Research ArticleRobust Tension Control of Strip for 5-Stand Tandem Cold Mills

Behrooz Shafiei1 Mohsen Ekramian2 and Khoshnam Shojaei3

1 Department of Automation Engineering Tandem Cold Rolling Mill Mobarakeh Steel Company Isfahan Iran2Department of Electrical Engineering Faculty of Engineering University of Isfahan Isfahan Iran3Department of Electrical Engineering Najafabad Branch Islamic Azad University Isfahan Iran

Correspondence should be addressed to Behrooz Shafiei bh shmoyahoocom

Received 3 July 2014 Accepted 8 October 2014 Published 5 November 2014

Academic Editor Xiao He

Copyright copy 2014 Behrooz Shafiei et alThis 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

Tandem cold rolling process is a nonlinear complex system with external and internal uncertainties and significant disturbancesThe improvement in the quality of the final output depends on the control strategy of centerline thickness and interstand tensionThis paper focuses on interstand tension control problem in 5-stand tandem cold rolling mills Tension dynamics can be describedby a nominalmodel perturbed by parametric uncertainties In order to overcome themodel uncertainties and external disturbancessuboptimal 119867

infinand 120583 controllers are proposed and the Hankel-norm approximation is used to reduce the order of 120583 controller

The performance of the proposed controllers is demonstrated by some simulations

1 Introduction

The production of cold rolled coils typically consists ofseparate processes such as annealing pickling cold rollingand skin pass rolling The cold rolling of strip is one of themost important processes in the areas of manufacturing andprocessing of metals Tandem cold mill (TCM) is one typeof cold rolling processes that strip thickness reduction isachieved with a number of nonreversing standsThe produc-tion of this process can be applied for various industries suchas food packaging automobile manufacturing and homeappliance [1 2] Thus the improvement and developmentof new technologies of rolling have become important forresearchers engineers and manufacturers

Tension control is one of the most important processparameters in tandem cold mill (TCM) Changing roll veloc-ity stabilizes the interstand tension [3] Interstand tensionshould be maintained at acceptable levels Large transientvariations in tension may result in a dramatic and physicallydamaging failure of the mill through looping where the striptension falls to zero or tearing of the strip [4 5]

Various advanced methods have been developed to ana-lyze and control rolling mills [6ndash11] Among the previousinvestigations some notes focus on tension control problemin cold rollingmills [12 13]There are numerous uncertainties

and disturbances in the model of tandem cold rolling process[14] Thus the control strategy for cold rolling should berobust with respect to the uncertainties and disturbancesLinear quadratic control strategy (119867

2) guarantees the opti-

mality but it has weakness in the presence of uncertain-ties The suboptimal 119867

infinmethod and 120583 synthesisanalysis

method have been adopted as the robust techniques in manyapproaches and provide systematic design procedures ofrobust controllers for linear system [3 15 16]

To promote tension control of cold rolling mills twoadvanced robust control systems are proposed First a state-space description of full stands with parametric uncertaintyis derived for tension system with 5-stand tandem millsTaking into account the variation of strip thickness Youngrsquosmodulus forward slip and speed actuator time constant themodel of structured uncertainties is presented In the nextstep two robust controllers are designed by using suboptimal119867infin

controller method and 120583 synthesis The order of 120583controller is too large and this makes it necessary to applycontroller order reduction Simulation results show that the120583 robust controller and reduced-order controller have betterrobustness and disturbance attenuation performance than119867infin

controllerThis paper is organized as follows In Section 2 interstand

tension and relations between its parameters are introduced

Hindawi Publishing CorporationJournal of EngineeringVolume 2014 Article ID 409014 13 pageshttpdxdoiorg1011552014409014

2 Journal of Engineering

Stand i Stand i + 1

StripTensiometer

Tension reference

Tensioncontroller

Speedcontroller

Backup rollWork roll(driven)

ΔV

Figure 1 Schematic of conventional tension control for TCM

Stand 1 Stand 2 Stand 3 Stand 4 Stand 5

Tension controllerInterstandtensions

Tension references

Strip

Figure 2 The structure of the advanced tension control system

Then a state-space description is derived for tension systemand it is shown that the perturbation can be representedby parametric uncertainty In Section 3 suboptimal 119867

infin

control and 120583 synthesis method are applied to tension controlproblem and then the order of 120583 controller is reduced TheSection 4 is devoted to comparing robust stability robust per-formance and time domain responses of robust controllersSome simulation results are also given to show the proposedmethods in the presence of model uncertainties and externaldisturbances Finally the concluding remarks are presentedin Section 5

2 Interstand Strip Tension Overview

In tandem cold mill (TCM) strip passes through individualpairs of work rolls Each work roll is supported by a backuproll of a large diameter [2 14] In order to achieve goodtension control tensiometer is the reliable instrument tomeasure the tension between stands Figure 1 shows theconventional tension control by speed where subscript 119894

represents themill stand number for 119894th standThe differencebetween the strip speed at the input to stand 119894+1 and the stripspeed at the output of stand 119894 implies tension change

The structure of conventional controllers is usually basedon single-input-single-output (SISO) control loops whichlimits its capability for improvement To overcome thisproblem and improvement in the performance of tensioncontrol conventional methods are broken and two advancedtension control strategies based on robust strip tension areproposed Figure 2 shows the structure of MIMO tensioncontrol system First stand is chosen as a ldquopivot standrdquo andthe speed of this stand is not trimmed

In this section a state-space description of the nominalinterstand tension perturbed by parametric uncertainty ispresented to be adopted in controller design

21 Mathematical Model To obtain the state-space and out-put equations for interstand tension the theoretical systemequations are introduced here [14] Abbreviations sectiondescribes the symbols that appear in these equations

Journal of Engineering 3

1 minus 120593i+1

1 + fi

E Hi+1 BL

d

+ +

+minus

ΔVi+1

ΔVi

1S ΔTii+1

Figure 3 Block diagram of tension system between adjacent stands

120575Hi+1

MHi+1

Figure 4 Representation of uncertain parameter (119867119894+1) as LFTs

(1) Interstand tension is developed by applying Hookersquoslaw to a length of strip between successive mill stands and isgoverned by the following dynamic

119889Δ (119879119894119894+1

)

119889119905equiv Δ119894119894+1

=EBH119894+1(119881in119894+1 minus 119881out119894)

119871 (1)

(2) During rolling the work roll is elastically flattenedAt the entry to work roll the strip speed is less than theperipheral speed of the work roll and at the exit of the workroll speed is greater than the peripheral speed of the workroll The forward slip ratio and the backward slip ratio aredetermined by the following

(a) Forward slip ratio

119891 =119881out minus 119881

119881 (2)

(b) Backward slip ratio

120593 =119881 minus 119881in119881

(3)

(3) The relationship among the entry and exit thicknessand strip velocity at the entry and exit to stand is called vol-ume continuity and is determined by the following equation

ℎin times 119881in = ℎout times 119881out (4)(4)Thework roll drive equation is modeled (closed-loop)

and is approximated in order one based on typical data

119894=119880V119894

120591V119894minus119881119894

120591V119894 (5)

Equations (1) to (3) are combined to yield the followingequation

Δ119894119894+1

=EBH119894+1

119871((1 minus 120593

119894+1) Δ119881119894+1minus (1 + 119891

119894) Δ119881119894) (6)

According to (6) the tension system between two stands isdepicted in Figure 3

22 Uncertainties in Modeling Based on (5) and (6) vari-ations of four coefficients 119867

119894+1 119864 120593

119894 and 120591V119894 are con-

sidered as parametric uncertainty Parametric uncertaintyis represented by variations of certain system parameterswhich may adversely affect the stability and performance ofa control system In Figure 3 the four constant blocks can bereplaced by upper linear fractional transformation (LFT) Forinstance119867

119894+1may be represented as upper LFT in 119875

119867119894+1and

120575119867119894+1

(minus1 lt 120575119867119894+1

lt 1) and further 119867119894+1

is the so-callednominal value of 119867

119894+1 The LFT framework for 119867

119894+1depicts

in Figure 4

119867119894+1= 119867119894+1(1 + 119875

119867119894+1120575119867119894+1

) = 119865119906(119872119867119894+1

120575119867119894+1

)

119872119867119894+1

= [0 119867

119894+1

119875119867119894+1

119867119894+1

]

(7)

Similarly the other parameters can be represented as upperLFT in 120575

120578119894+1 120575120591V119894 and 120575

119864 It is supposed that 120578 = 1 minus 120593 for

simplification The parameters of full stand tension systemand their tolerances are given in Table 1 [13]

4 Journal of Engineering

Table 1 Nominal values and their tolerance

Parameter Nominal value Unit Uncertainty119864119899

21 times 105 Nmm2

plusmn100119879V119899 01 s plusmn2001198672

135 mm plusmn501198673

077 mm plusmn501198674

05 mm plusmn501198675

033 mm plusmn501205932

03 mdash plusmn1001205933

025 mdash plusmn1001205934

02 mdash plusmn1001205935

02 mdash plusmn100119871 45 m 0119861 1 m 0

23 State-Space Representation Based on (4)ndash(6) the state-space description of tension system with 5-stand yields

1198831015840

= 119860119883 + 1198611119880119906+ 1198612119880

119884119906= 1198621119883 + 119863

11119880119906+ 11986312119880

119884 = 1198622119883 + 119863

21119880119906+ 11986322119880

(8)

The state vector 119883 the control inputs 119880 and 119880119906 and the

output vectors 119884 and 119884119906are defined as

119883 = [Δ1198812 Δ12059012 Δ1198813 Δ12059023 Δ1198814 Δ12059034 Δ1198815 Δ12059045]T

119880 = [Δ119880V2 Δ119880V3 Δ119880V4 Δ119880V5]T

119880119906= [119880120591V2 1198801205782 1198801198642 1198801198672 119880120591V3 1198801205783 1198801198643 1198801198673

119880120591V4 1198801205784 1198801198644 1198801198674 119880120591V5 1198801205785 1198801198645 1198801198675]T

119884119906= [119910120591V2 1199101205782 1199101198642 1199101198672 119910120591V3 1199101205783 1199101198643 1199101198673

119910120591V4 1199101205784 1199101198644 1199101198674 119910120591V5 1199101205785 1199101198645 1199101198675]T

119884 = [Δ11987912 Δ11987923 Δ11987934 Δ11987945]T

(9)

119866Ten denotes the inputoutput dynamics of the tension sys-tem which takes into account the uncertainty of parametersThe state-space representation of119866Ten is (the systemmatricesare presented in Appendix A)

119866Ten =[

[

119860 1198611

1198612

11986211198631111986312

11986221198632111986322

]

]

(10)

The uncertain behavior of the original system can bedescribed by an upper LFT representation

119884 = 119865119880(119866Ten Δ)119880 (11)

With diagonal uncertainty matrix Δ = diag(120575V2 1205751205782 12057511986421205751198672 120575V5 1205751205785 1205751198645 1205751198675) as shown in Figure 5 It should

be noted that the unknown matrix Δ has a fixed structureand in general is a block diagonal Such uncertainty is calledstructured uncertainty

Δ

GTen

U Y

Figure 5 LFT representation of the tension system with uncertain-ties

GTension

Δ

d

Tensioncontroller Wp ep

minusWu

euG

Ref

Figure 6 The block diagram of closed system with performancerequirements

3 Robust Tension Controller Design

This section considers the design of tension control system for5-stand tandem mill For this system open loop and closed-loop system interconnections are introduced and robustcontrollers are designed

31 Requirements to Fulfill Design The aim of robust ten-sion controller design is to achieve and maintain the ten-sion between stands in the presence of disturbances anduncertainties The block diagram of the closed-loop systemincluding the feedback and controller as well as the elementsreflecting the model uncertainty and weighting functionsrelated to performance requirements is shown in Figure 6

In Figure 6 the rectangle with dashed line represents thetransfer matrix 119866 Inside the rectangle is the nominal modelof tension system and the uncertainty matrix Δ that includesthe model uncertainties The variables 119889

1 1198892 1198893 and 119889

4are

the disturbances on the system at the system outputs Theoutput weighting performance vector (119890

119901) control weighting

performance (119890119906) and disturbances vector (119889) are defined as

119890119901= [1198901199011 1198901199012 1198901199013 1198901199014]T

119890119906= [1198901199061 1198901199062 1198901199063 1198901199064]T

119889 = [1198891 1198892 1198893 1198894]T

(12)

In general weighting functions would be used to meet thedesign specifications Finding appropriate weighting func-tions is a crucial step in robust controller design and usuallyneeds a few trials The performance weighting function is

Journal of Engineering 5

103

102

101

100

10minus1

10minus2

10minus3

wminus1p1

Inverse performance weighting function

Frequency (rads)

Mag

nitu

de

10minus4 10minus3 10minus2 10minus1 100 101 102 103 104

Figure 7 Singular values of 1119908119901

101

100

10minus1

10minus5 100 105 1010

wu1

Control weighting function

Frequency (rads)

Mag

nitu

de

Figure 8 Singular values of 119908119906

usually low-pass filter and control weighting function is high-pass filter Design experience will help in choosing weightingfunctions In this design the performanceweighting function119908119901and the control weighting functions 119908

119906119882119901 and119882

119906are

chosen as [13]

119908119901= 0012

1198782

+ 28119878 + 100

051198782 + 40119878 + 001

119908119906= 5

119878 + 10000

119878 + 500000

119882119901= (

1199081199011

0 0 0

0 1199081199012

0 0

0 0 1199081199013

0

0 0 0 1199081199014

)

119882119906= (

1199081199061

0 0 0

0 1199081199062

0 0

0 0 1199081199063

0

0 0 0 1199081199064

)

(13)The singular values of 1119908

119901over the frequency range

[10minus4

104

] are shown in Figure 7 This weighting functionshows that for low frequencies the closed-loop system (thenominal as well as perturbed system) attenuates the outputdisturbance in the ratio of 100 to 001 In other words theeffect of a unit disturbance on the steady-state output shouldbe of the order 10minus2 This performance requirement becomesless stringent with increasing frequency

