25

0

5

10

15

20

25

minus5

0

5

10

15

20

03 04 0519995

20

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 5 Behavior of system under slidingmode controller using nonlinear surface defined by (33) and a smooth saturation function tanh(sdot)

To prove the efficiency of the observer and the fact thatthe estimated states based controller can also achieve accuratepositioning in presence of perturbation and uncertainty wewill proceed by a step-by-step proof

Step 1 Let 1198903 = 1199093 minus 1199113 and let 1198902 = 1199092 minus 1199112 from (3) and (39)the error dynamics are expressed as

1198903 = 1198902 +119889 (119905) minus 1198713 sign (1198903) (41)Thus if 1198713 is chosen such that

1198713 gt sup119905gt01198902 (119905) + 119889 (119905) (42)

then a sliding mode is established at the observer slidingsurface 1198903 = 0 within a finite timeMoreover at slidingmodewe obtain 1198903 = 0 and thus from (41) we have

0 = 1199092 minus 1199112 +119889 (119905) minus 1198713 sign (1198903) (43)that is

1199092 +119889 (119905) = 1199112 +1198713 sign (1198903) = 2 (44)

Now if the sliding surface 1205901 of the nonlinear surface basedSMC is expressed in terms of the estimated state 2

1205901 (2 1199093) = 2 +1198823 (1199093 minus1199093ref)

+1198822 sign (1199093 minus1199093ref) (45)

then by using (3) (44) and (45) the position dynamics aregiven by

3 = minus1198823 (1199093 minus1199093ref) minus1198822 sign (1199093 minus1199093ref) (46)

Therefore within a finite time we obtain 1199093 = 1199093ref

Step 2 Let 1198901 = 1199091 minus1199111 from (2) and (38) the error dynamicsare expressed as

1198902 =1119898119905

(1198781198901 minus 1198871198902 minus 1198871198713 sign (1198903) minus 1198961198971198903) minusΔ119896119897

119898119905

1199093

minus1198712 sign (2 minus 1199112) (47)

Mathematical Problems in Engineering 9

Thus if 1198712 is chosen such that

1198712 gt1119898119905

1003816100381610038161003816Δ1198961198971199093 + 1198781198901 + 119887119889max1003816100381610038161003816 (48)

then we can deduce that 1198902 + 119889 tend to zero and hence 1199112tend to 1199092 + 119889 That is the observer will estimate the velocitywith the constant perturbation Moreover when the errordynamics are sliding on 1198903 = 0 and 1198902 + 119889 = 0 then 1198902 = 0and we get

1 = 1199091 +119887

1198781198902 minus

Δ119896119897

1198781199093 (49)

That is the observer will estimate the differential pressurewith a constant difference proportional to the perturbationand the uncertainty Now if the controller sliding surface 120590 isexpressed using the observer state variables

120590 (119911) = 1198781 + (1198981199051198823 +1198981199051198822120575 (1199093 minus1199093ref) minus 119887) 2

minus 1198961198971199113 +1198981199051198821 sign (1205901 (2 1199093))

(50)

then at sliding mode of the controller (119911) = 0 the velocitydynamics are given by

2 = minus1198823 (1199092 +119889) minus1198821 sign (1205901 (2 1199093)) (51)

Hence the velocity of the system rod will tend to minus119889

Step 3 Finally the differential pressure error dynamics canbe roughly expressed as

1198901 = 1198911 (119909) minus1198911 (119911) minus 1198711 sign(1198901 +119887

1198781198902 minus

Δ119896119897

1198781199093) (52)

if the system is already sliding on the surfaces 1199093 = 1199093ref and1198902 = minus119889 then by choosing 1198711 gt |1198911(119909)minus1198911(119911)| the differentialpressure is estimated with the constant difference minus(119887119878)119889 +Δ119881119905Figure 6 shows the convergence of the observer state 119911

3to

1199093although they are starting fromdifferent initial conditions

indeed the observer is starting at rod position 10 cm whereasthe system is starting at the origin As expected from theabove analysis 1199112 tends to 1199092 + 119889 and 1199111 tends to 1199091 with aconstant difference of 33 bar The observer gains are chosenas 1198711 = 1000 1198712 = 100 and 1198713 = 1 In Figure 7 the observerand system behaviors are shown with the controller beingcalculated using the estimated states The observer initialposition is minus10 cm

5 Simulation Results Analysis

The controllers designed in this paper used the sliding modetheory which is the most used approach to deal with systemsrunning under uncertainty conditions However we haveseen in the second section that the first-order sliding modecontroller with a linear sliding surface is not robust withrespect to perturbation and mismatched uncertainty In factby using the fact that on the sliding surface the systembehavesin a similar way to a linear second-order system it can be

0

100

200

300

minus10123

0

10

20

30

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 6 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with states feedback

minus1000

100200300

minus20246

minus100

102030

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 7 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with estimated states feed-back

easily shown that as far as the slidingmotion is preserved thesystem is asymptotically stable if the closed loop eigenvaluesare chosen as in (11) and thus the steady state error due to theuncertainty is expressed by (12) Also the steady state errordue to the constant perturbation is given by (13)

Therefore the steady state error gets smaller as the closedloop eigenvalues have a larger amplitude This is illustratedby the simulation results delineated in Figure 2 where 120582 = 50and the total error is 246 cm

To circumvent the problem we have suggested a second-order sliding mode controller Applying this controller we

10 Mathematical Problems in Engineering

can notice that it achieves a better performance than the firstSMC In fact the second-order SMC outperforms the first-order SMC in sense of robustness with respect tomismatchedperturbation but it cannot guarantee the robustness withrespect to the constant perturbation The steady state errordue to the perturbation is equal to 119889(119905)ℎ as demonstratedpreviously When the perturbation 119889(119905) = 01 the erroris equal to 01 cm as delineated in Figure 3 If we increasethe value of ℎ we should also increase the value of thevariable structure gains1198821 and1198822 So we obtain a chatteringphenomenon

To overcome the problem of mismatched perturbationand uncertainty we have suggested a sliding mode controllerwith nonlinear surfaceThe idea is based on the backsteppingmethod and using a robust variable structure virtual con-troller The obtained results achieved robustness with respectto parameter uncertainty and perturbation The zoomedcurve on Figure 4 shows that we have attained our aim todrive the hydraulic servo system to a reference position butwith a chattering phenomenon So to attenuate this problemwe suggest substituting the discontinuous function sign(119909)with a smooth saturation function tanh(120583119909) for sufficientlyhigh positive value 120583 gt 0 Owing to this smooth function wecan achieve the reference value in presence of mismatcheduncertainty and perturbation with attenuated chattering

Finally in Section 4 we designed a slidingmode observerin order to estimate the required states The efficiency of thisobserver is proved mathematically and also by simulationspresented in Figures 6 and 7

Compared to other methods such as that presented in[17] which is based on adaptive approach to achieve positioncontrol we notice that our method provides faster responsesince the adaptive controller proposed therein attempts toestimate the mismatched nonlinear uncertainty Indeed therod makes a displacement of 30 cm within more than 08 swhereas with our method a maximum of 02 s will beneeded to make the same displacement In [21] the authorspresented a controller based on variable structure PID todrive the hydraulic system position Although fast responsehas been achieved which is comparable to our results thegood robustness was obtained since the uncertainties andthe perturbation were matched with the controller In ourcase the robustness against the mismatched character of theperturbation and uncertainty presented the main challengeto take In [26] authors obtained comparable results tothose presented in this paper using integral sliding modecontroller with realizable reference compensation applied toan asymmetric piston

6 Conclusion

In this paper we developed several controllers based onthe sliding mode theory Our aim was to control theposition of a hydraulic servo system piston in presence ofmismatched uncertainty and perturbation We have shownthat a first-order sliding mode controller did not achieveany robustness Next we have developed a second-ordersliding mode controller that has shown robustness withrespect to parametric uncertainty but was not robust to

the perturbation Finally we have suggested a sliding modecontroller based on a nonlinear sliding surface The designis based on the backstepping method where on each stepa variable structure virtual controller design leads to thedesign of the sliding surface This controller emphasized thechattering phenomenon due to the nested sliding modesAs a remedy we suggested substituting the discontinuousfunction with a smooth saturation function Eventually wehave designed a robust sliding observer in order to substitutethe unmeasured states with their estimates A step-by-stepproof has shown that the controller issued from the estimatedstates achieved the position tracking

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H E Merritt Hydraulic Control Systems John Wiley amp SonsNew York NY USA 1967

[2] M Feki and E Richard ldquoIncluding leakage flow in the ser-vovalve static modelrdquo International Journal of Modelling andSimulation vol 25 no 1 pp 4004ndash4009 2005

[3] A Alleyne R Liu and H Wright ldquoOn the limitations offorce tracking control for hydraulic active suspensionsrdquo inProceedings of the American Control Conference (ACC rsquo98) vol1 pp 43ndash47 June 1998

[4] Y Cheng andB L RDeMoor ldquoRobustness analysis and controlsystem design for a hydraulic servo systemrdquo IEEE Transactionson Control Systems Technology vol 2 no 3 pp 183ndash197 1994

[5] M Feki E Richard and F Gomes Almeida ldquoCommande eneffort drsquoun verin hydraulique par linearization entreesortierdquoin Journees Doctorales drsquoAutomatiques (JDA rsquo99) pp 181ndash184Nancy-France 1999

[6] H Hahn A Piepenbrink and K-D Leimbach ldquoInputoutputlinearization control of an electro servo-hydraulic actuatorrdquo inProceedings of the 3rd IEEE Conference on Control Applicationspp 995ndash1000 August 1994

[7] M Jovanovic ldquoNonlinear control of an electrohydraulic velocityservosystemrdquo in Proceedings of the 2002 American ControlConference vol 1 pp 588ndash593 2002

[8] G A Sohl and J E Bobrow ldquoExperiments and simulationson the nonlinear control of a hydraulic servosystemrdquo IEEETransactions on Control Systems Technology vol 7 no 2 pp238ndash247 1999

[9] A G Alleyne and R Liu ldquoSystematic control of a class ofnonlinear systems with application to electrohydraulic cylinderpressure controlrdquo IEEE Transactions on Control Systems Tech-nology vol 8 no 4 pp 623ndash634 2000

[10] M A Avila A G Loukianov and E N Sanchez ldquoElectro-hydraulic actuator trajectory trackingrdquo in Proceedings of theAmerican Control Conference (AAC rsquo04) vol 3 pp 2603ndash2608IEEE Boston Mass USA July 2004

[11] A Bonchis P I Corke D C Rye and Q P Ha ldquoVariable struc-ture methods in hydraulic servo systems controlrdquo Automaticavol 37 no 4 pp 589ndash595 2001

[12] M Jerouane and F Lamnabhi-Lagarrigue ldquoA new robust slidingmode controller for a hydraulic actuatorrdquo in Proceedings of the

Mathematical Problems in Engineering 11

40th IEEE Conference on Decision and Control (CDC rsquo01) vol 1pp 908ndash913 December 2001

[13] H-M Chen J-C Renn and J-P Su ldquoSlidingmode control withvarying boundary layers for an electro-hydraulic position servosystemrdquo The International Journal of Advanced ManufacturingTechnology vol 26 no 1-2 pp 117ndash123 2005

[14] J M Zheng S D Zhao and S G Wei ldquoApplication of self-tuning fuzzy PID controller for a SRM direct drive volumecontrol hydraulic pressrdquo Control Engineering Practice vol 17no 12 pp 1398ndash1404 2009 Special Section The 2007 IFACSymposium on Advances in Automotive Control

[15] A Aly ldquoPid parameters optimization using genetic algorithmtechnique for electrohydraulic servo control systemrdquo IntelligentControl and Automation vol 2 no 2 pp 69ndash76 2011

[16] B Yao F Bu and G T C Chiu ldquoNonlinear adaptive robustcontrol of electro-hydraulic servo systems with discontinuousprojectionsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control (CDC rsquo98) pp 2265ndash2270 December1998

[17] B Yao F Bu J Reedy and G T-C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[18] K K Ahn and Q T Dinh ldquoSelf-tuning of quantitative feedbacktheory for force control of an electro-hydraulic test machinerdquoControl Engineering Practice vol 17 no 11 pp 1291ndash1306 2009

[19] E Kolsi Gdoura M Feki and N Derbel ldquoSliding mode controlof a hydraulic servo system position using adaptive slidingsurface and adaptive gainrdquo International Journal of ModellingIdentification and Control vol 23 pp 211ndash219 2015

[20] C Guan and S Pan ldquoAdaptive sliding mode control of electro-hydraulic systemwith nonlinear unknown parametersrdquoControlEngineering Practice vol 16 no 11 pp 1275ndash1284 2008

[21] D Liu Z Tang and Z Pei ldquoVariable structure compensationPID control of asymmetrical hydraulic cylinder trajectorytrackingrdquo Mathematical Problems in Engineering vol 2015Article ID 890704 9 pages 2015

[22] A G Loukianov J Rivera Y V Orlov and E Y MoralesTeraoka ldquoRobust trajectory tracking for an electrohydraulicactuatorrdquo IEEE Transactions on Industrial Electronics vol 56no 9 pp 3523ndash3531 2009

[23] M Ye Q Wang and S Jiao ldquoRobust 1198672119867infin

control forthe electrohydraulic steering system of a four-wheel vehiclerdquoMathematical Problems in Engineering vol 2014 Article ID208019 12 pages 2014

[24] W Shen J Jiang H R Karimi and X Su ldquoObserver-basedrobust control for hydraulic velocity control systemrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 6891329 pages 2013

[25] E Kolsi-Gdoura M Feki and N Derbel ldquoSliding mode-based robust position control of an electrohydraulic systemrdquoin Proceedings of the 10th International Multi-Conference onSystems Signals and Devices (SSD rsquo13) pp 1ndash5 March 2013

[26] E Kolsi Gdoura M Feki and N Derbel ldquoControl of ahydraulic servo system using sliding mode with an integral andrealizable reference compensationrdquo Control Engineering andApplied Informatics vol 17 pp 111ndash119 2015

[27] A Levant ldquoSliding order and sliding accuracy in sliding modecontrolrdquo International Journal of Control vol 58 no 6 pp 1247ndash1263 1993

[28] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[29] M Djemai K Busawon K Benmansour and A MaroufldquoHigh-order sliding mode control of a DC motor drive viaa switched controlled multi-cellular converterrdquo InternationalJournal of Systems Science vol 42 no 11 pp 1869ndash1882 2011

[30] M Lavanya R M Brisilla and V Sankaranarayanan ldquoHigherorder sliding mode control of permanent magnet dc motorrdquoin Proceedings of the 12th International Workshop on VariableStructure Systems (VSS rsquo12) pp 226ndash230 January 2012

[31] S Laghrouche M Smaoui F Plestan and X Brun ldquoHigherorder sliding mode control based on optimal approach of anelectropneumatic actuatorrdquo International Journal of Control vol79 no 2 pp 119ndash131 2006

[32] A G Loukianov E Sanchez and C Lizalde ldquoForce trackingneural block control for an electro-hydraulic actuator viasecond-order slidingmoderdquo International Journal of Robust andNonlinear Control vol 18 no 3 pp 319ndash332 2008

[33] J Komsta N van Oijen and P Antoszkiewicz ldquoIntegral slidingmode compensator for load pressure control of die-cushioncylinder driverdquo Control Engineering Practice vol 21 no 5 pp708ndash718 2013

[34] M Feki Synthese de commandes et drsquoobservateurs pour lessystemes non-lineaires application aux systemes hydrauliques[PhD thesis] Universite de Metz Metz France 2001

[35] V V Filippov ldquoOn a scalar equation with discontinuous right-hand side and the uniqueness theoremrdquo Differential Equationsvol 38 no 10 pp 1435ndash1445 2002

[36] T Boukhobza M Djemai and J P Barbot ldquoNonlinear slidingobserver for systems in output and output derivative injectionformrdquo in Proceedings of the IFAC World Congress pp 299ndash305San Antonio Tex USA 1996

[37] H Saadaoui N Manamanni M Djemaı J P Barbot andT Floquet ldquoExact differentiation and sliding mode observersfor switched Lagrangian systemsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 65 no 5 pp 1050ndash1069 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

0 01 02 03

03

04 050

50

100

150

200

Time (s)0 01 02 03 04 05

minus1

0

1

2

3

4

Velo

city

(ms

)

Time (s)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

0

5

10

15

20

Posit

ion

(cm

)

minus5

0

5

10

15

20

25

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 2 The behavior of the system under sliding mode controller defined by (9) 120582 = 50

By defining 1199111 = 120590ℎ and 1199112 = ℎ achieving sliding mode120590ℎ= 0 is equivalent to the finite stabilization of the system

1 = 1199112

2 = 1198861 (119909 119905) + Δ1198861 (119909 119905) + 1198862 (119909) 119906(17)

which hence ideally yields to the sliding set120590ℎ= 0 and

ℎ= 0

Taking into account the practical considerations the slidingset is defined as follows [27]

Definition 1 Given the sliding variable 120590ℎ(119909 119905) the ldquoreal

second-order sliding setrdquo associated with ((1)ndash(3)) is definedas

119904119879119890= 119909isin

120594

1003816100381610038161003816120590ℎ1003816100381610038161003816

le 119896111987921198901003816100381610038161003816ℎ1003816100381610038161003816 le 1198962119879119890 (18)

where 119879119890is the finite sampling time (fixed at 119879

119890= 20 120583s in the

sequel) and 1198961 and 1198962 are positive constants

Definition 2 Consider the nonempty real second-order slid-ing set 119904

119879119890given in (18) and assume that it is locally an

integral set in the Filippov senseThe corresponding behaviorof system ((1)ndash(3)) satisfying (18) is called ldquoreal second-ordersliding moderdquo with respect to 120590

ℎ(119909 119905) [27]

The variable structure control law 119906 can be chosen asfollows

119906 =1

1198862 (119909)(minus1198861 (119909 119905) minus11987011199111 minus11987021199112 minus1198821 sign (1199111)

minus1198822sign (1199112)) (19)

where all the controller gains 1198701 11987021198821 and1198822 are strictlypositive Applying controller (19) yields to

[1

2] = [

0 1minus1198701 minus1198702

][1199111

1199112]

+[0 0minus1198821 minus1198822

][sign (1199111)sign (1199112)

] minus [01]Δ1198861 (119909 119905)

(20)

Mathematical Problems in Engineering 5

which can be written in the following compact form

= 119860119870119885+119860

119882sign (119911) minus 119861Δ119886 (119909 119905) (21)

where 119860119870= [

0 1minus1198701 minus1198702

] 119860119882= [

0 0minus1198821 minus1198822

] and 119861 = [ 01 ]We can determine the control parameters 1198701 and 1198702

such that the characteristic equation of 119860119870has two equal

eigenvalues at 119904 = minus120582

1198701 = 1205822

1198702 = 2120582

120582 gt 0

(22)

Since 119860119870is a Hurwitz matrix then it satisfies the Lyapunov

function119860119879119870119875+119875119860

119870= minus119876 where 119875 and119876 are some positive

definite matrices To determine the gains 1198821 and 1198822 wedefine the following Lyapunov function candidate for system(20)

119881 (119885) = 119885119879

119875119885 (23)

