Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 343851 9 pageshttpdxdoiorg1011552013343851

Research ArticleAdaptive Sliding Mode Robust Control for VirtualCompound-Axis Servo System

Yan Ren12 Zhenghua Liu2 Le Chang2 and Nuan Wen2

1 Information Engineering School Inner Mongolia University of Science and Technology Baotou 014010 China2 School of Automation Science and Electrical Engineering Beihang University Beijing 100191 China

Correspondence should be addressed to Yan Ren renyanry163com

Received 26 July 2013 Revised 12 October 2013 Accepted 20 October 2013

Academic Editor Xudong Zhao

Copyright copy 2013 Yan Ren et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A structure mode of virtual compound-axis servo system is proposed to improve the tracking accuracy of the ordinary optoelectrictracking platform It is based on the structure and principles of compound-axis servo system A hybrid position control schemecombining the PD controller and feed-forward controller is used in subsystem to track the tracking error of the main systemThispaper analyzes the influences of the equivalent disturbance in main system and proposes an adaptive sliding mode robust controlmethod based on the improved disturbance observer The sliding mode technique helps this disturbance observer to deal with theuncompensated disturbance in high frequency by making use of the rapid switching control value which is based on the subtleerror of disturbance estimation Besides the high-frequency chattering is alleviated effectively in this proposalThe effectiveness ofthe proposal is confirmed by experiments on optoelectric tracking platform

1 Introduction

Optoelectric tracking (OET) platform is applied to ensurethe stability of line of sight (LOS) and achieve the automatictracking of the maneuvering target under the maneuvermotion of carrier and the external disturbance When thetarget distance is farther the small deviation will lead to alarge change of the target position Accordingly the hightracking accuracy is very important in OET system Thecompound-axis system is a form of dual-dimension asso-ciation control system in the multivariable control systemsIt is demonstrated effectively to improve the accuracy andbandwidth of the OET system by inserting a fast steeringmirror (FSM) with the high resonant frequency in the largeinertia tracking frame of themain optical path [1] It has beenwidely applied in high-precision OET systems satellite laserranging laser communications and space remote sensingdetection and so forth [2ndash5] But there are some ordinaryOET systems without the FSM Therefore it is valuableto research how to improve the tracking accuracy of theordinary OET system

High-precision motion control is the key of OET devicewhich directly influences the accuracy of LOS tracking

As a typical kind of servo motor system the robustnessagainst system uncertainties and external disturbances isurgently required without considering mechanical factorsThe disturbance observer (DOB) approach has been widelyused as an effective robust method to compensate the distur-bance and parameter variations from both environment andsystem [6 7] But DOB only deals with the disturbances inthe low-frequency domain High-frequency components ofthe disturbances such as sudden changes in external forcesand Coulomb friction can degrade the control effect of aDOB based tracking control Therefore some researchershave tried to design the fuzzy disturbance observer [8 9]nonlinear disturbance observer [10 11] and extended stateobserver [12 13]

As one of the most significant approaches sliding modecontrol (SMC) has been proven to be an effective robust con-trol strategy for the systems with large uncertainties nonlin-earities and bounded external disturbances Consequentlysome researchers have actively developed and researchedSMC which is used in uncertain systems [14 15] time-delay systems [16 17] fuzzy systems [18 19] and Markovianjump systems [20 21] Besides a sliding mode disturbanceobserver (SDOB) was employed in [22] It can deal with

2 Mathematical Problems in Engineering

high-frequency disturbance through selecting appropriateswitching control value which is greater than the upperbound of the disturbance Meanwhile the switching controlvalue also causes chattering phenomenon For alleviatingchattering [23] proposed an SDOB with an adaptive law thatrequires only a small switching gain However this methodneeds a model of unknown disturbance which is difficult toobtain in engineering practice As a result some intelligentmethods have been devoted to estimate the upper bound [2425] However most of these intelligent units are not realizedeasily in engineering practice and they are not adequatelysensitive to the chattering of control input

In this paper the characteristics and structure com-pound-axis system are analyzed at first For improving thetracking accuracy of the ordinary OET servo system withoutthe FSM the design scheme and the realization means ofvirtual compound-axis are presented In this method theadaptive slidingmode controller (ASMC) is designed inmainsystem to reduce the tracking error and deal with high-frequency disturbance by making use of its switching controlvalue and virtual subsystem employs a hybrid control schemecombining the PD controller (PDC) and feed-forward con-troller (FFC) simultaneously The improved DOB compen-sates disturbance and helps to acquire a small switchinggain of ASMC to alleviate the chattering Experiments areimplemented to confirm the validity of this method

The rest of the paper is organized as follows Section 2introduces the OET servo system including compound-axissystem and virtual compound-axis system Section 3 analyzesthe compound control scheme based on virtual compound-axis servo system Then the structure and design of DOBbased on ASMC are proposed Experimental results areshown in Section 4 Finally the conclusion and future workare given in Section 5

2 OET Servo System

21 Compound-Axis OET Servo System The compound-axisservo system improves the tracking accuracy of optoelectricaltracking system greatly which is an effective method to lockthe beam and target on one point A typical compound-axis control system uses double detectors structure includ-ing the main-axis system and subaxis system The mainsystem mainly realizes the coarse tracking and its error isthe subsystem input The subsystem is applied to adjustthe tracking residuals of the main system to realize high-precision tracking

22 Virtual Compound-Axis OET Servo System Accordingto compound-axis control principle the conception of thevirtual compound-axis servo system is put forward in OETsystem without the FSM To simplify the problem this paperanalyses a uniaxial platform as an example at firstWe assumea coaxial virtual platform that has the same position withthe physical platform and a virtual platform owns the virtualtracking detector Virtual platform connects to the physicalplatform which connects to foundation bed The spatial

Virtual detector line of sightTracking detector line of sight

Tracking platform

Foundation bed

Virtual platform

120579ref

120579n e

e

m

Δe

120579

Target

Figure 1 Virtual compound-axis servo system

120579ref

120579n

n

minus

minus Virtualtrackingdetector

Tracking detector 120579p

120579

eminus120591s

K2

K1

P2

P1

Figure 2 Structure diagramof virtual compound-axis servo system

position is overlapped between the tracking detector andvirtual tracking detector

The relationship of all the parts of the virtual compound-axis servo system is shown in Figure 1 where 119890 denotes theLOS deviation of the tracking detector and the target Δ119890

denotes the LOS deviation of the virtual tracking detectorand the target and 119890

119898denotes the LOS deviation of the

tracking detector and the virtual tracking detector Thetracking platform tracks the target roughly while the virtualplatform tracks 119890 finely Therefore the cooperated trackingmode which is similar to the compound-axis servo system isrealized through the combination of main axis and subaxisThe tracking platform is in motion relative to foundation bedand the virtual platform is in motion relative to the trackingplatform in motion

The structure of the virtual compound-axis system isshown in Figure 2 where 119870

1(119904) 119870

2(119904) 119875

1(119904) and 119875

2(119904)

denote the tracking loop controller virtual loop controllerthe equivalent characteristic of the tracking platform and theequivalent characteristic of the virtual platform respectively119890minus120591119904 is the tracking detector delay 119899 is the measuring noise

120579ref is the maneuvering target position in inertia space 120579 isthe LOS position of the tracking platform 120579

119901is the LOS of

the tracking detector and 120579119899is the LOS of the virtual tracking

detectorThe delay factor only affects the phase-frequency char-

acteristic of the system but does not affect the gain of thesystem In order to simplify the analysis the delay factorand the transfer function of the virtual tracking detector are

Mathematical Problems in Engineering 3

ignoredThen the closed-loop transfer function of the systemis expressed as

119866 (119904)

=

1198701(119904) 1198751(119904) + 119870

2(119904) 1198752(119904) + 119870

1(119904) 1198751(119904) 1198702(119904) 1198752(119904)

[1 + 1198701(119904) 1198751(119904)] [1 + 119870

2(119904) 1198752(119904)]

(1)

The transfer functions of main system and subsystem aredescribed as follows respectively

1198661(119904) =

1198701(119904) 1198751(119904)

1 + 1198701(119904) 1198751(119904)

1198662(119904) =

1198702(119904) 1198752(119904)

1 + 1198702(119904) 1198752(119904)

(2)

From (1)-(2) the poles of the entire system are the polesof the main system and subsystem Only if the main systemand subsystem are stable the entire system is stable

Suppose 1198901(119904) as the error of the main system and 119890

2(119904)

as the error of the subsystem The error of subsystem can beobtained as

1198902(119904) =

1198901(119904)

1 + 1198702(119904) 1198752(119904)

=

120579ref (119904)

[1 + 1198701(119904) 1198751(119904)] [1 + 119870

2(119904) 1198752(119904)]

(3)

From (3) the tracking accuracy of the virtual compound-axis system can be improved by controlling the main systemand subsystem And the error of the system is the errorof subsystem Therefore the virtual compound-axis systemowns the high tracking accuracy

23 Virtual Compound-Axis Servo System ImplementationTwo problems should be solved as virtual compound-axisservo system

(1) problem of virtual objects property namely howto determine the transfer function of the virtualplatform

(2) problem of implementing the virtual compound-axisservo system namely how to achieve the coarse-finetracking mode on a single tracking platform

In order to implement virtual compound-axis servosystem its control structure diagram is shown in Figure 3where the delay of the tracking detector and virtual detectoris ignored In Figure 3 main system is tracking platformsubsystem is virtual tracking platform and 119890 is the inputsignal of the subsystem In order to simplify the design thetransfer function of virtual platform uses the nominal modelof the practical system As the tracking accuracy of virtualplatform is better than that of the tracking platform thetracking platform tracks virtual platform to decrease 119890

119898for

getting high tracking accuracy In Figure 3 1198701and 119870

2are

the position controllers of the main system and subsystem

P2 nominalplant

P1 plant

Subsystem

DOB120579ref minus

minus

e

Main system

K1 adaptivesliding mode

control

K2 PD+feed-forward

control

un(t)

uc(t)

d

u

em

120579n

120579

minus

Figure 3 Control realization diagram of virtual compound-axisservo system

In order to get the satisfactory control effect the ASMC isused in themain system and PDC and FFC are adopted in thesystem The output of PDC and FFC which can be regardedas a control component of the main system provides controlvariable to the actual plant

3 Compound Control Scheme of VirtualCompound-Axis Servo System

31 Controller Design of Subsystem Because a LOS stabi-lization servo system is driven by a DC torque motor thedynamics of the plant can be described as

119869

120579 + 119861

120579 = 119906 + 119891 (sdot) (4)

where 119869 119861 120579 and 119906 denote the inertia mass the damping theangular position response and the control input respectively119891(sdot) denotes the disturbance such as nonlinear friction theforce from the environment the carrier disturbance andunknown time-varying and nonlinear dynamics which isdifficult to model Consider that 119869

119898le 119869 le 119869

119872 119861119898

le 119861 le

119861119872

is satisfied where 119869119898 119869119872 119861119898 and 119861

119872are positive real

numberBy assembling the parameter mismatch external distur-

bance and unknown dynamics into an equivalent distur-bance 119889 (4) can be written as

119869119899

120579 + 119861119899

120579 = 119906 + 119889 (5)

where 119869119899and 119861

119899denote the nominal mass and the nominal

damping respectivelyThe equivalent disturbance is given by119889 = (119869

119899minus 119869)

120579 + (119861

119899minus 119861)

120579 + 119891(sdot)

Define a tracking error of the system as 119890 = 120579ref minus 120579where 120579ref is a given reference signal Using the structure ofvirtual compound-axis for system (5) the control value 119906 canbe described as

119906 = 119906119899+ 119906119888minus

119889 (6)

where 119906119899and 119906

119888are the control value of subsystem and main

system respectively 119889 is the estimated disturbance through

DOBWhen the equivalent disturbance and parameters meetrequirements the task is to design 119906

119888and 119906

119899to make 119890 rarr 0

4 Mathematical Problems in Engineering

wu

d P1 plant

P(s)

dDOBQ(s)

120585

un(t)

uc(t)

minus

minus +

+ ++

++

1205791

s

Pminus1n (s)

Figure 4 Basic diagram of DOB

The subsystem nominal model is described as follows

119869119899

120579 + 119861119899

120579 = 119906119899 (7)

The controller of subsystem uses a hybrid control schemecombining PDC and FFC which can be described as

119906119875119863

= 119896119875119890 + 119896119863

119890

119906FFC = 119869119899

120579ref + 119861

119899

120579ref

(8)

where 119896119875and 119896

119863denote proportional coefficient and differ-

ential coefficient respectivelyAn approximate method for difference to estimate the

reference values of velocity and acceleration is introducedThis method is expressed by [26]

120579ref =

119892119904

119904 + 119892

120579ref

120579ref = (

119892119904

119904 + 119892

)

2

120579ref =119892119904

119904 + 119892

120579ref

(9)

where 119904 is used to concatenate one low-pass filter whose cut-off frequency is 119892 gt 0

Thence the output of controller is written as (8)

119906119899= 119906119875119863

+ 119906FFC = 119896119875119890 + 119896119863

119890 + 119869119899

120579ref + 119861

119899

120579ref (10)

32 Disturbance Observer-BasedMain System Control In theOET system nonlinear dynamic and uncertain elements aredifficult to compensate by accurate model Robust closed-loop control method based on DOB which is widely appliedin the high-precision servo system has simple design processIt can inhibit the variety of external disturbances and param-eters effectively

In order to realize disturbances suppression the equalcompensation is introduced into control input by means ofthe estimation of the DOB improved in this paper The basicidea of DOB is shown in Figure 4 In Figure 4 119875(119904) 119875

119899(119904)

and 119876(119904) represent the velocity model of the actual plantthe nominal velocity model a filter respectively 119904 meansLaplace operator 119908 120585 and

119889 are the actual velocity outputthe measurement noise and the estimated disturbancesrespectively Let 119906 119889 and 120585 be system input The velocityresponse can be acquired on the basis of superpositionprinciple

119908 (119904) = 119866119880119882

(119904) 119906 (119904) + 119866119863119882

(119904) 119889 (119904) + 119866120585119882

(119904) 120585 (119904) (11)

where

119866119880119882

(119904) =

119875 (119904) 119875119899(119904)

119875119899(119904) + [119875 (119904) minus 119875

119899(119904)] 119876 (119904)

119866119863119882

(119904) =

119875 (119904) 119875119899(119904) [1 minus 119876 (119904)]

119875119899(119904) + [119875 (119904) minus 119875

119899(119904)] 119876 (119904)

119866120585119882

(119904) =

119875 (119904) 119876 (119904)

119875119899(119904) + [119875 (119904) minus 119875

119899(119904)] 119876 (119904)

(12)

Assume that the bandwidth of ideal filter 119876(119904) is 1198910 At

low frequencies there is 119876(119904) asymp 1 when frequency 119891 le 1198910

therefore 119866119880119882

(119904) asymp 119875119899(119904) 119866

119863119882(119904) asymp 0 and 119866

120585119882(119904) asymp 1

This means that DOB makes the characteristic of the actualplant approximately the same as that of the nominal model inlow-frequency domain Therefore the system has powerfulinhibiting effect against external disturbances DOB is verysensitive to low-frequency noise In practical applications itis necessary to consider that appropriate measures are takento reduce the low-frequency noise in the measurement of themotion state

In high-frequency domain if frequency 119891 gt 1198910 then

119876(119904) asymp 0 Consequently 119866119880119882

(119904) = 119866119863119882

(119904) = 119875(119904) and119866120585119882

(119904) = 0 This means that DOB has no inhibiting effectson external disturbances while having great inhibiting abilityagainst the high-frequency measurement noise

The simplified nominal velocity model is procured asfollows

119875119899(119904) =

1

119869119899119904 + 119861119899

(13)

119876(119904) design is significant in DOB design The designmethod of low-pass filter119876(119904)was proposed [27]The relativedegree of 119876(119904) must be equal to or greater than that of119875minus1

119899(119904) in order to make119876(119904)119875

minus1

119899(119904) regular According to the

nominal model shown in (13) the following is chosen for thisresearch which satisfies the property stated above

119876 (119904) =

119892

119904 + 119892

(14)

where 119892 is the cut-off frequency of the low-pass filter (LPF)in DOB Then the estimated disturbance is described as

119889 = 119876 (119904) 119889 =

119892

119904 + 119892

119889 (15)

Taking into account the physical realization of 119869119899119904

119889 isrepresented as follows

119889 = 119876 (119904) [(119869

119899119904 + 119861119899) 119908 minus 119906]

=

119892

119904 + 119892

[(119869119899119904 + 119861119899) 119908 minus 119906]

= 119869119899119892119908 minus

119892

119904 + 119892

[(119869119899119892 minus 119869119899119861119899) 119908 + 119906]

(16)

Therefore the improved DOB can be obtained as shownin Figure 5 The improved DOB has only one differentialelement simple structure and small calculation

Mathematical Problems in Engineering 5

wu

dun(t)

uc(t)

d

120579

Improved DOBJng

1

Js + B

Jng minus JnBn

1

s

P1 plant

minus

minus

minus

+

+

+

+

s

g

Figure 5 Structural diagram of improved DOB

The nonlinear function 119891(sdot) can be modeled by theimproved DOB system as

119891 (sdot) =119889 + 120575 (17)

where 120575 is the approximation error of the DOB Considering120575 is a bounded variable 120593 is defined to meet

|120575| lt 120593 (18)

where 120593 gt 0 120593 = 120593 minus 120593 and = minus

120593 are introduced

to assist in estimating the switching gain of the sliding modecontroller which is designed in Section 33

33 Adaptive Sliding Mode Controller Based Main SystemDesign Define the tracking error of model as 119890

119898= 120579 minus 120579

119899

where 120579119899is an output value of subsystem Define a sliding

mode 119911 as follows

119911 = 119890119898

+ 120582119890119898 120582 =

119861119899

119869119899

(19)

Then 119906119888is designed as

119906119888= minus119870119911 minus ℎ sdot sgn (119911) minus

119869119899minus 119869

119869119899

119906119899+ 119861

120579 minus 120582119869

120579 (20)

where 119870 is a positive constant gain and ℎ is a positiveswitching gain 119869 and 119861 are the estimation values of 119869 and119861 respectively sgn(119911) is a sign function that is defined asfollows

sgn (119911) =

1 119911 gt 0

0 119911 = 0

minus1 119911 lt 0

(21)

The adaptive law is adopted

= Proj

119869(minus1199031(

1

119869119899

119906119899minus 120582

120579) 119911)

= Proj

119861(minus1199032

120579119911)

= Proj

120593[1199033119911 sgn (119911)]

ℎ = 120593

(22)

where 1199031 1199032 and 119903

3are positive real constants The function

Proj∙(V) is defined as

Proj∙(V) =

V ∙119898

lt ∙ lt ∙119872

0 ∙ = ∙119872

and V gt 0

0 ∙ = ∙119898and V lt 0

(23)

where ∙119898is the minimum value of ∙ and ∙

119872is the maximum

value of ∙

Theorem 1 For system (4) if the condition |120575| lt 120593 is satisfied119911 exponentially decays to zero and 119890