Also the singular values of 119908119906over the frequency range

[10minus4 104] are shown in Figure 8

32 Open Loop and Closed-Loop System InterconnectionBefore designing the tension controller it is necessary to build

6 Journal of Engineering

Pertin1-16 Pertout1-16

sum

+

+

minus

sum

G

yc1-4

Wp

minusWu

ep1-4

eu1-4

ΔT1-4

Dist1-4

Ref = 0

Figure 9 Structure of the open loop system

Pertin1-16 Pertout1-16

Control1-4

ep1-4

eu1-4

ΔT1-4

Dist1-4 tension ic

Figure 10 Block diagram of open loop system

Pertin1-16 Pertout1-16

T1-4

U1-4

yc1-4

G

K

+

+

+minus

sum

sum

Dist1-4

Ref1-4

Figure 11 Structure of the closed-loop system

interconnection for open-loop and closed-loop system Thestructure of the open-loop system is represented in Figure 9In Figure 10 tension ic denotes the transfer function matrixof open loop system with 24 inputs and 28 outputs

The simulation of closed-loop system with designedcontrollers is based on the structure shown in Figure 11 Theweighting functions 119908

119901and 119908

119906are not included in the block

diagram for obvious reasons

33 119867infin

Controller The interconnection of nominal openloop system is denoted by hin ic as shown in Figure 12

The suboptimal 119867infin

controller minimizes the infinite-normof 119865119897(ℎ119894119899 119894119888 119896) overall stabilizing controller 119870 Note that

119865119897(ℎ119894119899 119894119888 119896) is the transfer function matrix of the nominal

closed-loop system from the disturbances to the errorsTo compute 119867

infinoptimal controller for LTI plant some

conditions are satisfied by plant Sometimes one of theseconditions is not satisfied and the algorithm for computingdoes not work [15 17] When this situation accrued otherrobust control methods are used although some tricks canbe used to satisfy conditions For example adding or sub-tracting epsilon may solve the problem in these conditions

Journal of Engineering 7

K

eDist hin ic

Figure 12 Closed-loop LFTs in119867infindesign

10minus1 100 101 102 103 10401

02

03

04

05

06

Frequency (rads)10minus1 100 101 102 103 104

Frequency (rads)

010

020304050607

Closed-loop complex 120583 controller number 3

Com

plex120583

Closed-loop mixed 120583 controller number 3

Mix

ed120583

Figure 13 120583 plot after third iteration

100

10minus1

10minus2

10minus4 10minus3 10minus2 10minus1 100 101 102 103 104

Robust stability

Frequency (rads)

120583-controllerReduced-order controller

Upp

er b

ound

of120583

Hinfin controller

Figure 14 Comparison of robust stability for three controllers

For this plant 119867infin

controller is numerically computed andfinally the obtained controller is of 20th orderTheminimumvalue of 120574 is 0101

34 120583 Controller The 120583 synthesis is introduced by twoiterative methods 119863-119870 and 120583-119870 iteration For this plant120583 synthesis is executed by the M-file dkit from the RobustControl Toolbox of Matlab which automates the procedureby using 119863-119870 iterations [15] After the third iteration

the maximum value of 120583 is 0567 which indicates that therobust performance has been achieved The final controllerobtained is of 104th order The 120583 plot of the closed-loopsystem is shown in Figure 13

The order of this controller is 104 which makes it difficultin implementation On the other hand high order controllerswill lead to high cost difficult commissioning poor relia-bility and potential problems in maintenance This is theweakness of 120583 controller in this design Usually the order

8 Journal of Engineering

10minus4 10minus3 10minus2 10minus1 100 101 102 103 1040

01

02

03

04

05

06

07

08

09

1Robust performance

Frequency (rads)Hinfin controller120583-controllerReduced-order controller

Upp

er b

ound

of120583

Figure 15 Comparison of robust performance for three controllers

0 10 20 30 40minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref1 (tension ST1-ST2)

Time (s)0 10 20 30 40

minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref2 (tension ST2-ST3)

Time (s)

0 10 20 30 40minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref3 (tension ST3-ST4)

Time (s)0 10 20 30 40

minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref4 (tension ST4-ST5)

Time (s)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Figure 16 Comparison of transient response to reference input

Journal of Engineering 9

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 1

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600Transient response to disturbance 2

Transient response to disturbance 4

Time (s)

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 3

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600

Time (s)

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Figure 17 Comparison of transient response to disturbance signal

can be reduced and reduction of the controller order maylead to deterioration of the closed-loop performance Variousmethods may be used to reduce the order and in this casestudy the Hankel-norm approximation is usedThe reduced-order controller is equal to 13 that leads to a controllermaintaining the robust stability and performance of theclosed-loop system (matrices for reduced-order controllerare presented in Appendix B)

4 Simulation Results

In the previous section robust controllers for tension con-trol system are introduced It is intended here to comparethe robust stability robust performance and time domainresponses of the proposed controllers Since the considereduncertainty is structured verification of the robust stabilityand performance needs the frequency response in terms of120583 values On the other hand controllers should be tested bymeans of 120583 analysis The robust stability and performancedo not maintain if the uncertainty is unstructured whichshows that the 120583 values give less conservative results if furtherinformation is known about the uncertainty

41 Robust Stability The frequency responses of the upperbound of 120583 for 119867

infincontroller and 120583 and reduced-order

controllers are shown in Figure 14Themaximum120583 values for119867infincontroller are equal to 07801 and the maximum 120583 values

for 120583 and reduced-order controllers are 0202 Thereforethe robust stability of control systems is achieved and thesystems with the 120583 controller and reduced-order controllersare characterized with better robust stability

42 Robust Performance The robust performance of controlsystems is plotted in Figure 15 The maximum 120583 valuesfor 119867

infincontroller are 099 for 120583 controller 0597 and for

reduced-order controller 069 which implies the robust 120583controller and reduced-order controller achieve better robustperformance than119867

infincontroller

43 Time Domain Simulation The transient responses of theclosed-loop system to the reference input and the disturbancesignal are shown in Figures 16 and 17 respectively The 120583 andthe reduced-order controllers give a better transient responsewith smaller overshoot than the119867

infincontroller In summary

10 Journal of Engineering

the 120583 and the reduced-order controllers lead to better trade-off between the requirements in terms of transient responsedisturbance rejection and robustness

5 Conclusion

In cold rolling the strip tension should be controlled forbetter quality of products For cold rolling with 5 standsthe state-space representation perturbed with parametricuncertainties is derived and further 119867

infincontroller and

120583 controller based on robust control theory are designedTension control is obtained by means of the proposed robustcontrollers in the presence of uncertainties and disturbancesThe order of the 120583 controller is large which requires thecontroller order reduction to preserve the closed-loop per-formance and robustness while making implementation ofthe controller easier andmore reliableThe simulation resultsshow that control systems achieve robust stability and robust

performance The comparison of the robust stability androbust performance for controllers as well as the comparisonof the corresponding transient responses shows that the 120583controller and its reduced-order are preferable although thereduced-order controller would be usually preferred dueto its lower order It should be mentioned that in suchimportant industrial process most commissioning personnelhave very little or no background in advanced control theoryespecially in the theory and application of119867

infinrobust control

techniques and are accustomed to single-input-single-output(SISO) proportional-integral (PI) type control loops Thesesimpler configurations are easy to tune at commissioningduring actual operation

Appendices

A The System Matrices

Consider

119860 =

[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

44200 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

minus44200 0 26000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 minus26800 0 18600 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 minus18400 0 12200 0

]]]]]]]]]]

]

1198612=

[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 10 0

0 0 0 0

0 0 0 10

0 0 0 0

]]]]]]]]]]]]

]

1198611=

[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 6280 003 00011 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 minus6220 0 0 0 3600 0017 00011 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 minus3600 0 0 0 2300 0011 0011 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 minus2300 0 0 0 1540 0073 0011

]]]]]]]]]]

]

1198621=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

07

147000

0

0

0

0

0

0

0

0

0

0

0

0

0

0

199000 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

0 0 075 0 0 0 0 0

minus257000 0 157000 0 0 0 0 0

minus197000 0 120000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 0 0 08 0 0 0

0 0 minus242000 0 168000 0 0 0

0 0 minus1210000 0 84000 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 0 0 08 0

0 0 0 0 minus253000 0 168000 0

0 0 0 0 minus84000 0 55000 0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

1198622=

[[[

[

0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

]]]

]

Journal of Engineering 11

11986311=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 21000 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 28300 0135 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 36700 0 0 0 21000 0 0 0 0 0 0 0 0 0 0

0 minus28200 0 0 0 16700 0077 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 minus32300 0 0 0 21000 0 0 0 0 0 0

0 0 0 0 0 minus16100 0 005 10500 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 minus31700 0 0 0 21000 0 0

0 0 0 0 0 0 0 0 0 minus10400 0 0 0 6930 0033 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986312=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 10

0 0 0 0

0 0 0 0

0 0 0 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(A1)

B Matrices for Reduced-Order Controller

Consider

ak =

[[[[[[[[[[[[[[[[[[[[

[

minus13534 9671 2455 minus1674 minus122 155 minus048 299 109 014 minus008 minus019 011

0 minus2734 523 1642 147 minus068 04 minus136 minus065 0005 007 01 minus0048

0 minus2872 minus277 03 minus1267 minus021 005 minus065 003 minus01 009 002 004

0 0 0 minus611 minus184 018 minus029 031 042 minus009 minus002 minus002 0018

0 0 0 145 minus662 minus008 019 minus011 minus006 minus002 0 0003 0

0 0 0 0 0 minus0008 006 minus002 minus0008 minus0001 0 0002 0001

0 0 0 0 0 minus006 minus0004 0008 0008 minus0002 0 0 0001

0 0 0 0 0 0 0 minus002 002 minus0004 minus0001 0 0

0 0 0 0 0 0 0 minus004 minus0008 0002 minus0003 0 0

0 0 0 0 0 0 0 0 0 minus0003 0 0 0

0 0 0 0 0 0 0 0 0 0 minus0002 0 0

0 0 0 0 0 0 0 0 0 0 0 minus0002 0

0 0 0 0 0 0 0 0 0 0 0 0 minus0002

]]]]]]]]]]]]]]]]]]]]

]

12 Journal of Engineering

bk =

[[[[[[[[[[[[[[[[[[[[

[

015 008 002 004

minus004 minus003 minus001 minus001

minus005 minus0002 minus002 0

minus0008 001 001 001

minus0006 0008 0003 minus001

0 0 0 0

0 0 0 0

minus0001 0 0 0

0 0 0 0

0 0 0 0

0001 0 0 0

0 0 0 minus0001

0 0 0 minus0001

]]]]]]]]]]]]]]]]]]]]

]

ck =[[[

[

minus0001 minus0006 0 0001 minus0001 0 0 0 0 0 0 0 0

006 minus002 minus0009 0001 0007 0 0 0 0 0 0 0001 0

0009 minus004 minus003 001 0 0 0 0 0 0 0001 0 0001

013 minus008 minus005 001 0 0 0001 minus0002 0 0 0001 minus0001 minus0001

]]]

]

dk =[[[

[

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

]]]

]

(B1)

Abbreviations

119881 Work roll linear velocity119880V Speed actuator input120591V Speed actuator time constant119881in Input strip velocity119881out Output strip velocityℎin Stand input thicknessℎout Stand output thickness119891 Forward slip ratio120593 Backward slip ratio119864 Youngrsquos modulus119861 Width of strip119867119894+1

Strip thickness between stand 119894 and stand 119894 + 1119871 Length between adjacent stands119879119894119894+1

Strip tension between stand 119894 and stand 119894 + 1

Conflict of Interests

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

References

[1] W S Levine Control System Applications Taylor amp FrancisBoca Raton Fla USA 2011

[2] J Pittner N S Samaras and M A Simaan ldquoA simple rollingmillmodel with linear quadratic optimal controllerrdquo inProceed-ings of the 37th IAS Annual Meeting and World Conference onIndustrial Applications of Electrical Energy pp 142ndash149 October2002

[3] H R Koofigar F Sheikholeslam and S Hosseinnia ldquoUnifiedgauge-tension control in cold rolling mills a robust regulationtechniquerdquo International Journal of Precision Engineering andManufacturing vol 12 no 3 pp 393ndash403 2011

[4] E J M Geddes and I Postlethwaite ldquoImprovements in productquality in tandem cold rolling using robust multivariablecontrolrdquo IEEE Transactions on Control Systems Technology vol6 no 2 pp 257ndash269 1998

[5] E J M Geddes and I Postlethwaite ldquoMultivariable control ofa high performance tandem cold rolling millrdquo in Proceedings ofthe International Conference on Control (CONTROL rsquo94) vol 1pp 202ndash207 Coventry UK March 1994

[6] E Arinton S Caraman and J Korbicz ldquoNeural networks formodelling and fault detection of the inter-stand strip tension ofa cold tandem millrdquo Control Engineering Practice vol 20 no 7pp 684ndash694 2012

[7] KMTakami JMahmoudi andEDahlquist ldquoAdaptive controlof cold rolling system in electrical strips production systemwithonline-offline predictorsrdquo International Journal of AdvancedManufacturing Technology vol 50 no 9ndash12 pp 917ndash930 2010

[8] G Pin V Francesconi F A Cuzzola and T Parisini ldquoAdaptivetask-space metal strip-flatness control in cold multi-roll millstandsrdquo Journal of Process Control vol 23 no 2 pp 108ndash1192013

[9] J Pittner and M A Simaan ldquoAn optimal control method forimprovement in tandem cold metal rollingrdquo in Proceedings ofthe IEEE 42nd AnnualMeeting Industry Applications Conference(IAS rsquo07) pp 382ndash389 September 2007

[10] J R Pittner Pointwise linear quadratic optimal control of atandem cold rolling mill [PhD thesis] School of EngineeringUniversity of Pittsburgh 2006

[11] L E Zarate ldquoPredictive model for the cold rolling processthrough sensitivity factors via neural networksrdquo Journal of the

Journal of Engineering 13

Brazilian Society ofMechanical Sciences and Engineering vol 28no 1 pp 111ndash117 2006

[12] S-H Song and S-K Sul ldquoA new tension controller for con-tinuous strip processing linerdquo IEEE Transactions on IndustryApplications vol 36 no 2 pp 633ndash639 2000

[13] X Zhang and Q Zhang ldquoRobust control of strip tension fortandem cold rolling millrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 2390ndash2393 Yantai China July2011

[14] J Pittner and M A Simaan Tandem Cold Metal Rolling MillControl Using Practical Advanced Methods Springer LondonUK 2011

[15] DGu PH Petkov andMKonstantinovRobust Control Designwith MATLAB Springer London UK 2005