Its time derivative in the direction of system (20) trajectoriesis given by

= 119879

119875119885+119885119879

119875

= minus119885119879

119876119885+ 2119885119879119875119860119882sign (119885) minus 2119885119879119875119861Δ119886 (119909 119905)

(24)

If we choose 119875221198821= 119875121198822then 119875119860

119882is a negative definite

matrix and we have

lt minus119885119879

119876119885+ 2120582max (119875119860119882) 1198851

+ 2 1198851 120582min (119875) 1198611 |Δ119886 (119909 119905)| lt 0(25)

provided that1198821 and1198822 are also chosen such that

minus120582max (119875119860119882) gt 120582min (119875)1003816100381610038161003816Δ1198861 (119909 119905)

1003816100381610038161003816 (26)

where 120582max(sdot) and 120582min(sdot) are respectively the largest andleast eigenvalues of the matrix

From Figure 3 we can notice that the second-orderSMC achieves a better performance than the first-orderSMC However the robustness with respect to the constantperturbation is not guaranteed Indeed in sliding mode wehave 120590

ℎ= 0 thus 1199092 = minusℎ(1199093 minus 1199093ref ) = minusℎ1198903 and hence on

the sliding set 120590ℎ= 0 we get

1198903 = minus ℎ1198903 +119889 (119905) (27)

which is a first-order nonautonomous system with a steadystate value equal to 119889(119905)ℎ Therefore for the constantperturbation 119889(119905) = 01 the steady state error is as expectedand delineated in Figure 3 1199093 minus 1199093ref = 01 cm Whenincreasing the value of ℎ the variable structure gains 1198821and1198822 should be increased thus accentuating the chatteringphenomenon which was attenuated by the use of the second-order sliding mode

The boundedness of 1199091 and 1199092 is ensured since lt 0 and119881(119885) depends on 1199111 and 1199112 which in turn depend on 1199091 1199092and 1199093

3 Sliding Mode Controller withNonlinear Surface

To overcome the problem of mismatched perturbation anduncertainty we suggest in this section designing a slidingsurface 120590(119909) based on the backstepping method and usingrobust variable structure virtual controller

In Section 21 the sliding surface 120590(119909) = 0 was obtainedfrom the design of the linear controller 1199091 = 119901(119909) forsubsystem ((2)-(3)) To ensure robustness with respect to theparametric uncertainty a variable structure virtual controller1199091 = 119901(119909) is designed by choosing a sliding surface 1205901(1199092 1199093)and requiring its attractivity by imposing

1 (1199092 1199093) = minus1198821 sign (1205901 (1199092 1199093)) (28)

Therefore the virtual controller 1199091 = 119901(119909) is obtained bysolving (28) and the sliding surface 120590(119909) = 1199091 minus 119901(119909)However when sliding on 1205901(1199092 1199093) is achieved we get1205901(1199092 1199093) = 0 which means 1199092 = V(1199093) and thus in slidingmode we have

3 = V (1199093) + 119889 (119905) (29)

Again to ensure asymptotic convergence of 1199093(119905) to 1199093ref asimple linear proportional term V(1199093) = minus119862(1199093 minus 1199093ref ) is notsatisfactory but a variable structure term can guarantee therequired convergence Indeed with

V (1199093) = minus1198823 (1199093 minus1199093ref) minus1198822 sign (1199093 minus1199093ref) (30)

where the gains are chosen such that1198822 gt 119889max and1198823 gt 0we may easily show the convergence of 1199093(119905) to 1199093ref as far asthe sliding surface

1205901 (1199092 1199093) = 1199092 +1198823 (1199093 minus1199093ref)

+1198822 sign (1199093 minus1199093ref)(31)

is attractiveWhen attempting to achieve the attractivity of 1205901(1199092 1199093)

by imposing (28) we face the attempt to differentiate a dis-continuous function Although this is mathematically impos-sible we can overcome the problem from an engineeringpoint of view by either replacing the discontinuous signumfunction with a smoother saturation function or directlyconsidering that the derivative of the signum function is theDirac impulse (120575(119909)) which is zero everywhere except atan isolated single point Indeed in engineering realizationusing a processor and discretization process the isolateddiscontinuities especially when they are few can easily beavoided and would not cause any problems Therefore from(28) we have

1119898119905

(1198781199091 minus 1198871199092 minus 1198961198971199093) +11988231199092 +11988221199092120575 (1199093 minus1199093ref)

= minus1198821 sign (1205901 (1199092 1199093))(32)

and the sliding surface 120590(119909) becomes

120590 (119909) = 1198781199091 + (1198981199051198823 +1198981199051198822120575 (1199093 minus1199093ref) minus 119887) 1199092

minus 1198961198971199093 +1198981199051198821 sign (1205901 (1199092 1199093))

(33)

6 Mathematical Problems in Engineering

0

50

100

150

200

250

minus05

0

05

1

15

2

25

3

0

5

10

15

20

25

minus5

0

5

10

15

20

03 04 0520

201

202

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 3 Behavior of system under second-order sliding mode controller (19) 120582 = 50 and ℎ = 100

Finally by imposing (119909) = minus119882sign(120590(119909)) the attractivitycondition 120590(119909)(119909) lt 0 is satisfied The choice of thesurface 120590(119909) and its imposed derivative lead to the followingcontroller

119906 (119909) =

119873119889(119909)

(4119861119878119896119881119905)radic119875119889minus 1199091

if 119873119889(119909) ge 0

119873119889(119909)

(4119861119878119896119881119905)radic119875119889+ 1199091

if 119873119889(119909) lt 0

(34)

with

119873119889(119909) = minus119882 sign (120590 (119909)) minus119898

1199051198821120575 (1205901 (119909))

+ (4119861119878120572119881119905

+119887119878

119898119905

minus 1198781198824)1199091

+(41198611198782

119881119905minus1198872

119898119905

+ 119896119897+ 1198871198824)1199092

+(1198961198971198824 minus

119887119896119897

119898119905

)1199093

(35)

where the derivative of the Dirac impulse is consideredas zero and the sliding gain 119882 should be chosen suchthat the attractivity occurs in presence of uncertainty andperturbation

119882 gt

10038161003816100381610038161003816100381610038161003816

119887119881119905

119878119898119905

Δ119896119897

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119881119905

1198781198823Δ119896119897

10038161003816100381610038161003816100381610038161003816

+1003816100381610038161003816119896119897119889max

1003816100381610038161003816 (36)

From Figure 4 we can notice clearly that we haveattained our aim to drive the hydraulic servo system tothe reference position Nevertheless due to the nested slid-ing modes the chattering phenomenon was emphasized

Mathematical Problems in Engineering 7

0

50

100

150

200

250

minus05

0

05

1

15

2

25

3

0

5

10

15

20

25

minus5

0

5

10

15

20

25

03 04 05199

20

201

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 4 Behavior of system under sliding mode controller using nonlinear surface defined by (33)

In order to attenuate it and to avoid the mathematicalproblem of taking the derivative of the signum function wehave used a smooth saturation function sign(119909) ≃ tanh(120583119909)for a sufficiently high positive value 120583 gt 0 The resultsdepicted in Figure 5 show that the use of the smoothfunction helped to achieve the reference value in presenceof mismatched uncertainty and perturbation with attenuatedchattering

4 Sliding Mode Observer Design

As we can notice the controller conceived in the foregoingsection uses all three state variables However measuring thedifferential pressureΔ119875 = 1199091 is a costly task and requires hightechnology procedure to avoid additional leakage To remedythis situation we propose in this section designing a slidingmode observer that may estimate the required states that arethen used to construct the controller

Let us consider the observer model given in (37) andinferred from the step-by-step observer presented in [36 37]

1 =4119861119881119905

(119896119906radic119875119889minus sign (119906) 1 minus

12057211 + 120574 |119906|

minus 1198782)

+1198711 sign (1 minus 1199111) (37)

2 =1119898119905

(1198781199111 minus 1198872 minus 1198961198971199113) + 1198712 sign (2 minus 1199112) (38)

3 = 1199112 +1198713 sign (1199093 minus z3) (39)

where 1198711 1198712 and 1198713 are the observer gain and 1 and 2 aredefined as

1 = 1199111 +119896119897

119878(1199093 minus 1199113) +

119898119905

1198781198712 sign (2 minus 1199112)

2 = 1199112 +1198713 sign (1199093 minus 1199113) (40)

8 Mathematical Problems in Engineering

0

50

100

150

200

250

minus05

0

05

1

15

2

25

0

5

10

15

20

25

minus5

0

5

10

15

20

03 04 0519995

20

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 5 Behavior of system under slidingmode controller using nonlinear surface defined by (33) and a smooth saturation function tanh(sdot)

To prove the efficiency of the observer and the fact thatthe estimated states based controller can also achieve accuratepositioning in presence of perturbation and uncertainty wewill proceed by a step-by-step proof

Step 1 Let 1198903 = 1199093 minus 1199113 and let 1198902 = 1199092 minus 1199112 from (3) and (39)the error dynamics are expressed as

1198903 = 1198902 +119889 (119905) minus 1198713 sign (1198903) (41)Thus if 1198713 is chosen such that

1198713 gt sup119905gt01198902 (119905) + 119889 (119905) (42)

then a sliding mode is established at the observer slidingsurface 1198903 = 0 within a finite timeMoreover at slidingmodewe obtain 1198903 = 0 and thus from (41) we have

0 = 1199092 minus 1199112 +119889 (119905) minus 1198713 sign (1198903) (43)that is

1199092 +119889 (119905) = 1199112 +1198713 sign (1198903) = 2 (44)

Now if the sliding surface 1205901 of the nonlinear surface basedSMC is expressed in terms of the estimated state 2

1205901 (2 1199093) = 2 +1198823 (1199093 minus1199093ref)

+1198822 sign (1199093 minus1199093ref) (45)

then by using (3) (44) and (45) the position dynamics aregiven by

3 = minus1198823 (1199093 minus1199093ref) minus1198822 sign (1199093 minus1199093ref) (46)

Therefore within a finite time we obtain 1199093 = 1199093ref

Step 2 Let 1198901 = 1199091 minus1199111 from (2) and (38) the error dynamicsare expressed as

1198902 =1119898119905

(1198781198901 minus 1198871198902 minus 1198871198713 sign (1198903) minus 1198961198971198903) minusΔ119896119897

119898119905

1199093

minus1198712 sign (2 minus 1199112) (47)

Mathematical Problems in Engineering 9

Thus if 1198712 is chosen such that

1198712 gt1119898119905

1003816100381610038161003816Δ1198961198971199093 + 1198781198901 + 119887119889max1003816100381610038161003816 (48)

then we can deduce that 1198902 + 119889 tend to zero and hence 1199112tend to 1199092 + 119889 That is the observer will estimate the velocitywith the constant perturbation Moreover when the errordynamics are sliding on 1198903 = 0 and 1198902 + 119889 = 0 then 1198902 = 0and we get

1 = 1199091 +119887

1198781198902 minus

Δ119896119897

1198781199093 (49)

That is the observer will estimate the differential pressurewith a constant difference proportional to the perturbationand the uncertainty Now if the controller sliding surface 120590 isexpressed using the observer state variables

120590 (119911) = 1198781 + (1198981199051198823 +1198981199051198822120575 (1199093 minus1199093ref) minus 119887) 2

minus 1198961198971199113 +1198981199051198821 sign (1205901 (2 1199093))

(50)

then at sliding mode of the controller (119911) = 0 the velocitydynamics are given by

2 = minus1198823 (1199092 +119889) minus1198821 sign (1205901 (2 1199093)) (51)

Hence the velocity of the system rod will tend to minus119889

Step 3 Finally the differential pressure error dynamics canbe roughly expressed as

1198901 = 1198911 (119909) minus1198911 (119911) minus 1198711 sign(1198901 +119887

1198781198902 minus

Δ119896119897

1198781199093) (52)

if the system is already sliding on the surfaces 1199093 = 1199093ref and1198902 = minus119889 then by choosing 1198711 gt |1198911(119909)minus1198911(119911)| the differentialpressure is estimated with the constant difference minus(119887119878)119889 +Δ119881119905Figure 6 shows the convergence of the observer state 119911

3to

1199093although they are starting fromdifferent initial conditions

indeed the observer is starting at rod position 10 cm whereasthe system is starting at the origin As expected from theabove analysis 1199112 tends to 1199092 + 119889 and 1199111 tends to 1199091 with aconstant difference of 33 bar The observer gains are chosenas 1198711 = 1000 1198712 = 100 and 1198713 = 1 In Figure 7 the observerand system behaviors are shown with the controller beingcalculated using the estimated states The observer initialposition is minus10 cm

5 Simulation Results Analysis

The controllers designed in this paper used the sliding modetheory which is the most used approach to deal with systemsrunning under uncertainty conditions However we haveseen in the second section that the first-order sliding modecontroller with a linear sliding surface is not robust withrespect to perturbation and mismatched uncertainty In factby using the fact that on the sliding surface the systembehavesin a similar way to a linear second-order system it can be

0

100

200

300

minus10123

0

10

20

30

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 6 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with states feedback

minus1000

100200300

minus20246

minus100

102030

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 7 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with estimated states feed-back

easily shown that as far as the slidingmotion is preserved thesystem is asymptotically stable if the closed loop eigenvaluesare chosen as in (11) and thus the steady state error due to theuncertainty is expressed by (12) Also the steady state errordue to the constant perturbation is given by (13)

Therefore the steady state error gets smaller as the closedloop eigenvalues have a larger amplitude This is illustratedby the simulation results delineated in Figure 2 where 120582 = 50and the total error is 246 cm

To circumvent the problem we have suggested a second-order sliding mode controller Applying this controller we

10 Mathematical Problems in Engineering

can notice that it achieves a better performance than the firstSMC In fact the second-order SMC outperforms the first-order SMC in sense of robustness with respect tomismatchedperturbation but it cannot guarantee the robustness withrespect to the constant perturbation The steady state errordue to the perturbation is equal to 119889(119905)ℎ as demonstratedpreviously When the perturbation 119889(119905) = 01 the erroris equal to 01 cm as delineated in Figure 3 If we increasethe value of ℎ we should also increase the value of thevariable structure gains1198821 and1198822 So we obtain a chatteringphenomenon

To overcome the problem of mismatched perturbationand uncertainty we have suggested a sliding mode controllerwith nonlinear surfaceThe idea is based on the backsteppingmethod and using a robust variable structure virtual con-troller The obtained results achieved robustness with respectto parameter uncertainty and perturbation The zoomedcurve on Figure 4 shows that we have attained our aim todrive the hydraulic servo system to a reference position butwith a chattering phenomenon So to attenuate this problemwe suggest substituting the discontinuous function sign(119909)with a smooth saturation function tanh(120583119909) for sufficientlyhigh positive value 120583 gt 0 Owing to this smooth function wecan achieve the reference value in presence of mismatcheduncertainty and perturbation with attenuated chattering

Finally in Section 4 we designed a slidingmode observerin order to estimate the required states The efficiency of thisobserver is proved mathematically and also by simulationspresented in Figures 6 and 7

Compared to other methods such as that presented in[17] which is based on adaptive approach to achieve positioncontrol we notice that our method provides faster responsesince the adaptive controller proposed therein attempts toestimate the mismatched nonlinear uncertainty Indeed therod makes a displacement of 30 cm within more than 08 swhereas with our method a maximum of 02 s will beneeded to make the same displacement In [21] the authorspresented a controller based on variable structure PID todrive the hydraulic system position Although fast responsehas been achieved which is comparable to our results thegood robustness was obtained since the uncertainties andthe perturbation were matched with the controller In ourcase the robustness against the mismatched character of theperturbation and uncertainty presented the main challengeto take In [26] authors obtained comparable results tothose presented in this paper using integral sliding modecontroller with realizable reference compensation applied toan asymmetric piston

6 Conclusion

In this paper we developed several controllers based onthe sliding mode theory Our aim was to control theposition of a hydraulic servo system piston in presence ofmismatched uncertainty and perturbation We have shownthat a first-order sliding mode controller did not achieveany robustness Next we have developed a second-ordersliding mode controller that has shown robustness withrespect to parametric uncertainty but was not robust to

the perturbation Finally we have suggested a sliding modecontroller based on a nonlinear sliding surface The designis based on the backstepping method where on each stepa variable structure virtual controller design leads to thedesign of the sliding surface This controller emphasized thechattering phenomenon due to the nested sliding modesAs a remedy we suggested substituting the discontinuousfunction with a smooth saturation function Eventually wehave designed a robust sliding observer in order to substitutethe unmeasured states with their estimates A step-by-stepproof has shown that the controller issued from the estimatedstates achieved the position tracking

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H E Merritt Hydraulic Control Systems John Wiley amp SonsNew York NY USA 1967

[2] M Feki and E Richard ldquoIncluding leakage flow in the ser-vovalve static modelrdquo International Journal of Modelling andSimulation vol 25 no 1 pp 4004ndash4009 2005

[3] A Alleyne R Liu and H Wright ldquoOn the limitations offorce tracking control for hydraulic active suspensionsrdquo inProceedings of the American Control Conference (ACC rsquo98) vol1 pp 43ndash47 June 1998

[4] Y Cheng andB L RDeMoor ldquoRobustness analysis and controlsystem design for a hydraulic servo systemrdquo IEEE Transactionson Control Systems Technology vol 2 no 3 pp 183ndash197 1994

[5] M Feki E Richard and F Gomes Almeida ldquoCommande eneffort drsquoun verin hydraulique par linearization entreesortierdquoin Journees Doctorales drsquoAutomatiques (JDA rsquo99) pp 181ndash184Nancy-France 1999

[6] H Hahn A Piepenbrink and K-D Leimbach ldquoInputoutputlinearization control of an electro servo-hydraulic actuatorrdquo inProceedings of the 3rd IEEE Conference on Control Applicationspp 995ndash1000 August 1994

[7] M Jovanovic ldquoNonlinear control of an electrohydraulic velocityservosystemrdquo in Proceedings of the 2002 American ControlConference vol 1 pp 588ndash593 2002

[8] G A Sohl and J E Bobrow ldquoExperiments and simulationson the nonlinear control of a hydraulic servosystemrdquo IEEETransactions on Control Systems Technology vol 7 no 2 pp238ndash247 1999

[9] A G Alleyne and R Liu ldquoSystematic control of a class ofnonlinear systems with application to electrohydraulic cylinderpressure controlrdquo IEEE Transactions on Control Systems Tech-nology vol 8 no 4 pp 623ndash634 2000

[10] M A Avila A G Loukianov and E N Sanchez ldquoElectro-hydraulic actuator trajectory trackingrdquo in Proceedings of theAmerican Control Conference (AAC rsquo04) vol 3 pp 2603ndash2608IEEE Boston Mass USA July 2004

[11] A Bonchis P I Corke D C Rye and Q P Ha ldquoVariable struc-ture methods in hydraulic servo systems controlrdquo Automaticavol 37 no 4 pp 589ndash595 2001

[12] M Jerouane and F Lamnabhi-Lagarrigue ldquoA new robust slidingmode controller for a hydraulic actuatorrdquo in Proceedings of the

Mathematical Problems in Engineering 11

40th IEEE Conference on Decision and Control (CDC rsquo01) vol 1pp 908ndash913 December 2001

