Research ArticleIterative Learning Control with Extended State Observer forTelescope System

Huaxiang Cai123 Yongmei Huang12 Junfeng Du12

Tao Tang12 Dan Zuo123 and Jinying Li12

1Chinese Academy of Sciences Institute of Optics and Electronics PO Box 350 Shuangliu Chengdu 610209 China2Key Laboratory of Optical Engineering Chinese Academy of Sciences Chengdu China3University of Chinese Academy of Sciences Beijing 100039 China

Correspondence should be addressed to Huaxiang Cai chx ioe163com

Received 4 March 2015 Revised 24 June 2015 Accepted 1 July 2015

Academic Editor Xinggang Yan

Copyright copy 2015 Huaxiang Cai et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An Iterative Learning Control (ILC) method with Extended State Observer (ESO) is proposed to enhance the tracking precisionof telescope Telescope systems usually suffer some uncertain nonlinear disturbances such as nonlinear friction and unknowndisturbances Thereby to ensure the tracking precision the ESO which can estimate system states (including parts of uncertainnonlinear disturbances) is introduced The nonlinear system is converted to an approximate linear system by making use of theESO Besides to make further improvement on the tracking precision we make use of the ILC method which can find an idealcontrol signal by the process of iterative learning Furthermore this control method theoretically guarantees a prescribed trackingperformance and final tracking accuracy Finally a few comparative experimental results show that the proposed control methodhas excellent performance for reducing the tracking error of telescope system

1 Introduction

A high performance telescope system has attracted lots ofattention since it is widely used in space applications suchas space observer [1] and satellite surveillance [2] Howeverit is inevitable that there are always some nonlinear charac-teristics in the telescope tracking control systems such asdeadzone [3] friction [4] and some uncertainly disturbances[5] Without doubt all the nonlinear features will deterioratethe tracking performance of systems Simultaneously theconventional linear control method (proportional-integral)has not guaranteed a higher tracking precision To solve theabove nonlinear problems there are many control methodsbeing proposed such as adaptive control [6] slide control [7]compensating control based on DOB [8] active disturbancerejection control (ADRC) [9] and iterative learning control(ILC) [10]

In recent years since the iterative learning control (ILC)needs too little knowledge of system dynamics it has receiveda great deal of attention [11] ILC is proposed by Arimoto

et al in 1984 [12] It is essentially a feedforward controlapproach that fully utilizes the previous control information[13] Over the past three decades it has been successfullyused in extensive research fields such as industrial robotics[14] manufacturing process [15] stochastic process controlsystem [16] and hysteretic system [17] Nevertheless theILC still has some of its inherent shortages It can onlyeliminate some disturbances which emerge repeatedly Ifthere are some nonlinear nonperiod disturbances involvedin the systems the sole ILC method will face difficulty toget a higher tracking precision The ADRC is different fromthe ILCwhich can estimate unknownnonlinear disturbancesinvolved in the systems As the core technology of ADRC theESO also needs to know little knowledge of system dynamicswhich only needs the information of systemrsquos input andoutputThereby it has beenwidely used inmany fields such asmissile control system [18] optical system [19] and other highprecision occasions [20] Even though the ESO has excellentestimation ability for system states its estimation ability is stillrestricted to the bandwidth of ESO

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 701510 8 pageshttpdxdoiorg1011552015701510

2 Mathematical Problems in Engineering

In this paper since there are some uncertain nonlineardisturbances in the telescope system which reduce severelythe tracking precision of system it is needed to eliminate thenonlinear disturbances and find the best input signal for thesystemUnder the circumstances an ILC techniquewith ESOis proposed which takes use of the advantages of the twocontrol methods (ILC and ESO) The function of ESO is toeliminate parts of uncertain nonlinear disturbances and totransform the nonlinear system into an approximate linearsystem And the function of ILC is to get the best input signalfor the approximate linear system In the ILC controllera PD-type learning algorithm with the forgetting factoris chosen The forgetting factor can balance the learningprecision and the robustness of system To illustrate theexcellent performance of the proposed control method threecomparative experiments will be given in this paper

This remainder of this paper is organized as followsSection 2 gets the dynamic models Section 3 introduces thecontroller design and the theory analysis The experimentresults are presented in Section 4 And some conclusions canbe found in Section 5

2 Dynamic Models

The tracking control system is made of brushless DC loadhigh precision encoder servo actuator and host computerThe goal is to make the system track the given referencemotion trajectory as far as possible The control structure isshown in Figure 1

In the telescope system since the electrical response of theactuator is very fast the current dynamics can be neglectedAnd the dynamics of the tracking system is

119889120579 (119905)

119889119905= 120596 (119905)

119889120596 (119905)

119889119905=

119906

119869minus]119869

120596 (119905) minus 119891 (119905 120596)

119906 = 119896119905119894 119869 = 119869

119898+ 119869119897

(1)

In (1) 119869119898and 119869119897are the motor inertia and the load inertia

respectively 120579 and 120596 represent for the load angle positionand the load angle velocity respectively 119906 represents thetorque imposed on the motor 119894 is the control input currentof the motor 119891(119905 120596) represents nonlinear disturbances (suchas the friction disturbance external disturbances and someunmodeled dynamics) ] is the viscous coefficient

Rewrite the dynamic model (1) as

119889120579 (119905)

119889119905= 120596 (119905)

119889120596 (119905)

119889119905= 1198870119906 + 1198911 (119905 120596)

119906 = 119896119905119894 119869 = 119869

119898+ 119869119897

(2)

1198911(119905 120596) = (1119869 minus 1198870)119906 minus (]119869)120596(119905) minus 119891(119905 120596) whichincludes the nonlinear disturbances minus119891(119905 120596) and the innerdisturbances (1119869 minus 1198870)119906 minus (]119869)120596(119905) Here 1198870 is a parameterthat can be adjusted in the controller

Position feedback

Encoder

LoadE axis

Power supplyMotor

Actuator120579ref Motion

controller

Figure 1 Architecture of tracking system

3 Controller Design

The aim of designing controller is eliminating the nonlineardisturbance 1198911(119905 120596) and enhancing the tracking precision ofsystem The ESO makes the nonlinear system transform intoan approximate linear system And the ILC with ESO gets theideal input control signal

In most of telescope systems the multiple-loops controlstructure is usually applied to guarantee the tracking preci-sion of systems which includes position loop velocity loopand current loop In this paper since the current dynamic hashigher frequency range than other loops it has been ignoredWe mainly analyze position loop and speed loop

31 Design PI Controller with ESO

311 Design the ESO Before designing the ESO we assumethat the disturbance 1198911(119905 120596) is continuous and differential

Assumption 1 The disturbance 1198911(119905 120596) is continuous anddifferential and ℎ = minus1198891198911(119905 120596)119889119905 Besides ℎ(119905) is boundednamely there is a positive constant 120573 to meet

120573 = sup 119905 gt 0 |ℎ (119905)| (3)

Consequently we extend the 1198911(119905 120596) as a separate state1199093 of system Then the original plant of system (2) can bedescribed as

1 (119905) = 1199092 (119905)

2 (119905) = 1199093 (119905) + 1198870119906 (119905)

3 (119905) = minus ℎ

(4)

where 119909 = [1199091 1199092 1199093]119879

= [120579 120596 1198911]119879

By (4) we can know that it is observable Hence an ESOcan be constructed as (5)

1199091 (119905) = 1199092 (119905) minus 12057301 [1199091 (119905) minus 1199091 (119905)]

1199092 (119905) = 1199093 (119905) minus 12057302 [1199091 (119905) minus 1199091 (119905)] + 1198870119906 (119905)

1199093 (119905) = minus 12057303 [1199091 (119905) minus 1199091 (119905)]

(5)

Mathematical Problems in Engineering 3

where 12057301 12057302 and 12057303 are the parameters that will bedesigned in the ESO

The aim of ESO is to make

1199091 (119905) 997888rarr 1199091 (119905)

1199092 (119905) 997888rarr 1199092 (119905)

1199093 (119905) 997888rarr 1198911 (119905 120596)

(6)

Let 120578119894

= 119909119894minus 119909119894 119894 = 1 2 3 denote the estimation error

subtracting (4) from (5) then we can obtain

120578 = 119860120578 + 119872ℎ (7)

where

120578 =[[

[

1205781

1205782

1205783

]]

]

119860 =[[

[

minus12057301 1 0minus12057302 0 1minus12057303 0 0

]]

]

119872 =[[

[

001

]]

]

(8)

Therefore if the matrix 119860 is Hurwitz the ESO willbe Bounded-Input Bounded-Output (BIBO) stable In thematrix 119860 the parameters 12057301 12057302 and 12057303 can be designedas [21]

12057301 = 31205960

12057302 = 312059620

12057303 = 12059630

(9)

where 1205960 is the bandwidth of ESO Equation (9) can promisethe matrix 119860 being Hurwitz

Lemma 2 If the ℎ(119905) is bounded there will be a positiveconstant 120577

119894gt 0 and a finite time 1198791 gt 0 such that

1003816100381610038161003816120578119894 (119905)1003816100381610038161003816 le 120577119894

120577119894= o(

11205961198880

)

119894 = 1 2 3 forall119905 ge 1198791

(10)

The proof can be found in the literature [22]

Remark 3 Equation (9) makes the ESO (5) a BIBO stableobserver The result of Lemma 2 shows that the ESO has anexcellent estimation ability for disturbances Its estimationability depends on the bandwidth 1205960 of ESO When thebandwidth 1205960 is big enough it can estimate accurately 1199091(119905)1199092(119905) and 1198911(119905 120596)

According to the analysis in Lemma 2 when designing anappropriate parameter1205960 it canmake 119900(1120596

119888

0) asymp 0 And therewill be |1199093minus1199093| = |1205783(119905)| le 119900(1120596

119888

0) namely the 1199093 asymp 1198911(119905 120596)So we can transform the input 119906 into 119906

120596 namely 119906 = (119906

120596minus

1199093)1198870Substituting the 119906 = (119906

120596minus 1199093)1198870 to (2) we can get

119889120579 (119905)

119889119905= 120596 (119905)

119889120596 (119905)

119889119905asymp 119906120596

(11)

312 Design Velocity Loop PI Controller The nonlinear sys-tem (2) is converted to an approximate linear integral system(11) So we can design a PI controller in the velocity loop

119906120596

= 119870120596

119901(120596

refminus 1199092) + 119870

120596

119894int

119905

0[120596

ref(120591) minus 1199092 (120591)] 119889120591 (12)

where 120596ref is the input signal of velocity loop and 119870

120596

119901 119870120596119894are

the proportion gain and the integral gain in the velocity loopcontroller respectively

313 Design Position Loop PI Controller Similarly we candesign a PI controller in the position loop

120596ref

= 119870120579

119901(120579

refminus 1199091) + 119870

120579

119894int

119905

0[120579

ref(120591) minus 1199091 (120591)] 119889120591 (13)

where 120579ref is the input signal of position loop and 119870

120596

119901 119870120596119894are

the proportion gain and the integral gain in the position loopcontroller respectively

32 Iterative Learning Controller with ESO Design From(11) we can see that the system is only an approximatelinear system Thereby to make further improvement on thetracking precision of system we will find an ideal input 119906

120596by

the method of iterative learning And it can make the output120596 of system close to the input 120596

ref as far as possible In thissection we will design an ILC to substitute the PI controllerin the velocity loop

According to (11) the state space equation of system canbe obtained as

(119905) = 119860119897119909 + 119861119906

120596

119910 = 119862119909

(14)

where 119909 = [11990911199092 ] = [ 120579

120596]119860119897= [ 0 1

0 0 ]119861 = [ 01 ] and119862 = [ 0 1 ]

Tomake the system track the desired reference trajectorywe assume that there is an ideal input signal meeting thecondition

Assumption 4 The ideal input signal 119906119889(119905) can make the

system track the desired reference trajectory 119910119889(119905)

119889

(119905) = 119860119897119909119889

(119905) + 119861119906119889

(119905)

119910119889

(119905) = 119862119909119889

(119905)

(15)

where 119909119889(119905) 119910

119889(119905) represent the ideal states of system

respectively

4 Mathematical Problems in Engineering

120579ref

x1

minus minus minus

e120579K120579p

+

+

+

++

x2

ek

Φ

Γd

dt

1 minus 120572

x3

Memory

ESO

Plant

uk

uk+11

b0

u

f

120579

b0

K120579i intd120591

120596ref

Figure 2 Schematic diagram of the proposed iterative learning control with ESO

Since system (14) will be tracked a period trajectory therewill be once input in every period Then the state spaceequation of system (14) can be rewritten

119896

(119905) = 119860119897119909119896

+ 119861119906119896

119910119896

= 119862119909119896

(16)

where 119896 is the times Equation (16) means the 119896th states ofsystem under the 119896th input 119906

119896

For system (16) a PD-type learning algorithm with theforgetting factor 120572 is selected The factor 120572 is introduced tomake a tradeoff between perfect learning and robustnesswhich can increase the robustness of ILC against noiseinitialization error and fluctuation of system dynamics [23]The selected iterative learning scheme is found in (17) whoseschematic diagram is depicted in Figure 2

119906119896+1 (119905) = (1minus 120572) 119906

119896(119905) + 1205721199060 (119905) + Φ (119890

119896(119905))

+ Γ ( 119890119896

(119905))

(17)

where Φ and Γ are the proportion gain matrix and deviationgain matrix respectively Φ isin R119898times119902 Γ isin R119898times119902 are bounded119896 is the times 119890

119896(119905) = 119910

119889(119905) minus 119910

119896(119905) is the iterative error in the

119896th times

33 Iterative Learning Controller Convergence Analysis

Theorem 5 If the system described by (16) satisfies assump-tions and uses the update law (18) Given a desired trajectory119910119889(119905) and an initial state 119909

119889(0) which are achievable if

(1minus 120572) 119868 minus Γ119862119861 le 120588 lt 1 (18)

then as 119896 rarr infin the iterative output 119910119896(119905) of system will

converge to the desired reference trajectory 119910119889(119905)

Proof Consider

119906119889

(119905) minus 119906119896+1 (119905) = 119906

119889(119905) minus (1minus 120572) 119906

119896(119905) minus 1205721199060 (119905)

minus Φ (119890119896

(119905)) minus Γ (119889 (119890119896

(119905))

119889119905)

= (1minus 120572) [119906119889

(119905) minus 119906119896

(119905)]

+ 120572 [119906119889

(119905) minus 1199060 (119905)]

minus Φ119862 [119909119889

(119905) minus 119909119896

(119905)]

minus Γ119862119860 [119909119889

(119905) minus 119909119896

(119905)]

minus Γ119862119861 [119906119889

(119905) minus 119906119896

(119905)]

= [(1minus 120572) 119868 minus Γ119862119861] [119906119889

(119905) minus 119906119896

(119905)]

minus (Φ119862 + Γ119862119860) [119909119889

(119905) minus 119909119896

(119905)]

(19)

In (19) by the state space equation (15) and (16) the119909119889(119905) minus 119909

119896(119905) can be transformed into

119909119889

(119905) minus 119909119896

(119905) = 119890119860119905

[119909119889

(0) minus 119909119896

(0)]

+ int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896

(120591)] 119889120591

(20)

Substituting (20) to (19) we can have

119906119889

(119905) minus 119906119896+1 (119905)

= [(1minus 120572) 119868 minus Γ119862119861] [119906119889

(119905) minus 119906119896

(119905)]

minus (Φ119862 + Γ119862119860) 119890119860119905

[119909119889

(0) minus 119909119896

(0)]

minus (Φ119862 + Γ119862119860) int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896

(120591)] 119889120591

(21)

Mathematical Problems in Engineering 5

Taking the norm sdot on both sides of (21) we obtain1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817

le (1minus 120572) 119868 minus Γ119862119861 sdot1003817100381710038171003817119906119889

(119905) minus 119906119896

(119905)1003817100381710038171003817

+ 119890119860119905

Φ119862 + Γ119862119860 sdot1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

+ Φ119862 + Γ119862119860

sdot int

119905

0119890119860(119905minus120591)

1198611003817100381710038171003817119906119889

(120591) minus 119906119896

(120591)1003817100381710038171003817 119889120591

(22)

Multiplying by 119890minus120582119905 120582 gt 119860 and we have

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817 119890minus120582119905

le (1minus 120572) 119868 minus Γ119862119861 sdot1003817100381710038171003817119906119889

(119905)

minus 119906119896

(119905)1003817100381710038171003817 119890minus120582119905

minus Φ119862 + Γ119862119860 sdot1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

sdot 119890(119860minus120582)119905

+ Φ119862 + Γ119862119860

sdot int

119905

0119890(119860minus120582)(119905minus120591)

