Research ArticleAutomatic Gauge Control in Rolling Process Based onMultiple Smith Predictor Models

Jiangyun Li1 Kang Wang1 and Yang Li2

1 School of Automation and Electrical Engineering University of Science and Technology Beijing Beijing 100083 China2 School of International Studies Communication University of China (CUC) Beijing 100024 China

Correspondence should be addressed to Kang Wang vangkang163com

Received 13 August 2014 Accepted 12 September 2014 Published 24 November 2014

Academic Editor Shen Yin

Copyright copy 2014 Jiangyun Li 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

Automatic rolling process is a high-speed system which always requires high-speed control and communication capabilitiesMeanwhile it is also a typical complex electromechanical system distributed control has become the mainstream of computercontrol system for rolling mill Generally the control system adopts the 2-level control structuremdashbasic automation (Level 1) andprocess control (Level 2)mdashto achieve the automatic gauge control In Level 1 there is always a certain distance between the rollgap of each stand and the thickness testing point leading to the time delay of gauge control Smith predictor is a method to copewith time-delay system but the practical feedback control based on traditional Smith predictor cannot get the ideal control resultbecause the time delay is hard to bemeasured precisely and in some situations it may vary in a certain range In this paper based onadaptive Smith predictor we employmultiple models to cover the uncertainties of time delayThe optimal model will be selected bythe proposed switchmechanism Simulations show that the proposedmultiple Smithmodelmethod exhibits excellent performancein improving the control result even for system with jumping time delay

1 Introduction

Because the control objects are electromechanical andhydraulic systems with small inertia and fast response auto-matic rolling is a fast process which requires high-speed con-trol and communication capabilities With the technologicaland industrial development requirements for high-qualitystrips and high-level automation are becoming critical Dis-tributed control has become the mainstream of computercontrol system for rolling mill which is a typical com-plex electromechanical system [1 2] Generally the controlsystem adopts the 2-level control structuremdashbasic automa-tion (Level 1) and process control (Level 2)mdashto achieveautomatic gauge control [3 4] Composed of process serverand high-speed communication network the process control(Level 2) can accomplish mathematical model calculatinginitial values setting and material tracking This level alwayshas a high demand on network bandwidth with the abilityto transfer store and share massive process information atypical network adopted is the industrial Ethernet Material

tracking and initial value settings of roll gap and roll forcein finish rolling AGC (automatic gauge control) process arecalculated through the mathematical model by the serversin Level 2 corresponding information is transferred to thebasic automation controllers through Ethernet shown as inFigure 1 The basic automation Level 1 accomplishes the fastcollection of process information and the output of controlresults controllers in this level are high-speed embeddedcontrollers or high-performance PLC Because of the timerequirement of fast loop control network communicationrates among controllers are also highly demanded Typicalnetworks adopted in this level are the GErsquos RFnet (reflectivememory network) and Siemensrsquo GDM (global data memory)network Figure 2 shows the basic automation networkdeployment of a tandem rolling line in which controllersin the rough rolling section finish rolling section and basicautomation are connected through RFnet In conclusionnetwork delay and packet loss in AGC will both influencethe manufacturing process and even lead to the productdisqualified in severe cases

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 872418 10 pageshttpdxdoiorg1011552014872418

2 Abstract and Applied Analysis

of coiler

Finishing rolling section

Main electrical room

Main electrical room

Armored fiberTwisted pairTwisted pair (2)

Hot rolling Ethernet

Furnace section

Coiling section

Main electrical room

Network switchNetwork switch

Network switch

Network switch

Network switchNetwork switch

Network switch

Network switch Network switch

Rough rolling section

Main electrical room

Network switch Network switch

Level 2

Level 1

of furnace sectionof rough rolling section

sectionof finishing rolling

120572120572120572120572

Figure 1 Ethernet network structure in a tandem rolling line

In practical AGC thickness control process two mainfactors will lead to the uncertain time delay for controlled sys-tem One is the certain distance between the thickness gaugeand rolling gap of each stand and the other is the essentiallyexisting time delay in network transmission Smith predictoris a method to deal with above problem but as the value ofsystem time delay is hard to be measured precisely the Smithpredictor method always cannot be carried out effectivelyin practice Adaptive Smith predictor model can cope withpartial model mismatch [5 6] However improper selectionof adaptive initial value may also cause the system responsewith bad transient process this situation may get worse forsystem with variant time delay because the single predictorcannot match all conditions with different time delay wellMultiplemodel adaptive control (MMAC) is a kind of tool foradaptive control wheremultiple elementmodels will be set upto cover the uncertainty of the parameter or structure of thesystem and a switching mechanism will be used to form thefinal multiple model controllerThis kind of controller can beused to improve the transient response [7] From the 1990s a

lot of MMAC based on continuous-time [8 9] and discrete-time [10] system have been given and a lot of application ofMMAC also can be seen [11 12]

In this paper we introduce MMAC into adaptive Smithpredictor to form the multiple model adaptive Smith predic-tor controlmethod so as to improve the control performanceAccording to the possible varying range of the controlledplantrsquos time delay multiple fixed and adaptive models areestablished to cover the plantrsquos uncertainties correspondingcontrollers are also set up the switch index function is pro-posed based on model error to decide the most appropriatecontroller for the system at every moment and select it asthe current controller Ultimately multiple adaptive Smithcontrollers are composed to improve the systemrsquos controlperformance

The following paper is organized as follows Section 2introduces the design principle of Smith predictor controlSection 3 presents the proposedmultiplemodel Smith predic-tor controller in detail Numerical simulations are conductedin Section 4 and Section 5 concludes this work

Abstract and Applied Analysis 3

Network switch Network switch

Level 2

Level 1

Armored fiberFiber jumpers

RM Fiber optic network

Rough rolling section

RFM5595

Coiling section

Furnace section

RFM5595

RFM5595

RFM5595

Finishing rolling

RFM5595

section

120572120572120572

Figure 2 RM fiber optic network in a tandem rolling line

2 Controller Design Principle Based onSmith Predictor

21 Fixed Smith Predictor Controller AGC thickness controlof rolling mill is a control system with time delay traditionalAGC control methods cannot get satisfying control perfor-mance [13 14]

According to control theory to cope with the time-delay problem we can take the Smith predictor schemeA prediction model is added into the AGC process as thefeedback loop which can predict the change of system outputand give feedback signal in advance so as to offset theoriginal system delay and make the characteristic equationof the whole closed loop system without time delay [15] Theprinciple of Smith predictor in AGC process [16] is shown inFigure 3

In Figure 3 ℎ119903(119905) and 119862(119905) are the preset value and

measured value of the exit thickness respectivelyΔℎ(119905) is thethickness difference Δ119878 is the regulation value of the rollinggap119866

119888(119904) is the controller119866(119904)eminus120591119904 is the transfer function of

controlled plant and 119866119898(119904)eminus120591119898119904 is the prediction model

hr(t) +

minusminus

ΔhGc(s)

ΔSG(s) eminus120591s C(t)

Gm(s) eminus120591119898s+

minus

Figure 3 AGC control process based on Smith predictor

When Smith predictor is not used the system transferfunction is

119862 (119904) =119866119888(119904) 119866 (119904) eminus120591119904

1 + 119866119888(119904) 119866 (119904) eminus120591119904

ℎ119903(119904) (1)

Corresponding characteristic equation is 1 + 119866119888(119904)119866(119904)

eminus120591119904 = 0As the pure time-delay part eminus120591119904 exists in both transfer

function and characteristic equation with the increase oftime delay 120591 the phase delay increases system stability

4 Abstract and Applied Analysis

Feedback

Feedforward

Adaptive mechanism

regulator

regulator

hr(t)

+

+

+

+

+

minus

minus

minus

Δh ΔSG(s) eminus120591s

C(t)

eminus120591119898s

Gc(s)

Gm(s)

Gm(s)

K(e t)

G998400m(t)ΔS

998400

e(t)

Gm(t)

F(e t)

C998400(t)

Figure 4 AGC control process based on adaptive Smith predictor

decreases and the control performance goes worse whilein the Smith predictor scheme when the prediction modelmatches the controlled plant perfectly that is 119866

119898(119904) = 119866(119904)

120591119898= 120591 we have 119866

119898(119904)eminus120591119898119904 = 119866(119904)eminus120591119904 and system output

becomes

119862 (119904)

=119866119888(119904) 119866 (119904) eminus120591119904

1 + 119866119888(119904) 119866 (119904) eminus120591119904 minus 119866

119888(119904) 119866119898(119904) eminus120591119898119904 + 119866

119888(119904) 119866119898(119904)

times ℎ119903(119904)

=119866119888(119904) 119866 (119904) eminus120591119904

1 + 119866119888(119904) 119866119898(119904)

ℎ119903(119904)

(2)

Corresponding characteristic equation becomes 1 +

119866119888(119904)119866119898(119904) = 0 without time-delay part so the system

stability will not be influenced by time delay 120591

22 Adaptive Smith Predictor Controller Practically condi-tion 119866

119898(119904) = 119866(119904) 120591

119898= 120591 cannot be satisfied easily [17]

proposes a more practical method in which adaptive adjust-ing law of feedback coefficient 119865(119905) and feedforward coef-ficient 119870(119905) is applied to guarantee the identification Smithpredictor 119866

119898(119904) = 119866(119904) under the condition 120591

119898= 120591 = 0 The

principle of adaptive Smith predictior is shown in Figure 4Consider 120591

119898= 120591 = 0 the transfer functions of the system

and Smith predictor model are given as

119866 (119904) =119896

119904 + 119886 119866

119898(119904) =

119896119898

119904 + 119886119898

(3)

Corresponding state equations can be got as

(119905) = minus119886119862 (119905) + 119896Δ119878 (119905)

119898(119905) = minus119886

119898119862 (119905) + 119896

119898Δ1198781015840

(119905)

(4)

For given generalized error

119890 (119905) = 119862 (119905) minus 119862119898(119905) (5)

Lynaponov function is set up as

V (119905) =1

2[119896119898119890 (119905)2

+1

1205781

1205952

(119905) +1

1205782

1205932

(119905)] (6)

where

120595 (119905) = 119886 minus 119886119898minus 119896119898119865 (119905)

120593 (119905) = 119896 minus 119896119898119870 (119905)

(7)

An adaptive control law

119865 (119905) = minusint 1205781119890 (119905) 119862

119898(119905) + 119865 (0)

119870 (119905) = int 1205782119890 (119905) Δ119878 (119905) + 119870 (0)

(8)

can be used to guarantee 119890(119905) rarr 0120595(119905) rarr 0 and 120593(119905) rarr 0

when 119905 rarr infin

Remark 1 The adaptive parameter identification mechanismcan be viewed as a kind of data-driven strategy [18 19]Information about process measurements and initial valuesis required in the condition of given prediction modelHowever unlike pure data-driven methods such as partialleast squares (PLS) iterative learning control (ILC) andmodel free adaptive control which only depend on theprocess measurements [20] this adaptive law is a model-dataintegrated method [21] which also needs the relative precisepredictionmodel so that the process information can be fullyutilized based on the model to realize the system identifica-tion

From Figure 4 this adaptive law can ensure119866119898(119904) = 119866(119904)

(where119866119898(119904) is an identification Smith predictor adjusted by

119865 119866119898 and 119870) the identification Smith predictor 119866

119898(119904) can

be used But in practice both 119866119898(119904) = 119866(119904) and 120591

119898= 120591 = 0

are always very hard to be satisfied because the precisemodelof the system cannot be setup and time delay of the systemcannot be measured accurately So above adaptive Smith pre-dictor is also very difficult to be applied Due to this situationthe following multiple adaptive Smith prediction modelcontroller will be considered

Abstract and Applied Analysis 5

Adaptive mechanism

Feedforwardregulator

Feedback regulator

+

+

++

minus

minus

hr(t)

minus

ΔhGc(s)

ΔSG(s)

C(t)

Gm(s)

eminus120591119898s

eminus120591119898s

eminusΔ120591sG998400(s)

C998400(t)

ΔS998400

e(t)

C998400m(t)

Gm(t)

F(e t)

Gm(s)

Smith predictor P1

Smith predictor Pi

Smith predictor Pn

e1(t)

ei(t)

en(t)

Switching

k(e t)

Figure 5 AGC control process based on multiple adaptive Smith prediction model

3 Multiple Smith Prediction Model Controller

Consider the rolling system in Figure 5 its transfer functionand time delay can be generally described by the followingmodel

119866 (119904) eminusΔ120591119904eminus120591119898119904 (9)

where 120591119898is the delay under normal working condition and

Δ120591 stands for the delay fluctuation caused by uncertain factorssuch as network transmission delay

Define

1198661015840

(119904) = 119866 (119904) eminusΔ120591119904 (10)

Consider the character of first order system we assumethat a first order systemwith small delay can be approximatedby a first order processwith different parameters [22] we have

1198661015840

(119904) asymp1198961015840

119904 + 1198861015840 (11)

Further we have the multiple model adaptive controlsystem (see Figure 5)

As in Figure 5 we use the Smith prediction model 119866119898(119904)

adjusted by 119865119866119898 and119870 to approximate the controlled plant

1198661015840

(119904) Consider the varying scope of Δ120591 parameters of 1198661015840(119904)will be uncertain or time varying so it is not appropriate todescribe the systemby single fixedmodel For the single adap-tive model if its initial parameters are not properly selectedmodel parameters will converge slowly Moreover if thereis drastic system parameter jump control performance willget worse So we take multiple models with different initialvalues to approximate the system model uncertainty

31 Multiple Adaptive Smith Model Controller Based onthe possible parameters variation range of 1198661015840(119904) multipleadaptive Smith predictors 119875

119894(119894 = 1 2 119899 119899 is the

number of models) with different initial values are designedFor original Smith prediction model 119866

119898(119904) corresponding

adaptive parameters of the feedforward and feedback loopsatisfy

119865119894(119905) = minusint 120578

1119890 (119905) 119862

1015840

119898(119905) + 119865

119894(0)

119870119894(119905) = int 120578

2119890 (119905) Δ119878 (119905) + 119870

119894(0)

(12)

where 119894 = 1 2 119899Define the switch index function

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905 (13)

where 119894 = 1 2 119899 120572 gt 0 and 120573 gt 0 And the model error

119890119894(119905) = 119862

1015840

(119905) minus 1198621015840

119898119894(119905) asymp 119862 (119905) minus 119862

119898119894(119905) (14)

At time 119905 calculate

119897 (1199050 119905) = argmin

119894isin12119898

119869119894(1199050 119905) (15)

and Smith predictor based on model 119897 will be switched intothe closed loop system

Remark 2 In practical AGC control process 1198621015840(119905) cannot bemeasured so as in (14) 119890

119894(119905) = 119888(119905)minus119888

119898(119905)will be used instead

of 1198621015840(119905) minus 1198621015840119898(119905) in practical control

32 Multiple Fixed Smith Prediction Model Controller Formultiple adaptive Smith prediction model controller parallelcomputation of multiple adaptive processes will lead to theincrease of calculation amount

If all the 119899models are selected to be fixed models that isfor 119894 = 1 2 119899

119865119894(119905) = 119865

119894(0) gt 0

119870119894(119905) = 119870

119894(0) gt 0

(16)

6 Abstract and Applied Analysis

With enough fixed models take (13) as the switch indexfunction calculation amount for multiple models can bedecreased drastically However without the stable characterof adaptive model system stability can hardly be guaranteedby pure multiple fixed models

33 Multiple Adaptive Smith Controllers with Multiple Fixedand Adaptive Models Multiple fixed models as in Section 32can reduce the amount of calculation but due to the lack ofadaptive parameter adjusting process system stability cannotbe easily obtained If in the 119899 models 119899 minus 1 are set up asfixed models and 1 model is adaptive model that is for 119894 =1 2 119899 minus 1

119865119894(119905) = 119865

119894(0) gt 0

119870119894(119905) = 119870

119894(0) gt 0

119865119899(119905) = minusint 120578

1119890119899(119905) 1198621015840

119898(119905) + 119865

119899(0)

119870119899(119905) = int 120578

2119890119899(119905) Δ119878 (119905) + 119870

119899(0)

(17)

The existence of multiple fixed models reduces theamount of calculation and the adaptivemodel guarantees thesystem stability so the control performance can be improvedas the case of multiple adaptive models

Theorem3 If the adaptive Smith controller in Section 22 (120591 =120591119898= 0) is applied to the AGC system model error 119890 can be

guaranteed to be bounded and to approach zero as 119905 rarr infinthat is lim

119905rarrinfin119890(119905) = 0

Proof From [17] for adaptive Smith predictionmodel takingderivative of Lynaponov function (6) we have

V (119905) = minus 1198961198981198861198902

(119905) + 120595 (119905) [1

1205781

(119905) minus 119896119898119890 (119905) 119862

119898(119905)]

+ 120593 (119905) [1

1205782

(119905) + 119896119898119890 (119905) Δ119878 (119905)]

(18)

from (7) and (8)

V (119905) = minus1198961198981198861198902

(119905) le 0 (19)

Consider 1205781gt 0 120578

2gt 0 119896 gt 0 and 119896

119898gt 0 so V(119905) gt 0

holds for all 119905 = 0From (19) V(119905) is bounded so 119890(119905) 120595(119905) and 120593(119905) are all

bounded Further we have

int

119905

1199050

1198902

(119905) 119889119905 lt infin 119890 (119905) isin 1198712 (20)

For

1015840

119898(119905) = minus119886

1198981198621015840

119898(119905) + 119896

119898Δ1198781015840

(119905)

Δ1198781015840

(119905) = 119870 (119905) Δ119878 (119905) minus 119865 (119905) 1198621015840

119898(119905)

1198621015840 (119905) = minus1198861198621015840

(119905) + 119896Δ119878 (119905)

119890 (119905) = 1198621015840

(119905) minus 1198621015840

119898(119905)

(21)

we have

119890 (119905) = minus119886119890 (119905) minus 120595 (119905) 1198621015840

119898(119905) + 120593 (119905) Δ119878 (119905) (22)

Consider Δ119878(119905) 1198621015840(119905) and 119890(119905) are bounded then 1198621015840119898(119905)

is bounded from (22) we have 119890(119905) isin 119871infin from (20) and the

Barbalet Lemma [23] we have lim119905rarrinfin

119890(119905) = 0

Theorem 4 Take (15) as the switch index function and selectmultiple adaptivemodels in Section 31 ormultiple fixedmodelsand one adaptive model in Section 33 as the multiple adaptiveSmith controllers For the AGC Smith controller shown as inFigure 5 each of these two kinds of controllers can cancel thefluctuation of system delay effectively and improve the controlperformance

Proof (1) For multiple model control composed of multipleadaptive Smith predictors since each Smith model has itsown adaptive mechanism so for each model we have

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905 lt infin

119894 = 1 2 119899

lim119905rarrinfin

119890119894(119905) = 0

(23)

Though there is model switching among multiple modelsbased on the switch index function it finally comes to

Δ119878 (119866119898119894(119904) minus 119866

1015840

(119904)) = 0 (24)

Uncertain part of time delay is canceled andmultiple modelsare used to improve the transient process

(2) For multiple model control composed of multiplefixed Smith predictors and one adaptive Smith predictor asto the adaptive Smith predictor

119869119899(1199050 119905) = 120572119890

2

119899(119905) + 120573int

119905

1199050

1198902

119899(119905) 119889119905 lt infin (25)

Consider the switch index function

119897 (1199050 119905) = argmin

119894isin12119899

119869119894(1199050 119905) (26)

For each fixed model

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905

119894 = 1 2 119899 minus 1

(27)

If

lim119905rarrinfin

119869119894(1199050 119905) = infin 119894 = 1 2 119899 minus 1 (28)

then finally the adaptive model will be selected and theadaptive Smith model matches the systemmodel completelythat is Δ119878(119866

119898119899(119904) minus 119866

1015840

(119904)) = 0

Abstract and Applied Analysis 7

If there is a fixed model 119894 satisfying

lim119905rarrinfin

119869119894(1199050 119905) = lim

119905rarrinfin

[1205721198902

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905] lt infin (29)

where 119894 = 1 2 119899 minus 1 from (21)

1015840

(119905) = minus1198861198621015840

(119905) + 119896Δ119878 (119905)

1015840

119898119894(119905) = minus [119886

119898+ 119896119898119865119894(0)] 119862

1015840

119898119894(119905) + 119896

119898119870119894(0) Δ119878 (119905)

119890119894(119905) = 119862

1015840

(119905) minus 1198621015840

119898119894(119905)

(30)

If the parameters of selected fixed model satisfy 119865119894(0) gt 0 or

119886119898+ 119896119898119865119894(0) gt 0 then 119862

1015840

119898119894is bounded and from (7) (29)

and (30) 119890119894(119905) 120595119894(119905) and 120593

119894(119905) are bounded the following

inequation also holds

119890119894(119905) = minus119886119890

119894(119905) minus 120595

119894(119905) 1198621015840

119898(119905) + 120593

119894(119905) Δ119878 (119905) lt infin (31)

then from (29) and (31) and the Barbalet Lemma we havelim119905rarrinfin

119890119895(119905) = 0

119895 isin 119895 = 119899 or 119895 = 119908 | lim119905rarrinfin

119869119908(1199050 119905) lt infin

119908 = 1 2 119899 minus 1

(32)

and it finally comes to

Δ119878 (119866119898119895(119904) minus 119866

1015840

(119904)) = 0 (33)

The multiple model adaptive Smith predictor still has thematching function

Remark 5 Multiple model control based on multiple fixedSmith models can improve the transient response but as itlacks the adaptive parameter regulation system stability canhardly be guaranteed

Remark 6 The existence of multiple models covering theparameter uncertain range of the controlled plant canimprove the systemrsquos transient response and improve thecontrol quality for system with jumping parameter

4 Simulation Research

A first order time delay system as follows will be simulatedfor rolling AGC system

119866 (119904) eminus120591119904 = 119896

119904 + 119886eminus120591119904 = 021

0053119904 + 1eminus035119904

= 1198661015840

(119904) eminus03119904(34)

where 1198661015840(119904) = (021(0053119904 + 1))eminus005119904The same PID controller as in [17] based on ITAE

criterion [24] will also be used that is

119866119888(119904) = 119870

119901(1 +

1

119879119894119904+

119879119889119904

1 + 11987911988911990410

)

= 223 (1 +1

02119904+

0034119904

00034119904 + 1)

(35)

12

1

08

06

04

02

00 5 10 15 20 25 30 35 40

1005

0995

1

39 395 40Out

put t

hick

nessC(t)

(mm

)