[13] H-M Chen J-C Renn and J-P Su ldquoSlidingmode control withvarying boundary layers for an electro-hydraulic position servosystemrdquo The International Journal of Advanced ManufacturingTechnology vol 26 no 1-2 pp 117ndash123 2005

[14] J M Zheng S D Zhao and S G Wei ldquoApplication of self-tuning fuzzy PID controller for a SRM direct drive volumecontrol hydraulic pressrdquo Control Engineering Practice vol 17no 12 pp 1398ndash1404 2009 Special Section The 2007 IFACSymposium on Advances in Automotive Control

[15] A Aly ldquoPid parameters optimization using genetic algorithmtechnique for electrohydraulic servo control systemrdquo IntelligentControl and Automation vol 2 no 2 pp 69ndash76 2011

[16] B Yao F Bu and G T C Chiu ldquoNonlinear adaptive robustcontrol of electro-hydraulic servo systems with discontinuousprojectionsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control (CDC rsquo98) pp 2265ndash2270 December1998

[17] B Yao F Bu J Reedy and G T-C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[18] K K Ahn and Q T Dinh ldquoSelf-tuning of quantitative feedbacktheory for force control of an electro-hydraulic test machinerdquoControl Engineering Practice vol 17 no 11 pp 1291ndash1306 2009

[19] E Kolsi Gdoura M Feki and N Derbel ldquoSliding mode controlof a hydraulic servo system position using adaptive slidingsurface and adaptive gainrdquo International Journal of ModellingIdentification and Control vol 23 pp 211ndash219 2015

[20] C Guan and S Pan ldquoAdaptive sliding mode control of electro-hydraulic systemwith nonlinear unknown parametersrdquoControlEngineering Practice vol 16 no 11 pp 1275ndash1284 2008

[21] D Liu Z Tang and Z Pei ldquoVariable structure compensationPID control of asymmetrical hydraulic cylinder trajectorytrackingrdquo Mathematical Problems in Engineering vol 2015Article ID 890704 9 pages 2015

[22] A G Loukianov J Rivera Y V Orlov and E Y MoralesTeraoka ldquoRobust trajectory tracking for an electrohydraulicactuatorrdquo IEEE Transactions on Industrial Electronics vol 56no 9 pp 3523ndash3531 2009

[23] M Ye Q Wang and S Jiao ldquoRobust 1198672119867infin

control forthe electrohydraulic steering system of a four-wheel vehiclerdquoMathematical Problems in Engineering vol 2014 Article ID208019 12 pages 2014

[24] W Shen J Jiang H R Karimi and X Su ldquoObserver-basedrobust control for hydraulic velocity control systemrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 6891329 pages 2013

[25] E Kolsi-Gdoura M Feki and N Derbel ldquoSliding mode-based robust position control of an electrohydraulic systemrdquoin Proceedings of the 10th International Multi-Conference onSystems Signals and Devices (SSD rsquo13) pp 1ndash5 March 2013

[26] E Kolsi Gdoura M Feki and N Derbel ldquoControl of ahydraulic servo system using sliding mode with an integral andrealizable reference compensationrdquo Control Engineering andApplied Informatics vol 17 pp 111ndash119 2015

[27] A Levant ldquoSliding order and sliding accuracy in sliding modecontrolrdquo International Journal of Control vol 58 no 6 pp 1247ndash1263 1993

[28] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[29] M Djemai K Busawon K Benmansour and A MaroufldquoHigh-order sliding mode control of a DC motor drive viaa switched controlled multi-cellular converterrdquo InternationalJournal of Systems Science vol 42 no 11 pp 1869ndash1882 2011

[30] M Lavanya R M Brisilla and V Sankaranarayanan ldquoHigherorder sliding mode control of permanent magnet dc motorrdquoin Proceedings of the 12th International Workshop on VariableStructure Systems (VSS rsquo12) pp 226ndash230 January 2012

[31] S Laghrouche M Smaoui F Plestan and X Brun ldquoHigherorder sliding mode control based on optimal approach of anelectropneumatic actuatorrdquo International Journal of Control vol79 no 2 pp 119ndash131 2006

[32] A G Loukianov E Sanchez and C Lizalde ldquoForce trackingneural block control for an electro-hydraulic actuator viasecond-order slidingmoderdquo International Journal of Robust andNonlinear Control vol 18 no 3 pp 319ndash332 2008

[33] J Komsta N van Oijen and P Antoszkiewicz ldquoIntegral slidingmode compensator for load pressure control of die-cushioncylinder driverdquo Control Engineering Practice vol 21 no 5 pp708ndash718 2013

[34] M Feki Synthese de commandes et drsquoobservateurs pour lessystemes non-lineaires application aux systemes hydrauliques[PhD thesis] Universite de Metz Metz France 2001

[35] V V Filippov ldquoOn a scalar equation with discontinuous right-hand side and the uniqueness theoremrdquo Differential Equationsvol 38 no 10 pp 1435ndash1445 2002

[36] T Boukhobza M Djemai and J P Barbot ldquoNonlinear slidingobserver for systems in output and output derivative injectionformrdquo in Proceedings of the IFAC World Congress pp 299ndash305San Antonio Tex USA 1996

[37] H Saadaoui N Manamanni M Djemaı J P Barbot andT Floquet ldquoExact differentiation and sliding mode observersfor switched Lagrangian systemsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 65 no 5 pp 1050ndash1069 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

which can be written in the following compact form

= 119860119870119885+119860

119882sign (119911) minus 119861Δ119886 (119909 119905) (21)

where 119860119870= [

0 1minus1198701 minus1198702

] 119860119882= [

0 0minus1198821 minus1198822

] and 119861 = [ 01 ]We can determine the control parameters 1198701 and 1198702

such that the characteristic equation of 119860119870has two equal

eigenvalues at 119904 = minus120582

1198701 = 1205822

1198702 = 2120582

120582 gt 0

(22)

Since 119860119870is a Hurwitz matrix then it satisfies the Lyapunov

function119860119879119870119875+119875119860

119870= minus119876 where 119875 and119876 are some positive

definite matrices To determine the gains 1198821 and 1198822 wedefine the following Lyapunov function candidate for system(20)

119881 (119885) = 119885119879

119875119885 (23)

Its time derivative in the direction of system (20) trajectoriesis given by

= 119879

119875119885+119885119879

119875

= minus119885119879

119876119885+ 2119885119879119875119860119882sign (119885) minus 2119885119879119875119861Δ119886 (119909 119905)

(24)

If we choose 119875221198821= 119875121198822then 119875119860

119882is a negative definite

matrix and we have

lt minus119885119879

119876119885+ 2120582max (119875119860119882) 1198851

+ 2 1198851 120582min (119875) 1198611 |Δ119886 (119909 119905)| lt 0(25)

provided that1198821 and1198822 are also chosen such that

minus120582max (119875119860119882) gt 120582min (119875)1003816100381610038161003816Δ1198861 (119909 119905)

1003816100381610038161003816 (26)

where 120582max(sdot) and 120582min(sdot) are respectively the largest andleast eigenvalues of the matrix

From Figure 3 we can notice that the second-orderSMC achieves a better performance than the first-orderSMC However the robustness with respect to the constantperturbation is not guaranteed Indeed in sliding mode wehave 120590

ℎ= 0 thus 1199092 = minusℎ(1199093 minus 1199093ref ) = minusℎ1198903 and hence on

the sliding set 120590ℎ= 0 we get

1198903 = minus ℎ1198903 +119889 (119905) (27)

which is a first-order nonautonomous system with a steadystate value equal to 119889(119905)ℎ Therefore for the constantperturbation 119889(119905) = 01 the steady state error is as expectedand delineated in Figure 3 1199093 minus 1199093ref = 01 cm Whenincreasing the value of ℎ the variable structure gains 1198821and1198822 should be increased thus accentuating the chatteringphenomenon which was attenuated by the use of the second-order sliding mode

The boundedness of 1199091 and 1199092 is ensured since lt 0 and119881(119885) depends on 1199111 and 1199112 which in turn depend on 1199091 1199092and 1199093

3 Sliding Mode Controller withNonlinear Surface

To overcome the problem of mismatched perturbation anduncertainty we suggest in this section designing a slidingsurface 120590(119909) based on the backstepping method and usingrobust variable structure virtual controller

In Section 21 the sliding surface 120590(119909) = 0 was obtainedfrom the design of the linear controller 1199091 = 119901(119909) forsubsystem ((2)-(3)) To ensure robustness with respect to theparametric uncertainty a variable structure virtual controller1199091 = 119901(119909) is designed by choosing a sliding surface 1205901(1199092 1199093)and requiring its attractivity by imposing

1 (1199092 1199093) = minus1198821 sign (1205901 (1199092 1199093)) (28)

Therefore the virtual controller 1199091 = 119901(119909) is obtained bysolving (28) and the sliding surface 120590(119909) = 1199091 minus 119901(119909)However when sliding on 1205901(1199092 1199093) is achieved we get1205901(1199092 1199093) = 0 which means 1199092 = V(1199093) and thus in slidingmode we have

3 = V (1199093) + 119889 (119905) (29)

Again to ensure asymptotic convergence of 1199093(119905) to 1199093ref asimple linear proportional term V(1199093) = minus119862(1199093 minus 1199093ref ) is notsatisfactory but a variable structure term can guarantee therequired convergence Indeed with

V (1199093) = minus1198823 (1199093 minus1199093ref) minus1198822 sign (1199093 minus1199093ref) (30)

where the gains are chosen such that1198822 gt 119889max and1198823 gt 0we may easily show the convergence of 1199093(119905) to 1199093ref as far asthe sliding surface

1205901 (1199092 1199093) = 1199092 +1198823 (1199093 minus1199093ref)

+1198822 sign (1199093 minus1199093ref)(31)

is attractiveWhen attempting to achieve the attractivity of 1205901(1199092 1199093)

by imposing (28) we face the attempt to differentiate a dis-continuous function Although this is mathematically impos-sible we can overcome the problem from an engineeringpoint of view by either replacing the discontinuous signumfunction with a smoother saturation function or directlyconsidering that the derivative of the signum function is theDirac impulse (120575(119909)) which is zero everywhere except atan isolated single point Indeed in engineering realizationusing a processor and discretization process the isolateddiscontinuities especially when they are few can easily beavoided and would not cause any problems Therefore from(28) we have

1119898119905

(1198781199091 minus 1198871199092 minus 1198961198971199093) +11988231199092 +11988221199092120575 (1199093 minus1199093ref)

= minus1198821 sign (1205901 (1199092 1199093))(32)

and the sliding surface 120590(119909) becomes

120590 (119909) = 1198781199091 + (1198981199051198823 +1198981199051198822120575 (1199093 minus1199093ref) minus 119887) 1199092

minus 1198961198971199093 +1198981199051198821 sign (1205901 (1199092 1199093))

(33)

6 Mathematical Problems in Engineering

0

50

100

150

200

250

minus05

0

05

1

15

2

25

3

0

5

10

15

20

25

minus5

0

5

10

15

20

03 04 0520

201

202

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 3 Behavior of system under second-order sliding mode controller (19) 120582 = 50 and ℎ = 100

Finally by imposing (119909) = minus119882sign(120590(119909)) the attractivitycondition 120590(119909)(119909) lt 0 is satisfied The choice of thesurface 120590(119909) and its imposed derivative lead to the followingcontroller

119906 (119909) =

119873119889(119909)

(4119861119878119896119881119905)radic119875119889minus 1199091

if 119873119889(119909) ge 0

119873119889(119909)

(4119861119878119896119881119905)radic119875119889+ 1199091

if 119873119889(119909) lt 0

(34)

with

119873119889(119909) = minus119882 sign (120590 (119909)) minus119898

1199051198821120575 (1205901 (119909))

+ (4119861119878120572119881119905

+119887119878

119898119905

minus 1198781198824)1199091

+(41198611198782

119881119905minus1198872

119898119905

+ 119896119897+ 1198871198824)1199092

+(1198961198971198824 minus

119887119896119897

119898119905

)1199093

(35)

where the derivative of the Dirac impulse is consideredas zero and the sliding gain 119882 should be chosen suchthat the attractivity occurs in presence of uncertainty andperturbation

119882 gt

10038161003816100381610038161003816100381610038161003816

119887119881119905

119878119898119905

Δ119896119897

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119881119905

1198781198823Δ119896119897

10038161003816100381610038161003816100381610038161003816

+1003816100381610038161003816119896119897119889max

1003816100381610038161003816 (36)

From Figure 4 we can notice clearly that we haveattained our aim to drive the hydraulic servo system tothe reference position Nevertheless due to the nested slid-ing modes the chattering phenomenon was emphasized

Mathematical Problems in Engineering 7

0

50

100

150

200

250

minus05

0

05

1

15

2

25

3

0

5

10

15

20

25

minus5

0

5

10

15

20

25

03 04 05199

20

201

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 4 Behavior of system under sliding mode controller using nonlinear surface defined by (33)

In order to attenuate it and to avoid the mathematicalproblem of taking the derivative of the signum function wehave used a smooth saturation function sign(119909) ≃ tanh(120583119909)for a sufficiently high positive value 120583 gt 0 The resultsdepicted in Figure 5 show that the use of the smoothfunction helped to achieve the reference value in presenceof mismatched uncertainty and perturbation with attenuatedchattering

4 Sliding Mode Observer Design

As we can notice the controller conceived in the foregoingsection uses all three state variables However measuring thedifferential pressureΔ119875 = 1199091 is a costly task and requires hightechnology procedure to avoid additional leakage To remedythis situation we propose in this section designing a slidingmode observer that may estimate the required states that arethen used to construct the controller

Let us consider the observer model given in (37) andinferred from the step-by-step observer presented in [36 37]

1 =4119861119881119905

(119896119906radic119875119889minus sign (119906) 1 minus

12057211 + 120574 |119906|

minus 1198782)

+1198711 sign (1 minus 1199111) (37)

2 =1119898119905

(1198781199111 minus 1198872 minus 1198961198971199113) + 1198712 sign (2 minus 1199112) (38)

3 = 1199112 +1198713 sign (1199093 minus z3) (39)

where 1198711 1198712 and 1198713 are the observer gain and 1 and 2 aredefined as

1 = 1199111 +119896119897

119878(1199093 minus 1199113) +

119898119905

1198781198712 sign (2 minus 1199112)

2 = 1199112 +1198713 sign (1199093 minus 1199113) (40)

8 Mathematical Problems in Engineering

0

50

100

150

200

250

minus05

0

05

1

15

2

25

0

5

10

15

20

25

minus5

0

5

10

15

20

03 04 0519995

20

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 5 Behavior of system under slidingmode controller using nonlinear surface defined by (33) and a smooth saturation function tanh(sdot)

To prove the efficiency of the observer and the fact thatthe estimated states based controller can also achieve accuratepositioning in presence of perturbation and uncertainty wewill proceed by a step-by-step proof

Step 1 Let 1198903 = 1199093 minus 1199113 and let 1198902 = 1199092 minus 1199112 from (3) and (39)the error dynamics are expressed as

1198903 = 1198902 +119889 (119905) minus 1198713 sign (1198903) (41)Thus if 1198713 is chosen such that

1198713 gt sup119905gt01198902 (119905) + 119889 (119905) (42)

then a sliding mode is established at the observer slidingsurface 1198903 = 0 within a finite timeMoreover at slidingmodewe obtain 1198903 = 0 and thus from (41) we have

0 = 1199092 minus 1199112 +119889 (119905) minus 1198713 sign (1198903) (43)that is

1199092 +119889 (119905) = 1199112 +1198713 sign (1198903) = 2 (44)

Now if the sliding surface 1205901 of the nonlinear surface basedSMC is expressed in terms of the estimated state 2

1205901 (2 1199093) = 2 +1198823 (1199093 minus1199093ref)

+1198822 sign (1199093 minus1199093ref) (45)

then by using (3) (44) and (45) the position dynamics aregiven by

3 = minus1198823 (1199093 minus1199093ref) minus1198822 sign (1199093 minus1199093ref) (46)

Therefore within a finite time we obtain 1199093 = 1199093ref

Step 2 Let 1198901 = 1199091 minus1199111 from (2) and (38) the error dynamicsare expressed as

1198902 =1119898119905

(1198781198901 minus 1198871198902 minus 1198871198713 sign (1198903) minus 1198961198971198903) minusΔ119896119897

119898119905

1199093

minus1198712 sign (2 minus 1199112) (47)

Mathematical Problems in Engineering 9

Thus if 1198712 is chosen such that

1198712 gt1119898119905

1003816100381610038161003816Δ1198961198971199093 + 1198781198901 + 119887119889max1003816100381610038161003816 (48)

then we can deduce that 1198902 + 119889 tend to zero and hence 1199112tend to 1199092 + 119889 That is the observer will estimate the velocitywith the constant perturbation Moreover when the errordynamics are sliding on 1198903 = 0 and 1198902 + 119889 = 0 then 1198902 = 0and we get

1 = 1199091 +119887

1198781198902 minus

Δ119896119897

1198781199093 (49)

That is the observer will estimate the differential pressurewith a constant difference proportional to the perturbationand the uncertainty Now if the controller sliding surface 120590 isexpressed using the observer state variables

120590 (119911) = 1198781 + (1198981199051198823 +1198981199051198822120575 (1199093 minus1199093ref) minus 119887) 2

minus 1198961198971199113 +1198981199051198821 sign (1205901 (2 1199093))

(50)

then at sliding mode of the controller (119911) = 0 the velocitydynamics are given by

2 = minus1198823 (1199092 +119889) minus1198821 sign (1205901 (2 1199093)) (51)

Hence the velocity of the system rod will tend to minus119889

Step 3 Finally the differential pressure error dynamics canbe roughly expressed as

1198901 = 1198911 (119909) minus1198911 (119911) minus 1198711 sign(1198901 +119887

1198781198902 minus

Δ119896119897

1198781199093) (52)

if the system is already sliding on the surfaces 1199093 = 1199093ref and1198902 = minus119889 then by choosing 1198711 gt |1198911(119909)minus1198911(119911)| the differentialpressure is estimated with the constant difference minus(119887119878)119889 +Δ119881119905Figure 6 shows the convergence of the observer state 119911

3to

1199093although they are starting fromdifferent initial conditions

indeed the observer is starting at rod position 10 cm whereasthe system is starting at the origin As expected from theabove analysis 1199112 tends to 1199092 + 119889 and 1199111 tends to 1199091 with aconstant difference of 33 bar The observer gains are chosenas 1198711 = 1000 1198712 = 100 and 1198713 = 1 In Figure 7 the observerand system behaviors are shown with the controller beingcalculated using the estimated states The observer initialposition is minus10 cm

5 Simulation Results Analysis

The controllers designed in this paper used the sliding modetheory which is the most used approach to deal with systemsrunning under uncertainty conditions However we haveseen in the second section that the first-order sliding modecontroller with a linear sliding surface is not robust withrespect to perturbation and mismatched uncertainty In factby using the fact that on the sliding surface the systembehavesin a similar way to a linear second-order system it can be

0

100

200

300

minus10123

0

10

20

30

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 6 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with states feedback

minus1000

100200300

minus20246

minus100

102030

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 7 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with estimated states feed-back

easily shown that as far as the slidingmotion is preserved thesystem is asymptotically stable if the closed loop eigenvaluesare chosen as in (11) and thus the steady state error due to theuncertainty is expressed by (12) Also the steady state errordue to the constant perturbation is given by (13)