119890minus120582120591

1198611003817100381710038171003817119906119889

(120591) minus 119906119896

(120591)1003817100381710038171003817 119889120591

le (1minus 120572) 119868 minus Γ119862119861 sdot1003817100381710038171003817119906119889

(119905) minus 119906119896

(119905)1003817100381710038171003817 119890minus120582119905

minus Φ119862

+ Γ119862119860 sdot1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817 119890(119860minus120582)119905

+Φ119862 + Γ119862119860 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

1003817100381710038171003817119906119889

(119905)

minus 119906119896

(119905)1003817100381710038171003817 119890minus120582119905

(23)

Definition 6 Define 120582 norm for a function ℎ [0 119879] rarr R119896

[24] then

ℎ (sdot)120582

≜ sup119905isin[0119879]

119890minus120582119905

ℎ (sdot) 997888rarr R119896 (24)

Then

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817120582 le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(1minus 120572) 119868 minus Γ119862119861

+Φ119862 + Γ119862119860 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

10038171003817100381710038171003817100381710038171003817100381710038171003817

sdot1003817100381710038171003817119906119889

(119905)

minus 119906119896

(119905)1003817100381710038171003817120582 minus Φ119862 + Γ119862119860 sdot

1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

(25)

Since the (1 minus 120572)119868 minus Γ119862119861 le 120588 lt 1 we can find a 120582 gt 119860

which makes

120588 =

10038171003817100381710038171003817100381710038171003817100381710038171003817

(1minus 120572) 119868 minus Γ119862119861

+Φ119862 + Γ119862119860 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

10038171003817100381710038171003817100381710038171003817100381710038171003817

lt 1

(26)

So we can get

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817120582 leΦ119862 + Γ119862119860

1 minus 120588

1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817 (27)

Remark 7 It shows that 119906119896converges to the 119906

119889of radius

(Φ119862 + Γ119862119860(1 minus 120588))119909119889(0) minus 119909

119896(0) with respect to the

120582 norm Besides we can know that the convergent radiusis bounded And the boundary depends on the initial inputstates of system

Consequently the tracking error of system will be

119890119896+1 (119905) = 119910

119889(119905) minus 119910

119896+1 (119905) = 119862 sdot 119890119860119905

[119909119889

(0) minus 119909119896

(0)]

+ int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896

(120591)] 119889120591

(28)

Similarly taking the norm sdot 120582on both sides of (28) we

can get

1003817100381710038171003817119890119896+1 (119905)

1003817100381710038171003817120582 =1003817100381710038171003817119910119889

(119905) minus 119910119896+1 (119905)

1003817100381710038171003817120582 le 119862

sdot

10038171003817100381710038171003817100381710038171003817119890119860119905

[119909119889

(0) minus 119909119896+1 (0)]

+ int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896+1 (120591)] 119889120591

10038171003817100381710038171003817100381710038171003817120582le 119862

sdot1003817100381710038171003817[119909119889

(0) minus 119909119896+1 (0)]

1003817100381710038171003817

+119862 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817120582

le 119862 sdot1003817100381710038171003817[119909119889

(0) minus 119909119896+1 (0)]

1003817100381710038171003817

+119862 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

sdotΦ119862 + Γ119862119860

1 minus 120588

1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

(29)

Specially the system usually can meet the initial condi-tion that is

119909119889

(0) = 119909119896

(0) 119896 = 1 2 3 (30)

That is said as

lim119896rarrinfin

1003817100381710038171003817119890119896+1 (119905)

1003817100381710038171003817120582 997888rarr 0 (31)

To sum up when the PD-type learning algorithm isapplied to the ILC with ESO the input 119906

119896(119905) of system can

converge to the ideal input 119906119889(119905) and the output of system

can converge to the ideal output 119910119889(119905)

4 Experiment Setup and Result

The verification platform is the rotary table of telescopesystem which consists of a DC motor a tracking loadan electrical driver a control system and a high precisionencoder whose accuracy is about plusmn0618 (arc-second) Thecontrol scheme includes two loops position loop and velocityloop

6 Mathematical Problems in Engineering

040302010

minus01minus02minus03minus04

0 10 20 30 40 50 60

Ang

le (d

eg)

t (s)

Reference signal

120572 = 314 (∘s2) 120596 = 10 (∘s)

Figure 3 The desired reference signal

The following three controllers are compared

(1) PI-PI this is the traditional Proportional-Integral (PI)controller In the velocity loop and position loop weuse two PI controllers The velocity signal can beobtained by differentiating the position signal

(2) PI-ESO in both of the position loop and the velocityloop the PI controllers are used but the velocitysignal will be gotten by the ESO The bandwidth 1205960of ESO is 1205960 = 100 The parameters of position looprsquosPI controller remain the same

(3) PI-ILC-ESO the PI controller is used in the positionloop and the ILC controller is used in the speed loopThe parameters of position looprsquos PI controller stillremain the same The velocity signal is also the signalwhich is gotten by the ESO To satisfy the condition(18) The factor 120572 and the deviation gain matrix Γ arechosen as 120572 = 02 and Γ = 05 By the state spaceequation of system (14) it will have

(1minus 120572) 119868 minus Γ119862119861 =

1003817100381710038171003817100381710038171003817100381710038171003817

(1minus 02) 119868 minus 05times [0 1] [01]

1003817100381710038171003817100381710038171003817100381710038171003817

= 03 lt 1

(32)

The three controllers are tested for a sinusoidal signalwhose velocity and acceleration are 120596 = 10 (

∘s) and 120572 =

314 (∘s2) The period of 119879 sinusoidal signal is 2 (s)

The desired reference trajectory is shown in Figure 3And the corresponding tracking performance under the threecontrollers is shown in Figures 4ndash6 As seen the PI-PIcontroller has a larger tracking error whose maximum valueis about 288410158401015840 Comparatively speaking the PI controllerwith ESO has a better tracking performance which illustratesthat ESO has an excellent estimation ability for disturbancesinvolved in the system Furthermore the proposed PI-ILC-ESO controller has the least tracking error than other controlmethods whose maximum value of error is about 200610158401015840This controller combines the advantages of ESO with theadvantages of ILC It makes use of ESO to eliminate most ofthe nonlinear disturbances Simultaneously the ILC gets thebest control input by learning previous control informationSome detailed results of the three controllers are shown inTable 1

0 10 20 30 40 50 60t (s)

PI-PI

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 4 The tracking error of PI-PI

0 10 20 30 40 50 60t (s)

PI-ESO

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 5 The tracking error of PI-ESO

0 10 20 30 40 50 60t (s)

PI-ILC-ESO

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 6 The tracking error of PI-ILC-ESO

Table 1 The experimental results of the three controllers

PI-PI PI-ESO PI-ILC-ESO|maximum error| 28843210158401015840 20998810158401015840 20059210158401015840

Root mean square error 9830110158401015840 6028210158401015840 5646810158401015840

From Table 1 it can be seen that the PI-ILC-ESO hasbetter performance than other control methods Howevercomparing the PI-ESO controller with the PI-ILC-ESO con-troller it can be seen that the effectiveness of using ILCis not obvious The maximum error reduces only to 09410158401015840and the root mean square error reduces to 03810158401015840 In factComparing with Figures 5 and 6 it can be found that theerrors of many position points exceed 1010158401015840 and the errorsare close to 2010158401015840 when the PI-ESO controller is used Butwhen the PI-ILC-ESO controller is used almost all the errorsof position points are near 1010158401015840 So the conclusion that thecomprehensive performance of PI-ILC-ESO is better than

Mathematical Problems in Engineering 7

other control methods can be gotten The ILC method helpsthe system get the best input signal Besides in Figures 5and 6 it can be found that the errors are evenly distributedComparing with the Figure 4 (PI-PI controller) both of thetwo controllers have reduced almost all the turning errorsof system The remaining errors are mostly caused by thesignal noise So it is difficult to reduce the remaining error andfurther improve the precision of system unless the problemof signal noise is solved

5 Conclusion

Uncertain nonlinear disturbances have been themajor factorswhich restrict the performance of tracking control systemsThis is because the systems will have a low tracking precisionwhen some uncertain nonlinear disturbances are induced inthe systems Therefore to reduce the influence introducedby the uncertain nonlinear disturbances an ILC methodwith ESO is proposed in this paper The ILC can get anexcellent input signal by learning previous control informa-tion It owns a better ability for eliminating some perioddisturbances Meanwhile an ESO is designed for estimatingsome uncertain nonlinear disturbances It compensates theshortage of the ILCrsquos disablement for nonperiod disturbancesIn addition the ESO can estimate an accurate velocity signaland supply the velocity as the feedback input signal ofiterative learning controller Furthermore the convergenceanalysis of the proposed control method guarantees therobustness of system Finally the experiment results showthat the proposed control method has excellent performancefor reducing the tracking error of telescope system

Conflict of Interests

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

References

[1] G G Valyavin V D Bychkov M V Yushkin et al ldquoHigh-resolution fiber-fed echelle spectrograph for the 6-m telescopeI Optical scheme arrangement and control systemrdquoAstrophys-ical Bulletin vol 69 no 2 pp 224ndash239 2014

[2] K Lu and Y Xia ldquoAdaptive attitude tracking control for rigidspacecraft with finite-time convergencerdquo Automatica vol 49no 12 pp 3591ndash3599 2013

[3] Y-J Sun ldquoComposite tracking control for generalized practicalsynchronization of Duffing-Holmes systems with parametermismatching unknown external excitation plant uncertaintiesand uncertain deadzone nonlinearitiesrdquo Abstract and AppliedAnalysis vol 2012 Article ID 640568 11 pages 2012

[4] Y Fu and T Chai ldquoSelf-tuning control with a filter and aneural compensator for a class of nonlinear systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 24no 5 pp 837ndash843 2013

[5] M M Bridges D M Dawson and J Hu ldquoAdaptive control fora class of direct drive robot manipulatorsrdquo International Journalof Adaptive Control and Signal Processing vol 10 no 4-5 pp417ndash441 1996

[6] A Tadayoni W-F Xie and B W Gordon ldquoAdaptive con-trol of harmonic drive with parameter varying friction usingstructurally dynamic wavelet networkrdquo International Journal ofControl Automation and Systems vol 9 no 1 pp 50ndash59 2011

[7] Y Alipouri and J Poshtan ldquoDesigning a robust minimum vari-ance controller using discrete slide mode controller approachrdquoISA Transactions vol 52 no 2 pp 291ndash299 2013

[8] B-Z Guo and F-F Jin ldquoSliding mode and active disturbancerejection control to stabilization of one-dimensional anti-stablewave equations subject to disturbance in boundary inputrdquo IEEETransactions on Automatic Control vol 58 no 5 pp 1269ndash12742013

[9] Y-S Lu and S-M Lin ldquoDisturbance-observer-based adaptivefeedforward cancellation of torque ripples in harmonic drivesystemsrdquo Electrical Engineering vol 90 no 2 pp 95ndash106 2007

[10] T Kavli ldquoFrequency domain synthesis of trajectory learningcontrollers for robot manipulatorsrdquo Journal of Robotic Systemsvol 9 no 5 pp 663ndash680 1992

[11] Y-C Wang and C-J Chien ldquoAn observer-based adaptiveiterative learning control using filtered-FNN design for roboticsystemsrdquo Advances in Mechanical Engineering vol 6 Article ID471418 2014

[12] S Arimoto S Kawamura and FMiyazaki ldquoBettering operationof Robots by learningrdquo Journal of Robotic Systems vol 1 no 2pp 123ndash140 1984

[13] J-X Xu D Huang V Venkataramanan and The Cat TuongHuynh ldquoExtreme precise motion tracking of piezoelectricpositioning stage using sampled-data iterative learning controlrdquoIEEE Transactions on Control Systems Technology vol 21 no 4pp 1432ndash1439 2013

[14] M Norrlof ldquoAn adaptive iterative learning control algorithmwith experiments on an industrial robotrdquo IEEE Transactions onRobotics and Automation vol 18 no 2 pp 245ndash251 2002

[15] J-X Xu Y Chen T H Lee and S Yamamoto ldquoTerminaliterative learning control with an application to RTPCVDthickness controlrdquo Automatica vol 35 no 9 pp 1535ndash15421999

[16] S S Saab ldquoA discrete-time stochastic learning control algo-rithmrdquo IEEE Transactions on Automatic Control vol 46 no 6pp 877ndash887 2001

[17] K K Leang and S Devasia ldquoDesign of hysteresis-compensatingiterative learning control for piezo-positioners application toatomic force microscopesrdquo Mechatronics vol 16 no 3-4 pp141ndash158 2006

[18] Z Zhu D Xu J Liu and Y Xia ldquoMissile guidance law basedon extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 60 no 12 pp 5882ndash5891 2013

[19] M Pizzocaro D Calonico C Calosso et al ldquoActive disturbancerejection control of temperature for ultrastable optical cavitiesrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 60 no 2 pp 273ndash280 2013

[20] J Yao Z Jiao and D Ma ldquoExtended-state-observer-basedoutput feedback nonlinear robust control of hydraulic systemswith backsteppingrdquo IEEE Transactions on Industrial Electronicsvol 61 no 11 pp 6285ndash6293 2014

[21] J Yao Z Jiao and D Ma ldquoAdaptive robust control of dc motorswith extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 61 no 7 pp 3630ndash3637 2014

[22] Q Zheng L Q Gao and Z Gao ldquoOn stability analysis ofactive disturbance rejection control for nonlinear time-varyingplants with unknown dynamicsrdquo inProceedings of the 46th IEEE

8 Mathematical Problems in Engineering

Conference on Decision and Control (CDC rsquo07) pp 3501ndash3506New Orleans La USA December 2007

[23] WQian S K Panda and J-X Xu ldquoTorque rippleminimizationin PM synchronous motors using iterative learning controlrdquoIEEE Transactions on Power Electronics vol 19 no 2 pp 272ndash279 2004

[24] G Heinzinger D Fenwick B Paden and F Miyazaki ldquoStabilityof learning control with disturbances and uncertain initialconditionsrdquo IEEETransactions onAutomatic Control vol 37 no1 pp 110ndash114 1992

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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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

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

In this paper since there are some uncertain nonlineardisturbances in the telescope system which reduce severelythe tracking precision of system it is needed to eliminate thenonlinear disturbances and find the best input signal for thesystemUnder the circumstances an ILC techniquewith ESOis proposed which takes use of the advantages of the twocontrol methods (ILC and ESO) The function of ESO is toeliminate parts of uncertain nonlinear disturbances and totransform the nonlinear system into an approximate linearsystem And the function of ILC is to get the best input signalfor the approximate linear system In the ILC controllera PD-type learning algorithm with the forgetting factoris chosen The forgetting factor can balance the learningprecision and the robustness of system To illustrate theexcellent performance of the proposed control method threecomparative experiments will be given in this paper

This remainder of this paper is organized as followsSection 2 gets the dynamic models Section 3 introduces thecontroller design and the theory analysis The experimentresults are presented in Section 4 And some conclusions canbe found in Section 5

2 Dynamic Models

The tracking control system is made of brushless DC loadhigh precision encoder servo actuator and host computerThe goal is to make the system track the given referencemotion trajectory as far as possible The control structure isshown in Figure 1

In the telescope system since the electrical response of theactuator is very fast the current dynamics can be neglectedAnd the dynamics of the tracking system is

119889120579 (119905)

119889119905= 120596 (119905)

119889120596 (119905)

119889119905=

119906

119869minus]119869

120596 (119905) minus 119891 (119905 120596)

119906 = 119896119905119894 119869 = 119869

119898+ 119869119897

(1)

In (1) 119869119898and 119869119897are the motor inertia and the load inertia

respectively 120579 and 120596 represent for the load angle positionand the load angle velocity respectively 119906 represents thetorque imposed on the motor 119894 is the control input currentof the motor 119891(119905 120596) represents nonlinear disturbances (suchas the friction disturbance external disturbances and someunmodeled dynamics) ] is the viscous coefficient

Rewrite the dynamic model (1) as

119889120579 (119905)

119889119905= 120596 (119905)

119889120596 (119905)

119889119905= 1198870119906 + 1198911 (119905 120596)

119906 = 119896119905119894 119869 = 119869

119898+ 119869119897

(2)

1198911(119905 120596) = (1119869 minus 1198870)119906 minus (]119869)120596(119905) minus 119891(119905 120596) whichincludes the nonlinear disturbances minus119891(119905 120596) and the innerdisturbances (1119869 minus 1198870)119906 minus (]119869)120596(119905) Here 1198870 is a parameterthat can be adjusted in the controller