Single adaptive Smith

Time t (s)

Figure 6 Single model adaptive Smith predictor

1005

0995

1

39 395 4000 5 10 15 20 25 30 35 40

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Multiple adaptive Smith

Time t (s)

Figure 7 Multiple model adaptive Smith predictor

10005

09995

1

39 395 40

0 5 10 15 20 25 30 35 400

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Multiple fixed Smith

Time t (s)

Figure 8 Multiple fixed Smith prediction model

The compensation Smith predictor will be designed as119866119898(119904) = 021(0053119904 + 1) 120591

119898= 03119904 And parameters of

119865(119905) and119870(119905) will be tuned as (8)

41 Multiple Model Smith Predictor for System with FixedTime Delay For single adaptive Smith prediction model letthe feedforward and feedback adaptive initial parameters be119865(0) = 35 and119870(0) = 50 in (8) the control result is shown inFigure 6 For multiple model adaptive control four adaptiveSmith predictionmodels are established corresponding feed-forward and feedback adaptive initial parameters in (12) are1198651(0) = 35 119865

2(0) = 30 119865

3(0) = 25 119865

4(0) = 18 and 119870

1(0) =

1198702(0) = 119870

3(0) = 119870

4(0) = 50 The control result is shown

in Figure 7

8 Abstract and Applied Analysis

Multiple fixed and one adaptive Smith

1005

0995

1

39 395 400 5 10 15 20 25 30 35 40

0

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Time t (s)

(a) System output

4

3

2

1Switc

hing

sequ

ence

I

Multiple fixed and one adaptive Smith

0 5 10 15 20 25 30 35 40

Time t (s)

(b) Switching sequence

Figure 9 Multiple fixed and single adaptive Smith prediction model

It can be seen in Figures 6 and 7 when multiple adaptiveSmith predictionmodels withmultiple groups of feedforwardand feedback adaptive initial parameters are applied to theAGC system the overshot and system response time aredecreased obviously compared with single Smith predictionmodel case Meanwhile when single model adaptive Smithpredictor is applied because model parameters match badlywith system parameters initial identification error is rel-atively big and the identification time goes relatively longcorrespondingly Since multiple model adaptive Smith pre-dictor adoptsmultiple adaptive initial parameters system canswitch to the closest matching model adaptively and conse-quently system identification error and identification timedecrease obviously

When four fixed Smith prediction models with initialadaptive feedback parameters 119865

1(0) = 35 119865

2(0) = 30

1198653(0) = 25 and 119865

4(0) = 18 and same initial feedforward

parameters 1198701(0) = 119870

2(0) = 119870

3(0) = 119870

4(0) = 50 in (16) are

applied the control result is shown in Figure 8 At the sametime the mixed multiple model Smith predictor with threefixed models whose initial feedback adaptive parameters are1198651(0) = 35 119865

2(0) = 30 and 119865

3(0) = 18 and an adaptive

model with initial value 1198654(0) = 18 the same feedforward

parameters 1198701(0) = 119870

2(0) = 119870

3(0) = 119870

4(0) = 50 are given

and the control results can be seen in Figure 9Comparedwithmultiple adaptive Smith predictormodel

the computing time can be reduced by using multiple fixedSmith predictor models but the stability cannot be guaran-teed because of the absence of adaptive adjustment FromFigure 8 suitable number of fixed models can be used toimprove the overshoot and response time but the steadyidentification error cannot be made into 0 at last In Figure 9the combination of fixed and adaptive Smith models willgive an improved transient response without the loss of finalproperty of system stability The model switching process isshown in Figure 9(b) the initial fixed model with 119865

1(0) = 35

is not the best one and fixed model 3 with 1198653(0) = 18 is

selected to control the system for a period of time finally thecontrol model is switched to the 4th adaptive Smith model

14

12

1

1

08

06

04

02

00 5 10 15 20 25

1004

1002

8 10 15 20 25

Change of time delay

Single adaptive SmithMultiple adaptive SmithSingle adaptive and multiple fixed Smith

Out

put t

hick

nessC(t)

(mm

)

Time t (s)

Figure 10 Multiple Smith prediction model for system with jump-ing time delay

42 Multiple Smith Prediction Model for System with JumpingTime Delay The rolling AGC system with jumping timedelay can be described as

119866 (119904) eminus120591119904 = 021

0053119904 + 1eminus120591119904

120591 = 03 0 lt 119905 lt 10

07 119905 ge 10

(36)

Take the single adaptive Smith predictionmodelmultipleadaptive Smith prediction model and the multiple mixedfixed-adaptive Smith prediction model with the same initialfeedback and feedforward parameters as in Section 41 andwe set 120591

119898= 025 Control results are shown in Figure 10

Abstract and Applied Analysis 9

When system time delay jumps the present multipleadaptive Smith prediction model and multiple mixed fixed-adaptive Smith prediction model can both guarantee thesystem stability Meanwhile compared with single adaptiveSmith prediction model system transient response of mul-tiple adaptive Smith prediction model has been improvedobviously no matter before or after the time delay changepoint

5 Conclusion

In the rolling AGC control process control systems undercomplex network condition are generally used Consider thetime delay of network transmission and thickness gauge isusually far away from the roll gap the transfer functionmodelof the controlled plant will always be described by a firstorder system with uncertain time delay Generally fixed andadaptive Smith predictors are the effective methods to solvethis problem but they all need the precisely known time delayvalue which is hard to be obtained This paper approximatesthe variation of time delay and the parameters in systemmodel by a first order process with different parametersMultiple Smith prediction model for AGC will be used toset up a MMAC to cope with the change range of time delayin rolling process The problem of regular Smith predictor issolved and the transient response of adaptive Smith predictoris improved

Conflict of Interests

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

Acknowledgments

This work was supposed by the Fundamental ResearchFunds for the Central Universities under Grant FRF-TP-12-005B the Program for New Century Excellent Talents inUniversities under Grant NCET-11-0578 and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (SRFDP) under Grant 20130006110008

References

[1] W Yang Y Sun C Cao C Wang and J Yang ldquoDistributedcomputer control system for hot narrow strip millrdquo Iron andSteel vol 28 no 8 pp 30ndash34 1993

[2] Y Q Wang M H Sun and W Zhang ldquoDistributed computercontrol in hydraulic AGC system of tandem cold rolling millrdquoMachine Tool and Hydraulics vol 35 no 3 pp 120ndash123 2007

[3] Y K Sun ldquoThe computer control system applied to rollingprocessrdquo Engineering Science vol 2 no 1 pp 73ndash76 2000

[4] Y Wang M Sun W Zhang Y Fang G Chen and Q ZhangldquoResearch on feedforward control system of tandem coldrolling millrsquos hydraulic AGC and its computer control strategyrdquoin Proceedings of the International Technology and Innova-tion Conference (ITIC rsquo06) pp 2093ndash2098 Hangzhou ChinaNovember 2006

[5] D Q Song Research about feedback AGC control stragety ofreversible cold rolling mill control systems based on adaptiveSmith predictor [PhD thesis] Central South University Chang-sha China 2009

[6] C L Lai and P L Hsu ldquoDesign the remote control systemwith the time-delay estimator and the adaptive smith predictorrdquoIEEE Transactions on Industrial Informatics vol 6 no 1 pp 73ndash80 2010

[7] K S Narendra and J Balakrishnan ldquoImproving transientresponse of adaptive control systems usingmultiple models andswitchingrdquo IEEE Transactions on Automatic Control vol 39 no9 pp 1861ndash1866 1994

[8] Z Han and K S Narendra ldquoNew concepts in adaptive controlusing multiple modelsrdquo IEEE Transactions on Automatic Con-trol vol 57 no 1 pp 78ndash89 2012

[9] K S Narendra and J Balakrishnan ldquoAdaptive control usingmultiple modelsrdquo IEEE Transactions on Automatic Control vol42 no 2 pp 171ndash187 1997

[10] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000

[11] B Chaudhuri R Majumder and B C Pal ldquoApplication ofmultiple-model adaptive control strategy for robust dampingof interarea oscillations in power systemrdquo IEEE Transactions onControl Systems Technology vol 12 no 5 pp 727ndash736 2004

[12] W T Chen and B D O Anderson ldquoA combined multiplemodel adaptive control scheme and its application to nonlinearsystemswith nonlinear parameterizationrdquo IEEETransactions onAutomatic Control vol 57 no 7 pp 1778ndash1782 2012

[13] K Ibrahim ldquoA new Smith predictor and controller for controlof processes with long dead timerdquo ISA Transactions vol 42 no1 pp 101ndash110 2003

[14] S-B Tan M-W Bao and J-C Liu ldquoResearch on applicationof smith predictor to monitor AGC in hot strip rolling millsrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo09) pp 4172ndash4175 Guilin China June 2009

[15] J C Liu S S Gu S F Zheng andH TMi ldquoApplication of smithprediction control strategy to AGC systemrdquo Iron and Steel vol33 no 10 pp 40ndash43 1998

[16] DMeng Y Jia J Du andF Yu ldquoLearning control for time-delaysystems with iteration-varying uncertainty a Smith predictor-based approachrdquo IET Control Theory amp Applications vol 4 no12 pp 2707ndash2718 2010

[17] X Li D Q Song S Y Yu andW H Gui ldquoFeedback automaticgauge control system using model reference adaptive Smithpredictorrdquo Control Theory and Applications vol 26 no 9 pp999ndash1003 2009

[18] S Yin S X Ding X C Xie andH Luo ldquoA review on basic data-driven approaches for industrial process monitoringrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6418ndash6428 2014

[19] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[20] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[21] S Yin X W Li H J Gao and O Kaynak ldquoData-based tech-niques focused on modern industry an overviewrdquo IEEE Trans-actions on Industrial Electronics 2014

10 Abstract and Applied Analysis

[22] S S Hu Automatic Control Theory Science Press BeijingChina 5th edition 2007

[23] K S Narendra and A M Annaswamy Stable Adaptive SystemsPrentice Hall Englewood Cliffs NJ USA 1989

[24] N Anne and L Jonathan ldquoVarying time delay Smith predictorprocess controllerrdquo ISA Transactions vol 25 no 3 pp 77ndash812004

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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

of coiler

Finishing rolling section

Main electrical room

Main electrical room

Armored fiberTwisted pairTwisted pair (2)

Hot rolling Ethernet

Furnace section

Coiling section

Main electrical room

Network switchNetwork switch

Network switch

Network switch

Network switchNetwork switch

Network switch

Network switch Network switch

Rough rolling section

Main electrical room

Network switch Network switch

Level 2

Level 1

of furnace sectionof rough rolling section

sectionof finishing rolling

120572120572120572120572

Figure 1 Ethernet network structure in a tandem rolling line

In practical AGC thickness control process two mainfactors will lead to the uncertain time delay for controlled sys-tem One is the certain distance between the thickness gaugeand rolling gap of each stand and the other is the essentiallyexisting time delay in network transmission Smith predictoris a method to deal with above problem but as the value ofsystem time delay is hard to be measured precisely the Smithpredictor method always cannot be carried out effectivelyin practice Adaptive Smith predictor model can cope withpartial model mismatch [5 6] However improper selectionof adaptive initial value may also cause the system responsewith bad transient process this situation may get worse forsystem with variant time delay because the single predictorcannot match all conditions with different time delay wellMultiplemodel adaptive control (MMAC) is a kind of tool foradaptive control wheremultiple elementmodels will be set upto cover the uncertainty of the parameter or structure of thesystem and a switching mechanism will be used to form thefinal multiple model controllerThis kind of controller can beused to improve the transient response [7] From the 1990s a

lot of MMAC based on continuous-time [8 9] and discrete-time [10] system have been given and a lot of application ofMMAC also can be seen [11 12]

In this paper we introduce MMAC into adaptive Smithpredictor to form the multiple model adaptive Smith predic-tor controlmethod so as to improve the control performanceAccording to the possible varying range of the controlledplantrsquos time delay multiple fixed and adaptive models areestablished to cover the plantrsquos uncertainties correspondingcontrollers are also set up the switch index function is pro-posed based on model error to decide the most appropriatecontroller for the system at every moment and select it asthe current controller Ultimately multiple adaptive Smithcontrollers are composed to improve the systemrsquos controlperformance

The following paper is organized as follows Section 2introduces the design principle of Smith predictor controlSection 3 presents the proposedmultiplemodel Smith predic-tor controller in detail Numerical simulations are conductedin Section 4 and Section 5 concludes this work

Abstract and Applied Analysis 3

Network switch Network switch

Level 2

Level 1

Armored fiberFiber jumpers

RM Fiber optic network

Rough rolling section

RFM5595

Coiling section

Furnace section

RFM5595

RFM5595

RFM5595

Finishing rolling

RFM5595

section

120572120572120572

Figure 2 RM fiber optic network in a tandem rolling line

2 Controller Design Principle Based onSmith Predictor

21 Fixed Smith Predictor Controller AGC thickness controlof rolling mill is a control system with time delay traditionalAGC control methods cannot get satisfying control perfor-mance [13 14]

According to control theory to cope with the time-delay problem we can take the Smith predictor schemeA prediction model is added into the AGC process as thefeedback loop which can predict the change of system outputand give feedback signal in advance so as to offset theoriginal system delay and make the characteristic equationof the whole closed loop system without time delay [15] Theprinciple of Smith predictor in AGC process [16] is shown inFigure 3

In Figure 3 ℎ119903(119905) and 119862(119905) are the preset value and

measured value of the exit thickness respectivelyΔℎ(119905) is thethickness difference Δ119878 is the regulation value of the rollinggap119866

119888(119904) is the controller119866(119904)eminus120591119904 is the transfer function of

controlled plant and 119866119898(119904)eminus120591119898119904 is the prediction model

hr(t) +

minusminus

ΔhGc(s)

ΔSG(s) eminus120591s C(t)

Gm(s) eminus120591119898s+

minus

Figure 3 AGC control process based on Smith predictor

When Smith predictor is not used the system transferfunction is

119862 (119904) =119866119888(119904) 119866 (119904) eminus120591119904

1 + 119866119888(119904) 119866 (119904) eminus120591119904

ℎ119903(119904) (1)

Corresponding characteristic equation is 1 + 119866119888(119904)119866(119904)

eminus120591119904 = 0As the pure time-delay part eminus120591119904 exists in both transfer

function and characteristic equation with the increase oftime delay 120591 the phase delay increases system stability

4 Abstract and Applied Analysis

Feedback

Feedforward

Adaptive mechanism

regulator

regulator

hr(t)

+

+

+

+

+

minus

minus

minus

Δh ΔSG(s) eminus120591s

C(t)

eminus120591119898s

Gc(s)

Gm(s)

Gm(s)

K(e t)

G998400m(t)ΔS

998400

e(t)

Gm(t)

F(e t)

C998400(t)

Figure 4 AGC control process based on adaptive Smith predictor

decreases and the control performance goes worse whilein the Smith predictor scheme when the prediction modelmatches the controlled plant perfectly that is 119866

119898(119904) = 119866(119904)

120591119898= 120591 we have 119866

119898(119904)eminus120591119898119904 = 119866(119904)eminus120591119904 and system output

becomes

119862 (119904)

=119866119888(119904) 119866 (119904) eminus120591119904

1 + 119866119888(119904) 119866 (119904) eminus120591119904 minus 119866

119888(119904) 119866119898(119904) eminus120591119898119904 + 119866

119888(119904) 119866119898(119904)

times ℎ119903(119904)

=119866119888(119904) 119866 (119904) eminus120591119904

1 + 119866119888(119904) 119866119898(119904)

ℎ119903(119904)

(2)

Corresponding characteristic equation becomes 1 +

119866119888(119904)119866119898(119904) = 0 without time-delay part so the system

stability will not be influenced by time delay 120591

22 Adaptive Smith Predictor Controller Practically condi-tion 119866

119898(119904) = 119866(119904) 120591

119898= 120591 cannot be satisfied easily [17]

proposes a more practical method in which adaptive adjust-ing law of feedback coefficient 119865(119905) and feedforward coef-ficient 119870(119905) is applied to guarantee the identification Smithpredictor 119866

119898(119904) = 119866(119904) under the condition 120591

119898= 120591 = 0 The

principle of adaptive Smith predictior is shown in Figure 4Consider 120591

119898= 120591 = 0 the transfer functions of the system

and Smith predictor model are given as

119866 (119904) =119896

119904 + 119886 119866

119898(119904) =

119896119898

119904 + 119886119898

(3)

Corresponding state equations can be got as

(119905) = minus119886119862 (119905) + 119896Δ119878 (119905)

119898(119905) = minus119886

119898119862 (119905) + 119896

119898Δ1198781015840

(119905)

(4)

For given generalized error

119890 (119905) = 119862 (119905) minus 119862119898(119905) (5)

Lynaponov function is set up as

V (119905) =1

2[119896119898119890 (119905)2

+1

1205781

1205952

(119905) +1

1205782

1205932

(119905)] (6)

where

120595 (119905) = 119886 minus 119886119898minus 119896119898119865 (119905)

120593 (119905) = 119896 minus 119896119898119870 (119905)

(7)

An adaptive control law

119865 (119905) = minusint 1205781119890 (119905) 119862

119898(119905) + 119865 (0)

119870 (119905) = int 1205782119890 (119905) Δ119878 (119905) + 119870 (0)

(8)

can be used to guarantee 119890(119905) rarr 0120595(119905) rarr 0 and 120593(119905) rarr 0

when 119905 rarr infin

Remark 1 The adaptive parameter identification mechanismcan be viewed as a kind of data-driven strategy [18 19]Information about process measurements and initial valuesis required in the condition of given prediction modelHowever unlike pure data-driven methods such as partialleast squares (PLS) iterative learning control (ILC) andmodel free adaptive control which only depend on theprocess measurements [20] this adaptive law is a model-dataintegrated method [21] which also needs the relative precisepredictionmodel so that the process information can be fullyutilized based on the model to realize the system identifica-tion

From Figure 4 this adaptive law can ensure119866119898(119904) = 119866(119904)

(where119866119898(119904) is an identification Smith predictor adjusted by

119865 119866119898 and 119870) the identification Smith predictor 119866

119898(119904) can

be used But in practice both 119866119898(119904) = 119866(119904) and 120591

119898= 120591 = 0

are always very hard to be satisfied because the precisemodelof the system cannot be setup and time delay of the systemcannot be measured accurately So above adaptive Smith pre-dictor is also very difficult to be applied Due to this situationthe following multiple adaptive Smith prediction modelcontroller will be considered

Abstract and Applied Analysis 5

Adaptive mechanism

Feedforwardregulator

Feedback regulator

+

+

++

minus

minus

hr(t)

minus

ΔhGc(s)

ΔSG(s)

C(t)

Gm(s)

eminus120591119898s

eminus120591119898s

eminusΔ120591sG998400(s)

C998400(t)

ΔS998400

e(t)

C998400m(t)

Gm(t)

F(e t)

Gm(s)

Smith predictor P1

Smith predictor Pi

Smith predictor Pn

e1(t)

ei(t)

en(t)

Switching

k(e t)

Figure 5 AGC control process based on multiple adaptive Smith prediction model

3 Multiple Smith Prediction Model Controller

Consider the rolling system in Figure 5 its transfer functionand time delay can be generally described by the followingmodel

119866 (119904) eminusΔ120591119904eminus120591119898119904 (9)

where 120591119898is the delay under normal working condition and

Δ120591 stands for the delay fluctuation caused by uncertain factorssuch as network transmission delay

Define

1198661015840

(119904) = 119866 (119904) eminusΔ120591119904 (10)

Consider the character of first order system we assumethat a first order systemwith small delay can be approximatedby a first order processwith different parameters [22] we have

1198661015840

(119904) asymp1198961015840

119904 + 1198861015840 (11)

Further we have the multiple model adaptive controlsystem (see Figure 5)

As in Figure 5 we use the Smith prediction model 119866119898(119904)

adjusted by 119865119866119898 and119870 to approximate the controlled plant

1198661015840

(119904) Consider the varying scope of Δ120591 parameters of 1198661015840(119904)will be uncertain or time varying so it is not appropriate todescribe the systemby single fixedmodel For the single adap-tive model if its initial parameters are not properly selectedmodel parameters will converge slowly Moreover if thereis drastic system parameter jump control performance willget worse So we take multiple models with different initialvalues to approximate the system model uncertainty

31 Multiple Adaptive Smith Model Controller Based onthe possible parameters variation range of 1198661015840(119904) multipleadaptive Smith predictors 119875

119894(119894 = 1 2 119899 119899 is the

number of models) with different initial values are designedFor original Smith prediction model 119866

119898(119904) corresponding

adaptive parameters of the feedforward and feedback loopsatisfy

119865119894(119905) = minusint 120578

1119890 (119905) 119862

1015840

119898(119905) + 119865

119894(0)

119870119894(119905) = int 120578

2119890 (119905) Δ119878 (119905) + 119870

119894(0)

(12)

where 119894 = 1 2 119899Define the switch index function

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905 (13)

where 119894 = 1 2 119899 120572 gt 0 and 120573 gt 0 And the model error

119890119894(119905) = 119862

1015840

(119905) minus 1198621015840

119898119894(119905) asymp 119862 (119905) minus 119862

119898119894(119905) (14)

At time 119905 calculate

119897 (1199050 119905) = argmin

119894isin12119898

119869119894(1199050 119905) (15)

and Smith predictor based on model 119897 will be switched intothe closed loop system

Remark 2 In practical AGC control process 1198621015840(119905) cannot bemeasured so as in (14) 119890

119894(119905) = 119888(119905)minus119888

119898(119905)will be used instead

of 1198621015840(119905) minus 1198621015840119898(119905) in practical control

32 Multiple Fixed Smith Prediction Model Controller Formultiple adaptive Smith prediction model controller parallelcomputation of multiple adaptive processes will lead to theincrease of calculation amount

If all the 119899models are selected to be fixed models that isfor 119894 = 1 2 119899

119865119894(119905) = 119865

119894(0) gt 0

119870119894(119905) = 119870

119894(0) gt 0

(16)

6 Abstract and Applied Analysis

With enough fixed models take (13) as the switch indexfunction calculation amount for multiple models can bedecreased drastically However without the stable characterof adaptive model system stability can hardly be guaranteedby pure multiple fixed models

33 Multiple Adaptive Smith Controllers with Multiple Fixedand Adaptive Models Multiple fixed models as in Section 32can reduce the amount of calculation but due to the lack ofadaptive parameter adjusting process system stability cannotbe easily obtained If in the 119899 models 119899 minus 1 are set up asfixed models and 1 model is adaptive model that is for 119894 =1 2 119899 minus 1