[16] Z Xiaofeng and ZQingdong ldquoRobust control of strip thicknessfor cold rolling millrdquo in Informatics in Control Automation andRobotics vol 133 of Lecture Notes in Electrical Engineering pp777ndash785 Springer Berlin Germany 2012

[17] K Zohu and J C Doyle Essentials of Robust Control PrenticeHall 1997

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International Journal of

2 Journal of Engineering

Stand i Stand i + 1

StripTensiometer

Tension reference

Tensioncontroller

Speedcontroller

Backup rollWork roll(driven)

ΔV

Figure 1 Schematic of conventional tension control for TCM

Stand 1 Stand 2 Stand 3 Stand 4 Stand 5

Tension controllerInterstandtensions

Tension references

Strip

Figure 2 The structure of the advanced tension control system

Then a state-space description is derived for tension systemand it is shown that the perturbation can be representedby parametric uncertainty In Section 3 suboptimal 119867

infin

control and 120583 synthesis method are applied to tension controlproblem and then the order of 120583 controller is reduced TheSection 4 is devoted to comparing robust stability robust per-formance and time domain responses of robust controllersSome simulation results are also given to show the proposedmethods in the presence of model uncertainties and externaldisturbances Finally the concluding remarks are presentedin Section 5

2 Interstand Strip Tension Overview

In tandem cold mill (TCM) strip passes through individualpairs of work rolls Each work roll is supported by a backuproll of a large diameter [2 14] In order to achieve goodtension control tensiometer is the reliable instrument tomeasure the tension between stands Figure 1 shows theconventional tension control by speed where subscript 119894

represents themill stand number for 119894th standThe differencebetween the strip speed at the input to stand 119894+1 and the stripspeed at the output of stand 119894 implies tension change

The structure of conventional controllers is usually basedon single-input-single-output (SISO) control loops whichlimits its capability for improvement To overcome thisproblem and improvement in the performance of tensioncontrol conventional methods are broken and two advancedtension control strategies based on robust strip tension areproposed Figure 2 shows the structure of MIMO tensioncontrol system First stand is chosen as a ldquopivot standrdquo andthe speed of this stand is not trimmed

In this section a state-space description of the nominalinterstand tension perturbed by parametric uncertainty ispresented to be adopted in controller design

21 Mathematical Model To obtain the state-space and out-put equations for interstand tension the theoretical systemequations are introduced here [14] Abbreviations sectiondescribes the symbols that appear in these equations

Journal of Engineering 3

1 minus 120593i+1

1 + fi

E Hi+1 BL

d

+ +

+minus

ΔVi+1

ΔVi

1S ΔTii+1

Figure 3 Block diagram of tension system between adjacent stands

120575Hi+1

MHi+1

Figure 4 Representation of uncertain parameter (119867119894+1) as LFTs

(1) Interstand tension is developed by applying Hookersquoslaw to a length of strip between successive mill stands and isgoverned by the following dynamic

119889Δ (119879119894119894+1

)

119889119905equiv Δ119894119894+1

=EBH119894+1(119881in119894+1 minus 119881out119894)

119871 (1)

(2) During rolling the work roll is elastically flattenedAt the entry to work roll the strip speed is less than theperipheral speed of the work roll and at the exit of the workroll speed is greater than the peripheral speed of the workroll The forward slip ratio and the backward slip ratio aredetermined by the following

(a) Forward slip ratio

119891 =119881out minus 119881

119881 (2)

(b) Backward slip ratio

120593 =119881 minus 119881in119881

(3)

(3) The relationship among the entry and exit thicknessand strip velocity at the entry and exit to stand is called vol-ume continuity and is determined by the following equation

ℎin times 119881in = ℎout times 119881out (4)(4)Thework roll drive equation is modeled (closed-loop)

and is approximated in order one based on typical data

119894=119880V119894

120591V119894minus119881119894

120591V119894 (5)

Equations (1) to (3) are combined to yield the followingequation

Δ119894119894+1

=EBH119894+1

119871((1 minus 120593

119894+1) Δ119881119894+1minus (1 + 119891

119894) Δ119881119894) (6)

According to (6) the tension system between two stands isdepicted in Figure 3

22 Uncertainties in Modeling Based on (5) and (6) vari-ations of four coefficients 119867

119894+1 119864 120593

119894 and 120591V119894 are con-

sidered as parametric uncertainty Parametric uncertaintyis represented by variations of certain system parameterswhich may adversely affect the stability and performance ofa control system In Figure 3 the four constant blocks can bereplaced by upper linear fractional transformation (LFT) Forinstance119867

119894+1may be represented as upper LFT in 119875

119867119894+1and

120575119867119894+1

(minus1 lt 120575119867119894+1

lt 1) and further 119867119894+1

is the so-callednominal value of 119867

119894+1 The LFT framework for 119867

119894+1depicts

in Figure 4

119867119894+1= 119867119894+1(1 + 119875

119867119894+1120575119867119894+1

) = 119865119906(119872119867119894+1

120575119867119894+1

)

119872119867119894+1

= [0 119867

119894+1

119875119867119894+1

119867119894+1

]

(7)

Similarly the other parameters can be represented as upperLFT in 120575

120578119894+1 120575120591V119894 and 120575

119864 It is supposed that 120578 = 1 minus 120593 for

simplification The parameters of full stand tension systemand their tolerances are given in Table 1 [13]

4 Journal of Engineering

Table 1 Nominal values and their tolerance

Parameter Nominal value Unit Uncertainty119864119899

21 times 105 Nmm2

plusmn100119879V119899 01 s plusmn2001198672

135 mm plusmn501198673

077 mm plusmn501198674

05 mm plusmn501198675

033 mm plusmn501205932

03 mdash plusmn1001205933

025 mdash plusmn1001205934

02 mdash plusmn1001205935

02 mdash plusmn100119871 45 m 0119861 1 m 0

23 State-Space Representation Based on (4)ndash(6) the state-space description of tension system with 5-stand yields

1198831015840

= 119860119883 + 1198611119880119906+ 1198612119880

119884119906= 1198621119883 + 119863

11119880119906+ 11986312119880

119884 = 1198622119883 + 119863

21119880119906+ 11986322119880

(8)

The state vector 119883 the control inputs 119880 and 119880119906 and the

output vectors 119884 and 119884119906are defined as

119883 = [Δ1198812 Δ12059012 Δ1198813 Δ12059023 Δ1198814 Δ12059034 Δ1198815 Δ12059045]T

119880 = [Δ119880V2 Δ119880V3 Δ119880V4 Δ119880V5]T

119880119906= [119880120591V2 1198801205782 1198801198642 1198801198672 119880120591V3 1198801205783 1198801198643 1198801198673

119880120591V4 1198801205784 1198801198644 1198801198674 119880120591V5 1198801205785 1198801198645 1198801198675]T

119884119906= [119910120591V2 1199101205782 1199101198642 1199101198672 119910120591V3 1199101205783 1199101198643 1199101198673

119910120591V4 1199101205784 1199101198644 1199101198674 119910120591V5 1199101205785 1199101198645 1199101198675]T

119884 = [Δ11987912 Δ11987923 Δ11987934 Δ11987945]T

(9)

119866Ten denotes the inputoutput dynamics of the tension sys-tem which takes into account the uncertainty of parametersThe state-space representation of119866Ten is (the systemmatricesare presented in Appendix A)

119866Ten =[

[

119860 1198611

1198612

11986211198631111986312

11986221198632111986322

]

]

(10)

The uncertain behavior of the original system can bedescribed by an upper LFT representation

119884 = 119865119880(119866Ten Δ)119880 (11)

With diagonal uncertainty matrix Δ = diag(120575V2 1205751205782 12057511986421205751198672 120575V5 1205751205785 1205751198645 1205751198675) as shown in Figure 5 It should

be noted that the unknown matrix Δ has a fixed structureand in general is a block diagonal Such uncertainty is calledstructured uncertainty

Δ

GTen

U Y

Figure 5 LFT representation of the tension system with uncertain-ties

GTension

Δ

d

Tensioncontroller Wp ep

minusWu

euG

Ref

Figure 6 The block diagram of closed system with performancerequirements

3 Robust Tension Controller Design

This section considers the design of tension control system for5-stand tandem mill For this system open loop and closed-loop system interconnections are introduced and robustcontrollers are designed

31 Requirements to Fulfill Design The aim of robust ten-sion controller design is to achieve and maintain the ten-sion between stands in the presence of disturbances anduncertainties The block diagram of the closed-loop systemincluding the feedback and controller as well as the elementsreflecting the model uncertainty and weighting functionsrelated to performance requirements is shown in Figure 6

In Figure 6 the rectangle with dashed line represents thetransfer matrix 119866 Inside the rectangle is the nominal modelof tension system and the uncertainty matrix Δ that includesthe model uncertainties The variables 119889

1 1198892 1198893 and 119889

4are

the disturbances on the system at the system outputs Theoutput weighting performance vector (119890

119901) control weighting

performance (119890119906) and disturbances vector (119889) are defined as

119890119901= [1198901199011 1198901199012 1198901199013 1198901199014]T

119890119906= [1198901199061 1198901199062 1198901199063 1198901199064]T

119889 = [1198891 1198892 1198893 1198894]T

(12)

In general weighting functions would be used to meet thedesign specifications Finding appropriate weighting func-tions is a crucial step in robust controller design and usuallyneeds a few trials The performance weighting function is

Journal of Engineering 5

103

102

101

100

10minus1

10minus2

10minus3

wminus1p1

Inverse performance weighting function

Frequency (rads)

Mag

nitu

de

10minus4 10minus3 10minus2 10minus1 100 101 102 103 104

Figure 7 Singular values of 1119908119901

101

100

10minus1

10minus5 100 105 1010

wu1

Control weighting function

Frequency (rads)

Mag

nitu

de

Figure 8 Singular values of 119908119906

usually low-pass filter and control weighting function is high-pass filter Design experience will help in choosing weightingfunctions In this design the performanceweighting function119908119901and the control weighting functions 119908

119906119882119901 and119882

119906are

chosen as [13]

119908119901= 0012

1198782

+ 28119878 + 100

051198782 + 40119878 + 001

119908119906= 5

119878 + 10000

119878 + 500000

119882119901= (

1199081199011

0 0 0

0 1199081199012

0 0

0 0 1199081199013

0

0 0 0 1199081199014

)

119882119906= (

1199081199061

0 0 0

0 1199081199062

0 0

0 0 1199081199063

0

0 0 0 1199081199064

)

(13)The singular values of 1119908

119901over the frequency range

[10minus4

104

] are shown in Figure 7 This weighting functionshows that for low frequencies the closed-loop system (thenominal as well as perturbed system) attenuates the outputdisturbance in the ratio of 100 to 001 In other words theeffect of a unit disturbance on the steady-state output shouldbe of the order 10minus2 This performance requirement becomesless stringent with increasing frequency

Also the singular values of 119908119906over the frequency range

[10minus4 104] are shown in Figure 8

32 Open Loop and Closed-Loop System InterconnectionBefore designing the tension controller it is necessary to build

6 Journal of Engineering

Pertin1-16 Pertout1-16

sum

+

+

minus

sum

G

yc1-4

Wp

minusWu

ep1-4

eu1-4

ΔT1-4

Dist1-4

Ref = 0

Figure 9 Structure of the open loop system

Pertin1-16 Pertout1-16

Control1-4

ep1-4

eu1-4

ΔT1-4

Dist1-4 tension ic

Figure 10 Block diagram of open loop system

Pertin1-16 Pertout1-16

T1-4

U1-4

yc1-4

G

K

+

+

+minus

sum

sum

Dist1-4

Ref1-4

Figure 11 Structure of the closed-loop system

interconnection for open-loop and closed-loop system Thestructure of the open-loop system is represented in Figure 9In Figure 10 tension ic denotes the transfer function matrixof open loop system with 24 inputs and 28 outputs

The simulation of closed-loop system with designedcontrollers is based on the structure shown in Figure 11 Theweighting functions 119908

119901and 119908

119906are not included in the block

diagram for obvious reasons

33 119867infin

Controller The interconnection of nominal openloop system is denoted by hin ic as shown in Figure 12

The suboptimal 119867infin

controller minimizes the infinite-normof 119865119897(ℎ119894119899 119894119888 119896) overall stabilizing controller 119870 Note that

119865119897(ℎ119894119899 119894119888 119896) is the transfer function matrix of the nominal

closed-loop system from the disturbances to the errorsTo compute 119867

infinoptimal controller for LTI plant some

conditions are satisfied by plant Sometimes one of theseconditions is not satisfied and the algorithm for computingdoes not work [15 17] When this situation accrued otherrobust control methods are used although some tricks canbe used to satisfy conditions For example adding or sub-tracting epsilon may solve the problem in these conditions

Journal of Engineering 7

K

eDist hin ic

Figure 12 Closed-loop LFTs in119867infindesign

10minus1 100 101 102 103 10401

02

03

04

05

06

Frequency (rads)10minus1 100 101 102 103 104

Frequency (rads)

010

020304050607

Closed-loop complex 120583 controller number 3

Com

plex120583

Closed-loop mixed 120583 controller number 3

Mix

ed120583

Figure 13 120583 plot after third iteration

100

10minus1

10minus2

10minus4 10minus3 10minus2 10minus1 100 101 102 103 104

Robust stability

Frequency (rads)

120583-controllerReduced-order controller

Upp

er b

ound

of120583

Hinfin controller

Figure 14 Comparison of robust stability for three controllers

For this plant 119867infin

controller is numerically computed andfinally the obtained controller is of 20th orderTheminimumvalue of 120574 is 0101

34 120583 Controller The 120583 synthesis is introduced by twoiterative methods 119863-119870 and 120583-119870 iteration For this plant120583 synthesis is executed by the M-file dkit from the RobustControl Toolbox of Matlab which automates the procedureby using 119863-119870 iterations [15] After the third iteration

the maximum value of 120583 is 0567 which indicates that therobust performance has been achieved The final controllerobtained is of 104th order The 120583 plot of the closed-loopsystem is shown in Figure 13

The order of this controller is 104 which makes it difficultin implementation On the other hand high order controllerswill lead to high cost difficult commissioning poor relia-bility and potential problems in maintenance This is theweakness of 120583 controller in this design Usually the order

8 Journal of Engineering

10minus4 10minus3 10minus2 10minus1 100 101 102 103 1040

01

02

03

04

05

06

07

08

09

1Robust performance

Frequency (rads)Hinfin controller120583-controllerReduced-order controller

Upp

er b

ound

of120583

Figure 15 Comparison of robust performance for three controllers

0 10 20 30 40minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref1 (tension ST1-ST2)