Therefore the steady state error gets smaller as the closedloop eigenvalues have a larger amplitude This is illustratedby the simulation results delineated in Figure 2 where 120582 = 50and the total error is 246 cm

To circumvent the problem we have suggested a second-order sliding mode controller Applying this controller we

10 Mathematical Problems in Engineering

can notice that it achieves a better performance than the firstSMC In fact the second-order SMC outperforms the first-order SMC in sense of robustness with respect tomismatchedperturbation but it cannot guarantee the robustness withrespect to the constant perturbation The steady state errordue to the perturbation is equal to 119889(119905)ℎ as demonstratedpreviously When the perturbation 119889(119905) = 01 the erroris equal to 01 cm as delineated in Figure 3 If we increasethe value of ℎ we should also increase the value of thevariable structure gains1198821 and1198822 So we obtain a chatteringphenomenon

To overcome the problem of mismatched perturbationand uncertainty we have suggested a sliding mode controllerwith nonlinear surfaceThe idea is based on the backsteppingmethod and using a robust variable structure virtual con-troller The obtained results achieved robustness with respectto parameter uncertainty and perturbation The zoomedcurve on Figure 4 shows that we have attained our aim todrive the hydraulic servo system to a reference position butwith a chattering phenomenon So to attenuate this problemwe suggest substituting the discontinuous function sign(119909)with a smooth saturation function tanh(120583119909) for sufficientlyhigh positive value 120583 gt 0 Owing to this smooth function wecan achieve the reference value in presence of mismatcheduncertainty and perturbation with attenuated chattering

Finally in Section 4 we designed a slidingmode observerin order to estimate the required states The efficiency of thisobserver is proved mathematically and also by simulationspresented in Figures 6 and 7

Compared to other methods such as that presented in[17] which is based on adaptive approach to achieve positioncontrol we notice that our method provides faster responsesince the adaptive controller proposed therein attempts toestimate the mismatched nonlinear uncertainty Indeed therod makes a displacement of 30 cm within more than 08 swhereas with our method a maximum of 02 s will beneeded to make the same displacement In [21] the authorspresented a controller based on variable structure PID todrive the hydraulic system position Although fast responsehas been achieved which is comparable to our results thegood robustness was obtained since the uncertainties andthe perturbation were matched with the controller In ourcase the robustness against the mismatched character of theperturbation and uncertainty presented the main challengeto take In [26] authors obtained comparable results tothose presented in this paper using integral sliding modecontroller with realizable reference compensation applied toan asymmetric piston

6 Conclusion

In this paper we developed several controllers based onthe sliding mode theory Our aim was to control theposition of a hydraulic servo system piston in presence ofmismatched uncertainty and perturbation We have shownthat a first-order sliding mode controller did not achieveany robustness Next we have developed a second-ordersliding mode controller that has shown robustness withrespect to parametric uncertainty but was not robust to

the perturbation Finally we have suggested a sliding modecontroller based on a nonlinear sliding surface The designis based on the backstepping method where on each stepa variable structure virtual controller design leads to thedesign of the sliding surface This controller emphasized thechattering phenomenon due to the nested sliding modesAs a remedy we suggested substituting the discontinuousfunction with a smooth saturation function Eventually wehave designed a robust sliding observer in order to substitutethe unmeasured states with their estimates A step-by-stepproof has shown that the controller issued from the estimatedstates achieved the position tracking

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H E Merritt Hydraulic Control Systems John Wiley amp SonsNew York NY USA 1967

[2] M Feki and E Richard ldquoIncluding leakage flow in the ser-vovalve static modelrdquo International Journal of Modelling andSimulation vol 25 no 1 pp 4004ndash4009 2005

[3] A Alleyne R Liu and H Wright ldquoOn the limitations offorce tracking control for hydraulic active suspensionsrdquo inProceedings of the American Control Conference (ACC rsquo98) vol1 pp 43ndash47 June 1998

[4] Y Cheng andB L RDeMoor ldquoRobustness analysis and controlsystem design for a hydraulic servo systemrdquo IEEE Transactionson Control Systems Technology vol 2 no 3 pp 183ndash197 1994

[5] M Feki E Richard and F Gomes Almeida ldquoCommande eneffort drsquoun verin hydraulique par linearization entreesortierdquoin Journees Doctorales drsquoAutomatiques (JDA rsquo99) pp 181ndash184Nancy-France 1999

[6] H Hahn A Piepenbrink and K-D Leimbach ldquoInputoutputlinearization control of an electro servo-hydraulic actuatorrdquo inProceedings of the 3rd IEEE Conference on Control Applicationspp 995ndash1000 August 1994

[7] M Jovanovic ldquoNonlinear control of an electrohydraulic velocityservosystemrdquo in Proceedings of the 2002 American ControlConference vol 1 pp 588ndash593 2002

[8] G A Sohl and J E Bobrow ldquoExperiments and simulationson the nonlinear control of a hydraulic servosystemrdquo IEEETransactions on Control Systems Technology vol 7 no 2 pp238ndash247 1999

[9] A G Alleyne and R Liu ldquoSystematic control of a class ofnonlinear systems with application to electrohydraulic cylinderpressure controlrdquo IEEE Transactions on Control Systems Tech-nology vol 8 no 4 pp 623ndash634 2000

[10] M A Avila A G Loukianov and E N Sanchez ldquoElectro-hydraulic actuator trajectory trackingrdquo in Proceedings of theAmerican Control Conference (AAC rsquo04) vol 3 pp 2603ndash2608IEEE Boston Mass USA July 2004

[11] A Bonchis P I Corke D C Rye and Q P Ha ldquoVariable struc-ture methods in hydraulic servo systems controlrdquo Automaticavol 37 no 4 pp 589ndash595 2001

[12] M Jerouane and F Lamnabhi-Lagarrigue ldquoA new robust slidingmode controller for a hydraulic actuatorrdquo in Proceedings of the

Mathematical Problems in Engineering 11

40th IEEE Conference on Decision and Control (CDC rsquo01) vol 1pp 908ndash913 December 2001

[13] H-M Chen J-C Renn and J-P Su ldquoSlidingmode control withvarying boundary layers for an electro-hydraulic position servosystemrdquo The International Journal of Advanced ManufacturingTechnology vol 26 no 1-2 pp 117ndash123 2005

[14] J M Zheng S D Zhao and S G Wei ldquoApplication of self-tuning fuzzy PID controller for a SRM direct drive volumecontrol hydraulic pressrdquo Control Engineering Practice vol 17no 12 pp 1398ndash1404 2009 Special Section The 2007 IFACSymposium on Advances in Automotive Control

[15] A Aly ldquoPid parameters optimization using genetic algorithmtechnique for electrohydraulic servo control systemrdquo IntelligentControl and Automation vol 2 no 2 pp 69ndash76 2011

[16] B Yao F Bu and G T C Chiu ldquoNonlinear adaptive robustcontrol of electro-hydraulic servo systems with discontinuousprojectionsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control (CDC rsquo98) pp 2265ndash2270 December1998

[17] B Yao F Bu J Reedy and G T-C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[18] K K Ahn and Q T Dinh ldquoSelf-tuning of quantitative feedbacktheory for force control of an electro-hydraulic test machinerdquoControl Engineering Practice vol 17 no 11 pp 1291ndash1306 2009

[19] E Kolsi Gdoura M Feki and N Derbel ldquoSliding mode controlof a hydraulic servo system position using adaptive slidingsurface and adaptive gainrdquo International Journal of ModellingIdentification and Control vol 23 pp 211ndash219 2015

[20] C Guan and S Pan ldquoAdaptive sliding mode control of electro-hydraulic systemwith nonlinear unknown parametersrdquoControlEngineering Practice vol 16 no 11 pp 1275ndash1284 2008

[21] D Liu Z Tang and Z Pei ldquoVariable structure compensationPID control of asymmetrical hydraulic cylinder trajectorytrackingrdquo Mathematical Problems in Engineering vol 2015Article ID 890704 9 pages 2015

[22] A G Loukianov J Rivera Y V Orlov and E Y MoralesTeraoka ldquoRobust trajectory tracking for an electrohydraulicactuatorrdquo IEEE Transactions on Industrial Electronics vol 56no 9 pp 3523ndash3531 2009

[23] M Ye Q Wang and S Jiao ldquoRobust 1198672119867infin

control forthe electrohydraulic steering system of a four-wheel vehiclerdquoMathematical Problems in Engineering vol 2014 Article ID208019 12 pages 2014

[24] W Shen J Jiang H R Karimi and X Su ldquoObserver-basedrobust control for hydraulic velocity control systemrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 6891329 pages 2013

[25] E Kolsi-Gdoura M Feki and N Derbel ldquoSliding mode-based robust position control of an electrohydraulic systemrdquoin Proceedings of the 10th International Multi-Conference onSystems Signals and Devices (SSD rsquo13) pp 1ndash5 March 2013

[26] E Kolsi Gdoura M Feki and N Derbel ldquoControl of ahydraulic servo system using sliding mode with an integral andrealizable reference compensationrdquo Control Engineering andApplied Informatics vol 17 pp 111ndash119 2015

[27] A Levant ldquoSliding order and sliding accuracy in sliding modecontrolrdquo International Journal of Control vol 58 no 6 pp 1247ndash1263 1993

[28] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[29] M Djemai K Busawon K Benmansour and A MaroufldquoHigh-order sliding mode control of a DC motor drive viaa switched controlled multi-cellular converterrdquo InternationalJournal of Systems Science vol 42 no 11 pp 1869ndash1882 2011

[30] M Lavanya R M Brisilla and V Sankaranarayanan ldquoHigherorder sliding mode control of permanent magnet dc motorrdquoin Proceedings of the 12th International Workshop on VariableStructure Systems (VSS rsquo12) pp 226ndash230 January 2012

[31] S Laghrouche M Smaoui F Plestan and X Brun ldquoHigherorder sliding mode control based on optimal approach of anelectropneumatic actuatorrdquo International Journal of Control vol79 no 2 pp 119ndash131 2006

[32] A G Loukianov E Sanchez and C Lizalde ldquoForce trackingneural block control for an electro-hydraulic actuator viasecond-order slidingmoderdquo International Journal of Robust andNonlinear Control vol 18 no 3 pp 319ndash332 2008

[33] J Komsta N van Oijen and P Antoszkiewicz ldquoIntegral slidingmode compensator for load pressure control of die-cushioncylinder driverdquo Control Engineering Practice vol 21 no 5 pp708ndash718 2013

[34] M Feki Synthese de commandes et drsquoobservateurs pour lessystemes non-lineaires application aux systemes hydrauliques[PhD thesis] Universite de Metz Metz France 2001

[35] V V Filippov ldquoOn a scalar equation with discontinuous right-hand side and the uniqueness theoremrdquo Differential Equationsvol 38 no 10 pp 1435ndash1445 2002

[36] T Boukhobza M Djemai and J P Barbot ldquoNonlinear slidingobserver for systems in output and output derivative injectionformrdquo in Proceedings of the IFAC World Congress pp 299ndash305San Antonio Tex USA 1996

[37] H Saadaoui N Manamanni M Djemaı J P Barbot andT Floquet ldquoExact differentiation and sliding mode observersfor switched Lagrangian systemsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 65 no 5 pp 1050ndash1069 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0

50

100

150

200

250

minus05

0

05

1

15

2

25

3

0

5

10

15

20

25

minus5

0

5

10

15

20

03 04 0520

201

202

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 3 Behavior of system under second-order sliding mode controller (19) 120582 = 50 and ℎ = 100

Finally by imposing (119909) = minus119882sign(120590(119909)) the attractivitycondition 120590(119909)(119909) lt 0 is satisfied The choice of thesurface 120590(119909) and its imposed derivative lead to the followingcontroller

119906 (119909) =

119873119889(119909)

(4119861119878119896119881119905)radic119875119889minus 1199091

if 119873119889(119909) ge 0

119873119889(119909)

(4119861119878119896119881119905)radic119875119889+ 1199091

if 119873119889(119909) lt 0

(34)

with

119873119889(119909) = minus119882 sign (120590 (119909)) minus119898

1199051198821120575 (1205901 (119909))

+ (4119861119878120572119881119905

+119887119878

119898119905

minus 1198781198824)1199091

+(41198611198782

119881119905minus1198872

119898119905

+ 119896119897+ 1198871198824)1199092

+(1198961198971198824 minus

119887119896119897

119898119905

)1199093

(35)

where the derivative of the Dirac impulse is consideredas zero and the sliding gain 119882 should be chosen suchthat the attractivity occurs in presence of uncertainty andperturbation

119882 gt

10038161003816100381610038161003816100381610038161003816

119887119881119905

119878119898119905

Δ119896119897

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119881119905

1198781198823Δ119896119897

10038161003816100381610038161003816100381610038161003816

+1003816100381610038161003816119896119897119889max

1003816100381610038161003816 (36)

From Figure 4 we can notice clearly that we haveattained our aim to drive the hydraulic servo system tothe reference position Nevertheless due to the nested slid-ing modes the chattering phenomenon was emphasized

Mathematical Problems in Engineering 7

0

50

100

150

200

250

minus05

0

05

1

15

2

25

3

0

5

10

15

20

25

minus5

0

5

10

15

20

25

03 04 05199

20

201

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 4 Behavior of system under sliding mode controller using nonlinear surface defined by (33)

In order to attenuate it and to avoid the mathematicalproblem of taking the derivative of the signum function wehave used a smooth saturation function sign(119909) ≃ tanh(120583119909)for a sufficiently high positive value 120583 gt 0 The resultsdepicted in Figure 5 show that the use of the smoothfunction helped to achieve the reference value in presenceof mismatched uncertainty and perturbation with attenuatedchattering

4 Sliding Mode Observer Design

As we can notice the controller conceived in the foregoingsection uses all three state variables However measuring thedifferential pressureΔ119875 = 1199091 is a costly task and requires hightechnology procedure to avoid additional leakage To remedythis situation we propose in this section designing a slidingmode observer that may estimate the required states that arethen used to construct the controller

Let us consider the observer model given in (37) andinferred from the step-by-step observer presented in [36 37]

1 =4119861119881119905

(119896119906radic119875119889minus sign (119906) 1 minus

12057211 + 120574 |119906|

minus 1198782)

+1198711 sign (1 minus 1199111) (37)

2 =1119898119905

(1198781199111 minus 1198872 minus 1198961198971199113) + 1198712 sign (2 minus 1199112) (38)

3 = 1199112 +1198713 sign (1199093 minus z3) (39)

where 1198711 1198712 and 1198713 are the observer gain and 1 and 2 aredefined as

1 = 1199111 +119896119897

119878(1199093 minus 1199113) +

119898119905

1198781198712 sign (2 minus 1199112)

2 = 1199112 +1198713 sign (1199093 minus 1199113) (40)

8 Mathematical Problems in Engineering

0

50

100

150

200

250

minus05

0

05

1

15

2

25

0

5

10

15

20

25

minus5

0

5

10

15

20

03 04 0519995

20

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 5 Behavior of system under slidingmode controller using nonlinear surface defined by (33) and a smooth saturation function tanh(sdot)

To prove the efficiency of the observer and the fact thatthe estimated states based controller can also achieve accuratepositioning in presence of perturbation and uncertainty wewill proceed by a step-by-step proof

Step 1 Let 1198903 = 1199093 minus 1199113 and let 1198902 = 1199092 minus 1199112 from (3) and (39)the error dynamics are expressed as

1198903 = 1198902 +119889 (119905) minus 1198713 sign (1198903) (41)Thus if 1198713 is chosen such that

1198713 gt sup119905gt01198902 (119905) + 119889 (119905) (42)

then a sliding mode is established at the observer slidingsurface 1198903 = 0 within a finite timeMoreover at slidingmodewe obtain 1198903 = 0 and thus from (41) we have

0 = 1199092 minus 1199112 +119889 (119905) minus 1198713 sign (1198903) (43)that is

1199092 +119889 (119905) = 1199112 +1198713 sign (1198903) = 2 (44)

Now if the sliding surface 1205901 of the nonlinear surface basedSMC is expressed in terms of the estimated state 2

1205901 (2 1199093) = 2 +1198823 (1199093 minus1199093ref)

+1198822 sign (1199093 minus1199093ref) (45)

then by using (3) (44) and (45) the position dynamics aregiven by

3 = minus1198823 (1199093 minus1199093ref) minus1198822 sign (1199093 minus1199093ref) (46)

Therefore within a finite time we obtain 1199093 = 1199093ref

Step 2 Let 1198901 = 1199091 minus1199111 from (2) and (38) the error dynamicsare expressed as

1198902 =1119898119905

(1198781198901 minus 1198871198902 minus 1198871198713 sign (1198903) minus 1198961198971198903) minusΔ119896119897

119898119905

1199093

minus1198712 sign (2 minus 1199112) (47)

Mathematical Problems in Engineering 9

Thus if 1198712 is chosen such that

1198712 gt1119898119905

1003816100381610038161003816Δ1198961198971199093 + 1198781198901 + 119887119889max1003816100381610038161003816 (48)

then we can deduce that 1198902 + 119889 tend to zero and hence 1199112tend to 1199092 + 119889 That is the observer will estimate the velocitywith the constant perturbation Moreover when the errordynamics are sliding on 1198903 = 0 and 1198902 + 119889 = 0 then 1198902 = 0and we get

1 = 1199091 +119887

1198781198902 minus

Δ119896119897

1198781199093 (49)

That is the observer will estimate the differential pressurewith a constant difference proportional to the perturbationand the uncertainty Now if the controller sliding surface 120590 isexpressed using the observer state variables

120590 (119911) = 1198781 + (1198981199051198823 +1198981199051198822120575 (1199093 minus1199093ref) minus 119887) 2

minus 1198961198971199113 +1198981199051198821 sign (1205901 (2 1199093))

(50)

then at sliding mode of the controller (119911) = 0 the velocitydynamics are given by

2 = minus1198823 (1199092 +119889) minus1198821 sign (1205901 (2 1199093)) (51)

Hence the velocity of the system rod will tend to minus119889

Step 3 Finally the differential pressure error dynamics canbe roughly expressed as

1198901 = 1198911 (119909) minus1198911 (119911) minus 1198711 sign(1198901 +119887

1198781198902 minus

Δ119896119897

1198781199093) (52)

if the system is already sliding on the surfaces 1199093 = 1199093ref and1198902 = minus119889 then by choosing 1198711 gt |1198911(119909)minus1198911(119911)| the differentialpressure is estimated with the constant difference minus(119887119878)119889 +Δ119881119905Figure 6 shows the convergence of the observer state 119911

3to

1199093although they are starting fromdifferent initial conditions

indeed the observer is starting at rod position 10 cm whereasthe system is starting at the origin As expected from theabove analysis 1199112 tends to 1199092 + 119889 and 1199111 tends to 1199091 with aconstant difference of 33 bar The observer gains are chosenas 1198711 = 1000 1198712 = 100 and 1198713 = 1 In Figure 7 the observerand system behaviors are shown with the controller beingcalculated using the estimated states The observer initialposition is minus10 cm