Position feedback

Encoder

LoadE axis

Power supplyMotor

Actuator120579ref Motion

controller

Figure 1 Architecture of tracking system

3 Controller Design

The aim of designing controller is eliminating the nonlineardisturbance 1198911(119905 120596) and enhancing the tracking precision ofsystem The ESO makes the nonlinear system transform intoan approximate linear system And the ILC with ESO gets theideal input control signal

In most of telescope systems the multiple-loops controlstructure is usually applied to guarantee the tracking preci-sion of systems which includes position loop velocity loopand current loop In this paper since the current dynamic hashigher frequency range than other loops it has been ignoredWe mainly analyze position loop and speed loop

31 Design PI Controller with ESO

311 Design the ESO Before designing the ESO we assumethat the disturbance 1198911(119905 120596) is continuous and differential

Assumption 1 The disturbance 1198911(119905 120596) is continuous anddifferential and ℎ = minus1198891198911(119905 120596)119889119905 Besides ℎ(119905) is boundednamely there is a positive constant 120573 to meet

120573 = sup 119905 gt 0 |ℎ (119905)| (3)

Consequently we extend the 1198911(119905 120596) as a separate state1199093 of system Then the original plant of system (2) can bedescribed as

1 (119905) = 1199092 (119905)

2 (119905) = 1199093 (119905) + 1198870119906 (119905)

3 (119905) = minus ℎ

(4)

where 119909 = [1199091 1199092 1199093]119879

= [120579 120596 1198911]119879

By (4) we can know that it is observable Hence an ESOcan be constructed as (5)

1199091 (119905) = 1199092 (119905) minus 12057301 [1199091 (119905) minus 1199091 (119905)]

1199092 (119905) = 1199093 (119905) minus 12057302 [1199091 (119905) minus 1199091 (119905)] + 1198870119906 (119905)

1199093 (119905) = minus 12057303 [1199091 (119905) minus 1199091 (119905)]

(5)

Mathematical Problems in Engineering 3

where 12057301 12057302 and 12057303 are the parameters that will bedesigned in the ESO

The aim of ESO is to make

1199091 (119905) 997888rarr 1199091 (119905)

1199092 (119905) 997888rarr 1199092 (119905)

1199093 (119905) 997888rarr 1198911 (119905 120596)

(6)

Let 120578119894

= 119909119894minus 119909119894 119894 = 1 2 3 denote the estimation error

subtracting (4) from (5) then we can obtain

120578 = 119860120578 + 119872ℎ (7)

where

120578 =[[

[

1205781

1205782

1205783

]]

]

119860 =[[

[

minus12057301 1 0minus12057302 0 1minus12057303 0 0

]]

]

119872 =[[

[

001

]]

]

(8)

Therefore if the matrix 119860 is Hurwitz the ESO willbe Bounded-Input Bounded-Output (BIBO) stable In thematrix 119860 the parameters 12057301 12057302 and 12057303 can be designedas [21]

12057301 = 31205960

12057302 = 312059620

12057303 = 12059630

(9)

where 1205960 is the bandwidth of ESO Equation (9) can promisethe matrix 119860 being Hurwitz

Lemma 2 If the ℎ(119905) is bounded there will be a positiveconstant 120577

119894gt 0 and a finite time 1198791 gt 0 such that

1003816100381610038161003816120578119894 (119905)1003816100381610038161003816 le 120577119894

120577119894= o(

11205961198880

)

119894 = 1 2 3 forall119905 ge 1198791

(10)

The proof can be found in the literature [22]

Remark 3 Equation (9) makes the ESO (5) a BIBO stableobserver The result of Lemma 2 shows that the ESO has anexcellent estimation ability for disturbances Its estimationability depends on the bandwidth 1205960 of ESO When thebandwidth 1205960 is big enough it can estimate accurately 1199091(119905)1199092(119905) and 1198911(119905 120596)

According to the analysis in Lemma 2 when designing anappropriate parameter1205960 it canmake 119900(1120596

119888

0) asymp 0 And therewill be |1199093minus1199093| = |1205783(119905)| le 119900(1120596

119888

0) namely the 1199093 asymp 1198911(119905 120596)So we can transform the input 119906 into 119906

120596 namely 119906 = (119906

120596minus

1199093)1198870Substituting the 119906 = (119906

120596minus 1199093)1198870 to (2) we can get

119889120579 (119905)

119889119905= 120596 (119905)

119889120596 (119905)

119889119905asymp 119906120596

(11)

312 Design Velocity Loop PI Controller The nonlinear sys-tem (2) is converted to an approximate linear integral system(11) So we can design a PI controller in the velocity loop

119906120596

= 119870120596

119901(120596

refminus 1199092) + 119870

120596

119894int

119905

0[120596

ref(120591) minus 1199092 (120591)] 119889120591 (12)

where 120596ref is the input signal of velocity loop and 119870

120596

119901 119870120596119894are

the proportion gain and the integral gain in the velocity loopcontroller respectively

313 Design Position Loop PI Controller Similarly we candesign a PI controller in the position loop

120596ref

= 119870120579

119901(120579

refminus 1199091) + 119870

120579

119894int

119905

0[120579

ref(120591) minus 1199091 (120591)] 119889120591 (13)

where 120579ref is the input signal of position loop and 119870

120596

119901 119870120596119894are

the proportion gain and the integral gain in the position loopcontroller respectively

32 Iterative Learning Controller with ESO Design From(11) we can see that the system is only an approximatelinear system Thereby to make further improvement on thetracking precision of system we will find an ideal input 119906

120596by

the method of iterative learning And it can make the output120596 of system close to the input 120596

ref as far as possible In thissection we will design an ILC to substitute the PI controllerin the velocity loop

According to (11) the state space equation of system canbe obtained as

(119905) = 119860119897119909 + 119861119906

120596

119910 = 119862119909

(14)

where 119909 = [11990911199092 ] = [ 120579

120596]119860119897= [ 0 1

0 0 ]119861 = [ 01 ] and119862 = [ 0 1 ]

Tomake the system track the desired reference trajectorywe assume that there is an ideal input signal meeting thecondition

Assumption 4 The ideal input signal 119906119889(119905) can make the

system track the desired reference trajectory 119910119889(119905)

119889

(119905) = 119860119897119909119889

(119905) + 119861119906119889

(119905)

119910119889

(119905) = 119862119909119889

(119905)

(15)

where 119909119889(119905) 119910

119889(119905) represent the ideal states of system

respectively

4 Mathematical Problems in Engineering

120579ref

x1

minus minus minus

e120579K120579p

+

+

+

++

x2

ek

Φ

Γd

dt

1 minus 120572

x3

Memory

ESO

Plant

uk

uk+11

b0

u

f

120579

b0

K120579i intd120591

120596ref

Figure 2 Schematic diagram of the proposed iterative learning control with ESO

Since system (14) will be tracked a period trajectory therewill be once input in every period Then the state spaceequation of system (14) can be rewritten

119896

(119905) = 119860119897119909119896

+ 119861119906119896

119910119896

= 119862119909119896

(16)

where 119896 is the times Equation (16) means the 119896th states ofsystem under the 119896th input 119906

119896

For system (16) a PD-type learning algorithm with theforgetting factor 120572 is selected The factor 120572 is introduced tomake a tradeoff between perfect learning and robustnesswhich can increase the robustness of ILC against noiseinitialization error and fluctuation of system dynamics [23]The selected iterative learning scheme is found in (17) whoseschematic diagram is depicted in Figure 2

119906119896+1 (119905) = (1minus 120572) 119906

119896(119905) + 1205721199060 (119905) + Φ (119890

119896(119905))

+ Γ ( 119890119896

(119905))

(17)

where Φ and Γ are the proportion gain matrix and deviationgain matrix respectively Φ isin R119898times119902 Γ isin R119898times119902 are bounded119896 is the times 119890

119896(119905) = 119910

119889(119905) minus 119910

119896(119905) is the iterative error in the

119896th times

33 Iterative Learning Controller Convergence Analysis

Theorem 5 If the system described by (16) satisfies assump-tions and uses the update law (18) Given a desired trajectory119910119889(119905) and an initial state 119909

119889(0) which are achievable if

(1minus 120572) 119868 minus Γ119862119861 le 120588 lt 1 (18)

then as 119896 rarr infin the iterative output 119910119896(119905) of system will

converge to the desired reference trajectory 119910119889(119905)

Proof Consider

119906119889

(119905) minus 119906119896+1 (119905) = 119906

119889(119905) minus (1minus 120572) 119906

119896(119905) minus 1205721199060 (119905)

minus Φ (119890119896

(119905)) minus Γ (119889 (119890119896

(119905))

119889119905)

= (1minus 120572) [119906119889

(119905) minus 119906119896

(119905)]

+ 120572 [119906119889

(119905) minus 1199060 (119905)]

minus Φ119862 [119909119889

(119905) minus 119909119896

(119905)]

minus Γ119862119860 [119909119889

(119905) minus 119909119896

(119905)]

minus Γ119862119861 [119906119889

(119905) minus 119906119896

(119905)]

= [(1minus 120572) 119868 minus Γ119862119861] [119906119889

(119905) minus 119906119896

(119905)]

minus (Φ119862 + Γ119862119860) [119909119889

(119905) minus 119909119896

(119905)]

(19)

In (19) by the state space equation (15) and (16) the119909119889(119905) minus 119909

119896(119905) can be transformed into

119909119889

(119905) minus 119909119896

(119905) = 119890119860119905

[119909119889

(0) minus 119909119896

(0)]

+ int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896

(120591)] 119889120591

(20)

Substituting (20) to (19) we can have

119906119889

(119905) minus 119906119896+1 (119905)

= [(1minus 120572) 119868 minus Γ119862119861] [119906119889

(119905) minus 119906119896

(119905)]

minus (Φ119862 + Γ119862119860) 119890119860119905

[119909119889

(0) minus 119909119896

(0)]

minus (Φ119862 + Γ119862119860) int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896

(120591)] 119889120591

(21)

Mathematical Problems in Engineering 5

Taking the norm sdot on both sides of (21) we obtain1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817

le (1minus 120572) 119868 minus Γ119862119861 sdot1003817100381710038171003817119906119889

(119905) minus 119906119896

(119905)1003817100381710038171003817

+ 119890119860119905

Φ119862 + Γ119862119860 sdot1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

+ Φ119862 + Γ119862119860

sdot int

119905

0119890119860(119905minus120591)

1198611003817100381710038171003817119906119889

(120591) minus 119906119896

(120591)1003817100381710038171003817 119889120591

(22)

Multiplying by 119890minus120582119905 120582 gt 119860 and we have

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817 119890minus120582119905

le (1minus 120572) 119868 minus Γ119862119861 sdot1003817100381710038171003817119906119889

(119905)

minus 119906119896

(119905)1003817100381710038171003817 119890minus120582119905

minus Φ119862 + Γ119862119860 sdot1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

sdot 119890(119860minus120582)119905

+ Φ119862 + Γ119862119860

sdot int

119905

0119890(119860minus120582)(119905minus120591)

119890minus120582120591

1198611003817100381710038171003817119906119889

(120591) minus 119906119896

(120591)1003817100381710038171003817 119889120591

le (1minus 120572) 119868 minus Γ119862119861 sdot1003817100381710038171003817119906119889

(119905) minus 119906119896

(119905)1003817100381710038171003817 119890minus120582119905

minus Φ119862

+ Γ119862119860 sdot1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817 119890(119860minus120582)119905

+Φ119862 + Γ119862119860 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

1003817100381710038171003817119906119889

(119905)

minus 119906119896

(119905)1003817100381710038171003817 119890minus120582119905

(23)

Definition 6 Define 120582 norm for a function ℎ [0 119879] rarr R119896

[24] then

ℎ (sdot)120582

≜ sup119905isin[0119879]

119890minus120582119905

ℎ (sdot) 997888rarr R119896 (24)

Then

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817120582 le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(1minus 120572) 119868 minus Γ119862119861

+Φ119862 + Γ119862119860 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

10038171003817100381710038171003817100381710038171003817100381710038171003817

sdot1003817100381710038171003817119906119889

(119905)

minus 119906119896

(119905)1003817100381710038171003817120582 minus Φ119862 + Γ119862119860 sdot

1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

(25)

Since the (1 minus 120572)119868 minus Γ119862119861 le 120588 lt 1 we can find a 120582 gt 119860

which makes

120588 =

10038171003817100381710038171003817100381710038171003817100381710038171003817

(1minus 120572) 119868 minus Γ119862119861

+Φ119862 + Γ119862119860 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

10038171003817100381710038171003817100381710038171003817100381710038171003817

lt 1

(26)

So we can get

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817120582 leΦ119862 + Γ119862119860

1 minus 120588

1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817 (27)

Remark 7 It shows that 119906119896converges to the 119906

119889of radius

(Φ119862 + Γ119862119860(1 minus 120588))119909119889(0) minus 119909

119896(0) with respect to the

120582 norm Besides we can know that the convergent radiusis bounded And the boundary depends on the initial inputstates of system

Consequently the tracking error of system will be

119890119896+1 (119905) = 119910

119889(119905) minus 119910

119896+1 (119905) = 119862 sdot 119890119860119905

[119909119889

(0) minus 119909119896

(0)]

+ int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896

(120591)] 119889120591

(28)

Similarly taking the norm sdot 120582on both sides of (28) we

can get

1003817100381710038171003817119890119896+1 (119905)

1003817100381710038171003817120582 =1003817100381710038171003817119910119889

(119905) minus 119910119896+1 (119905)

1003817100381710038171003817120582 le 119862

sdot

10038171003817100381710038171003817100381710038171003817119890119860119905

[119909119889

(0) minus 119909119896+1 (0)]

+ int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896+1 (120591)] 119889120591

10038171003817100381710038171003817100381710038171003817120582le 119862

sdot1003817100381710038171003817[119909119889

(0) minus 119909119896+1 (0)]

1003817100381710038171003817

+119862 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817120582

le 119862 sdot1003817100381710038171003817[119909119889

(0) minus 119909119896+1 (0)]

1003817100381710038171003817

+119862 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

sdotΦ119862 + Γ119862119860

1 minus 120588

1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

(29)

Specially the system usually can meet the initial condi-tion that is

119909119889

(0) = 119909119896

(0) 119896 = 1 2 3 (30)

That is said as

lim119896rarrinfin

1003817100381710038171003817119890119896+1 (119905)

1003817100381710038171003817120582 997888rarr 0 (31)

To sum up when the PD-type learning algorithm isapplied to the ILC with ESO the input 119906

119896(119905) of system can

converge to the ideal input 119906119889(119905) and the output of system

can converge to the ideal output 119910119889(119905)

4 Experiment Setup and Result

The verification platform is the rotary table of telescopesystem which consists of a DC motor a tracking loadan electrical driver a control system and a high precisionencoder whose accuracy is about plusmn0618 (arc-second) Thecontrol scheme includes two loops position loop and velocityloop

6 Mathematical Problems in Engineering

040302010

minus01minus02minus03minus04

0 10 20 30 40 50 60

Ang

le (d

eg)

t (s)

Reference signal

120572 = 314 (∘s2) 120596 = 10 (∘s)

Figure 3 The desired reference signal

The following three controllers are compared

(1) PI-PI this is the traditional Proportional-Integral (PI)controller In the velocity loop and position loop weuse two PI controllers The velocity signal can beobtained by differentiating the position signal

(2) PI-ESO in both of the position loop and the velocityloop the PI controllers are used but the velocitysignal will be gotten by the ESO The bandwidth 1205960of ESO is 1205960 = 100 The parameters of position looprsquosPI controller remain the same

(3) PI-ILC-ESO the PI controller is used in the positionloop and the ILC controller is used in the speed loopThe parameters of position looprsquos PI controller stillremain the same The velocity signal is also the signalwhich is gotten by the ESO To satisfy the condition(18) The factor 120572 and the deviation gain matrix Γ arechosen as 120572 = 02 and Γ = 05 By the state spaceequation of system (14) it will have

(1minus 120572) 119868 minus Γ119862119861 =

1003817100381710038171003817100381710038171003817100381710038171003817

(1minus 02) 119868 minus 05times [0 1] [01]

1003817100381710038171003817100381710038171003817100381710038171003817

= 03 lt 1

(32)

The three controllers are tested for a sinusoidal signalwhose velocity and acceleration are 120596 = 10 (

∘s) and 120572 =

314 (∘s2) The period of 119879 sinusoidal signal is 2 (s)