Time (s)0 10 20 30 40

minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref2 (tension ST2-ST3)

Time (s)

0 10 20 30 40minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref3 (tension ST3-ST4)

Time (s)0 10 20 30 40

minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref4 (tension ST4-ST5)

Time (s)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Figure 16 Comparison of transient response to reference input

Journal of Engineering 9

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 1

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600Transient response to disturbance 2

Transient response to disturbance 4

Time (s)

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 3

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600

Time (s)

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Figure 17 Comparison of transient response to disturbance signal

can be reduced and reduction of the controller order maylead to deterioration of the closed-loop performance Variousmethods may be used to reduce the order and in this casestudy the Hankel-norm approximation is usedThe reduced-order controller is equal to 13 that leads to a controllermaintaining the robust stability and performance of theclosed-loop system (matrices for reduced-order controllerare presented in Appendix B)

4 Simulation Results

In the previous section robust controllers for tension con-trol system are introduced It is intended here to comparethe robust stability robust performance and time domainresponses of the proposed controllers Since the considereduncertainty is structured verification of the robust stabilityand performance needs the frequency response in terms of120583 values On the other hand controllers should be tested bymeans of 120583 analysis The robust stability and performancedo not maintain if the uncertainty is unstructured whichshows that the 120583 values give less conservative results if furtherinformation is known about the uncertainty

41 Robust Stability The frequency responses of the upperbound of 120583 for 119867

infincontroller and 120583 and reduced-order

controllers are shown in Figure 14Themaximum120583 values for119867infincontroller are equal to 07801 and the maximum 120583 values

for 120583 and reduced-order controllers are 0202 Thereforethe robust stability of control systems is achieved and thesystems with the 120583 controller and reduced-order controllersare characterized with better robust stability

42 Robust Performance The robust performance of controlsystems is plotted in Figure 15 The maximum 120583 valuesfor 119867

infincontroller are 099 for 120583 controller 0597 and for

reduced-order controller 069 which implies the robust 120583controller and reduced-order controller achieve better robustperformance than119867

infincontroller

43 Time Domain Simulation The transient responses of theclosed-loop system to the reference input and the disturbancesignal are shown in Figures 16 and 17 respectively The 120583 andthe reduced-order controllers give a better transient responsewith smaller overshoot than the119867

infincontroller In summary

10 Journal of Engineering

the 120583 and the reduced-order controllers lead to better trade-off between the requirements in terms of transient responsedisturbance rejection and robustness

5 Conclusion

In cold rolling the strip tension should be controlled forbetter quality of products For cold rolling with 5 standsthe state-space representation perturbed with parametricuncertainties is derived and further 119867

infincontroller and

120583 controller based on robust control theory are designedTension control is obtained by means of the proposed robustcontrollers in the presence of uncertainties and disturbancesThe order of the 120583 controller is large which requires thecontroller order reduction to preserve the closed-loop per-formance and robustness while making implementation ofthe controller easier andmore reliableThe simulation resultsshow that control systems achieve robust stability and robust

performance The comparison of the robust stability androbust performance for controllers as well as the comparisonof the corresponding transient responses shows that the 120583controller and its reduced-order are preferable although thereduced-order controller would be usually preferred dueto its lower order It should be mentioned that in suchimportant industrial process most commissioning personnelhave very little or no background in advanced control theoryespecially in the theory and application of119867

infinrobust control

techniques and are accustomed to single-input-single-output(SISO) proportional-integral (PI) type control loops Thesesimpler configurations are easy to tune at commissioningduring actual operation

Appendices

A The System Matrices

Consider

119860 =

[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

44200 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

minus44200 0 26000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 minus26800 0 18600 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 minus18400 0 12200 0

]]]]]]]]]]

]

1198612=

[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 10 0

0 0 0 0

0 0 0 10

0 0 0 0

]]]]]]]]]]]]

]

1198611=

[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 6280 003 00011 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 minus6220 0 0 0 3600 0017 00011 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 minus3600 0 0 0 2300 0011 0011 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 minus2300 0 0 0 1540 0073 0011

]]]]]]]]]]

]

1198621=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

07

147000

0

0

0

0

0

0

0

0

0

0

0

0

0

0

199000 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

0 0 075 0 0 0 0 0

minus257000 0 157000 0 0 0 0 0

minus197000 0 120000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 0 0 08 0 0 0

0 0 minus242000 0 168000 0 0 0

0 0 minus1210000 0 84000 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 0 0 08 0

0 0 0 0 minus253000 0 168000 0

0 0 0 0 minus84000 0 55000 0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

1198622=

[[[

[

0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

]]]

]

Journal of Engineering 11

11986311=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 21000 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 28300 0135 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 36700 0 0 0 21000 0 0 0 0 0 0 0 0 0 0

0 minus28200 0 0 0 16700 0077 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 minus32300 0 0 0 21000 0 0 0 0 0 0

0 0 0 0 0 minus16100 0 005 10500 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 minus31700 0 0 0 21000 0 0

0 0 0 0 0 0 0 0 0 minus10400 0 0 0 6930 0033 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986312=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 10

0 0 0 0

0 0 0 0

0 0 0 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(A1)

B Matrices for Reduced-Order Controller

Consider

ak =

[[[[[[[[[[[[[[[[[[[[

[

minus13534 9671 2455 minus1674 minus122 155 minus048 299 109 014 minus008 minus019 011

0 minus2734 523 1642 147 minus068 04 minus136 minus065 0005 007 01 minus0048

0 minus2872 minus277 03 minus1267 minus021 005 minus065 003 minus01 009 002 004

0 0 0 minus611 minus184 018 minus029 031 042 minus009 minus002 minus002 0018

0 0 0 145 minus662 minus008 019 minus011 minus006 minus002 0 0003 0

0 0 0 0 0 minus0008 006 minus002 minus0008 minus0001 0 0002 0001

0 0 0 0 0 minus006 minus0004 0008 0008 minus0002 0 0 0001

0 0 0 0 0 0 0 minus002 002 minus0004 minus0001 0 0

0 0 0 0 0 0 0 minus004 minus0008 0002 minus0003 0 0

0 0 0 0 0 0 0 0 0 minus0003 0 0 0

0 0 0 0 0 0 0 0 0 0 minus0002 0 0

0 0 0 0 0 0 0 0 0 0 0 minus0002 0

0 0 0 0 0 0 0 0 0 0 0 0 minus0002

]]]]]]]]]]]]]]]]]]]]

]

12 Journal of Engineering

bk =

[[[[[[[[[[[[[[[[[[[[

[

015 008 002 004

minus004 minus003 minus001 minus001

minus005 minus0002 minus002 0

minus0008 001 001 001

minus0006 0008 0003 minus001

0 0 0 0

0 0 0 0

minus0001 0 0 0

0 0 0 0

0 0 0 0

0001 0 0 0

0 0 0 minus0001

0 0 0 minus0001

]]]]]]]]]]]]]]]]]]]]

]

ck =[[[

[

minus0001 minus0006 0 0001 minus0001 0 0 0 0 0 0 0 0

006 minus002 minus0009 0001 0007 0 0 0 0 0 0 0001 0

0009 minus004 minus003 001 0 0 0 0 0 0 0001 0 0001

013 minus008 minus005 001 0 0 0001 minus0002 0 0 0001 minus0001 minus0001

]]]

]

dk =[[[

[

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

]]]

]

(B1)

Abbreviations

119881 Work roll linear velocity119880V Speed actuator input120591V Speed actuator time constant119881in Input strip velocity119881out Output strip velocityℎin Stand input thicknessℎout Stand output thickness119891 Forward slip ratio120593 Backward slip ratio119864 Youngrsquos modulus119861 Width of strip119867119894+1

Strip thickness between stand 119894 and stand 119894 + 1119871 Length between adjacent stands119879119894119894+1

Strip tension between stand 119894 and stand 119894 + 1

Conflict of Interests

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

References

[1] W S Levine Control System Applications Taylor amp FrancisBoca Raton Fla USA 2011

[2] J Pittner N S Samaras and M A Simaan ldquoA simple rollingmillmodel with linear quadratic optimal controllerrdquo inProceed-ings of the 37th IAS Annual Meeting and World Conference onIndustrial Applications of Electrical Energy pp 142ndash149 October2002

[3] H R Koofigar F Sheikholeslam and S Hosseinnia ldquoUnifiedgauge-tension control in cold rolling mills a robust regulationtechniquerdquo International Journal of Precision Engineering andManufacturing vol 12 no 3 pp 393ndash403 2011

[4] E J M Geddes and I Postlethwaite ldquoImprovements in productquality in tandem cold rolling using robust multivariablecontrolrdquo IEEE Transactions on Control Systems Technology vol6 no 2 pp 257ndash269 1998

[5] E J M Geddes and I Postlethwaite ldquoMultivariable control ofa high performance tandem cold rolling millrdquo in Proceedings ofthe International Conference on Control (CONTROL rsquo94) vol 1pp 202ndash207 Coventry UK March 1994

[6] E Arinton S Caraman and J Korbicz ldquoNeural networks formodelling and fault detection of the inter-stand strip tension ofa cold tandem millrdquo Control Engineering Practice vol 20 no 7pp 684ndash694 2012

[7] KMTakami JMahmoudi andEDahlquist ldquoAdaptive controlof cold rolling system in electrical strips production systemwithonline-offline predictorsrdquo International Journal of AdvancedManufacturing Technology vol 50 no 9ndash12 pp 917ndash930 2010

[8] G Pin V Francesconi F A Cuzzola and T Parisini ldquoAdaptivetask-space metal strip-flatness control in cold multi-roll millstandsrdquo Journal of Process Control vol 23 no 2 pp 108ndash1192013

[9] J Pittner and M A Simaan ldquoAn optimal control method forimprovement in tandem cold metal rollingrdquo in Proceedings ofthe IEEE 42nd AnnualMeeting Industry Applications Conference(IAS rsquo07) pp 382ndash389 September 2007

[10] J R Pittner Pointwise linear quadratic optimal control of atandem cold rolling mill [PhD thesis] School of EngineeringUniversity of Pittsburgh 2006

[11] L E Zarate ldquoPredictive model for the cold rolling processthrough sensitivity factors via neural networksrdquo Journal of the

Journal of Engineering 13

Brazilian Society ofMechanical Sciences and Engineering vol 28no 1 pp 111ndash117 2006

[12] S-H Song and S-K Sul ldquoA new tension controller for con-tinuous strip processing linerdquo IEEE Transactions on IndustryApplications vol 36 no 2 pp 633ndash639 2000

[13] X Zhang and Q Zhang ldquoRobust control of strip tension fortandem cold rolling millrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 2390ndash2393 Yantai China July2011

[14] J Pittner and M A Simaan Tandem Cold Metal Rolling MillControl Using Practical Advanced Methods Springer LondonUK 2011

[15] DGu PH Petkov andMKonstantinovRobust Control Designwith MATLAB Springer London UK 2005

[16] Z Xiaofeng and ZQingdong ldquoRobust control of strip thicknessfor cold rolling millrdquo in Informatics in Control Automation andRobotics vol 133 of Lecture Notes in Electrical Engineering pp777ndash785 Springer Berlin Germany 2012

[17] K Zohu and J C Doyle Essentials of Robust Control PrenticeHall 1997

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Journal of Engineering 3

1 minus 120593i+1

1 + fi

E Hi+1 BL

d

+ +

+minus

ΔVi+1

ΔVi

1S ΔTii+1

Figure 3 Block diagram of tension system between adjacent stands

120575Hi+1

MHi+1

Figure 4 Representation of uncertain parameter (119867119894+1) as LFTs

(1) Interstand tension is developed by applying Hookersquoslaw to a length of strip between successive mill stands and isgoverned by the following dynamic

119889Δ (119879119894119894+1

)

119889119905equiv Δ119894119894+1

=EBH119894+1(119881in119894+1 minus 119881out119894)

119871 (1)

(2) During rolling the work roll is elastically flattenedAt the entry to work roll the strip speed is less than theperipheral speed of the work roll and at the exit of the workroll speed is greater than the peripheral speed of the workroll The forward slip ratio and the backward slip ratio aredetermined by the following

(a) Forward slip ratio

119891 =119881out minus 119881

119881 (2)

(b) Backward slip ratio

120593 =119881 minus 119881in119881

(3)

(3) The relationship among the entry and exit thicknessand strip velocity at the entry and exit to stand is called vol-ume continuity and is determined by the following equation

ℎin times 119881in = ℎout times 119881out (4)(4)Thework roll drive equation is modeled (closed-loop)

and is approximated in order one based on typical data

119894=119880V119894

120591V119894minus119881119894

120591V119894 (5)

Equations (1) to (3) are combined to yield the followingequation

Δ119894119894+1

=EBH119894+1

119871((1 minus 120593

119894+1) Δ119881119894+1minus (1 + 119891

119894) Δ119881119894) (6)

According to (6) the tension system between two stands isdepicted in Figure 3

22 Uncertainties in Modeling Based on (5) and (6) vari-ations of four coefficients 119867

119894+1 119864 120593

119894 and 120591V119894 are con-

sidered as parametric uncertainty Parametric uncertaintyis represented by variations of certain system parameterswhich may adversely affect the stability and performance ofa control system In Figure 3 the four constant blocks can bereplaced by upper linear fractional transformation (LFT) Forinstance119867

119894+1may be represented as upper LFT in 119875

119867119894+1and

120575119867119894+1

(minus1 lt 120575119867119894+1

lt 1) and further 119867119894+1

is the so-callednominal value of 119867

119894+1 The LFT framework for 119867

119894+1depicts

in Figure 4

119867119894+1= 119867119894+1(1 + 119875

119867119894+1120575119867119894+1

) = 119865119906(119872119867119894+1

120575119867119894+1

)

119872119867119894+1

= [0 119867

119894+1

119875119867119894+1

119867119894+1

]

(7)

Similarly the other parameters can be represented as upperLFT in 120575

120578119894+1 120575120591V119894 and 120575

119864 It is supposed that 120578 = 1 minus 120593 for

simplification The parameters of full stand tension systemand their tolerances are given in Table 1 [13]