119898asymptotically decays to

zero using the control law as (10) (15) (20) and (22) The statevariables of the system are bounded meanwhile

Proof Define a positive-definite Lyapunov candidate

119881 (119911) =

1

2

1198691199112

+

1

21199031

1198692

+

1

21199032

1198612

+

1

21199033

1205932

(24)

119869 = 119869 [(

120579 minus

120579119899) + 120582 (

120579 minus

120579119899)]

= (119869

120579 + 119861

120579) minus

119869

119869119899

(119869119899

120579119899+ 119861119899

120579119899) minus 119861

120579 + 120582119869

120579

= 119906119888+ 119891 (sdot) minus

119889 + 119906119899minus

119869

119869119899

119906119899minus 119861

120579 + 120582119869

120579

= 119906119888+ 120575 +

119869119899minus 119869

119869119899

119906119899minus 119861

120579 + 120582119869

120579

(25)

Define 119869 = 119869 minus 119869 119861 = 119861 minus 119861 then = minus

= minus

Substitute (20) into (25)

119869 = minus 119870119911 minus ℎ sdot sgn (119911) minus

119869119899minus 119869

119869119899

119906119899

+ 119861

120579 minus 120582119869

120579 +

119869119899minus 119869

119869119899

119906119899minus 119861

120579 + 120582119869

120579 + 120575

= minus 119870119911 minus ℎ sdot sgn (119911) + (119869 minus 119869) (

1

119869119899

119906119899minus 120582

120579)

+ (119861 minus 119861)

120579 + 120575

= minus 119870119911 minus ℎ sdot sgn (119911) minus 119869 (

1

119869119899

119906119899minus 120582

120579) minus 119861

120579 + 120575

(119911) = minus 1198701199112

minus ℎ sdot 119911 sgn (119911) minus 119869119911 (

1

119869119899

119906119899minus 120582

120579)

minus 119861119911

120579 minus

1

1199031

119869

minus

1

1199032

119861

minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus ℎ sdot 119911 sgn (119911) minus 119869 [119911(

1

119869119899

119906119899minus 120582

120579) +

1

1199031

]

minus 119861(119911

120579 +

1

1199032

) minus

1

1199033

120593

+ 119911120575

(26)

6 Mathematical Problems in Engineering

According to (22) the time derivative of Lyapunovfunction becomes

(119911) = minus 1198701199112

minus 120593 sdot 119911 sgn (119911) minus |119911| 120593 + 119911120575

le minus 1198701199112

minus 120593 |119911| + |119911| |120575|

= minus 1198701199112

minus |119911| (120593 minus |120575|)

(27)

Since |120575| lt 120593 then |119911|(120593 minus |120575|) gt 0 Therefore (119911) le minus1198961199112 is

metAccording to (24) there is

1199112

(119905) le 119911(0)2 exp(minus

119870

119869

119905) (28)

which implies that 119911 exponentially decays to zero Thenaccording to the definition of 119911 = 119890

119898+120582119890119898 119890119898asymptotically

decays to zero If 119890119898(0) and 119890

119898(0) are bounded then 119890

119898(119905)

and 119890119898(119905) are bounded to arbitrary 119905 gt 0 because of 119911(119905)

being uniformly bounded If 120579119899(0) and

120579119899(0) are bounded

then 120579119899(119905) and

120579119899(119905) are bounded According to 119890

119898= 120579 minus 120579

119899

120579 and 120579 are bounded Consequently the state variables of the

system are boundedTheorem 1 is proved

In this method the switching gain ℎ is only requiredto be greater than |120575| which is a small variable Thisdesign alleviates the chattering phenomenon of sliding modecontroller In engineering practice since sign function sgn(sdot)in control law (20) can cause the frequent switching of thecontrol variable and result in the output chattering it is easyto damage the power amplifier In order to avoid frequentswitching of the control output the sgn(sdot) is replaced by thesaturation function (29) to weaken the chattering further

sat(119909

Δ

) =

1 119909 gt Δ

1

Δ

119909 |119909| le Δ

minus1 119909 lt minusΔ

(29)

where Δ is positive real constant namely the width of aboundary layer

Introducing sat(sdot) control law (20) becomes

119906119888= minus119870119911 minus ℎ sdot sat(ℎ119911

4120576

) minus

119869119899minus 119869

119869119899

119906119899+ 119861

120579 minus 120582119869

120579 (30)

where 120576 is a small positive constant Then there is thefollowingTheorem 2

Theorem 2 For system (4) if the condition |120575| lt 120593 is satisfiedand the control law as (10) (15) (22) and (30) is adopted then

(1) 119911 exponentially decays to zero and lim119905rarrinfin

|119911(119905)| le

radic120576119870 is met(2) 119890119898asymptotically decays to zero and lim

119905rarrinfin|119890119898(119905)| le

(1120582)radic120576119870 is met(3) 119890119898(119905) asymptotically converges and lim

119905rarrinfin| 119890119898(119905)| le

2radic120576119870 is met

Proof Select the positive-definite function119881(119911) as (24) thenits time derivative is calculated as

(119911) = minus 1198701199112

minus ℎ sdot 119911 sat(ℎ119911

4120576

) minus 119869119911(

1

119869119899

119906119899minus 120582

120579)

minus 119861119911

120579 minus

1

1199031

119869

minus

1

1199032

119861

minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus ℎ sdot 119911 sat(ℎ119911

4120576

) minus 119869 [119911(

1

119869119899

119906119899minus 120582

120579) +

1

1199031

]

minus 119861(119911

120579 +

1

1199032

) minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus 120593119911 sat(ℎ119911

4120576

) minus 120593119911 sat(ℎ119911

4120576

) + |119911| 120575

= minus 1198701199112

minus 120593119911 sat(ℎ119911

4120576

) + |119911| 120575

(31)

Since |119911| ge 4120576ℎ sat(ℎ1199114120576) = sgn(119911) According to theproof of Theorem 1 when 120593 ge |120575| (119911) le minus119896119911

2 is metTherefore 119911 exponentially converges until

lim119905rarrinfin

|119911 (119905)| lt

4120576

ℎ

(32)

If |119911| lt 4120576ℎ then sat(ℎ1199114120576) = ℎ1199114120576 (119911) is describedas

(119911) le minus 1198701199112

minus 120593

ℎ|119911|2

4120576

+ |119911| |120575|

le minus 1198701199112

minus

1205932

|119911|2

4120576

+ 120593 |119911|

le minus 1198701199112

minus

1

120576

(

1

2

120593 |119911| minus 120576)

2

+ 120576

le minus 1198701199112

+ 120576 le minus

2119870

119869

119881 + 120576

(33)

Consequently

119881 (119905) le expminus(2119870119869)119905119881 (0) +

119869120576

2119870

(1 minus expminus(2119870119869)119905) (34)

From (34) yield to

lim119905rarrinfin

|119911 (119905)| le radic

120576

119870

(35)

It is similar to Theorem 1 that the state variables arebounded in this system From (35) 119911 asymptotically con-verges to the bounded region According to (19) 119890

119898(119905) and

119890119898(119905) are satisfied as follows

lim119905rarrinfin

1003816100381610038161003816119890119898

(119905)1003816100381610038161003816le

1

120582

radic

120576

119870

lim119905rarrinfin

1003816100381610038161003816119890119898

(119905)1003816100381610038161003816le 2radic

120576

119870

(36)

Mathematical Problems in Engineering 7Ph

ase (

deg)

Log

mag

nitu

de

Actual plantNominal plant

100 101 102 103 104

102

100

10minus2

10minus4

minus400minus300minus200minus100

0

Frequency (rads)

100 101 102 103 104

Frequency (rads)

Figure 6 Fitting curves for frequency characteristics of actual plantand nominal model

which means that 119890119898(119905) and 119890

119898(119905) asymptotically converge to

the bounded regionTheorem 2 is proved

4 Experiment Results

This section investigates the feasibility and effectiveness ofthe proposed compound controller based on the structureof the virtual compound-axis servo system by experimentsThe tracking experiments are implemented through usingsome type of OET platform The position sensor of thedevice uses the optical-electrical encoder with resolution of00007 degrees The control algorithm program is writtenwith C language based on Windows-RTX real-time systemin an industrial computer which adopts Advantech IPC610 and connects with the servo drivers by a 16-bit DAconvertor of PCI bus The control cycle is 0001 s It is worthmentioning that the adaptive online adjustment will consumelots of computational resource With the development of theadvanced technologies on computer the problem of lots ofcomputation has been solved The proposed adaptive slidingcontroller is proved to be feasible in a practical system by thisexperimental platform

Based on what has been mentioned above the pitch axisis chosen herein to verify the method because each axisof the OET platform can be designed independently Theparameters of the nominal model are identified as 119869

119899=

0000125 kg and 119861119899

= 0003125N sdot s sdot mminus1 by a white noisefrequency sweep method The fitting curves for frequencycharacteristics of actual plant and nominal model are shownin Figure 6

The experiments are separated into two parts First theexperiment is completed under the proposed compound con-trol scheme in which sign function uses saturation functionSecond in order to verify the effect of the proposedmethod acomparative experiment is achieved by the traditional controlschemewith PDC + FFC + DOBOther parameters are givenas follows 119896

119875= 045 119896

119863= 012 119870 = 004 119903

1= 001 119903

2=

0001 1199033= 0005 and 120576 = 00002 Considering the modeling

mismatch and the robustness of the system the value ofparameter 119892 in improved DOB is chosen as 1000 rads

Figure 7 compares the tracking error and the controlvalue of the two control schemes when the reference inputsignal is a sinusoidal signal where the amplitude is 2 degreesand frequency is 05HzThe results indicate that the proposedcontrol scheme could achieve a better position tracking per-formance than traditional control laws a maximum trackingerror decreases from 0052 degrees to 0023 degrees withthe help of the ASMC ASMC achieves the adjustment to119890119898

better From Figure 7(a) we can see that the trackingerror reaches the maximum value at zero velocity and thetracking error of the proposed control scheme is smallerCompared with traditional control scheme the compoundcontrol scheme based on the virtual compound-axis systemhas better tracking performance and robustness The distur-bance is sufficiently compensated by using improved DOBwhich simplifies design process and decreases calculationtime Meanwhile the accuracy of the system is guaranteedby introducing ASMC to adjust 119890

119898again whose gain is

greatly related to the upper bound on the error of thedisturbance estimation and the reduction of switching gainhelps to alleviate the chattering Figure 7(b) illustrates that thecontroller output curve of the compound control scheme issmoother than that of traditional control scheme

In order to test the dynamic performance of the trackingobject further the tracking command signal is employed assin(2120587 sdot 3119905) Figure 8(a) shows that the maximum trackingerror decreases from 014 degrees to 006 degrees underthe reference signal Compared to the traditional controlscheme the systemwith compound control schemehas betterdynamic performanceThe testing standard of OET platformstipulates that the maximum tracking error cannot exceed10 of the amplitude of the reference signal in working bandObviously the tracking errors of Figures 7(a) and 8(a) meetthis standard under the proposed method Because it has aswitching control the adaptive sliding mode technique canreject the impact of nonlinear disturbance in high-frequencydomain which is not compensated by the DOB From thispoint of view the proposed control scheme enhances thesystem robustness Furthermore according to Figures 7(b)and 8(b) the control value of the proposed method is farless than the limit of the DA converter The control valuedoes not exhibit obvious chattering phenomenon becauseASMC with the improved DOB can alleviate the chatteringphenomenon Consequently the proposed scheme not onlyensures strong robustness against system uncertainties andsmall tracking error but also suppresses the high-frequencychattering at control input effectively The results indicatethat the compound controller can be reliably performed inpractical application

5 Conclusions

To obtain the high performance and good robustness forordinary OET system without the FSM the design methodof virtual compound-axis servo system is proposed in thispaperThe proposed compound control scheme based on the

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8Time (s)

minus006

minus004

minus002

0

002

004

006

Compound control schemeScheme with PDC + FFC + DOB

Trac

king

erro

r (∘)

(a) Comparison of tracking error curves

0 1 2 3 4 5 6 7 8Time (s)

Compound control schemeScheme with PDC + FFC + DOB

minus04minus03minus02minus01

001

020304

Con

trol v

alue

(V)

(b) Comparison of control value

Figure 7 Comparison curves under two schemes with sinusoidal input (119860 = 2∘ 119891 = 05Hz)

0 1 2 3 4 5 6 7 8 9 10minus02

minus01

0

01

02

Time (s)

Compound control schemeScheme with PDC + FFC + DOB

Trac

king

erro

r (∘)

(a) Comparison of tracking error curves

0 1 2 3 4 5 6 7 8 9 10minus03minus02minus01

001020304

Time (s)

Compound control schemeScheme with PDC + FFC + DOB

Con

trol v

alue

(V)

(b) Comparison of control value

Figure 8 Comparison curves under two schemes with sinusoidal input (119860 = 1∘ 119891 = 3Hz)

structure of the virtual compound-axis system can maintainthe robustness of OET system against various disturbancesin the whole control period and then the tracking errorof OET system can be limited within an expected levelin the existence of sustained uncertainties Although thevirtual compound-axis does not perform as well as the realcompound-axis it can also improve the tracking precisionof the OET system to some extent Therefore the proposedmethod can be used as a new control technology applied inOET system In addition because OET system is a typicalservo motion system the theoretic results are able to beextended to other relational fields

However the velocity and acceleration command signalin FFC cannot be measured directly and easily in practiceTherefore some more excellent differential methods shouldbe considered in further work Simultaneously in order totrack the maneuvering target with higher speed furtherstudies aimed at better performance at higher frequenciesare needed Besides with the development of advancedcomputer and information technologies massive amountof measurement data can be utilized to extract the usefulinformation about the current state of the OET process As

a result the excellent method of the process monitoring andfault diagnosis such as data-driven techniques [28] can beused as references in future work

Conflict of Interests

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

Acknowledgments

This work was supported by the National Basic Research Pro-gram (Grant no 2012CB821200) and the Scientific Researchfor Colleges and Universities of Inner Mongolia (Grant noNJZY13143) in China

References

[1] W Thanmas ldquoDigital laser ranging and tracking using acompound axis servomechanismrdquo Applied Optics vol 5 no 4pp 497ndash505 1966

Mathematical Problems in Engineering 9

[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008

[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)

[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004

[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007

[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007

[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010

[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002

[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013

[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008

[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004

[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012

[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012

[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013

[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004

[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008

[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010

[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007

[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012

[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)

[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009

[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995

[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012

[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011

[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994

[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012

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

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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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

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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

high-frequency disturbance through selecting appropriateswitching control value which is greater than the upperbound of the disturbance Meanwhile the switching controlvalue also causes chattering phenomenon For alleviatingchattering [23] proposed an SDOB with an adaptive law thatrequires only a small switching gain However this methodneeds a model of unknown disturbance which is difficult toobtain in engineering practice As a result some intelligentmethods have been devoted to estimate the upper bound [2425] However most of these intelligent units are not realizedeasily in engineering practice and they are not adequatelysensitive to the chattering of control input

In this paper the characteristics and structure com-pound-axis system are analyzed at first For improving thetracking accuracy of the ordinary OET servo system withoutthe FSM the design scheme and the realization means ofvirtual compound-axis are presented In this method theadaptive slidingmode controller (ASMC) is designed inmainsystem to reduce the tracking error and deal with high-frequency disturbance by making use of its switching controlvalue and virtual subsystem employs a hybrid control schemecombining the PD controller (PDC) and feed-forward con-troller (FFC) simultaneously The improved DOB compen-sates disturbance and helps to acquire a small switchinggain of ASMC to alleviate the chattering Experiments areimplemented to confirm the validity of this method

The rest of the paper is organized as follows Section 2introduces the OET servo system including compound-axissystem and virtual compound-axis system Section 3 analyzesthe compound control scheme based on virtual compound-axis servo system Then the structure and design of DOBbased on ASMC are proposed Experimental results areshown in Section 4 Finally the conclusion and future workare given in Section 5

2 OET Servo System

21 Compound-Axis OET Servo System The compound-axisservo system improves the tracking accuracy of optoelectricaltracking system greatly which is an effective method to lockthe beam and target on one point A typical compound-axis control system uses double detectors structure includ-ing the main-axis system and subaxis system The mainsystem mainly realizes the coarse tracking and its error isthe subsystem input The subsystem is applied to adjustthe tracking residuals of the main system to realize high-precision tracking

22 Virtual Compound-Axis OET Servo System Accordingto compound-axis control principle the conception of thevirtual compound-axis servo system is put forward in OETsystem without the FSM To simplify the problem this paperanalyses a uniaxial platform as an example at firstWe assumea coaxial virtual platform that has the same position withthe physical platform and a virtual platform owns the virtualtracking detector Virtual platform connects to the physicalplatform which connects to foundation bed The spatial

Virtual detector line of sightTracking detector line of sight

Tracking platform

Foundation bed

Virtual platform

120579ref

120579n e

e

m

Δe

120579

Target

Figure 1 Virtual compound-axis servo system

120579ref

120579n

n

minus

minus Virtualtrackingdetector

Tracking detector 120579p

120579

eminus120591s

K2

K1

P2

P1

Figure 2 Structure diagramof virtual compound-axis servo system

position is overlapped between the tracking detector andvirtual tracking detector

The relationship of all the parts of the virtual compound-axis servo system is shown in Figure 1 where 119890 denotes theLOS deviation of the tracking detector and the target Δ119890

denotes the LOS deviation of the virtual tracking detectorand the target and 119890

119898denotes the LOS deviation of the

tracking detector and the virtual tracking detector Thetracking platform tracks the target roughly while the virtualplatform tracks 119890 finely Therefore the cooperated trackingmode which is similar to the compound-axis servo system isrealized through the combination of main axis and subaxisThe tracking platform is in motion relative to foundation bedand the virtual platform is in motion relative to the trackingplatform in motion

The structure of the virtual compound-axis system isshown in Figure 2 where 119870

1(119904) 119870

2(119904) 119875

1(119904) and 119875

2(119904)

denote the tracking loop controller virtual loop controllerthe equivalent characteristic of the tracking platform and theequivalent characteristic of the virtual platform respectively119890minus120591119904 is the tracking detector delay 119899 is the measuring noise

120579ref is the maneuvering target position in inertia space 120579 isthe LOS position of the tracking platform 120579

119901is the LOS of

the tracking detector and 120579119899is the LOS of the virtual tracking

detectorThe delay factor only affects the phase-frequency char-

acteristic of the system but does not affect the gain of thesystem In order to simplify the analysis the delay factorand the transfer function of the virtual tracking detector are