5 Simulation Results Analysis

The controllers designed in this paper used the sliding modetheory which is the most used approach to deal with systemsrunning under uncertainty conditions However we haveseen in the second section that the first-order sliding modecontroller with a linear sliding surface is not robust withrespect to perturbation and mismatched uncertainty In factby using the fact that on the sliding surface the systembehavesin a similar way to a linear second-order system it can be

0

100

200

300

minus10123

0

10

20

30

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 6 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with states feedback

minus1000

100200300

minus20246

minus100

102030

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 7 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with estimated states feed-back

easily shown that as far as the slidingmotion is preserved thesystem is asymptotically stable if the closed loop eigenvaluesare chosen as in (11) and thus the steady state error due to theuncertainty is expressed by (12) Also the steady state errordue to the constant perturbation is given by (13)

Therefore the steady state error gets smaller as the closedloop eigenvalues have a larger amplitude This is illustratedby the simulation results delineated in Figure 2 where 120582 = 50and the total error is 246 cm

To circumvent the problem we have suggested a second-order sliding mode controller Applying this controller we

10 Mathematical Problems in Engineering

can notice that it achieves a better performance than the firstSMC In fact the second-order SMC outperforms the first-order SMC in sense of robustness with respect tomismatchedperturbation but it cannot guarantee the robustness withrespect to the constant perturbation The steady state errordue to the perturbation is equal to 119889(119905)ℎ as demonstratedpreviously When the perturbation 119889(119905) = 01 the erroris equal to 01 cm as delineated in Figure 3 If we increasethe value of ℎ we should also increase the value of thevariable structure gains1198821 and1198822 So we obtain a chatteringphenomenon

To overcome the problem of mismatched perturbationand uncertainty we have suggested a sliding mode controllerwith nonlinear surfaceThe idea is based on the backsteppingmethod and using a robust variable structure virtual con-troller The obtained results achieved robustness with respectto parameter uncertainty and perturbation The zoomedcurve on Figure 4 shows that we have attained our aim todrive the hydraulic servo system to a reference position butwith a chattering phenomenon So to attenuate this problemwe suggest substituting the discontinuous function sign(119909)with a smooth saturation function tanh(120583119909) for sufficientlyhigh positive value 120583 gt 0 Owing to this smooth function wecan achieve the reference value in presence of mismatcheduncertainty and perturbation with attenuated chattering

Finally in Section 4 we designed a slidingmode observerin order to estimate the required states The efficiency of thisobserver is proved mathematically and also by simulationspresented in Figures 6 and 7

Compared to other methods such as that presented in[17] which is based on adaptive approach to achieve positioncontrol we notice that our method provides faster responsesince the adaptive controller proposed therein attempts toestimate the mismatched nonlinear uncertainty Indeed therod makes a displacement of 30 cm within more than 08 swhereas with our method a maximum of 02 s will beneeded to make the same displacement In [21] the authorspresented a controller based on variable structure PID todrive the hydraulic system position Although fast responsehas been achieved which is comparable to our results thegood robustness was obtained since the uncertainties andthe perturbation were matched with the controller In ourcase the robustness against the mismatched character of theperturbation and uncertainty presented the main challengeto take In [26] authors obtained comparable results tothose presented in this paper using integral sliding modecontroller with realizable reference compensation applied toan asymmetric piston

6 Conclusion

In this paper we developed several controllers based onthe sliding mode theory Our aim was to control theposition of a hydraulic servo system piston in presence ofmismatched uncertainty and perturbation We have shownthat a first-order sliding mode controller did not achieveany robustness Next we have developed a second-ordersliding mode controller that has shown robustness withrespect to parametric uncertainty but was not robust to

the perturbation Finally we have suggested a sliding modecontroller based on a nonlinear sliding surface The designis based on the backstepping method where on each stepa variable structure virtual controller design leads to thedesign of the sliding surface This controller emphasized thechattering phenomenon due to the nested sliding modesAs a remedy we suggested substituting the discontinuousfunction with a smooth saturation function Eventually wehave designed a robust sliding observer in order to substitutethe unmeasured states with their estimates A step-by-stepproof has shown that the controller issued from the estimatedstates achieved the position tracking

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H E Merritt Hydraulic Control Systems John Wiley amp SonsNew York NY USA 1967

[2] M Feki and E Richard ldquoIncluding leakage flow in the ser-vovalve static modelrdquo International Journal of Modelling andSimulation vol 25 no 1 pp 4004ndash4009 2005

[3] A Alleyne R Liu and H Wright ldquoOn the limitations offorce tracking control for hydraulic active suspensionsrdquo inProceedings of the American Control Conference (ACC rsquo98) vol1 pp 43ndash47 June 1998

[4] Y Cheng andB L RDeMoor ldquoRobustness analysis and controlsystem design for a hydraulic servo systemrdquo IEEE Transactionson Control Systems Technology vol 2 no 3 pp 183ndash197 1994

[5] M Feki E Richard and F Gomes Almeida ldquoCommande eneffort drsquoun verin hydraulique par linearization entreesortierdquoin Journees Doctorales drsquoAutomatiques (JDA rsquo99) pp 181ndash184Nancy-France 1999

[6] H Hahn A Piepenbrink and K-D Leimbach ldquoInputoutputlinearization control of an electro servo-hydraulic actuatorrdquo inProceedings of the 3rd IEEE Conference on Control Applicationspp 995ndash1000 August 1994

[7] M Jovanovic ldquoNonlinear control of an electrohydraulic velocityservosystemrdquo in Proceedings of the 2002 American ControlConference vol 1 pp 588ndash593 2002

[8] G A Sohl and J E Bobrow ldquoExperiments and simulationson the nonlinear control of a hydraulic servosystemrdquo IEEETransactions on Control Systems Technology vol 7 no 2 pp238ndash247 1999

[9] A G Alleyne and R Liu ldquoSystematic control of a class ofnonlinear systems with application to electrohydraulic cylinderpressure controlrdquo IEEE Transactions on Control Systems Tech-nology vol 8 no 4 pp 623ndash634 2000

[10] M A Avila A G Loukianov and E N Sanchez ldquoElectro-hydraulic actuator trajectory trackingrdquo in Proceedings of theAmerican Control Conference (AAC rsquo04) vol 3 pp 2603ndash2608IEEE Boston Mass USA July 2004

[11] A Bonchis P I Corke D C Rye and Q P Ha ldquoVariable struc-ture methods in hydraulic servo systems controlrdquo Automaticavol 37 no 4 pp 589ndash595 2001

[12] M Jerouane and F Lamnabhi-Lagarrigue ldquoA new robust slidingmode controller for a hydraulic actuatorrdquo in Proceedings of the

Mathematical Problems in Engineering 11

40th IEEE Conference on Decision and Control (CDC rsquo01) vol 1pp 908ndash913 December 2001

[13] H-M Chen J-C Renn and J-P Su ldquoSlidingmode control withvarying boundary layers for an electro-hydraulic position servosystemrdquo The International Journal of Advanced ManufacturingTechnology vol 26 no 1-2 pp 117ndash123 2005

[14] J M Zheng S D Zhao and S G Wei ldquoApplication of self-tuning fuzzy PID controller for a SRM direct drive volumecontrol hydraulic pressrdquo Control Engineering Practice vol 17no 12 pp 1398ndash1404 2009 Special Section The 2007 IFACSymposium on Advances in Automotive Control

[15] A Aly ldquoPid parameters optimization using genetic algorithmtechnique for electrohydraulic servo control systemrdquo IntelligentControl and Automation vol 2 no 2 pp 69ndash76 2011

[16] B Yao F Bu and G T C Chiu ldquoNonlinear adaptive robustcontrol of electro-hydraulic servo systems with discontinuousprojectionsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control (CDC rsquo98) pp 2265ndash2270 December1998

[17] B Yao F Bu J Reedy and G T-C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[18] K K Ahn and Q T Dinh ldquoSelf-tuning of quantitative feedbacktheory for force control of an electro-hydraulic test machinerdquoControl Engineering Practice vol 17 no 11 pp 1291ndash1306 2009

[19] E Kolsi Gdoura M Feki and N Derbel ldquoSliding mode controlof a hydraulic servo system position using adaptive slidingsurface and adaptive gainrdquo International Journal of ModellingIdentification and Control vol 23 pp 211ndash219 2015

[20] C Guan and S Pan ldquoAdaptive sliding mode control of electro-hydraulic systemwith nonlinear unknown parametersrdquoControlEngineering Practice vol 16 no 11 pp 1275ndash1284 2008

[21] D Liu Z Tang and Z Pei ldquoVariable structure compensationPID control of asymmetrical hydraulic cylinder trajectorytrackingrdquo Mathematical Problems in Engineering vol 2015Article ID 890704 9 pages 2015

[22] A G Loukianov J Rivera Y V Orlov and E Y MoralesTeraoka ldquoRobust trajectory tracking for an electrohydraulicactuatorrdquo IEEE Transactions on Industrial Electronics vol 56no 9 pp 3523ndash3531 2009

[23] M Ye Q Wang and S Jiao ldquoRobust 1198672119867infin

control forthe electrohydraulic steering system of a four-wheel vehiclerdquoMathematical Problems in Engineering vol 2014 Article ID208019 12 pages 2014

[24] W Shen J Jiang H R Karimi and X Su ldquoObserver-basedrobust control for hydraulic velocity control systemrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 6891329 pages 2013

[25] E Kolsi-Gdoura M Feki and N Derbel ldquoSliding mode-based robust position control of an electrohydraulic systemrdquoin Proceedings of the 10th International Multi-Conference onSystems Signals and Devices (SSD rsquo13) pp 1ndash5 March 2013

[26] E Kolsi Gdoura M Feki and N Derbel ldquoControl of ahydraulic servo system using sliding mode with an integral andrealizable reference compensationrdquo Control Engineering andApplied Informatics vol 17 pp 111ndash119 2015

[27] A Levant ldquoSliding order and sliding accuracy in sliding modecontrolrdquo International Journal of Control vol 58 no 6 pp 1247ndash1263 1993

[28] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[29] M Djemai K Busawon K Benmansour and A MaroufldquoHigh-order sliding mode control of a DC motor drive viaa switched controlled multi-cellular converterrdquo InternationalJournal of Systems Science vol 42 no 11 pp 1869ndash1882 2011

[30] M Lavanya R M Brisilla and V Sankaranarayanan ldquoHigherorder sliding mode control of permanent magnet dc motorrdquoin Proceedings of the 12th International Workshop on VariableStructure Systems (VSS rsquo12) pp 226ndash230 January 2012

[31] S Laghrouche M Smaoui F Plestan and X Brun ldquoHigherorder sliding mode control based on optimal approach of anelectropneumatic actuatorrdquo International Journal of Control vol79 no 2 pp 119ndash131 2006

[32] A G Loukianov E Sanchez and C Lizalde ldquoForce trackingneural block control for an electro-hydraulic actuator viasecond-order slidingmoderdquo International Journal of Robust andNonlinear Control vol 18 no 3 pp 319ndash332 2008

[33] J Komsta N van Oijen and P Antoszkiewicz ldquoIntegral slidingmode compensator for load pressure control of die-cushioncylinder driverdquo Control Engineering Practice vol 21 no 5 pp708ndash718 2013

[34] M Feki Synthese de commandes et drsquoobservateurs pour lessystemes non-lineaires application aux systemes hydrauliques[PhD thesis] Universite de Metz Metz France 2001

[35] V V Filippov ldquoOn a scalar equation with discontinuous right-hand side and the uniqueness theoremrdquo Differential Equationsvol 38 no 10 pp 1435ndash1445 2002

[36] T Boukhobza M Djemai and J P Barbot ldquoNonlinear slidingobserver for systems in output and output derivative injectionformrdquo in Proceedings of the IFAC World Congress pp 299ndash305San Antonio Tex USA 1996

[37] H Saadaoui N Manamanni M Djemaı J P Barbot andT Floquet ldquoExact differentiation and sliding mode observersfor switched Lagrangian systemsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 65 no 5 pp 1050ndash1069 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0

50

100

150

200

250

minus05

0

05

1

15

2

25

3

0

5

10

15

20

25

minus5

0

5

10

15

20

25

03 04 05199

20

201

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 4 Behavior of system under sliding mode controller using nonlinear surface defined by (33)

In order to attenuate it and to avoid the mathematicalproblem of taking the derivative of the signum function wehave used a smooth saturation function sign(119909) ≃ tanh(120583119909)for a sufficiently high positive value 120583 gt 0 The resultsdepicted in Figure 5 show that the use of the smoothfunction helped to achieve the reference value in presenceof mismatched uncertainty and perturbation with attenuatedchattering

4 Sliding Mode Observer Design

As we can notice the controller conceived in the foregoingsection uses all three state variables However measuring thedifferential pressureΔ119875 = 1199091 is a costly task and requires hightechnology procedure to avoid additional leakage To remedythis situation we propose in this section designing a slidingmode observer that may estimate the required states that arethen used to construct the controller

Let us consider the observer model given in (37) andinferred from the step-by-step observer presented in [36 37]

1 =4119861119881119905

(119896119906radic119875119889minus sign (119906) 1 minus

12057211 + 120574 |119906|

minus 1198782)

+1198711 sign (1 minus 1199111) (37)

2 =1119898119905

(1198781199111 minus 1198872 minus 1198961198971199113) + 1198712 sign (2 minus 1199112) (38)

3 = 1199112 +1198713 sign (1199093 minus z3) (39)

where 1198711 1198712 and 1198713 are the observer gain and 1 and 2 aredefined as

1 = 1199111 +119896119897

119878(1199093 minus 1199113) +

119898119905

1198781198712 sign (2 minus 1199112)

2 = 1199112 +1198713 sign (1199093 minus 1199113) (40)

8 Mathematical Problems in Engineering

0

50

100

150

200

250

minus05

0

05

1

15

2

25

0

5

10

15

20

25

minus5

0

5

10

15

20

03 04 0519995

20

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 5 Behavior of system under slidingmode controller using nonlinear surface defined by (33) and a smooth saturation function tanh(sdot)

To prove the efficiency of the observer and the fact thatthe estimated states based controller can also achieve accuratepositioning in presence of perturbation and uncertainty wewill proceed by a step-by-step proof

Step 1 Let 1198903 = 1199093 minus 1199113 and let 1198902 = 1199092 minus 1199112 from (3) and (39)the error dynamics are expressed as

1198903 = 1198902 +119889 (119905) minus 1198713 sign (1198903) (41)Thus if 1198713 is chosen such that

1198713 gt sup119905gt01198902 (119905) + 119889 (119905) (42)

then a sliding mode is established at the observer slidingsurface 1198903 = 0 within a finite timeMoreover at slidingmodewe obtain 1198903 = 0 and thus from (41) we have

0 = 1199092 minus 1199112 +119889 (119905) minus 1198713 sign (1198903) (43)that is

1199092 +119889 (119905) = 1199112 +1198713 sign (1198903) = 2 (44)

Now if the sliding surface 1205901 of the nonlinear surface basedSMC is expressed in terms of the estimated state 2

1205901 (2 1199093) = 2 +1198823 (1199093 minus1199093ref)

+1198822 sign (1199093 minus1199093ref) (45)

then by using (3) (44) and (45) the position dynamics aregiven by

3 = minus1198823 (1199093 minus1199093ref) minus1198822 sign (1199093 minus1199093ref) (46)

Therefore within a finite time we obtain 1199093 = 1199093ref

Step 2 Let 1198901 = 1199091 minus1199111 from (2) and (38) the error dynamicsare expressed as

1198902 =1119898119905

(1198781198901 minus 1198871198902 minus 1198871198713 sign (1198903) minus 1198961198971198903) minusΔ119896119897

119898119905

1199093

minus1198712 sign (2 minus 1199112) (47)

Mathematical Problems in Engineering 9

Thus if 1198712 is chosen such that

1198712 gt1119898119905

1003816100381610038161003816Δ1198961198971199093 + 1198781198901 + 119887119889max1003816100381610038161003816 (48)

then we can deduce that 1198902 + 119889 tend to zero and hence 1199112tend to 1199092 + 119889 That is the observer will estimate the velocitywith the constant perturbation Moreover when the errordynamics are sliding on 1198903 = 0 and 1198902 + 119889 = 0 then 1198902 = 0and we get

1 = 1199091 +119887

1198781198902 minus

Δ119896119897

1198781199093 (49)

That is the observer will estimate the differential pressurewith a constant difference proportional to the perturbationand the uncertainty Now if the controller sliding surface 120590 isexpressed using the observer state variables

120590 (119911) = 1198781 + (1198981199051198823 +1198981199051198822120575 (1199093 minus1199093ref) minus 119887) 2

minus 1198961198971199113 +1198981199051198821 sign (1205901 (2 1199093))

(50)

then at sliding mode of the controller (119911) = 0 the velocitydynamics are given by

2 = minus1198823 (1199092 +119889) minus1198821 sign (1205901 (2 1199093)) (51)

Hence the velocity of the system rod will tend to minus119889

Step 3 Finally the differential pressure error dynamics canbe roughly expressed as

1198901 = 1198911 (119909) minus1198911 (119911) minus 1198711 sign(1198901 +119887

1198781198902 minus

Δ119896119897

1198781199093) (52)

if the system is already sliding on the surfaces 1199093 = 1199093ref and1198902 = minus119889 then by choosing 1198711 gt |1198911(119909)minus1198911(119911)| the differentialpressure is estimated with the constant difference minus(119887119878)119889 +Δ119881119905Figure 6 shows the convergence of the observer state 119911

3to

1199093although they are starting fromdifferent initial conditions

indeed the observer is starting at rod position 10 cm whereasthe system is starting at the origin As expected from theabove analysis 1199112 tends to 1199092 + 119889 and 1199111 tends to 1199091 with aconstant difference of 33 bar The observer gains are chosenas 1198711 = 1000 1198712 = 100 and 1198713 = 1 In Figure 7 the observerand system behaviors are shown with the controller beingcalculated using the estimated states The observer initialposition is minus10 cm

5 Simulation Results Analysis

The controllers designed in this paper used the sliding modetheory which is the most used approach to deal with systemsrunning under uncertainty conditions However we haveseen in the second section that the first-order sliding modecontroller with a linear sliding surface is not robust withrespect to perturbation and mismatched uncertainty In factby using the fact that on the sliding surface the systembehavesin a similar way to a linear second-order system it can be

0

100

200

300

minus10123

0

10

20

30

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 6 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with states feedback

minus1000

100200300

minus20246

minus100

102030

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 7 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with estimated states feed-back

easily shown that as far as the slidingmotion is preserved thesystem is asymptotically stable if the closed loop eigenvaluesare chosen as in (11) and thus the steady state error due to theuncertainty is expressed by (12) Also the steady state errordue to the constant perturbation is given by (13)

Therefore the steady state error gets smaller as the closedloop eigenvalues have a larger amplitude This is illustratedby the simulation results delineated in Figure 2 where 120582 = 50and the total error is 246 cm

To circumvent the problem we have suggested a second-order sliding mode controller Applying this controller we

10 Mathematical Problems in Engineering

can notice that it achieves a better performance than the firstSMC In fact the second-order SMC outperforms the first-order SMC in sense of robustness with respect tomismatchedperturbation but it cannot guarantee the robustness withrespect to the constant perturbation The steady state errordue to the perturbation is equal to 119889(119905)ℎ as demonstratedpreviously When the perturbation 119889(119905) = 01 the erroris equal to 01 cm as delineated in Figure 3 If we increasethe value of ℎ we should also increase the value of thevariable structure gains1198821 and1198822 So we obtain a chatteringphenomenon