119865119894(119905) = 119865

119894(0) gt 0

119870119894(119905) = 119870

119894(0) gt 0

119865119899(119905) = minusint 120578

1119890119899(119905) 1198621015840

119898(119905) + 119865

119899(0)

119870119899(119905) = int 120578

2119890119899(119905) Δ119878 (119905) + 119870

119899(0)

(17)

The existence of multiple fixed models reduces theamount of calculation and the adaptivemodel guarantees thesystem stability so the control performance can be improvedas the case of multiple adaptive models

Theorem3 If the adaptive Smith controller in Section 22 (120591 =120591119898= 0) is applied to the AGC system model error 119890 can be

guaranteed to be bounded and to approach zero as 119905 rarr infinthat is lim

119905rarrinfin119890(119905) = 0

Proof From [17] for adaptive Smith predictionmodel takingderivative of Lynaponov function (6) we have

V (119905) = minus 1198961198981198861198902

(119905) + 120595 (119905) [1

1205781

(119905) minus 119896119898119890 (119905) 119862

119898(119905)]

+ 120593 (119905) [1

1205782

(119905) + 119896119898119890 (119905) Δ119878 (119905)]

(18)

from (7) and (8)

V (119905) = minus1198961198981198861198902

(119905) le 0 (19)

Consider 1205781gt 0 120578

2gt 0 119896 gt 0 and 119896

119898gt 0 so V(119905) gt 0

holds for all 119905 = 0From (19) V(119905) is bounded so 119890(119905) 120595(119905) and 120593(119905) are all

bounded Further we have

int

119905

1199050

1198902

(119905) 119889119905 lt infin 119890 (119905) isin 1198712 (20)

For

1015840

119898(119905) = minus119886

1198981198621015840

119898(119905) + 119896

119898Δ1198781015840

(119905)

Δ1198781015840

(119905) = 119870 (119905) Δ119878 (119905) minus 119865 (119905) 1198621015840

119898(119905)

1198621015840 (119905) = minus1198861198621015840

(119905) + 119896Δ119878 (119905)

119890 (119905) = 1198621015840

(119905) minus 1198621015840

119898(119905)

(21)

we have

119890 (119905) = minus119886119890 (119905) minus 120595 (119905) 1198621015840

119898(119905) + 120593 (119905) Δ119878 (119905) (22)

Consider Δ119878(119905) 1198621015840(119905) and 119890(119905) are bounded then 1198621015840119898(119905)

is bounded from (22) we have 119890(119905) isin 119871infin from (20) and the

Barbalet Lemma [23] we have lim119905rarrinfin

119890(119905) = 0

Theorem 4 Take (15) as the switch index function and selectmultiple adaptivemodels in Section 31 ormultiple fixedmodelsand one adaptive model in Section 33 as the multiple adaptiveSmith controllers For the AGC Smith controller shown as inFigure 5 each of these two kinds of controllers can cancel thefluctuation of system delay effectively and improve the controlperformance

Proof (1) For multiple model control composed of multipleadaptive Smith predictors since each Smith model has itsown adaptive mechanism so for each model we have

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905 lt infin

119894 = 1 2 119899

lim119905rarrinfin

119890119894(119905) = 0

(23)

Though there is model switching among multiple modelsbased on the switch index function it finally comes to

Δ119878 (119866119898119894(119904) minus 119866

1015840

(119904)) = 0 (24)

Uncertain part of time delay is canceled andmultiple modelsare used to improve the transient process

(2) For multiple model control composed of multiplefixed Smith predictors and one adaptive Smith predictor asto the adaptive Smith predictor

119869119899(1199050 119905) = 120572119890

2

119899(119905) + 120573int

119905

1199050

1198902

119899(119905) 119889119905 lt infin (25)

Consider the switch index function

119897 (1199050 119905) = argmin

119894isin12119899

119869119894(1199050 119905) (26)

For each fixed model

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905

119894 = 1 2 119899 minus 1

(27)

If

lim119905rarrinfin

119869119894(1199050 119905) = infin 119894 = 1 2 119899 minus 1 (28)

then finally the adaptive model will be selected and theadaptive Smith model matches the systemmodel completelythat is Δ119878(119866

119898119899(119904) minus 119866

1015840

(119904)) = 0

Abstract and Applied Analysis 7

If there is a fixed model 119894 satisfying

lim119905rarrinfin

119869119894(1199050 119905) = lim

119905rarrinfin

[1205721198902

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905] lt infin (29)

where 119894 = 1 2 119899 minus 1 from (21)

1015840

(119905) = minus1198861198621015840

(119905) + 119896Δ119878 (119905)

1015840

119898119894(119905) = minus [119886

119898+ 119896119898119865119894(0)] 119862

1015840

119898119894(119905) + 119896

119898119870119894(0) Δ119878 (119905)

119890119894(119905) = 119862

1015840

(119905) minus 1198621015840

119898119894(119905)

(30)

If the parameters of selected fixed model satisfy 119865119894(0) gt 0 or

119886119898+ 119896119898119865119894(0) gt 0 then 119862

1015840

119898119894is bounded and from (7) (29)

and (30) 119890119894(119905) 120595119894(119905) and 120593

119894(119905) are bounded the following

inequation also holds

119890119894(119905) = minus119886119890

119894(119905) minus 120595

119894(119905) 1198621015840

119898(119905) + 120593

119894(119905) Δ119878 (119905) lt infin (31)

then from (29) and (31) and the Barbalet Lemma we havelim119905rarrinfin

119890119895(119905) = 0

119895 isin 119895 = 119899 or 119895 = 119908 | lim119905rarrinfin

119869119908(1199050 119905) lt infin

119908 = 1 2 119899 minus 1

(32)

and it finally comes to

Δ119878 (119866119898119895(119904) minus 119866

1015840

(119904)) = 0 (33)

The multiple model adaptive Smith predictor still has thematching function

Remark 5 Multiple model control based on multiple fixedSmith models can improve the transient response but as itlacks the adaptive parameter regulation system stability canhardly be guaranteed

Remark 6 The existence of multiple models covering theparameter uncertain range of the controlled plant canimprove the systemrsquos transient response and improve thecontrol quality for system with jumping parameter

4 Simulation Research

A first order time delay system as follows will be simulatedfor rolling AGC system

119866 (119904) eminus120591119904 = 119896

119904 + 119886eminus120591119904 = 021

0053119904 + 1eminus035119904

= 1198661015840

(119904) eminus03119904(34)

where 1198661015840(119904) = (021(0053119904 + 1))eminus005119904The same PID controller as in [17] based on ITAE

criterion [24] will also be used that is

119866119888(119904) = 119870

119901(1 +

1

119879119894119904+

119879119889119904

1 + 11987911988911990410

)

= 223 (1 +1

02119904+

0034119904

00034119904 + 1)

(35)

12

1

08

06

04

02

00 5 10 15 20 25 30 35 40

1005

0995

1

39 395 40Out

put t

hick

nessC(t)

(mm

)

Single adaptive Smith

Time t (s)

Figure 6 Single model adaptive Smith predictor

1005

0995

1

39 395 4000 5 10 15 20 25 30 35 40

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Multiple adaptive Smith

Time t (s)

Figure 7 Multiple model adaptive Smith predictor

10005

09995

1

39 395 40

0 5 10 15 20 25 30 35 400

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Multiple fixed Smith

Time t (s)

Figure 8 Multiple fixed Smith prediction model

The compensation Smith predictor will be designed as119866119898(119904) = 021(0053119904 + 1) 120591

119898= 03119904 And parameters of

119865(119905) and119870(119905) will be tuned as (8)

41 Multiple Model Smith Predictor for System with FixedTime Delay For single adaptive Smith prediction model letthe feedforward and feedback adaptive initial parameters be119865(0) = 35 and119870(0) = 50 in (8) the control result is shown inFigure 6 For multiple model adaptive control four adaptiveSmith predictionmodels are established corresponding feed-forward and feedback adaptive initial parameters in (12) are1198651(0) = 35 119865

2(0) = 30 119865

3(0) = 25 119865

4(0) = 18 and 119870

1(0) =

1198702(0) = 119870

3(0) = 119870

4(0) = 50 The control result is shown

in Figure 7

8 Abstract and Applied Analysis

Multiple fixed and one adaptive Smith

1005

0995

1

39 395 400 5 10 15 20 25 30 35 40

0

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Time t (s)

(a) System output

4

3

2

1Switc

hing

sequ

ence

I

Multiple fixed and one adaptive Smith

0 5 10 15 20 25 30 35 40

Time t (s)

(b) Switching sequence

Figure 9 Multiple fixed and single adaptive Smith prediction model

It can be seen in Figures 6 and 7 when multiple adaptiveSmith predictionmodels withmultiple groups of feedforwardand feedback adaptive initial parameters are applied to theAGC system the overshot and system response time aredecreased obviously compared with single Smith predictionmodel case Meanwhile when single model adaptive Smithpredictor is applied because model parameters match badlywith system parameters initial identification error is rel-atively big and the identification time goes relatively longcorrespondingly Since multiple model adaptive Smith pre-dictor adoptsmultiple adaptive initial parameters system canswitch to the closest matching model adaptively and conse-quently system identification error and identification timedecrease obviously

When four fixed Smith prediction models with initialadaptive feedback parameters 119865

1(0) = 35 119865

2(0) = 30

1198653(0) = 25 and 119865

4(0) = 18 and same initial feedforward

parameters 1198701(0) = 119870

2(0) = 119870

3(0) = 119870

4(0) = 50 in (16) are

applied the control result is shown in Figure 8 At the sametime the mixed multiple model Smith predictor with threefixed models whose initial feedback adaptive parameters are1198651(0) = 35 119865

2(0) = 30 and 119865

3(0) = 18 and an adaptive

model with initial value 1198654(0) = 18 the same feedforward

parameters 1198701(0) = 119870

2(0) = 119870

3(0) = 119870

4(0) = 50 are given

and the control results can be seen in Figure 9Comparedwithmultiple adaptive Smith predictormodel

the computing time can be reduced by using multiple fixedSmith predictor models but the stability cannot be guaran-teed because of the absence of adaptive adjustment FromFigure 8 suitable number of fixed models can be used toimprove the overshoot and response time but the steadyidentification error cannot be made into 0 at last In Figure 9the combination of fixed and adaptive Smith models willgive an improved transient response without the loss of finalproperty of system stability The model switching process isshown in Figure 9(b) the initial fixed model with 119865

1(0) = 35

is not the best one and fixed model 3 with 1198653(0) = 18 is

selected to control the system for a period of time finally thecontrol model is switched to the 4th adaptive Smith model

14

12

1

1

08

06

04

02

00 5 10 15 20 25

1004

1002

8 10 15 20 25

Change of time delay

Single adaptive SmithMultiple adaptive SmithSingle adaptive and multiple fixed Smith

Out

put t

hick

nessC(t)

(mm

)

Time t (s)

Figure 10 Multiple Smith prediction model for system with jump-ing time delay

42 Multiple Smith Prediction Model for System with JumpingTime Delay The rolling AGC system with jumping timedelay can be described as

119866 (119904) eminus120591119904 = 021

0053119904 + 1eminus120591119904

120591 = 03 0 lt 119905 lt 10

07 119905 ge 10

(36)

Take the single adaptive Smith predictionmodelmultipleadaptive Smith prediction model and the multiple mixedfixed-adaptive Smith prediction model with the same initialfeedback and feedforward parameters as in Section 41 andwe set 120591

119898= 025 Control results are shown in Figure 10

Abstract and Applied Analysis 9

When system time delay jumps the present multipleadaptive Smith prediction model and multiple mixed fixed-adaptive Smith prediction model can both guarantee thesystem stability Meanwhile compared with single adaptiveSmith prediction model system transient response of mul-tiple adaptive Smith prediction model has been improvedobviously no matter before or after the time delay changepoint

5 Conclusion

In the rolling AGC control process control systems undercomplex network condition are generally used Consider thetime delay of network transmission and thickness gauge isusually far away from the roll gap the transfer functionmodelof the controlled plant will always be described by a firstorder system with uncertain time delay Generally fixed andadaptive Smith predictors are the effective methods to solvethis problem but they all need the precisely known time delayvalue which is hard to be obtained This paper approximatesthe variation of time delay and the parameters in systemmodel by a first order process with different parametersMultiple Smith prediction model for AGC will be used toset up a MMAC to cope with the change range of time delayin rolling process The problem of regular Smith predictor issolved and the transient response of adaptive Smith predictoris improved

Conflict of Interests

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

Acknowledgments

This work was supposed by the Fundamental ResearchFunds for the Central Universities under Grant FRF-TP-12-005B the Program for New Century Excellent Talents inUniversities under Grant NCET-11-0578 and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (SRFDP) under Grant 20130006110008

References

[1] W Yang Y Sun C Cao C Wang and J Yang ldquoDistributedcomputer control system for hot narrow strip millrdquo Iron andSteel vol 28 no 8 pp 30ndash34 1993

[2] Y Q Wang M H Sun and W Zhang ldquoDistributed computercontrol in hydraulic AGC system of tandem cold rolling millrdquoMachine Tool and Hydraulics vol 35 no 3 pp 120ndash123 2007

[3] Y K Sun ldquoThe computer control system applied to rollingprocessrdquo Engineering Science vol 2 no 1 pp 73ndash76 2000

[4] Y Wang M Sun W Zhang Y Fang G Chen and Q ZhangldquoResearch on feedforward control system of tandem coldrolling millrsquos hydraulic AGC and its computer control strategyrdquoin Proceedings of the International Technology and Innova-tion Conference (ITIC rsquo06) pp 2093ndash2098 Hangzhou ChinaNovember 2006

[5] D Q Song Research about feedback AGC control stragety ofreversible cold rolling mill control systems based on adaptiveSmith predictor [PhD thesis] Central South University Chang-sha China 2009

[6] C L Lai and P L Hsu ldquoDesign the remote control systemwith the time-delay estimator and the adaptive smith predictorrdquoIEEE Transactions on Industrial Informatics vol 6 no 1 pp 73ndash80 2010

[7] K S Narendra and J Balakrishnan ldquoImproving transientresponse of adaptive control systems usingmultiple models andswitchingrdquo IEEE Transactions on Automatic Control vol 39 no9 pp 1861ndash1866 1994

[8] Z Han and K S Narendra ldquoNew concepts in adaptive controlusing multiple modelsrdquo IEEE Transactions on Automatic Con-trol vol 57 no 1 pp 78ndash89 2012

[9] K S Narendra and J Balakrishnan ldquoAdaptive control usingmultiple modelsrdquo IEEE Transactions on Automatic Control vol42 no 2 pp 171ndash187 1997

[10] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000

[11] B Chaudhuri R Majumder and B C Pal ldquoApplication ofmultiple-model adaptive control strategy for robust dampingof interarea oscillations in power systemrdquo IEEE Transactions onControl Systems Technology vol 12 no 5 pp 727ndash736 2004

[12] W T Chen and B D O Anderson ldquoA combined multiplemodel adaptive control scheme and its application to nonlinearsystemswith nonlinear parameterizationrdquo IEEETransactions onAutomatic Control vol 57 no 7 pp 1778ndash1782 2012

[13] K Ibrahim ldquoA new Smith predictor and controller for controlof processes with long dead timerdquo ISA Transactions vol 42 no1 pp 101ndash110 2003

[14] S-B Tan M-W Bao and J-C Liu ldquoResearch on applicationof smith predictor to monitor AGC in hot strip rolling millsrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo09) pp 4172ndash4175 Guilin China June 2009

[15] J C Liu S S Gu S F Zheng andH TMi ldquoApplication of smithprediction control strategy to AGC systemrdquo Iron and Steel vol33 no 10 pp 40ndash43 1998

[16] DMeng Y Jia J Du andF Yu ldquoLearning control for time-delaysystems with iteration-varying uncertainty a Smith predictor-based approachrdquo IET Control Theory amp Applications vol 4 no12 pp 2707ndash2718 2010

[17] X Li D Q Song S Y Yu andW H Gui ldquoFeedback automaticgauge control system using model reference adaptive Smithpredictorrdquo Control Theory and Applications vol 26 no 9 pp999ndash1003 2009

[18] S Yin S X Ding X C Xie andH Luo ldquoA review on basic data-driven approaches for industrial process monitoringrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6418ndash6428 2014

[19] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[20] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[21] S Yin X W Li H J Gao and O Kaynak ldquoData-based tech-niques focused on modern industry an overviewrdquo IEEE Trans-actions on Industrial Electronics 2014

10 Abstract and Applied Analysis

[22] S S Hu Automatic Control Theory Science Press BeijingChina 5th edition 2007

[23] K S Narendra and A M Annaswamy Stable Adaptive SystemsPrentice Hall Englewood Cliffs NJ USA 1989

[24] N Anne and L Jonathan ldquoVarying time delay Smith predictorprocess controllerrdquo ISA Transactions vol 25 no 3 pp 77ndash812004

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

Abstract and Applied Analysis 3

Network switch Network switch

Level 2

Level 1

Armored fiberFiber jumpers

RM Fiber optic network

Rough rolling section

RFM5595

Coiling section

Furnace section

RFM5595

RFM5595

RFM5595

Finishing rolling

RFM5595

section

120572120572120572

Figure 2 RM fiber optic network in a tandem rolling line

2 Controller Design Principle Based onSmith Predictor

21 Fixed Smith Predictor Controller AGC thickness controlof rolling mill is a control system with time delay traditionalAGC control methods cannot get satisfying control perfor-mance [13 14]

According to control theory to cope with the time-delay problem we can take the Smith predictor schemeA prediction model is added into the AGC process as thefeedback loop which can predict the change of system outputand give feedback signal in advance so as to offset theoriginal system delay and make the characteristic equationof the whole closed loop system without time delay [15] Theprinciple of Smith predictor in AGC process [16] is shown inFigure 3

In Figure 3 ℎ119903(119905) and 119862(119905) are the preset value and

measured value of the exit thickness respectivelyΔℎ(119905) is thethickness difference Δ119878 is the regulation value of the rollinggap119866

119888(119904) is the controller119866(119904)eminus120591119904 is the transfer function of

controlled plant and 119866119898(119904)eminus120591119898119904 is the prediction model

hr(t) +

minusminus

ΔhGc(s)

ΔSG(s) eminus120591s C(t)

Gm(s) eminus120591119898s+

minus

Figure 3 AGC control process based on Smith predictor

When Smith predictor is not used the system transferfunction is

119862 (119904) =119866119888(119904) 119866 (119904) eminus120591119904

1 + 119866119888(119904) 119866 (119904) eminus120591119904

ℎ119903(119904) (1)

Corresponding characteristic equation is 1 + 119866119888(119904)119866(119904)

eminus120591119904 = 0As the pure time-delay part eminus120591119904 exists in both transfer

function and characteristic equation with the increase oftime delay 120591 the phase delay increases system stability

4 Abstract and Applied Analysis

Feedback

Feedforward

Adaptive mechanism

regulator

regulator

hr(t)

+

+

+

+

+

minus

minus

minus

Δh ΔSG(s) eminus120591s

C(t)

eminus120591119898s

Gc(s)

Gm(s)

Gm(s)

K(e t)

G998400m(t)ΔS

998400

e(t)

Gm(t)

F(e t)

C998400(t)

Figure 4 AGC control process based on adaptive Smith predictor

decreases and the control performance goes worse whilein the Smith predictor scheme when the prediction modelmatches the controlled plant perfectly that is 119866

119898(119904) = 119866(119904)

120591119898= 120591 we have 119866

119898(119904)eminus120591119898119904 = 119866(119904)eminus120591119904 and system output

becomes

119862 (119904)

=119866119888(119904) 119866 (119904) eminus120591119904

1 + 119866119888(119904) 119866 (119904) eminus120591119904 minus 119866

119888(119904) 119866119898(119904) eminus120591119898119904 + 119866

119888(119904) 119866119898(119904)

times ℎ119903(119904)

=119866119888(119904) 119866 (119904) eminus120591119904

1 + 119866119888(119904) 119866119898(119904)

ℎ119903(119904)

(2)

Corresponding characteristic equation becomes 1 +

119866119888(119904)119866119898(119904) = 0 without time-delay part so the system

stability will not be influenced by time delay 120591

22 Adaptive Smith Predictor Controller Practically condi-tion 119866

119898(119904) = 119866(119904) 120591

119898= 120591 cannot be satisfied easily [17]

proposes a more practical method in which adaptive adjust-ing law of feedback coefficient 119865(119905) and feedforward coef-ficient 119870(119905) is applied to guarantee the identification Smithpredictor 119866

119898(119904) = 119866(119904) under the condition 120591

119898= 120591 = 0 The

principle of adaptive Smith predictior is shown in Figure 4Consider 120591

119898= 120591 = 0 the transfer functions of the system

and Smith predictor model are given as

119866 (119904) =119896

119904 + 119886 119866

119898(119904) =

119896119898

119904 + 119886119898

(3)

Corresponding state equations can be got as

(119905) = minus119886119862 (119905) + 119896Δ119878 (119905)

119898(119905) = minus119886

119898119862 (119905) + 119896

119898Δ1198781015840

(119905)

(4)

For given generalized error

119890 (119905) = 119862 (119905) minus 119862119898(119905) (5)

Lynaponov function is set up as

V (119905) =1

2[119896119898119890 (119905)2

+1

1205781

1205952

(119905) +1

1205782

1205932

(119905)] (6)

where

120595 (119905) = 119886 minus 119886119898minus 119896119898119865 (119905)

120593 (119905) = 119896 minus 119896119898119870 (119905)

(7)

An adaptive control law

119865 (119905) = minusint 1205781119890 (119905) 119862

119898(119905) + 119865 (0)

119870 (119905) = int 1205782119890 (119905) Δ119878 (119905) + 119870 (0)

(8)

can be used to guarantee 119890(119905) rarr 0120595(119905) rarr 0 and 120593(119905) rarr 0

when 119905 rarr infin

Remark 1 The adaptive parameter identification mechanismcan be viewed as a kind of data-driven strategy [18 19]Information about process measurements and initial valuesis required in the condition of given prediction modelHowever unlike pure data-driven methods such as partialleast squares (PLS) iterative learning control (ILC) andmodel free adaptive control which only depend on theprocess measurements [20] this adaptive law is a model-dataintegrated method [21] which also needs the relative precisepredictionmodel so that the process information can be fullyutilized based on the model to realize the system identifica-tion

From Figure 4 this adaptive law can ensure119866119898(119904) = 119866(119904)