Mathematical Problems in Engineering 3

ignoredThen the closed-loop transfer function of the systemis expressed as

119866 (119904)

=

1198701(119904) 1198751(119904) + 119870

2(119904) 1198752(119904) + 119870

1(119904) 1198751(119904) 1198702(119904) 1198752(119904)

[1 + 1198701(119904) 1198751(119904)] [1 + 119870

2(119904) 1198752(119904)]

(1)

The transfer functions of main system and subsystem aredescribed as follows respectively

1198661(119904) =

1198701(119904) 1198751(119904)

1 + 1198701(119904) 1198751(119904)

1198662(119904) =

1198702(119904) 1198752(119904)

1 + 1198702(119904) 1198752(119904)

(2)

From (1)-(2) the poles of the entire system are the polesof the main system and subsystem Only if the main systemand subsystem are stable the entire system is stable

Suppose 1198901(119904) as the error of the main system and 119890

2(119904)

as the error of the subsystem The error of subsystem can beobtained as

1198902(119904) =

1198901(119904)

1 + 1198702(119904) 1198752(119904)

=

120579ref (119904)

[1 + 1198701(119904) 1198751(119904)] [1 + 119870

2(119904) 1198752(119904)]

(3)

From (3) the tracking accuracy of the virtual compound-axis system can be improved by controlling the main systemand subsystem And the error of the system is the errorof subsystem Therefore the virtual compound-axis systemowns the high tracking accuracy

23 Virtual Compound-Axis Servo System ImplementationTwo problems should be solved as virtual compound-axisservo system

(1) problem of virtual objects property namely howto determine the transfer function of the virtualplatform

(2) problem of implementing the virtual compound-axisservo system namely how to achieve the coarse-finetracking mode on a single tracking platform

In order to implement virtual compound-axis servosystem its control structure diagram is shown in Figure 3where the delay of the tracking detector and virtual detectoris ignored In Figure 3 main system is tracking platformsubsystem is virtual tracking platform and 119890 is the inputsignal of the subsystem In order to simplify the design thetransfer function of virtual platform uses the nominal modelof the practical system As the tracking accuracy of virtualplatform is better than that of the tracking platform thetracking platform tracks virtual platform to decrease 119890

119898for

getting high tracking accuracy In Figure 3 1198701and 119870

2are

the position controllers of the main system and subsystem

P2 nominalplant

P1 plant

Subsystem

DOB120579ref minus

minus

e

Main system

K1 adaptivesliding mode

control

K2 PD+feed-forward

control

un(t)

uc(t)

d

u

em

120579n

120579

minus

Figure 3 Control realization diagram of virtual compound-axisservo system

In order to get the satisfactory control effect the ASMC isused in themain system and PDC and FFC are adopted in thesystem The output of PDC and FFC which can be regardedas a control component of the main system provides controlvariable to the actual plant

3 Compound Control Scheme of VirtualCompound-Axis Servo System

31 Controller Design of Subsystem Because a LOS stabi-lization servo system is driven by a DC torque motor thedynamics of the plant can be described as

119869

120579 + 119861

120579 = 119906 + 119891 (sdot) (4)

where 119869 119861 120579 and 119906 denote the inertia mass the damping theangular position response and the control input respectively119891(sdot) denotes the disturbance such as nonlinear friction theforce from the environment the carrier disturbance andunknown time-varying and nonlinear dynamics which isdifficult to model Consider that 119869

119898le 119869 le 119869

119872 119861119898

le 119861 le

119861119872

is satisfied where 119869119898 119869119872 119861119898 and 119861

119872are positive real

numberBy assembling the parameter mismatch external distur-

bance and unknown dynamics into an equivalent distur-bance 119889 (4) can be written as

119869119899

120579 + 119861119899

120579 = 119906 + 119889 (5)

where 119869119899and 119861

119899denote the nominal mass and the nominal

damping respectivelyThe equivalent disturbance is given by119889 = (119869

119899minus 119869)

120579 + (119861

119899minus 119861)

120579 + 119891(sdot)

Define a tracking error of the system as 119890 = 120579ref minus 120579where 120579ref is a given reference signal Using the structure ofvirtual compound-axis for system (5) the control value 119906 canbe described as

119906 = 119906119899+ 119906119888minus

119889 (6)

where 119906119899and 119906

119888are the control value of subsystem and main

system respectively 119889 is the estimated disturbance through

DOBWhen the equivalent disturbance and parameters meetrequirements the task is to design 119906

119888and 119906

119899to make 119890 rarr 0

4 Mathematical Problems in Engineering

wu

d P1 plant

P(s)

dDOBQ(s)

120585

un(t)

uc(t)

minus

minus +

+ ++

++

1205791

s

Pminus1n (s)

Figure 4 Basic diagram of DOB

The subsystem nominal model is described as follows

119869119899

120579 + 119861119899

120579 = 119906119899 (7)

The controller of subsystem uses a hybrid control schemecombining PDC and FFC which can be described as

119906119875119863

= 119896119875119890 + 119896119863

119890

119906FFC = 119869119899

120579ref + 119861

119899

120579ref

(8)

where 119896119875and 119896

119863denote proportional coefficient and differ-

ential coefficient respectivelyAn approximate method for difference to estimate the

reference values of velocity and acceleration is introducedThis method is expressed by [26]

120579ref =

119892119904

119904 + 119892

120579ref

120579ref = (

119892119904

119904 + 119892

)

2

120579ref =119892119904

119904 + 119892

120579ref

(9)

where 119904 is used to concatenate one low-pass filter whose cut-off frequency is 119892 gt 0

Thence the output of controller is written as (8)

119906119899= 119906119875119863

+ 119906FFC = 119896119875119890 + 119896119863

119890 + 119869119899

120579ref + 119861

119899

120579ref (10)

32 Disturbance Observer-BasedMain System Control In theOET system nonlinear dynamic and uncertain elements aredifficult to compensate by accurate model Robust closed-loop control method based on DOB which is widely appliedin the high-precision servo system has simple design processIt can inhibit the variety of external disturbances and param-eters effectively

In order to realize disturbances suppression the equalcompensation is introduced into control input by means ofthe estimation of the DOB improved in this paper The basicidea of DOB is shown in Figure 4 In Figure 4 119875(119904) 119875

119899(119904)

and 119876(119904) represent the velocity model of the actual plantthe nominal velocity model a filter respectively 119904 meansLaplace operator 119908 120585 and

119889 are the actual velocity outputthe measurement noise and the estimated disturbancesrespectively Let 119906 119889 and 120585 be system input The velocityresponse can be acquired on the basis of superpositionprinciple

119908 (119904) = 119866119880119882

(119904) 119906 (119904) + 119866119863119882

(119904) 119889 (119904) + 119866120585119882

(119904) 120585 (119904) (11)

where

119866119880119882

(119904) =

119875 (119904) 119875119899(119904)

119875119899(119904) + [119875 (119904) minus 119875

119899(119904)] 119876 (119904)

119866119863119882

(119904) =

119875 (119904) 119875119899(119904) [1 minus 119876 (119904)]

119875119899(119904) + [119875 (119904) minus 119875

119899(119904)] 119876 (119904)

119866120585119882

(119904) =

119875 (119904) 119876 (119904)

119875119899(119904) + [119875 (119904) minus 119875

119899(119904)] 119876 (119904)

(12)

Assume that the bandwidth of ideal filter 119876(119904) is 1198910 At

low frequencies there is 119876(119904) asymp 1 when frequency 119891 le 1198910

therefore 119866119880119882

(119904) asymp 119875119899(119904) 119866

119863119882(119904) asymp 0 and 119866

120585119882(119904) asymp 1

This means that DOB makes the characteristic of the actualplant approximately the same as that of the nominal model inlow-frequency domain Therefore the system has powerfulinhibiting effect against external disturbances DOB is verysensitive to low-frequency noise In practical applications itis necessary to consider that appropriate measures are takento reduce the low-frequency noise in the measurement of themotion state

In high-frequency domain if frequency 119891 gt 1198910 then

119876(119904) asymp 0 Consequently 119866119880119882

(119904) = 119866119863119882

(119904) = 119875(119904) and119866120585119882

(119904) = 0 This means that DOB has no inhibiting effectson external disturbances while having great inhibiting abilityagainst the high-frequency measurement noise

The simplified nominal velocity model is procured asfollows

119875119899(119904) =

1

119869119899119904 + 119861119899

(13)

119876(119904) design is significant in DOB design The designmethod of low-pass filter119876(119904)was proposed [27]The relativedegree of 119876(119904) must be equal to or greater than that of119875minus1

119899(119904) in order to make119876(119904)119875

minus1

119899(119904) regular According to the

nominal model shown in (13) the following is chosen for thisresearch which satisfies the property stated above

119876 (119904) =

119892

119904 + 119892

(14)

where 119892 is the cut-off frequency of the low-pass filter (LPF)in DOB Then the estimated disturbance is described as

119889 = 119876 (119904) 119889 =

119892

119904 + 119892

119889 (15)

Taking into account the physical realization of 119869119899119904

119889 isrepresented as follows

119889 = 119876 (119904) [(119869

119899119904 + 119861119899) 119908 minus 119906]

=

119892

119904 + 119892

[(119869119899119904 + 119861119899) 119908 minus 119906]

= 119869119899119892119908 minus

119892

119904 + 119892

[(119869119899119892 minus 119869119899119861119899) 119908 + 119906]

(16)

Therefore the improved DOB can be obtained as shownin Figure 5 The improved DOB has only one differentialelement simple structure and small calculation

Mathematical Problems in Engineering 5

wu

dun(t)

uc(t)

d

120579

Improved DOBJng

1

Js + B

Jng minus JnBn

1

s

P1 plant

minus

minus

minus

+

+

+

+

s

g

Figure 5 Structural diagram of improved DOB

The nonlinear function 119891(sdot) can be modeled by theimproved DOB system as

119891 (sdot) =119889 + 120575 (17)

where 120575 is the approximation error of the DOB Considering120575 is a bounded variable 120593 is defined to meet

|120575| lt 120593 (18)

where 120593 gt 0 120593 = 120593 minus 120593 and = minus

120593 are introduced

to assist in estimating the switching gain of the sliding modecontroller which is designed in Section 33

33 Adaptive Sliding Mode Controller Based Main SystemDesign Define the tracking error of model as 119890

119898= 120579 minus 120579

119899

where 120579119899is an output value of subsystem Define a sliding

mode 119911 as follows

119911 = 119890119898

+ 120582119890119898 120582 =

119861119899

119869119899

(19)

Then 119906119888is designed as

119906119888= minus119870119911 minus ℎ sdot sgn (119911) minus

119869119899minus 119869

119869119899

119906119899+ 119861

120579 minus 120582119869

120579 (20)

where 119870 is a positive constant gain and ℎ is a positiveswitching gain 119869 and 119861 are the estimation values of 119869 and119861 respectively sgn(119911) is a sign function that is defined asfollows

sgn (119911) =

1 119911 gt 0

0 119911 = 0

minus1 119911 lt 0

(21)

The adaptive law is adopted

= Proj

119869(minus1199031(

1

119869119899

119906119899minus 120582

120579) 119911)

= Proj

119861(minus1199032

120579119911)

= Proj

120593[1199033119911 sgn (119911)]

ℎ = 120593

(22)

where 1199031 1199032 and 119903

3are positive real constants The function

Proj∙(V) is defined as

Proj∙(V) =

V ∙119898

lt ∙ lt ∙119872

0 ∙ = ∙119872

and V gt 0

0 ∙ = ∙119898and V lt 0

(23)

where ∙119898is the minimum value of ∙ and ∙

119872is the maximum

value of ∙

Theorem 1 For system (4) if the condition |120575| lt 120593 is satisfied119911 exponentially decays to zero and 119890

119898asymptotically decays to

zero using the control law as (10) (15) (20) and (22) The statevariables of the system are bounded meanwhile

Proof Define a positive-definite Lyapunov candidate

119881 (119911) =

1

2

1198691199112

+

1

21199031

1198692

+

1

21199032

1198612

+

1

21199033

1205932

(24)

119869 = 119869 [(

120579 minus

120579119899) + 120582 (

120579 minus

120579119899)]

= (119869

120579 + 119861

120579) minus

119869

119869119899

(119869119899

120579119899+ 119861119899

120579119899) minus 119861

120579 + 120582119869

120579

= 119906119888+ 119891 (sdot) minus

119889 + 119906119899minus

119869

119869119899

119906119899minus 119861

120579 + 120582119869

120579

= 119906119888+ 120575 +

119869119899minus 119869

119869119899

119906119899minus 119861

120579 + 120582119869

120579

(25)

Define 119869 = 119869 minus 119869 119861 = 119861 minus 119861 then = minus

= minus

Substitute (20) into (25)

119869 = minus 119870119911 minus ℎ sdot sgn (119911) minus

119869119899minus 119869

119869119899

119906119899

+ 119861

120579 minus 120582119869

120579 +

119869119899minus 119869

119869119899

119906119899minus 119861

120579 + 120582119869

120579 + 120575

= minus 119870119911 minus ℎ sdot sgn (119911) + (119869 minus 119869) (

1

119869119899

119906119899minus 120582

120579)

+ (119861 minus 119861)

120579 + 120575

= minus 119870119911 minus ℎ sdot sgn (119911) minus 119869 (

1

119869119899

119906119899minus 120582

120579) minus 119861

120579 + 120575

(119911) = minus 1198701199112

minus ℎ sdot 119911 sgn (119911) minus 119869119911 (

1

119869119899

119906119899minus 120582

120579)

minus 119861119911

120579 minus

1

1199031

119869

minus

1

1199032

119861

minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus ℎ sdot 119911 sgn (119911) minus 119869 [119911(

1

119869119899

119906119899minus 120582

120579) +

1

1199031

]

minus 119861(119911

120579 +

1

1199032

) minus

1

1199033

120593

+ 119911120575

(26)

6 Mathematical Problems in Engineering

According to (22) the time derivative of Lyapunovfunction becomes

(119911) = minus 1198701199112

minus 120593 sdot 119911 sgn (119911) minus |119911| 120593 + 119911120575

le minus 1198701199112

minus 120593 |119911| + |119911| |120575|

= minus 1198701199112

minus |119911| (120593 minus |120575|)

(27)

Since |120575| lt 120593 then |119911|(120593 minus |120575|) gt 0 Therefore (119911) le minus1198961199112 is

metAccording to (24) there is

1199112

(119905) le 119911(0)2 exp(minus

119870

119869

119905) (28)

which implies that 119911 exponentially decays to zero Thenaccording to the definition of 119911 = 119890

119898+120582119890119898 119890119898asymptotically

decays to zero If 119890119898(0) and 119890

119898(0) are bounded then 119890

119898(119905)

and 119890119898(119905) are bounded to arbitrary 119905 gt 0 because of 119911(119905)

being uniformly bounded If 120579119899(0) and

120579119899(0) are bounded

then 120579119899(119905) and

120579119899(119905) are bounded According to 119890

119898= 120579 minus 120579

119899

120579 and 120579 are bounded Consequently the state variables of the

system are boundedTheorem 1 is proved

In this method the switching gain ℎ is only requiredto be greater than |120575| which is a small variable Thisdesign alleviates the chattering phenomenon of sliding modecontroller In engineering practice since sign function sgn(sdot)in control law (20) can cause the frequent switching of thecontrol variable and result in the output chattering it is easyto damage the power amplifier In order to avoid frequentswitching of the control output the sgn(sdot) is replaced by thesaturation function (29) to weaken the chattering further

sat(119909

Δ

) =

1 119909 gt Δ

1

Δ

119909 |119909| le Δ

minus1 119909 lt minusΔ

(29)

where Δ is positive real constant namely the width of aboundary layer

Introducing sat(sdot) control law (20) becomes

119906119888= minus119870119911 minus ℎ sdot sat(ℎ119911

4120576

) minus

119869119899minus 119869

119869119899

119906119899+ 119861

120579 minus 120582119869

120579 (30)

where 120576 is a small positive constant Then there is thefollowingTheorem 2

Theorem 2 For system (4) if the condition |120575| lt 120593 is satisfiedand the control law as (10) (15) (22) and (30) is adopted then

(1) 119911 exponentially decays to zero and lim119905rarrinfin

|119911(119905)| le

radic120576119870 is met(2) 119890119898asymptotically decays to zero and lim

119905rarrinfin|119890119898(119905)| le

(1120582)radic120576119870 is met(3) 119890119898(119905) asymptotically converges and lim

119905rarrinfin| 119890119898(119905)| le

2radic120576119870 is met

Proof Select the positive-definite function119881(119911) as (24) thenits time derivative is calculated as

(119911) = minus 1198701199112

minus ℎ sdot 119911 sat(ℎ119911

4120576

) minus 119869119911(

1

119869119899

119906119899minus 120582

120579)

minus 119861119911

120579 minus

1

1199031

119869

minus

1

1199032

119861

minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus ℎ sdot 119911 sat(ℎ119911

4120576

) minus 119869 [119911(

1

119869119899

119906119899minus 120582

120579) +

1

1199031

]

minus 119861(119911

120579 +

1

1199032

) minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus 120593119911 sat(ℎ119911

4120576

) minus 120593119911 sat(ℎ119911

4120576

) + |119911| 120575

= minus 1198701199112

minus 120593119911 sat(ℎ119911

4120576

) + |119911| 120575

(31)

Since |119911| ge 4120576ℎ sat(ℎ1199114120576) = sgn(119911) According to theproof of Theorem 1 when 120593 ge |120575| (119911) le minus119896119911

2 is metTherefore 119911 exponentially converges until

lim119905rarrinfin

|119911 (119905)| lt

4120576

ℎ

(32)

If |119911| lt 4120576ℎ then sat(ℎ1199114120576) = ℎ1199114120576 (119911) is describedas

(119911) le minus 1198701199112

minus 120593

ℎ|119911|2

4120576

+ |119911| |120575|

le minus 1198701199112

minus

1205932

|119911|2

4120576

+ 120593 |119911|

le minus 1198701199112

minus

1

120576

(

1

2

120593 |119911| minus 120576)

2

+ 120576

le minus 1198701199112

+ 120576 le minus

2119870

119869

119881 + 120576

(33)

Consequently

119881 (119905) le expminus(2119870119869)119905119881 (0) +

119869120576

2119870

(1 minus expminus(2119870119869)119905) (34)

From (34) yield to

lim119905rarrinfin

|119911 (119905)| le radic

120576

119870

(35)

It is similar to Theorem 1 that the state variables arebounded in this system From (35) 119911 asymptotically con-verges to the bounded region According to (19) 119890

119898(119905) and

119890119898(119905) are satisfied as follows

lim119905rarrinfin

1003816100381610038161003816119890119898

(119905)1003816100381610038161003816le

1

120582

radic

120576

119870

lim119905rarrinfin

1003816100381610038161003816119890119898

(119905)1003816100381610038161003816le 2radic

120576

119870

(36)