4 Journal of Engineering

Table 1 Nominal values and their tolerance

Parameter Nominal value Unit Uncertainty119864119899

21 times 105 Nmm2

plusmn100119879V119899 01 s plusmn2001198672

135 mm plusmn501198673

077 mm plusmn501198674

05 mm plusmn501198675

033 mm plusmn501205932

03 mdash plusmn1001205933

025 mdash plusmn1001205934

02 mdash plusmn1001205935

02 mdash plusmn100119871 45 m 0119861 1 m 0

23 State-Space Representation Based on (4)ndash(6) the state-space description of tension system with 5-stand yields

1198831015840

= 119860119883 + 1198611119880119906+ 1198612119880

119884119906= 1198621119883 + 119863

11119880119906+ 11986312119880

119884 = 1198622119883 + 119863

21119880119906+ 11986322119880

(8)

The state vector 119883 the control inputs 119880 and 119880119906 and the

output vectors 119884 and 119884119906are defined as

119883 = [Δ1198812 Δ12059012 Δ1198813 Δ12059023 Δ1198814 Δ12059034 Δ1198815 Δ12059045]T

119880 = [Δ119880V2 Δ119880V3 Δ119880V4 Δ119880V5]T

119880119906= [119880120591V2 1198801205782 1198801198642 1198801198672 119880120591V3 1198801205783 1198801198643 1198801198673

119880120591V4 1198801205784 1198801198644 1198801198674 119880120591V5 1198801205785 1198801198645 1198801198675]T

119884119906= [119910120591V2 1199101205782 1199101198642 1199101198672 119910120591V3 1199101205783 1199101198643 1199101198673

119910120591V4 1199101205784 1199101198644 1199101198674 119910120591V5 1199101205785 1199101198645 1199101198675]T

119884 = [Δ11987912 Δ11987923 Δ11987934 Δ11987945]T

(9)

119866Ten denotes the inputoutput dynamics of the tension sys-tem which takes into account the uncertainty of parametersThe state-space representation of119866Ten is (the systemmatricesare presented in Appendix A)

119866Ten =[

[

119860 1198611

1198612

11986211198631111986312

11986221198632111986322

]

]

(10)

The uncertain behavior of the original system can bedescribed by an upper LFT representation

119884 = 119865119880(119866Ten Δ)119880 (11)

With diagonal uncertainty matrix Δ = diag(120575V2 1205751205782 12057511986421205751198672 120575V5 1205751205785 1205751198645 1205751198675) as shown in Figure 5 It should

be noted that the unknown matrix Δ has a fixed structureand in general is a block diagonal Such uncertainty is calledstructured uncertainty

Δ

GTen

U Y

Figure 5 LFT representation of the tension system with uncertain-ties

GTension

Δ

d

Tensioncontroller Wp ep

minusWu

euG

Ref

Figure 6 The block diagram of closed system with performancerequirements

3 Robust Tension Controller Design

This section considers the design of tension control system for5-stand tandem mill For this system open loop and closed-loop system interconnections are introduced and robustcontrollers are designed

31 Requirements to Fulfill Design The aim of robust ten-sion controller design is to achieve and maintain the ten-sion between stands in the presence of disturbances anduncertainties The block diagram of the closed-loop systemincluding the feedback and controller as well as the elementsreflecting the model uncertainty and weighting functionsrelated to performance requirements is shown in Figure 6

In Figure 6 the rectangle with dashed line represents thetransfer matrix 119866 Inside the rectangle is the nominal modelof tension system and the uncertainty matrix Δ that includesthe model uncertainties The variables 119889

1 1198892 1198893 and 119889

4are

the disturbances on the system at the system outputs Theoutput weighting performance vector (119890

119901) control weighting

performance (119890119906) and disturbances vector (119889) are defined as

119890119901= [1198901199011 1198901199012 1198901199013 1198901199014]T

119890119906= [1198901199061 1198901199062 1198901199063 1198901199064]T

119889 = [1198891 1198892 1198893 1198894]T

(12)

In general weighting functions would be used to meet thedesign specifications Finding appropriate weighting func-tions is a crucial step in robust controller design and usuallyneeds a few trials The performance weighting function is

Journal of Engineering 5

103

102

101

100

10minus1

10minus2

10minus3

wminus1p1

Inverse performance weighting function

Frequency (rads)

Mag

nitu

de

10minus4 10minus3 10minus2 10minus1 100 101 102 103 104

Figure 7 Singular values of 1119908119901

101

100

10minus1

10minus5 100 105 1010

wu1

Control weighting function

Frequency (rads)

Mag

nitu

de

Figure 8 Singular values of 119908119906

usually low-pass filter and control weighting function is high-pass filter Design experience will help in choosing weightingfunctions In this design the performanceweighting function119908119901and the control weighting functions 119908

119906119882119901 and119882

119906are

chosen as [13]

119908119901= 0012

1198782

+ 28119878 + 100

051198782 + 40119878 + 001

119908119906= 5

119878 + 10000

119878 + 500000

119882119901= (

1199081199011

0 0 0

0 1199081199012

0 0

0 0 1199081199013

0

0 0 0 1199081199014

)

119882119906= (

1199081199061

0 0 0

0 1199081199062

0 0

0 0 1199081199063

0

0 0 0 1199081199064

)

(13)The singular values of 1119908

119901over the frequency range

[10minus4

104

] are shown in Figure 7 This weighting functionshows that for low frequencies the closed-loop system (thenominal as well as perturbed system) attenuates the outputdisturbance in the ratio of 100 to 001 In other words theeffect of a unit disturbance on the steady-state output shouldbe of the order 10minus2 This performance requirement becomesless stringent with increasing frequency

Also the singular values of 119908119906over the frequency range

[10minus4 104] are shown in Figure 8

32 Open Loop and Closed-Loop System InterconnectionBefore designing the tension controller it is necessary to build

6 Journal of Engineering

Pertin1-16 Pertout1-16

sum

+

+

minus

sum

G

yc1-4

Wp

minusWu

ep1-4

eu1-4

ΔT1-4

Dist1-4

Ref = 0

Figure 9 Structure of the open loop system

Pertin1-16 Pertout1-16

Control1-4

ep1-4

eu1-4

ΔT1-4

Dist1-4 tension ic

Figure 10 Block diagram of open loop system

Pertin1-16 Pertout1-16

T1-4

U1-4

yc1-4

G

K

+

+

+minus

sum

sum

Dist1-4

Ref1-4

Figure 11 Structure of the closed-loop system

interconnection for open-loop and closed-loop system Thestructure of the open-loop system is represented in Figure 9In Figure 10 tension ic denotes the transfer function matrixof open loop system with 24 inputs and 28 outputs

The simulation of closed-loop system with designedcontrollers is based on the structure shown in Figure 11 Theweighting functions 119908

119901and 119908

119906are not included in the block

diagram for obvious reasons

33 119867infin

Controller The interconnection of nominal openloop system is denoted by hin ic as shown in Figure 12

The suboptimal 119867infin

controller minimizes the infinite-normof 119865119897(ℎ119894119899 119894119888 119896) overall stabilizing controller 119870 Note that

119865119897(ℎ119894119899 119894119888 119896) is the transfer function matrix of the nominal

closed-loop system from the disturbances to the errorsTo compute 119867

infinoptimal controller for LTI plant some

conditions are satisfied by plant Sometimes one of theseconditions is not satisfied and the algorithm for computingdoes not work [15 17] When this situation accrued otherrobust control methods are used although some tricks canbe used to satisfy conditions For example adding or sub-tracting epsilon may solve the problem in these conditions

Journal of Engineering 7

K

eDist hin ic

Figure 12 Closed-loop LFTs in119867infindesign

10minus1 100 101 102 103 10401

02

03

04

05

06

Frequency (rads)10minus1 100 101 102 103 104

Frequency (rads)

010

020304050607

Closed-loop complex 120583 controller number 3

Com

plex120583

Closed-loop mixed 120583 controller number 3

Mix

ed120583

Figure 13 120583 plot after third iteration

100

10minus1

10minus2

10minus4 10minus3 10minus2 10minus1 100 101 102 103 104

Robust stability

Frequency (rads)

120583-controllerReduced-order controller

Upp

er b

ound

of120583

Hinfin controller

Figure 14 Comparison of robust stability for three controllers

For this plant 119867infin

controller is numerically computed andfinally the obtained controller is of 20th orderTheminimumvalue of 120574 is 0101

34 120583 Controller The 120583 synthesis is introduced by twoiterative methods 119863-119870 and 120583-119870 iteration For this plant120583 synthesis is executed by the M-file dkit from the RobustControl Toolbox of Matlab which automates the procedureby using 119863-119870 iterations [15] After the third iteration

the maximum value of 120583 is 0567 which indicates that therobust performance has been achieved The final controllerobtained is of 104th order The 120583 plot of the closed-loopsystem is shown in Figure 13

The order of this controller is 104 which makes it difficultin implementation On the other hand high order controllerswill lead to high cost difficult commissioning poor relia-bility and potential problems in maintenance This is theweakness of 120583 controller in this design Usually the order

8 Journal of Engineering

10minus4 10minus3 10minus2 10minus1 100 101 102 103 1040

01

02

03

04

05

06

07

08

09

1Robust performance

Frequency (rads)Hinfin controller120583-controllerReduced-order controller

Upp

er b

ound

of120583

Figure 15 Comparison of robust performance for three controllers

0 10 20 30 40minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref1 (tension ST1-ST2)

Time (s)0 10 20 30 40

minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref2 (tension ST2-ST3)

Time (s)

0 10 20 30 40minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref3 (tension ST3-ST4)

Time (s)0 10 20 30 40

minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref4 (tension ST4-ST5)

Time (s)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Figure 16 Comparison of transient response to reference input

Journal of Engineering 9

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 1

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600Transient response to disturbance 2

Transient response to disturbance 4

Time (s)

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 3

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600

Time (s)

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Figure 17 Comparison of transient response to disturbance signal

can be reduced and reduction of the controller order maylead to deterioration of the closed-loop performance Variousmethods may be used to reduce the order and in this casestudy the Hankel-norm approximation is usedThe reduced-order controller is equal to 13 that leads to a controllermaintaining the robust stability and performance of theclosed-loop system (matrices for reduced-order controllerare presented in Appendix B)

4 Simulation Results

In the previous section robust controllers for tension con-trol system are introduced It is intended here to comparethe robust stability robust performance and time domainresponses of the proposed controllers Since the considereduncertainty is structured verification of the robust stabilityand performance needs the frequency response in terms of120583 values On the other hand controllers should be tested bymeans of 120583 analysis The robust stability and performancedo not maintain if the uncertainty is unstructured whichshows that the 120583 values give less conservative results if furtherinformation is known about the uncertainty

41 Robust Stability The frequency responses of the upperbound of 120583 for 119867

infincontroller and 120583 and reduced-order

controllers are shown in Figure 14Themaximum120583 values for119867infincontroller are equal to 07801 and the maximum 120583 values

for 120583 and reduced-order controllers are 0202 Thereforethe robust stability of control systems is achieved and thesystems with the 120583 controller and reduced-order controllersare characterized with better robust stability

42 Robust Performance The robust performance of controlsystems is plotted in Figure 15 The maximum 120583 valuesfor 119867

infincontroller are 099 for 120583 controller 0597 and for

reduced-order controller 069 which implies the robust 120583controller and reduced-order controller achieve better robustperformance than119867

infincontroller

43 Time Domain Simulation The transient responses of theclosed-loop system to the reference input and the disturbancesignal are shown in Figures 16 and 17 respectively The 120583 andthe reduced-order controllers give a better transient responsewith smaller overshoot than the119867

infincontroller In summary

10 Journal of Engineering

the 120583 and the reduced-order controllers lead to better trade-off between the requirements in terms of transient responsedisturbance rejection and robustness

5 Conclusion

In cold rolling the strip tension should be controlled forbetter quality of products For cold rolling with 5 standsthe state-space representation perturbed with parametricuncertainties is derived and further 119867

infincontroller and

120583 controller based on robust control theory are designedTension control is obtained by means of the proposed robustcontrollers in the presence of uncertainties and disturbancesThe order of the 120583 controller is large which requires thecontroller order reduction to preserve the closed-loop per-formance and robustness while making implementation ofthe controller easier andmore reliableThe simulation resultsshow that control systems achieve robust stability and robust

performance The comparison of the robust stability androbust performance for controllers as well as the comparisonof the corresponding transient responses shows that the 120583controller and its reduced-order are preferable although thereduced-order controller would be usually preferred dueto its lower order It should be mentioned that in suchimportant industrial process most commissioning personnelhave very little or no background in advanced control theoryespecially in the theory and application of119867

infinrobust control

techniques and are accustomed to single-input-single-output(SISO) proportional-integral (PI) type control loops Thesesimpler configurations are easy to tune at commissioningduring actual operation

Appendices

A The System Matrices

Consider

119860 =

[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

44200 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

minus44200 0 26000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 minus26800 0 18600 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 minus18400 0 12200 0

]]]]]]]]]]

]

1198612=

[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 10 0

0 0 0 0

0 0 0 10

0 0 0 0

]]]]]]]]]]]]

]

1198611=

[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 6280 003 00011 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 minus6220 0 0 0 3600 0017 00011 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 minus3600 0 0 0 2300 0011 0011 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 minus2300 0 0 0 1540 0073 0011

]]]]]]]]]]

]

1198621=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

07

147000

0

0

0

0

0

0

0

0

0

0

0

0

0

0

199000 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

0 0 075 0 0 0 0 0

minus257000 0 157000 0 0 0 0 0

minus197000 0 120000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 0 0 08 0 0 0

0 0 minus242000 0 168000 0 0 0

0 0 minus1210000 0 84000 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 0 0 08 0

0 0 0 0 minus253000 0 168000 0

0 0 0 0 minus84000 0 55000 0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

1198622=

[[[

[

0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

]]]

]

Journal of Engineering 11

11986311=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 21000 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 28300 0135 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 36700 0 0 0 21000 0 0 0 0 0 0 0 0 0 0

0 minus28200 0 0 0 16700 0077 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 minus32300 0 0 0 21000 0 0 0 0 0 0

0 0 0 0 0 minus16100 0 005 10500 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 minus31700 0 0 0 21000 0 0

0 0 0 0 0 0 0 0 0 minus10400 0 0 0 6930 0033 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986312=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 10

0 0 0 0

0 0 0 0

0 0 0 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(A1)

B Matrices for Reduced-Order Controller

Consider

ak =

[[[[[[[[[[[[[[[[[[[[

[

minus13534 9671 2455 minus1674 minus122 155 minus048 299 109 014 minus008 minus019 011