To overcome the problem of mismatched perturbationand uncertainty we have suggested a sliding mode controllerwith nonlinear surfaceThe idea is based on the backsteppingmethod and using a robust variable structure virtual con-troller The obtained results achieved robustness with respectto parameter uncertainty and perturbation The zoomedcurve on Figure 4 shows that we have attained our aim todrive the hydraulic servo system to a reference position butwith a chattering phenomenon So to attenuate this problemwe suggest substituting the discontinuous function sign(119909)with a smooth saturation function tanh(120583119909) for sufficientlyhigh positive value 120583 gt 0 Owing to this smooth function wecan achieve the reference value in presence of mismatcheduncertainty and perturbation with attenuated chattering

Finally in Section 4 we designed a slidingmode observerin order to estimate the required states The efficiency of thisobserver is proved mathematically and also by simulationspresented in Figures 6 and 7

Compared to other methods such as that presented in[17] which is based on adaptive approach to achieve positioncontrol we notice that our method provides faster responsesince the adaptive controller proposed therein attempts toestimate the mismatched nonlinear uncertainty Indeed therod makes a displacement of 30 cm within more than 08 swhereas with our method a maximum of 02 s will beneeded to make the same displacement In [21] the authorspresented a controller based on variable structure PID todrive the hydraulic system position Although fast responsehas been achieved which is comparable to our results thegood robustness was obtained since the uncertainties andthe perturbation were matched with the controller In ourcase the robustness against the mismatched character of theperturbation and uncertainty presented the main challengeto take In [26] authors obtained comparable results tothose presented in this paper using integral sliding modecontroller with realizable reference compensation applied toan asymmetric piston

6 Conclusion

In this paper we developed several controllers based onthe sliding mode theory Our aim was to control theposition of a hydraulic servo system piston in presence ofmismatched uncertainty and perturbation We have shownthat a first-order sliding mode controller did not achieveany robustness Next we have developed a second-ordersliding mode controller that has shown robustness withrespect to parametric uncertainty but was not robust to

the perturbation Finally we have suggested a sliding modecontroller based on a nonlinear sliding surface The designis based on the backstepping method where on each stepa variable structure virtual controller design leads to thedesign of the sliding surface This controller emphasized thechattering phenomenon due to the nested sliding modesAs a remedy we suggested substituting the discontinuousfunction with a smooth saturation function Eventually wehave designed a robust sliding observer in order to substitutethe unmeasured states with their estimates A step-by-stepproof has shown that the controller issued from the estimatedstates achieved the position tracking

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H E Merritt Hydraulic Control Systems John Wiley amp SonsNew York NY USA 1967

[2] M Feki and E Richard ldquoIncluding leakage flow in the ser-vovalve static modelrdquo International Journal of Modelling andSimulation vol 25 no 1 pp 4004ndash4009 2005

[3] A Alleyne R Liu and H Wright ldquoOn the limitations offorce tracking control for hydraulic active suspensionsrdquo inProceedings of the American Control Conference (ACC rsquo98) vol1 pp 43ndash47 June 1998

[4] Y Cheng andB L RDeMoor ldquoRobustness analysis and controlsystem design for a hydraulic servo systemrdquo IEEE Transactionson Control Systems Technology vol 2 no 3 pp 183ndash197 1994

[5] M Feki E Richard and F Gomes Almeida ldquoCommande eneffort drsquoun verin hydraulique par linearization entreesortierdquoin Journees Doctorales drsquoAutomatiques (JDA rsquo99) pp 181ndash184Nancy-France 1999

[6] H Hahn A Piepenbrink and K-D Leimbach ldquoInputoutputlinearization control of an electro servo-hydraulic actuatorrdquo inProceedings of the 3rd IEEE Conference on Control Applicationspp 995ndash1000 August 1994

[7] M Jovanovic ldquoNonlinear control of an electrohydraulic velocityservosystemrdquo in Proceedings of the 2002 American ControlConference vol 1 pp 588ndash593 2002

[8] G A Sohl and J E Bobrow ldquoExperiments and simulationson the nonlinear control of a hydraulic servosystemrdquo IEEETransactions on Control Systems Technology vol 7 no 2 pp238ndash247 1999

[9] A G Alleyne and R Liu ldquoSystematic control of a class ofnonlinear systems with application to electrohydraulic cylinderpressure controlrdquo IEEE Transactions on Control Systems Tech-nology vol 8 no 4 pp 623ndash634 2000

[10] M A Avila A G Loukianov and E N Sanchez ldquoElectro-hydraulic actuator trajectory trackingrdquo in Proceedings of theAmerican Control Conference (AAC rsquo04) vol 3 pp 2603ndash2608IEEE Boston Mass USA July 2004

[11] A Bonchis P I Corke D C Rye and Q P Ha ldquoVariable struc-ture methods in hydraulic servo systems controlrdquo Automaticavol 37 no 4 pp 589ndash595 2001

[12] M Jerouane and F Lamnabhi-Lagarrigue ldquoA new robust slidingmode controller for a hydraulic actuatorrdquo in Proceedings of the

Mathematical Problems in Engineering 11

40th IEEE Conference on Decision and Control (CDC rsquo01) vol 1pp 908ndash913 December 2001

[13] H-M Chen J-C Renn and J-P Su ldquoSlidingmode control withvarying boundary layers for an electro-hydraulic position servosystemrdquo The International Journal of Advanced ManufacturingTechnology vol 26 no 1-2 pp 117ndash123 2005

[14] J M Zheng S D Zhao and S G Wei ldquoApplication of self-tuning fuzzy PID controller for a SRM direct drive volumecontrol hydraulic pressrdquo Control Engineering Practice vol 17no 12 pp 1398ndash1404 2009 Special Section The 2007 IFACSymposium on Advances in Automotive Control

[15] A Aly ldquoPid parameters optimization using genetic algorithmtechnique for electrohydraulic servo control systemrdquo IntelligentControl and Automation vol 2 no 2 pp 69ndash76 2011

[16] B Yao F Bu and G T C Chiu ldquoNonlinear adaptive robustcontrol of electro-hydraulic servo systems with discontinuousprojectionsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control (CDC rsquo98) pp 2265ndash2270 December1998

[17] B Yao F Bu J Reedy and G T-C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[18] K K Ahn and Q T Dinh ldquoSelf-tuning of quantitative feedbacktheory for force control of an electro-hydraulic test machinerdquoControl Engineering Practice vol 17 no 11 pp 1291ndash1306 2009

[19] E Kolsi Gdoura M Feki and N Derbel ldquoSliding mode controlof a hydraulic servo system position using adaptive slidingsurface and adaptive gainrdquo International Journal of ModellingIdentification and Control vol 23 pp 211ndash219 2015

[20] C Guan and S Pan ldquoAdaptive sliding mode control of electro-hydraulic systemwith nonlinear unknown parametersrdquoControlEngineering Practice vol 16 no 11 pp 1275ndash1284 2008

[21] D Liu Z Tang and Z Pei ldquoVariable structure compensationPID control of asymmetrical hydraulic cylinder trajectorytrackingrdquo Mathematical Problems in Engineering vol 2015Article ID 890704 9 pages 2015

[22] A G Loukianov J Rivera Y V Orlov and E Y MoralesTeraoka ldquoRobust trajectory tracking for an electrohydraulicactuatorrdquo IEEE Transactions on Industrial Electronics vol 56no 9 pp 3523ndash3531 2009

[23] M Ye Q Wang and S Jiao ldquoRobust 1198672119867infin

control forthe electrohydraulic steering system of a four-wheel vehiclerdquoMathematical Problems in Engineering vol 2014 Article ID208019 12 pages 2014

[24] W Shen J Jiang H R Karimi and X Su ldquoObserver-basedrobust control for hydraulic velocity control systemrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 6891329 pages 2013

[25] E Kolsi-Gdoura M Feki and N Derbel ldquoSliding mode-based robust position control of an electrohydraulic systemrdquoin Proceedings of the 10th International Multi-Conference onSystems Signals and Devices (SSD rsquo13) pp 1ndash5 March 2013

[26] E Kolsi Gdoura M Feki and N Derbel ldquoControl of ahydraulic servo system using sliding mode with an integral andrealizable reference compensationrdquo Control Engineering andApplied Informatics vol 17 pp 111ndash119 2015

[27] A Levant ldquoSliding order and sliding accuracy in sliding modecontrolrdquo International Journal of Control vol 58 no 6 pp 1247ndash1263 1993

[28] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[29] M Djemai K Busawon K Benmansour and A MaroufldquoHigh-order sliding mode control of a DC motor drive viaa switched controlled multi-cellular converterrdquo InternationalJournal of Systems Science vol 42 no 11 pp 1869ndash1882 2011

[30] M Lavanya R M Brisilla and V Sankaranarayanan ldquoHigherorder sliding mode control of permanent magnet dc motorrdquoin Proceedings of the 12th International Workshop on VariableStructure Systems (VSS rsquo12) pp 226ndash230 January 2012

[31] S Laghrouche M Smaoui F Plestan and X Brun ldquoHigherorder sliding mode control based on optimal approach of anelectropneumatic actuatorrdquo International Journal of Control vol79 no 2 pp 119ndash131 2006

[32] A G Loukianov E Sanchez and C Lizalde ldquoForce trackingneural block control for an electro-hydraulic actuator viasecond-order slidingmoderdquo International Journal of Robust andNonlinear Control vol 18 no 3 pp 319ndash332 2008

[33] J Komsta N van Oijen and P Antoszkiewicz ldquoIntegral slidingmode compensator for load pressure control of die-cushioncylinder driverdquo Control Engineering Practice vol 21 no 5 pp708ndash718 2013

[34] M Feki Synthese de commandes et drsquoobservateurs pour lessystemes non-lineaires application aux systemes hydrauliques[PhD thesis] Universite de Metz Metz France 2001

[35] V V Filippov ldquoOn a scalar equation with discontinuous right-hand side and the uniqueness theoremrdquo Differential Equationsvol 38 no 10 pp 1435ndash1445 2002

[36] T Boukhobza M Djemai and J P Barbot ldquoNonlinear slidingobserver for systems in output and output derivative injectionformrdquo in Proceedings of the IFAC World Congress pp 299ndash305San Antonio Tex USA 1996

[37] H Saadaoui N Manamanni M Djemaı J P Barbot andT Floquet ldquoExact differentiation and sliding mode observersfor switched Lagrangian systemsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 65 no 5 pp 1050ndash1069 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0

50

100

150

200

250

minus05

0

05

1

15

2

25

0

5

10

15

20

25

minus5

0

5

10

15

20

03 04 0519995

20

0 01 02 03

03

04 05 0 01 02 03 04 05

Velo

city

(ms

)

0 01 02 04 05Time (s)

0 01 02 03 04 05Time (s)

Time (s) Time (s)

Posit

ion

(cm

)

u(t)

(mA

)

Δ p

ress

ure (

bar)

Figure 5 Behavior of system under slidingmode controller using nonlinear surface defined by (33) and a smooth saturation function tanh(sdot)

To prove the efficiency of the observer and the fact thatthe estimated states based controller can also achieve accuratepositioning in presence of perturbation and uncertainty wewill proceed by a step-by-step proof

Step 1 Let 1198903 = 1199093 minus 1199113 and let 1198902 = 1199092 minus 1199112 from (3) and (39)the error dynamics are expressed as

1198903 = 1198902 +119889 (119905) minus 1198713 sign (1198903) (41)Thus if 1198713 is chosen such that

1198713 gt sup119905gt01198902 (119905) + 119889 (119905) (42)

then a sliding mode is established at the observer slidingsurface 1198903 = 0 within a finite timeMoreover at slidingmodewe obtain 1198903 = 0 and thus from (41) we have

0 = 1199092 minus 1199112 +119889 (119905) minus 1198713 sign (1198903) (43)that is

1199092 +119889 (119905) = 1199112 +1198713 sign (1198903) = 2 (44)

Now if the sliding surface 1205901 of the nonlinear surface basedSMC is expressed in terms of the estimated state 2

1205901 (2 1199093) = 2 +1198823 (1199093 minus1199093ref)

+1198822 sign (1199093 minus1199093ref) (45)

then by using (3) (44) and (45) the position dynamics aregiven by

3 = minus1198823 (1199093 minus1199093ref) minus1198822 sign (1199093 minus1199093ref) (46)

Therefore within a finite time we obtain 1199093 = 1199093ref

Step 2 Let 1198901 = 1199091 minus1199111 from (2) and (38) the error dynamicsare expressed as

1198902 =1119898119905

(1198781198901 minus 1198871198902 minus 1198871198713 sign (1198903) minus 1198961198971198903) minusΔ119896119897

119898119905

1199093

minus1198712 sign (2 minus 1199112) (47)

Mathematical Problems in Engineering 9

Thus if 1198712 is chosen such that

1198712 gt1119898119905

1003816100381610038161003816Δ1198961198971199093 + 1198781198901 + 119887119889max1003816100381610038161003816 (48)

then we can deduce that 1198902 + 119889 tend to zero and hence 1199112tend to 1199092 + 119889 That is the observer will estimate the velocitywith the constant perturbation Moreover when the errordynamics are sliding on 1198903 = 0 and 1198902 + 119889 = 0 then 1198902 = 0and we get

1 = 1199091 +119887

1198781198902 minus

Δ119896119897

1198781199093 (49)

That is the observer will estimate the differential pressurewith a constant difference proportional to the perturbationand the uncertainty Now if the controller sliding surface 120590 isexpressed using the observer state variables

120590 (119911) = 1198781 + (1198981199051198823 +1198981199051198822120575 (1199093 minus1199093ref) minus 119887) 2

minus 1198961198971199113 +1198981199051198821 sign (1205901 (2 1199093))

(50)

then at sliding mode of the controller (119911) = 0 the velocitydynamics are given by

2 = minus1198823 (1199092 +119889) minus1198821 sign (1205901 (2 1199093)) (51)

Hence the velocity of the system rod will tend to minus119889

Step 3 Finally the differential pressure error dynamics canbe roughly expressed as

1198901 = 1198911 (119909) minus1198911 (119911) minus 1198711 sign(1198901 +119887

1198781198902 minus

Δ119896119897

1198781199093) (52)

if the system is already sliding on the surfaces 1199093 = 1199093ref and1198902 = minus119889 then by choosing 1198711 gt |1198911(119909)minus1198911(119911)| the differentialpressure is estimated with the constant difference minus(119887119878)119889 +Δ119881119905Figure 6 shows the convergence of the observer state 119911

3to

1199093although they are starting fromdifferent initial conditions

indeed the observer is starting at rod position 10 cm whereasthe system is starting at the origin As expected from theabove analysis 1199112 tends to 1199092 + 119889 and 1199111 tends to 1199091 with aconstant difference of 33 bar The observer gains are chosenas 1198711 = 1000 1198712 = 100 and 1198713 = 1 In Figure 7 the observerand system behaviors are shown with the controller beingcalculated using the estimated states The observer initialposition is minus10 cm

5 Simulation Results Analysis

The controllers designed in this paper used the sliding modetheory which is the most used approach to deal with systemsrunning under uncertainty conditions However we haveseen in the second section that the first-order sliding modecontroller with a linear sliding surface is not robust withrespect to perturbation and mismatched uncertainty In factby using the fact that on the sliding surface the systembehavesin a similar way to a linear second-order system it can be

0

100

200

300

minus10123

0

10

20

30

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 6 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with states feedback

minus1000

100200300

minus20246

minus100

102030

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 7 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with estimated states feed-back

easily shown that as far as the slidingmotion is preserved thesystem is asymptotically stable if the closed loop eigenvaluesare chosen as in (11) and thus the steady state error due to theuncertainty is expressed by (12) Also the steady state errordue to the constant perturbation is given by (13)

Therefore the steady state error gets smaller as the closedloop eigenvalues have a larger amplitude This is illustratedby the simulation results delineated in Figure 2 where 120582 = 50and the total error is 246 cm

To circumvent the problem we have suggested a second-order sliding mode controller Applying this controller we

10 Mathematical Problems in Engineering

can notice that it achieves a better performance than the firstSMC In fact the second-order SMC outperforms the first-order SMC in sense of robustness with respect tomismatchedperturbation but it cannot guarantee the robustness withrespect to the constant perturbation The steady state errordue to the perturbation is equal to 119889(119905)ℎ as demonstratedpreviously When the perturbation 119889(119905) = 01 the erroris equal to 01 cm as delineated in Figure 3 If we increasethe value of ℎ we should also increase the value of thevariable structure gains1198821 and1198822 So we obtain a chatteringphenomenon

To overcome the problem of mismatched perturbationand uncertainty we have suggested a sliding mode controllerwith nonlinear surfaceThe idea is based on the backsteppingmethod and using a robust variable structure virtual con-troller The obtained results achieved robustness with respectto parameter uncertainty and perturbation The zoomedcurve on Figure 4 shows that we have attained our aim todrive the hydraulic servo system to a reference position butwith a chattering phenomenon So to attenuate this problemwe suggest substituting the discontinuous function sign(119909)with a smooth saturation function tanh(120583119909) for sufficientlyhigh positive value 120583 gt 0 Owing to this smooth function wecan achieve the reference value in presence of mismatcheduncertainty and perturbation with attenuated chattering

Finally in Section 4 we designed a slidingmode observerin order to estimate the required states The efficiency of thisobserver is proved mathematically and also by simulationspresented in Figures 6 and 7

Compared to other methods such as that presented in[17] which is based on adaptive approach to achieve positioncontrol we notice that our method provides faster responsesince the adaptive controller proposed therein attempts toestimate the mismatched nonlinear uncertainty Indeed therod makes a displacement of 30 cm within more than 08 swhereas with our method a maximum of 02 s will beneeded to make the same displacement In [21] the authorspresented a controller based on variable structure PID todrive the hydraulic system position Although fast responsehas been achieved which is comparable to our results thegood robustness was obtained since the uncertainties andthe perturbation were matched with the controller In ourcase the robustness against the mismatched character of theperturbation and uncertainty presented the main challengeto take In [26] authors obtained comparable results tothose presented in this paper using integral sliding modecontroller with realizable reference compensation applied toan asymmetric piston

6 Conclusion

In this paper we developed several controllers based onthe sliding mode theory Our aim was to control theposition of a hydraulic servo system piston in presence ofmismatched uncertainty and perturbation We have shownthat a first-order sliding mode controller did not achieveany robustness Next we have developed a second-ordersliding mode controller that has shown robustness withrespect to parametric uncertainty but was not robust to

the perturbation Finally we have suggested a sliding modecontroller based on a nonlinear sliding surface The designis based on the backstepping method where on each stepa variable structure virtual controller design leads to thedesign of the sliding surface This controller emphasized thechattering phenomenon due to the nested sliding modesAs a remedy we suggested substituting the discontinuousfunction with a smooth saturation function Eventually wehave designed a robust sliding observer in order to substitutethe unmeasured states with their estimates A step-by-stepproof has shown that the controller issued from the estimatedstates achieved the position tracking

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H E Merritt Hydraulic Control Systems John Wiley amp SonsNew York NY USA 1967