Mathematical Problems in Engineering 7Ph

ase (

deg)

Log

mag

nitu

de

Actual plantNominal plant

100 101 102 103 104

102

100

10minus2

10minus4

minus400minus300minus200minus100

0

Frequency (rads)

100 101 102 103 104

Frequency (rads)

Figure 6 Fitting curves for frequency characteristics of actual plantand nominal model

which means that 119890119898(119905) and 119890

119898(119905) asymptotically converge to

the bounded regionTheorem 2 is proved

4 Experiment Results

This section investigates the feasibility and effectiveness ofthe proposed compound controller based on the structureof the virtual compound-axis servo system by experimentsThe tracking experiments are implemented through usingsome type of OET platform The position sensor of thedevice uses the optical-electrical encoder with resolution of00007 degrees The control algorithm program is writtenwith C language based on Windows-RTX real-time systemin an industrial computer which adopts Advantech IPC610 and connects with the servo drivers by a 16-bit DAconvertor of PCI bus The control cycle is 0001 s It is worthmentioning that the adaptive online adjustment will consumelots of computational resource With the development of theadvanced technologies on computer the problem of lots ofcomputation has been solved The proposed adaptive slidingcontroller is proved to be feasible in a practical system by thisexperimental platform

Based on what has been mentioned above the pitch axisis chosen herein to verify the method because each axisof the OET platform can be designed independently Theparameters of the nominal model are identified as 119869

119899=

0000125 kg and 119861119899

= 0003125N sdot s sdot mminus1 by a white noisefrequency sweep method The fitting curves for frequencycharacteristics of actual plant and nominal model are shownin Figure 6

The experiments are separated into two parts First theexperiment is completed under the proposed compound con-trol scheme in which sign function uses saturation functionSecond in order to verify the effect of the proposedmethod acomparative experiment is achieved by the traditional controlschemewith PDC + FFC + DOBOther parameters are givenas follows 119896

119875= 045 119896

119863= 012 119870 = 004 119903

1= 001 119903

2=

0001 1199033= 0005 and 120576 = 00002 Considering the modeling

mismatch and the robustness of the system the value ofparameter 119892 in improved DOB is chosen as 1000 rads

Figure 7 compares the tracking error and the controlvalue of the two control schemes when the reference inputsignal is a sinusoidal signal where the amplitude is 2 degreesand frequency is 05HzThe results indicate that the proposedcontrol scheme could achieve a better position tracking per-formance than traditional control laws a maximum trackingerror decreases from 0052 degrees to 0023 degrees withthe help of the ASMC ASMC achieves the adjustment to119890119898

better From Figure 7(a) we can see that the trackingerror reaches the maximum value at zero velocity and thetracking error of the proposed control scheme is smallerCompared with traditional control scheme the compoundcontrol scheme based on the virtual compound-axis systemhas better tracking performance and robustness The distur-bance is sufficiently compensated by using improved DOBwhich simplifies design process and decreases calculationtime Meanwhile the accuracy of the system is guaranteedby introducing ASMC to adjust 119890

119898again whose gain is

greatly related to the upper bound on the error of thedisturbance estimation and the reduction of switching gainhelps to alleviate the chattering Figure 7(b) illustrates that thecontroller output curve of the compound control scheme issmoother than that of traditional control scheme

In order to test the dynamic performance of the trackingobject further the tracking command signal is employed assin(2120587 sdot 3119905) Figure 8(a) shows that the maximum trackingerror decreases from 014 degrees to 006 degrees underthe reference signal Compared to the traditional controlscheme the systemwith compound control schemehas betterdynamic performanceThe testing standard of OET platformstipulates that the maximum tracking error cannot exceed10 of the amplitude of the reference signal in working bandObviously the tracking errors of Figures 7(a) and 8(a) meetthis standard under the proposed method Because it has aswitching control the adaptive sliding mode technique canreject the impact of nonlinear disturbance in high-frequencydomain which is not compensated by the DOB From thispoint of view the proposed control scheme enhances thesystem robustness Furthermore according to Figures 7(b)and 8(b) the control value of the proposed method is farless than the limit of the DA converter The control valuedoes not exhibit obvious chattering phenomenon becauseASMC with the improved DOB can alleviate the chatteringphenomenon Consequently the proposed scheme not onlyensures strong robustness against system uncertainties andsmall tracking error but also suppresses the high-frequencychattering at control input effectively The results indicatethat the compound controller can be reliably performed inpractical application

5 Conclusions

To obtain the high performance and good robustness forordinary OET system without the FSM the design methodof virtual compound-axis servo system is proposed in thispaperThe proposed compound control scheme based on the

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8Time (s)

minus006

minus004

minus002

0

002

004

006

Compound control schemeScheme with PDC + FFC + DOB

Trac

king

erro

r (∘)

(a) Comparison of tracking error curves

0 1 2 3 4 5 6 7 8Time (s)

Compound control schemeScheme with PDC + FFC + DOB

minus04minus03minus02minus01

001

020304

Con

trol v

alue

(V)

(b) Comparison of control value

Figure 7 Comparison curves under two schemes with sinusoidal input (119860 = 2∘ 119891 = 05Hz)

0 1 2 3 4 5 6 7 8 9 10minus02

minus01

0

01

02

Time (s)

Compound control schemeScheme with PDC + FFC + DOB

Trac

king

erro

r (∘)

(a) Comparison of tracking error curves

0 1 2 3 4 5 6 7 8 9 10minus03minus02minus01

001020304

Time (s)

Compound control schemeScheme with PDC + FFC + DOB

Con

trol v

alue

(V)

(b) Comparison of control value

Figure 8 Comparison curves under two schemes with sinusoidal input (119860 = 1∘ 119891 = 3Hz)

structure of the virtual compound-axis system can maintainthe robustness of OET system against various disturbancesin the whole control period and then the tracking errorof OET system can be limited within an expected levelin the existence of sustained uncertainties Although thevirtual compound-axis does not perform as well as the realcompound-axis it can also improve the tracking precisionof the OET system to some extent Therefore the proposedmethod can be used as a new control technology applied inOET system In addition because OET system is a typicalservo motion system the theoretic results are able to beextended to other relational fields

However the velocity and acceleration command signalin FFC cannot be measured directly and easily in practiceTherefore some more excellent differential methods shouldbe considered in further work Simultaneously in order totrack the maneuvering target with higher speed furtherstudies aimed at better performance at higher frequenciesare needed Besides with the development of advancedcomputer and information technologies massive amountof measurement data can be utilized to extract the usefulinformation about the current state of the OET process As

a result the excellent method of the process monitoring andfault diagnosis such as data-driven techniques [28] can beused as references in future work

Conflict of Interests

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

Acknowledgments

This work was supported by the National Basic Research Pro-gram (Grant no 2012CB821200) and the Scientific Researchfor Colleges and Universities of Inner Mongolia (Grant noNJZY13143) in China

References

[1] W Thanmas ldquoDigital laser ranging and tracking using acompound axis servomechanismrdquo Applied Optics vol 5 no 4pp 497ndash505 1966

Mathematical Problems in Engineering 9

[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008

[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)

[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004

[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007

[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007

[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010

[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002

[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013

[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008

[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004

[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012

[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012

[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013

[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004

[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008

[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010

[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007

[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012

[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)

[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009

[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995

[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012

[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011

[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994

[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012

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Mathematical Problems in Engineering

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

ignoredThen the closed-loop transfer function of the systemis expressed as

119866 (119904)

=

1198701(119904) 1198751(119904) + 119870

2(119904) 1198752(119904) + 119870

1(119904) 1198751(119904) 1198702(119904) 1198752(119904)

[1 + 1198701(119904) 1198751(119904)] [1 + 119870

2(119904) 1198752(119904)]

(1)

The transfer functions of main system and subsystem aredescribed as follows respectively

1198661(119904) =

1198701(119904) 1198751(119904)

1 + 1198701(119904) 1198751(119904)

1198662(119904) =

1198702(119904) 1198752(119904)

1 + 1198702(119904) 1198752(119904)

(2)

From (1)-(2) the poles of the entire system are the polesof the main system and subsystem Only if the main systemand subsystem are stable the entire system is stable

Suppose 1198901(119904) as the error of the main system and 119890

2(119904)

as the error of the subsystem The error of subsystem can beobtained as

1198902(119904) =

1198901(119904)

1 + 1198702(119904) 1198752(119904)

=

120579ref (119904)

[1 + 1198701(119904) 1198751(119904)] [1 + 119870

2(119904) 1198752(119904)]

(3)

From (3) the tracking accuracy of the virtual compound-axis system can be improved by controlling the main systemand subsystem And the error of the system is the errorof subsystem Therefore the virtual compound-axis systemowns the high tracking accuracy

23 Virtual Compound-Axis Servo System ImplementationTwo problems should be solved as virtual compound-axisservo system

(1) problem of virtual objects property namely howto determine the transfer function of the virtualplatform

(2) problem of implementing the virtual compound-axisservo system namely how to achieve the coarse-finetracking mode on a single tracking platform

In order to implement virtual compound-axis servosystem its control structure diagram is shown in Figure 3where the delay of the tracking detector and virtual detectoris ignored In Figure 3 main system is tracking platformsubsystem is virtual tracking platform and 119890 is the inputsignal of the subsystem In order to simplify the design thetransfer function of virtual platform uses the nominal modelof the practical system As the tracking accuracy of virtualplatform is better than that of the tracking platform thetracking platform tracks virtual platform to decrease 119890

119898for

getting high tracking accuracy In Figure 3 1198701and 119870

2are

the position controllers of the main system and subsystem

P2 nominalplant

P1 plant

Subsystem

DOB120579ref minus

minus

e

Main system

K1 adaptivesliding mode

control

K2 PD+feed-forward

control

un(t)

uc(t)

d

u

em

120579n

120579

minus

Figure 3 Control realization diagram of virtual compound-axisservo system

In order to get the satisfactory control effect the ASMC isused in themain system and PDC and FFC are adopted in thesystem The output of PDC and FFC which can be regardedas a control component of the main system provides controlvariable to the actual plant

3 Compound Control Scheme of VirtualCompound-Axis Servo System

31 Controller Design of Subsystem Because a LOS stabi-lization servo system is driven by a DC torque motor thedynamics of the plant can be described as

119869

120579 + 119861

120579 = 119906 + 119891 (sdot) (4)

where 119869 119861 120579 and 119906 denote the inertia mass the damping theangular position response and the control input respectively119891(sdot) denotes the disturbance such as nonlinear friction theforce from the environment the carrier disturbance andunknown time-varying and nonlinear dynamics which isdifficult to model Consider that 119869

119898le 119869 le 119869

119872 119861119898

le 119861 le

119861119872

is satisfied where 119869119898 119869119872 119861119898 and 119861

119872are positive real

numberBy assembling the parameter mismatch external distur-

bance and unknown dynamics into an equivalent distur-bance 119889 (4) can be written as

119869119899

120579 + 119861119899

120579 = 119906 + 119889 (5)

where 119869119899and 119861

119899denote the nominal mass and the nominal

damping respectivelyThe equivalent disturbance is given by119889 = (119869

119899minus 119869)

120579 + (119861

119899minus 119861)

120579 + 119891(sdot)

Define a tracking error of the system as 119890 = 120579ref minus 120579where 120579ref is a given reference signal Using the structure ofvirtual compound-axis for system (5) the control value 119906 canbe described as

119906 = 119906119899+ 119906119888minus

119889 (6)

where 119906119899and 119906

119888are the control value of subsystem and main

system respectively 119889 is the estimated disturbance through

DOBWhen the equivalent disturbance and parameters meetrequirements the task is to design 119906

119888and 119906

119899to make 119890 rarr 0

4 Mathematical Problems in Engineering

wu

d P1 plant

P(s)

dDOBQ(s)

120585

un(t)

uc(t)

minus

minus +

+ ++

++

1205791

s

Pminus1n (s)

Figure 4 Basic diagram of DOB

The subsystem nominal model is described as follows

119869119899

120579 + 119861119899

120579 = 119906119899 (7)

The controller of subsystem uses a hybrid control schemecombining PDC and FFC which can be described as

119906119875119863

= 119896119875119890 + 119896119863

119890

119906FFC = 119869119899

120579ref + 119861

119899

120579ref

(8)

where 119896119875and 119896

119863denote proportional coefficient and differ-

ential coefficient respectivelyAn approximate method for difference to estimate the

reference values of velocity and acceleration is introducedThis method is expressed by [26]

120579ref =

119892119904

119904 + 119892

120579ref

120579ref = (

119892119904

119904 + 119892

)

2

120579ref =119892119904

119904 + 119892

120579ref

(9)

where 119904 is used to concatenate one low-pass filter whose cut-off frequency is 119892 gt 0

Thence the output of controller is written as (8)

119906119899= 119906119875119863

+ 119906FFC = 119896119875119890 + 119896119863

119890 + 119869119899

120579ref + 119861

119899

120579ref (10)

32 Disturbance Observer-BasedMain System Control In theOET system nonlinear dynamic and uncertain elements aredifficult to compensate by accurate model Robust closed-loop control method based on DOB which is widely appliedin the high-precision servo system has simple design processIt can inhibit the variety of external disturbances and param-eters effectively

In order to realize disturbances suppression the equalcompensation is introduced into control input by means ofthe estimation of the DOB improved in this paper The basicidea of DOB is shown in Figure 4 In Figure 4 119875(119904) 119875

119899(119904)

and 119876(119904) represent the velocity model of the actual plantthe nominal velocity model a filter respectively 119904 meansLaplace operator 119908 120585 and

119889 are the actual velocity outputthe measurement noise and the estimated disturbancesrespectively Let 119906 119889 and 120585 be system input The velocityresponse can be acquired on the basis of superpositionprinciple

119908 (119904) = 119866119880119882

(119904) 119906 (119904) + 119866119863119882

(119904) 119889 (119904) + 119866120585119882

(119904) 120585 (119904) (11)

where

119866119880119882

(119904) =

119875 (119904) 119875119899(119904)

119875119899(119904) + [119875 (119904) minus 119875

119899(119904)] 119876 (119904)

119866119863119882

(119904) =

119875 (119904) 119875119899(119904) [1 minus 119876 (119904)]

119875119899(119904) + [119875 (119904) minus 119875

119899(119904)] 119876 (119904)

119866120585119882

(119904) =

119875 (119904) 119876 (119904)

119875119899(119904) + [119875 (119904) minus 119875

119899(119904)] 119876 (119904)

(12)

Assume that the bandwidth of ideal filter 119876(119904) is 1198910 At

low frequencies there is 119876(119904) asymp 1 when frequency 119891 le 1198910

therefore 119866119880119882

(119904) asymp 119875119899(119904) 119866

119863119882(119904) asymp 0 and 119866

120585119882(119904) asymp 1

This means that DOB makes the characteristic of the actualplant approximately the same as that of the nominal model inlow-frequency domain Therefore the system has powerfulinhibiting effect against external disturbances DOB is verysensitive to low-frequency noise In practical applications itis necessary to consider that appropriate measures are takento reduce the low-frequency noise in the measurement of themotion state

In high-frequency domain if frequency 119891 gt 1198910 then

119876(119904) asymp 0 Consequently 119866119880119882

(119904) = 119866119863119882

(119904) = 119875(119904) and119866120585119882

(119904) = 0 This means that DOB has no inhibiting effectson external disturbances while having great inhibiting abilityagainst the high-frequency measurement noise

The simplified nominal velocity model is procured asfollows

119875119899(119904) =

1

119869119899119904 + 119861119899

(13)

119876(119904) design is significant in DOB design The designmethod of low-pass filter119876(119904)was proposed [27]The relativedegree of 119876(119904) must be equal to or greater than that of119875minus1

119899(119904) in order to make119876(119904)119875

minus1

119899(119904) regular According to the

nominal model shown in (13) the following is chosen for thisresearch which satisfies the property stated above

119876 (119904) =

119892

119904 + 119892

(14)

where 119892 is the cut-off frequency of the low-pass filter (LPF)in DOB Then the estimated disturbance is described as

119889 = 119876 (119904) 119889 =

119892

119904 + 119892

119889 (15)

Taking into account the physical realization of 119869119899119904

119889 isrepresented as follows

119889 = 119876 (119904) [(119869

119899119904 + 119861119899) 119908 minus 119906]

=

119892

119904 + 119892

[(119869119899119904 + 119861119899) 119908 minus 119906]

= 119869119899119892119908 minus

119892

119904 + 119892

[(119869119899119892 minus 119869119899119861119899) 119908 + 119906]

(16)

Therefore the improved DOB can be obtained as shownin Figure 5 The improved DOB has only one differentialelement simple structure and small calculation

Mathematical Problems in Engineering 5

wu

dun(t)

uc(t)

d

120579

Improved DOBJng

1

Js + B

Jng minus JnBn

1

s

P1 plant

minus

minus

minus

+

+

+

+

s

g

Figure 5 Structural diagram of improved DOB

The nonlinear function 119891(sdot) can be modeled by theimproved DOB system as

119891 (sdot) =119889 + 120575 (17)

where 120575 is the approximation error of the DOB Considering120575 is a bounded variable 120593 is defined to meet

|120575| lt 120593 (18)

where 120593 gt 0 120593 = 120593 minus 120593 and = minus

120593 are introduced

to assist in estimating the switching gain of the sliding modecontroller which is designed in Section 33

33 Adaptive Sliding Mode Controller Based Main SystemDesign Define the tracking error of model as 119890

119898= 120579 minus 120579

119899

where 120579119899is an output value of subsystem Define a sliding

mode 119911 as follows

119911 = 119890119898

+ 120582119890119898 120582 =

119861119899

119869119899

(19)

Then 119906119888is designed as

119906119888= minus119870119911 minus ℎ sdot sgn (119911) minus

119869119899minus 119869

119869119899

119906119899+ 119861

120579 minus 120582119869

120579 (20)

where 119870 is a positive constant gain and ℎ is a positiveswitching gain 119869 and 119861 are the estimation values of 119869 and119861 respectively sgn(119911) is a sign function that is defined asfollows

sgn (119911) =

1 119911 gt 0

0 119911 = 0

minus1 119911 lt 0

(21)

The adaptive law is adopted

= Proj

119869(minus1199031(

1

119869119899

119906119899minus 120582

120579) 119911)

= Proj

119861(minus1199032

120579119911)

= Proj

120593[1199033119911 sgn (119911)]

ℎ = 120593

(22)

where 1199031 1199032 and 119903

3are positive real constants The function

Proj∙(V) is defined as

Proj∙(V) =

V ∙119898

lt ∙ lt ∙119872

0 ∙ = ∙119872

and V gt 0

0 ∙ = ∙119898and V lt 0

(23)

where ∙119898is the minimum value of ∙ and ∙

119872is the maximum

value of ∙

Theorem 1 For system (4) if the condition |120575| lt 120593 is satisfied119911 exponentially decays to zero and 119890