0 minus2734 523 1642 147 minus068 04 minus136 minus065 0005 007 01 minus0048

0 minus2872 minus277 03 minus1267 minus021 005 minus065 003 minus01 009 002 004

0 0 0 minus611 minus184 018 minus029 031 042 minus009 minus002 minus002 0018

0 0 0 145 minus662 minus008 019 minus011 minus006 minus002 0 0003 0

0 0 0 0 0 minus0008 006 minus002 minus0008 minus0001 0 0002 0001

0 0 0 0 0 minus006 minus0004 0008 0008 minus0002 0 0 0001

0 0 0 0 0 0 0 minus002 002 minus0004 minus0001 0 0

0 0 0 0 0 0 0 minus004 minus0008 0002 minus0003 0 0

0 0 0 0 0 0 0 0 0 minus0003 0 0 0

0 0 0 0 0 0 0 0 0 0 minus0002 0 0

0 0 0 0 0 0 0 0 0 0 0 minus0002 0

0 0 0 0 0 0 0 0 0 0 0 0 minus0002

]]]]]]]]]]]]]]]]]]]]

]

12 Journal of Engineering

bk =

[[[[[[[[[[[[[[[[[[[[

[

015 008 002 004

minus004 minus003 minus001 minus001

minus005 minus0002 minus002 0

minus0008 001 001 001

minus0006 0008 0003 minus001

0 0 0 0

0 0 0 0

minus0001 0 0 0

0 0 0 0

0 0 0 0

0001 0 0 0

0 0 0 minus0001

0 0 0 minus0001

]]]]]]]]]]]]]]]]]]]]

]

ck =[[[

[

minus0001 minus0006 0 0001 minus0001 0 0 0 0 0 0 0 0

006 minus002 minus0009 0001 0007 0 0 0 0 0 0 0001 0

0009 minus004 minus003 001 0 0 0 0 0 0 0001 0 0001

013 minus008 minus005 001 0 0 0001 minus0002 0 0 0001 minus0001 minus0001

]]]

]

dk =[[[

[

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

]]]

]

(B1)

Abbreviations

119881 Work roll linear velocity119880V Speed actuator input120591V Speed actuator time constant119881in Input strip velocity119881out Output strip velocityℎin Stand input thicknessℎout Stand output thickness119891 Forward slip ratio120593 Backward slip ratio119864 Youngrsquos modulus119861 Width of strip119867119894+1

Strip thickness between stand 119894 and stand 119894 + 1119871 Length between adjacent stands119879119894119894+1

Strip tension between stand 119894 and stand 119894 + 1

Conflict of Interests

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

References

[1] W S Levine Control System Applications Taylor amp FrancisBoca Raton Fla USA 2011

[2] J Pittner N S Samaras and M A Simaan ldquoA simple rollingmillmodel with linear quadratic optimal controllerrdquo inProceed-ings of the 37th IAS Annual Meeting and World Conference onIndustrial Applications of Electrical Energy pp 142ndash149 October2002

[3] H R Koofigar F Sheikholeslam and S Hosseinnia ldquoUnifiedgauge-tension control in cold rolling mills a robust regulationtechniquerdquo International Journal of Precision Engineering andManufacturing vol 12 no 3 pp 393ndash403 2011

[4] E J M Geddes and I Postlethwaite ldquoImprovements in productquality in tandem cold rolling using robust multivariablecontrolrdquo IEEE Transactions on Control Systems Technology vol6 no 2 pp 257ndash269 1998

[5] E J M Geddes and I Postlethwaite ldquoMultivariable control ofa high performance tandem cold rolling millrdquo in Proceedings ofthe International Conference on Control (CONTROL rsquo94) vol 1pp 202ndash207 Coventry UK March 1994

[6] E Arinton S Caraman and J Korbicz ldquoNeural networks formodelling and fault detection of the inter-stand strip tension ofa cold tandem millrdquo Control Engineering Practice vol 20 no 7pp 684ndash694 2012

[7] KMTakami JMahmoudi andEDahlquist ldquoAdaptive controlof cold rolling system in electrical strips production systemwithonline-offline predictorsrdquo International Journal of AdvancedManufacturing Technology vol 50 no 9ndash12 pp 917ndash930 2010

[8] G Pin V Francesconi F A Cuzzola and T Parisini ldquoAdaptivetask-space metal strip-flatness control in cold multi-roll millstandsrdquo Journal of Process Control vol 23 no 2 pp 108ndash1192013

[9] J Pittner and M A Simaan ldquoAn optimal control method forimprovement in tandem cold metal rollingrdquo in Proceedings ofthe IEEE 42nd AnnualMeeting Industry Applications Conference(IAS rsquo07) pp 382ndash389 September 2007

[10] J R Pittner Pointwise linear quadratic optimal control of atandem cold rolling mill [PhD thesis] School of EngineeringUniversity of Pittsburgh 2006

[11] L E Zarate ldquoPredictive model for the cold rolling processthrough sensitivity factors via neural networksrdquo Journal of the

Journal of Engineering 13

Brazilian Society ofMechanical Sciences and Engineering vol 28no 1 pp 111ndash117 2006

[12] S-H Song and S-K Sul ldquoA new tension controller for con-tinuous strip processing linerdquo IEEE Transactions on IndustryApplications vol 36 no 2 pp 633ndash639 2000

[13] X Zhang and Q Zhang ldquoRobust control of strip tension fortandem cold rolling millrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 2390ndash2393 Yantai China July2011

[14] J Pittner and M A Simaan Tandem Cold Metal Rolling MillControl Using Practical Advanced Methods Springer LondonUK 2011

[15] DGu PH Petkov andMKonstantinovRobust Control Designwith MATLAB Springer London UK 2005

[16] Z Xiaofeng and ZQingdong ldquoRobust control of strip thicknessfor cold rolling millrdquo in Informatics in Control Automation andRobotics vol 133 of Lecture Notes in Electrical Engineering pp777ndash785 Springer Berlin Germany 2012

[17] K Zohu and J C Doyle Essentials of Robust Control PrenticeHall 1997

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International Journal of

4 Journal of Engineering

Table 1 Nominal values and their tolerance

Parameter Nominal value Unit Uncertainty119864119899

21 times 105 Nmm2

plusmn100119879V119899 01 s plusmn2001198672

135 mm plusmn501198673

077 mm plusmn501198674

05 mm plusmn501198675

033 mm plusmn501205932

03 mdash plusmn1001205933

025 mdash plusmn1001205934

02 mdash plusmn1001205935

02 mdash plusmn100119871 45 m 0119861 1 m 0

23 State-Space Representation Based on (4)ndash(6) the state-space description of tension system with 5-stand yields

1198831015840

= 119860119883 + 1198611119880119906+ 1198612119880

119884119906= 1198621119883 + 119863

11119880119906+ 11986312119880

119884 = 1198622119883 + 119863

21119880119906+ 11986322119880

(8)

The state vector 119883 the control inputs 119880 and 119880119906 and the

output vectors 119884 and 119884119906are defined as

119883 = [Δ1198812 Δ12059012 Δ1198813 Δ12059023 Δ1198814 Δ12059034 Δ1198815 Δ12059045]T

119880 = [Δ119880V2 Δ119880V3 Δ119880V4 Δ119880V5]T

119880119906= [119880120591V2 1198801205782 1198801198642 1198801198672 119880120591V3 1198801205783 1198801198643 1198801198673

119880120591V4 1198801205784 1198801198644 1198801198674 119880120591V5 1198801205785 1198801198645 1198801198675]T

119884119906= [119910120591V2 1199101205782 1199101198642 1199101198672 119910120591V3 1199101205783 1199101198643 1199101198673

119910120591V4 1199101205784 1199101198644 1199101198674 119910120591V5 1199101205785 1199101198645 1199101198675]T

119884 = [Δ11987912 Δ11987923 Δ11987934 Δ11987945]T

(9)

119866Ten denotes the inputoutput dynamics of the tension sys-tem which takes into account the uncertainty of parametersThe state-space representation of119866Ten is (the systemmatricesare presented in Appendix A)

119866Ten =[

[

119860 1198611

1198612

11986211198631111986312

11986221198632111986322

]

]

(10)

The uncertain behavior of the original system can bedescribed by an upper LFT representation

119884 = 119865119880(119866Ten Δ)119880 (11)

With diagonal uncertainty matrix Δ = diag(120575V2 1205751205782 12057511986421205751198672 120575V5 1205751205785 1205751198645 1205751198675) as shown in Figure 5 It should

be noted that the unknown matrix Δ has a fixed structureand in general is a block diagonal Such uncertainty is calledstructured uncertainty

Δ

GTen

U Y

Figure 5 LFT representation of the tension system with uncertain-ties

GTension

Δ

d

Tensioncontroller Wp ep

minusWu

euG

Ref

Figure 6 The block diagram of closed system with performancerequirements

3 Robust Tension Controller Design

This section considers the design of tension control system for5-stand tandem mill For this system open loop and closed-loop system interconnections are introduced and robustcontrollers are designed

31 Requirements to Fulfill Design The aim of robust ten-sion controller design is to achieve and maintain the ten-sion between stands in the presence of disturbances anduncertainties The block diagram of the closed-loop systemincluding the feedback and controller as well as the elementsreflecting the model uncertainty and weighting functionsrelated to performance requirements is shown in Figure 6

In Figure 6 the rectangle with dashed line represents thetransfer matrix 119866 Inside the rectangle is the nominal modelof tension system and the uncertainty matrix Δ that includesthe model uncertainties The variables 119889

1 1198892 1198893 and 119889

4are

the disturbances on the system at the system outputs Theoutput weighting performance vector (119890

119901) control weighting

performance (119890119906) and disturbances vector (119889) are defined as

119890119901= [1198901199011 1198901199012 1198901199013 1198901199014]T

119890119906= [1198901199061 1198901199062 1198901199063 1198901199064]T

119889 = [1198891 1198892 1198893 1198894]T

(12)

In general weighting functions would be used to meet thedesign specifications Finding appropriate weighting func-tions is a crucial step in robust controller design and usuallyneeds a few trials The performance weighting function is

Journal of Engineering 5

103

102

101

100

10minus1

10minus2

10minus3

wminus1p1

Inverse performance weighting function

Frequency (rads)

Mag

nitu

de

10minus4 10minus3 10minus2 10minus1 100 101 102 103 104

Figure 7 Singular values of 1119908119901

101

100

10minus1

10minus5 100 105 1010

wu1

Control weighting function

Frequency (rads)

Mag

nitu

de

Figure 8 Singular values of 119908119906

usually low-pass filter and control weighting function is high-pass filter Design experience will help in choosing weightingfunctions In this design the performanceweighting function119908119901and the control weighting functions 119908

119906119882119901 and119882

119906are

chosen as [13]

119908119901= 0012

1198782

+ 28119878 + 100

051198782 + 40119878 + 001

119908119906= 5

119878 + 10000

119878 + 500000

119882119901= (

1199081199011

0 0 0

0 1199081199012

0 0

0 0 1199081199013

0

0 0 0 1199081199014

)

119882119906= (

1199081199061

0 0 0

0 1199081199062

0 0

0 0 1199081199063

0

0 0 0 1199081199064

)

(13)The singular values of 1119908

119901over the frequency range

[10minus4

104

] are shown in Figure 7 This weighting functionshows that for low frequencies the closed-loop system (thenominal as well as perturbed system) attenuates the outputdisturbance in the ratio of 100 to 001 In other words theeffect of a unit disturbance on the steady-state output shouldbe of the order 10minus2 This performance requirement becomesless stringent with increasing frequency

Also the singular values of 119908119906over the frequency range

[10minus4 104] are shown in Figure 8

32 Open Loop and Closed-Loop System InterconnectionBefore designing the tension controller it is necessary to build

6 Journal of Engineering

Pertin1-16 Pertout1-16

sum

+

+

minus

sum

G

yc1-4

Wp

minusWu

ep1-4

eu1-4

ΔT1-4

Dist1-4

Ref = 0

Figure 9 Structure of the open loop system

Pertin1-16 Pertout1-16

Control1-4

ep1-4

eu1-4

ΔT1-4

Dist1-4 tension ic

Figure 10 Block diagram of open loop system

Pertin1-16 Pertout1-16

T1-4

U1-4

yc1-4

G

K

+

+

+minus

sum

sum

Dist1-4

Ref1-4

Figure 11 Structure of the closed-loop system

interconnection for open-loop and closed-loop system Thestructure of the open-loop system is represented in Figure 9In Figure 10 tension ic denotes the transfer function matrixof open loop system with 24 inputs and 28 outputs

The simulation of closed-loop system with designedcontrollers is based on the structure shown in Figure 11 Theweighting functions 119908

119901and 119908

119906are not included in the block

diagram for obvious reasons

33 119867infin

Controller The interconnection of nominal openloop system is denoted by hin ic as shown in Figure 12

The suboptimal 119867infin

controller minimizes the infinite-normof 119865119897(ℎ119894119899 119894119888 119896) overall stabilizing controller 119870 Note that

119865119897(ℎ119894119899 119894119888 119896) is the transfer function matrix of the nominal

closed-loop system from the disturbances to the errorsTo compute 119867

infinoptimal controller for LTI plant some

conditions are satisfied by plant Sometimes one of theseconditions is not satisfied and the algorithm for computingdoes not work [15 17] When this situation accrued otherrobust control methods are used although some tricks canbe used to satisfy conditions For example adding or sub-tracting epsilon may solve the problem in these conditions

Journal of Engineering 7

K

eDist hin ic

Figure 12 Closed-loop LFTs in119867infindesign

10minus1 100 101 102 103 10401

02

03

04

05

06

Frequency (rads)10minus1 100 101 102 103 104

Frequency (rads)

010

020304050607

Closed-loop complex 120583 controller number 3

Com

plex120583

Closed-loop mixed 120583 controller number 3

Mix

ed120583

Figure 13 120583 plot after third iteration

100

10minus1

10minus2

10minus4 10minus3 10minus2 10minus1 100 101 102 103 104

Robust stability

Frequency (rads)

120583-controllerReduced-order controller

Upp

er b

ound

of120583

Hinfin controller

Figure 14 Comparison of robust stability for three controllers

For this plant 119867infin

controller is numerically computed andfinally the obtained controller is of 20th orderTheminimumvalue of 120574 is 0101