(where119866119898(119904) is an identification Smith predictor adjusted by

119865 119866119898 and 119870) the identification Smith predictor 119866

119898(119904) can

be used But in practice both 119866119898(119904) = 119866(119904) and 120591

119898= 120591 = 0

are always very hard to be satisfied because the precisemodelof the system cannot be setup and time delay of the systemcannot be measured accurately So above adaptive Smith pre-dictor is also very difficult to be applied Due to this situationthe following multiple adaptive Smith prediction modelcontroller will be considered

Abstract and Applied Analysis 5

Adaptive mechanism

Feedforwardregulator

Feedback regulator

+

+

++

minus

minus

hr(t)

minus

ΔhGc(s)

ΔSG(s)

C(t)

Gm(s)

eminus120591119898s

eminus120591119898s

eminusΔ120591sG998400(s)

C998400(t)

ΔS998400

e(t)

C998400m(t)

Gm(t)

F(e t)

Gm(s)

Smith predictor P1

Smith predictor Pi

Smith predictor Pn

e1(t)

ei(t)

en(t)

Switching

k(e t)

Figure 5 AGC control process based on multiple adaptive Smith prediction model

3 Multiple Smith Prediction Model Controller

Consider the rolling system in Figure 5 its transfer functionand time delay can be generally described by the followingmodel

119866 (119904) eminusΔ120591119904eminus120591119898119904 (9)

where 120591119898is the delay under normal working condition and

Δ120591 stands for the delay fluctuation caused by uncertain factorssuch as network transmission delay

Define

1198661015840

(119904) = 119866 (119904) eminusΔ120591119904 (10)

Consider the character of first order system we assumethat a first order systemwith small delay can be approximatedby a first order processwith different parameters [22] we have

1198661015840

(119904) asymp1198961015840

119904 + 1198861015840 (11)

Further we have the multiple model adaptive controlsystem (see Figure 5)

As in Figure 5 we use the Smith prediction model 119866119898(119904)

adjusted by 119865119866119898 and119870 to approximate the controlled plant

1198661015840

(119904) Consider the varying scope of Δ120591 parameters of 1198661015840(119904)will be uncertain or time varying so it is not appropriate todescribe the systemby single fixedmodel For the single adap-tive model if its initial parameters are not properly selectedmodel parameters will converge slowly Moreover if thereis drastic system parameter jump control performance willget worse So we take multiple models with different initialvalues to approximate the system model uncertainty

31 Multiple Adaptive Smith Model Controller Based onthe possible parameters variation range of 1198661015840(119904) multipleadaptive Smith predictors 119875

119894(119894 = 1 2 119899 119899 is the

number of models) with different initial values are designedFor original Smith prediction model 119866

119898(119904) corresponding

adaptive parameters of the feedforward and feedback loopsatisfy

119865119894(119905) = minusint 120578

1119890 (119905) 119862

1015840

119898(119905) + 119865

119894(0)

119870119894(119905) = int 120578

2119890 (119905) Δ119878 (119905) + 119870

119894(0)

(12)

where 119894 = 1 2 119899Define the switch index function

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905 (13)

where 119894 = 1 2 119899 120572 gt 0 and 120573 gt 0 And the model error

119890119894(119905) = 119862

1015840

(119905) minus 1198621015840

119898119894(119905) asymp 119862 (119905) minus 119862

119898119894(119905) (14)

At time 119905 calculate

119897 (1199050 119905) = argmin

119894isin12119898

119869119894(1199050 119905) (15)

and Smith predictor based on model 119897 will be switched intothe closed loop system

Remark 2 In practical AGC control process 1198621015840(119905) cannot bemeasured so as in (14) 119890

119894(119905) = 119888(119905)minus119888

119898(119905)will be used instead

of 1198621015840(119905) minus 1198621015840119898(119905) in practical control

32 Multiple Fixed Smith Prediction Model Controller Formultiple adaptive Smith prediction model controller parallelcomputation of multiple adaptive processes will lead to theincrease of calculation amount

If all the 119899models are selected to be fixed models that isfor 119894 = 1 2 119899

119865119894(119905) = 119865

119894(0) gt 0

119870119894(119905) = 119870

119894(0) gt 0

(16)

6 Abstract and Applied Analysis

With enough fixed models take (13) as the switch indexfunction calculation amount for multiple models can bedecreased drastically However without the stable characterof adaptive model system stability can hardly be guaranteedby pure multiple fixed models

33 Multiple Adaptive Smith Controllers with Multiple Fixedand Adaptive Models Multiple fixed models as in Section 32can reduce the amount of calculation but due to the lack ofadaptive parameter adjusting process system stability cannotbe easily obtained If in the 119899 models 119899 minus 1 are set up asfixed models and 1 model is adaptive model that is for 119894 =1 2 119899 minus 1

119865119894(119905) = 119865

119894(0) gt 0

119870119894(119905) = 119870

119894(0) gt 0

119865119899(119905) = minusint 120578

1119890119899(119905) 1198621015840

119898(119905) + 119865

119899(0)

119870119899(119905) = int 120578

2119890119899(119905) Δ119878 (119905) + 119870

119899(0)

(17)

The existence of multiple fixed models reduces theamount of calculation and the adaptivemodel guarantees thesystem stability so the control performance can be improvedas the case of multiple adaptive models

Theorem3 If the adaptive Smith controller in Section 22 (120591 =120591119898= 0) is applied to the AGC system model error 119890 can be

guaranteed to be bounded and to approach zero as 119905 rarr infinthat is lim

119905rarrinfin119890(119905) = 0

Proof From [17] for adaptive Smith predictionmodel takingderivative of Lynaponov function (6) we have

V (119905) = minus 1198961198981198861198902

(119905) + 120595 (119905) [1

1205781

(119905) minus 119896119898119890 (119905) 119862

119898(119905)]

+ 120593 (119905) [1

1205782

(119905) + 119896119898119890 (119905) Δ119878 (119905)]

(18)

from (7) and (8)

V (119905) = minus1198961198981198861198902

(119905) le 0 (19)

Consider 1205781gt 0 120578

2gt 0 119896 gt 0 and 119896

119898gt 0 so V(119905) gt 0

holds for all 119905 = 0From (19) V(119905) is bounded so 119890(119905) 120595(119905) and 120593(119905) are all

bounded Further we have

int

119905

1199050

1198902

(119905) 119889119905 lt infin 119890 (119905) isin 1198712 (20)

For

1015840

119898(119905) = minus119886

1198981198621015840

119898(119905) + 119896

119898Δ1198781015840

(119905)

Δ1198781015840

(119905) = 119870 (119905) Δ119878 (119905) minus 119865 (119905) 1198621015840

119898(119905)

1198621015840 (119905) = minus1198861198621015840

(119905) + 119896Δ119878 (119905)

119890 (119905) = 1198621015840

(119905) minus 1198621015840

119898(119905)

(21)

we have

119890 (119905) = minus119886119890 (119905) minus 120595 (119905) 1198621015840

119898(119905) + 120593 (119905) Δ119878 (119905) (22)

Consider Δ119878(119905) 1198621015840(119905) and 119890(119905) are bounded then 1198621015840119898(119905)

is bounded from (22) we have 119890(119905) isin 119871infin from (20) and the

Barbalet Lemma [23] we have lim119905rarrinfin

119890(119905) = 0

Theorem 4 Take (15) as the switch index function and selectmultiple adaptivemodels in Section 31 ormultiple fixedmodelsand one adaptive model in Section 33 as the multiple adaptiveSmith controllers For the AGC Smith controller shown as inFigure 5 each of these two kinds of controllers can cancel thefluctuation of system delay effectively and improve the controlperformance

Proof (1) For multiple model control composed of multipleadaptive Smith predictors since each Smith model has itsown adaptive mechanism so for each model we have

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905 lt infin

119894 = 1 2 119899

lim119905rarrinfin

119890119894(119905) = 0

(23)

Though there is model switching among multiple modelsbased on the switch index function it finally comes to

Δ119878 (119866119898119894(119904) minus 119866

1015840

(119904)) = 0 (24)

Uncertain part of time delay is canceled andmultiple modelsare used to improve the transient process

(2) For multiple model control composed of multiplefixed Smith predictors and one adaptive Smith predictor asto the adaptive Smith predictor

119869119899(1199050 119905) = 120572119890

2

119899(119905) + 120573int

119905

1199050

1198902

119899(119905) 119889119905 lt infin (25)

Consider the switch index function

119897 (1199050 119905) = argmin

119894isin12119899

119869119894(1199050 119905) (26)

For each fixed model

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905

119894 = 1 2 119899 minus 1

(27)

If

lim119905rarrinfin

119869119894(1199050 119905) = infin 119894 = 1 2 119899 minus 1 (28)

then finally the adaptive model will be selected and theadaptive Smith model matches the systemmodel completelythat is Δ119878(119866

119898119899(119904) minus 119866

1015840

(119904)) = 0

Abstract and Applied Analysis 7

If there is a fixed model 119894 satisfying

lim119905rarrinfin

119869119894(1199050 119905) = lim

119905rarrinfin

[1205721198902

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905] lt infin (29)

where 119894 = 1 2 119899 minus 1 from (21)

1015840

(119905) = minus1198861198621015840

(119905) + 119896Δ119878 (119905)

1015840

119898119894(119905) = minus [119886

119898+ 119896119898119865119894(0)] 119862

1015840

119898119894(119905) + 119896

119898119870119894(0) Δ119878 (119905)

119890119894(119905) = 119862

1015840

(119905) minus 1198621015840

119898119894(119905)

(30)

If the parameters of selected fixed model satisfy 119865119894(0) gt 0 or

119886119898+ 119896119898119865119894(0) gt 0 then 119862

1015840

119898119894is bounded and from (7) (29)

and (30) 119890119894(119905) 120595119894(119905) and 120593

119894(119905) are bounded the following

inequation also holds

119890119894(119905) = minus119886119890

119894(119905) minus 120595

119894(119905) 1198621015840

119898(119905) + 120593

119894(119905) Δ119878 (119905) lt infin (31)

then from (29) and (31) and the Barbalet Lemma we havelim119905rarrinfin

119890119895(119905) = 0

119895 isin 119895 = 119899 or 119895 = 119908 | lim119905rarrinfin

119869119908(1199050 119905) lt infin

119908 = 1 2 119899 minus 1

(32)

and it finally comes to

Δ119878 (119866119898119895(119904) minus 119866

1015840

(119904)) = 0 (33)

The multiple model adaptive Smith predictor still has thematching function

Remark 5 Multiple model control based on multiple fixedSmith models can improve the transient response but as itlacks the adaptive parameter regulation system stability canhardly be guaranteed

Remark 6 The existence of multiple models covering theparameter uncertain range of the controlled plant canimprove the systemrsquos transient response and improve thecontrol quality for system with jumping parameter

4 Simulation Research

A first order time delay system as follows will be simulatedfor rolling AGC system

119866 (119904) eminus120591119904 = 119896

119904 + 119886eminus120591119904 = 021

0053119904 + 1eminus035119904

= 1198661015840

(119904) eminus03119904(34)

where 1198661015840(119904) = (021(0053119904 + 1))eminus005119904The same PID controller as in [17] based on ITAE

criterion [24] will also be used that is

119866119888(119904) = 119870

119901(1 +

1

119879119894119904+

119879119889119904

1 + 11987911988911990410

)

= 223 (1 +1

02119904+

0034119904

00034119904 + 1)

(35)

12

1

08

06

04

02

00 5 10 15 20 25 30 35 40

1005

0995

1

39 395 40Out

put t

hick

nessC(t)

(mm

)

Single adaptive Smith

Time t (s)

Figure 6 Single model adaptive Smith predictor

1005

0995

1

39 395 4000 5 10 15 20 25 30 35 40

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Multiple adaptive Smith

Time t (s)

Figure 7 Multiple model adaptive Smith predictor

10005

09995

1

39 395 40

0 5 10 15 20 25 30 35 400

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Multiple fixed Smith

Time t (s)

Figure 8 Multiple fixed Smith prediction model

The compensation Smith predictor will be designed as119866119898(119904) = 021(0053119904 + 1) 120591

119898= 03119904 And parameters of

119865(119905) and119870(119905) will be tuned as (8)

41 Multiple Model Smith Predictor for System with FixedTime Delay For single adaptive Smith prediction model letthe feedforward and feedback adaptive initial parameters be119865(0) = 35 and119870(0) = 50 in (8) the control result is shown inFigure 6 For multiple model adaptive control four adaptiveSmith predictionmodels are established corresponding feed-forward and feedback adaptive initial parameters in (12) are1198651(0) = 35 119865

2(0) = 30 119865

3(0) = 25 119865

4(0) = 18 and 119870

1(0) =

1198702(0) = 119870

3(0) = 119870

4(0) = 50 The control result is shown

in Figure 7

8 Abstract and Applied Analysis

Multiple fixed and one adaptive Smith

1005

0995

1

39 395 400 5 10 15 20 25 30 35 40

0

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Time t (s)

(a) System output

4

3

2

1Switc

hing

sequ

ence

I

Multiple fixed and one adaptive Smith

0 5 10 15 20 25 30 35 40

Time t (s)

(b) Switching sequence

Figure 9 Multiple fixed and single adaptive Smith prediction model

It can be seen in Figures 6 and 7 when multiple adaptiveSmith predictionmodels withmultiple groups of feedforwardand feedback adaptive initial parameters are applied to theAGC system the overshot and system response time aredecreased obviously compared with single Smith predictionmodel case Meanwhile when single model adaptive Smithpredictor is applied because model parameters match badlywith system parameters initial identification error is rel-atively big and the identification time goes relatively longcorrespondingly Since multiple model adaptive Smith pre-dictor adoptsmultiple adaptive initial parameters system canswitch to the closest matching model adaptively and conse-quently system identification error and identification timedecrease obviously

When four fixed Smith prediction models with initialadaptive feedback parameters 119865

1(0) = 35 119865

2(0) = 30

1198653(0) = 25 and 119865

4(0) = 18 and same initial feedforward

parameters 1198701(0) = 119870

2(0) = 119870

3(0) = 119870

4(0) = 50 in (16) are

applied the control result is shown in Figure 8 At the sametime the mixed multiple model Smith predictor with threefixed models whose initial feedback adaptive parameters are1198651(0) = 35 119865

2(0) = 30 and 119865

3(0) = 18 and an adaptive

model with initial value 1198654(0) = 18 the same feedforward

parameters 1198701(0) = 119870

2(0) = 119870

3(0) = 119870

4(0) = 50 are given

and the control results can be seen in Figure 9Comparedwithmultiple adaptive Smith predictormodel

the computing time can be reduced by using multiple fixedSmith predictor models but the stability cannot be guaran-teed because of the absence of adaptive adjustment FromFigure 8 suitable number of fixed models can be used toimprove the overshoot and response time but the steadyidentification error cannot be made into 0 at last In Figure 9the combination of fixed and adaptive Smith models willgive an improved transient response without the loss of finalproperty of system stability The model switching process isshown in Figure 9(b) the initial fixed model with 119865

1(0) = 35

is not the best one and fixed model 3 with 1198653(0) = 18 is

selected to control the system for a period of time finally thecontrol model is switched to the 4th adaptive Smith model

14

12

1

1

08

06

04

02

00 5 10 15 20 25

1004

1002

8 10 15 20 25

Change of time delay

Single adaptive SmithMultiple adaptive SmithSingle adaptive and multiple fixed Smith

Out

put t

hick

nessC(t)

(mm

)

Time t (s)

Figure 10 Multiple Smith prediction model for system with jump-ing time delay

42 Multiple Smith Prediction Model for System with JumpingTime Delay The rolling AGC system with jumping timedelay can be described as

119866 (119904) eminus120591119904 = 021

0053119904 + 1eminus120591119904

120591 = 03 0 lt 119905 lt 10

07 119905 ge 10

(36)

Take the single adaptive Smith predictionmodelmultipleadaptive Smith prediction model and the multiple mixedfixed-adaptive Smith prediction model with the same initialfeedback and feedforward parameters as in Section 41 andwe set 120591

119898= 025 Control results are shown in Figure 10

Abstract and Applied Analysis 9

When system time delay jumps the present multipleadaptive Smith prediction model and multiple mixed fixed-adaptive Smith prediction model can both guarantee thesystem stability Meanwhile compared with single adaptiveSmith prediction model system transient response of mul-tiple adaptive Smith prediction model has been improvedobviously no matter before or after the time delay changepoint

5 Conclusion

In the rolling AGC control process control systems undercomplex network condition are generally used Consider thetime delay of network transmission and thickness gauge isusually far away from the roll gap the transfer functionmodelof the controlled plant will always be described by a firstorder system with uncertain time delay Generally fixed andadaptive Smith predictors are the effective methods to solvethis problem but they all need the precisely known time delayvalue which is hard to be obtained This paper approximatesthe variation of time delay and the parameters in systemmodel by a first order process with different parametersMultiple Smith prediction model for AGC will be used toset up a MMAC to cope with the change range of time delayin rolling process The problem of regular Smith predictor issolved and the transient response of adaptive Smith predictoris improved

Conflict of Interests

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

Acknowledgments

This work was supposed by the Fundamental ResearchFunds for the Central Universities under Grant FRF-TP-12-005B the Program for New Century Excellent Talents inUniversities under Grant NCET-11-0578 and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (SRFDP) under Grant 20130006110008

References

[1] W Yang Y Sun C Cao C Wang and J Yang ldquoDistributedcomputer control system for hot narrow strip millrdquo Iron andSteel vol 28 no 8 pp 30ndash34 1993

[2] Y Q Wang M H Sun and W Zhang ldquoDistributed computercontrol in hydraulic AGC system of tandem cold rolling millrdquoMachine Tool and Hydraulics vol 35 no 3 pp 120ndash123 2007

[3] Y K Sun ldquoThe computer control system applied to rollingprocessrdquo Engineering Science vol 2 no 1 pp 73ndash76 2000

[4] Y Wang M Sun W Zhang Y Fang G Chen and Q ZhangldquoResearch on feedforward control system of tandem coldrolling millrsquos hydraulic AGC and its computer control strategyrdquoin Proceedings of the International Technology and Innova-tion Conference (ITIC rsquo06) pp 2093ndash2098 Hangzhou ChinaNovember 2006

[5] D Q Song Research about feedback AGC control stragety ofreversible cold rolling mill control systems based on adaptiveSmith predictor [PhD thesis] Central South University Chang-sha China 2009

[6] C L Lai and P L Hsu ldquoDesign the remote control systemwith the time-delay estimator and the adaptive smith predictorrdquoIEEE Transactions on Industrial Informatics vol 6 no 1 pp 73ndash80 2010

[7] K S Narendra and J Balakrishnan ldquoImproving transientresponse of adaptive control systems usingmultiple models andswitchingrdquo IEEE Transactions on Automatic Control vol 39 no9 pp 1861ndash1866 1994

[8] Z Han and K S Narendra ldquoNew concepts in adaptive controlusing multiple modelsrdquo IEEE Transactions on Automatic Con-trol vol 57 no 1 pp 78ndash89 2012

[9] K S Narendra and J Balakrishnan ldquoAdaptive control usingmultiple modelsrdquo IEEE Transactions on Automatic Control vol42 no 2 pp 171ndash187 1997

[10] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000

[11] B Chaudhuri R Majumder and B C Pal ldquoApplication ofmultiple-model adaptive control strategy for robust dampingof interarea oscillations in power systemrdquo IEEE Transactions onControl Systems Technology vol 12 no 5 pp 727ndash736 2004

[12] W T Chen and B D O Anderson ldquoA combined multiplemodel adaptive control scheme and its application to nonlinearsystemswith nonlinear parameterizationrdquo IEEETransactions onAutomatic Control vol 57 no 7 pp 1778ndash1782 2012

[13] K Ibrahim ldquoA new Smith predictor and controller for controlof processes with long dead timerdquo ISA Transactions vol 42 no1 pp 101ndash110 2003

[14] S-B Tan M-W Bao and J-C Liu ldquoResearch on applicationof smith predictor to monitor AGC in hot strip rolling millsrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo09) pp 4172ndash4175 Guilin China June 2009

[15] J C Liu S S Gu S F Zheng andH TMi ldquoApplication of smithprediction control strategy to AGC systemrdquo Iron and Steel vol33 no 10 pp 40ndash43 1998

[16] DMeng Y Jia J Du andF Yu ldquoLearning control for time-delaysystems with iteration-varying uncertainty a Smith predictor-based approachrdquo IET Control Theory amp Applications vol 4 no12 pp 2707ndash2718 2010

[17] X Li D Q Song S Y Yu andW H Gui ldquoFeedback automaticgauge control system using model reference adaptive Smithpredictorrdquo Control Theory and Applications vol 26 no 9 pp999ndash1003 2009

[18] S Yin S X Ding X C Xie andH Luo ldquoA review on basic data-driven approaches for industrial process monitoringrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6418ndash6428 2014

[19] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[20] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[21] S Yin X W Li H J Gao and O Kaynak ldquoData-based tech-niques focused on modern industry an overviewrdquo IEEE Trans-actions on Industrial Electronics 2014

10 Abstract and Applied Analysis

[22] S S Hu Automatic Control Theory Science Press BeijingChina 5th edition 2007

[23] K S Narendra and A M Annaswamy Stable Adaptive SystemsPrentice Hall Englewood Cliffs NJ USA 1989

[24] N Anne and L Jonathan ldquoVarying time delay Smith predictorprocess controllerrdquo ISA Transactions vol 25 no 3 pp 77ndash812004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Feedback

Feedforward

Adaptive mechanism

regulator

regulator

hr(t)

+

+

+

+

+

minus

minus

minus

Δh ΔSG(s) eminus120591s

C(t)

eminus120591119898s

Gc(s)

Gm(s)

Gm(s)

K(e t)

G998400m(t)ΔS

998400

e(t)

Gm(t)

F(e t)

C998400(t)

Figure 4 AGC control process based on adaptive Smith predictor

decreases and the control performance goes worse whilein the Smith predictor scheme when the prediction modelmatches the controlled plant perfectly that is 119866

119898(119904) = 119866(119904)

120591119898= 120591 we have 119866

119898(119904)eminus120591119898119904 = 119866(119904)eminus120591119904 and system output

becomes

119862 (119904)

=119866119888(119904) 119866 (119904) eminus120591119904

1 + 119866119888(119904) 119866 (119904) eminus120591119904 minus 119866

119888(119904) 119866119898(119904) eminus120591119898119904 + 119866

119888(119904) 119866119898(119904)

times ℎ119903(119904)