119898asymptotically decays to

zero using the control law as (10) (15) (20) and (22) The statevariables of the system are bounded meanwhile

Proof Define a positive-definite Lyapunov candidate

119881 (119911) =

1

2

1198691199112

+

1

21199031

1198692

+

1

21199032

1198612

+

1

21199033

1205932

(24)

119869 = 119869 [(

120579 minus

120579119899) + 120582 (

120579 minus

120579119899)]

= (119869

120579 + 119861

120579) minus

119869

119869119899

(119869119899

120579119899+ 119861119899

120579119899) minus 119861

120579 + 120582119869

120579

= 119906119888+ 119891 (sdot) minus

119889 + 119906119899minus

119869

119869119899

119906119899minus 119861

120579 + 120582119869

120579

= 119906119888+ 120575 +

119869119899minus 119869

119869119899

119906119899minus 119861

120579 + 120582119869

120579

(25)

Define 119869 = 119869 minus 119869 119861 = 119861 minus 119861 then = minus

= minus

Substitute (20) into (25)

119869 = minus 119870119911 minus ℎ sdot sgn (119911) minus

119869119899minus 119869

119869119899

119906119899

+ 119861

120579 minus 120582119869

120579 +

119869119899minus 119869

119869119899

119906119899minus 119861

120579 + 120582119869

120579 + 120575

= minus 119870119911 minus ℎ sdot sgn (119911) + (119869 minus 119869) (

1

119869119899

119906119899minus 120582

120579)

+ (119861 minus 119861)

120579 + 120575

= minus 119870119911 minus ℎ sdot sgn (119911) minus 119869 (

1

119869119899

119906119899minus 120582

120579) minus 119861

120579 + 120575

(119911) = minus 1198701199112

minus ℎ sdot 119911 sgn (119911) minus 119869119911 (

1

119869119899

119906119899minus 120582

120579)

minus 119861119911

120579 minus

1

1199031

119869

minus

1

1199032

119861

minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus ℎ sdot 119911 sgn (119911) minus 119869 [119911(

1

119869119899

119906119899minus 120582

120579) +

1

1199031

]

minus 119861(119911

120579 +

1

1199032

) minus

1

1199033

120593

+ 119911120575

(26)

6 Mathematical Problems in Engineering

According to (22) the time derivative of Lyapunovfunction becomes

(119911) = minus 1198701199112

minus 120593 sdot 119911 sgn (119911) minus |119911| 120593 + 119911120575

le minus 1198701199112

minus 120593 |119911| + |119911| |120575|

= minus 1198701199112

minus |119911| (120593 minus |120575|)

(27)

Since |120575| lt 120593 then |119911|(120593 minus |120575|) gt 0 Therefore (119911) le minus1198961199112 is

metAccording to (24) there is

1199112

(119905) le 119911(0)2 exp(minus

119870

119869

119905) (28)

which implies that 119911 exponentially decays to zero Thenaccording to the definition of 119911 = 119890

119898+120582119890119898 119890119898asymptotically

decays to zero If 119890119898(0) and 119890

119898(0) are bounded then 119890

119898(119905)

and 119890119898(119905) are bounded to arbitrary 119905 gt 0 because of 119911(119905)

being uniformly bounded If 120579119899(0) and

120579119899(0) are bounded

then 120579119899(119905) and

120579119899(119905) are bounded According to 119890

119898= 120579 minus 120579

119899

120579 and 120579 are bounded Consequently the state variables of the

system are boundedTheorem 1 is proved

In this method the switching gain ℎ is only requiredto be greater than |120575| which is a small variable Thisdesign alleviates the chattering phenomenon of sliding modecontroller In engineering practice since sign function sgn(sdot)in control law (20) can cause the frequent switching of thecontrol variable and result in the output chattering it is easyto damage the power amplifier In order to avoid frequentswitching of the control output the sgn(sdot) is replaced by thesaturation function (29) to weaken the chattering further

sat(119909

Δ

) =

1 119909 gt Δ

1

Δ

119909 |119909| le Δ

minus1 119909 lt minusΔ

(29)

where Δ is positive real constant namely the width of aboundary layer

Introducing sat(sdot) control law (20) becomes

119906119888= minus119870119911 minus ℎ sdot sat(ℎ119911

4120576

) minus

119869119899minus 119869

119869119899

119906119899+ 119861

120579 minus 120582119869

120579 (30)

where 120576 is a small positive constant Then there is thefollowingTheorem 2

Theorem 2 For system (4) if the condition |120575| lt 120593 is satisfiedand the control law as (10) (15) (22) and (30) is adopted then

(1) 119911 exponentially decays to zero and lim119905rarrinfin

|119911(119905)| le

radic120576119870 is met(2) 119890119898asymptotically decays to zero and lim

119905rarrinfin|119890119898(119905)| le

(1120582)radic120576119870 is met(3) 119890119898(119905) asymptotically converges and lim

119905rarrinfin| 119890119898(119905)| le

2radic120576119870 is met

Proof Select the positive-definite function119881(119911) as (24) thenits time derivative is calculated as

(119911) = minus 1198701199112

minus ℎ sdot 119911 sat(ℎ119911

4120576

) minus 119869119911(

1

119869119899

119906119899minus 120582

120579)

minus 119861119911

120579 minus

1

1199031

119869

minus

1

1199032

119861

minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus ℎ sdot 119911 sat(ℎ119911

4120576

) minus 119869 [119911(

1

119869119899

119906119899minus 120582

120579) +

1

1199031

]

minus 119861(119911

120579 +

1

1199032

) minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus 120593119911 sat(ℎ119911

4120576

) minus 120593119911 sat(ℎ119911

4120576

) + |119911| 120575

= minus 1198701199112

minus 120593119911 sat(ℎ119911

4120576

) + |119911| 120575

(31)

Since |119911| ge 4120576ℎ sat(ℎ1199114120576) = sgn(119911) According to theproof of Theorem 1 when 120593 ge |120575| (119911) le minus119896119911

2 is metTherefore 119911 exponentially converges until

lim119905rarrinfin

|119911 (119905)| lt

4120576

ℎ

(32)

If |119911| lt 4120576ℎ then sat(ℎ1199114120576) = ℎ1199114120576 (119911) is describedas

(119911) le minus 1198701199112

minus 120593

ℎ|119911|2

4120576

+ |119911| |120575|

le minus 1198701199112

minus

1205932

|119911|2

4120576

+ 120593 |119911|

le minus 1198701199112

minus

1

120576

(

1

2

120593 |119911| minus 120576)

2

+ 120576

le minus 1198701199112

+ 120576 le minus

2119870

119869

119881 + 120576

(33)

Consequently

119881 (119905) le expminus(2119870119869)119905119881 (0) +

119869120576

2119870

(1 minus expminus(2119870119869)119905) (34)

From (34) yield to

lim119905rarrinfin

|119911 (119905)| le radic

120576

119870

(35)

It is similar to Theorem 1 that the state variables arebounded in this system From (35) 119911 asymptotically con-verges to the bounded region According to (19) 119890

119898(119905) and

119890119898(119905) are satisfied as follows

lim119905rarrinfin

1003816100381610038161003816119890119898

(119905)1003816100381610038161003816le

1

120582

radic

120576

119870

lim119905rarrinfin

1003816100381610038161003816119890119898

(119905)1003816100381610038161003816le 2radic

120576

119870

(36)

Mathematical Problems in Engineering 7Ph

ase (

deg)

Log

mag

nitu

de

Actual plantNominal plant

100 101 102 103 104

102

100

10minus2

10minus4

minus400minus300minus200minus100

0

Frequency (rads)

100 101 102 103 104

Frequency (rads)

Figure 6 Fitting curves for frequency characteristics of actual plantand nominal model

which means that 119890119898(119905) and 119890

119898(119905) asymptotically converge to

the bounded regionTheorem 2 is proved

4 Experiment Results

This section investigates the feasibility and effectiveness ofthe proposed compound controller based on the structureof the virtual compound-axis servo system by experimentsThe tracking experiments are implemented through usingsome type of OET platform The position sensor of thedevice uses the optical-electrical encoder with resolution of00007 degrees The control algorithm program is writtenwith C language based on Windows-RTX real-time systemin an industrial computer which adopts Advantech IPC610 and connects with the servo drivers by a 16-bit DAconvertor of PCI bus The control cycle is 0001 s It is worthmentioning that the adaptive online adjustment will consumelots of computational resource With the development of theadvanced technologies on computer the problem of lots ofcomputation has been solved The proposed adaptive slidingcontroller is proved to be feasible in a practical system by thisexperimental platform

Based on what has been mentioned above the pitch axisis chosen herein to verify the method because each axisof the OET platform can be designed independently Theparameters of the nominal model are identified as 119869

119899=

0000125 kg and 119861119899

= 0003125N sdot s sdot mminus1 by a white noisefrequency sweep method The fitting curves for frequencycharacteristics of actual plant and nominal model are shownin Figure 6

The experiments are separated into two parts First theexperiment is completed under the proposed compound con-trol scheme in which sign function uses saturation functionSecond in order to verify the effect of the proposedmethod acomparative experiment is achieved by the traditional controlschemewith PDC + FFC + DOBOther parameters are givenas follows 119896

119875= 045 119896

119863= 012 119870 = 004 119903

1= 001 119903

2=

0001 1199033= 0005 and 120576 = 00002 Considering the modeling

mismatch and the robustness of the system the value ofparameter 119892 in improved DOB is chosen as 1000 rads

Figure 7 compares the tracking error and the controlvalue of the two control schemes when the reference inputsignal is a sinusoidal signal where the amplitude is 2 degreesand frequency is 05HzThe results indicate that the proposedcontrol scheme could achieve a better position tracking per-formance than traditional control laws a maximum trackingerror decreases from 0052 degrees to 0023 degrees withthe help of the ASMC ASMC achieves the adjustment to119890119898

better From Figure 7(a) we can see that the trackingerror reaches the maximum value at zero velocity and thetracking error of the proposed control scheme is smallerCompared with traditional control scheme the compoundcontrol scheme based on the virtual compound-axis systemhas better tracking performance and robustness The distur-bance is sufficiently compensated by using improved DOBwhich simplifies design process and decreases calculationtime Meanwhile the accuracy of the system is guaranteedby introducing ASMC to adjust 119890

119898again whose gain is

greatly related to the upper bound on the error of thedisturbance estimation and the reduction of switching gainhelps to alleviate the chattering Figure 7(b) illustrates that thecontroller output curve of the compound control scheme issmoother than that of traditional control scheme

In order to test the dynamic performance of the trackingobject further the tracking command signal is employed assin(2120587 sdot 3119905) Figure 8(a) shows that the maximum trackingerror decreases from 014 degrees to 006 degrees underthe reference signal Compared to the traditional controlscheme the systemwith compound control schemehas betterdynamic performanceThe testing standard of OET platformstipulates that the maximum tracking error cannot exceed10 of the amplitude of the reference signal in working bandObviously the tracking errors of Figures 7(a) and 8(a) meetthis standard under the proposed method Because it has aswitching control the adaptive sliding mode technique canreject the impact of nonlinear disturbance in high-frequencydomain which is not compensated by the DOB From thispoint of view the proposed control scheme enhances thesystem robustness Furthermore according to Figures 7(b)and 8(b) the control value of the proposed method is farless than the limit of the DA converter The control valuedoes not exhibit obvious chattering phenomenon becauseASMC with the improved DOB can alleviate the chatteringphenomenon Consequently the proposed scheme not onlyensures strong robustness against system uncertainties andsmall tracking error but also suppresses the high-frequencychattering at control input effectively The results indicatethat the compound controller can be reliably performed inpractical application

5 Conclusions

To obtain the high performance and good robustness forordinary OET system without the FSM the design methodof virtual compound-axis servo system is proposed in thispaperThe proposed compound control scheme based on the

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8Time (s)

minus006

minus004

minus002

0

002

004

006

Compound control schemeScheme with PDC + FFC + DOB

Trac

king

erro

r (∘)

(a) Comparison of tracking error curves

0 1 2 3 4 5 6 7 8Time (s)

Compound control schemeScheme with PDC + FFC + DOB

minus04minus03minus02minus01

001

020304

Con

trol v

alue

(V)

(b) Comparison of control value

Figure 7 Comparison curves under two schemes with sinusoidal input (119860 = 2∘ 119891 = 05Hz)

0 1 2 3 4 5 6 7 8 9 10minus02

minus01

0

01

02

Time (s)

Compound control schemeScheme with PDC + FFC + DOB

Trac

king

erro

r (∘)

(a) Comparison of tracking error curves

0 1 2 3 4 5 6 7 8 9 10minus03minus02minus01

001020304

Time (s)

Compound control schemeScheme with PDC + FFC + DOB

Con

trol v

alue

(V)

(b) Comparison of control value

Figure 8 Comparison curves under two schemes with sinusoidal input (119860 = 1∘ 119891 = 3Hz)

structure of the virtual compound-axis system can maintainthe robustness of OET system against various disturbancesin the whole control period and then the tracking errorof OET system can be limited within an expected levelin the existence of sustained uncertainties Although thevirtual compound-axis does not perform as well as the realcompound-axis it can also improve the tracking precisionof the OET system to some extent Therefore the proposedmethod can be used as a new control technology applied inOET system In addition because OET system is a typicalservo motion system the theoretic results are able to beextended to other relational fields

However the velocity and acceleration command signalin FFC cannot be measured directly and easily in practiceTherefore some more excellent differential methods shouldbe considered in further work Simultaneously in order totrack the maneuvering target with higher speed furtherstudies aimed at better performance at higher frequenciesare needed Besides with the development of advancedcomputer and information technologies massive amountof measurement data can be utilized to extract the usefulinformation about the current state of the OET process As

a result the excellent method of the process monitoring andfault diagnosis such as data-driven techniques [28] can beused as references in future work

Conflict of Interests

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

Acknowledgments

This work was supported by the National Basic Research Pro-gram (Grant no 2012CB821200) and the Scientific Researchfor Colleges and Universities of Inner Mongolia (Grant noNJZY13143) in China

References

[1] W Thanmas ldquoDigital laser ranging and tracking using acompound axis servomechanismrdquo Applied Optics vol 5 no 4pp 497ndash505 1966

Mathematical Problems in Engineering 9

[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008

[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)

[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004

[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007

[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007

[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010

[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002

[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013

[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008

[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004

[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012

[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012

[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013

[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004

[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008

[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010

[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007

[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012

[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)

[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009

[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995

[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012

[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011

[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994

[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012

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

wu

d P1 plant

P(s)

dDOBQ(s)

120585

un(t)

uc(t)

minus

minus +

+ ++

++

1205791

s

Pminus1n (s)

Figure 4 Basic diagram of DOB

The subsystem nominal model is described as follows

119869119899

120579 + 119861119899

120579 = 119906119899 (7)

The controller of subsystem uses a hybrid control schemecombining PDC and FFC which can be described as

119906119875119863

= 119896119875119890 + 119896119863

119890

119906FFC = 119869119899

120579ref + 119861

119899

120579ref

(8)

where 119896119875and 119896

119863denote proportional coefficient and differ-

ential coefficient respectivelyAn approximate method for difference to estimate the

reference values of velocity and acceleration is introducedThis method is expressed by [26]

120579ref =

119892119904

119904 + 119892

120579ref

120579ref = (

119892119904

119904 + 119892

)

2

120579ref =119892119904

119904 + 119892

120579ref

(9)

where 119904 is used to concatenate one low-pass filter whose cut-off frequency is 119892 gt 0

Thence the output of controller is written as (8)

119906119899= 119906119875119863

+ 119906FFC = 119896119875119890 + 119896119863

119890 + 119869119899

120579ref + 119861

119899

120579ref (10)

32 Disturbance Observer-BasedMain System Control In theOET system nonlinear dynamic and uncertain elements aredifficult to compensate by accurate model Robust closed-loop control method based on DOB which is widely appliedin the high-precision servo system has simple design processIt can inhibit the variety of external disturbances and param-eters effectively

In order to realize disturbances suppression the equalcompensation is introduced into control input by means ofthe estimation of the DOB improved in this paper The basicidea of DOB is shown in Figure 4 In Figure 4 119875(119904) 119875

119899(119904)

and 119876(119904) represent the velocity model of the actual plantthe nominal velocity model a filter respectively 119904 meansLaplace operator 119908 120585 and

119889 are the actual velocity outputthe measurement noise and the estimated disturbancesrespectively Let 119906 119889 and 120585 be system input The velocityresponse can be acquired on the basis of superpositionprinciple

119908 (119904) = 119866119880119882

(119904) 119906 (119904) + 119866119863119882

(119904) 119889 (119904) + 119866120585119882

(119904) 120585 (119904) (11)

where

119866119880119882

(119904) =

119875 (119904) 119875119899(119904)

119875119899(119904) + [119875 (119904) minus 119875

119899(119904)] 119876 (119904)

119866119863119882

(119904) =

119875 (119904) 119875119899(119904) [1 minus 119876 (119904)]

119875119899(119904) + [119875 (119904) minus 119875

119899(119904)] 119876 (119904)

119866120585119882

(119904) =

119875 (119904) 119876 (119904)

119875119899(119904) + [119875 (119904) minus 119875

119899(119904)] 119876 (119904)

(12)

Assume that the bandwidth of ideal filter 119876(119904) is 1198910 At

low frequencies there is 119876(119904) asymp 1 when frequency 119891 le 1198910

therefore 119866119880119882

(119904) asymp 119875119899(119904) 119866

119863119882(119904) asymp 0 and 119866

120585119882(119904) asymp 1

This means that DOB makes the characteristic of the actualplant approximately the same as that of the nominal model inlow-frequency domain Therefore the system has powerfulinhibiting effect against external disturbances DOB is verysensitive to low-frequency noise In practical applications itis necessary to consider that appropriate measures are takento reduce the low-frequency noise in the measurement of themotion state

In high-frequency domain if frequency 119891 gt 1198910 then

119876(119904) asymp 0 Consequently 119866119880119882

(119904) = 119866119863119882

(119904) = 119875(119904) and119866120585119882

(119904) = 0 This means that DOB has no inhibiting effectson external disturbances while having great inhibiting abilityagainst the high-frequency measurement noise

The simplified nominal velocity model is procured asfollows

119875119899(119904) =

1

119869119899119904 + 119861119899

(13)

119876(119904) design is significant in DOB design The designmethod of low-pass filter119876(119904)was proposed [27]The relativedegree of 119876(119904) must be equal to or greater than that of119875minus1

119899(119904) in order to make119876(119904)119875

minus1

119899(119904) regular According to the

nominal model shown in (13) the following is chosen for thisresearch which satisfies the property stated above

119876 (119904) =

119892

119904 + 119892

(14)

where 119892 is the cut-off frequency of the low-pass filter (LPF)in DOB Then the estimated disturbance is described as