[2] M Feki and E Richard ldquoIncluding leakage flow in the ser-vovalve static modelrdquo International Journal of Modelling andSimulation vol 25 no 1 pp 4004ndash4009 2005

[3] A Alleyne R Liu and H Wright ldquoOn the limitations offorce tracking control for hydraulic active suspensionsrdquo inProceedings of the American Control Conference (ACC rsquo98) vol1 pp 43ndash47 June 1998

[4] Y Cheng andB L RDeMoor ldquoRobustness analysis and controlsystem design for a hydraulic servo systemrdquo IEEE Transactionson Control Systems Technology vol 2 no 3 pp 183ndash197 1994

[5] M Feki E Richard and F Gomes Almeida ldquoCommande eneffort drsquoun verin hydraulique par linearization entreesortierdquoin Journees Doctorales drsquoAutomatiques (JDA rsquo99) pp 181ndash184Nancy-France 1999

[6] H Hahn A Piepenbrink and K-D Leimbach ldquoInputoutputlinearization control of an electro servo-hydraulic actuatorrdquo inProceedings of the 3rd IEEE Conference on Control Applicationspp 995ndash1000 August 1994

[7] M Jovanovic ldquoNonlinear control of an electrohydraulic velocityservosystemrdquo in Proceedings of the 2002 American ControlConference vol 1 pp 588ndash593 2002

[8] G A Sohl and J E Bobrow ldquoExperiments and simulationson the nonlinear control of a hydraulic servosystemrdquo IEEETransactions on Control Systems Technology vol 7 no 2 pp238ndash247 1999

[9] A G Alleyne and R Liu ldquoSystematic control of a class ofnonlinear systems with application to electrohydraulic cylinderpressure controlrdquo IEEE Transactions on Control Systems Tech-nology vol 8 no 4 pp 623ndash634 2000

[10] M A Avila A G Loukianov and E N Sanchez ldquoElectro-hydraulic actuator trajectory trackingrdquo in Proceedings of theAmerican Control Conference (AAC rsquo04) vol 3 pp 2603ndash2608IEEE Boston Mass USA July 2004

[11] A Bonchis P I Corke D C Rye and Q P Ha ldquoVariable struc-ture methods in hydraulic servo systems controlrdquo Automaticavol 37 no 4 pp 589ndash595 2001

[12] M Jerouane and F Lamnabhi-Lagarrigue ldquoA new robust slidingmode controller for a hydraulic actuatorrdquo in Proceedings of the

Mathematical Problems in Engineering 11

40th IEEE Conference on Decision and Control (CDC rsquo01) vol 1pp 908ndash913 December 2001

[13] H-M Chen J-C Renn and J-P Su ldquoSlidingmode control withvarying boundary layers for an electro-hydraulic position servosystemrdquo The International Journal of Advanced ManufacturingTechnology vol 26 no 1-2 pp 117ndash123 2005

[14] J M Zheng S D Zhao and S G Wei ldquoApplication of self-tuning fuzzy PID controller for a SRM direct drive volumecontrol hydraulic pressrdquo Control Engineering Practice vol 17no 12 pp 1398ndash1404 2009 Special Section The 2007 IFACSymposium on Advances in Automotive Control

[15] A Aly ldquoPid parameters optimization using genetic algorithmtechnique for electrohydraulic servo control systemrdquo IntelligentControl and Automation vol 2 no 2 pp 69ndash76 2011

[16] B Yao F Bu and G T C Chiu ldquoNonlinear adaptive robustcontrol of electro-hydraulic servo systems with discontinuousprojectionsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control (CDC rsquo98) pp 2265ndash2270 December1998

[17] B Yao F Bu J Reedy and G T-C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[18] K K Ahn and Q T Dinh ldquoSelf-tuning of quantitative feedbacktheory for force control of an electro-hydraulic test machinerdquoControl Engineering Practice vol 17 no 11 pp 1291ndash1306 2009

[19] E Kolsi Gdoura M Feki and N Derbel ldquoSliding mode controlof a hydraulic servo system position using adaptive slidingsurface and adaptive gainrdquo International Journal of ModellingIdentification and Control vol 23 pp 211ndash219 2015

[20] C Guan and S Pan ldquoAdaptive sliding mode control of electro-hydraulic systemwith nonlinear unknown parametersrdquoControlEngineering Practice vol 16 no 11 pp 1275ndash1284 2008

[21] D Liu Z Tang and Z Pei ldquoVariable structure compensationPID control of asymmetrical hydraulic cylinder trajectorytrackingrdquo Mathematical Problems in Engineering vol 2015Article ID 890704 9 pages 2015

[22] A G Loukianov J Rivera Y V Orlov and E Y MoralesTeraoka ldquoRobust trajectory tracking for an electrohydraulicactuatorrdquo IEEE Transactions on Industrial Electronics vol 56no 9 pp 3523ndash3531 2009

[23] M Ye Q Wang and S Jiao ldquoRobust 1198672119867infin

control forthe electrohydraulic steering system of a four-wheel vehiclerdquoMathematical Problems in Engineering vol 2014 Article ID208019 12 pages 2014

[24] W Shen J Jiang H R Karimi and X Su ldquoObserver-basedrobust control for hydraulic velocity control systemrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 6891329 pages 2013

[25] E Kolsi-Gdoura M Feki and N Derbel ldquoSliding mode-based robust position control of an electrohydraulic systemrdquoin Proceedings of the 10th International Multi-Conference onSystems Signals and Devices (SSD rsquo13) pp 1ndash5 March 2013

[26] E Kolsi Gdoura M Feki and N Derbel ldquoControl of ahydraulic servo system using sliding mode with an integral andrealizable reference compensationrdquo Control Engineering andApplied Informatics vol 17 pp 111ndash119 2015

[27] A Levant ldquoSliding order and sliding accuracy in sliding modecontrolrdquo International Journal of Control vol 58 no 6 pp 1247ndash1263 1993

[28] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[29] M Djemai K Busawon K Benmansour and A MaroufldquoHigh-order sliding mode control of a DC motor drive viaa switched controlled multi-cellular converterrdquo InternationalJournal of Systems Science vol 42 no 11 pp 1869ndash1882 2011

[30] M Lavanya R M Brisilla and V Sankaranarayanan ldquoHigherorder sliding mode control of permanent magnet dc motorrdquoin Proceedings of the 12th International Workshop on VariableStructure Systems (VSS rsquo12) pp 226ndash230 January 2012

[31] S Laghrouche M Smaoui F Plestan and X Brun ldquoHigherorder sliding mode control based on optimal approach of anelectropneumatic actuatorrdquo International Journal of Control vol79 no 2 pp 119ndash131 2006

[32] A G Loukianov E Sanchez and C Lizalde ldquoForce trackingneural block control for an electro-hydraulic actuator viasecond-order slidingmoderdquo International Journal of Robust andNonlinear Control vol 18 no 3 pp 319ndash332 2008

[33] J Komsta N van Oijen and P Antoszkiewicz ldquoIntegral slidingmode compensator for load pressure control of die-cushioncylinder driverdquo Control Engineering Practice vol 21 no 5 pp708ndash718 2013

[34] M Feki Synthese de commandes et drsquoobservateurs pour lessystemes non-lineaires application aux systemes hydrauliques[PhD thesis] Universite de Metz Metz France 2001

[35] V V Filippov ldquoOn a scalar equation with discontinuous right-hand side and the uniqueness theoremrdquo Differential Equationsvol 38 no 10 pp 1435ndash1445 2002

[36] T Boukhobza M Djemai and J P Barbot ldquoNonlinear slidingobserver for systems in output and output derivative injectionformrdquo in Proceedings of the IFAC World Congress pp 299ndash305San Antonio Tex USA 1996

[37] H Saadaoui N Manamanni M Djemaı J P Barbot andT Floquet ldquoExact differentiation and sliding mode observersfor switched Lagrangian systemsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 65 no 5 pp 1050ndash1069 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Thus if 1198712 is chosen such that

1198712 gt1119898119905

1003816100381610038161003816Δ1198961198971199093 + 1198781198901 + 119887119889max1003816100381610038161003816 (48)

then we can deduce that 1198902 + 119889 tend to zero and hence 1199112tend to 1199092 + 119889 That is the observer will estimate the velocitywith the constant perturbation Moreover when the errordynamics are sliding on 1198903 = 0 and 1198902 + 119889 = 0 then 1198902 = 0and we get

1 = 1199091 +119887

1198781198902 minus

Δ119896119897

1198781199093 (49)

That is the observer will estimate the differential pressurewith a constant difference proportional to the perturbationand the uncertainty Now if the controller sliding surface 120590 isexpressed using the observer state variables

120590 (119911) = 1198781 + (1198981199051198823 +1198981199051198822120575 (1199093 minus1199093ref) minus 119887) 2

minus 1198961198971199113 +1198981199051198821 sign (1205901 (2 1199093))

(50)

then at sliding mode of the controller (119911) = 0 the velocitydynamics are given by

2 = minus1198823 (1199092 +119889) minus1198821 sign (1205901 (2 1199093)) (51)

Hence the velocity of the system rod will tend to minus119889

Step 3 Finally the differential pressure error dynamics canbe roughly expressed as

1198901 = 1198911 (119909) minus1198911 (119911) minus 1198711 sign(1198901 +119887

1198781198902 minus

Δ119896119897

1198781199093) (52)

if the system is already sliding on the surfaces 1199093 = 1199093ref and1198902 = minus119889 then by choosing 1198711 gt |1198911(119909)minus1198911(119911)| the differentialpressure is estimated with the constant difference minus(119887119878)119889 +Δ119881119905Figure 6 shows the convergence of the observer state 119911

3to

1199093although they are starting fromdifferent initial conditions

indeed the observer is starting at rod position 10 cm whereasthe system is starting at the origin As expected from theabove analysis 1199112 tends to 1199092 + 119889 and 1199111 tends to 1199091 with aconstant difference of 33 bar The observer gains are chosenas 1198711 = 1000 1198712 = 100 and 1198713 = 1 In Figure 7 the observerand system behaviors are shown with the controller beingcalculated using the estimated states The observer initialposition is minus10 cm

5 Simulation Results Analysis

The controllers designed in this paper used the sliding modetheory which is the most used approach to deal with systemsrunning under uncertainty conditions However we haveseen in the second section that the first-order sliding modecontroller with a linear sliding surface is not robust withrespect to perturbation and mismatched uncertainty In factby using the fact that on the sliding surface the systembehavesin a similar way to a linear second-order system it can be

0

100

200

300

minus10123

0

10

20

30

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 6 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with states feedback

minus1000

100200300

minus20246

minus100

102030

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

0 005 01 015 02 025 03 035 04 045 05Time (s)

Velo

city

(ms

)Po

sitio

n (c

m)

Δ p

ress

ure (

bar)

Figure 7 Behavior of the observer and the hydraulic servo systemcontrolled with sliding mode controller with estimated states feed-back

easily shown that as far as the slidingmotion is preserved thesystem is asymptotically stable if the closed loop eigenvaluesare chosen as in (11) and thus the steady state error due to theuncertainty is expressed by (12) Also the steady state errordue to the constant perturbation is given by (13)

Therefore the steady state error gets smaller as the closedloop eigenvalues have a larger amplitude This is illustratedby the simulation results delineated in Figure 2 where 120582 = 50and the total error is 246 cm

To circumvent the problem we have suggested a second-order sliding mode controller Applying this controller we

10 Mathematical Problems in Engineering

can notice that it achieves a better performance than the firstSMC In fact the second-order SMC outperforms the first-order SMC in sense of robustness with respect tomismatchedperturbation but it cannot guarantee the robustness withrespect to the constant perturbation The steady state errordue to the perturbation is equal to 119889(119905)ℎ as demonstratedpreviously When the perturbation 119889(119905) = 01 the erroris equal to 01 cm as delineated in Figure 3 If we increasethe value of ℎ we should also increase the value of thevariable structure gains1198821 and1198822 So we obtain a chatteringphenomenon

To overcome the problem of mismatched perturbationand uncertainty we have suggested a sliding mode controllerwith nonlinear surfaceThe idea is based on the backsteppingmethod and using a robust variable structure virtual con-troller The obtained results achieved robustness with respectto parameter uncertainty and perturbation The zoomedcurve on Figure 4 shows that we have attained our aim todrive the hydraulic servo system to a reference position butwith a chattering phenomenon So to attenuate this problemwe suggest substituting the discontinuous function sign(119909)with a smooth saturation function tanh(120583119909) for sufficientlyhigh positive value 120583 gt 0 Owing to this smooth function wecan achieve the reference value in presence of mismatcheduncertainty and perturbation with attenuated chattering

Finally in Section 4 we designed a slidingmode observerin order to estimate the required states The efficiency of thisobserver is proved mathematically and also by simulationspresented in Figures 6 and 7

Compared to other methods such as that presented in[17] which is based on adaptive approach to achieve positioncontrol we notice that our method provides faster responsesince the adaptive controller proposed therein attempts toestimate the mismatched nonlinear uncertainty Indeed therod makes a displacement of 30 cm within more than 08 swhereas with our method a maximum of 02 s will beneeded to make the same displacement In [21] the authorspresented a controller based on variable structure PID todrive the hydraulic system position Although fast responsehas been achieved which is comparable to our results thegood robustness was obtained since the uncertainties andthe perturbation were matched with the controller In ourcase the robustness against the mismatched character of theperturbation and uncertainty presented the main challengeto take In [26] authors obtained comparable results tothose presented in this paper using integral sliding modecontroller with realizable reference compensation applied toan asymmetric piston

6 Conclusion

In this paper we developed several controllers based onthe sliding mode theory Our aim was to control theposition of a hydraulic servo system piston in presence ofmismatched uncertainty and perturbation We have shownthat a first-order sliding mode controller did not achieveany robustness Next we have developed a second-ordersliding mode controller that has shown robustness withrespect to parametric uncertainty but was not robust to

the perturbation Finally we have suggested a sliding modecontroller based on a nonlinear sliding surface The designis based on the backstepping method where on each stepa variable structure virtual controller design leads to thedesign of the sliding surface This controller emphasized thechattering phenomenon due to the nested sliding modesAs a remedy we suggested substituting the discontinuousfunction with a smooth saturation function Eventually wehave designed a robust sliding observer in order to substitutethe unmeasured states with their estimates A step-by-stepproof has shown that the controller issued from the estimatedstates achieved the position tracking

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H E Merritt Hydraulic Control Systems John Wiley amp SonsNew York NY USA 1967

[2] M Feki and E Richard ldquoIncluding leakage flow in the ser-vovalve static modelrdquo International Journal of Modelling andSimulation vol 25 no 1 pp 4004ndash4009 2005

[3] A Alleyne R Liu and H Wright ldquoOn the limitations offorce tracking control for hydraulic active suspensionsrdquo inProceedings of the American Control Conference (ACC rsquo98) vol1 pp 43ndash47 June 1998

[4] Y Cheng andB L RDeMoor ldquoRobustness analysis and controlsystem design for a hydraulic servo systemrdquo IEEE Transactionson Control Systems Technology vol 2 no 3 pp 183ndash197 1994

[5] M Feki E Richard and F Gomes Almeida ldquoCommande eneffort drsquoun verin hydraulique par linearization entreesortierdquoin Journees Doctorales drsquoAutomatiques (JDA rsquo99) pp 181ndash184Nancy-France 1999

[6] H Hahn A Piepenbrink and K-D Leimbach ldquoInputoutputlinearization control of an electro servo-hydraulic actuatorrdquo inProceedings of the 3rd IEEE Conference on Control Applicationspp 995ndash1000 August 1994

[7] M Jovanovic ldquoNonlinear control of an electrohydraulic velocityservosystemrdquo in Proceedings of the 2002 American ControlConference vol 1 pp 588ndash593 2002

[8] G A Sohl and J E Bobrow ldquoExperiments and simulationson the nonlinear control of a hydraulic servosystemrdquo IEEETransactions on Control Systems Technology vol 7 no 2 pp238ndash247 1999

[9] A G Alleyne and R Liu ldquoSystematic control of a class ofnonlinear systems with application to electrohydraulic cylinderpressure controlrdquo IEEE Transactions on Control Systems Tech-nology vol 8 no 4 pp 623ndash634 2000

[10] M A Avila A G Loukianov and E N Sanchez ldquoElectro-hydraulic actuator trajectory trackingrdquo in Proceedings of theAmerican Control Conference (AAC rsquo04) vol 3 pp 2603ndash2608IEEE Boston Mass USA July 2004

[11] A Bonchis P I Corke D C Rye and Q P Ha ldquoVariable struc-ture methods in hydraulic servo systems controlrdquo Automaticavol 37 no 4 pp 589ndash595 2001

[12] M Jerouane and F Lamnabhi-Lagarrigue ldquoA new robust slidingmode controller for a hydraulic actuatorrdquo in Proceedings of the

Mathematical Problems in Engineering 11

40th IEEE Conference on Decision and Control (CDC rsquo01) vol 1pp 908ndash913 December 2001

[13] H-M Chen J-C Renn and J-P Su ldquoSlidingmode control withvarying boundary layers for an electro-hydraulic position servosystemrdquo The International Journal of Advanced ManufacturingTechnology vol 26 no 1-2 pp 117ndash123 2005

[14] J M Zheng S D Zhao and S G Wei ldquoApplication of self-tuning fuzzy PID controller for a SRM direct drive volumecontrol hydraulic pressrdquo Control Engineering Practice vol 17no 12 pp 1398ndash1404 2009 Special Section The 2007 IFACSymposium on Advances in Automotive Control

[15] A Aly ldquoPid parameters optimization using genetic algorithmtechnique for electrohydraulic servo control systemrdquo IntelligentControl and Automation vol 2 no 2 pp 69ndash76 2011

[16] B Yao F Bu and G T C Chiu ldquoNonlinear adaptive robustcontrol of electro-hydraulic servo systems with discontinuousprojectionsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control (CDC rsquo98) pp 2265ndash2270 December1998

[17] B Yao F Bu J Reedy and G T-C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[18] K K Ahn and Q T Dinh ldquoSelf-tuning of quantitative feedbacktheory for force control of an electro-hydraulic test machinerdquoControl Engineering Practice vol 17 no 11 pp 1291ndash1306 2009

[19] E Kolsi Gdoura M Feki and N Derbel ldquoSliding mode controlof a hydraulic servo system position using adaptive slidingsurface and adaptive gainrdquo International Journal of ModellingIdentification and Control vol 23 pp 211ndash219 2015

[20] C Guan and S Pan ldquoAdaptive sliding mode control of electro-hydraulic systemwith nonlinear unknown parametersrdquoControlEngineering Practice vol 16 no 11 pp 1275ndash1284 2008

[21] D Liu Z Tang and Z Pei ldquoVariable structure compensationPID control of asymmetrical hydraulic cylinder trajectorytrackingrdquo Mathematical Problems in Engineering vol 2015Article ID 890704 9 pages 2015

[22] A G Loukianov J Rivera Y V Orlov and E Y MoralesTeraoka ldquoRobust trajectory tracking for an electrohydraulicactuatorrdquo IEEE Transactions on Industrial Electronics vol 56no 9 pp 3523ndash3531 2009