The desired reference trajectory is shown in Figure 3And the corresponding tracking performance under the threecontrollers is shown in Figures 4ndash6 As seen the PI-PIcontroller has a larger tracking error whose maximum valueis about 288410158401015840 Comparatively speaking the PI controllerwith ESO has a better tracking performance which illustratesthat ESO has an excellent estimation ability for disturbancesinvolved in the system Furthermore the proposed PI-ILC-ESO controller has the least tracking error than other controlmethods whose maximum value of error is about 200610158401015840This controller combines the advantages of ESO with theadvantages of ILC It makes use of ESO to eliminate most ofthe nonlinear disturbances Simultaneously the ILC gets thebest control input by learning previous control informationSome detailed results of the three controllers are shown inTable 1

0 10 20 30 40 50 60t (s)

PI-PI

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 4 The tracking error of PI-PI

0 10 20 30 40 50 60t (s)

PI-ESO

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 5 The tracking error of PI-ESO

0 10 20 30 40 50 60t (s)

PI-ILC-ESO

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 6 The tracking error of PI-ILC-ESO

Table 1 The experimental results of the three controllers

PI-PI PI-ESO PI-ILC-ESO|maximum error| 28843210158401015840 20998810158401015840 20059210158401015840

Root mean square error 9830110158401015840 6028210158401015840 5646810158401015840

From Table 1 it can be seen that the PI-ILC-ESO hasbetter performance than other control methods Howevercomparing the PI-ESO controller with the PI-ILC-ESO con-troller it can be seen that the effectiveness of using ILCis not obvious The maximum error reduces only to 09410158401015840and the root mean square error reduces to 03810158401015840 In factComparing with Figures 5 and 6 it can be found that theerrors of many position points exceed 1010158401015840 and the errorsare close to 2010158401015840 when the PI-ESO controller is used Butwhen the PI-ILC-ESO controller is used almost all the errorsof position points are near 1010158401015840 So the conclusion that thecomprehensive performance of PI-ILC-ESO is better than

Mathematical Problems in Engineering 7

other control methods can be gotten The ILC method helpsthe system get the best input signal Besides in Figures 5and 6 it can be found that the errors are evenly distributedComparing with the Figure 4 (PI-PI controller) both of thetwo controllers have reduced almost all the turning errorsof system The remaining errors are mostly caused by thesignal noise So it is difficult to reduce the remaining error andfurther improve the precision of system unless the problemof signal noise is solved

5 Conclusion

Uncertain nonlinear disturbances have been themajor factorswhich restrict the performance of tracking control systemsThis is because the systems will have a low tracking precisionwhen some uncertain nonlinear disturbances are induced inthe systems Therefore to reduce the influence introducedby the uncertain nonlinear disturbances an ILC methodwith ESO is proposed in this paper The ILC can get anexcellent input signal by learning previous control informa-tion It owns a better ability for eliminating some perioddisturbances Meanwhile an ESO is designed for estimatingsome uncertain nonlinear disturbances It compensates theshortage of the ILCrsquos disablement for nonperiod disturbancesIn addition the ESO can estimate an accurate velocity signaland supply the velocity as the feedback input signal ofiterative learning controller Furthermore the convergenceanalysis of the proposed control method guarantees therobustness of system Finally the experiment results showthat the proposed control method has excellent performancefor reducing the tracking error of telescope system

Conflict of Interests

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

References

[1] G G Valyavin V D Bychkov M V Yushkin et al ldquoHigh-resolution fiber-fed echelle spectrograph for the 6-m telescopeI Optical scheme arrangement and control systemrdquoAstrophys-ical Bulletin vol 69 no 2 pp 224ndash239 2014

[2] K Lu and Y Xia ldquoAdaptive attitude tracking control for rigidspacecraft with finite-time convergencerdquo Automatica vol 49no 12 pp 3591ndash3599 2013

[3] Y-J Sun ldquoComposite tracking control for generalized practicalsynchronization of Duffing-Holmes systems with parametermismatching unknown external excitation plant uncertaintiesand uncertain deadzone nonlinearitiesrdquo Abstract and AppliedAnalysis vol 2012 Article ID 640568 11 pages 2012

[4] Y Fu and T Chai ldquoSelf-tuning control with a filter and aneural compensator for a class of nonlinear systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 24no 5 pp 837ndash843 2013

[5] M M Bridges D M Dawson and J Hu ldquoAdaptive control fora class of direct drive robot manipulatorsrdquo International Journalof Adaptive Control and Signal Processing vol 10 no 4-5 pp417ndash441 1996

[6] A Tadayoni W-F Xie and B W Gordon ldquoAdaptive con-trol of harmonic drive with parameter varying friction usingstructurally dynamic wavelet networkrdquo International Journal ofControl Automation and Systems vol 9 no 1 pp 50ndash59 2011

[7] Y Alipouri and J Poshtan ldquoDesigning a robust minimum vari-ance controller using discrete slide mode controller approachrdquoISA Transactions vol 52 no 2 pp 291ndash299 2013

[8] B-Z Guo and F-F Jin ldquoSliding mode and active disturbancerejection control to stabilization of one-dimensional anti-stablewave equations subject to disturbance in boundary inputrdquo IEEETransactions on Automatic Control vol 58 no 5 pp 1269ndash12742013

[9] Y-S Lu and S-M Lin ldquoDisturbance-observer-based adaptivefeedforward cancellation of torque ripples in harmonic drivesystemsrdquo Electrical Engineering vol 90 no 2 pp 95ndash106 2007

[10] T Kavli ldquoFrequency domain synthesis of trajectory learningcontrollers for robot manipulatorsrdquo Journal of Robotic Systemsvol 9 no 5 pp 663ndash680 1992

[11] Y-C Wang and C-J Chien ldquoAn observer-based adaptiveiterative learning control using filtered-FNN design for roboticsystemsrdquo Advances in Mechanical Engineering vol 6 Article ID471418 2014

[12] S Arimoto S Kawamura and FMiyazaki ldquoBettering operationof Robots by learningrdquo Journal of Robotic Systems vol 1 no 2pp 123ndash140 1984

[13] J-X Xu D Huang V Venkataramanan and The Cat TuongHuynh ldquoExtreme precise motion tracking of piezoelectricpositioning stage using sampled-data iterative learning controlrdquoIEEE Transactions on Control Systems Technology vol 21 no 4pp 1432ndash1439 2013

[14] M Norrlof ldquoAn adaptive iterative learning control algorithmwith experiments on an industrial robotrdquo IEEE Transactions onRobotics and Automation vol 18 no 2 pp 245ndash251 2002

[15] J-X Xu Y Chen T H Lee and S Yamamoto ldquoTerminaliterative learning control with an application to RTPCVDthickness controlrdquo Automatica vol 35 no 9 pp 1535ndash15421999

[16] S S Saab ldquoA discrete-time stochastic learning control algo-rithmrdquo IEEE Transactions on Automatic Control vol 46 no 6pp 877ndash887 2001

[17] K K Leang and S Devasia ldquoDesign of hysteresis-compensatingiterative learning control for piezo-positioners application toatomic force microscopesrdquo Mechatronics vol 16 no 3-4 pp141ndash158 2006

[18] Z Zhu D Xu J Liu and Y Xia ldquoMissile guidance law basedon extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 60 no 12 pp 5882ndash5891 2013

[19] M Pizzocaro D Calonico C Calosso et al ldquoActive disturbancerejection control of temperature for ultrastable optical cavitiesrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 60 no 2 pp 273ndash280 2013

[20] J Yao Z Jiao and D Ma ldquoExtended-state-observer-basedoutput feedback nonlinear robust control of hydraulic systemswith backsteppingrdquo IEEE Transactions on Industrial Electronicsvol 61 no 11 pp 6285ndash6293 2014

[21] J Yao Z Jiao and D Ma ldquoAdaptive robust control of dc motorswith extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 61 no 7 pp 3630ndash3637 2014

[22] Q Zheng L Q Gao and Z Gao ldquoOn stability analysis ofactive disturbance rejection control for nonlinear time-varyingplants with unknown dynamicsrdquo inProceedings of the 46th IEEE

8 Mathematical Problems in Engineering

Conference on Decision and Control (CDC rsquo07) pp 3501ndash3506New Orleans La USA December 2007

[23] WQian S K Panda and J-X Xu ldquoTorque rippleminimizationin PM synchronous motors using iterative learning controlrdquoIEEE Transactions on Power Electronics vol 19 no 2 pp 272ndash279 2004

[24] G Heinzinger D Fenwick B Paden and F Miyazaki ldquoStabilityof learning control with disturbances and uncertain initialconditionsrdquo IEEETransactions onAutomatic Control vol 37 no1 pp 110ndash114 1992

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 3

where 12057301 12057302 and 12057303 are the parameters that will bedesigned in the ESO

The aim of ESO is to make

1199091 (119905) 997888rarr 1199091 (119905)

1199092 (119905) 997888rarr 1199092 (119905)

1199093 (119905) 997888rarr 1198911 (119905 120596)

(6)

Let 120578119894

= 119909119894minus 119909119894 119894 = 1 2 3 denote the estimation error

subtracting (4) from (5) then we can obtain

120578 = 119860120578 + 119872ℎ (7)

where

120578 =[[

[

1205781

1205782

1205783

]]

]

119860 =[[

[

minus12057301 1 0minus12057302 0 1minus12057303 0 0

]]

]

119872 =[[

[

001

]]

]

(8)

Therefore if the matrix 119860 is Hurwitz the ESO willbe Bounded-Input Bounded-Output (BIBO) stable In thematrix 119860 the parameters 12057301 12057302 and 12057303 can be designedas [21]

12057301 = 31205960

12057302 = 312059620

12057303 = 12059630

(9)

where 1205960 is the bandwidth of ESO Equation (9) can promisethe matrix 119860 being Hurwitz

Lemma 2 If the ℎ(119905) is bounded there will be a positiveconstant 120577

119894gt 0 and a finite time 1198791 gt 0 such that

1003816100381610038161003816120578119894 (119905)1003816100381610038161003816 le 120577119894

120577119894= o(

11205961198880

)

119894 = 1 2 3 forall119905 ge 1198791

(10)

The proof can be found in the literature [22]

Remark 3 Equation (9) makes the ESO (5) a BIBO stableobserver The result of Lemma 2 shows that the ESO has anexcellent estimation ability for disturbances Its estimationability depends on the bandwidth 1205960 of ESO When thebandwidth 1205960 is big enough it can estimate accurately 1199091(119905)1199092(119905) and 1198911(119905 120596)

According to the analysis in Lemma 2 when designing anappropriate parameter1205960 it canmake 119900(1120596

119888

0) asymp 0 And therewill be |1199093minus1199093| = |1205783(119905)| le 119900(1120596

119888

0) namely the 1199093 asymp 1198911(119905 120596)So we can transform the input 119906 into 119906

120596 namely 119906 = (119906

120596minus

1199093)1198870Substituting the 119906 = (119906

120596minus 1199093)1198870 to (2) we can get

119889120579 (119905)

119889119905= 120596 (119905)

119889120596 (119905)

119889119905asymp 119906120596

(11)

312 Design Velocity Loop PI Controller The nonlinear sys-tem (2) is converted to an approximate linear integral system(11) So we can design a PI controller in the velocity loop

119906120596

= 119870120596

119901(120596

refminus 1199092) + 119870

120596

119894int

119905

0[120596

ref(120591) minus 1199092 (120591)] 119889120591 (12)

where 120596ref is the input signal of velocity loop and 119870

120596

119901 119870120596119894are

the proportion gain and the integral gain in the velocity loopcontroller respectively

313 Design Position Loop PI Controller Similarly we candesign a PI controller in the position loop

120596ref

= 119870120579

119901(120579

refminus 1199091) + 119870

120579

119894int

119905

0[120579

ref(120591) minus 1199091 (120591)] 119889120591 (13)

where 120579ref is the input signal of position loop and 119870

120596

119901 119870120596119894are

the proportion gain and the integral gain in the position loopcontroller respectively

32 Iterative Learning Controller with ESO Design From(11) we can see that the system is only an approximatelinear system Thereby to make further improvement on thetracking precision of system we will find an ideal input 119906

120596by

the method of iterative learning And it can make the output120596 of system close to the input 120596

ref as far as possible In thissection we will design an ILC to substitute the PI controllerin the velocity loop

According to (11) the state space equation of system canbe obtained as

(119905) = 119860119897119909 + 119861119906

120596

119910 = 119862119909

(14)

where 119909 = [11990911199092 ] = [ 120579

120596]119860119897= [ 0 1

0 0 ]119861 = [ 01 ] and119862 = [ 0 1 ]

Tomake the system track the desired reference trajectorywe assume that there is an ideal input signal meeting thecondition

Assumption 4 The ideal input signal 119906119889(119905) can make the

system track the desired reference trajectory 119910119889(119905)

119889

(119905) = 119860119897119909119889

(119905) + 119861119906119889

(119905)

119910119889

(119905) = 119862119909119889

(119905)

(15)

where 119909119889(119905) 119910

119889(119905) represent the ideal states of system

respectively

4 Mathematical Problems in Engineering

120579ref

x1

minus minus minus

e120579K120579p

+

+

+

++

x2

ek

Φ

Γd

dt

1 minus 120572

x3

Memory

ESO

Plant

uk

uk+11

b0

u

f

120579

b0

K120579i intd120591

120596ref

Figure 2 Schematic diagram of the proposed iterative learning control with ESO

Since system (14) will be tracked a period trajectory therewill be once input in every period Then the state spaceequation of system (14) can be rewritten

119896

(119905) = 119860119897119909119896

+ 119861119906119896

119910119896

= 119862119909119896

(16)

where 119896 is the times Equation (16) means the 119896th states ofsystem under the 119896th input 119906

119896

For system (16) a PD-type learning algorithm with theforgetting factor 120572 is selected The factor 120572 is introduced tomake a tradeoff between perfect learning and robustnesswhich can increase the robustness of ILC against noiseinitialization error and fluctuation of system dynamics [23]The selected iterative learning scheme is found in (17) whoseschematic diagram is depicted in Figure 2

119906119896+1 (119905) = (1minus 120572) 119906

119896(119905) + 1205721199060 (119905) + Φ (119890

119896(119905))

+ Γ ( 119890119896

(119905))

(17)

where Φ and Γ are the proportion gain matrix and deviationgain matrix respectively Φ isin R119898times119902 Γ isin R119898times119902 are bounded119896 is the times 119890

119896(119905) = 119910

119889(119905) minus 119910

119896(119905) is the iterative error in the

119896th times

33 Iterative Learning Controller Convergence Analysis

Theorem 5 If the system described by (16) satisfies assump-tions and uses the update law (18) Given a desired trajectory119910119889(119905) and an initial state 119909

119889(0) which are achievable if

(1minus 120572) 119868 minus Γ119862119861 le 120588 lt 1 (18)

then as 119896 rarr infin the iterative output 119910119896(119905) of system will

converge to the desired reference trajectory 119910119889(119905)

Proof Consider

119906119889

(119905) minus 119906119896+1 (119905) = 119906

119889(119905) minus (1minus 120572) 119906

119896(119905) minus 1205721199060 (119905)

minus Φ (119890119896

(119905)) minus Γ (119889 (119890119896

(119905))

119889119905)

= (1minus 120572) [119906119889

(119905) minus 119906119896

(119905)]

+ 120572 [119906119889

(119905) minus 1199060 (119905)]

minus Φ119862 [119909119889

(119905) minus 119909119896

(119905)]

minus Γ119862119860 [119909119889

(119905) minus 119909119896

(119905)]

minus Γ119862119861 [119906119889

(119905) minus 119906119896

(119905)]

= [(1minus 120572) 119868 minus Γ119862119861] [119906119889

(119905) minus 119906119896

(119905)]

minus (Φ119862 + Γ119862119860) [119909119889

(119905) minus 119909119896

(119905)]

(19)

In (19) by the state space equation (15) and (16) the119909119889(119905) minus 119909

119896(119905) can be transformed into

119909119889

(119905) minus 119909119896

(119905) = 119890119860119905

[119909119889

(0) minus 119909119896

(0)]

+ int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896

(120591)] 119889120591

(20)

Substituting (20) to (19) we can have

119906119889

(119905) minus 119906119896+1 (119905)

= [(1minus 120572) 119868 minus Γ119862119861] [119906119889

(119905) minus 119906119896

(119905)]

minus (Φ119862 + Γ119862119860) 119890119860119905

[119909119889

(0) minus 119909119896

(0)]

minus (Φ119862 + Γ119862119860) int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896

(120591)] 119889120591

(21)

Mathematical Problems in Engineering 5

Taking the norm sdot on both sides of (21) we obtain1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817

le (1minus 120572) 119868 minus Γ119862119861 sdot1003817100381710038171003817119906119889