119889 = 119876 (119904) 119889 =

119892

119904 + 119892

119889 (15)

Taking into account the physical realization of 119869119899119904

119889 isrepresented as follows

119889 = 119876 (119904) [(119869

119899119904 + 119861119899) 119908 minus 119906]

=

119892

119904 + 119892

[(119869119899119904 + 119861119899) 119908 minus 119906]

= 119869119899119892119908 minus

119892

119904 + 119892

[(119869119899119892 minus 119869119899119861119899) 119908 + 119906]

(16)

Therefore the improved DOB can be obtained as shownin Figure 5 The improved DOB has only one differentialelement simple structure and small calculation

Mathematical Problems in Engineering 5

wu

dun(t)

uc(t)

d

120579

Improved DOBJng

1

Js + B

Jng minus JnBn

1

s

P1 plant

minus

minus

minus

+

+

+

+

s

g

Figure 5 Structural diagram of improved DOB

The nonlinear function 119891(sdot) can be modeled by theimproved DOB system as

119891 (sdot) =119889 + 120575 (17)

where 120575 is the approximation error of the DOB Considering120575 is a bounded variable 120593 is defined to meet

|120575| lt 120593 (18)

where 120593 gt 0 120593 = 120593 minus 120593 and = minus

120593 are introduced

to assist in estimating the switching gain of the sliding modecontroller which is designed in Section 33

33 Adaptive Sliding Mode Controller Based Main SystemDesign Define the tracking error of model as 119890

119898= 120579 minus 120579

119899

where 120579119899is an output value of subsystem Define a sliding

mode 119911 as follows

119911 = 119890119898

+ 120582119890119898 120582 =

119861119899

119869119899

(19)

Then 119906119888is designed as

119906119888= minus119870119911 minus ℎ sdot sgn (119911) minus

119869119899minus 119869

119869119899

119906119899+ 119861

120579 minus 120582119869

120579 (20)

where 119870 is a positive constant gain and ℎ is a positiveswitching gain 119869 and 119861 are the estimation values of 119869 and119861 respectively sgn(119911) is a sign function that is defined asfollows

sgn (119911) =

1 119911 gt 0

0 119911 = 0

minus1 119911 lt 0

(21)

The adaptive law is adopted

= Proj

119869(minus1199031(

1

119869119899

119906119899minus 120582

120579) 119911)

= Proj

119861(minus1199032

120579119911)

= Proj

120593[1199033119911 sgn (119911)]

ℎ = 120593

(22)

where 1199031 1199032 and 119903

3are positive real constants The function

Proj∙(V) is defined as

Proj∙(V) =

V ∙119898

lt ∙ lt ∙119872

0 ∙ = ∙119872

and V gt 0

0 ∙ = ∙119898and V lt 0

(23)

where ∙119898is the minimum value of ∙ and ∙

119872is the maximum

value of ∙

Theorem 1 For system (4) if the condition |120575| lt 120593 is satisfied119911 exponentially decays to zero and 119890

119898asymptotically decays to

zero using the control law as (10) (15) (20) and (22) The statevariables of the system are bounded meanwhile

Proof Define a positive-definite Lyapunov candidate

119881 (119911) =

1

2

1198691199112

+

1

21199031

1198692

+

1

21199032

1198612

+

1

21199033

1205932

(24)

119869 = 119869 [(

120579 minus

120579119899) + 120582 (

120579 minus

120579119899)]

= (119869

120579 + 119861

120579) minus

119869

119869119899

(119869119899

120579119899+ 119861119899

120579119899) minus 119861

120579 + 120582119869

120579

= 119906119888+ 119891 (sdot) minus

119889 + 119906119899minus

119869

119869119899

119906119899minus 119861

120579 + 120582119869

120579

= 119906119888+ 120575 +

119869119899minus 119869

119869119899

119906119899minus 119861

120579 + 120582119869

120579

(25)

Define 119869 = 119869 minus 119869 119861 = 119861 minus 119861 then = minus

= minus

Substitute (20) into (25)

119869 = minus 119870119911 minus ℎ sdot sgn (119911) minus

119869119899minus 119869

119869119899

119906119899

+ 119861

120579 minus 120582119869

120579 +

119869119899minus 119869

119869119899

119906119899minus 119861

120579 + 120582119869

120579 + 120575

= minus 119870119911 minus ℎ sdot sgn (119911) + (119869 minus 119869) (

1

119869119899

119906119899minus 120582

120579)

+ (119861 minus 119861)

120579 + 120575

= minus 119870119911 minus ℎ sdot sgn (119911) minus 119869 (

1

119869119899

119906119899minus 120582

120579) minus 119861

120579 + 120575

(119911) = minus 1198701199112

minus ℎ sdot 119911 sgn (119911) minus 119869119911 (

1

119869119899

119906119899minus 120582

120579)

minus 119861119911

120579 minus

1

1199031

119869

minus

1

1199032

119861

minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus ℎ sdot 119911 sgn (119911) minus 119869 [119911(

1

119869119899

119906119899minus 120582

120579) +

1

1199031

]

minus 119861(119911

120579 +

1

1199032

) minus

1

1199033

120593

+ 119911120575

(26)

6 Mathematical Problems in Engineering

According to (22) the time derivative of Lyapunovfunction becomes

(119911) = minus 1198701199112

minus 120593 sdot 119911 sgn (119911) minus |119911| 120593 + 119911120575

le minus 1198701199112

minus 120593 |119911| + |119911| |120575|

= minus 1198701199112

minus |119911| (120593 minus |120575|)

(27)

Since |120575| lt 120593 then |119911|(120593 minus |120575|) gt 0 Therefore (119911) le minus1198961199112 is

metAccording to (24) there is

1199112

(119905) le 119911(0)2 exp(minus

119870

119869

119905) (28)

which implies that 119911 exponentially decays to zero Thenaccording to the definition of 119911 = 119890

119898+120582119890119898 119890119898asymptotically

decays to zero If 119890119898(0) and 119890

119898(0) are bounded then 119890

119898(119905)

and 119890119898(119905) are bounded to arbitrary 119905 gt 0 because of 119911(119905)

being uniformly bounded If 120579119899(0) and

120579119899(0) are bounded

then 120579119899(119905) and

120579119899(119905) are bounded According to 119890

119898= 120579 minus 120579

119899

120579 and 120579 are bounded Consequently the state variables of the

system are boundedTheorem 1 is proved

In this method the switching gain ℎ is only requiredto be greater than |120575| which is a small variable Thisdesign alleviates the chattering phenomenon of sliding modecontroller In engineering practice since sign function sgn(sdot)in control law (20) can cause the frequent switching of thecontrol variable and result in the output chattering it is easyto damage the power amplifier In order to avoid frequentswitching of the control output the sgn(sdot) is replaced by thesaturation function (29) to weaken the chattering further

sat(119909

Δ

) =

1 119909 gt Δ

1

Δ

119909 |119909| le Δ

minus1 119909 lt minusΔ

(29)

where Δ is positive real constant namely the width of aboundary layer

Introducing sat(sdot) control law (20) becomes

119906119888= minus119870119911 minus ℎ sdot sat(ℎ119911

4120576

) minus

119869119899minus 119869

119869119899

119906119899+ 119861

120579 minus 120582119869

120579 (30)

where 120576 is a small positive constant Then there is thefollowingTheorem 2

Theorem 2 For system (4) if the condition |120575| lt 120593 is satisfiedand the control law as (10) (15) (22) and (30) is adopted then

(1) 119911 exponentially decays to zero and lim119905rarrinfin

|119911(119905)| le

radic120576119870 is met(2) 119890119898asymptotically decays to zero and lim

119905rarrinfin|119890119898(119905)| le

(1120582)radic120576119870 is met(3) 119890119898(119905) asymptotically converges and lim

119905rarrinfin| 119890119898(119905)| le

2radic120576119870 is met

Proof Select the positive-definite function119881(119911) as (24) thenits time derivative is calculated as

(119911) = minus 1198701199112

minus ℎ sdot 119911 sat(ℎ119911

4120576

) minus 119869119911(

1

119869119899

119906119899minus 120582

120579)

minus 119861119911

120579 minus

1

1199031

119869

minus

1

1199032

119861

minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus ℎ sdot 119911 sat(ℎ119911

4120576

) minus 119869 [119911(

1

119869119899

119906119899minus 120582

120579) +

1

1199031

]

minus 119861(119911

120579 +

1

1199032

) minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus 120593119911 sat(ℎ119911

4120576

) minus 120593119911 sat(ℎ119911

4120576

) + |119911| 120575

= minus 1198701199112

minus 120593119911 sat(ℎ119911

4120576

) + |119911| 120575

(31)

Since |119911| ge 4120576ℎ sat(ℎ1199114120576) = sgn(119911) According to theproof of Theorem 1 when 120593 ge |120575| (119911) le minus119896119911

2 is metTherefore 119911 exponentially converges until

lim119905rarrinfin

|119911 (119905)| lt

4120576

ℎ

(32)

If |119911| lt 4120576ℎ then sat(ℎ1199114120576) = ℎ1199114120576 (119911) is describedas

(119911) le minus 1198701199112

minus 120593

ℎ|119911|2

4120576

+ |119911| |120575|

le minus 1198701199112

minus

1205932

|119911|2

4120576

+ 120593 |119911|

le minus 1198701199112

minus

1

120576

(

1

2

120593 |119911| minus 120576)

2

+ 120576

le minus 1198701199112

+ 120576 le minus

2119870

119869

119881 + 120576

(33)

Consequently

119881 (119905) le expminus(2119870119869)119905119881 (0) +

119869120576

2119870

(1 minus expminus(2119870119869)119905) (34)

From (34) yield to

lim119905rarrinfin

|119911 (119905)| le radic

120576

119870

(35)

It is similar to Theorem 1 that the state variables arebounded in this system From (35) 119911 asymptotically con-verges to the bounded region According to (19) 119890

119898(119905) and

119890119898(119905) are satisfied as follows

lim119905rarrinfin

1003816100381610038161003816119890119898

(119905)1003816100381610038161003816le

1

120582

radic

120576

119870

lim119905rarrinfin

1003816100381610038161003816119890119898

(119905)1003816100381610038161003816le 2radic

120576

119870

(36)

Mathematical Problems in Engineering 7Ph

ase (

deg)

Log

mag

nitu

de

Actual plantNominal plant

100 101 102 103 104

102

100

10minus2

10minus4

minus400minus300minus200minus100

0

Frequency (rads)

100 101 102 103 104

Frequency (rads)

Figure 6 Fitting curves for frequency characteristics of actual plantand nominal model

which means that 119890119898(119905) and 119890

119898(119905) asymptotically converge to

the bounded regionTheorem 2 is proved

4 Experiment Results

This section investigates the feasibility and effectiveness ofthe proposed compound controller based on the structureof the virtual compound-axis servo system by experimentsThe tracking experiments are implemented through usingsome type of OET platform The position sensor of thedevice uses the optical-electrical encoder with resolution of00007 degrees The control algorithm program is writtenwith C language based on Windows-RTX real-time systemin an industrial computer which adopts Advantech IPC610 and connects with the servo drivers by a 16-bit DAconvertor of PCI bus The control cycle is 0001 s It is worthmentioning that the adaptive online adjustment will consumelots of computational resource With the development of theadvanced technologies on computer the problem of lots ofcomputation has been solved The proposed adaptive slidingcontroller is proved to be feasible in a practical system by thisexperimental platform

Based on what has been mentioned above the pitch axisis chosen herein to verify the method because each axisof the OET platform can be designed independently Theparameters of the nominal model are identified as 119869

119899=

0000125 kg and 119861119899

= 0003125N sdot s sdot mminus1 by a white noisefrequency sweep method The fitting curves for frequencycharacteristics of actual plant and nominal model are shownin Figure 6

The experiments are separated into two parts First theexperiment is completed under the proposed compound con-trol scheme in which sign function uses saturation functionSecond in order to verify the effect of the proposedmethod acomparative experiment is achieved by the traditional controlschemewith PDC + FFC + DOBOther parameters are givenas follows 119896

119875= 045 119896

119863= 012 119870 = 004 119903

1= 001 119903

2=

0001 1199033= 0005 and 120576 = 00002 Considering the modeling

mismatch and the robustness of the system the value ofparameter 119892 in improved DOB is chosen as 1000 rads

Figure 7 compares the tracking error and the controlvalue of the two control schemes when the reference inputsignal is a sinusoidal signal where the amplitude is 2 degreesand frequency is 05HzThe results indicate that the proposedcontrol scheme could achieve a better position tracking per-formance than traditional control laws a maximum trackingerror decreases from 0052 degrees to 0023 degrees withthe help of the ASMC ASMC achieves the adjustment to119890119898

better From Figure 7(a) we can see that the trackingerror reaches the maximum value at zero velocity and thetracking error of the proposed control scheme is smallerCompared with traditional control scheme the compoundcontrol scheme based on the virtual compound-axis systemhas better tracking performance and robustness The distur-bance is sufficiently compensated by using improved DOBwhich simplifies design process and decreases calculationtime Meanwhile the accuracy of the system is guaranteedby introducing ASMC to adjust 119890

119898again whose gain is

greatly related to the upper bound on the error of thedisturbance estimation and the reduction of switching gainhelps to alleviate the chattering Figure 7(b) illustrates that thecontroller output curve of the compound control scheme issmoother than that of traditional control scheme

In order to test the dynamic performance of the trackingobject further the tracking command signal is employed assin(2120587 sdot 3119905) Figure 8(a) shows that the maximum trackingerror decreases from 014 degrees to 006 degrees underthe reference signal Compared to the traditional controlscheme the systemwith compound control schemehas betterdynamic performanceThe testing standard of OET platformstipulates that the maximum tracking error cannot exceed10 of the amplitude of the reference signal in working bandObviously the tracking errors of Figures 7(a) and 8(a) meetthis standard under the proposed method Because it has aswitching control the adaptive sliding mode technique canreject the impact of nonlinear disturbance in high-frequencydomain which is not compensated by the DOB From thispoint of view the proposed control scheme enhances thesystem robustness Furthermore according to Figures 7(b)and 8(b) the control value of the proposed method is farless than the limit of the DA converter The control valuedoes not exhibit obvious chattering phenomenon becauseASMC with the improved DOB can alleviate the chatteringphenomenon Consequently the proposed scheme not onlyensures strong robustness against system uncertainties andsmall tracking error but also suppresses the high-frequencychattering at control input effectively The results indicatethat the compound controller can be reliably performed inpractical application

5 Conclusions

To obtain the high performance and good robustness forordinary OET system without the FSM the design methodof virtual compound-axis servo system is proposed in thispaperThe proposed compound control scheme based on the

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8Time (s)

minus006

minus004

minus002

0

002

004

006

Compound control schemeScheme with PDC + FFC + DOB

Trac

king

erro

r (∘)

(a) Comparison of tracking error curves

0 1 2 3 4 5 6 7 8Time (s)

Compound control schemeScheme with PDC + FFC + DOB

minus04minus03minus02minus01

001

020304

Con

trol v

alue

(V)

(b) Comparison of control value

Figure 7 Comparison curves under two schemes with sinusoidal input (119860 = 2∘ 119891 = 05Hz)

0 1 2 3 4 5 6 7 8 9 10minus02

minus01

0

01

02

Time (s)

Compound control schemeScheme with PDC + FFC + DOB

Trac

king

erro

r (∘)

(a) Comparison of tracking error curves

0 1 2 3 4 5 6 7 8 9 10minus03minus02minus01

001020304

Time (s)

Compound control schemeScheme with PDC + FFC + DOB

Con

trol v

alue

(V)

(b) Comparison of control value

Figure 8 Comparison curves under two schemes with sinusoidal input (119860 = 1∘ 119891 = 3Hz)

structure of the virtual compound-axis system can maintainthe robustness of OET system against various disturbancesin the whole control period and then the tracking errorof OET system can be limited within an expected levelin the existence of sustained uncertainties Although thevirtual compound-axis does not perform as well as the realcompound-axis it can also improve the tracking precisionof the OET system to some extent Therefore the proposedmethod can be used as a new control technology applied inOET system In addition because OET system is a typicalservo motion system the theoretic results are able to beextended to other relational fields

However the velocity and acceleration command signalin FFC cannot be measured directly and easily in practiceTherefore some more excellent differential methods shouldbe considered in further work Simultaneously in order totrack the maneuvering target with higher speed furtherstudies aimed at better performance at higher frequenciesare needed Besides with the development of advancedcomputer and information technologies massive amountof measurement data can be utilized to extract the usefulinformation about the current state of the OET process As

a result the excellent method of the process monitoring andfault diagnosis such as data-driven techniques [28] can beused as references in future work

Conflict of Interests

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

Acknowledgments

This work was supported by the National Basic Research Pro-gram (Grant no 2012CB821200) and the Scientific Researchfor Colleges and Universities of Inner Mongolia (Grant noNJZY13143) in China

References

[1] W Thanmas ldquoDigital laser ranging and tracking using acompound axis servomechanismrdquo Applied Optics vol 5 no 4pp 497ndash505 1966

Mathematical Problems in Engineering 9

[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008

[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)

[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004

[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007

[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007

[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010

[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002

[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013

[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008

[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004

[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012

[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012

[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013

[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004

[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008

[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010

[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007

[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012

[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)

[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009

[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995

[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012

[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011

[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994

[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012

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

wu

dun(t)

uc(t)

d

120579

Improved DOBJng

1

Js + B

Jng minus JnBn

1

s

P1 plant

minus

minus

minus

+

+

+

+

s

g

Figure 5 Structural diagram of improved DOB

The nonlinear function 119891(sdot) can be modeled by theimproved DOB system as

119891 (sdot) =119889 + 120575 (17)

where 120575 is the approximation error of the DOB Considering120575 is a bounded variable 120593 is defined to meet

|120575| lt 120593 (18)

where 120593 gt 0 120593 = 120593 minus 120593 and = minus

120593 are introduced

to assist in estimating the switching gain of the sliding modecontroller which is designed in Section 33

33 Adaptive Sliding Mode Controller Based Main SystemDesign Define the tracking error of model as 119890

119898= 120579 minus 120579

119899

where 120579119899is an output value of subsystem Define a sliding

mode 119911 as follows

119911 = 119890119898

+ 120582119890119898 120582 =

119861119899

119869119899

(19)

Then 119906119888is designed as

119906119888= minus119870119911 minus ℎ sdot sgn (119911) minus

119869119899minus 119869

119869119899

119906119899+ 119861

120579 minus 120582119869

120579 (20)

where 119870 is a positive constant gain and ℎ is a positiveswitching gain 119869 and 119861 are the estimation values of 119869 and119861 respectively sgn(119911) is a sign function that is defined asfollows

sgn (119911) =

1 119911 gt 0

0 119911 = 0

minus1 119911 lt 0

(21)