=119866119888(119904) 119866 (119904) eminus120591119904

1 + 119866119888(119904) 119866119898(119904)

ℎ119903(119904)

(2)

Corresponding characteristic equation becomes 1 +

119866119888(119904)119866119898(119904) = 0 without time-delay part so the system

stability will not be influenced by time delay 120591

22 Adaptive Smith Predictor Controller Practically condi-tion 119866

119898(119904) = 119866(119904) 120591

119898= 120591 cannot be satisfied easily [17]

proposes a more practical method in which adaptive adjust-ing law of feedback coefficient 119865(119905) and feedforward coef-ficient 119870(119905) is applied to guarantee the identification Smithpredictor 119866

119898(119904) = 119866(119904) under the condition 120591

119898= 120591 = 0 The

principle of adaptive Smith predictior is shown in Figure 4Consider 120591

119898= 120591 = 0 the transfer functions of the system

and Smith predictor model are given as

119866 (119904) =119896

119904 + 119886 119866

119898(119904) =

119896119898

119904 + 119886119898

(3)

Corresponding state equations can be got as

(119905) = minus119886119862 (119905) + 119896Δ119878 (119905)

119898(119905) = minus119886

119898119862 (119905) + 119896

119898Δ1198781015840

(119905)

(4)

For given generalized error

119890 (119905) = 119862 (119905) minus 119862119898(119905) (5)

Lynaponov function is set up as

V (119905) =1

2[119896119898119890 (119905)2

+1

1205781

1205952

(119905) +1

1205782

1205932

(119905)] (6)

where

120595 (119905) = 119886 minus 119886119898minus 119896119898119865 (119905)

120593 (119905) = 119896 minus 119896119898119870 (119905)

(7)

An adaptive control law

119865 (119905) = minusint 1205781119890 (119905) 119862

119898(119905) + 119865 (0)

119870 (119905) = int 1205782119890 (119905) Δ119878 (119905) + 119870 (0)

(8)

can be used to guarantee 119890(119905) rarr 0120595(119905) rarr 0 and 120593(119905) rarr 0

when 119905 rarr infin

Remark 1 The adaptive parameter identification mechanismcan be viewed as a kind of data-driven strategy [18 19]Information about process measurements and initial valuesis required in the condition of given prediction modelHowever unlike pure data-driven methods such as partialleast squares (PLS) iterative learning control (ILC) andmodel free adaptive control which only depend on theprocess measurements [20] this adaptive law is a model-dataintegrated method [21] which also needs the relative precisepredictionmodel so that the process information can be fullyutilized based on the model to realize the system identifica-tion

From Figure 4 this adaptive law can ensure119866119898(119904) = 119866(119904)

(where119866119898(119904) is an identification Smith predictor adjusted by

119865 119866119898 and 119870) the identification Smith predictor 119866

119898(119904) can

be used But in practice both 119866119898(119904) = 119866(119904) and 120591

119898= 120591 = 0

are always very hard to be satisfied because the precisemodelof the system cannot be setup and time delay of the systemcannot be measured accurately So above adaptive Smith pre-dictor is also very difficult to be applied Due to this situationthe following multiple adaptive Smith prediction modelcontroller will be considered

Abstract and Applied Analysis 5

Adaptive mechanism

Feedforwardregulator

Feedback regulator

+

+

++

minus

minus

hr(t)

minus

ΔhGc(s)

ΔSG(s)

C(t)

Gm(s)

eminus120591119898s

eminus120591119898s

eminusΔ120591sG998400(s)

C998400(t)

ΔS998400

e(t)

C998400m(t)

Gm(t)

F(e t)

Gm(s)

Smith predictor P1

Smith predictor Pi

Smith predictor Pn

e1(t)

ei(t)

en(t)

Switching

k(e t)

Figure 5 AGC control process based on multiple adaptive Smith prediction model

3 Multiple Smith Prediction Model Controller

Consider the rolling system in Figure 5 its transfer functionand time delay can be generally described by the followingmodel

119866 (119904) eminusΔ120591119904eminus120591119898119904 (9)

where 120591119898is the delay under normal working condition and

Δ120591 stands for the delay fluctuation caused by uncertain factorssuch as network transmission delay

Define

1198661015840

(119904) = 119866 (119904) eminusΔ120591119904 (10)

Consider the character of first order system we assumethat a first order systemwith small delay can be approximatedby a first order processwith different parameters [22] we have

1198661015840

(119904) asymp1198961015840

119904 + 1198861015840 (11)

Further we have the multiple model adaptive controlsystem (see Figure 5)

As in Figure 5 we use the Smith prediction model 119866119898(119904)

adjusted by 119865119866119898 and119870 to approximate the controlled plant

1198661015840

(119904) Consider the varying scope of Δ120591 parameters of 1198661015840(119904)will be uncertain or time varying so it is not appropriate todescribe the systemby single fixedmodel For the single adap-tive model if its initial parameters are not properly selectedmodel parameters will converge slowly Moreover if thereis drastic system parameter jump control performance willget worse So we take multiple models with different initialvalues to approximate the system model uncertainty

31 Multiple Adaptive Smith Model Controller Based onthe possible parameters variation range of 1198661015840(119904) multipleadaptive Smith predictors 119875

119894(119894 = 1 2 119899 119899 is the

number of models) with different initial values are designedFor original Smith prediction model 119866

119898(119904) corresponding

adaptive parameters of the feedforward and feedback loopsatisfy

119865119894(119905) = minusint 120578

1119890 (119905) 119862

1015840

119898(119905) + 119865

119894(0)

119870119894(119905) = int 120578

2119890 (119905) Δ119878 (119905) + 119870

119894(0)

(12)

where 119894 = 1 2 119899Define the switch index function

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905 (13)

where 119894 = 1 2 119899 120572 gt 0 and 120573 gt 0 And the model error

119890119894(119905) = 119862

1015840

(119905) minus 1198621015840

119898119894(119905) asymp 119862 (119905) minus 119862

119898119894(119905) (14)

At time 119905 calculate

119897 (1199050 119905) = argmin

119894isin12119898

119869119894(1199050 119905) (15)

and Smith predictor based on model 119897 will be switched intothe closed loop system

Remark 2 In practical AGC control process 1198621015840(119905) cannot bemeasured so as in (14) 119890

119894(119905) = 119888(119905)minus119888

119898(119905)will be used instead

of 1198621015840(119905) minus 1198621015840119898(119905) in practical control

32 Multiple Fixed Smith Prediction Model Controller Formultiple adaptive Smith prediction model controller parallelcomputation of multiple adaptive processes will lead to theincrease of calculation amount

If all the 119899models are selected to be fixed models that isfor 119894 = 1 2 119899

119865119894(119905) = 119865

119894(0) gt 0

119870119894(119905) = 119870

119894(0) gt 0

(16)

6 Abstract and Applied Analysis

With enough fixed models take (13) as the switch indexfunction calculation amount for multiple models can bedecreased drastically However without the stable characterof adaptive model system stability can hardly be guaranteedby pure multiple fixed models

33 Multiple Adaptive Smith Controllers with Multiple Fixedand Adaptive Models Multiple fixed models as in Section 32can reduce the amount of calculation but due to the lack ofadaptive parameter adjusting process system stability cannotbe easily obtained If in the 119899 models 119899 minus 1 are set up asfixed models and 1 model is adaptive model that is for 119894 =1 2 119899 minus 1

119865119894(119905) = 119865

119894(0) gt 0

119870119894(119905) = 119870

119894(0) gt 0

119865119899(119905) = minusint 120578

1119890119899(119905) 1198621015840

119898(119905) + 119865

119899(0)

119870119899(119905) = int 120578

2119890119899(119905) Δ119878 (119905) + 119870

119899(0)

(17)

The existence of multiple fixed models reduces theamount of calculation and the adaptivemodel guarantees thesystem stability so the control performance can be improvedas the case of multiple adaptive models

Theorem3 If the adaptive Smith controller in Section 22 (120591 =120591119898= 0) is applied to the AGC system model error 119890 can be

guaranteed to be bounded and to approach zero as 119905 rarr infinthat is lim

119905rarrinfin119890(119905) = 0

Proof From [17] for adaptive Smith predictionmodel takingderivative of Lynaponov function (6) we have

V (119905) = minus 1198961198981198861198902

(119905) + 120595 (119905) [1

1205781

(119905) minus 119896119898119890 (119905) 119862

119898(119905)]

+ 120593 (119905) [1

1205782

(119905) + 119896119898119890 (119905) Δ119878 (119905)]

(18)

from (7) and (8)

V (119905) = minus1198961198981198861198902

(119905) le 0 (19)

Consider 1205781gt 0 120578

2gt 0 119896 gt 0 and 119896

119898gt 0 so V(119905) gt 0

holds for all 119905 = 0From (19) V(119905) is bounded so 119890(119905) 120595(119905) and 120593(119905) are all

bounded Further we have

int

119905

1199050

1198902

(119905) 119889119905 lt infin 119890 (119905) isin 1198712 (20)

For

1015840

119898(119905) = minus119886

1198981198621015840

119898(119905) + 119896

119898Δ1198781015840

(119905)

Δ1198781015840

(119905) = 119870 (119905) Δ119878 (119905) minus 119865 (119905) 1198621015840

119898(119905)

1198621015840 (119905) = minus1198861198621015840

(119905) + 119896Δ119878 (119905)

119890 (119905) = 1198621015840

(119905) minus 1198621015840

119898(119905)

(21)

we have

119890 (119905) = minus119886119890 (119905) minus 120595 (119905) 1198621015840

119898(119905) + 120593 (119905) Δ119878 (119905) (22)

Consider Δ119878(119905) 1198621015840(119905) and 119890(119905) are bounded then 1198621015840119898(119905)

is bounded from (22) we have 119890(119905) isin 119871infin from (20) and the

Barbalet Lemma [23] we have lim119905rarrinfin

119890(119905) = 0

Theorem 4 Take (15) as the switch index function and selectmultiple adaptivemodels in Section 31 ormultiple fixedmodelsand one adaptive model in Section 33 as the multiple adaptiveSmith controllers For the AGC Smith controller shown as inFigure 5 each of these two kinds of controllers can cancel thefluctuation of system delay effectively and improve the controlperformance

Proof (1) For multiple model control composed of multipleadaptive Smith predictors since each Smith model has itsown adaptive mechanism so for each model we have

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905 lt infin

119894 = 1 2 119899

lim119905rarrinfin

119890119894(119905) = 0

(23)

Though there is model switching among multiple modelsbased on the switch index function it finally comes to

Δ119878 (119866119898119894(119904) minus 119866

1015840

(119904)) = 0 (24)

Uncertain part of time delay is canceled andmultiple modelsare used to improve the transient process

(2) For multiple model control composed of multiplefixed Smith predictors and one adaptive Smith predictor asto the adaptive Smith predictor

119869119899(1199050 119905) = 120572119890

2

119899(119905) + 120573int

119905

1199050

1198902

119899(119905) 119889119905 lt infin (25)

Consider the switch index function

119897 (1199050 119905) = argmin

119894isin12119899

119869119894(1199050 119905) (26)

For each fixed model

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905

119894 = 1 2 119899 minus 1

(27)

If

lim119905rarrinfin

119869119894(1199050 119905) = infin 119894 = 1 2 119899 minus 1 (28)

then finally the adaptive model will be selected and theadaptive Smith model matches the systemmodel completelythat is Δ119878(119866

119898119899(119904) minus 119866

1015840

(119904)) = 0

Abstract and Applied Analysis 7

If there is a fixed model 119894 satisfying

lim119905rarrinfin

119869119894(1199050 119905) = lim

119905rarrinfin

[1205721198902

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905] lt infin (29)

where 119894 = 1 2 119899 minus 1 from (21)

1015840

(119905) = minus1198861198621015840

(119905) + 119896Δ119878 (119905)

1015840

119898119894(119905) = minus [119886

119898+ 119896119898119865119894(0)] 119862

1015840

119898119894(119905) + 119896

119898119870119894(0) Δ119878 (119905)

119890119894(119905) = 119862

1015840

(119905) minus 1198621015840

119898119894(119905)

(30)

If the parameters of selected fixed model satisfy 119865119894(0) gt 0 or

119886119898+ 119896119898119865119894(0) gt 0 then 119862

1015840

119898119894is bounded and from (7) (29)

and (30) 119890119894(119905) 120595119894(119905) and 120593

119894(119905) are bounded the following

inequation also holds

119890119894(119905) = minus119886119890

119894(119905) minus 120595

119894(119905) 1198621015840

119898(119905) + 120593

119894(119905) Δ119878 (119905) lt infin (31)

then from (29) and (31) and the Barbalet Lemma we havelim119905rarrinfin

119890119895(119905) = 0

119895 isin 119895 = 119899 or 119895 = 119908 | lim119905rarrinfin

119869119908(1199050 119905) lt infin

119908 = 1 2 119899 minus 1

(32)

and it finally comes to

Δ119878 (119866119898119895(119904) minus 119866

1015840

(119904)) = 0 (33)

The multiple model adaptive Smith predictor still has thematching function

Remark 5 Multiple model control based on multiple fixedSmith models can improve the transient response but as itlacks the adaptive parameter regulation system stability canhardly be guaranteed

Remark 6 The existence of multiple models covering theparameter uncertain range of the controlled plant canimprove the systemrsquos transient response and improve thecontrol quality for system with jumping parameter

4 Simulation Research

A first order time delay system as follows will be simulatedfor rolling AGC system

119866 (119904) eminus120591119904 = 119896

119904 + 119886eminus120591119904 = 021

0053119904 + 1eminus035119904

= 1198661015840

(119904) eminus03119904(34)

where 1198661015840(119904) = (021(0053119904 + 1))eminus005119904The same PID controller as in [17] based on ITAE

criterion [24] will also be used that is

119866119888(119904) = 119870

119901(1 +

1

119879119894119904+

119879119889119904

1 + 11987911988911990410

)

= 223 (1 +1

02119904+

0034119904

00034119904 + 1)

(35)

12

1

08

06

04

02

00 5 10 15 20 25 30 35 40

1005

0995

1

39 395 40Out

put t

hick

nessC(t)

(mm

)

Single adaptive Smith

Time t (s)

Figure 6 Single model adaptive Smith predictor

1005

0995

1

39 395 4000 5 10 15 20 25 30 35 40

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Multiple adaptive Smith

Time t (s)

Figure 7 Multiple model adaptive Smith predictor

10005

09995

1

39 395 40

0 5 10 15 20 25 30 35 400

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Multiple fixed Smith

Time t (s)

Figure 8 Multiple fixed Smith prediction model

The compensation Smith predictor will be designed as119866119898(119904) = 021(0053119904 + 1) 120591

119898= 03119904 And parameters of

119865(119905) and119870(119905) will be tuned as (8)

41 Multiple Model Smith Predictor for System with FixedTime Delay For single adaptive Smith prediction model letthe feedforward and feedback adaptive initial parameters be119865(0) = 35 and119870(0) = 50 in (8) the control result is shown inFigure 6 For multiple model adaptive control four adaptiveSmith predictionmodels are established corresponding feed-forward and feedback adaptive initial parameters in (12) are1198651(0) = 35 119865

2(0) = 30 119865

3(0) = 25 119865

4(0) = 18 and 119870

1(0) =

1198702(0) = 119870

3(0) = 119870

4(0) = 50 The control result is shown

in Figure 7

8 Abstract and Applied Analysis

Multiple fixed and one adaptive Smith

1005

0995

1

39 395 400 5 10 15 20 25 30 35 40

0

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Time t (s)

(a) System output

4

3

2

1Switc

hing

sequ

ence

I

Multiple fixed and one adaptive Smith

0 5 10 15 20 25 30 35 40

Time t (s)

(b) Switching sequence

Figure 9 Multiple fixed and single adaptive Smith prediction model

It can be seen in Figures 6 and 7 when multiple adaptiveSmith predictionmodels withmultiple groups of feedforwardand feedback adaptive initial parameters are applied to theAGC system the overshot and system response time aredecreased obviously compared with single Smith predictionmodel case Meanwhile when single model adaptive Smithpredictor is applied because model parameters match badlywith system parameters initial identification error is rel-atively big and the identification time goes relatively longcorrespondingly Since multiple model adaptive Smith pre-dictor adoptsmultiple adaptive initial parameters system canswitch to the closest matching model adaptively and conse-quently system identification error and identification timedecrease obviously

When four fixed Smith prediction models with initialadaptive feedback parameters 119865

1(0) = 35 119865

2(0) = 30

1198653(0) = 25 and 119865

4(0) = 18 and same initial feedforward

parameters 1198701(0) = 119870

2(0) = 119870

3(0) = 119870

4(0) = 50 in (16) are

applied the control result is shown in Figure 8 At the sametime the mixed multiple model Smith predictor with threefixed models whose initial feedback adaptive parameters are1198651(0) = 35 119865

2(0) = 30 and 119865

3(0) = 18 and an adaptive

model with initial value 1198654(0) = 18 the same feedforward

parameters 1198701(0) = 119870

2(0) = 119870

3(0) = 119870

4(0) = 50 are given

and the control results can be seen in Figure 9Comparedwithmultiple adaptive Smith predictormodel

the computing time can be reduced by using multiple fixedSmith predictor models but the stability cannot be guaran-teed because of the absence of adaptive adjustment FromFigure 8 suitable number of fixed models can be used toimprove the overshoot and response time but the steadyidentification error cannot be made into 0 at last In Figure 9the combination of fixed and adaptive Smith models willgive an improved transient response without the loss of finalproperty of system stability The model switching process isshown in Figure 9(b) the initial fixed model with 119865

1(0) = 35

is not the best one and fixed model 3 with 1198653(0) = 18 is

selected to control the system for a period of time finally thecontrol model is switched to the 4th adaptive Smith model

14

12

1

1

08

06

04

02

00 5 10 15 20 25

1004

1002

8 10 15 20 25

Change of time delay

Single adaptive SmithMultiple adaptive SmithSingle adaptive and multiple fixed Smith

Out

put t

hick

nessC(t)

(mm

)

Time t (s)

Figure 10 Multiple Smith prediction model for system with jump-ing time delay

42 Multiple Smith Prediction Model for System with JumpingTime Delay The rolling AGC system with jumping timedelay can be described as

119866 (119904) eminus120591119904 = 021

0053119904 + 1eminus120591119904

120591 = 03 0 lt 119905 lt 10

07 119905 ge 10

(36)

Take the single adaptive Smith predictionmodelmultipleadaptive Smith prediction model and the multiple mixedfixed-adaptive Smith prediction model with the same initialfeedback and feedforward parameters as in Section 41 andwe set 120591

119898= 025 Control results are shown in Figure 10

Abstract and Applied Analysis 9

When system time delay jumps the present multipleadaptive Smith prediction model and multiple mixed fixed-adaptive Smith prediction model can both guarantee thesystem stability Meanwhile compared with single adaptiveSmith prediction model system transient response of mul-tiple adaptive Smith prediction model has been improvedobviously no matter before or after the time delay changepoint

5 Conclusion

In the rolling AGC control process control systems undercomplex network condition are generally used Consider thetime delay of network transmission and thickness gauge isusually far away from the roll gap the transfer functionmodelof the controlled plant will always be described by a firstorder system with uncertain time delay Generally fixed andadaptive Smith predictors are the effective methods to solvethis problem but they all need the precisely known time delayvalue which is hard to be obtained This paper approximatesthe variation of time delay and the parameters in systemmodel by a first order process with different parametersMultiple Smith prediction model for AGC will be used toset up a MMAC to cope with the change range of time delayin rolling process The problem of regular Smith predictor issolved and the transient response of adaptive Smith predictoris improved

Conflict of Interests

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

Acknowledgments

This work was supposed by the Fundamental ResearchFunds for the Central Universities under Grant FRF-TP-12-005B the Program for New Century Excellent Talents inUniversities under Grant NCET-11-0578 and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (SRFDP) under Grant 20130006110008

References

[1] W Yang Y Sun C Cao C Wang and J Yang ldquoDistributedcomputer control system for hot narrow strip millrdquo Iron andSteel vol 28 no 8 pp 30ndash34 1993

[2] Y Q Wang M H Sun and W Zhang ldquoDistributed computercontrol in hydraulic AGC system of tandem cold rolling millrdquoMachine Tool and Hydraulics vol 35 no 3 pp 120ndash123 2007

[3] Y K Sun ldquoThe computer control system applied to rollingprocessrdquo Engineering Science vol 2 no 1 pp 73ndash76 2000

[4] Y Wang M Sun W Zhang Y Fang G Chen and Q ZhangldquoResearch on feedforward control system of tandem coldrolling millrsquos hydraulic AGC and its computer control strategyrdquoin Proceedings of the International Technology and Innova-tion Conference (ITIC rsquo06) pp 2093ndash2098 Hangzhou ChinaNovember 2006

[5] D Q Song Research about feedback AGC control stragety ofreversible cold rolling mill control systems based on adaptiveSmith predictor [PhD thesis] Central South University Chang-sha China 2009

[6] C L Lai and P L Hsu ldquoDesign the remote control systemwith the time-delay estimator and the adaptive smith predictorrdquoIEEE Transactions on Industrial Informatics vol 6 no 1 pp 73ndash80 2010

[7] K S Narendra and J Balakrishnan ldquoImproving transientresponse of adaptive control systems usingmultiple models andswitchingrdquo IEEE Transactions on Automatic Control vol 39 no9 pp 1861ndash1866 1994

[8] Z Han and K S Narendra ldquoNew concepts in adaptive controlusing multiple modelsrdquo IEEE Transactions on Automatic Con-trol vol 57 no 1 pp 78ndash89 2012

[9] K S Narendra and J Balakrishnan ldquoAdaptive control usingmultiple modelsrdquo IEEE Transactions on Automatic Control vol42 no 2 pp 171ndash187 1997

[10] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000

[11] B Chaudhuri R Majumder and B C Pal ldquoApplication ofmultiple-model adaptive control strategy for robust dampingof interarea oscillations in power systemrdquo IEEE Transactions onControl Systems Technology vol 12 no 5 pp 727ndash736 2004

[12] W T Chen and B D O Anderson ldquoA combined multiplemodel adaptive control scheme and its application to nonlinearsystemswith nonlinear parameterizationrdquo IEEETransactions onAutomatic Control vol 57 no 7 pp 1778ndash1782 2012

[13] K Ibrahim ldquoA new Smith predictor and controller for controlof processes with long dead timerdquo ISA Transactions vol 42 no1 pp 101ndash110 2003