(119905) minus 119906119896

(119905)1003817100381710038171003817

+ 119890119860119905

Φ119862 + Γ119862119860 sdot1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

+ Φ119862 + Γ119862119860

sdot int

119905

0119890119860(119905minus120591)

1198611003817100381710038171003817119906119889

(120591) minus 119906119896

(120591)1003817100381710038171003817 119889120591

(22)

Multiplying by 119890minus120582119905 120582 gt 119860 and we have

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817 119890minus120582119905

le (1minus 120572) 119868 minus Γ119862119861 sdot1003817100381710038171003817119906119889

(119905)

minus 119906119896

(119905)1003817100381710038171003817 119890minus120582119905

minus Φ119862 + Γ119862119860 sdot1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

sdot 119890(119860minus120582)119905

+ Φ119862 + Γ119862119860

sdot int

119905

0119890(119860minus120582)(119905minus120591)

119890minus120582120591

1198611003817100381710038171003817119906119889

(120591) minus 119906119896

(120591)1003817100381710038171003817 119889120591

le (1minus 120572) 119868 minus Γ119862119861 sdot1003817100381710038171003817119906119889

(119905) minus 119906119896

(119905)1003817100381710038171003817 119890minus120582119905

minus Φ119862

+ Γ119862119860 sdot1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817 119890(119860minus120582)119905

+Φ119862 + Γ119862119860 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

1003817100381710038171003817119906119889

(119905)

minus 119906119896

(119905)1003817100381710038171003817 119890minus120582119905

(23)

Definition 6 Define 120582 norm for a function ℎ [0 119879] rarr R119896

[24] then

ℎ (sdot)120582

≜ sup119905isin[0119879]

119890minus120582119905

ℎ (sdot) 997888rarr R119896 (24)

Then

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817120582 le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(1minus 120572) 119868 minus Γ119862119861

+Φ119862 + Γ119862119860 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

10038171003817100381710038171003817100381710038171003817100381710038171003817

sdot1003817100381710038171003817119906119889

(119905)

minus 119906119896

(119905)1003817100381710038171003817120582 minus Φ119862 + Γ119862119860 sdot

1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

(25)

Since the (1 minus 120572)119868 minus Γ119862119861 le 120588 lt 1 we can find a 120582 gt 119860

which makes

120588 =

10038171003817100381710038171003817100381710038171003817100381710038171003817

(1minus 120572) 119868 minus Γ119862119861

+Φ119862 + Γ119862119860 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

10038171003817100381710038171003817100381710038171003817100381710038171003817

lt 1

(26)

So we can get

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817120582 leΦ119862 + Γ119862119860

1 minus 120588

1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817 (27)

Remark 7 It shows that 119906119896converges to the 119906

119889of radius

(Φ119862 + Γ119862119860(1 minus 120588))119909119889(0) minus 119909

119896(0) with respect to the

120582 norm Besides we can know that the convergent radiusis bounded And the boundary depends on the initial inputstates of system

Consequently the tracking error of system will be

119890119896+1 (119905) = 119910

119889(119905) minus 119910

119896+1 (119905) = 119862 sdot 119890119860119905

[119909119889

(0) minus 119909119896

(0)]

+ int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896

(120591)] 119889120591

(28)

Similarly taking the norm sdot 120582on both sides of (28) we

can get

1003817100381710038171003817119890119896+1 (119905)

1003817100381710038171003817120582 =1003817100381710038171003817119910119889

(119905) minus 119910119896+1 (119905)

1003817100381710038171003817120582 le 119862

sdot

10038171003817100381710038171003817100381710038171003817119890119860119905

[119909119889

(0) minus 119909119896+1 (0)]

+ int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896+1 (120591)] 119889120591

10038171003817100381710038171003817100381710038171003817120582le 119862

sdot1003817100381710038171003817[119909119889

(0) minus 119909119896+1 (0)]

1003817100381710038171003817

+119862 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817120582

le 119862 sdot1003817100381710038171003817[119909119889

(0) minus 119909119896+1 (0)]

1003817100381710038171003817

+119862 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

sdotΦ119862 + Γ119862119860

1 minus 120588

1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

(29)

Specially the system usually can meet the initial condi-tion that is

119909119889

(0) = 119909119896

(0) 119896 = 1 2 3 (30)

That is said as

lim119896rarrinfin

1003817100381710038171003817119890119896+1 (119905)

1003817100381710038171003817120582 997888rarr 0 (31)

To sum up when the PD-type learning algorithm isapplied to the ILC with ESO the input 119906

119896(119905) of system can

converge to the ideal input 119906119889(119905) and the output of system

can converge to the ideal output 119910119889(119905)

4 Experiment Setup and Result

The verification platform is the rotary table of telescopesystem which consists of a DC motor a tracking loadan electrical driver a control system and a high precisionencoder whose accuracy is about plusmn0618 (arc-second) Thecontrol scheme includes two loops position loop and velocityloop

6 Mathematical Problems in Engineering

040302010

minus01minus02minus03minus04

0 10 20 30 40 50 60

Ang

le (d

eg)

t (s)

Reference signal

120572 = 314 (∘s2) 120596 = 10 (∘s)

Figure 3 The desired reference signal

The following three controllers are compared

(1) PI-PI this is the traditional Proportional-Integral (PI)controller In the velocity loop and position loop weuse two PI controllers The velocity signal can beobtained by differentiating the position signal

(2) PI-ESO in both of the position loop and the velocityloop the PI controllers are used but the velocitysignal will be gotten by the ESO The bandwidth 1205960of ESO is 1205960 = 100 The parameters of position looprsquosPI controller remain the same

(3) PI-ILC-ESO the PI controller is used in the positionloop and the ILC controller is used in the speed loopThe parameters of position looprsquos PI controller stillremain the same The velocity signal is also the signalwhich is gotten by the ESO To satisfy the condition(18) The factor 120572 and the deviation gain matrix Γ arechosen as 120572 = 02 and Γ = 05 By the state spaceequation of system (14) it will have

(1minus 120572) 119868 minus Γ119862119861 =

1003817100381710038171003817100381710038171003817100381710038171003817

(1minus 02) 119868 minus 05times [0 1] [01]

1003817100381710038171003817100381710038171003817100381710038171003817

= 03 lt 1

(32)

The three controllers are tested for a sinusoidal signalwhose velocity and acceleration are 120596 = 10 (

∘s) and 120572 =

314 (∘s2) The period of 119879 sinusoidal signal is 2 (s)

The desired reference trajectory is shown in Figure 3And the corresponding tracking performance under the threecontrollers is shown in Figures 4ndash6 As seen the PI-PIcontroller has a larger tracking error whose maximum valueis about 288410158401015840 Comparatively speaking the PI controllerwith ESO has a better tracking performance which illustratesthat ESO has an excellent estimation ability for disturbancesinvolved in the system Furthermore the proposed PI-ILC-ESO controller has the least tracking error than other controlmethods whose maximum value of error is about 200610158401015840This controller combines the advantages of ESO with theadvantages of ILC It makes use of ESO to eliminate most ofthe nonlinear disturbances Simultaneously the ILC gets thebest control input by learning previous control informationSome detailed results of the three controllers are shown inTable 1

0 10 20 30 40 50 60t (s)

PI-PI

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 4 The tracking error of PI-PI

0 10 20 30 40 50 60t (s)

PI-ESO

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 5 The tracking error of PI-ESO

0 10 20 30 40 50 60t (s)

PI-ILC-ESO

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 6 The tracking error of PI-ILC-ESO

Table 1 The experimental results of the three controllers

PI-PI PI-ESO PI-ILC-ESO|maximum error| 28843210158401015840 20998810158401015840 20059210158401015840

Root mean square error 9830110158401015840 6028210158401015840 5646810158401015840

From Table 1 it can be seen that the PI-ILC-ESO hasbetter performance than other control methods Howevercomparing the PI-ESO controller with the PI-ILC-ESO con-troller it can be seen that the effectiveness of using ILCis not obvious The maximum error reduces only to 09410158401015840and the root mean square error reduces to 03810158401015840 In factComparing with Figures 5 and 6 it can be found that theerrors of many position points exceed 1010158401015840 and the errorsare close to 2010158401015840 when the PI-ESO controller is used Butwhen the PI-ILC-ESO controller is used almost all the errorsof position points are near 1010158401015840 So the conclusion that thecomprehensive performance of PI-ILC-ESO is better than

Mathematical Problems in Engineering 7

other control methods can be gotten The ILC method helpsthe system get the best input signal Besides in Figures 5and 6 it can be found that the errors are evenly distributedComparing with the Figure 4 (PI-PI controller) both of thetwo controllers have reduced almost all the turning errorsof system The remaining errors are mostly caused by thesignal noise So it is difficult to reduce the remaining error andfurther improve the precision of system unless the problemof signal noise is solved

5 Conclusion

Uncertain nonlinear disturbances have been themajor factorswhich restrict the performance of tracking control systemsThis is because the systems will have a low tracking precisionwhen some uncertain nonlinear disturbances are induced inthe systems Therefore to reduce the influence introducedby the uncertain nonlinear disturbances an ILC methodwith ESO is proposed in this paper The ILC can get anexcellent input signal by learning previous control informa-tion It owns a better ability for eliminating some perioddisturbances Meanwhile an ESO is designed for estimatingsome uncertain nonlinear disturbances It compensates theshortage of the ILCrsquos disablement for nonperiod disturbancesIn addition the ESO can estimate an accurate velocity signaland supply the velocity as the feedback input signal ofiterative learning controller Furthermore the convergenceanalysis of the proposed control method guarantees therobustness of system Finally the experiment results showthat the proposed control method has excellent performancefor reducing the tracking error of telescope system

Conflict of Interests

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

References

[1] G G Valyavin V D Bychkov M V Yushkin et al ldquoHigh-resolution fiber-fed echelle spectrograph for the 6-m telescopeI Optical scheme arrangement and control systemrdquoAstrophys-ical Bulletin vol 69 no 2 pp 224ndash239 2014

[2] K Lu and Y Xia ldquoAdaptive attitude tracking control for rigidspacecraft with finite-time convergencerdquo Automatica vol 49no 12 pp 3591ndash3599 2013

[3] Y-J Sun ldquoComposite tracking control for generalized practicalsynchronization of Duffing-Holmes systems with parametermismatching unknown external excitation plant uncertaintiesand uncertain deadzone nonlinearitiesrdquo Abstract and AppliedAnalysis vol 2012 Article ID 640568 11 pages 2012

[4] Y Fu and T Chai ldquoSelf-tuning control with a filter and aneural compensator for a class of nonlinear systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 24no 5 pp 837ndash843 2013

[5] M M Bridges D M Dawson and J Hu ldquoAdaptive control fora class of direct drive robot manipulatorsrdquo International Journalof Adaptive Control and Signal Processing vol 10 no 4-5 pp417ndash441 1996

[6] A Tadayoni W-F Xie and B W Gordon ldquoAdaptive con-trol of harmonic drive with parameter varying friction usingstructurally dynamic wavelet networkrdquo International Journal ofControl Automation and Systems vol 9 no 1 pp 50ndash59 2011

[7] Y Alipouri and J Poshtan ldquoDesigning a robust minimum vari-ance controller using discrete slide mode controller approachrdquoISA Transactions vol 52 no 2 pp 291ndash299 2013

[8] B-Z Guo and F-F Jin ldquoSliding mode and active disturbancerejection control to stabilization of one-dimensional anti-stablewave equations subject to disturbance in boundary inputrdquo IEEETransactions on Automatic Control vol 58 no 5 pp 1269ndash12742013

[9] Y-S Lu and S-M Lin ldquoDisturbance-observer-based adaptivefeedforward cancellation of torque ripples in harmonic drivesystemsrdquo Electrical Engineering vol 90 no 2 pp 95ndash106 2007

[10] T Kavli ldquoFrequency domain synthesis of trajectory learningcontrollers for robot manipulatorsrdquo Journal of Robotic Systemsvol 9 no 5 pp 663ndash680 1992

[11] Y-C Wang and C-J Chien ldquoAn observer-based adaptiveiterative learning control using filtered-FNN design for roboticsystemsrdquo Advances in Mechanical Engineering vol 6 Article ID471418 2014

[12] S Arimoto S Kawamura and FMiyazaki ldquoBettering operationof Robots by learningrdquo Journal of Robotic Systems vol 1 no 2pp 123ndash140 1984

[13] J-X Xu D Huang V Venkataramanan and The Cat TuongHuynh ldquoExtreme precise motion tracking of piezoelectricpositioning stage using sampled-data iterative learning controlrdquoIEEE Transactions on Control Systems Technology vol 21 no 4pp 1432ndash1439 2013

[14] M Norrlof ldquoAn adaptive iterative learning control algorithmwith experiments on an industrial robotrdquo IEEE Transactions onRobotics and Automation vol 18 no 2 pp 245ndash251 2002

[15] J-X Xu Y Chen T H Lee and S Yamamoto ldquoTerminaliterative learning control with an application to RTPCVDthickness controlrdquo Automatica vol 35 no 9 pp 1535ndash15421999

[16] S S Saab ldquoA discrete-time stochastic learning control algo-rithmrdquo IEEE Transactions on Automatic Control vol 46 no 6pp 877ndash887 2001

[17] K K Leang and S Devasia ldquoDesign of hysteresis-compensatingiterative learning control for piezo-positioners application toatomic force microscopesrdquo Mechatronics vol 16 no 3-4 pp141ndash158 2006

[18] Z Zhu D Xu J Liu and Y Xia ldquoMissile guidance law basedon extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 60 no 12 pp 5882ndash5891 2013

[19] M Pizzocaro D Calonico C Calosso et al ldquoActive disturbancerejection control of temperature for ultrastable optical cavitiesrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 60 no 2 pp 273ndash280 2013

[20] J Yao Z Jiao and D Ma ldquoExtended-state-observer-basedoutput feedback nonlinear robust control of hydraulic systemswith backsteppingrdquo IEEE Transactions on Industrial Electronicsvol 61 no 11 pp 6285ndash6293 2014

[21] J Yao Z Jiao and D Ma ldquoAdaptive robust control of dc motorswith extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 61 no 7 pp 3630ndash3637 2014

[22] Q Zheng L Q Gao and Z Gao ldquoOn stability analysis ofactive disturbance rejection control for nonlinear time-varyingplants with unknown dynamicsrdquo inProceedings of the 46th IEEE

8 Mathematical Problems in Engineering

Conference on Decision and Control (CDC rsquo07) pp 3501ndash3506New Orleans La USA December 2007

[23] WQian S K Panda and J-X Xu ldquoTorque rippleminimizationin PM synchronous motors using iterative learning controlrdquoIEEE Transactions on Power Electronics vol 19 no 2 pp 272ndash279 2004

[24] G Heinzinger D Fenwick B Paden and F Miyazaki ldquoStabilityof learning control with disturbances and uncertain initialconditionsrdquo IEEETransactions onAutomatic Control vol 37 no1 pp 110ndash114 1992

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

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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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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

4 Mathematical Problems in Engineering

120579ref

x1

minus minus minus

e120579K120579p

+

+

+

++

x2

ek

Φ

Γd

dt

1 minus 120572

x3

Memory

ESO

Plant

uk

uk+11

b0

u

f

120579

b0

K120579i intd120591

120596ref

Figure 2 Schematic diagram of the proposed iterative learning control with ESO

Since system (14) will be tracked a period trajectory therewill be once input in every period Then the state spaceequation of system (14) can be rewritten

119896

(119905) = 119860119897119909119896

+ 119861119906119896

119910119896

= 119862119909119896

(16)

where 119896 is the times Equation (16) means the 119896th states ofsystem under the 119896th input 119906

119896

For system (16) a PD-type learning algorithm with theforgetting factor 120572 is selected The factor 120572 is introduced tomake a tradeoff between perfect learning and robustnesswhich can increase the robustness of ILC against noiseinitialization error and fluctuation of system dynamics [23]The selected iterative learning scheme is found in (17) whoseschematic diagram is depicted in Figure 2

119906119896+1 (119905) = (1minus 120572) 119906

119896(119905) + 1205721199060 (119905) + Φ (119890

119896(119905))

+ Γ ( 119890119896

(119905))

(17)

where Φ and Γ are the proportion gain matrix and deviationgain matrix respectively Φ isin R119898times119902 Γ isin R119898times119902 are bounded119896 is the times 119890

119896(119905) = 119910

119889(119905) minus 119910

119896(119905) is the iterative error in the

119896th times

33 Iterative Learning Controller Convergence Analysis

Theorem 5 If the system described by (16) satisfies assump-tions and uses the update law (18) Given a desired trajectory119910119889(119905) and an initial state 119909