The adaptive law is adopted

= Proj

119869(minus1199031(

1

119869119899

119906119899minus 120582

120579) 119911)

= Proj

119861(minus1199032

120579119911)

= Proj

120593[1199033119911 sgn (119911)]

ℎ = 120593

(22)

where 1199031 1199032 and 119903

3are positive real constants The function

Proj∙(V) is defined as

Proj∙(V) =

V ∙119898

lt ∙ lt ∙119872

0 ∙ = ∙119872

and V gt 0

0 ∙ = ∙119898and V lt 0

(23)

where ∙119898is the minimum value of ∙ and ∙

119872is the maximum

value of ∙

Theorem 1 For system (4) if the condition |120575| lt 120593 is satisfied119911 exponentially decays to zero and 119890

119898asymptotically decays to

zero using the control law as (10) (15) (20) and (22) The statevariables of the system are bounded meanwhile

Proof Define a positive-definite Lyapunov candidate

119881 (119911) =

1

2

1198691199112

+

1

21199031

1198692

+

1

21199032

1198612

+

1

21199033

1205932

(24)

119869 = 119869 [(

120579 minus

120579119899) + 120582 (

120579 minus

120579119899)]

= (119869

120579 + 119861

120579) minus

119869

119869119899

(119869119899

120579119899+ 119861119899

120579119899) minus 119861

120579 + 120582119869

120579

= 119906119888+ 119891 (sdot) minus

119889 + 119906119899minus

119869

119869119899

119906119899minus 119861

120579 + 120582119869

120579

= 119906119888+ 120575 +

119869119899minus 119869

119869119899

119906119899minus 119861

120579 + 120582119869

120579

(25)

Define 119869 = 119869 minus 119869 119861 = 119861 minus 119861 then = minus

= minus

Substitute (20) into (25)

119869 = minus 119870119911 minus ℎ sdot sgn (119911) minus

119869119899minus 119869

119869119899

119906119899

+ 119861

120579 minus 120582119869

120579 +

119869119899minus 119869

119869119899

119906119899minus 119861

120579 + 120582119869

120579 + 120575

= minus 119870119911 minus ℎ sdot sgn (119911) + (119869 minus 119869) (

1

119869119899

119906119899minus 120582

120579)

+ (119861 minus 119861)

120579 + 120575

= minus 119870119911 minus ℎ sdot sgn (119911) minus 119869 (

1

119869119899

119906119899minus 120582

120579) minus 119861

120579 + 120575

(119911) = minus 1198701199112

minus ℎ sdot 119911 sgn (119911) minus 119869119911 (

1

119869119899

119906119899minus 120582

120579)

minus 119861119911

120579 minus

1

1199031

119869

minus

1

1199032

119861

minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus ℎ sdot 119911 sgn (119911) minus 119869 [119911(

1

119869119899

119906119899minus 120582

120579) +

1

1199031

]

minus 119861(119911

120579 +

1

1199032

) minus

1

1199033

120593

+ 119911120575

(26)

6 Mathematical Problems in Engineering

According to (22) the time derivative of Lyapunovfunction becomes

(119911) = minus 1198701199112

minus 120593 sdot 119911 sgn (119911) minus |119911| 120593 + 119911120575

le minus 1198701199112

minus 120593 |119911| + |119911| |120575|

= minus 1198701199112

minus |119911| (120593 minus |120575|)

(27)

Since |120575| lt 120593 then |119911|(120593 minus |120575|) gt 0 Therefore (119911) le minus1198961199112 is

metAccording to (24) there is

1199112

(119905) le 119911(0)2 exp(minus

119870

119869

119905) (28)

which implies that 119911 exponentially decays to zero Thenaccording to the definition of 119911 = 119890

119898+120582119890119898 119890119898asymptotically

decays to zero If 119890119898(0) and 119890

119898(0) are bounded then 119890

119898(119905)

and 119890119898(119905) are bounded to arbitrary 119905 gt 0 because of 119911(119905)

being uniformly bounded If 120579119899(0) and

120579119899(0) are bounded

then 120579119899(119905) and

120579119899(119905) are bounded According to 119890

119898= 120579 minus 120579

119899

120579 and 120579 are bounded Consequently the state variables of the

system are boundedTheorem 1 is proved

In this method the switching gain ℎ is only requiredto be greater than |120575| which is a small variable Thisdesign alleviates the chattering phenomenon of sliding modecontroller In engineering practice since sign function sgn(sdot)in control law (20) can cause the frequent switching of thecontrol variable and result in the output chattering it is easyto damage the power amplifier In order to avoid frequentswitching of the control output the sgn(sdot) is replaced by thesaturation function (29) to weaken the chattering further

sat(119909

Δ

) =

1 119909 gt Δ

1

Δ

119909 |119909| le Δ

minus1 119909 lt minusΔ

(29)

where Δ is positive real constant namely the width of aboundary layer

Introducing sat(sdot) control law (20) becomes

119906119888= minus119870119911 minus ℎ sdot sat(ℎ119911

4120576

) minus

119869119899minus 119869

119869119899

119906119899+ 119861

120579 minus 120582119869

120579 (30)

where 120576 is a small positive constant Then there is thefollowingTheorem 2

Theorem 2 For system (4) if the condition |120575| lt 120593 is satisfiedand the control law as (10) (15) (22) and (30) is adopted then

(1) 119911 exponentially decays to zero and lim119905rarrinfin

|119911(119905)| le

radic120576119870 is met(2) 119890119898asymptotically decays to zero and lim

119905rarrinfin|119890119898(119905)| le

(1120582)radic120576119870 is met(3) 119890119898(119905) asymptotically converges and lim

119905rarrinfin| 119890119898(119905)| le

2radic120576119870 is met

Proof Select the positive-definite function119881(119911) as (24) thenits time derivative is calculated as

(119911) = minus 1198701199112

minus ℎ sdot 119911 sat(ℎ119911

4120576

) minus 119869119911(

1

119869119899

119906119899minus 120582

120579)

minus 119861119911

120579 minus

1

1199031

119869

minus

1

1199032

119861

minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus ℎ sdot 119911 sat(ℎ119911

4120576

) minus 119869 [119911(

1

119869119899

119906119899minus 120582

120579) +

1

1199031

]

minus 119861(119911

120579 +

1

1199032

) minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus 120593119911 sat(ℎ119911

4120576

) minus 120593119911 sat(ℎ119911

4120576

) + |119911| 120575

= minus 1198701199112

minus 120593119911 sat(ℎ119911

4120576

) + |119911| 120575

(31)

Since |119911| ge 4120576ℎ sat(ℎ1199114120576) = sgn(119911) According to theproof of Theorem 1 when 120593 ge |120575| (119911) le minus119896119911

2 is metTherefore 119911 exponentially converges until

lim119905rarrinfin

|119911 (119905)| lt

4120576

ℎ

(32)

If |119911| lt 4120576ℎ then sat(ℎ1199114120576) = ℎ1199114120576 (119911) is describedas

(119911) le minus 1198701199112

minus 120593

ℎ|119911|2

4120576

+ |119911| |120575|

le minus 1198701199112

minus

1205932

|119911|2

4120576

+ 120593 |119911|

le minus 1198701199112

minus

1

120576

(

1

2

120593 |119911| minus 120576)

2

+ 120576

le minus 1198701199112

+ 120576 le minus

2119870

119869

119881 + 120576

(33)

Consequently

119881 (119905) le expminus(2119870119869)119905119881 (0) +

119869120576

2119870

(1 minus expminus(2119870119869)119905) (34)

From (34) yield to

lim119905rarrinfin

|119911 (119905)| le radic

120576

119870

(35)

It is similar to Theorem 1 that the state variables arebounded in this system From (35) 119911 asymptotically con-verges to the bounded region According to (19) 119890

119898(119905) and

119890119898(119905) are satisfied as follows

lim119905rarrinfin

1003816100381610038161003816119890119898

(119905)1003816100381610038161003816le

1

120582

radic

120576

119870

lim119905rarrinfin

1003816100381610038161003816119890119898

(119905)1003816100381610038161003816le 2radic

120576

119870

(36)

Mathematical Problems in Engineering 7Ph

ase (

deg)

Log

mag

nitu

de

Actual plantNominal plant

100 101 102 103 104

102

100

10minus2

10minus4

minus400minus300minus200minus100

0

Frequency (rads)

100 101 102 103 104

Frequency (rads)

Figure 6 Fitting curves for frequency characteristics of actual plantand nominal model

which means that 119890119898(119905) and 119890

119898(119905) asymptotically converge to

the bounded regionTheorem 2 is proved

4 Experiment Results

This section investigates the feasibility and effectiveness ofthe proposed compound controller based on the structureof the virtual compound-axis servo system by experimentsThe tracking experiments are implemented through usingsome type of OET platform The position sensor of thedevice uses the optical-electrical encoder with resolution of00007 degrees The control algorithm program is writtenwith C language based on Windows-RTX real-time systemin an industrial computer which adopts Advantech IPC610 and connects with the servo drivers by a 16-bit DAconvertor of PCI bus The control cycle is 0001 s It is worthmentioning that the adaptive online adjustment will consumelots of computational resource With the development of theadvanced technologies on computer the problem of lots ofcomputation has been solved The proposed adaptive slidingcontroller is proved to be feasible in a practical system by thisexperimental platform

Based on what has been mentioned above the pitch axisis chosen herein to verify the method because each axisof the OET platform can be designed independently Theparameters of the nominal model are identified as 119869

119899=

0000125 kg and 119861119899

= 0003125N sdot s sdot mminus1 by a white noisefrequency sweep method The fitting curves for frequencycharacteristics of actual plant and nominal model are shownin Figure 6

The experiments are separated into two parts First theexperiment is completed under the proposed compound con-trol scheme in which sign function uses saturation functionSecond in order to verify the effect of the proposedmethod acomparative experiment is achieved by the traditional controlschemewith PDC + FFC + DOBOther parameters are givenas follows 119896

119875= 045 119896

119863= 012 119870 = 004 119903

1= 001 119903

2=

0001 1199033= 0005 and 120576 = 00002 Considering the modeling

mismatch and the robustness of the system the value ofparameter 119892 in improved DOB is chosen as 1000 rads

Figure 7 compares the tracking error and the controlvalue of the two control schemes when the reference inputsignal is a sinusoidal signal where the amplitude is 2 degreesand frequency is 05HzThe results indicate that the proposedcontrol scheme could achieve a better position tracking per-formance than traditional control laws a maximum trackingerror decreases from 0052 degrees to 0023 degrees withthe help of the ASMC ASMC achieves the adjustment to119890119898

better From Figure 7(a) we can see that the trackingerror reaches the maximum value at zero velocity and thetracking error of the proposed control scheme is smallerCompared with traditional control scheme the compoundcontrol scheme based on the virtual compound-axis systemhas better tracking performance and robustness The distur-bance is sufficiently compensated by using improved DOBwhich simplifies design process and decreases calculationtime Meanwhile the accuracy of the system is guaranteedby introducing ASMC to adjust 119890

119898again whose gain is

greatly related to the upper bound on the error of thedisturbance estimation and the reduction of switching gainhelps to alleviate the chattering Figure 7(b) illustrates that thecontroller output curve of the compound control scheme issmoother than that of traditional control scheme

In order to test the dynamic performance of the trackingobject further the tracking command signal is employed assin(2120587 sdot 3119905) Figure 8(a) shows that the maximum trackingerror decreases from 014 degrees to 006 degrees underthe reference signal Compared to the traditional controlscheme the systemwith compound control schemehas betterdynamic performanceThe testing standard of OET platformstipulates that the maximum tracking error cannot exceed10 of the amplitude of the reference signal in working bandObviously the tracking errors of Figures 7(a) and 8(a) meetthis standard under the proposed method Because it has aswitching control the adaptive sliding mode technique canreject the impact of nonlinear disturbance in high-frequencydomain which is not compensated by the DOB From thispoint of view the proposed control scheme enhances thesystem robustness Furthermore according to Figures 7(b)and 8(b) the control value of the proposed method is farless than the limit of the DA converter The control valuedoes not exhibit obvious chattering phenomenon becauseASMC with the improved DOB can alleviate the chatteringphenomenon Consequently the proposed scheme not onlyensures strong robustness against system uncertainties andsmall tracking error but also suppresses the high-frequencychattering at control input effectively The results indicatethat the compound controller can be reliably performed inpractical application

5 Conclusions

To obtain the high performance and good robustness forordinary OET system without the FSM the design methodof virtual compound-axis servo system is proposed in thispaperThe proposed compound control scheme based on the

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8Time (s)

minus006

minus004

minus002

0

002

004

006

Compound control schemeScheme with PDC + FFC + DOB

Trac

king

erro

r (∘)

(a) Comparison of tracking error curves

0 1 2 3 4 5 6 7 8Time (s)

Compound control schemeScheme with PDC + FFC + DOB

minus04minus03minus02minus01

001

020304

Con

trol v

alue

(V)

(b) Comparison of control value

Figure 7 Comparison curves under two schemes with sinusoidal input (119860 = 2∘ 119891 = 05Hz)

0 1 2 3 4 5 6 7 8 9 10minus02

minus01

0

01

02

Time (s)

Compound control schemeScheme with PDC + FFC + DOB

Trac

king

erro

r (∘)

(a) Comparison of tracking error curves

0 1 2 3 4 5 6 7 8 9 10minus03minus02minus01

001020304

Time (s)

Compound control schemeScheme with PDC + FFC + DOB

Con

trol v

alue

(V)

(b) Comparison of control value

Figure 8 Comparison curves under two schemes with sinusoidal input (119860 = 1∘ 119891 = 3Hz)

structure of the virtual compound-axis system can maintainthe robustness of OET system against various disturbancesin the whole control period and then the tracking errorof OET system can be limited within an expected levelin the existence of sustained uncertainties Although thevirtual compound-axis does not perform as well as the realcompound-axis it can also improve the tracking precisionof the OET system to some extent Therefore the proposedmethod can be used as a new control technology applied inOET system In addition because OET system is a typicalservo motion system the theoretic results are able to beextended to other relational fields

However the velocity and acceleration command signalin FFC cannot be measured directly and easily in practiceTherefore some more excellent differential methods shouldbe considered in further work Simultaneously in order totrack the maneuvering target with higher speed furtherstudies aimed at better performance at higher frequenciesare needed Besides with the development of advancedcomputer and information technologies massive amountof measurement data can be utilized to extract the usefulinformation about the current state of the OET process As

a result the excellent method of the process monitoring andfault diagnosis such as data-driven techniques [28] can beused as references in future work

Conflict of Interests

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

Acknowledgments

This work was supported by the National Basic Research Pro-gram (Grant no 2012CB821200) and the Scientific Researchfor Colleges and Universities of Inner Mongolia (Grant noNJZY13143) in China

References

[1] W Thanmas ldquoDigital laser ranging and tracking using acompound axis servomechanismrdquo Applied Optics vol 5 no 4pp 497ndash505 1966

Mathematical Problems in Engineering 9

[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008

[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)

[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004

[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007

[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007

[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010

[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002

[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013

[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008

[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004

[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012

[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012

[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013

[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004

[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008

[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010

[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007

[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012

[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)

[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009

[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995

[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012

[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011

[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994

[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012

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

According to (22) the time derivative of Lyapunovfunction becomes

(119911) = minus 1198701199112

minus 120593 sdot 119911 sgn (119911) minus |119911| 120593 + 119911120575

le minus 1198701199112

minus 120593 |119911| + |119911| |120575|

= minus 1198701199112

minus |119911| (120593 minus |120575|)

(27)

Since |120575| lt 120593 then |119911|(120593 minus |120575|) gt 0 Therefore (119911) le minus1198961199112 is

metAccording to (24) there is

1199112

(119905) le 119911(0)2 exp(minus

119870

119869

119905) (28)

which implies that 119911 exponentially decays to zero Thenaccording to the definition of 119911 = 119890

119898+120582119890119898 119890119898asymptotically

decays to zero If 119890119898(0) and 119890

119898(0) are bounded then 119890

119898(119905)

and 119890119898(119905) are bounded to arbitrary 119905 gt 0 because of 119911(119905)

being uniformly bounded If 120579119899(0) and

120579119899(0) are bounded

then 120579119899(119905) and

120579119899(119905) are bounded According to 119890

119898= 120579 minus 120579

119899

120579 and 120579 are bounded Consequently the state variables of the

system are boundedTheorem 1 is proved

In this method the switching gain ℎ is only requiredto be greater than |120575| which is a small variable Thisdesign alleviates the chattering phenomenon of sliding modecontroller In engineering practice since sign function sgn(sdot)in control law (20) can cause the frequent switching of thecontrol variable and result in the output chattering it is easyto damage the power amplifier In order to avoid frequentswitching of the control output the sgn(sdot) is replaced by thesaturation function (29) to weaken the chattering further

sat(119909

Δ

) =

1 119909 gt Δ

1

Δ

119909 |119909| le Δ

minus1 119909 lt minusΔ

(29)

where Δ is positive real constant namely the width of aboundary layer

Introducing sat(sdot) control law (20) becomes

119906119888= minus119870119911 minus ℎ sdot sat(ℎ119911

4120576

) minus

119869119899minus 119869

119869119899

119906119899+ 119861

120579 minus 120582119869

120579 (30)

where 120576 is a small positive constant Then there is thefollowingTheorem 2

Theorem 2 For system (4) if the condition |120575| lt 120593 is satisfiedand the control law as (10) (15) (22) and (30) is adopted then

(1) 119911 exponentially decays to zero and lim119905rarrinfin

|119911(119905)| le

radic120576119870 is met(2) 119890119898asymptotically decays to zero and lim

119905rarrinfin|119890119898(119905)| le

(1120582)radic120576119870 is met(3) 119890119898(119905) asymptotically converges and lim

119905rarrinfin| 119890119898(119905)| le

2radic120576119870 is met

Proof Select the positive-definite function119881(119911) as (24) thenits time derivative is calculated as

(119911) = minus 1198701199112

minus ℎ sdot 119911 sat(ℎ119911

4120576

) minus 119869119911(

1

119869119899

119906119899minus 120582

120579)

minus 119861119911

120579 minus

1

1199031

119869

minus

1

1199032

119861

minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus ℎ sdot 119911 sat(ℎ119911

4120576

) minus 119869 [119911(

1

119869119899

119906119899minus 120582

120579) +

1

1199031

]

minus 119861(119911

120579 +

1

1199032

) minus

1

1199033

120593

+ 119911120575

= minus 1198701199112

minus 120593119911 sat(ℎ119911

4120576

) minus 120593119911 sat(ℎ119911

4120576

) + |119911| 120575

= minus 1198701199112

minus 120593119911 sat(ℎ119911

4120576

) + |119911| 120575

(31)

Since |119911| ge 4120576ℎ sat(ℎ1199114120576) = sgn(119911) According to theproof of Theorem 1 when 120593 ge |120575| (119911) le minus119896119911