[14] S-B Tan M-W Bao and J-C Liu ldquoResearch on applicationof smith predictor to monitor AGC in hot strip rolling millsrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo09) pp 4172ndash4175 Guilin China June 2009

[15] J C Liu S S Gu S F Zheng andH TMi ldquoApplication of smithprediction control strategy to AGC systemrdquo Iron and Steel vol33 no 10 pp 40ndash43 1998

[16] DMeng Y Jia J Du andF Yu ldquoLearning control for time-delaysystems with iteration-varying uncertainty a Smith predictor-based approachrdquo IET Control Theory amp Applications vol 4 no12 pp 2707ndash2718 2010

[17] X Li D Q Song S Y Yu andW H Gui ldquoFeedback automaticgauge control system using model reference adaptive Smithpredictorrdquo Control Theory and Applications vol 26 no 9 pp999ndash1003 2009

[18] S Yin S X Ding X C Xie andH Luo ldquoA review on basic data-driven approaches for industrial process monitoringrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6418ndash6428 2014

[19] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[20] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[21] S Yin X W Li H J Gao and O Kaynak ldquoData-based tech-niques focused on modern industry an overviewrdquo IEEE Trans-actions on Industrial Electronics 2014

10 Abstract and Applied Analysis

[22] S S Hu Automatic Control Theory Science Press BeijingChina 5th edition 2007

[23] K S Narendra and A M Annaswamy Stable Adaptive SystemsPrentice Hall Englewood Cliffs NJ USA 1989

[24] N Anne and L Jonathan ldquoVarying time delay Smith predictorprocess controllerrdquo ISA Transactions vol 25 no 3 pp 77ndash812004

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Adaptive mechanism

Feedforwardregulator

Feedback regulator

+

+

++

minus

minus

hr(t)

minus

ΔhGc(s)

ΔSG(s)

C(t)

Gm(s)

eminus120591119898s

eminus120591119898s

eminusΔ120591sG998400(s)

C998400(t)

ΔS998400

e(t)

C998400m(t)

Gm(t)

F(e t)

Gm(s)

Smith predictor P1

Smith predictor Pi

Smith predictor Pn

e1(t)

ei(t)

en(t)

Switching

k(e t)

Figure 5 AGC control process based on multiple adaptive Smith prediction model

3 Multiple Smith Prediction Model Controller

Consider the rolling system in Figure 5 its transfer functionand time delay can be generally described by the followingmodel

119866 (119904) eminusΔ120591119904eminus120591119898119904 (9)

where 120591119898is the delay under normal working condition and

Δ120591 stands for the delay fluctuation caused by uncertain factorssuch as network transmission delay

Define

1198661015840

(119904) = 119866 (119904) eminusΔ120591119904 (10)

Consider the character of first order system we assumethat a first order systemwith small delay can be approximatedby a first order processwith different parameters [22] we have

1198661015840

(119904) asymp1198961015840

119904 + 1198861015840 (11)

Further we have the multiple model adaptive controlsystem (see Figure 5)

As in Figure 5 we use the Smith prediction model 119866119898(119904)

adjusted by 119865119866119898 and119870 to approximate the controlled plant

1198661015840

(119904) Consider the varying scope of Δ120591 parameters of 1198661015840(119904)will be uncertain or time varying so it is not appropriate todescribe the systemby single fixedmodel For the single adap-tive model if its initial parameters are not properly selectedmodel parameters will converge slowly Moreover if thereis drastic system parameter jump control performance willget worse So we take multiple models with different initialvalues to approximate the system model uncertainty

31 Multiple Adaptive Smith Model Controller Based onthe possible parameters variation range of 1198661015840(119904) multipleadaptive Smith predictors 119875

119894(119894 = 1 2 119899 119899 is the

number of models) with different initial values are designedFor original Smith prediction model 119866

119898(119904) corresponding

adaptive parameters of the feedforward and feedback loopsatisfy

119865119894(119905) = minusint 120578

1119890 (119905) 119862

1015840

119898(119905) + 119865

119894(0)

119870119894(119905) = int 120578

2119890 (119905) Δ119878 (119905) + 119870

119894(0)

(12)

where 119894 = 1 2 119899Define the switch index function

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905 (13)

where 119894 = 1 2 119899 120572 gt 0 and 120573 gt 0 And the model error

119890119894(119905) = 119862

1015840

(119905) minus 1198621015840

119898119894(119905) asymp 119862 (119905) minus 119862

119898119894(119905) (14)

At time 119905 calculate

119897 (1199050 119905) = argmin

119894isin12119898

119869119894(1199050 119905) (15)

and Smith predictor based on model 119897 will be switched intothe closed loop system

Remark 2 In practical AGC control process 1198621015840(119905) cannot bemeasured so as in (14) 119890

119894(119905) = 119888(119905)minus119888

119898(119905)will be used instead

of 1198621015840(119905) minus 1198621015840119898(119905) in practical control

32 Multiple Fixed Smith Prediction Model Controller Formultiple adaptive Smith prediction model controller parallelcomputation of multiple adaptive processes will lead to theincrease of calculation amount

If all the 119899models are selected to be fixed models that isfor 119894 = 1 2 119899

119865119894(119905) = 119865

119894(0) gt 0

119870119894(119905) = 119870

119894(0) gt 0

(16)

6 Abstract and Applied Analysis

With enough fixed models take (13) as the switch indexfunction calculation amount for multiple models can bedecreased drastically However without the stable characterof adaptive model system stability can hardly be guaranteedby pure multiple fixed models

33 Multiple Adaptive Smith Controllers with Multiple Fixedand Adaptive Models Multiple fixed models as in Section 32can reduce the amount of calculation but due to the lack ofadaptive parameter adjusting process system stability cannotbe easily obtained If in the 119899 models 119899 minus 1 are set up asfixed models and 1 model is adaptive model that is for 119894 =1 2 119899 minus 1

119865119894(119905) = 119865

119894(0) gt 0

119870119894(119905) = 119870

119894(0) gt 0

119865119899(119905) = minusint 120578

1119890119899(119905) 1198621015840

119898(119905) + 119865

119899(0)

119870119899(119905) = int 120578

2119890119899(119905) Δ119878 (119905) + 119870

119899(0)

(17)

The existence of multiple fixed models reduces theamount of calculation and the adaptivemodel guarantees thesystem stability so the control performance can be improvedas the case of multiple adaptive models

Theorem3 If the adaptive Smith controller in Section 22 (120591 =120591119898= 0) is applied to the AGC system model error 119890 can be

guaranteed to be bounded and to approach zero as 119905 rarr infinthat is lim

119905rarrinfin119890(119905) = 0

Proof From [17] for adaptive Smith predictionmodel takingderivative of Lynaponov function (6) we have

V (119905) = minus 1198961198981198861198902

(119905) + 120595 (119905) [1

1205781

(119905) minus 119896119898119890 (119905) 119862

119898(119905)]

+ 120593 (119905) [1

1205782

(119905) + 119896119898119890 (119905) Δ119878 (119905)]

(18)

from (7) and (8)

V (119905) = minus1198961198981198861198902

(119905) le 0 (19)

Consider 1205781gt 0 120578

2gt 0 119896 gt 0 and 119896

119898gt 0 so V(119905) gt 0

holds for all 119905 = 0From (19) V(119905) is bounded so 119890(119905) 120595(119905) and 120593(119905) are all

bounded Further we have

int

119905

1199050

1198902

(119905) 119889119905 lt infin 119890 (119905) isin 1198712 (20)

For

1015840

119898(119905) = minus119886

1198981198621015840

119898(119905) + 119896

119898Δ1198781015840

(119905)

Δ1198781015840

(119905) = 119870 (119905) Δ119878 (119905) minus 119865 (119905) 1198621015840

119898(119905)

1198621015840 (119905) = minus1198861198621015840

(119905) + 119896Δ119878 (119905)

119890 (119905) = 1198621015840

(119905) minus 1198621015840

119898(119905)

(21)

we have

119890 (119905) = minus119886119890 (119905) minus 120595 (119905) 1198621015840

119898(119905) + 120593 (119905) Δ119878 (119905) (22)

Consider Δ119878(119905) 1198621015840(119905) and 119890(119905) are bounded then 1198621015840119898(119905)

is bounded from (22) we have 119890(119905) isin 119871infin from (20) and the

Barbalet Lemma [23] we have lim119905rarrinfin

119890(119905) = 0

Theorem 4 Take (15) as the switch index function and selectmultiple adaptivemodels in Section 31 ormultiple fixedmodelsand one adaptive model in Section 33 as the multiple adaptiveSmith controllers For the AGC Smith controller shown as inFigure 5 each of these two kinds of controllers can cancel thefluctuation of system delay effectively and improve the controlperformance

Proof (1) For multiple model control composed of multipleadaptive Smith predictors since each Smith model has itsown adaptive mechanism so for each model we have

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905 lt infin

119894 = 1 2 119899

lim119905rarrinfin

119890119894(119905) = 0

(23)

Though there is model switching among multiple modelsbased on the switch index function it finally comes to

Δ119878 (119866119898119894(119904) minus 119866

1015840

(119904)) = 0 (24)

Uncertain part of time delay is canceled andmultiple modelsare used to improve the transient process

(2) For multiple model control composed of multiplefixed Smith predictors and one adaptive Smith predictor asto the adaptive Smith predictor

119869119899(1199050 119905) = 120572119890

2

119899(119905) + 120573int

119905

1199050

1198902

119899(119905) 119889119905 lt infin (25)

Consider the switch index function

119897 (1199050 119905) = argmin

119894isin12119899

119869119894(1199050 119905) (26)

For each fixed model

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905

119894 = 1 2 119899 minus 1

(27)

If

lim119905rarrinfin

119869119894(1199050 119905) = infin 119894 = 1 2 119899 minus 1 (28)

then finally the adaptive model will be selected and theadaptive Smith model matches the systemmodel completelythat is Δ119878(119866

119898119899(119904) minus 119866

1015840

(119904)) = 0

Abstract and Applied Analysis 7

If there is a fixed model 119894 satisfying

lim119905rarrinfin

119869119894(1199050 119905) = lim

119905rarrinfin

[1205721198902

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905] lt infin (29)

where 119894 = 1 2 119899 minus 1 from (21)

1015840

(119905) = minus1198861198621015840

(119905) + 119896Δ119878 (119905)

1015840

119898119894(119905) = minus [119886

119898+ 119896119898119865119894(0)] 119862

1015840

119898119894(119905) + 119896

119898119870119894(0) Δ119878 (119905)

119890119894(119905) = 119862

1015840

(119905) minus 1198621015840

119898119894(119905)

(30)

If the parameters of selected fixed model satisfy 119865119894(0) gt 0 or

119886119898+ 119896119898119865119894(0) gt 0 then 119862

1015840

119898119894is bounded and from (7) (29)

and (30) 119890119894(119905) 120595119894(119905) and 120593

119894(119905) are bounded the following

inequation also holds

119890119894(119905) = minus119886119890

119894(119905) minus 120595

119894(119905) 1198621015840

119898(119905) + 120593

119894(119905) Δ119878 (119905) lt infin (31)

then from (29) and (31) and the Barbalet Lemma we havelim119905rarrinfin

119890119895(119905) = 0

119895 isin 119895 = 119899 or 119895 = 119908 | lim119905rarrinfin

119869119908(1199050 119905) lt infin

119908 = 1 2 119899 minus 1

(32)

and it finally comes to

Δ119878 (119866119898119895(119904) minus 119866

1015840

(119904)) = 0 (33)

The multiple model adaptive Smith predictor still has thematching function

Remark 5 Multiple model control based on multiple fixedSmith models can improve the transient response but as itlacks the adaptive parameter regulation system stability canhardly be guaranteed

Remark 6 The existence of multiple models covering theparameter uncertain range of the controlled plant canimprove the systemrsquos transient response and improve thecontrol quality for system with jumping parameter

4 Simulation Research

A first order time delay system as follows will be simulatedfor rolling AGC system

119866 (119904) eminus120591119904 = 119896

119904 + 119886eminus120591119904 = 021

0053119904 + 1eminus035119904

= 1198661015840

(119904) eminus03119904(34)

where 1198661015840(119904) = (021(0053119904 + 1))eminus005119904The same PID controller as in [17] based on ITAE

criterion [24] will also be used that is

119866119888(119904) = 119870

119901(1 +

1

119879119894119904+

119879119889119904

1 + 11987911988911990410

)

= 223 (1 +1

02119904+

0034119904

00034119904 + 1)

(35)

12

1

08

06

04

02

00 5 10 15 20 25 30 35 40

1005

0995

1

39 395 40Out

put t

hick

nessC(t)

(mm

)

Single adaptive Smith

Time t (s)

Figure 6 Single model adaptive Smith predictor

1005

0995

1

39 395 4000 5 10 15 20 25 30 35 40

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Multiple adaptive Smith

Time t (s)

Figure 7 Multiple model adaptive Smith predictor

10005

09995

1

39 395 40

0 5 10 15 20 25 30 35 400

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Multiple fixed Smith

Time t (s)

Figure 8 Multiple fixed Smith prediction model

The compensation Smith predictor will be designed as119866119898(119904) = 021(0053119904 + 1) 120591

119898= 03119904 And parameters of

119865(119905) and119870(119905) will be tuned as (8)

41 Multiple Model Smith Predictor for System with FixedTime Delay For single adaptive Smith prediction model letthe feedforward and feedback adaptive initial parameters be119865(0) = 35 and119870(0) = 50 in (8) the control result is shown inFigure 6 For multiple model adaptive control four adaptiveSmith predictionmodels are established corresponding feed-forward and feedback adaptive initial parameters in (12) are1198651(0) = 35 119865

2(0) = 30 119865

3(0) = 25 119865

4(0) = 18 and 119870

1(0) =

1198702(0) = 119870

3(0) = 119870

4(0) = 50 The control result is shown

in Figure 7

8 Abstract and Applied Analysis

Multiple fixed and one adaptive Smith

1005

0995

1

39 395 400 5 10 15 20 25 30 35 40

0

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Time t (s)

(a) System output

4

3

2

1Switc

hing

sequ

ence

I

Multiple fixed and one adaptive Smith

0 5 10 15 20 25 30 35 40

Time t (s)

(b) Switching sequence

Figure 9 Multiple fixed and single adaptive Smith prediction model

It can be seen in Figures 6 and 7 when multiple adaptiveSmith predictionmodels withmultiple groups of feedforwardand feedback adaptive initial parameters are applied to theAGC system the overshot and system response time aredecreased obviously compared with single Smith predictionmodel case Meanwhile when single model adaptive Smithpredictor is applied because model parameters match badlywith system parameters initial identification error is rel-atively big and the identification time goes relatively longcorrespondingly Since multiple model adaptive Smith pre-dictor adoptsmultiple adaptive initial parameters system canswitch to the closest matching model adaptively and conse-quently system identification error and identification timedecrease obviously

When four fixed Smith prediction models with initialadaptive feedback parameters 119865

1(0) = 35 119865

2(0) = 30

1198653(0) = 25 and 119865

4(0) = 18 and same initial feedforward

parameters 1198701(0) = 119870

2(0) = 119870

3(0) = 119870

4(0) = 50 in (16) are

applied the control result is shown in Figure 8 At the sametime the mixed multiple model Smith predictor with threefixed models whose initial feedback adaptive parameters are1198651(0) = 35 119865

2(0) = 30 and 119865

3(0) = 18 and an adaptive

model with initial value 1198654(0) = 18 the same feedforward

parameters 1198701(0) = 119870

2(0) = 119870

3(0) = 119870

4(0) = 50 are given

and the control results can be seen in Figure 9Comparedwithmultiple adaptive Smith predictormodel

the computing time can be reduced by using multiple fixedSmith predictor models but the stability cannot be guaran-teed because of the absence of adaptive adjustment FromFigure 8 suitable number of fixed models can be used toimprove the overshoot and response time but the steadyidentification error cannot be made into 0 at last In Figure 9the combination of fixed and adaptive Smith models willgive an improved transient response without the loss of finalproperty of system stability The model switching process isshown in Figure 9(b) the initial fixed model with 119865

1(0) = 35

is not the best one and fixed model 3 with 1198653(0) = 18 is

selected to control the system for a period of time finally thecontrol model is switched to the 4th adaptive Smith model

14

12

1

1

08

06

04

02

00 5 10 15 20 25

1004

1002

8 10 15 20 25

Change of time delay

Single adaptive SmithMultiple adaptive SmithSingle adaptive and multiple fixed Smith

Out

put t

hick

nessC(t)

(mm

)

Time t (s)

Figure 10 Multiple Smith prediction model for system with jump-ing time delay

42 Multiple Smith Prediction Model for System with JumpingTime Delay The rolling AGC system with jumping timedelay can be described as

119866 (119904) eminus120591119904 = 021

0053119904 + 1eminus120591119904

120591 = 03 0 lt 119905 lt 10

07 119905 ge 10

(36)

Take the single adaptive Smith predictionmodelmultipleadaptive Smith prediction model and the multiple mixedfixed-adaptive Smith prediction model with the same initialfeedback and feedforward parameters as in Section 41 andwe set 120591

119898= 025 Control results are shown in Figure 10

Abstract and Applied Analysis 9

When system time delay jumps the present multipleadaptive Smith prediction model and multiple mixed fixed-adaptive Smith prediction model can both guarantee thesystem stability Meanwhile compared with single adaptiveSmith prediction model system transient response of mul-tiple adaptive Smith prediction model has been improvedobviously no matter before or after the time delay changepoint

5 Conclusion

In the rolling AGC control process control systems undercomplex network condition are generally used Consider thetime delay of network transmission and thickness gauge isusually far away from the roll gap the transfer functionmodelof the controlled plant will always be described by a firstorder system with uncertain time delay Generally fixed andadaptive Smith predictors are the effective methods to solvethis problem but they all need the precisely known time delayvalue which is hard to be obtained This paper approximatesthe variation of time delay and the parameters in systemmodel by a first order process with different parametersMultiple Smith prediction model for AGC will be used toset up a MMAC to cope with the change range of time delayin rolling process The problem of regular Smith predictor issolved and the transient response of adaptive Smith predictoris improved

Conflict of Interests

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

Acknowledgments

This work was supposed by the Fundamental ResearchFunds for the Central Universities under Grant FRF-TP-12-005B the Program for New Century Excellent Talents inUniversities under Grant NCET-11-0578 and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (SRFDP) under Grant 20130006110008

References

[1] W Yang Y Sun C Cao C Wang and J Yang ldquoDistributedcomputer control system for hot narrow strip millrdquo Iron andSteel vol 28 no 8 pp 30ndash34 1993

[2] Y Q Wang M H Sun and W Zhang ldquoDistributed computercontrol in hydraulic AGC system of tandem cold rolling millrdquoMachine Tool and Hydraulics vol 35 no 3 pp 120ndash123 2007

[3] Y K Sun ldquoThe computer control system applied to rollingprocessrdquo Engineering Science vol 2 no 1 pp 73ndash76 2000

[4] Y Wang M Sun W Zhang Y Fang G Chen and Q ZhangldquoResearch on feedforward control system of tandem coldrolling millrsquos hydraulic AGC and its computer control strategyrdquoin Proceedings of the International Technology and Innova-tion Conference (ITIC rsquo06) pp 2093ndash2098 Hangzhou ChinaNovember 2006

[5] D Q Song Research about feedback AGC control stragety ofreversible cold rolling mill control systems based on adaptiveSmith predictor [PhD thesis] Central South University Chang-sha China 2009

[6] C L Lai and P L Hsu ldquoDesign the remote control systemwith the time-delay estimator and the adaptive smith predictorrdquoIEEE Transactions on Industrial Informatics vol 6 no 1 pp 73ndash80 2010

[7] K S Narendra and J Balakrishnan ldquoImproving transientresponse of adaptive control systems usingmultiple models andswitchingrdquo IEEE Transactions on Automatic Control vol 39 no9 pp 1861ndash1866 1994

[8] Z Han and K S Narendra ldquoNew concepts in adaptive controlusing multiple modelsrdquo IEEE Transactions on Automatic Con-trol vol 57 no 1 pp 78ndash89 2012

[9] K S Narendra and J Balakrishnan ldquoAdaptive control usingmultiple modelsrdquo IEEE Transactions on Automatic Control vol42 no 2 pp 171ndash187 1997

[10] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000

[11] B Chaudhuri R Majumder and B C Pal ldquoApplication ofmultiple-model adaptive control strategy for robust dampingof interarea oscillations in power systemrdquo IEEE Transactions onControl Systems Technology vol 12 no 5 pp 727ndash736 2004

[12] W T Chen and B D O Anderson ldquoA combined multiplemodel adaptive control scheme and its application to nonlinearsystemswith nonlinear parameterizationrdquo IEEETransactions onAutomatic Control vol 57 no 7 pp 1778ndash1782 2012

[13] K Ibrahim ldquoA new Smith predictor and controller for controlof processes with long dead timerdquo ISA Transactions vol 42 no1 pp 101ndash110 2003

[14] S-B Tan M-W Bao and J-C Liu ldquoResearch on applicationof smith predictor to monitor AGC in hot strip rolling millsrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo09) pp 4172ndash4175 Guilin China June 2009

[15] J C Liu S S Gu S F Zheng andH TMi ldquoApplication of smithprediction control strategy to AGC systemrdquo Iron and Steel vol33 no 10 pp 40ndash43 1998

[16] DMeng Y Jia J Du andF Yu ldquoLearning control for time-delaysystems with iteration-varying uncertainty a Smith predictor-based approachrdquo IET Control Theory amp Applications vol 4 no12 pp 2707ndash2718 2010

[17] X Li D Q Song S Y Yu andW H Gui ldquoFeedback automaticgauge control system using model reference adaptive Smithpredictorrdquo Control Theory and Applications vol 26 no 9 pp999ndash1003 2009

[18] S Yin S X Ding X C Xie andH Luo ldquoA review on basic data-driven approaches for industrial process monitoringrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6418ndash6428 2014

[19] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[20] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[21] S Yin X W Li H J Gao and O Kaynak ldquoData-based tech-niques focused on modern industry an overviewrdquo IEEE Trans-actions on Industrial Electronics 2014

10 Abstract and Applied Analysis

[22] S S Hu Automatic Control Theory Science Press BeijingChina 5th edition 2007

[23] K S Narendra and A M Annaswamy Stable Adaptive SystemsPrentice Hall Englewood Cliffs NJ USA 1989

[24] N Anne and L Jonathan ldquoVarying time delay Smith predictorprocess controllerrdquo ISA Transactions vol 25 no 3 pp 77ndash812004