119889(0) which are achievable if

(1minus 120572) 119868 minus Γ119862119861 le 120588 lt 1 (18)

then as 119896 rarr infin the iterative output 119910119896(119905) of system will

converge to the desired reference trajectory 119910119889(119905)

Proof Consider

119906119889

(119905) minus 119906119896+1 (119905) = 119906

119889(119905) minus (1minus 120572) 119906

119896(119905) minus 1205721199060 (119905)

minus Φ (119890119896

(119905)) minus Γ (119889 (119890119896

(119905))

119889119905)

= (1minus 120572) [119906119889

(119905) minus 119906119896

(119905)]

+ 120572 [119906119889

(119905) minus 1199060 (119905)]

minus Φ119862 [119909119889

(119905) minus 119909119896

(119905)]

minus Γ119862119860 [119909119889

(119905) minus 119909119896

(119905)]

minus Γ119862119861 [119906119889

(119905) minus 119906119896

(119905)]

= [(1minus 120572) 119868 minus Γ119862119861] [119906119889

(119905) minus 119906119896

(119905)]

minus (Φ119862 + Γ119862119860) [119909119889

(119905) minus 119909119896

(119905)]

(19)

In (19) by the state space equation (15) and (16) the119909119889(119905) minus 119909

119896(119905) can be transformed into

119909119889

(119905) minus 119909119896

(119905) = 119890119860119905

[119909119889

(0) minus 119909119896

(0)]

+ int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896

(120591)] 119889120591

(20)

Substituting (20) to (19) we can have

119906119889

(119905) minus 119906119896+1 (119905)

= [(1minus 120572) 119868 minus Γ119862119861] [119906119889

(119905) minus 119906119896

(119905)]

minus (Φ119862 + Γ119862119860) 119890119860119905

[119909119889

(0) minus 119909119896

(0)]

minus (Φ119862 + Γ119862119860) int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896

(120591)] 119889120591

(21)

Mathematical Problems in Engineering 5

Taking the norm sdot on both sides of (21) we obtain1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817

le (1minus 120572) 119868 minus Γ119862119861 sdot1003817100381710038171003817119906119889

(119905) minus 119906119896

(119905)1003817100381710038171003817

+ 119890119860119905

Φ119862 + Γ119862119860 sdot1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

+ Φ119862 + Γ119862119860

sdot int

119905

0119890119860(119905minus120591)

1198611003817100381710038171003817119906119889

(120591) minus 119906119896

(120591)1003817100381710038171003817 119889120591

(22)

Multiplying by 119890minus120582119905 120582 gt 119860 and we have

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817 119890minus120582119905

le (1minus 120572) 119868 minus Γ119862119861 sdot1003817100381710038171003817119906119889

(119905)

minus 119906119896

(119905)1003817100381710038171003817 119890minus120582119905

minus Φ119862 + Γ119862119860 sdot1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

sdot 119890(119860minus120582)119905

+ Φ119862 + Γ119862119860

sdot int

119905

0119890(119860minus120582)(119905minus120591)

119890minus120582120591

1198611003817100381710038171003817119906119889

(120591) minus 119906119896

(120591)1003817100381710038171003817 119889120591

le (1minus 120572) 119868 minus Γ119862119861 sdot1003817100381710038171003817119906119889

(119905) minus 119906119896

(119905)1003817100381710038171003817 119890minus120582119905

minus Φ119862

+ Γ119862119860 sdot1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817 119890(119860minus120582)119905

+Φ119862 + Γ119862119860 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

1003817100381710038171003817119906119889

(119905)

minus 119906119896

(119905)1003817100381710038171003817 119890minus120582119905

(23)

Definition 6 Define 120582 norm for a function ℎ [0 119879] rarr R119896

[24] then

ℎ (sdot)120582

≜ sup119905isin[0119879]

119890minus120582119905

ℎ (sdot) 997888rarr R119896 (24)

Then

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817120582 le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(1minus 120572) 119868 minus Γ119862119861

+Φ119862 + Γ119862119860 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

10038171003817100381710038171003817100381710038171003817100381710038171003817

sdot1003817100381710038171003817119906119889

(119905)

minus 119906119896

(119905)1003817100381710038171003817120582 minus Φ119862 + Γ119862119860 sdot

1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

(25)

Since the (1 minus 120572)119868 minus Γ119862119861 le 120588 lt 1 we can find a 120582 gt 119860

which makes

120588 =

10038171003817100381710038171003817100381710038171003817100381710038171003817

(1minus 120572) 119868 minus Γ119862119861

+Φ119862 + Γ119862119860 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

10038171003817100381710038171003817100381710038171003817100381710038171003817

lt 1

(26)

So we can get

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817120582 leΦ119862 + Γ119862119860

1 minus 120588

1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817 (27)

Remark 7 It shows that 119906119896converges to the 119906

119889of radius

(Φ119862 + Γ119862119860(1 minus 120588))119909119889(0) minus 119909

119896(0) with respect to the

120582 norm Besides we can know that the convergent radiusis bounded And the boundary depends on the initial inputstates of system

Consequently the tracking error of system will be

119890119896+1 (119905) = 119910

119889(119905) minus 119910

119896+1 (119905) = 119862 sdot 119890119860119905

[119909119889

(0) minus 119909119896

(0)]

+ int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896

(120591)] 119889120591

(28)

Similarly taking the norm sdot 120582on both sides of (28) we

can get

1003817100381710038171003817119890119896+1 (119905)

1003817100381710038171003817120582 =1003817100381710038171003817119910119889

(119905) minus 119910119896+1 (119905)

1003817100381710038171003817120582 le 119862

sdot

10038171003817100381710038171003817100381710038171003817119890119860119905

[119909119889

(0) minus 119909119896+1 (0)]

+ int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896+1 (120591)] 119889120591

10038171003817100381710038171003817100381710038171003817120582le 119862

sdot1003817100381710038171003817[119909119889

(0) minus 119909119896+1 (0)]

1003817100381710038171003817

+119862 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817120582

le 119862 sdot1003817100381710038171003817[119909119889

(0) minus 119909119896+1 (0)]

1003817100381710038171003817

+119862 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

sdotΦ119862 + Γ119862119860

1 minus 120588

1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

(29)

Specially the system usually can meet the initial condi-tion that is

119909119889

(0) = 119909119896

(0) 119896 = 1 2 3 (30)

That is said as

lim119896rarrinfin

1003817100381710038171003817119890119896+1 (119905)

1003817100381710038171003817120582 997888rarr 0 (31)

To sum up when the PD-type learning algorithm isapplied to the ILC with ESO the input 119906

119896(119905) of system can

converge to the ideal input 119906119889(119905) and the output of system

can converge to the ideal output 119910119889(119905)

4 Experiment Setup and Result

The verification platform is the rotary table of telescopesystem which consists of a DC motor a tracking loadan electrical driver a control system and a high precisionencoder whose accuracy is about plusmn0618 (arc-second) Thecontrol scheme includes two loops position loop and velocityloop

6 Mathematical Problems in Engineering

040302010

minus01minus02minus03minus04

0 10 20 30 40 50 60

Ang

le (d

eg)

t (s)

Reference signal

120572 = 314 (∘s2) 120596 = 10 (∘s)

Figure 3 The desired reference signal

The following three controllers are compared

(1) PI-PI this is the traditional Proportional-Integral (PI)controller In the velocity loop and position loop weuse two PI controllers The velocity signal can beobtained by differentiating the position signal

(2) PI-ESO in both of the position loop and the velocityloop the PI controllers are used but the velocitysignal will be gotten by the ESO The bandwidth 1205960of ESO is 1205960 = 100 The parameters of position looprsquosPI controller remain the same

(3) PI-ILC-ESO the PI controller is used in the positionloop and the ILC controller is used in the speed loopThe parameters of position looprsquos PI controller stillremain the same The velocity signal is also the signalwhich is gotten by the ESO To satisfy the condition(18) The factor 120572 and the deviation gain matrix Γ arechosen as 120572 = 02 and Γ = 05 By the state spaceequation of system (14) it will have

(1minus 120572) 119868 minus Γ119862119861 =

1003817100381710038171003817100381710038171003817100381710038171003817

(1minus 02) 119868 minus 05times [0 1] [01]

1003817100381710038171003817100381710038171003817100381710038171003817

= 03 lt 1

(32)

The three controllers are tested for a sinusoidal signalwhose velocity and acceleration are 120596 = 10 (

∘s) and 120572 =

314 (∘s2) The period of 119879 sinusoidal signal is 2 (s)

The desired reference trajectory is shown in Figure 3And the corresponding tracking performance under the threecontrollers is shown in Figures 4ndash6 As seen the PI-PIcontroller has a larger tracking error whose maximum valueis about 288410158401015840 Comparatively speaking the PI controllerwith ESO has a better tracking performance which illustratesthat ESO has an excellent estimation ability for disturbancesinvolved in the system Furthermore the proposed PI-ILC-ESO controller has the least tracking error than other controlmethods whose maximum value of error is about 200610158401015840This controller combines the advantages of ESO with theadvantages of ILC It makes use of ESO to eliminate most ofthe nonlinear disturbances Simultaneously the ILC gets thebest control input by learning previous control informationSome detailed results of the three controllers are shown inTable 1

0 10 20 30 40 50 60t (s)

PI-PI

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 4 The tracking error of PI-PI

0 10 20 30 40 50 60t (s)

PI-ESO

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 5 The tracking error of PI-ESO

0 10 20 30 40 50 60t (s)

PI-ILC-ESO

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 6 The tracking error of PI-ILC-ESO

Table 1 The experimental results of the three controllers

PI-PI PI-ESO PI-ILC-ESO|maximum error| 28843210158401015840 20998810158401015840 20059210158401015840

Root mean square error 9830110158401015840 6028210158401015840 5646810158401015840

From Table 1 it can be seen that the PI-ILC-ESO hasbetter performance than other control methods Howevercomparing the PI-ESO controller with the PI-ILC-ESO con-troller it can be seen that the effectiveness of using ILCis not obvious The maximum error reduces only to 09410158401015840and the root mean square error reduces to 03810158401015840 In factComparing with Figures 5 and 6 it can be found that theerrors of many position points exceed 1010158401015840 and the errorsare close to 2010158401015840 when the PI-ESO controller is used Butwhen the PI-ILC-ESO controller is used almost all the errorsof position points are near 1010158401015840 So the conclusion that thecomprehensive performance of PI-ILC-ESO is better than

Mathematical Problems in Engineering 7

other control methods can be gotten The ILC method helpsthe system get the best input signal Besides in Figures 5and 6 it can be found that the errors are evenly distributedComparing with the Figure 4 (PI-PI controller) both of thetwo controllers have reduced almost all the turning errorsof system The remaining errors are mostly caused by thesignal noise So it is difficult to reduce the remaining error andfurther improve the precision of system unless the problemof signal noise is solved

5 Conclusion

Uncertain nonlinear disturbances have been themajor factorswhich restrict the performance of tracking control systemsThis is because the systems will have a low tracking precisionwhen some uncertain nonlinear disturbances are induced inthe systems Therefore to reduce the influence introducedby the uncertain nonlinear disturbances an ILC methodwith ESO is proposed in this paper The ILC can get anexcellent input signal by learning previous control informa-tion It owns a better ability for eliminating some perioddisturbances Meanwhile an ESO is designed for estimatingsome uncertain nonlinear disturbances It compensates theshortage of the ILCrsquos disablement for nonperiod disturbancesIn addition the ESO can estimate an accurate velocity signaland supply the velocity as the feedback input signal ofiterative learning controller Furthermore the convergenceanalysis of the proposed control method guarantees therobustness of system Finally the experiment results showthat the proposed control method has excellent performancefor reducing the tracking error of telescope system

Conflict of Interests

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

References

[1] G G Valyavin V D Bychkov M V Yushkin et al ldquoHigh-resolution fiber-fed echelle spectrograph for the 6-m telescopeI Optical scheme arrangement and control systemrdquoAstrophys-ical Bulletin vol 69 no 2 pp 224ndash239 2014

[2] K Lu and Y Xia ldquoAdaptive attitude tracking control for rigidspacecraft with finite-time convergencerdquo Automatica vol 49no 12 pp 3591ndash3599 2013

[3] Y-J Sun ldquoComposite tracking control for generalized practicalsynchronization of Duffing-Holmes systems with parametermismatching unknown external excitation plant uncertaintiesand uncertain deadzone nonlinearitiesrdquo Abstract and AppliedAnalysis vol 2012 Article ID 640568 11 pages 2012

[4] Y Fu and T Chai ldquoSelf-tuning control with a filter and aneural compensator for a class of nonlinear systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 24no 5 pp 837ndash843 2013

[5] M M Bridges D M Dawson and J Hu ldquoAdaptive control fora class of direct drive robot manipulatorsrdquo International Journalof Adaptive Control and Signal Processing vol 10 no 4-5 pp417ndash441 1996

[6] A Tadayoni W-F Xie and B W Gordon ldquoAdaptive con-trol of harmonic drive with parameter varying friction usingstructurally dynamic wavelet networkrdquo International Journal ofControl Automation and Systems vol 9 no 1 pp 50ndash59 2011

[7] Y Alipouri and J Poshtan ldquoDesigning a robust minimum vari-ance controller using discrete slide mode controller approachrdquoISA Transactions vol 52 no 2 pp 291ndash299 2013

[8] B-Z Guo and F-F Jin ldquoSliding mode and active disturbancerejection control to stabilization of one-dimensional anti-stablewave equations subject to disturbance in boundary inputrdquo IEEETransactions on Automatic Control vol 58 no 5 pp 1269ndash12742013

[9] Y-S Lu and S-M Lin ldquoDisturbance-observer-based adaptivefeedforward cancellation of torque ripples in harmonic drivesystemsrdquo Electrical Engineering vol 90 no 2 pp 95ndash106 2007

[10] T Kavli ldquoFrequency domain synthesis of trajectory learningcontrollers for robot manipulatorsrdquo Journal of Robotic Systemsvol 9 no 5 pp 663ndash680 1992

[11] Y-C Wang and C-J Chien ldquoAn observer-based adaptiveiterative learning control using filtered-FNN design for roboticsystemsrdquo Advances in Mechanical Engineering vol 6 Article ID471418 2014

[12] S Arimoto S Kawamura and FMiyazaki ldquoBettering operationof Robots by learningrdquo Journal of Robotic Systems vol 1 no 2pp 123ndash140 1984

[13] J-X Xu D Huang V Venkataramanan and The Cat TuongHuynh ldquoExtreme precise motion tracking of piezoelectricpositioning stage using sampled-data iterative learning controlrdquoIEEE Transactions on Control Systems Technology vol 21 no 4pp 1432ndash1439 2013

[14] M Norrlof ldquoAn adaptive iterative learning control algorithmwith experiments on an industrial robotrdquo IEEE Transactions onRobotics and Automation vol 18 no 2 pp 245ndash251 2002

[15] J-X Xu Y Chen T H Lee and S Yamamoto ldquoTerminaliterative learning control with an application to RTPCVDthickness controlrdquo Automatica vol 35 no 9 pp 1535ndash15421999

[16] S S Saab ldquoA discrete-time stochastic learning control algo-rithmrdquo IEEE Transactions on Automatic Control vol 46 no 6pp 877ndash887 2001

[17] K K Leang and S Devasia ldquoDesign of hysteresis-compensatingiterative learning control for piezo-positioners application toatomic force microscopesrdquo Mechatronics vol 16 no 3-4 pp141ndash158 2006

[18] Z Zhu D Xu J Liu and Y Xia ldquoMissile guidance law basedon extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 60 no 12 pp 5882ndash5891 2013

[19] M Pizzocaro D Calonico C Calosso et al ldquoActive disturbancerejection control of temperature for ultrastable optical cavitiesrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 60 no 2 pp 273ndash280 2013

[20] J Yao Z Jiao and D Ma ldquoExtended-state-observer-basedoutput feedback nonlinear robust control of hydraulic systemswith backsteppingrdquo IEEE Transactions on Industrial Electronicsvol 61 no 11 pp 6285ndash6293 2014

[21] J Yao Z Jiao and D Ma ldquoAdaptive robust control of dc motorswith extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 61 no 7 pp 3630ndash3637 2014

[22] Q Zheng L Q Gao and Z Gao ldquoOn stability analysis ofactive disturbance rejection control for nonlinear time-varyingplants with unknown dynamicsrdquo inProceedings of the 46th IEEE

8 Mathematical Problems in Engineering

Conference on Decision and Control (CDC rsquo07) pp 3501ndash3506New Orleans La USA December 2007