2 is metTherefore 119911 exponentially converges until

lim119905rarrinfin

|119911 (119905)| lt

4120576

ℎ

(32)

If |119911| lt 4120576ℎ then sat(ℎ1199114120576) = ℎ1199114120576 (119911) is describedas

(119911) le minus 1198701199112

minus 120593

ℎ|119911|2

4120576

+ |119911| |120575|

le minus 1198701199112

minus

1205932

|119911|2

4120576

+ 120593 |119911|

le minus 1198701199112

minus

1

120576

(

1

2

120593 |119911| minus 120576)

2

+ 120576

le minus 1198701199112

+ 120576 le minus

2119870

119869

119881 + 120576

(33)

Consequently

119881 (119905) le expminus(2119870119869)119905119881 (0) +

119869120576

2119870

(1 minus expminus(2119870119869)119905) (34)

From (34) yield to

lim119905rarrinfin

|119911 (119905)| le radic

120576

119870

(35)

It is similar to Theorem 1 that the state variables arebounded in this system From (35) 119911 asymptotically con-verges to the bounded region According to (19) 119890

119898(119905) and

119890119898(119905) are satisfied as follows

lim119905rarrinfin

1003816100381610038161003816119890119898

(119905)1003816100381610038161003816le

1

120582

radic

120576

119870

lim119905rarrinfin

1003816100381610038161003816119890119898

(119905)1003816100381610038161003816le 2radic

120576

119870

(36)

Mathematical Problems in Engineering 7Ph

ase (

deg)

Log

mag

nitu

de

Actual plantNominal plant

100 101 102 103 104

102

100

10minus2

10minus4

minus400minus300minus200minus100

0

Frequency (rads)

100 101 102 103 104

Frequency (rads)

Figure 6 Fitting curves for frequency characteristics of actual plantand nominal model

which means that 119890119898(119905) and 119890

119898(119905) asymptotically converge to

the bounded regionTheorem 2 is proved

4 Experiment Results

This section investigates the feasibility and effectiveness ofthe proposed compound controller based on the structureof the virtual compound-axis servo system by experimentsThe tracking experiments are implemented through usingsome type of OET platform The position sensor of thedevice uses the optical-electrical encoder with resolution of00007 degrees The control algorithm program is writtenwith C language based on Windows-RTX real-time systemin an industrial computer which adopts Advantech IPC610 and connects with the servo drivers by a 16-bit DAconvertor of PCI bus The control cycle is 0001 s It is worthmentioning that the adaptive online adjustment will consumelots of computational resource With the development of theadvanced technologies on computer the problem of lots ofcomputation has been solved The proposed adaptive slidingcontroller is proved to be feasible in a practical system by thisexperimental platform

Based on what has been mentioned above the pitch axisis chosen herein to verify the method because each axisof the OET platform can be designed independently Theparameters of the nominal model are identified as 119869

119899=

0000125 kg and 119861119899

= 0003125N sdot s sdot mminus1 by a white noisefrequency sweep method The fitting curves for frequencycharacteristics of actual plant and nominal model are shownin Figure 6

The experiments are separated into two parts First theexperiment is completed under the proposed compound con-trol scheme in which sign function uses saturation functionSecond in order to verify the effect of the proposedmethod acomparative experiment is achieved by the traditional controlschemewith PDC + FFC + DOBOther parameters are givenas follows 119896

119875= 045 119896

119863= 012 119870 = 004 119903

1= 001 119903

2=

0001 1199033= 0005 and 120576 = 00002 Considering the modeling

mismatch and the robustness of the system the value ofparameter 119892 in improved DOB is chosen as 1000 rads

Figure 7 compares the tracking error and the controlvalue of the two control schemes when the reference inputsignal is a sinusoidal signal where the amplitude is 2 degreesand frequency is 05HzThe results indicate that the proposedcontrol scheme could achieve a better position tracking per-formance than traditional control laws a maximum trackingerror decreases from 0052 degrees to 0023 degrees withthe help of the ASMC ASMC achieves the adjustment to119890119898

better From Figure 7(a) we can see that the trackingerror reaches the maximum value at zero velocity and thetracking error of the proposed control scheme is smallerCompared with traditional control scheme the compoundcontrol scheme based on the virtual compound-axis systemhas better tracking performance and robustness The distur-bance is sufficiently compensated by using improved DOBwhich simplifies design process and decreases calculationtime Meanwhile the accuracy of the system is guaranteedby introducing ASMC to adjust 119890

119898again whose gain is

greatly related to the upper bound on the error of thedisturbance estimation and the reduction of switching gainhelps to alleviate the chattering Figure 7(b) illustrates that thecontroller output curve of the compound control scheme issmoother than that of traditional control scheme

In order to test the dynamic performance of the trackingobject further the tracking command signal is employed assin(2120587 sdot 3119905) Figure 8(a) shows that the maximum trackingerror decreases from 014 degrees to 006 degrees underthe reference signal Compared to the traditional controlscheme the systemwith compound control schemehas betterdynamic performanceThe testing standard of OET platformstipulates that the maximum tracking error cannot exceed10 of the amplitude of the reference signal in working bandObviously the tracking errors of Figures 7(a) and 8(a) meetthis standard under the proposed method Because it has aswitching control the adaptive sliding mode technique canreject the impact of nonlinear disturbance in high-frequencydomain which is not compensated by the DOB From thispoint of view the proposed control scheme enhances thesystem robustness Furthermore according to Figures 7(b)and 8(b) the control value of the proposed method is farless than the limit of the DA converter The control valuedoes not exhibit obvious chattering phenomenon becauseASMC with the improved DOB can alleviate the chatteringphenomenon Consequently the proposed scheme not onlyensures strong robustness against system uncertainties andsmall tracking error but also suppresses the high-frequencychattering at control input effectively The results indicatethat the compound controller can be reliably performed inpractical application

5 Conclusions

To obtain the high performance and good robustness forordinary OET system without the FSM the design methodof virtual compound-axis servo system is proposed in thispaperThe proposed compound control scheme based on the

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8Time (s)

minus006

minus004

minus002

0

002

004

006

Compound control schemeScheme with PDC + FFC + DOB

Trac

king

erro

r (∘)

(a) Comparison of tracking error curves

0 1 2 3 4 5 6 7 8Time (s)

Compound control schemeScheme with PDC + FFC + DOB

minus04minus03minus02minus01

001

020304

Con

trol v

alue

(V)

(b) Comparison of control value

Figure 7 Comparison curves under two schemes with sinusoidal input (119860 = 2∘ 119891 = 05Hz)

0 1 2 3 4 5 6 7 8 9 10minus02

minus01

0

01

02

Time (s)

Compound control schemeScheme with PDC + FFC + DOB

Trac

king

erro

r (∘)

(a) Comparison of tracking error curves

0 1 2 3 4 5 6 7 8 9 10minus03minus02minus01

001020304

Time (s)

Compound control schemeScheme with PDC + FFC + DOB

Con

trol v

alue

(V)

(b) Comparison of control value

Figure 8 Comparison curves under two schemes with sinusoidal input (119860 = 1∘ 119891 = 3Hz)

structure of the virtual compound-axis system can maintainthe robustness of OET system against various disturbancesin the whole control period and then the tracking errorof OET system can be limited within an expected levelin the existence of sustained uncertainties Although thevirtual compound-axis does not perform as well as the realcompound-axis it can also improve the tracking precisionof the OET system to some extent Therefore the proposedmethod can be used as a new control technology applied inOET system In addition because OET system is a typicalservo motion system the theoretic results are able to beextended to other relational fields

However the velocity and acceleration command signalin FFC cannot be measured directly and easily in practiceTherefore some more excellent differential methods shouldbe considered in further work Simultaneously in order totrack the maneuvering target with higher speed furtherstudies aimed at better performance at higher frequenciesare needed Besides with the development of advancedcomputer and information technologies massive amountof measurement data can be utilized to extract the usefulinformation about the current state of the OET process As

a result the excellent method of the process monitoring andfault diagnosis such as data-driven techniques [28] can beused as references in future work

Conflict of Interests

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

Acknowledgments

This work was supported by the National Basic Research Pro-gram (Grant no 2012CB821200) and the Scientific Researchfor Colleges and Universities of Inner Mongolia (Grant noNJZY13143) in China

References

[1] W Thanmas ldquoDigital laser ranging and tracking using acompound axis servomechanismrdquo Applied Optics vol 5 no 4pp 497ndash505 1966

Mathematical Problems in Engineering 9

[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008

[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)

[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004

[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007

[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007

[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010

[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002

[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013

[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008

[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004

[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012

[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012

[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013

[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004

[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008

[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010

[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007

[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012

[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)

[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009

[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995

[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012

[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011

[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994

[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012

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 7Ph

ase (

deg)

Log

mag

nitu

de

Actual plantNominal plant

100 101 102 103 104

102

100

10minus2

10minus4

minus400minus300minus200minus100

0

Frequency (rads)

100 101 102 103 104

Frequency (rads)

Figure 6 Fitting curves for frequency characteristics of actual plantand nominal model

which means that 119890119898(119905) and 119890

119898(119905) asymptotically converge to

the bounded regionTheorem 2 is proved

4 Experiment Results

This section investigates the feasibility and effectiveness ofthe proposed compound controller based on the structureof the virtual compound-axis servo system by experimentsThe tracking experiments are implemented through usingsome type of OET platform The position sensor of thedevice uses the optical-electrical encoder with resolution of00007 degrees The control algorithm program is writtenwith C language based on Windows-RTX real-time systemin an industrial computer which adopts Advantech IPC610 and connects with the servo drivers by a 16-bit DAconvertor of PCI bus The control cycle is 0001 s It is worthmentioning that the adaptive online adjustment will consumelots of computational resource With the development of theadvanced technologies on computer the problem of lots ofcomputation has been solved The proposed adaptive slidingcontroller is proved to be feasible in a practical system by thisexperimental platform

Based on what has been mentioned above the pitch axisis chosen herein to verify the method because each axisof the OET platform can be designed independently Theparameters of the nominal model are identified as 119869

119899=

0000125 kg and 119861119899

= 0003125N sdot s sdot mminus1 by a white noisefrequency sweep method The fitting curves for frequencycharacteristics of actual plant and nominal model are shownin Figure 6

The experiments are separated into two parts First theexperiment is completed under the proposed compound con-trol scheme in which sign function uses saturation functionSecond in order to verify the effect of the proposedmethod acomparative experiment is achieved by the traditional controlschemewith PDC + FFC + DOBOther parameters are givenas follows 119896

119875= 045 119896

119863= 012 119870 = 004 119903

1= 001 119903

2=

0001 1199033= 0005 and 120576 = 00002 Considering the modeling

mismatch and the robustness of the system the value ofparameter 119892 in improved DOB is chosen as 1000 rads

Figure 7 compares the tracking error and the controlvalue of the two control schemes when the reference inputsignal is a sinusoidal signal where the amplitude is 2 degreesand frequency is 05HzThe results indicate that the proposedcontrol scheme could achieve a better position tracking per-formance than traditional control laws a maximum trackingerror decreases from 0052 degrees to 0023 degrees withthe help of the ASMC ASMC achieves the adjustment to119890119898

better From Figure 7(a) we can see that the trackingerror reaches the maximum value at zero velocity and thetracking error of the proposed control scheme is smallerCompared with traditional control scheme the compoundcontrol scheme based on the virtual compound-axis systemhas better tracking performance and robustness The distur-bance is sufficiently compensated by using improved DOBwhich simplifies design process and decreases calculationtime Meanwhile the accuracy of the system is guaranteedby introducing ASMC to adjust 119890

119898again whose gain is

greatly related to the upper bound on the error of thedisturbance estimation and the reduction of switching gainhelps to alleviate the chattering Figure 7(b) illustrates that thecontroller output curve of the compound control scheme issmoother than that of traditional control scheme

In order to test the dynamic performance of the trackingobject further the tracking command signal is employed assin(2120587 sdot 3119905) Figure 8(a) shows that the maximum trackingerror decreases from 014 degrees to 006 degrees underthe reference signal Compared to the traditional controlscheme the systemwith compound control schemehas betterdynamic performanceThe testing standard of OET platformstipulates that the maximum tracking error cannot exceed10 of the amplitude of the reference signal in working bandObviously the tracking errors of Figures 7(a) and 8(a) meetthis standard under the proposed method Because it has aswitching control the adaptive sliding mode technique canreject the impact of nonlinear disturbance in high-frequencydomain which is not compensated by the DOB From thispoint of view the proposed control scheme enhances thesystem robustness Furthermore according to Figures 7(b)and 8(b) the control value of the proposed method is farless than the limit of the DA converter The control valuedoes not exhibit obvious chattering phenomenon becauseASMC with the improved DOB can alleviate the chatteringphenomenon Consequently the proposed scheme not onlyensures strong robustness against system uncertainties andsmall tracking error but also suppresses the high-frequencychattering at control input effectively The results indicatethat the compound controller can be reliably performed inpractical application

5 Conclusions

To obtain the high performance and good robustness forordinary OET system without the FSM the design methodof virtual compound-axis servo system is proposed in thispaperThe proposed compound control scheme based on the

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8Time (s)

minus006

minus004

minus002

0

002

004

006

Compound control schemeScheme with PDC + FFC + DOB

Trac

king

erro

r (∘)

(a) Comparison of tracking error curves

0 1 2 3 4 5 6 7 8Time (s)

Compound control schemeScheme with PDC + FFC + DOB

minus04minus03minus02minus01

001

020304

Con

trol v

alue

(V)

(b) Comparison of control value

Figure 7 Comparison curves under two schemes with sinusoidal input (119860 = 2∘ 119891 = 05Hz)

0 1 2 3 4 5 6 7 8 9 10minus02

minus01

0

01

02

Time (s)

Compound control schemeScheme with PDC + FFC + DOB

Trac

king

erro

r (∘)

(a) Comparison of tracking error curves

0 1 2 3 4 5 6 7 8 9 10minus03minus02minus01

001020304

Time (s)

Compound control schemeScheme with PDC + FFC + DOB

Con

trol v

alue

(V)

(b) Comparison of control value

Figure 8 Comparison curves under two schemes with sinusoidal input (119860 = 1∘ 119891 = 3Hz)

structure of the virtual compound-axis system can maintainthe robustness of OET system against various disturbancesin the whole control period and then the tracking errorof OET system can be limited within an expected levelin the existence of sustained uncertainties Although thevirtual compound-axis does not perform as well as the realcompound-axis it can also improve the tracking precisionof the OET system to some extent Therefore the proposedmethod can be used as a new control technology applied inOET system In addition because OET system is a typicalservo motion system the theoretic results are able to beextended to other relational fields

However the velocity and acceleration command signalin FFC cannot be measured directly and easily in practiceTherefore some more excellent differential methods shouldbe considered in further work Simultaneously in order totrack the maneuvering target with higher speed furtherstudies aimed at better performance at higher frequenciesare needed Besides with the development of advancedcomputer and information technologies massive amountof measurement data can be utilized to extract the usefulinformation about the current state of the OET process As

a result the excellent method of the process monitoring andfault diagnosis such as data-driven techniques [28] can beused as references in future work

Conflict of Interests

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

Acknowledgments

This work was supported by the National Basic Research Pro-gram (Grant no 2012CB821200) and the Scientific Researchfor Colleges and Universities of Inner Mongolia (Grant noNJZY13143) in China

References

[1] W Thanmas ldquoDigital laser ranging and tracking using acompound axis servomechanismrdquo Applied Optics vol 5 no 4pp 497ndash505 1966

Mathematical Problems in Engineering 9

[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008

[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)

[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004

[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007

[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007

[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010

[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002

[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013

[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008

[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004

[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012

[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012

[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013

[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004

[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008

[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010

[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007

[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012

[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)

[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009

[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995

[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012

[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011

[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994

[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012

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 1 2 3 4 5 6 7 8Time (s)

minus006

minus004

minus002

0

002

004

006

Compound control schemeScheme with PDC + FFC + DOB

Trac

king

erro

r (∘)

(a) Comparison of tracking error curves

0 1 2 3 4 5 6 7 8Time (s)

Compound control schemeScheme with PDC + FFC + DOB

minus04minus03minus02minus01

001

020304

Con

trol v

alue

(V)

(b) Comparison of control value

Figure 7 Comparison curves under two schemes with sinusoidal input (119860 = 2∘ 119891 = 05Hz)

0 1 2 3 4 5 6 7 8 9 10minus02

minus01

0

01

02

Time (s)

Compound control schemeScheme with PDC + FFC + DOB

Trac

king

erro

r (∘)

(a) Comparison of tracking error curves

0 1 2 3 4 5 6 7 8 9 10minus03minus02minus01

001020304

Time (s)

Compound control schemeScheme with PDC + FFC + DOB

Con

trol v

alue

(V)

(b) Comparison of control value

Figure 8 Comparison curves under two schemes with sinusoidal input (119860 = 1∘ 119891 = 3Hz)

structure of the virtual compound-axis system can maintainthe robustness of OET system against various disturbancesin the whole control period and then the tracking errorof OET system can be limited within an expected levelin the existence of sustained uncertainties Although thevirtual compound-axis does not perform as well as the realcompound-axis it can also improve the tracking precisionof the OET system to some extent Therefore the proposedmethod can be used as a new control technology applied inOET system In addition because OET system is a typicalservo motion system the theoretic results are able to beextended to other relational fields

However the velocity and acceleration command signalin FFC cannot be measured directly and easily in practiceTherefore some more excellent differential methods shouldbe considered in further work Simultaneously in order totrack the maneuvering target with higher speed furtherstudies aimed at better performance at higher frequenciesare needed Besides with the development of advancedcomputer and information technologies massive amountof measurement data can be utilized to extract the usefulinformation about the current state of the OET process As

a result the excellent method of the process monitoring andfault diagnosis such as data-driven techniques [28] can beused as references in future work

Conflict of Interests

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

Acknowledgments

This work was supported by the National Basic Research Pro-gram (Grant no 2012CB821200) and the Scientific Researchfor Colleges and Universities of Inner Mongolia (Grant noNJZY13143) in China

References

[1] W Thanmas ldquoDigital laser ranging and tracking using acompound axis servomechanismrdquo Applied Optics vol 5 no 4pp 497ndash505 1966

Mathematical Problems in Engineering 9

[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008

[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)

[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004

[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007

[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007

[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010

[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002

[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013

[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008

[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004

[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012

[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012

[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013

[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004

[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008

[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010

[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007

[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012

[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)

[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009

[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995

[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012

[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011

[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994

[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012

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

[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008

[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)

[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004

[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007

[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007

[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010

[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002

[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013

[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008

[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004

[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012

[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012

[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013

[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004

[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008

[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010

[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007

[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012

[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)

[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009

[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995

[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012

[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011

[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994

[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012

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

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