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

With enough fixed models take (13) as the switch indexfunction calculation amount for multiple models can bedecreased drastically However without the stable characterof adaptive model system stability can hardly be guaranteedby pure multiple fixed models

33 Multiple Adaptive Smith Controllers with Multiple Fixedand Adaptive Models Multiple fixed models as in Section 32can reduce the amount of calculation but due to the lack ofadaptive parameter adjusting process system stability cannotbe easily obtained If in the 119899 models 119899 minus 1 are set up asfixed models and 1 model is adaptive model that is for 119894 =1 2 119899 minus 1

119865119894(119905) = 119865

119894(0) gt 0

119870119894(119905) = 119870

119894(0) gt 0

119865119899(119905) = minusint 120578

1119890119899(119905) 1198621015840

119898(119905) + 119865

119899(0)

119870119899(119905) = int 120578

2119890119899(119905) Δ119878 (119905) + 119870

119899(0)

(17)

The existence of multiple fixed models reduces theamount of calculation and the adaptivemodel guarantees thesystem stability so the control performance can be improvedas the case of multiple adaptive models

Theorem3 If the adaptive Smith controller in Section 22 (120591 =120591119898= 0) is applied to the AGC system model error 119890 can be

guaranteed to be bounded and to approach zero as 119905 rarr infinthat is lim

119905rarrinfin119890(119905) = 0

Proof From [17] for adaptive Smith predictionmodel takingderivative of Lynaponov function (6) we have

V (119905) = minus 1198961198981198861198902

(119905) + 120595 (119905) [1

1205781

(119905) minus 119896119898119890 (119905) 119862

119898(119905)]

+ 120593 (119905) [1

1205782

(119905) + 119896119898119890 (119905) Δ119878 (119905)]

(18)

from (7) and (8)

V (119905) = minus1198961198981198861198902

(119905) le 0 (19)

Consider 1205781gt 0 120578

2gt 0 119896 gt 0 and 119896

119898gt 0 so V(119905) gt 0

holds for all 119905 = 0From (19) V(119905) is bounded so 119890(119905) 120595(119905) and 120593(119905) are all

bounded Further we have

int

119905

1199050

1198902

(119905) 119889119905 lt infin 119890 (119905) isin 1198712 (20)

For

1015840

119898(119905) = minus119886

1198981198621015840

119898(119905) + 119896

119898Δ1198781015840

(119905)

Δ1198781015840

(119905) = 119870 (119905) Δ119878 (119905) minus 119865 (119905) 1198621015840

119898(119905)

1198621015840 (119905) = minus1198861198621015840

(119905) + 119896Δ119878 (119905)

119890 (119905) = 1198621015840

(119905) minus 1198621015840

119898(119905)

(21)

we have

119890 (119905) = minus119886119890 (119905) minus 120595 (119905) 1198621015840

119898(119905) + 120593 (119905) Δ119878 (119905) (22)

Consider Δ119878(119905) 1198621015840(119905) and 119890(119905) are bounded then 1198621015840119898(119905)

is bounded from (22) we have 119890(119905) isin 119871infin from (20) and the

Barbalet Lemma [23] we have lim119905rarrinfin

119890(119905) = 0

Theorem 4 Take (15) as the switch index function and selectmultiple adaptivemodels in Section 31 ormultiple fixedmodelsand one adaptive model in Section 33 as the multiple adaptiveSmith controllers For the AGC Smith controller shown as inFigure 5 each of these two kinds of controllers can cancel thefluctuation of system delay effectively and improve the controlperformance

Proof (1) For multiple model control composed of multipleadaptive Smith predictors since each Smith model has itsown adaptive mechanism so for each model we have

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905 lt infin

119894 = 1 2 119899

lim119905rarrinfin

119890119894(119905) = 0

(23)

Though there is model switching among multiple modelsbased on the switch index function it finally comes to

Δ119878 (119866119898119894(119904) minus 119866

1015840

(119904)) = 0 (24)

Uncertain part of time delay is canceled andmultiple modelsare used to improve the transient process

(2) For multiple model control composed of multiplefixed Smith predictors and one adaptive Smith predictor asto the adaptive Smith predictor

119869119899(1199050 119905) = 120572119890

2

119899(119905) + 120573int

119905

1199050

1198902

119899(119905) 119889119905 lt infin (25)

Consider the switch index function

119897 (1199050 119905) = argmin

119894isin12119899

119869119894(1199050 119905) (26)

For each fixed model

119869119894(1199050 119905) = 120572119890

2

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905

119894 = 1 2 119899 minus 1

(27)

If

lim119905rarrinfin

119869119894(1199050 119905) = infin 119894 = 1 2 119899 minus 1 (28)

then finally the adaptive model will be selected and theadaptive Smith model matches the systemmodel completelythat is Δ119878(119866

119898119899(119904) minus 119866

1015840

(119904)) = 0

Abstract and Applied Analysis 7

If there is a fixed model 119894 satisfying

lim119905rarrinfin

119869119894(1199050 119905) = lim

119905rarrinfin

[1205721198902

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905] lt infin (29)

where 119894 = 1 2 119899 minus 1 from (21)

1015840

(119905) = minus1198861198621015840

(119905) + 119896Δ119878 (119905)

1015840

119898119894(119905) = minus [119886

119898+ 119896119898119865119894(0)] 119862

1015840

119898119894(119905) + 119896

119898119870119894(0) Δ119878 (119905)

119890119894(119905) = 119862

1015840

(119905) minus 1198621015840

119898119894(119905)

(30)

If the parameters of selected fixed model satisfy 119865119894(0) gt 0 or

119886119898+ 119896119898119865119894(0) gt 0 then 119862

1015840

119898119894is bounded and from (7) (29)

and (30) 119890119894(119905) 120595119894(119905) and 120593

119894(119905) are bounded the following

inequation also holds

119890119894(119905) = minus119886119890

119894(119905) minus 120595

119894(119905) 1198621015840

119898(119905) + 120593

119894(119905) Δ119878 (119905) lt infin (31)

then from (29) and (31) and the Barbalet Lemma we havelim119905rarrinfin

119890119895(119905) = 0

119895 isin 119895 = 119899 or 119895 = 119908 | lim119905rarrinfin

119869119908(1199050 119905) lt infin

119908 = 1 2 119899 minus 1

(32)

and it finally comes to

Δ119878 (119866119898119895(119904) minus 119866

1015840

(119904)) = 0 (33)

The multiple model adaptive Smith predictor still has thematching function

Remark 5 Multiple model control based on multiple fixedSmith models can improve the transient response but as itlacks the adaptive parameter regulation system stability canhardly be guaranteed

Remark 6 The existence of multiple models covering theparameter uncertain range of the controlled plant canimprove the systemrsquos transient response and improve thecontrol quality for system with jumping parameter

4 Simulation Research

A first order time delay system as follows will be simulatedfor rolling AGC system

119866 (119904) eminus120591119904 = 119896

119904 + 119886eminus120591119904 = 021

0053119904 + 1eminus035119904

= 1198661015840

(119904) eminus03119904(34)

where 1198661015840(119904) = (021(0053119904 + 1))eminus005119904The same PID controller as in [17] based on ITAE

criterion [24] will also be used that is

119866119888(119904) = 119870

119901(1 +

1

119879119894119904+

119879119889119904

1 + 11987911988911990410

)

= 223 (1 +1

02119904+

0034119904

00034119904 + 1)

(35)

12

1

08

06

04

02

00 5 10 15 20 25 30 35 40

1005

0995

1

39 395 40Out

put t

hick

nessC(t)

(mm

)

Single adaptive Smith

Time t (s)

Figure 6 Single model adaptive Smith predictor

1005

0995

1

39 395 4000 5 10 15 20 25 30 35 40

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Multiple adaptive Smith

Time t (s)

Figure 7 Multiple model adaptive Smith predictor

10005

09995

1

39 395 40

0 5 10 15 20 25 30 35 400

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Multiple fixed Smith

Time t (s)

Figure 8 Multiple fixed Smith prediction model

The compensation Smith predictor will be designed as119866119898(119904) = 021(0053119904 + 1) 120591

119898= 03119904 And parameters of

119865(119905) and119870(119905) will be tuned as (8)

41 Multiple Model Smith Predictor for System with FixedTime Delay For single adaptive Smith prediction model letthe feedforward and feedback adaptive initial parameters be119865(0) = 35 and119870(0) = 50 in (8) the control result is shown inFigure 6 For multiple model adaptive control four adaptiveSmith predictionmodels are established corresponding feed-forward and feedback adaptive initial parameters in (12) are1198651(0) = 35 119865

2(0) = 30 119865

3(0) = 25 119865

4(0) = 18 and 119870

1(0) =

1198702(0) = 119870

3(0) = 119870

4(0) = 50 The control result is shown

in Figure 7

8 Abstract and Applied Analysis

Multiple fixed and one adaptive Smith

1005

0995

1

39 395 400 5 10 15 20 25 30 35 40

0

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Time t (s)

(a) System output

4

3

2

1Switc

hing

sequ

ence

I

Multiple fixed and one adaptive Smith

0 5 10 15 20 25 30 35 40

Time t (s)

(b) Switching sequence

Figure 9 Multiple fixed and single adaptive Smith prediction model

It can be seen in Figures 6 and 7 when multiple adaptiveSmith predictionmodels withmultiple groups of feedforwardand feedback adaptive initial parameters are applied to theAGC system the overshot and system response time aredecreased obviously compared with single Smith predictionmodel case Meanwhile when single model adaptive Smithpredictor is applied because model parameters match badlywith system parameters initial identification error is rel-atively big and the identification time goes relatively longcorrespondingly Since multiple model adaptive Smith pre-dictor adoptsmultiple adaptive initial parameters system canswitch to the closest matching model adaptively and conse-quently system identification error and identification timedecrease obviously

When four fixed Smith prediction models with initialadaptive feedback parameters 119865

1(0) = 35 119865

2(0) = 30

1198653(0) = 25 and 119865

4(0) = 18 and same initial feedforward

parameters 1198701(0) = 119870

2(0) = 119870

3(0) = 119870

4(0) = 50 in (16) are

applied the control result is shown in Figure 8 At the sametime the mixed multiple model Smith predictor with threefixed models whose initial feedback adaptive parameters are1198651(0) = 35 119865

2(0) = 30 and 119865

3(0) = 18 and an adaptive

model with initial value 1198654(0) = 18 the same feedforward

parameters 1198701(0) = 119870

2(0) = 119870

3(0) = 119870

4(0) = 50 are given

and the control results can be seen in Figure 9Comparedwithmultiple adaptive Smith predictormodel

the computing time can be reduced by using multiple fixedSmith predictor models but the stability cannot be guaran-teed because of the absence of adaptive adjustment FromFigure 8 suitable number of fixed models can be used toimprove the overshoot and response time but the steadyidentification error cannot be made into 0 at last In Figure 9the combination of fixed and adaptive Smith models willgive an improved transient response without the loss of finalproperty of system stability The model switching process isshown in Figure 9(b) the initial fixed model with 119865

1(0) = 35

is not the best one and fixed model 3 with 1198653(0) = 18 is

selected to control the system for a period of time finally thecontrol model is switched to the 4th adaptive Smith model

14

12

1

1

08

06

04

02

00 5 10 15 20 25

1004

1002

8 10 15 20 25

Change of time delay

Single adaptive SmithMultiple adaptive SmithSingle adaptive and multiple fixed Smith

Out

put t

hick

nessC(t)

(mm

)

Time t (s)

Figure 10 Multiple Smith prediction model for system with jump-ing time delay

42 Multiple Smith Prediction Model for System with JumpingTime Delay The rolling AGC system with jumping timedelay can be described as

119866 (119904) eminus120591119904 = 021

0053119904 + 1eminus120591119904

120591 = 03 0 lt 119905 lt 10

07 119905 ge 10

(36)

Take the single adaptive Smith predictionmodelmultipleadaptive Smith prediction model and the multiple mixedfixed-adaptive Smith prediction model with the same initialfeedback and feedforward parameters as in Section 41 andwe set 120591

119898= 025 Control results are shown in Figure 10

Abstract and Applied Analysis 9

When system time delay jumps the present multipleadaptive Smith prediction model and multiple mixed fixed-adaptive Smith prediction model can both guarantee thesystem stability Meanwhile compared with single adaptiveSmith prediction model system transient response of mul-tiple adaptive Smith prediction model has been improvedobviously no matter before or after the time delay changepoint

5 Conclusion

In the rolling AGC control process control systems undercomplex network condition are generally used Consider thetime delay of network transmission and thickness gauge isusually far away from the roll gap the transfer functionmodelof the controlled plant will always be described by a firstorder system with uncertain time delay Generally fixed andadaptive Smith predictors are the effective methods to solvethis problem but they all need the precisely known time delayvalue which is hard to be obtained This paper approximatesthe variation of time delay and the parameters in systemmodel by a first order process with different parametersMultiple Smith prediction model for AGC will be used toset up a MMAC to cope with the change range of time delayin rolling process The problem of regular Smith predictor issolved and the transient response of adaptive Smith predictoris improved

Conflict of Interests

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

Acknowledgments

This work was supposed by the Fundamental ResearchFunds for the Central Universities under Grant FRF-TP-12-005B the Program for New Century Excellent Talents inUniversities under Grant NCET-11-0578 and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (SRFDP) under Grant 20130006110008

References

[1] W Yang Y Sun C Cao C Wang and J Yang ldquoDistributedcomputer control system for hot narrow strip millrdquo Iron andSteel vol 28 no 8 pp 30ndash34 1993

[2] Y Q Wang M H Sun and W Zhang ldquoDistributed computercontrol in hydraulic AGC system of tandem cold rolling millrdquoMachine Tool and Hydraulics vol 35 no 3 pp 120ndash123 2007

[3] Y K Sun ldquoThe computer control system applied to rollingprocessrdquo Engineering Science vol 2 no 1 pp 73ndash76 2000

[4] Y Wang M Sun W Zhang Y Fang G Chen and Q ZhangldquoResearch on feedforward control system of tandem coldrolling millrsquos hydraulic AGC and its computer control strategyrdquoin Proceedings of the International Technology and Innova-tion Conference (ITIC rsquo06) pp 2093ndash2098 Hangzhou ChinaNovember 2006

[5] D Q Song Research about feedback AGC control stragety ofreversible cold rolling mill control systems based on adaptiveSmith predictor [PhD thesis] Central South University Chang-sha China 2009

[6] C L Lai and P L Hsu ldquoDesign the remote control systemwith the time-delay estimator and the adaptive smith predictorrdquoIEEE Transactions on Industrial Informatics vol 6 no 1 pp 73ndash80 2010

[7] K S Narendra and J Balakrishnan ldquoImproving transientresponse of adaptive control systems usingmultiple models andswitchingrdquo IEEE Transactions on Automatic Control vol 39 no9 pp 1861ndash1866 1994

[8] Z Han and K S Narendra ldquoNew concepts in adaptive controlusing multiple modelsrdquo IEEE Transactions on Automatic Con-trol vol 57 no 1 pp 78ndash89 2012

[9] K S Narendra and J Balakrishnan ldquoAdaptive control usingmultiple modelsrdquo IEEE Transactions on Automatic Control vol42 no 2 pp 171ndash187 1997

[10] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000

[11] B Chaudhuri R Majumder and B C Pal ldquoApplication ofmultiple-model adaptive control strategy for robust dampingof interarea oscillations in power systemrdquo IEEE Transactions onControl Systems Technology vol 12 no 5 pp 727ndash736 2004

[12] W T Chen and B D O Anderson ldquoA combined multiplemodel adaptive control scheme and its application to nonlinearsystemswith nonlinear parameterizationrdquo IEEETransactions onAutomatic Control vol 57 no 7 pp 1778ndash1782 2012

[13] K Ibrahim ldquoA new Smith predictor and controller for controlof processes with long dead timerdquo ISA Transactions vol 42 no1 pp 101ndash110 2003

[14] S-B Tan M-W Bao and J-C Liu ldquoResearch on applicationof smith predictor to monitor AGC in hot strip rolling millsrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo09) pp 4172ndash4175 Guilin China June 2009

[15] J C Liu S S Gu S F Zheng andH TMi ldquoApplication of smithprediction control strategy to AGC systemrdquo Iron and Steel vol33 no 10 pp 40ndash43 1998

[16] DMeng Y Jia J Du andF Yu ldquoLearning control for time-delaysystems with iteration-varying uncertainty a Smith predictor-based approachrdquo IET Control Theory amp Applications vol 4 no12 pp 2707ndash2718 2010

[17] X Li D Q Song S Y Yu andW H Gui ldquoFeedback automaticgauge control system using model reference adaptive Smithpredictorrdquo Control Theory and Applications vol 26 no 9 pp999ndash1003 2009

[18] S Yin S X Ding X C Xie andH Luo ldquoA review on basic data-driven approaches for industrial process monitoringrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6418ndash6428 2014

[19] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[20] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[21] S Yin X W Li H J Gao and O Kaynak ldquoData-based tech-niques focused on modern industry an overviewrdquo IEEE Trans-actions on Industrial Electronics 2014

10 Abstract and Applied Analysis

[22] S S Hu Automatic Control Theory Science Press BeijingChina 5th edition 2007

[23] K S Narendra and A M Annaswamy Stable Adaptive SystemsPrentice Hall Englewood Cliffs NJ USA 1989

[24] N Anne and L Jonathan ldquoVarying time delay Smith predictorprocess controllerrdquo ISA Transactions vol 25 no 3 pp 77ndash812004

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

Abstract and Applied Analysis 7

If there is a fixed model 119894 satisfying

lim119905rarrinfin

119869119894(1199050 119905) = lim

119905rarrinfin

[1205721198902

119894(119905) + 120573int

119905

1199050

1198902

119894(119905) 119889119905] lt infin (29)

where 119894 = 1 2 119899 minus 1 from (21)

1015840

(119905) = minus1198861198621015840

(119905) + 119896Δ119878 (119905)

1015840

119898119894(119905) = minus [119886

119898+ 119896119898119865119894(0)] 119862

1015840

119898119894(119905) + 119896

119898119870119894(0) Δ119878 (119905)

119890119894(119905) = 119862

1015840

(119905) minus 1198621015840

119898119894(119905)

(30)

If the parameters of selected fixed model satisfy 119865119894(0) gt 0 or

119886119898+ 119896119898119865119894(0) gt 0 then 119862

1015840

119898119894is bounded and from (7) (29)

and (30) 119890119894(119905) 120595119894(119905) and 120593

119894(119905) are bounded the following

inequation also holds

119890119894(119905) = minus119886119890

119894(119905) minus 120595

119894(119905) 1198621015840

119898(119905) + 120593

119894(119905) Δ119878 (119905) lt infin (31)

then from (29) and (31) and the Barbalet Lemma we havelim119905rarrinfin

119890119895(119905) = 0

119895 isin 119895 = 119899 or 119895 = 119908 | lim119905rarrinfin

119869119908(1199050 119905) lt infin

119908 = 1 2 119899 minus 1

(32)

and it finally comes to

Δ119878 (119866119898119895(119904) minus 119866

1015840

(119904)) = 0 (33)

The multiple model adaptive Smith predictor still has thematching function

Remark 5 Multiple model control based on multiple fixedSmith models can improve the transient response but as itlacks the adaptive parameter regulation system stability canhardly be guaranteed

Remark 6 The existence of multiple models covering theparameter uncertain range of the controlled plant canimprove the systemrsquos transient response and improve thecontrol quality for system with jumping parameter

4 Simulation Research

A first order time delay system as follows will be simulatedfor rolling AGC system

119866 (119904) eminus120591119904 = 119896

119904 + 119886eminus120591119904 = 021

0053119904 + 1eminus035119904

= 1198661015840

(119904) eminus03119904(34)

where 1198661015840(119904) = (021(0053119904 + 1))eminus005119904The same PID controller as in [17] based on ITAE

criterion [24] will also be used that is

119866119888(119904) = 119870

119901(1 +

1

119879119894119904+

119879119889119904

1 + 11987911988911990410

)

= 223 (1 +1

02119904+

0034119904

00034119904 + 1)

(35)

12

1

08

06

04

02

00 5 10 15 20 25 30 35 40

1005

0995

1

39 395 40Out

put t

hick

nessC(t)

(mm

)

Single adaptive Smith

Time t (s)

Figure 6 Single model adaptive Smith predictor

1005

0995

1

39 395 4000 5 10 15 20 25 30 35 40

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Multiple adaptive Smith

Time t (s)

Figure 7 Multiple model adaptive Smith predictor

10005

09995

1

39 395 40

0 5 10 15 20 25 30 35 400

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Multiple fixed Smith

Time t (s)

Figure 8 Multiple fixed Smith prediction model

The compensation Smith predictor will be designed as119866119898(119904) = 021(0053119904 + 1) 120591

119898= 03119904 And parameters of

119865(119905) and119870(119905) will be tuned as (8)

41 Multiple Model Smith Predictor for System with FixedTime Delay For single adaptive Smith prediction model letthe feedforward and feedback adaptive initial parameters be119865(0) = 35 and119870(0) = 50 in (8) the control result is shown inFigure 6 For multiple model adaptive control four adaptiveSmith predictionmodels are established corresponding feed-forward and feedback adaptive initial parameters in (12) are1198651(0) = 35 119865

2(0) = 30 119865

3(0) = 25 119865

4(0) = 18 and 119870

1(0) =

1198702(0) = 119870

3(0) = 119870

4(0) = 50 The control result is shown

in Figure 7

8 Abstract and Applied Analysis

Multiple fixed and one adaptive Smith

1005

0995

1

39 395 400 5 10 15 20 25 30 35 40

0

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Time t (s)

(a) System output

4

3

2

1Switc

hing

sequ

ence

I

Multiple fixed and one adaptive Smith

0 5 10 15 20 25 30 35 40

Time t (s)

(b) Switching sequence

Figure 9 Multiple fixed and single adaptive Smith prediction model

It can be seen in Figures 6 and 7 when multiple adaptiveSmith predictionmodels withmultiple groups of feedforwardand feedback adaptive initial parameters are applied to theAGC system the overshot and system response time aredecreased obviously compared with single Smith predictionmodel case Meanwhile when single model adaptive Smithpredictor is applied because model parameters match badlywith system parameters initial identification error is rel-atively big and the identification time goes relatively longcorrespondingly Since multiple model adaptive Smith pre-dictor adoptsmultiple adaptive initial parameters system canswitch to the closest matching model adaptively and conse-quently system identification error and identification timedecrease obviously

When four fixed Smith prediction models with initialadaptive feedback parameters 119865