34 120583 Controller The 120583 synthesis is introduced by twoiterative methods 119863-119870 and 120583-119870 iteration For this plant120583 synthesis is executed by the M-file dkit from the RobustControl Toolbox of Matlab which automates the procedureby using 119863-119870 iterations [15] After the third iteration

the maximum value of 120583 is 0567 which indicates that therobust performance has been achieved The final controllerobtained is of 104th order The 120583 plot of the closed-loopsystem is shown in Figure 13

The order of this controller is 104 which makes it difficultin implementation On the other hand high order controllerswill lead to high cost difficult commissioning poor relia-bility and potential problems in maintenance This is theweakness of 120583 controller in this design Usually the order

8 Journal of Engineering

10minus4 10minus3 10minus2 10minus1 100 101 102 103 1040

01

02

03

04

05

06

07

08

09

1Robust performance

Frequency (rads)Hinfin controller120583-controllerReduced-order controller

Upp

er b

ound

of120583

Figure 15 Comparison of robust performance for three controllers

0 10 20 30 40minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref1 (tension ST1-ST2)

Time (s)0 10 20 30 40

minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref2 (tension ST2-ST3)

Time (s)

0 10 20 30 40minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref3 (tension ST3-ST4)

Time (s)0 10 20 30 40

minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref4 (tension ST4-ST5)

Time (s)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Figure 16 Comparison of transient response to reference input

Journal of Engineering 9

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 1

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600Transient response to disturbance 2

Transient response to disturbance 4

Time (s)

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 3

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600

Time (s)

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Figure 17 Comparison of transient response to disturbance signal

can be reduced and reduction of the controller order maylead to deterioration of the closed-loop performance Variousmethods may be used to reduce the order and in this casestudy the Hankel-norm approximation is usedThe reduced-order controller is equal to 13 that leads to a controllermaintaining the robust stability and performance of theclosed-loop system (matrices for reduced-order controllerare presented in Appendix B)

4 Simulation Results

In the previous section robust controllers for tension con-trol system are introduced It is intended here to comparethe robust stability robust performance and time domainresponses of the proposed controllers Since the considereduncertainty is structured verification of the robust stabilityand performance needs the frequency response in terms of120583 values On the other hand controllers should be tested bymeans of 120583 analysis The robust stability and performancedo not maintain if the uncertainty is unstructured whichshows that the 120583 values give less conservative results if furtherinformation is known about the uncertainty

41 Robust Stability The frequency responses of the upperbound of 120583 for 119867

infincontroller and 120583 and reduced-order

controllers are shown in Figure 14Themaximum120583 values for119867infincontroller are equal to 07801 and the maximum 120583 values

for 120583 and reduced-order controllers are 0202 Thereforethe robust stability of control systems is achieved and thesystems with the 120583 controller and reduced-order controllersare characterized with better robust stability

42 Robust Performance The robust performance of controlsystems is plotted in Figure 15 The maximum 120583 valuesfor 119867

infincontroller are 099 for 120583 controller 0597 and for

reduced-order controller 069 which implies the robust 120583controller and reduced-order controller achieve better robustperformance than119867

infincontroller

43 Time Domain Simulation The transient responses of theclosed-loop system to the reference input and the disturbancesignal are shown in Figures 16 and 17 respectively The 120583 andthe reduced-order controllers give a better transient responsewith smaller overshoot than the119867

infincontroller In summary

10 Journal of Engineering

the 120583 and the reduced-order controllers lead to better trade-off between the requirements in terms of transient responsedisturbance rejection and robustness

5 Conclusion

In cold rolling the strip tension should be controlled forbetter quality of products For cold rolling with 5 standsthe state-space representation perturbed with parametricuncertainties is derived and further 119867

infincontroller and

120583 controller based on robust control theory are designedTension control is obtained by means of the proposed robustcontrollers in the presence of uncertainties and disturbancesThe order of the 120583 controller is large which requires thecontroller order reduction to preserve the closed-loop per-formance and robustness while making implementation ofthe controller easier andmore reliableThe simulation resultsshow that control systems achieve robust stability and robust

performance The comparison of the robust stability androbust performance for controllers as well as the comparisonof the corresponding transient responses shows that the 120583controller and its reduced-order are preferable although thereduced-order controller would be usually preferred dueto its lower order It should be mentioned that in suchimportant industrial process most commissioning personnelhave very little or no background in advanced control theoryespecially in the theory and application of119867

infinrobust control

techniques and are accustomed to single-input-single-output(SISO) proportional-integral (PI) type control loops Thesesimpler configurations are easy to tune at commissioningduring actual operation

Appendices

A The System Matrices

Consider

119860 =

[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

44200 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

minus44200 0 26000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 minus26800 0 18600 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 minus18400 0 12200 0

]]]]]]]]]]

]

1198612=

[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 10 0

0 0 0 0

0 0 0 10

0 0 0 0

]]]]]]]]]]]]

]

1198611=

[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 6280 003 00011 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 minus6220 0 0 0 3600 0017 00011 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 minus3600 0 0 0 2300 0011 0011 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 minus2300 0 0 0 1540 0073 0011

]]]]]]]]]]

]

1198621=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

07

147000

0

0

0

0

0

0

0

0

0

0

0

0

0

0

199000 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

0 0 075 0 0 0 0 0

minus257000 0 157000 0 0 0 0 0

minus197000 0 120000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 0 0 08 0 0 0

0 0 minus242000 0 168000 0 0 0

0 0 minus1210000 0 84000 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 0 0 08 0

0 0 0 0 minus253000 0 168000 0

0 0 0 0 minus84000 0 55000 0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

1198622=

[[[

[

0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

]]]

]

Journal of Engineering 11

11986311=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 21000 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 28300 0135 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 36700 0 0 0 21000 0 0 0 0 0 0 0 0 0 0

0 minus28200 0 0 0 16700 0077 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 minus32300 0 0 0 21000 0 0 0 0 0 0

0 0 0 0 0 minus16100 0 005 10500 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 minus31700 0 0 0 21000 0 0

0 0 0 0 0 0 0 0 0 minus10400 0 0 0 6930 0033 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986312=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 10

0 0 0 0

0 0 0 0

0 0 0 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(A1)

B Matrices for Reduced-Order Controller

Consider

ak =

[[[[[[[[[[[[[[[[[[[[

[

minus13534 9671 2455 minus1674 minus122 155 minus048 299 109 014 minus008 minus019 011

0 minus2734 523 1642 147 minus068 04 minus136 minus065 0005 007 01 minus0048

0 minus2872 minus277 03 minus1267 minus021 005 minus065 003 minus01 009 002 004

0 0 0 minus611 minus184 018 minus029 031 042 minus009 minus002 minus002 0018

0 0 0 145 minus662 minus008 019 minus011 minus006 minus002 0 0003 0

0 0 0 0 0 minus0008 006 minus002 minus0008 minus0001 0 0002 0001

0 0 0 0 0 minus006 minus0004 0008 0008 minus0002 0 0 0001

0 0 0 0 0 0 0 minus002 002 minus0004 minus0001 0 0

0 0 0 0 0 0 0 minus004 minus0008 0002 minus0003 0 0

0 0 0 0 0 0 0 0 0 minus0003 0 0 0

0 0 0 0 0 0 0 0 0 0 minus0002 0 0

0 0 0 0 0 0 0 0 0 0 0 minus0002 0

0 0 0 0 0 0 0 0 0 0 0 0 minus0002

]]]]]]]]]]]]]]]]]]]]

]

12 Journal of Engineering

bk =

[[[[[[[[[[[[[[[[[[[[

[

015 008 002 004

minus004 minus003 minus001 minus001

minus005 minus0002 minus002 0

minus0008 001 001 001

minus0006 0008 0003 minus001

0 0 0 0

0 0 0 0

minus0001 0 0 0

0 0 0 0

0 0 0 0

0001 0 0 0

0 0 0 minus0001

0 0 0 minus0001

]]]]]]]]]]]]]]]]]]]]

]

ck =[[[

[

minus0001 minus0006 0 0001 minus0001 0 0 0 0 0 0 0 0

006 minus002 minus0009 0001 0007 0 0 0 0 0 0 0001 0

0009 minus004 minus003 001 0 0 0 0 0 0 0001 0 0001

013 minus008 minus005 001 0 0 0001 minus0002 0 0 0001 minus0001 minus0001

]]]

]

dk =[[[

[

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

]]]

]

(B1)

Abbreviations

119881 Work roll linear velocity119880V Speed actuator input120591V Speed actuator time constant119881in Input strip velocity119881out Output strip velocityℎin Stand input thicknessℎout Stand output thickness119891 Forward slip ratio120593 Backward slip ratio119864 Youngrsquos modulus119861 Width of strip119867119894+1

Strip thickness between stand 119894 and stand 119894 + 1119871 Length between adjacent stands119879119894119894+1

Strip tension between stand 119894 and stand 119894 + 1

Conflict of Interests

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

References

[1] W S Levine Control System Applications Taylor amp FrancisBoca Raton Fla USA 2011

[2] J Pittner N S Samaras and M A Simaan ldquoA simple rollingmillmodel with linear quadratic optimal controllerrdquo inProceed-ings of the 37th IAS Annual Meeting and World Conference onIndustrial Applications of Electrical Energy pp 142ndash149 October2002

[3] H R Koofigar F Sheikholeslam and S Hosseinnia ldquoUnifiedgauge-tension control in cold rolling mills a robust regulationtechniquerdquo International Journal of Precision Engineering andManufacturing vol 12 no 3 pp 393ndash403 2011

[4] E J M Geddes and I Postlethwaite ldquoImprovements in productquality in tandem cold rolling using robust multivariablecontrolrdquo IEEE Transactions on Control Systems Technology vol6 no 2 pp 257ndash269 1998

[5] E J M Geddes and I Postlethwaite ldquoMultivariable control ofa high performance tandem cold rolling millrdquo in Proceedings ofthe International Conference on Control (CONTROL rsquo94) vol 1pp 202ndash207 Coventry UK March 1994

[6] E Arinton S Caraman and J Korbicz ldquoNeural networks formodelling and fault detection of the inter-stand strip tension ofa cold tandem millrdquo Control Engineering Practice vol 20 no 7pp 684ndash694 2012

[7] KMTakami JMahmoudi andEDahlquist ldquoAdaptive controlof cold rolling system in electrical strips production systemwithonline-offline predictorsrdquo International Journal of AdvancedManufacturing Technology vol 50 no 9ndash12 pp 917ndash930 2010

[8] G Pin V Francesconi F A Cuzzola and T Parisini ldquoAdaptivetask-space metal strip-flatness control in cold multi-roll millstandsrdquo Journal of Process Control vol 23 no 2 pp 108ndash1192013

[9] J Pittner and M A Simaan ldquoAn optimal control method forimprovement in tandem cold metal rollingrdquo in Proceedings ofthe IEEE 42nd AnnualMeeting Industry Applications Conference(IAS rsquo07) pp 382ndash389 September 2007

[10] J R Pittner Pointwise linear quadratic optimal control of atandem cold rolling mill [PhD thesis] School of EngineeringUniversity of Pittsburgh 2006

[11] L E Zarate ldquoPredictive model for the cold rolling processthrough sensitivity factors via neural networksrdquo Journal of the

Journal of Engineering 13

Brazilian Society ofMechanical Sciences and Engineering vol 28no 1 pp 111ndash117 2006

[12] S-H Song and S-K Sul ldquoA new tension controller for con-tinuous strip processing linerdquo IEEE Transactions on IndustryApplications vol 36 no 2 pp 633ndash639 2000

[13] X Zhang and Q Zhang ldquoRobust control of strip tension fortandem cold rolling millrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 2390ndash2393 Yantai China July2011

[14] J Pittner and M A Simaan Tandem Cold Metal Rolling MillControl Using Practical Advanced Methods Springer LondonUK 2011

[15] DGu PH Petkov andMKonstantinovRobust Control Designwith MATLAB Springer London UK 2005

[16] Z Xiaofeng and ZQingdong ldquoRobust control of strip thicknessfor cold rolling millrdquo in Informatics in Control Automation andRobotics vol 133 of Lecture Notes in Electrical Engineering pp777ndash785 Springer Berlin Germany 2012

[17] K Zohu and J C Doyle Essentials of Robust Control PrenticeHall 1997

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DistributedSensor Networks

International Journal of

Journal of Engineering 5

103

102

101

100

10minus1

10minus2

10minus3

wminus1p1

Inverse performance weighting function

Frequency (rads)

Mag

nitu

de

10minus4 10minus3 10minus2 10minus1 100 101 102 103 104

Figure 7 Singular values of 1119908119901

101

100

10minus1

10minus5 100 105 1010

wu1

Control weighting function

Frequency (rads)

Mag

nitu

de

Figure 8 Singular values of 119908119906

usually low-pass filter and control weighting function is high-pass filter Design experience will help in choosing weightingfunctions In this design the performanceweighting function119908119901and the control weighting functions 119908

119906119882119901 and119882

119906are

chosen as [13]

119908119901= 0012

1198782

+ 28119878 + 100

051198782 + 40119878 + 001

119908119906= 5

119878 + 10000

119878 + 500000

119882119901= (

1199081199011

0 0 0

0 1199081199012

0 0

0 0 1199081199013

0

0 0 0 1199081199014

)

119882119906= (

1199081199061

0 0 0

0 1199081199062

0 0

0 0 1199081199063

0

0 0 0 1199081199064

)

(13)The singular values of 1119908

119901over the frequency range

[10minus4

104

] are shown in Figure 7 This weighting functionshows that for low frequencies the closed-loop system (thenominal as well as perturbed system) attenuates the outputdisturbance in the ratio of 100 to 001 In other words theeffect of a unit disturbance on the steady-state output shouldbe of the order 10minus2 This performance requirement becomesless stringent with increasing frequency

Also the singular values of 119908119906over the frequency range

[10minus4 104] are shown in Figure 8

32 Open Loop and Closed-Loop System InterconnectionBefore designing the tension controller it is necessary to build

6 Journal of Engineering

Pertin1-16 Pertout1-16

sum

+

+

minus

sum

G

yc1-4

Wp

minusWu

ep1-4

eu1-4

ΔT1-4

Dist1-4

Ref = 0

Figure 9 Structure of the open loop system

Pertin1-16 Pertout1-16

Control1-4

ep1-4

eu1-4

ΔT1-4

Dist1-4 tension ic

Figure 10 Block diagram of open loop system

Pertin1-16 Pertout1-16

T1-4

U1-4

yc1-4

G

K

+

+

+minus

sum

sum

Dist1-4

Ref1-4

Figure 11 Structure of the closed-loop system

interconnection for open-loop and closed-loop system Thestructure of the open-loop system is represented in Figure 9In Figure 10 tension ic denotes the transfer function matrixof open loop system with 24 inputs and 28 outputs