[23] WQian S K Panda and J-X Xu ldquoTorque rippleminimizationin PM synchronous motors using iterative learning controlrdquoIEEE Transactions on Power Electronics vol 19 no 2 pp 272ndash279 2004

[24] G Heinzinger D Fenwick B Paden and F Miyazaki ldquoStabilityof learning control with disturbances and uncertain initialconditionsrdquo IEEETransactions onAutomatic Control vol 37 no1 pp 110ndash114 1992

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

Taking the norm sdot on both sides of (21) we obtain1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817

le (1minus 120572) 119868 minus Γ119862119861 sdot1003817100381710038171003817119906119889

(119905) minus 119906119896

(119905)1003817100381710038171003817

+ 119890119860119905

Φ119862 + Γ119862119860 sdot1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

+ Φ119862 + Γ119862119860

sdot int

119905

0119890119860(119905minus120591)

1198611003817100381710038171003817119906119889

(120591) minus 119906119896

(120591)1003817100381710038171003817 119889120591

(22)

Multiplying by 119890minus120582119905 120582 gt 119860 and we have

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817 119890minus120582119905

le (1minus 120572) 119868 minus Γ119862119861 sdot1003817100381710038171003817119906119889

(119905)

minus 119906119896

(119905)1003817100381710038171003817 119890minus120582119905

minus Φ119862 + Γ119862119860 sdot1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

sdot 119890(119860minus120582)119905

+ Φ119862 + Γ119862119860

sdot int

119905

0119890(119860minus120582)(119905minus120591)

119890minus120582120591

1198611003817100381710038171003817119906119889

(120591) minus 119906119896

(120591)1003817100381710038171003817 119889120591

le (1minus 120572) 119868 minus Γ119862119861 sdot1003817100381710038171003817119906119889

(119905) minus 119906119896

(119905)1003817100381710038171003817 119890minus120582119905

minus Φ119862

+ Γ119862119860 sdot1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817 119890(119860minus120582)119905

+Φ119862 + Γ119862119860 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

1003817100381710038171003817119906119889

(119905)

minus 119906119896

(119905)1003817100381710038171003817 119890minus120582119905

(23)

Definition 6 Define 120582 norm for a function ℎ [0 119879] rarr R119896

[24] then

ℎ (sdot)120582

≜ sup119905isin[0119879]

119890minus120582119905

ℎ (sdot) 997888rarr R119896 (24)

Then

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817120582 le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(1minus 120572) 119868 minus Γ119862119861

+Φ119862 + Γ119862119860 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

10038171003817100381710038171003817100381710038171003817100381710038171003817

sdot1003817100381710038171003817119906119889

(119905)

minus 119906119896

(119905)1003817100381710038171003817120582 minus Φ119862 + Γ119862119860 sdot

1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

(25)

Since the (1 minus 120572)119868 minus Γ119862119861 le 120588 lt 1 we can find a 120582 gt 119860

which makes

120588 =

10038171003817100381710038171003817100381710038171003817100381710038171003817

(1minus 120572) 119868 minus Γ119862119861

+Φ119862 + Γ119862119860 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

10038171003817100381710038171003817100381710038171003817100381710038171003817

lt 1

(26)

So we can get

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817120582 leΦ119862 + Γ119862119860

1 minus 120588

1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817 (27)

Remark 7 It shows that 119906119896converges to the 119906

119889of radius

(Φ119862 + Γ119862119860(1 minus 120588))119909119889(0) minus 119909

119896(0) with respect to the

120582 norm Besides we can know that the convergent radiusis bounded And the boundary depends on the initial inputstates of system

Consequently the tracking error of system will be

119890119896+1 (119905) = 119910

119889(119905) minus 119910

119896+1 (119905) = 119862 sdot 119890119860119905

[119909119889

(0) minus 119909119896

(0)]

+ int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896

(120591)] 119889120591

(28)

Similarly taking the norm sdot 120582on both sides of (28) we

can get

1003817100381710038171003817119890119896+1 (119905)

1003817100381710038171003817120582 =1003817100381710038171003817119910119889

(119905) minus 119910119896+1 (119905)

1003817100381710038171003817120582 le 119862

sdot

10038171003817100381710038171003817100381710038171003817119890119860119905

[119909119889

(0) minus 119909119896+1 (0)]

+ int

119905

0119890119860(119905minus120591)

119861 [119906119889

(120591) minus 119906119896+1 (120591)] 119889120591

10038171003817100381710038171003817100381710038171003817120582le 119862

sdot1003817100381710038171003817[119909119889

(0) minus 119909119896+1 (0)]

1003817100381710038171003817

+119862 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

1003817100381710038171003817119906119889

(119905) minus 119906119896+1 (119905)

1003817100381710038171003817120582

le 119862 sdot1003817100381710038171003817[119909119889

(0) minus 119909119896+1 (0)]

1003817100381710038171003817

+119862 sdot 119861 sdot (1 minus 119890

(119860minus120582)119905)

120582 minus 119860

sdotΦ119862 + Γ119862119860

1 minus 120588

1003817100381710038171003817119909119889

(0) minus 119909119896

(0)1003817100381710038171003817

(29)

Specially the system usually can meet the initial condi-tion that is

119909119889

(0) = 119909119896

(0) 119896 = 1 2 3 (30)

That is said as

lim119896rarrinfin

1003817100381710038171003817119890119896+1 (119905)

1003817100381710038171003817120582 997888rarr 0 (31)

To sum up when the PD-type learning algorithm isapplied to the ILC with ESO the input 119906

119896(119905) of system can

converge to the ideal input 119906119889(119905) and the output of system

can converge to the ideal output 119910119889(119905)

4 Experiment Setup and Result

The verification platform is the rotary table of telescopesystem which consists of a DC motor a tracking loadan electrical driver a control system and a high precisionencoder whose accuracy is about plusmn0618 (arc-second) Thecontrol scheme includes two loops position loop and velocityloop

6 Mathematical Problems in Engineering

040302010

minus01minus02minus03minus04

0 10 20 30 40 50 60

Ang

le (d

eg)

t (s)

Reference signal

120572 = 314 (∘s2) 120596 = 10 (∘s)

Figure 3 The desired reference signal

The following three controllers are compared

(1) PI-PI this is the traditional Proportional-Integral (PI)controller In the velocity loop and position loop weuse two PI controllers The velocity signal can beobtained by differentiating the position signal

(2) PI-ESO in both of the position loop and the velocityloop the PI controllers are used but the velocitysignal will be gotten by the ESO The bandwidth 1205960of ESO is 1205960 = 100 The parameters of position looprsquosPI controller remain the same

(3) PI-ILC-ESO the PI controller is used in the positionloop and the ILC controller is used in the speed loopThe parameters of position looprsquos PI controller stillremain the same The velocity signal is also the signalwhich is gotten by the ESO To satisfy the condition(18) The factor 120572 and the deviation gain matrix Γ arechosen as 120572 = 02 and Γ = 05 By the state spaceequation of system (14) it will have

(1minus 120572) 119868 minus Γ119862119861 =

1003817100381710038171003817100381710038171003817100381710038171003817

(1minus 02) 119868 minus 05times [0 1] [01]

1003817100381710038171003817100381710038171003817100381710038171003817

= 03 lt 1

(32)

The three controllers are tested for a sinusoidal signalwhose velocity and acceleration are 120596 = 10 (

∘s) and 120572 =

314 (∘s2) The period of 119879 sinusoidal signal is 2 (s)

The desired reference trajectory is shown in Figure 3And the corresponding tracking performance under the threecontrollers is shown in Figures 4ndash6 As seen the PI-PIcontroller has a larger tracking error whose maximum valueis about 288410158401015840 Comparatively speaking the PI controllerwith ESO has a better tracking performance which illustratesthat ESO has an excellent estimation ability for disturbancesinvolved in the system Furthermore the proposed PI-ILC-ESO controller has the least tracking error than other controlmethods whose maximum value of error is about 200610158401015840This controller combines the advantages of ESO with theadvantages of ILC It makes use of ESO to eliminate most ofthe nonlinear disturbances Simultaneously the ILC gets thebest control input by learning previous control informationSome detailed results of the three controllers are shown inTable 1

0 10 20 30 40 50 60t (s)

PI-PI

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 4 The tracking error of PI-PI

0 10 20 30 40 50 60t (s)

PI-ESO

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 5 The tracking error of PI-ESO

0 10 20 30 40 50 60t (s)

PI-ILC-ESO

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 6 The tracking error of PI-ILC-ESO

Table 1 The experimental results of the three controllers

PI-PI PI-ESO PI-ILC-ESO|maximum error| 28843210158401015840 20998810158401015840 20059210158401015840

Root mean square error 9830110158401015840 6028210158401015840 5646810158401015840

From Table 1 it can be seen that the PI-ILC-ESO hasbetter performance than other control methods Howevercomparing the PI-ESO controller with the PI-ILC-ESO con-troller it can be seen that the effectiveness of using ILCis not obvious The maximum error reduces only to 09410158401015840and the root mean square error reduces to 03810158401015840 In factComparing with Figures 5 and 6 it can be found that theerrors of many position points exceed 1010158401015840 and the errorsare close to 2010158401015840 when the PI-ESO controller is used Butwhen the PI-ILC-ESO controller is used almost all the errorsof position points are near 1010158401015840 So the conclusion that thecomprehensive performance of PI-ILC-ESO is better than

Mathematical Problems in Engineering 7

other control methods can be gotten The ILC method helpsthe system get the best input signal Besides in Figures 5and 6 it can be found that the errors are evenly distributedComparing with the Figure 4 (PI-PI controller) both of thetwo controllers have reduced almost all the turning errorsof system The remaining errors are mostly caused by thesignal noise So it is difficult to reduce the remaining error andfurther improve the precision of system unless the problemof signal noise is solved

5 Conclusion

Uncertain nonlinear disturbances have been themajor factorswhich restrict the performance of tracking control systemsThis is because the systems will have a low tracking precisionwhen some uncertain nonlinear disturbances are induced inthe systems Therefore to reduce the influence introducedby the uncertain nonlinear disturbances an ILC methodwith ESO is proposed in this paper The ILC can get anexcellent input signal by learning previous control informa-tion It owns a better ability for eliminating some perioddisturbances Meanwhile an ESO is designed for estimatingsome uncertain nonlinear disturbances It compensates theshortage of the ILCrsquos disablement for nonperiod disturbancesIn addition the ESO can estimate an accurate velocity signaland supply the velocity as the feedback input signal ofiterative learning controller Furthermore the convergenceanalysis of the proposed control method guarantees therobustness of system Finally the experiment results showthat the proposed control method has excellent performancefor reducing the tracking error of telescope system

Conflict of Interests

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

References

[1] G G Valyavin V D Bychkov M V Yushkin et al ldquoHigh-resolution fiber-fed echelle spectrograph for the 6-m telescopeI Optical scheme arrangement and control systemrdquoAstrophys-ical Bulletin vol 69 no 2 pp 224ndash239 2014

[2] K Lu and Y Xia ldquoAdaptive attitude tracking control for rigidspacecraft with finite-time convergencerdquo Automatica vol 49no 12 pp 3591ndash3599 2013

[3] Y-J Sun ldquoComposite tracking control for generalized practicalsynchronization of Duffing-Holmes systems with parametermismatching unknown external excitation plant uncertaintiesand uncertain deadzone nonlinearitiesrdquo Abstract and AppliedAnalysis vol 2012 Article ID 640568 11 pages 2012

[4] Y Fu and T Chai ldquoSelf-tuning control with a filter and aneural compensator for a class of nonlinear systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 24no 5 pp 837ndash843 2013

[5] M M Bridges D M Dawson and J Hu ldquoAdaptive control fora class of direct drive robot manipulatorsrdquo International Journalof Adaptive Control and Signal Processing vol 10 no 4-5 pp417ndash441 1996

[6] A Tadayoni W-F Xie and B W Gordon ldquoAdaptive con-trol of harmonic drive with parameter varying friction usingstructurally dynamic wavelet networkrdquo International Journal ofControl Automation and Systems vol 9 no 1 pp 50ndash59 2011

[7] Y Alipouri and J Poshtan ldquoDesigning a robust minimum vari-ance controller using discrete slide mode controller approachrdquoISA Transactions vol 52 no 2 pp 291ndash299 2013

[8] B-Z Guo and F-F Jin ldquoSliding mode and active disturbancerejection control to stabilization of one-dimensional anti-stablewave equations subject to disturbance in boundary inputrdquo IEEETransactions on Automatic Control vol 58 no 5 pp 1269ndash12742013

[9] Y-S Lu and S-M Lin ldquoDisturbance-observer-based adaptivefeedforward cancellation of torque ripples in harmonic drivesystemsrdquo Electrical Engineering vol 90 no 2 pp 95ndash106 2007

[10] T Kavli ldquoFrequency domain synthesis of trajectory learningcontrollers for robot manipulatorsrdquo Journal of Robotic Systemsvol 9 no 5 pp 663ndash680 1992

[11] Y-C Wang and C-J Chien ldquoAn observer-based adaptiveiterative learning control using filtered-FNN design for roboticsystemsrdquo Advances in Mechanical Engineering vol 6 Article ID471418 2014

[12] S Arimoto S Kawamura and FMiyazaki ldquoBettering operationof Robots by learningrdquo Journal of Robotic Systems vol 1 no 2pp 123ndash140 1984

[13] J-X Xu D Huang V Venkataramanan and The Cat TuongHuynh ldquoExtreme precise motion tracking of piezoelectricpositioning stage using sampled-data iterative learning controlrdquoIEEE Transactions on Control Systems Technology vol 21 no 4pp 1432ndash1439 2013

[14] M Norrlof ldquoAn adaptive iterative learning control algorithmwith experiments on an industrial robotrdquo IEEE Transactions onRobotics and Automation vol 18 no 2 pp 245ndash251 2002

[15] J-X Xu Y Chen T H Lee and S Yamamoto ldquoTerminaliterative learning control with an application to RTPCVDthickness controlrdquo Automatica vol 35 no 9 pp 1535ndash15421999

[16] S S Saab ldquoA discrete-time stochastic learning control algo-rithmrdquo IEEE Transactions on Automatic Control vol 46 no 6pp 877ndash887 2001

[17] K K Leang and S Devasia ldquoDesign of hysteresis-compensatingiterative learning control for piezo-positioners application toatomic force microscopesrdquo Mechatronics vol 16 no 3-4 pp141ndash158 2006

[18] Z Zhu D Xu J Liu and Y Xia ldquoMissile guidance law basedon extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 60 no 12 pp 5882ndash5891 2013

[19] M Pizzocaro D Calonico C Calosso et al ldquoActive disturbancerejection control of temperature for ultrastable optical cavitiesrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 60 no 2 pp 273ndash280 2013

[20] J Yao Z Jiao and D Ma ldquoExtended-state-observer-basedoutput feedback nonlinear robust control of hydraulic systemswith backsteppingrdquo IEEE Transactions on Industrial Electronicsvol 61 no 11 pp 6285ndash6293 2014

[21] J Yao Z Jiao and D Ma ldquoAdaptive robust control of dc motorswith extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 61 no 7 pp 3630ndash3637 2014

[22] Q Zheng L Q Gao and Z Gao ldquoOn stability analysis ofactive disturbance rejection control for nonlinear time-varyingplants with unknown dynamicsrdquo inProceedings of the 46th IEEE

8 Mathematical Problems in Engineering

Conference on Decision and Control (CDC rsquo07) pp 3501ndash3506New Orleans La USA December 2007

[23] WQian S K Panda and J-X Xu ldquoTorque rippleminimizationin PM synchronous motors using iterative learning controlrdquoIEEE Transactions on Power Electronics vol 19 no 2 pp 272ndash279 2004

[24] G Heinzinger D Fenwick B Paden and F Miyazaki ldquoStabilityof learning control with disturbances and uncertain initialconditionsrdquo IEEETransactions onAutomatic Control vol 37 no1 pp 110ndash114 1992

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

040302010

minus01minus02minus03minus04

0 10 20 30 40 50 60

Ang

le (d

eg)

t (s)

Reference signal

120572 = 314 (∘s2) 120596 = 10 (∘s)

Figure 3 The desired reference signal

The following three controllers are compared

(1) PI-PI this is the traditional Proportional-Integral (PI)controller In the velocity loop and position loop weuse two PI controllers The velocity signal can beobtained by differentiating the position signal

(2) PI-ESO in both of the position loop and the velocityloop the PI controllers are used but the velocitysignal will be gotten by the ESO The bandwidth 1205960of ESO is 1205960 = 100 The parameters of position looprsquosPI controller remain the same