1(0) = 35 119865

2(0) = 30

1198653(0) = 25 and 119865

4(0) = 18 and same initial feedforward

parameters 1198701(0) = 119870

2(0) = 119870

3(0) = 119870

4(0) = 50 in (16) are

applied the control result is shown in Figure 8 At the sametime the mixed multiple model Smith predictor with threefixed models whose initial feedback adaptive parameters are1198651(0) = 35 119865

2(0) = 30 and 119865

3(0) = 18 and an adaptive

model with initial value 1198654(0) = 18 the same feedforward

parameters 1198701(0) = 119870

2(0) = 119870

3(0) = 119870

4(0) = 50 are given

and the control results can be seen in Figure 9Comparedwithmultiple adaptive Smith predictormodel

the computing time can be reduced by using multiple fixedSmith predictor models but the stability cannot be guaran-teed because of the absence of adaptive adjustment FromFigure 8 suitable number of fixed models can be used toimprove the overshoot and response time but the steadyidentification error cannot be made into 0 at last In Figure 9the combination of fixed and adaptive Smith models willgive an improved transient response without the loss of finalproperty of system stability The model switching process isshown in Figure 9(b) the initial fixed model with 119865

1(0) = 35

is not the best one and fixed model 3 with 1198653(0) = 18 is

selected to control the system for a period of time finally thecontrol model is switched to the 4th adaptive Smith model

14

12

1

1

08

06

04

02

00 5 10 15 20 25

1004

1002

8 10 15 20 25

Change of time delay

Single adaptive SmithMultiple adaptive SmithSingle adaptive and multiple fixed Smith

Out

put t

hick

nessC(t)

(mm

)

Time t (s)

Figure 10 Multiple Smith prediction model for system with jump-ing time delay

42 Multiple Smith Prediction Model for System with JumpingTime Delay The rolling AGC system with jumping timedelay can be described as

119866 (119904) eminus120591119904 = 021

0053119904 + 1eminus120591119904

120591 = 03 0 lt 119905 lt 10

07 119905 ge 10

(36)

Take the single adaptive Smith predictionmodelmultipleadaptive Smith prediction model and the multiple mixedfixed-adaptive Smith prediction model with the same initialfeedback and feedforward parameters as in Section 41 andwe set 120591

119898= 025 Control results are shown in Figure 10

Abstract and Applied Analysis 9

When system time delay jumps the present multipleadaptive Smith prediction model and multiple mixed fixed-adaptive Smith prediction model can both guarantee thesystem stability Meanwhile compared with single adaptiveSmith prediction model system transient response of mul-tiple adaptive Smith prediction model has been improvedobviously no matter before or after the time delay changepoint

5 Conclusion

In the rolling AGC control process control systems undercomplex network condition are generally used Consider thetime delay of network transmission and thickness gauge isusually far away from the roll gap the transfer functionmodelof the controlled plant will always be described by a firstorder system with uncertain time delay Generally fixed andadaptive Smith predictors are the effective methods to solvethis problem but they all need the precisely known time delayvalue which is hard to be obtained This paper approximatesthe variation of time delay and the parameters in systemmodel by a first order process with different parametersMultiple Smith prediction model for AGC will be used toset up a MMAC to cope with the change range of time delayin rolling process The problem of regular Smith predictor issolved and the transient response of adaptive Smith predictoris improved

Conflict of Interests

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

Acknowledgments

This work was supposed by the Fundamental ResearchFunds for the Central Universities under Grant FRF-TP-12-005B the Program for New Century Excellent Talents inUniversities under Grant NCET-11-0578 and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (SRFDP) under Grant 20130006110008

References

[1] W Yang Y Sun C Cao C Wang and J Yang ldquoDistributedcomputer control system for hot narrow strip millrdquo Iron andSteel vol 28 no 8 pp 30ndash34 1993

[2] Y Q Wang M H Sun and W Zhang ldquoDistributed computercontrol in hydraulic AGC system of tandem cold rolling millrdquoMachine Tool and Hydraulics vol 35 no 3 pp 120ndash123 2007

[3] Y K Sun ldquoThe computer control system applied to rollingprocessrdquo Engineering Science vol 2 no 1 pp 73ndash76 2000

[4] Y Wang M Sun W Zhang Y Fang G Chen and Q ZhangldquoResearch on feedforward control system of tandem coldrolling millrsquos hydraulic AGC and its computer control strategyrdquoin Proceedings of the International Technology and Innova-tion Conference (ITIC rsquo06) pp 2093ndash2098 Hangzhou ChinaNovember 2006

[5] D Q Song Research about feedback AGC control stragety ofreversible cold rolling mill control systems based on adaptiveSmith predictor [PhD thesis] Central South University Chang-sha China 2009

[6] C L Lai and P L Hsu ldquoDesign the remote control systemwith the time-delay estimator and the adaptive smith predictorrdquoIEEE Transactions on Industrial Informatics vol 6 no 1 pp 73ndash80 2010

[7] K S Narendra and J Balakrishnan ldquoImproving transientresponse of adaptive control systems usingmultiple models andswitchingrdquo IEEE Transactions on Automatic Control vol 39 no9 pp 1861ndash1866 1994

[8] Z Han and K S Narendra ldquoNew concepts in adaptive controlusing multiple modelsrdquo IEEE Transactions on Automatic Con-trol vol 57 no 1 pp 78ndash89 2012

[9] K S Narendra and J Balakrishnan ldquoAdaptive control usingmultiple modelsrdquo IEEE Transactions on Automatic Control vol42 no 2 pp 171ndash187 1997

[10] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000

[11] B Chaudhuri R Majumder and B C Pal ldquoApplication ofmultiple-model adaptive control strategy for robust dampingof interarea oscillations in power systemrdquo IEEE Transactions onControl Systems Technology vol 12 no 5 pp 727ndash736 2004

[12] W T Chen and B D O Anderson ldquoA combined multiplemodel adaptive control scheme and its application to nonlinearsystemswith nonlinear parameterizationrdquo IEEETransactions onAutomatic Control vol 57 no 7 pp 1778ndash1782 2012

[13] K Ibrahim ldquoA new Smith predictor and controller for controlof processes with long dead timerdquo ISA Transactions vol 42 no1 pp 101ndash110 2003

[14] S-B Tan M-W Bao and J-C Liu ldquoResearch on applicationof smith predictor to monitor AGC in hot strip rolling millsrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo09) pp 4172ndash4175 Guilin China June 2009

[15] J C Liu S S Gu S F Zheng andH TMi ldquoApplication of smithprediction control strategy to AGC systemrdquo Iron and Steel vol33 no 10 pp 40ndash43 1998

[16] DMeng Y Jia J Du andF Yu ldquoLearning control for time-delaysystems with iteration-varying uncertainty a Smith predictor-based approachrdquo IET Control Theory amp Applications vol 4 no12 pp 2707ndash2718 2010

[17] X Li D Q Song S Y Yu andW H Gui ldquoFeedback automaticgauge control system using model reference adaptive Smithpredictorrdquo Control Theory and Applications vol 26 no 9 pp999ndash1003 2009

[18] S Yin S X Ding X C Xie andH Luo ldquoA review on basic data-driven approaches for industrial process monitoringrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6418ndash6428 2014

[19] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[20] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[21] S Yin X W Li H J Gao and O Kaynak ldquoData-based tech-niques focused on modern industry an overviewrdquo IEEE Trans-actions on Industrial Electronics 2014

10 Abstract and Applied Analysis

[22] S S Hu Automatic Control Theory Science Press BeijingChina 5th edition 2007

[23] K S Narendra and A M Annaswamy Stable Adaptive SystemsPrentice Hall Englewood Cliffs NJ USA 1989

[24] N Anne and L Jonathan ldquoVarying time delay Smith predictorprocess controllerrdquo ISA Transactions vol 25 no 3 pp 77ndash812004

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 Abstract and Applied Analysis

Multiple fixed and one adaptive Smith

1005

0995

1

39 395 400 5 10 15 20 25 30 35 40

0

12

1

08

06

04

02

Out

put t

hick

nessC(t)

(mm

)

Time t (s)

(a) System output

4

3

2

1Switc

hing

sequ

ence

I

Multiple fixed and one adaptive Smith

0 5 10 15 20 25 30 35 40

Time t (s)

(b) Switching sequence

Figure 9 Multiple fixed and single adaptive Smith prediction model

It can be seen in Figures 6 and 7 when multiple adaptiveSmith predictionmodels withmultiple groups of feedforwardand feedback adaptive initial parameters are applied to theAGC system the overshot and system response time aredecreased obviously compared with single Smith predictionmodel case Meanwhile when single model adaptive Smithpredictor is applied because model parameters match badlywith system parameters initial identification error is rel-atively big and the identification time goes relatively longcorrespondingly Since multiple model adaptive Smith pre-dictor adoptsmultiple adaptive initial parameters system canswitch to the closest matching model adaptively and conse-quently system identification error and identification timedecrease obviously

When four fixed Smith prediction models with initialadaptive feedback parameters 119865

1(0) = 35 119865

2(0) = 30

1198653(0) = 25 and 119865

4(0) = 18 and same initial feedforward

parameters 1198701(0) = 119870

2(0) = 119870

3(0) = 119870

4(0) = 50 in (16) are

applied the control result is shown in Figure 8 At the sametime the mixed multiple model Smith predictor with threefixed models whose initial feedback adaptive parameters are1198651(0) = 35 119865

2(0) = 30 and 119865

3(0) = 18 and an adaptive

model with initial value 1198654(0) = 18 the same feedforward

parameters 1198701(0) = 119870

2(0) = 119870

3(0) = 119870

4(0) = 50 are given

and the control results can be seen in Figure 9Comparedwithmultiple adaptive Smith predictormodel

the computing time can be reduced by using multiple fixedSmith predictor models but the stability cannot be guaran-teed because of the absence of adaptive adjustment FromFigure 8 suitable number of fixed models can be used toimprove the overshoot and response time but the steadyidentification error cannot be made into 0 at last In Figure 9the combination of fixed and adaptive Smith models willgive an improved transient response without the loss of finalproperty of system stability The model switching process isshown in Figure 9(b) the initial fixed model with 119865

1(0) = 35

is not the best one and fixed model 3 with 1198653(0) = 18 is

selected to control the system for a period of time finally thecontrol model is switched to the 4th adaptive Smith model

14

12

1

1

08

06

04

02

00 5 10 15 20 25

1004

1002

8 10 15 20 25

Change of time delay

Single adaptive SmithMultiple adaptive SmithSingle adaptive and multiple fixed Smith

Out

put t

hick

nessC(t)

(mm

)

Time t (s)

Figure 10 Multiple Smith prediction model for system with jump-ing time delay

42 Multiple Smith Prediction Model for System with JumpingTime Delay The rolling AGC system with jumping timedelay can be described as

119866 (119904) eminus120591119904 = 021

0053119904 + 1eminus120591119904

120591 = 03 0 lt 119905 lt 10

07 119905 ge 10

(36)

Take the single adaptive Smith predictionmodelmultipleadaptive Smith prediction model and the multiple mixedfixed-adaptive Smith prediction model with the same initialfeedback and feedforward parameters as in Section 41 andwe set 120591

119898= 025 Control results are shown in Figure 10

Abstract and Applied Analysis 9

When system time delay jumps the present multipleadaptive Smith prediction model and multiple mixed fixed-adaptive Smith prediction model can both guarantee thesystem stability Meanwhile compared with single adaptiveSmith prediction model system transient response of mul-tiple adaptive Smith prediction model has been improvedobviously no matter before or after the time delay changepoint

5 Conclusion

In the rolling AGC control process control systems undercomplex network condition are generally used Consider thetime delay of network transmission and thickness gauge isusually far away from the roll gap the transfer functionmodelof the controlled plant will always be described by a firstorder system with uncertain time delay Generally fixed andadaptive Smith predictors are the effective methods to solvethis problem but they all need the precisely known time delayvalue which is hard to be obtained This paper approximatesthe variation of time delay and the parameters in systemmodel by a first order process with different parametersMultiple Smith prediction model for AGC will be used toset up a MMAC to cope with the change range of time delayin rolling process The problem of regular Smith predictor issolved and the transient response of adaptive Smith predictoris improved

Conflict of Interests

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

Acknowledgments

This work was supposed by the Fundamental ResearchFunds for the Central Universities under Grant FRF-TP-12-005B the Program for New Century Excellent Talents inUniversities under Grant NCET-11-0578 and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (SRFDP) under Grant 20130006110008

References

[1] W Yang Y Sun C Cao C Wang and J Yang ldquoDistributedcomputer control system for hot narrow strip millrdquo Iron andSteel vol 28 no 8 pp 30ndash34 1993

[2] Y Q Wang M H Sun and W Zhang ldquoDistributed computercontrol in hydraulic AGC system of tandem cold rolling millrdquoMachine Tool and Hydraulics vol 35 no 3 pp 120ndash123 2007

[3] Y K Sun ldquoThe computer control system applied to rollingprocessrdquo Engineering Science vol 2 no 1 pp 73ndash76 2000

[4] Y Wang M Sun W Zhang Y Fang G Chen and Q ZhangldquoResearch on feedforward control system of tandem coldrolling millrsquos hydraulic AGC and its computer control strategyrdquoin Proceedings of the International Technology and Innova-tion Conference (ITIC rsquo06) pp 2093ndash2098 Hangzhou ChinaNovember 2006

[5] D Q Song Research about feedback AGC control stragety ofreversible cold rolling mill control systems based on adaptiveSmith predictor [PhD thesis] Central South University Chang-sha China 2009

[6] C L Lai and P L Hsu ldquoDesign the remote control systemwith the time-delay estimator and the adaptive smith predictorrdquoIEEE Transactions on Industrial Informatics vol 6 no 1 pp 73ndash80 2010

[7] K S Narendra and J Balakrishnan ldquoImproving transientresponse of adaptive control systems usingmultiple models andswitchingrdquo IEEE Transactions on Automatic Control vol 39 no9 pp 1861ndash1866 1994

[8] Z Han and K S Narendra ldquoNew concepts in adaptive controlusing multiple modelsrdquo IEEE Transactions on Automatic Con-trol vol 57 no 1 pp 78ndash89 2012

[9] K S Narendra and J Balakrishnan ldquoAdaptive control usingmultiple modelsrdquo IEEE Transactions on Automatic Control vol42 no 2 pp 171ndash187 1997

[10] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000

[11] B Chaudhuri R Majumder and B C Pal ldquoApplication ofmultiple-model adaptive control strategy for robust dampingof interarea oscillations in power systemrdquo IEEE Transactions onControl Systems Technology vol 12 no 5 pp 727ndash736 2004

[12] W T Chen and B D O Anderson ldquoA combined multiplemodel adaptive control scheme and its application to nonlinearsystemswith nonlinear parameterizationrdquo IEEETransactions onAutomatic Control vol 57 no 7 pp 1778ndash1782 2012

[13] K Ibrahim ldquoA new Smith predictor and controller for controlof processes with long dead timerdquo ISA Transactions vol 42 no1 pp 101ndash110 2003

[14] S-B Tan M-W Bao and J-C Liu ldquoResearch on applicationof smith predictor to monitor AGC in hot strip rolling millsrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo09) pp 4172ndash4175 Guilin China June 2009

[15] J C Liu S S Gu S F Zheng andH TMi ldquoApplication of smithprediction control strategy to AGC systemrdquo Iron and Steel vol33 no 10 pp 40ndash43 1998

[16] DMeng Y Jia J Du andF Yu ldquoLearning control for time-delaysystems with iteration-varying uncertainty a Smith predictor-based approachrdquo IET Control Theory amp Applications vol 4 no12 pp 2707ndash2718 2010

[17] X Li D Q Song S Y Yu andW H Gui ldquoFeedback automaticgauge control system using model reference adaptive Smithpredictorrdquo Control Theory and Applications vol 26 no 9 pp999ndash1003 2009

[18] S Yin S X Ding X C Xie andH Luo ldquoA review on basic data-driven approaches for industrial process monitoringrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6418ndash6428 2014

[19] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[20] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[21] S Yin X W Li H J Gao and O Kaynak ldquoData-based tech-niques focused on modern industry an overviewrdquo IEEE Trans-actions on Industrial Electronics 2014

10 Abstract and Applied Analysis

[22] S S Hu Automatic Control Theory Science Press BeijingChina 5th edition 2007

[23] K S Narendra and A M Annaswamy Stable Adaptive SystemsPrentice Hall Englewood Cliffs NJ USA 1989

[24] N Anne and L Jonathan ldquoVarying time delay Smith predictorprocess controllerrdquo ISA Transactions vol 25 no 3 pp 77ndash812004

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

Abstract and Applied Analysis 9

When system time delay jumps the present multipleadaptive Smith prediction model and multiple mixed fixed-adaptive Smith prediction model can both guarantee thesystem stability Meanwhile compared with single adaptiveSmith prediction model system transient response of mul-tiple adaptive Smith prediction model has been improvedobviously no matter before or after the time delay changepoint

5 Conclusion

In the rolling AGC control process control systems undercomplex network condition are generally used Consider thetime delay of network transmission and thickness gauge isusually far away from the roll gap the transfer functionmodelof the controlled plant will always be described by a firstorder system with uncertain time delay Generally fixed andadaptive Smith predictors are the effective methods to solvethis problem but they all need the precisely known time delayvalue which is hard to be obtained This paper approximatesthe variation of time delay and the parameters in systemmodel by a first order process with different parametersMultiple Smith prediction model for AGC will be used toset up a MMAC to cope with the change range of time delayin rolling process The problem of regular Smith predictor issolved and the transient response of adaptive Smith predictoris improved

Conflict of Interests

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

Acknowledgments

This work was supposed by the Fundamental ResearchFunds for the Central Universities under Grant FRF-TP-12-005B the Program for New Century Excellent Talents inUniversities under Grant NCET-11-0578 and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (SRFDP) under Grant 20130006110008

References

[1] W Yang Y Sun C Cao C Wang and J Yang ldquoDistributedcomputer control system for hot narrow strip millrdquo Iron andSteel vol 28 no 8 pp 30ndash34 1993

[2] Y Q Wang M H Sun and W Zhang ldquoDistributed computercontrol in hydraulic AGC system of tandem cold rolling millrdquoMachine Tool and Hydraulics vol 35 no 3 pp 120ndash123 2007

[3] Y K Sun ldquoThe computer control system applied to rollingprocessrdquo Engineering Science vol 2 no 1 pp 73ndash76 2000

[4] Y Wang M Sun W Zhang Y Fang G Chen and Q ZhangldquoResearch on feedforward control system of tandem coldrolling millrsquos hydraulic AGC and its computer control strategyrdquoin Proceedings of the International Technology and Innova-tion Conference (ITIC rsquo06) pp 2093ndash2098 Hangzhou ChinaNovember 2006

[5] D Q Song Research about feedback AGC control stragety ofreversible cold rolling mill control systems based on adaptiveSmith predictor [PhD thesis] Central South University Chang-sha China 2009

[6] C L Lai and P L Hsu ldquoDesign the remote control systemwith the time-delay estimator and the adaptive smith predictorrdquoIEEE Transactions on Industrial Informatics vol 6 no 1 pp 73ndash80 2010

[7] K S Narendra and J Balakrishnan ldquoImproving transientresponse of adaptive control systems usingmultiple models andswitchingrdquo IEEE Transactions on Automatic Control vol 39 no9 pp 1861ndash1866 1994

[8] Z Han and K S Narendra ldquoNew concepts in adaptive controlusing multiple modelsrdquo IEEE Transactions on Automatic Con-trol vol 57 no 1 pp 78ndash89 2012

[9] K S Narendra and J Balakrishnan ldquoAdaptive control usingmultiple modelsrdquo IEEE Transactions on Automatic Control vol42 no 2 pp 171ndash187 1997

[10] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000

[11] B Chaudhuri R Majumder and B C Pal ldquoApplication ofmultiple-model adaptive control strategy for robust dampingof interarea oscillations in power systemrdquo IEEE Transactions onControl Systems Technology vol 12 no 5 pp 727ndash736 2004

[12] W T Chen and B D O Anderson ldquoA combined multiplemodel adaptive control scheme and its application to nonlinearsystemswith nonlinear parameterizationrdquo IEEETransactions onAutomatic Control vol 57 no 7 pp 1778ndash1782 2012

[13] K Ibrahim ldquoA new Smith predictor and controller for controlof processes with long dead timerdquo ISA Transactions vol 42 no1 pp 101ndash110 2003

[14] S-B Tan M-W Bao and J-C Liu ldquoResearch on applicationof smith predictor to monitor AGC in hot strip rolling millsrdquoin Proceedings of the Chinese Control and Decision Conference(CCDC rsquo09) pp 4172ndash4175 Guilin China June 2009

[15] J C Liu S S Gu S F Zheng andH TMi ldquoApplication of smithprediction control strategy to AGC systemrdquo Iron and Steel vol33 no 10 pp 40ndash43 1998

[16] DMeng Y Jia J Du andF Yu ldquoLearning control for time-delaysystems with iteration-varying uncertainty a Smith predictor-based approachrdquo IET Control Theory amp Applications vol 4 no12 pp 2707ndash2718 2010

[17] X Li D Q Song S Y Yu andW H Gui ldquoFeedback automaticgauge control system using model reference adaptive Smithpredictorrdquo Control Theory and Applications vol 26 no 9 pp999ndash1003 2009

[18] S Yin S X Ding X C Xie andH Luo ldquoA review on basic data-driven approaches for industrial process monitoringrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6418ndash6428 2014

[19] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[20] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[21] S Yin X W Li H J Gao and O Kaynak ldquoData-based tech-niques focused on modern industry an overviewrdquo IEEE Trans-actions on Industrial Electronics 2014

10 Abstract and Applied Analysis

[22] S S Hu Automatic Control Theory Science Press BeijingChina 5th edition 2007

[23] K S Narendra and A M Annaswamy Stable Adaptive SystemsPrentice Hall Englewood Cliffs NJ USA 1989

[24] N Anne and L Jonathan ldquoVarying time delay Smith predictorprocess controllerrdquo ISA Transactions vol 25 no 3 pp 77ndash812004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

[22] S S Hu Automatic Control Theory Science Press BeijingChina 5th edition 2007

[23] K S Narendra and A M Annaswamy Stable Adaptive SystemsPrentice Hall Englewood Cliffs NJ USA 1989

[24] N Anne and L Jonathan ldquoVarying time delay Smith predictorprocess controllerrdquo ISA Transactions vol 25 no 3 pp 77ndash812004

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