[23] M Ye Q Wang and S Jiao ldquoRobust 1198672119867infin

control forthe electrohydraulic steering system of a four-wheel vehiclerdquoMathematical Problems in Engineering vol 2014 Article ID208019 12 pages 2014

[24] W Shen J Jiang H R Karimi and X Su ldquoObserver-basedrobust control for hydraulic velocity control systemrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 6891329 pages 2013

[25] E Kolsi-Gdoura M Feki and N Derbel ldquoSliding mode-based robust position control of an electrohydraulic systemrdquoin Proceedings of the 10th International Multi-Conference onSystems Signals and Devices (SSD rsquo13) pp 1ndash5 March 2013

[26] E Kolsi Gdoura M Feki and N Derbel ldquoControl of ahydraulic servo system using sliding mode with an integral andrealizable reference compensationrdquo Control Engineering andApplied Informatics vol 17 pp 111ndash119 2015

[27] A Levant ldquoSliding order and sliding accuracy in sliding modecontrolrdquo International Journal of Control vol 58 no 6 pp 1247ndash1263 1993

[28] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[29] M Djemai K Busawon K Benmansour and A MaroufldquoHigh-order sliding mode control of a DC motor drive viaa switched controlled multi-cellular converterrdquo InternationalJournal of Systems Science vol 42 no 11 pp 1869ndash1882 2011

[30] M Lavanya R M Brisilla and V Sankaranarayanan ldquoHigherorder sliding mode control of permanent magnet dc motorrdquoin Proceedings of the 12th International Workshop on VariableStructure Systems (VSS rsquo12) pp 226ndash230 January 2012

[31] S Laghrouche M Smaoui F Plestan and X Brun ldquoHigherorder sliding mode control based on optimal approach of anelectropneumatic actuatorrdquo International Journal of Control vol79 no 2 pp 119ndash131 2006

[32] A G Loukianov E Sanchez and C Lizalde ldquoForce trackingneural block control for an electro-hydraulic actuator viasecond-order slidingmoderdquo International Journal of Robust andNonlinear Control vol 18 no 3 pp 319ndash332 2008

[33] J Komsta N van Oijen and P Antoszkiewicz ldquoIntegral slidingmode compensator for load pressure control of die-cushioncylinder driverdquo Control Engineering Practice vol 21 no 5 pp708ndash718 2013

[34] M Feki Synthese de commandes et drsquoobservateurs pour lessystemes non-lineaires application aux systemes hydrauliques[PhD thesis] Universite de Metz Metz France 2001

[35] V V Filippov ldquoOn a scalar equation with discontinuous right-hand side and the uniqueness theoremrdquo Differential Equationsvol 38 no 10 pp 1435ndash1445 2002

[36] T Boukhobza M Djemai and J P Barbot ldquoNonlinear slidingobserver for systems in output and output derivative injectionformrdquo in Proceedings of the IFAC World Congress pp 299ndash305San Antonio Tex USA 1996

[37] H Saadaoui N Manamanni M Djemaı J P Barbot andT Floquet ldquoExact differentiation and sliding mode observersfor switched Lagrangian systemsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 65 no 5 pp 1050ndash1069 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

can notice that it achieves a better performance than the firstSMC In fact the second-order SMC outperforms the first-order SMC in sense of robustness with respect tomismatchedperturbation but it cannot guarantee the robustness withrespect to the constant perturbation The steady state errordue to the perturbation is equal to 119889(119905)ℎ as demonstratedpreviously When the perturbation 119889(119905) = 01 the erroris equal to 01 cm as delineated in Figure 3 If we increasethe value of ℎ we should also increase the value of thevariable structure gains1198821 and1198822 So we obtain a chatteringphenomenon

To overcome the problem of mismatched perturbationand uncertainty we have suggested a sliding mode controllerwith nonlinear surfaceThe idea is based on the backsteppingmethod and using a robust variable structure virtual con-troller The obtained results achieved robustness with respectto parameter uncertainty and perturbation The zoomedcurve on Figure 4 shows that we have attained our aim todrive the hydraulic servo system to a reference position butwith a chattering phenomenon So to attenuate this problemwe suggest substituting the discontinuous function sign(119909)with a smooth saturation function tanh(120583119909) for sufficientlyhigh positive value 120583 gt 0 Owing to this smooth function wecan achieve the reference value in presence of mismatcheduncertainty and perturbation with attenuated chattering

Finally in Section 4 we designed a slidingmode observerin order to estimate the required states The efficiency of thisobserver is proved mathematically and also by simulationspresented in Figures 6 and 7

Compared to other methods such as that presented in[17] which is based on adaptive approach to achieve positioncontrol we notice that our method provides faster responsesince the adaptive controller proposed therein attempts toestimate the mismatched nonlinear uncertainty Indeed therod makes a displacement of 30 cm within more than 08 swhereas with our method a maximum of 02 s will beneeded to make the same displacement In [21] the authorspresented a controller based on variable structure PID todrive the hydraulic system position Although fast responsehas been achieved which is comparable to our results thegood robustness was obtained since the uncertainties andthe perturbation were matched with the controller In ourcase the robustness against the mismatched character of theperturbation and uncertainty presented the main challengeto take In [26] authors obtained comparable results tothose presented in this paper using integral sliding modecontroller with realizable reference compensation applied toan asymmetric piston

6 Conclusion

In this paper we developed several controllers based onthe sliding mode theory Our aim was to control theposition of a hydraulic servo system piston in presence ofmismatched uncertainty and perturbation We have shownthat a first-order sliding mode controller did not achieveany robustness Next we have developed a second-ordersliding mode controller that has shown robustness withrespect to parametric uncertainty but was not robust to

the perturbation Finally we have suggested a sliding modecontroller based on a nonlinear sliding surface The designis based on the backstepping method where on each stepa variable structure virtual controller design leads to thedesign of the sliding surface This controller emphasized thechattering phenomenon due to the nested sliding modesAs a remedy we suggested substituting the discontinuousfunction with a smooth saturation function Eventually wehave designed a robust sliding observer in order to substitutethe unmeasured states with their estimates A step-by-stepproof has shown that the controller issued from the estimatedstates achieved the position tracking

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H E Merritt Hydraulic Control Systems John Wiley amp SonsNew York NY USA 1967

[2] M Feki and E Richard ldquoIncluding leakage flow in the ser-vovalve static modelrdquo International Journal of Modelling andSimulation vol 25 no 1 pp 4004ndash4009 2005

[3] A Alleyne R Liu and H Wright ldquoOn the limitations offorce tracking control for hydraulic active suspensionsrdquo inProceedings of the American Control Conference (ACC rsquo98) vol1 pp 43ndash47 June 1998

[4] Y Cheng andB L RDeMoor ldquoRobustness analysis and controlsystem design for a hydraulic servo systemrdquo IEEE Transactionson Control Systems Technology vol 2 no 3 pp 183ndash197 1994

[5] M Feki E Richard and F Gomes Almeida ldquoCommande eneffort drsquoun verin hydraulique par linearization entreesortierdquoin Journees Doctorales drsquoAutomatiques (JDA rsquo99) pp 181ndash184Nancy-France 1999

[6] H Hahn A Piepenbrink and K-D Leimbach ldquoInputoutputlinearization control of an electro servo-hydraulic actuatorrdquo inProceedings of the 3rd IEEE Conference on Control Applicationspp 995ndash1000 August 1994

[7] M Jovanovic ldquoNonlinear control of an electrohydraulic velocityservosystemrdquo in Proceedings of the 2002 American ControlConference vol 1 pp 588ndash593 2002

[8] G A Sohl and J E Bobrow ldquoExperiments and simulationson the nonlinear control of a hydraulic servosystemrdquo IEEETransactions on Control Systems Technology vol 7 no 2 pp238ndash247 1999

[9] A G Alleyne and R Liu ldquoSystematic control of a class ofnonlinear systems with application to electrohydraulic cylinderpressure controlrdquo IEEE Transactions on Control Systems Tech-nology vol 8 no 4 pp 623ndash634 2000

[10] M A Avila A G Loukianov and E N Sanchez ldquoElectro-hydraulic actuator trajectory trackingrdquo in Proceedings of theAmerican Control Conference (AAC rsquo04) vol 3 pp 2603ndash2608IEEE Boston Mass USA July 2004

[11] A Bonchis P I Corke D C Rye and Q P Ha ldquoVariable struc-ture methods in hydraulic servo systems controlrdquo Automaticavol 37 no 4 pp 589ndash595 2001

[12] M Jerouane and F Lamnabhi-Lagarrigue ldquoA new robust slidingmode controller for a hydraulic actuatorrdquo in Proceedings of the

Mathematical Problems in Engineering 11

40th IEEE Conference on Decision and Control (CDC rsquo01) vol 1pp 908ndash913 December 2001

[13] H-M Chen J-C Renn and J-P Su ldquoSlidingmode control withvarying boundary layers for an electro-hydraulic position servosystemrdquo The International Journal of Advanced ManufacturingTechnology vol 26 no 1-2 pp 117ndash123 2005

[14] J M Zheng S D Zhao and S G Wei ldquoApplication of self-tuning fuzzy PID controller for a SRM direct drive volumecontrol hydraulic pressrdquo Control Engineering Practice vol 17no 12 pp 1398ndash1404 2009 Special Section The 2007 IFACSymposium on Advances in Automotive Control

[15] A Aly ldquoPid parameters optimization using genetic algorithmtechnique for electrohydraulic servo control systemrdquo IntelligentControl and Automation vol 2 no 2 pp 69ndash76 2011

[16] B Yao F Bu and G T C Chiu ldquoNonlinear adaptive robustcontrol of electro-hydraulic servo systems with discontinuousprojectionsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control (CDC rsquo98) pp 2265ndash2270 December1998

[17] B Yao F Bu J Reedy and G T-C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[18] K K Ahn and Q T Dinh ldquoSelf-tuning of quantitative feedbacktheory for force control of an electro-hydraulic test machinerdquoControl Engineering Practice vol 17 no 11 pp 1291ndash1306 2009

[19] E Kolsi Gdoura M Feki and N Derbel ldquoSliding mode controlof a hydraulic servo system position using adaptive slidingsurface and adaptive gainrdquo International Journal of ModellingIdentification and Control vol 23 pp 211ndash219 2015

[20] C Guan and S Pan ldquoAdaptive sliding mode control of electro-hydraulic systemwith nonlinear unknown parametersrdquoControlEngineering Practice vol 16 no 11 pp 1275ndash1284 2008

[21] D Liu Z Tang and Z Pei ldquoVariable structure compensationPID control of asymmetrical hydraulic cylinder trajectorytrackingrdquo Mathematical Problems in Engineering vol 2015Article ID 890704 9 pages 2015

[22] A G Loukianov J Rivera Y V Orlov and E Y MoralesTeraoka ldquoRobust trajectory tracking for an electrohydraulicactuatorrdquo IEEE Transactions on Industrial Electronics vol 56no 9 pp 3523ndash3531 2009

[23] M Ye Q Wang and S Jiao ldquoRobust 1198672119867infin

control forthe electrohydraulic steering system of a four-wheel vehiclerdquoMathematical Problems in Engineering vol 2014 Article ID208019 12 pages 2014

[24] W Shen J Jiang H R Karimi and X Su ldquoObserver-basedrobust control for hydraulic velocity control systemrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 6891329 pages 2013

[25] E Kolsi-Gdoura M Feki and N Derbel ldquoSliding mode-based robust position control of an electrohydraulic systemrdquoin Proceedings of the 10th International Multi-Conference onSystems Signals and Devices (SSD rsquo13) pp 1ndash5 March 2013

[26] E Kolsi Gdoura M Feki and N Derbel ldquoControl of ahydraulic servo system using sliding mode with an integral andrealizable reference compensationrdquo Control Engineering andApplied Informatics vol 17 pp 111ndash119 2015

[27] A Levant ldquoSliding order and sliding accuracy in sliding modecontrolrdquo International Journal of Control vol 58 no 6 pp 1247ndash1263 1993

[28] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[29] M Djemai K Busawon K Benmansour and A MaroufldquoHigh-order sliding mode control of a DC motor drive viaa switched controlled multi-cellular converterrdquo InternationalJournal of Systems Science vol 42 no 11 pp 1869ndash1882 2011

[30] M Lavanya R M Brisilla and V Sankaranarayanan ldquoHigherorder sliding mode control of permanent magnet dc motorrdquoin Proceedings of the 12th International Workshop on VariableStructure Systems (VSS rsquo12) pp 226ndash230 January 2012

[31] S Laghrouche M Smaoui F Plestan and X Brun ldquoHigherorder sliding mode control based on optimal approach of anelectropneumatic actuatorrdquo International Journal of Control vol79 no 2 pp 119ndash131 2006

[32] A G Loukianov E Sanchez and C Lizalde ldquoForce trackingneural block control for an electro-hydraulic actuator viasecond-order slidingmoderdquo International Journal of Robust andNonlinear Control vol 18 no 3 pp 319ndash332 2008

[33] J Komsta N van Oijen and P Antoszkiewicz ldquoIntegral slidingmode compensator for load pressure control of die-cushioncylinder driverdquo Control Engineering Practice vol 21 no 5 pp708ndash718 2013

[34] M Feki Synthese de commandes et drsquoobservateurs pour lessystemes non-lineaires application aux systemes hydrauliques[PhD thesis] Universite de Metz Metz France 2001

[35] V V Filippov ldquoOn a scalar equation with discontinuous right-hand side and the uniqueness theoremrdquo Differential Equationsvol 38 no 10 pp 1435ndash1445 2002

[36] T Boukhobza M Djemai and J P Barbot ldquoNonlinear slidingobserver for systems in output and output derivative injectionformrdquo in Proceedings of the IFAC World Congress pp 299ndash305San Antonio Tex USA 1996

[37] H Saadaoui N Manamanni M Djemaı J P Barbot andT Floquet ldquoExact differentiation and sliding mode observersfor switched Lagrangian systemsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 65 no 5 pp 1050ndash1069 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

40th IEEE Conference on Decision and Control (CDC rsquo01) vol 1pp 908ndash913 December 2001

[13] H-M Chen J-C Renn and J-P Su ldquoSlidingmode control withvarying boundary layers for an electro-hydraulic position servosystemrdquo The International Journal of Advanced ManufacturingTechnology vol 26 no 1-2 pp 117ndash123 2005

[14] J M Zheng S D Zhao and S G Wei ldquoApplication of self-tuning fuzzy PID controller for a SRM direct drive volumecontrol hydraulic pressrdquo Control Engineering Practice vol 17no 12 pp 1398ndash1404 2009 Special Section The 2007 IFACSymposium on Advances in Automotive Control

[15] A Aly ldquoPid parameters optimization using genetic algorithmtechnique for electrohydraulic servo control systemrdquo IntelligentControl and Automation vol 2 no 2 pp 69ndash76 2011

[16] B Yao F Bu and G T C Chiu ldquoNonlinear adaptive robustcontrol of electro-hydraulic servo systems with discontinuousprojectionsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control (CDC rsquo98) pp 2265ndash2270 December1998

[17] B Yao F Bu J Reedy and G T-C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[18] K K Ahn and Q T Dinh ldquoSelf-tuning of quantitative feedbacktheory for force control of an electro-hydraulic test machinerdquoControl Engineering Practice vol 17 no 11 pp 1291ndash1306 2009

[19] E Kolsi Gdoura M Feki and N Derbel ldquoSliding mode controlof a hydraulic servo system position using adaptive slidingsurface and adaptive gainrdquo International Journal of ModellingIdentification and Control vol 23 pp 211ndash219 2015

[20] C Guan and S Pan ldquoAdaptive sliding mode control of electro-hydraulic systemwith nonlinear unknown parametersrdquoControlEngineering Practice vol 16 no 11 pp 1275ndash1284 2008

[21] D Liu Z Tang and Z Pei ldquoVariable structure compensationPID control of asymmetrical hydraulic cylinder trajectorytrackingrdquo Mathematical Problems in Engineering vol 2015Article ID 890704 9 pages 2015

[22] A G Loukianov J Rivera Y V Orlov and E Y MoralesTeraoka ldquoRobust trajectory tracking for an electrohydraulicactuatorrdquo IEEE Transactions on Industrial Electronics vol 56no 9 pp 3523ndash3531 2009

[23] M Ye Q Wang and S Jiao ldquoRobust 1198672119867infin

control forthe electrohydraulic steering system of a four-wheel vehiclerdquoMathematical Problems in Engineering vol 2014 Article ID208019 12 pages 2014

[24] W Shen J Jiang H R Karimi and X Su ldquoObserver-basedrobust control for hydraulic velocity control systemrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 6891329 pages 2013

[25] E Kolsi-Gdoura M Feki and N Derbel ldquoSliding mode-based robust position control of an electrohydraulic systemrdquoin Proceedings of the 10th International Multi-Conference onSystems Signals and Devices (SSD rsquo13) pp 1ndash5 March 2013

[26] E Kolsi Gdoura M Feki and N Derbel ldquoControl of ahydraulic servo system using sliding mode with an integral andrealizable reference compensationrdquo Control Engineering andApplied Informatics vol 17 pp 111ndash119 2015

[27] A Levant ldquoSliding order and sliding accuracy in sliding modecontrolrdquo International Journal of Control vol 58 no 6 pp 1247ndash1263 1993

[28] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[29] M Djemai K Busawon K Benmansour and A MaroufldquoHigh-order sliding mode control of a DC motor drive viaa switched controlled multi-cellular converterrdquo InternationalJournal of Systems Science vol 42 no 11 pp 1869ndash1882 2011

[30] M Lavanya R M Brisilla and V Sankaranarayanan ldquoHigherorder sliding mode control of permanent magnet dc motorrdquoin Proceedings of the 12th International Workshop on VariableStructure Systems (VSS rsquo12) pp 226ndash230 January 2012

[31] S Laghrouche M Smaoui F Plestan and X Brun ldquoHigherorder sliding mode control based on optimal approach of anelectropneumatic actuatorrdquo International Journal of Control vol79 no 2 pp 119ndash131 2006

[32] A G Loukianov E Sanchez and C Lizalde ldquoForce trackingneural block control for an electro-hydraulic actuator viasecond-order slidingmoderdquo International Journal of Robust andNonlinear Control vol 18 no 3 pp 319ndash332 2008

[33] J Komsta N van Oijen and P Antoszkiewicz ldquoIntegral slidingmode compensator for load pressure control of die-cushioncylinder driverdquo Control Engineering Practice vol 21 no 5 pp708ndash718 2013

[34] M Feki Synthese de commandes et drsquoobservateurs pour lessystemes non-lineaires application aux systemes hydrauliques[PhD thesis] Universite de Metz Metz France 2001

[35] V V Filippov ldquoOn a scalar equation with discontinuous right-hand side and the uniqueness theoremrdquo Differential Equationsvol 38 no 10 pp 1435ndash1445 2002

[36] T Boukhobza M Djemai and J P Barbot ldquoNonlinear slidingobserver for systems in output and output derivative injectionformrdquo in Proceedings of the IFAC World Congress pp 299ndash305San Antonio Tex USA 1996

[37] H Saadaoui N Manamanni M Djemaı J P Barbot andT Floquet ldquoExact differentiation and sliding mode observersfor switched Lagrangian systemsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 65 no 5 pp 1050ndash1069 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Stochastic AnalysisInternational Journal of