(3) PI-ILC-ESO the PI controller is used in the positionloop and the ILC controller is used in the speed loopThe parameters of position looprsquos PI controller stillremain the same The velocity signal is also the signalwhich is gotten by the ESO To satisfy the condition(18) The factor 120572 and the deviation gain matrix Γ arechosen as 120572 = 02 and Γ = 05 By the state spaceequation of system (14) it will have

(1minus 120572) 119868 minus Γ119862119861 =

1003817100381710038171003817100381710038171003817100381710038171003817

(1minus 02) 119868 minus 05times [0 1] [01]

1003817100381710038171003817100381710038171003817100381710038171003817

= 03 lt 1

(32)

The three controllers are tested for a sinusoidal signalwhose velocity and acceleration are 120596 = 10 (

∘s) and 120572 =

314 (∘s2) The period of 119879 sinusoidal signal is 2 (s)

The desired reference trajectory is shown in Figure 3And the corresponding tracking performance under the threecontrollers is shown in Figures 4ndash6 As seen the PI-PIcontroller has a larger tracking error whose maximum valueis about 288410158401015840 Comparatively speaking the PI controllerwith ESO has a better tracking performance which illustratesthat ESO has an excellent estimation ability for disturbancesinvolved in the system Furthermore the proposed PI-ILC-ESO controller has the least tracking error than other controlmethods whose maximum value of error is about 200610158401015840This controller combines the advantages of ESO with theadvantages of ILC It makes use of ESO to eliminate most ofthe nonlinear disturbances Simultaneously the ILC gets thebest control input by learning previous control informationSome detailed results of the three controllers are shown inTable 1

0 10 20 30 40 50 60t (s)

PI-PI

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 4 The tracking error of PI-PI

0 10 20 30 40 50 60t (s)

PI-ESO

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 5 The tracking error of PI-ESO

0 10 20 30 40 50 60t (s)

PI-ILC-ESO

30

20

10

0

minus10

minus20

minus30Ang

le er

ror (

arc-

seco

nd)

Figure 6 The tracking error of PI-ILC-ESO

Table 1 The experimental results of the three controllers

PI-PI PI-ESO PI-ILC-ESO|maximum error| 28843210158401015840 20998810158401015840 20059210158401015840

Root mean square error 9830110158401015840 6028210158401015840 5646810158401015840

From Table 1 it can be seen that the PI-ILC-ESO hasbetter performance than other control methods Howevercomparing the PI-ESO controller with the PI-ILC-ESO con-troller it can be seen that the effectiveness of using ILCis not obvious The maximum error reduces only to 09410158401015840and the root mean square error reduces to 03810158401015840 In factComparing with Figures 5 and 6 it can be found that theerrors of many position points exceed 1010158401015840 and the errorsare close to 2010158401015840 when the PI-ESO controller is used Butwhen the PI-ILC-ESO controller is used almost all the errorsof position points are near 1010158401015840 So the conclusion that thecomprehensive performance of PI-ILC-ESO is better than

Mathematical Problems in Engineering 7

other control methods can be gotten The ILC method helpsthe system get the best input signal Besides in Figures 5and 6 it can be found that the errors are evenly distributedComparing with the Figure 4 (PI-PI controller) both of thetwo controllers have reduced almost all the turning errorsof system The remaining errors are mostly caused by thesignal noise So it is difficult to reduce the remaining error andfurther improve the precision of system unless the problemof signal noise is solved

5 Conclusion

Uncertain nonlinear disturbances have been themajor factorswhich restrict the performance of tracking control systemsThis is because the systems will have a low tracking precisionwhen some uncertain nonlinear disturbances are induced inthe systems Therefore to reduce the influence introducedby the uncertain nonlinear disturbances an ILC methodwith ESO is proposed in this paper The ILC can get anexcellent input signal by learning previous control informa-tion It owns a better ability for eliminating some perioddisturbances Meanwhile an ESO is designed for estimatingsome uncertain nonlinear disturbances It compensates theshortage of the ILCrsquos disablement for nonperiod disturbancesIn addition the ESO can estimate an accurate velocity signaland supply the velocity as the feedback input signal ofiterative learning controller Furthermore the convergenceanalysis of the proposed control method guarantees therobustness of system Finally the experiment results showthat the proposed control method has excellent performancefor reducing the tracking error of telescope system

Conflict of Interests

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

References

[1] G G Valyavin V D Bychkov M V Yushkin et al ldquoHigh-resolution fiber-fed echelle spectrograph for the 6-m telescopeI Optical scheme arrangement and control systemrdquoAstrophys-ical Bulletin vol 69 no 2 pp 224ndash239 2014

[2] K Lu and Y Xia ldquoAdaptive attitude tracking control for rigidspacecraft with finite-time convergencerdquo Automatica vol 49no 12 pp 3591ndash3599 2013

[3] Y-J Sun ldquoComposite tracking control for generalized practicalsynchronization of Duffing-Holmes systems with parametermismatching unknown external excitation plant uncertaintiesand uncertain deadzone nonlinearitiesrdquo Abstract and AppliedAnalysis vol 2012 Article ID 640568 11 pages 2012

[4] Y Fu and T Chai ldquoSelf-tuning control with a filter and aneural compensator for a class of nonlinear systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 24no 5 pp 837ndash843 2013

[5] M M Bridges D M Dawson and J Hu ldquoAdaptive control fora class of direct drive robot manipulatorsrdquo International Journalof Adaptive Control and Signal Processing vol 10 no 4-5 pp417ndash441 1996

[6] A Tadayoni W-F Xie and B W Gordon ldquoAdaptive con-trol of harmonic drive with parameter varying friction usingstructurally dynamic wavelet networkrdquo International Journal ofControl Automation and Systems vol 9 no 1 pp 50ndash59 2011

[7] Y Alipouri and J Poshtan ldquoDesigning a robust minimum vari-ance controller using discrete slide mode controller approachrdquoISA Transactions vol 52 no 2 pp 291ndash299 2013

[8] B-Z Guo and F-F Jin ldquoSliding mode and active disturbancerejection control to stabilization of one-dimensional anti-stablewave equations subject to disturbance in boundary inputrdquo IEEETransactions on Automatic Control vol 58 no 5 pp 1269ndash12742013

[9] Y-S Lu and S-M Lin ldquoDisturbance-observer-based adaptivefeedforward cancellation of torque ripples in harmonic drivesystemsrdquo Electrical Engineering vol 90 no 2 pp 95ndash106 2007

[10] T Kavli ldquoFrequency domain synthesis of trajectory learningcontrollers for robot manipulatorsrdquo Journal of Robotic Systemsvol 9 no 5 pp 663ndash680 1992

[11] Y-C Wang and C-J Chien ldquoAn observer-based adaptiveiterative learning control using filtered-FNN design for roboticsystemsrdquo Advances in Mechanical Engineering vol 6 Article ID471418 2014

[12] S Arimoto S Kawamura and FMiyazaki ldquoBettering operationof Robots by learningrdquo Journal of Robotic Systems vol 1 no 2pp 123ndash140 1984

[13] J-X Xu D Huang V Venkataramanan and The Cat TuongHuynh ldquoExtreme precise motion tracking of piezoelectricpositioning stage using sampled-data iterative learning controlrdquoIEEE Transactions on Control Systems Technology vol 21 no 4pp 1432ndash1439 2013

[14] M Norrlof ldquoAn adaptive iterative learning control algorithmwith experiments on an industrial robotrdquo IEEE Transactions onRobotics and Automation vol 18 no 2 pp 245ndash251 2002

[15] J-X Xu Y Chen T H Lee and S Yamamoto ldquoTerminaliterative learning control with an application to RTPCVDthickness controlrdquo Automatica vol 35 no 9 pp 1535ndash15421999

[16] S S Saab ldquoA discrete-time stochastic learning control algo-rithmrdquo IEEE Transactions on Automatic Control vol 46 no 6pp 877ndash887 2001

[17] K K Leang and S Devasia ldquoDesign of hysteresis-compensatingiterative learning control for piezo-positioners application toatomic force microscopesrdquo Mechatronics vol 16 no 3-4 pp141ndash158 2006

[18] Z Zhu D Xu J Liu and Y Xia ldquoMissile guidance law basedon extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 60 no 12 pp 5882ndash5891 2013

[19] M Pizzocaro D Calonico C Calosso et al ldquoActive disturbancerejection control of temperature for ultrastable optical cavitiesrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 60 no 2 pp 273ndash280 2013

[20] J Yao Z Jiao and D Ma ldquoExtended-state-observer-basedoutput feedback nonlinear robust control of hydraulic systemswith backsteppingrdquo IEEE Transactions on Industrial Electronicsvol 61 no 11 pp 6285ndash6293 2014

[21] J Yao Z Jiao and D Ma ldquoAdaptive robust control of dc motorswith extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 61 no 7 pp 3630ndash3637 2014

[22] Q Zheng L Q Gao and Z Gao ldquoOn stability analysis ofactive disturbance rejection control for nonlinear time-varyingplants with unknown dynamicsrdquo inProceedings of the 46th IEEE

8 Mathematical Problems in Engineering

Conference on Decision and Control (CDC rsquo07) pp 3501ndash3506New Orleans La USA December 2007

[23] WQian S K Panda and J-X Xu ldquoTorque rippleminimizationin PM synchronous motors using iterative learning controlrdquoIEEE Transactions on Power Electronics vol 19 no 2 pp 272ndash279 2004

[24] G Heinzinger D Fenwick B Paden and F Miyazaki ldquoStabilityof learning control with disturbances and uncertain initialconditionsrdquo IEEETransactions onAutomatic Control vol 37 no1 pp 110ndash114 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

other control methods can be gotten The ILC method helpsthe system get the best input signal Besides in Figures 5and 6 it can be found that the errors are evenly distributedComparing with the Figure 4 (PI-PI controller) both of thetwo controllers have reduced almost all the turning errorsof system The remaining errors are mostly caused by thesignal noise So it is difficult to reduce the remaining error andfurther improve the precision of system unless the problemof signal noise is solved

5 Conclusion

Uncertain nonlinear disturbances have been themajor factorswhich restrict the performance of tracking control systemsThis is because the systems will have a low tracking precisionwhen some uncertain nonlinear disturbances are induced inthe systems Therefore to reduce the influence introducedby the uncertain nonlinear disturbances an ILC methodwith ESO is proposed in this paper The ILC can get anexcellent input signal by learning previous control informa-tion It owns a better ability for eliminating some perioddisturbances Meanwhile an ESO is designed for estimatingsome uncertain nonlinear disturbances It compensates theshortage of the ILCrsquos disablement for nonperiod disturbancesIn addition the ESO can estimate an accurate velocity signaland supply the velocity as the feedback input signal ofiterative learning controller Furthermore the convergenceanalysis of the proposed control method guarantees therobustness of system Finally the experiment results showthat the proposed control method has excellent performancefor reducing the tracking error of telescope system

Conflict of Interests

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

References

[1] G G Valyavin V D Bychkov M V Yushkin et al ldquoHigh-resolution fiber-fed echelle spectrograph for the 6-m telescopeI Optical scheme arrangement and control systemrdquoAstrophys-ical Bulletin vol 69 no 2 pp 224ndash239 2014

[2] K Lu and Y Xia ldquoAdaptive attitude tracking control for rigidspacecraft with finite-time convergencerdquo Automatica vol 49no 12 pp 3591ndash3599 2013

[3] Y-J Sun ldquoComposite tracking control for generalized practicalsynchronization of Duffing-Holmes systems with parametermismatching unknown external excitation plant uncertaintiesand uncertain deadzone nonlinearitiesrdquo Abstract and AppliedAnalysis vol 2012 Article ID 640568 11 pages 2012

[4] Y Fu and T Chai ldquoSelf-tuning control with a filter and aneural compensator for a class of nonlinear systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 24no 5 pp 837ndash843 2013

[5] M M Bridges D M Dawson and J Hu ldquoAdaptive control fora class of direct drive robot manipulatorsrdquo International Journalof Adaptive Control and Signal Processing vol 10 no 4-5 pp417ndash441 1996

[6] A Tadayoni W-F Xie and B W Gordon ldquoAdaptive con-trol of harmonic drive with parameter varying friction usingstructurally dynamic wavelet networkrdquo International Journal ofControl Automation and Systems vol 9 no 1 pp 50ndash59 2011

[7] Y Alipouri and J Poshtan ldquoDesigning a robust minimum vari-ance controller using discrete slide mode controller approachrdquoISA Transactions vol 52 no 2 pp 291ndash299 2013

[8] B-Z Guo and F-F Jin ldquoSliding mode and active disturbancerejection control to stabilization of one-dimensional anti-stablewave equations subject to disturbance in boundary inputrdquo IEEETransactions on Automatic Control vol 58 no 5 pp 1269ndash12742013

[9] Y-S Lu and S-M Lin ldquoDisturbance-observer-based adaptivefeedforward cancellation of torque ripples in harmonic drivesystemsrdquo Electrical Engineering vol 90 no 2 pp 95ndash106 2007

[10] T Kavli ldquoFrequency domain synthesis of trajectory learningcontrollers for robot manipulatorsrdquo Journal of Robotic Systemsvol 9 no 5 pp 663ndash680 1992

[11] Y-C Wang and C-J Chien ldquoAn observer-based adaptiveiterative learning control using filtered-FNN design for roboticsystemsrdquo Advances in Mechanical Engineering vol 6 Article ID471418 2014

[12] S Arimoto S Kawamura and FMiyazaki ldquoBettering operationof Robots by learningrdquo Journal of Robotic Systems vol 1 no 2pp 123ndash140 1984

[13] J-X Xu D Huang V Venkataramanan and The Cat TuongHuynh ldquoExtreme precise motion tracking of piezoelectricpositioning stage using sampled-data iterative learning controlrdquoIEEE Transactions on Control Systems Technology vol 21 no 4pp 1432ndash1439 2013

[14] M Norrlof ldquoAn adaptive iterative learning control algorithmwith experiments on an industrial robotrdquo IEEE Transactions onRobotics and Automation vol 18 no 2 pp 245ndash251 2002

[15] J-X Xu Y Chen T H Lee and S Yamamoto ldquoTerminaliterative learning control with an application to RTPCVDthickness controlrdquo Automatica vol 35 no 9 pp 1535ndash15421999

[16] S S Saab ldquoA discrete-time stochastic learning control algo-rithmrdquo IEEE Transactions on Automatic Control vol 46 no 6pp 877ndash887 2001

[17] K K Leang and S Devasia ldquoDesign of hysteresis-compensatingiterative learning control for piezo-positioners application toatomic force microscopesrdquo Mechatronics vol 16 no 3-4 pp141ndash158 2006

[18] Z Zhu D Xu J Liu and Y Xia ldquoMissile guidance law basedon extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 60 no 12 pp 5882ndash5891 2013

[19] M Pizzocaro D Calonico C Calosso et al ldquoActive disturbancerejection control of temperature for ultrastable optical cavitiesrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 60 no 2 pp 273ndash280 2013

[20] J Yao Z Jiao and D Ma ldquoExtended-state-observer-basedoutput feedback nonlinear robust control of hydraulic systemswith backsteppingrdquo IEEE Transactions on Industrial Electronicsvol 61 no 11 pp 6285ndash6293 2014

[21] J Yao Z Jiao and D Ma ldquoAdaptive robust control of dc motorswith extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 61 no 7 pp 3630ndash3637 2014

[22] Q Zheng L Q Gao and Z Gao ldquoOn stability analysis ofactive disturbance rejection control for nonlinear time-varyingplants with unknown dynamicsrdquo inProceedings of the 46th IEEE

8 Mathematical Problems in Engineering

Conference on Decision and Control (CDC rsquo07) pp 3501ndash3506New Orleans La USA December 2007

[23] WQian S K Panda and J-X Xu ldquoTorque rippleminimizationin PM synchronous motors using iterative learning controlrdquoIEEE Transactions on Power Electronics vol 19 no 2 pp 272ndash279 2004

[24] G Heinzinger D Fenwick B Paden and F Miyazaki ldquoStabilityof learning control with disturbances and uncertain initialconditionsrdquo IEEETransactions onAutomatic Control vol 37 no1 pp 110ndash114 1992

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

Conference on Decision and Control (CDC rsquo07) pp 3501ndash3506New Orleans La USA December 2007

[23] WQian S K Panda and J-X Xu ldquoTorque rippleminimizationin PM synchronous motors using iterative learning controlrdquoIEEE Transactions on Power Electronics vol 19 no 2 pp 272ndash279 2004

[24] G Heinzinger D Fenwick B Paden and F Miyazaki ldquoStabilityof learning control with disturbances and uncertain initialconditionsrdquo IEEETransactions onAutomatic Control vol 37 no1 pp 110ndash114 1992

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