The simulation of closed-loop system with designedcontrollers is based on the structure shown in Figure 11 Theweighting functions 119908

119901and 119908

119906are not included in the block

diagram for obvious reasons

33 119867infin

Controller The interconnection of nominal openloop system is denoted by hin ic as shown in Figure 12

The suboptimal 119867infin

controller minimizes the infinite-normof 119865119897(ℎ119894119899 119894119888 119896) overall stabilizing controller 119870 Note that

119865119897(ℎ119894119899 119894119888 119896) is the transfer function matrix of the nominal

closed-loop system from the disturbances to the errorsTo compute 119867

infinoptimal controller for LTI plant some

conditions are satisfied by plant Sometimes one of theseconditions is not satisfied and the algorithm for computingdoes not work [15 17] When this situation accrued otherrobust control methods are used although some tricks canbe used to satisfy conditions For example adding or sub-tracting epsilon may solve the problem in these conditions

Journal of Engineering 7

K

eDist hin ic

Figure 12 Closed-loop LFTs in119867infindesign

10minus1 100 101 102 103 10401

02

03

04

05

06

Frequency (rads)10minus1 100 101 102 103 104

Frequency (rads)

010

020304050607

Closed-loop complex 120583 controller number 3

Com

plex120583

Closed-loop mixed 120583 controller number 3

Mix

ed120583

Figure 13 120583 plot after third iteration

100

10minus1

10minus2

10minus4 10minus3 10minus2 10minus1 100 101 102 103 104

Robust stability

Frequency (rads)

120583-controllerReduced-order controller

Upp

er b

ound

of120583

Hinfin controller

Figure 14 Comparison of robust stability for three controllers

For this plant 119867infin

controller is numerically computed andfinally the obtained controller is of 20th orderTheminimumvalue of 120574 is 0101

34 120583 Controller The 120583 synthesis is introduced by twoiterative methods 119863-119870 and 120583-119870 iteration For this plant120583 synthesis is executed by the M-file dkit from the RobustControl Toolbox of Matlab which automates the procedureby using 119863-119870 iterations [15] After the third iteration

the maximum value of 120583 is 0567 which indicates that therobust performance has been achieved The final controllerobtained is of 104th order The 120583 plot of the closed-loopsystem is shown in Figure 13

The order of this controller is 104 which makes it difficultin implementation On the other hand high order controllerswill lead to high cost difficult commissioning poor relia-bility and potential problems in maintenance This is theweakness of 120583 controller in this design Usually the order

8 Journal of Engineering

10minus4 10minus3 10minus2 10minus1 100 101 102 103 1040

01

02

03

04

05

06

07

08

09

1Robust performance

Frequency (rads)Hinfin controller120583-controllerReduced-order controller

Upp

er b

ound

of120583

Figure 15 Comparison of robust performance for three controllers

0 10 20 30 40minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref1 (tension ST1-ST2)

Time (s)0 10 20 30 40

minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref2 (tension ST2-ST3)

Time (s)

0 10 20 30 40minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref3 (tension ST3-ST4)

Time (s)0 10 20 30 40

minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref4 (tension ST4-ST5)

Time (s)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Figure 16 Comparison of transient response to reference input

Journal of Engineering 9

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 1

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600Transient response to disturbance 2

Transient response to disturbance 4

Time (s)

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 3

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600

Time (s)

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Figure 17 Comparison of transient response to disturbance signal

can be reduced and reduction of the controller order maylead to deterioration of the closed-loop performance Variousmethods may be used to reduce the order and in this casestudy the Hankel-norm approximation is usedThe reduced-order controller is equal to 13 that leads to a controllermaintaining the robust stability and performance of theclosed-loop system (matrices for reduced-order controllerare presented in Appendix B)

4 Simulation Results

In the previous section robust controllers for tension con-trol system are introduced It is intended here to comparethe robust stability robust performance and time domainresponses of the proposed controllers Since the considereduncertainty is structured verification of the robust stabilityand performance needs the frequency response in terms of120583 values On the other hand controllers should be tested bymeans of 120583 analysis The robust stability and performancedo not maintain if the uncertainty is unstructured whichshows that the 120583 values give less conservative results if furtherinformation is known about the uncertainty

41 Robust Stability The frequency responses of the upperbound of 120583 for 119867

infincontroller and 120583 and reduced-order

controllers are shown in Figure 14Themaximum120583 values for119867infincontroller are equal to 07801 and the maximum 120583 values

for 120583 and reduced-order controllers are 0202 Thereforethe robust stability of control systems is achieved and thesystems with the 120583 controller and reduced-order controllersare characterized with better robust stability

42 Robust Performance The robust performance of controlsystems is plotted in Figure 15 The maximum 120583 valuesfor 119867

infincontroller are 099 for 120583 controller 0597 and for

reduced-order controller 069 which implies the robust 120583controller and reduced-order controller achieve better robustperformance than119867

infincontroller

43 Time Domain Simulation The transient responses of theclosed-loop system to the reference input and the disturbancesignal are shown in Figures 16 and 17 respectively The 120583 andthe reduced-order controllers give a better transient responsewith smaller overshoot than the119867

infincontroller In summary

10 Journal of Engineering

the 120583 and the reduced-order controllers lead to better trade-off between the requirements in terms of transient responsedisturbance rejection and robustness

5 Conclusion

In cold rolling the strip tension should be controlled forbetter quality of products For cold rolling with 5 standsthe state-space representation perturbed with parametricuncertainties is derived and further 119867

infincontroller and

120583 controller based on robust control theory are designedTension control is obtained by means of the proposed robustcontrollers in the presence of uncertainties and disturbancesThe order of the 120583 controller is large which requires thecontroller order reduction to preserve the closed-loop per-formance and robustness while making implementation ofthe controller easier andmore reliableThe simulation resultsshow that control systems achieve robust stability and robust

performance The comparison of the robust stability androbust performance for controllers as well as the comparisonof the corresponding transient responses shows that the 120583controller and its reduced-order are preferable although thereduced-order controller would be usually preferred dueto its lower order It should be mentioned that in suchimportant industrial process most commissioning personnelhave very little or no background in advanced control theoryespecially in the theory and application of119867

infinrobust control

techniques and are accustomed to single-input-single-output(SISO) proportional-integral (PI) type control loops Thesesimpler configurations are easy to tune at commissioningduring actual operation

Appendices

A The System Matrices

Consider

119860 =

[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

44200 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

minus44200 0 26000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 minus26800 0 18600 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 minus18400 0 12200 0

]]]]]]]]]]

]

1198612=

[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 10 0

0 0 0 0

0 0 0 10

0 0 0 0

]]]]]]]]]]]]

]

1198611=

[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 6280 003 00011 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 minus6220 0 0 0 3600 0017 00011 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 minus3600 0 0 0 2300 0011 0011 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 minus2300 0 0 0 1540 0073 0011

]]]]]]]]]]

]

1198621=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

07

147000

0

0

0

0

0

0

0

0

0

0

0

0

0

0

199000 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

0 0 075 0 0 0 0 0

minus257000 0 157000 0 0 0 0 0

minus197000 0 120000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 0 0 08 0 0 0

0 0 minus242000 0 168000 0 0 0

0 0 minus1210000 0 84000 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 0 0 08 0

0 0 0 0 minus253000 0 168000 0

0 0 0 0 minus84000 0 55000 0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

1198622=

[[[

[

0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

]]]

]

Journal of Engineering 11

11986311=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 21000 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 28300 0135 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 36700 0 0 0 21000 0 0 0 0 0 0 0 0 0 0

0 minus28200 0 0 0 16700 0077 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 minus32300 0 0 0 21000 0 0 0 0 0 0

0 0 0 0 0 minus16100 0 005 10500 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 minus31700 0 0 0 21000 0 0

0 0 0 0 0 0 0 0 0 minus10400 0 0 0 6930 0033 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986312=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 10

0 0 0 0

0 0 0 0

0 0 0 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(A1)

B Matrices for Reduced-Order Controller

Consider

ak =

[[[[[[[[[[[[[[[[[[[[

[

minus13534 9671 2455 minus1674 minus122 155 minus048 299 109 014 minus008 minus019 011

0 minus2734 523 1642 147 minus068 04 minus136 minus065 0005 007 01 minus0048

0 minus2872 minus277 03 minus1267 minus021 005 minus065 003 minus01 009 002 004

0 0 0 minus611 minus184 018 minus029 031 042 minus009 minus002 minus002 0018

0 0 0 145 minus662 minus008 019 minus011 minus006 minus002 0 0003 0

0 0 0 0 0 minus0008 006 minus002 minus0008 minus0001 0 0002 0001

0 0 0 0 0 minus006 minus0004 0008 0008 minus0002 0 0 0001

0 0 0 0 0 0 0 minus002 002 minus0004 minus0001 0 0

0 0 0 0 0 0 0 minus004 minus0008 0002 minus0003 0 0

0 0 0 0 0 0 0 0 0 minus0003 0 0 0

0 0 0 0 0 0 0 0 0 0 minus0002 0 0

0 0 0 0 0 0 0 0 0 0 0 minus0002 0

0 0 0 0 0 0 0 0 0 0 0 0 minus0002

]]]]]]]]]]]]]]]]]]]]

]

12 Journal of Engineering

bk =

[[[[[[[[[[[[[[[[[[[[

[

015 008 002 004

minus004 minus003 minus001 minus001

minus005 minus0002 minus002 0

minus0008 001 001 001

minus0006 0008 0003 minus001

0 0 0 0

0 0 0 0

minus0001 0 0 0

0 0 0 0

0 0 0 0

0001 0 0 0

0 0 0 minus0001

0 0 0 minus0001

]]]]]]]]]]]]]]]]]]]]

]

ck =[[[

[

minus0001 minus0006 0 0001 minus0001 0 0 0 0 0 0 0 0

006 minus002 minus0009 0001 0007 0 0 0 0 0 0 0001 0

0009 minus004 minus003 001 0 0 0 0 0 0 0001 0 0001

013 minus008 minus005 001 0 0 0001 minus0002 0 0 0001 minus0001 minus0001

]]]

]

dk =[[[

[

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

]]]

]

(B1)

Abbreviations

119881 Work roll linear velocity119880V Speed actuator input120591V Speed actuator time constant119881in Input strip velocity119881out Output strip velocityℎin Stand input thicknessℎout Stand output thickness119891 Forward slip ratio120593 Backward slip ratio119864 Youngrsquos modulus119861 Width of strip119867119894+1

Strip thickness between stand 119894 and stand 119894 + 1119871 Length between adjacent stands119879119894119894+1

Strip tension between stand 119894 and stand 119894 + 1

Conflict of Interests

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

References

[1] W S Levine Control System Applications Taylor amp FrancisBoca Raton Fla USA 2011

[2] J Pittner N S Samaras and M A Simaan ldquoA simple rollingmillmodel with linear quadratic optimal controllerrdquo inProceed-ings of the 37th IAS Annual Meeting and World Conference onIndustrial Applications of Electrical Energy pp 142ndash149 October2002

[3] H R Koofigar F Sheikholeslam and S Hosseinnia ldquoUnifiedgauge-tension control in cold rolling mills a robust regulationtechniquerdquo International Journal of Precision Engineering andManufacturing vol 12 no 3 pp 393ndash403 2011

[4] E J M Geddes and I Postlethwaite ldquoImprovements in productquality in tandem cold rolling using robust multivariablecontrolrdquo IEEE Transactions on Control Systems Technology vol6 no 2 pp 257ndash269 1998

[5] E J M Geddes and I Postlethwaite ldquoMultivariable control ofa high performance tandem cold rolling millrdquo in Proceedings ofthe International Conference on Control (CONTROL rsquo94) vol 1pp 202ndash207 Coventry UK March 1994

[6] E Arinton S Caraman and J Korbicz ldquoNeural networks formodelling and fault detection of the inter-stand strip tension ofa cold tandem millrdquo Control Engineering Practice vol 20 no 7pp 684ndash694 2012

[7] KMTakami JMahmoudi andEDahlquist ldquoAdaptive controlof cold rolling system in electrical strips production systemwithonline-offline predictorsrdquo International Journal of AdvancedManufacturing Technology vol 50 no 9ndash12 pp 917ndash930 2010

[8] G Pin V Francesconi F A Cuzzola and T Parisini ldquoAdaptivetask-space metal strip-flatness control in cold multi-roll millstandsrdquo Journal of Process Control vol 23 no 2 pp 108ndash1192013

[9] J Pittner and M A Simaan ldquoAn optimal control method forimprovement in tandem cold metal rollingrdquo in Proceedings ofthe IEEE 42nd AnnualMeeting Industry Applications Conference(IAS rsquo07) pp 382ndash389 September 2007

[10] J R Pittner Pointwise linear quadratic optimal control of atandem cold rolling mill [PhD thesis] School of EngineeringUniversity of Pittsburgh 2006

[11] L E Zarate ldquoPredictive model for the cold rolling processthrough sensitivity factors via neural networksrdquo Journal of the

Journal of Engineering 13

Brazilian Society ofMechanical Sciences and Engineering vol 28no 1 pp 111ndash117 2006

[12] S-H Song and S-K Sul ldquoA new tension controller for con-tinuous strip processing linerdquo IEEE Transactions on IndustryApplications vol 36 no 2 pp 633ndash639 2000

[13] X Zhang and Q Zhang ldquoRobust control of strip tension fortandem cold rolling millrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 2390ndash2393 Yantai China July2011

[14] J Pittner and M A Simaan Tandem Cold Metal Rolling MillControl Using Practical Advanced Methods Springer LondonUK 2011

[15] DGu PH Petkov andMKonstantinovRobust Control Designwith MATLAB Springer London UK 2005

[16] Z Xiaofeng and ZQingdong ldquoRobust control of strip thicknessfor cold rolling millrdquo in Informatics in Control Automation andRobotics vol 133 of Lecture Notes in Electrical Engineering pp777ndash785 Springer Berlin Germany 2012

[17] K Zohu and J C Doyle Essentials of Robust Control PrenticeHall 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Journal of Engineering

Pertin1-16 Pertout1-16

sum

+

+

minus

sum

G

yc1-4

Wp

minusWu

ep1-4

eu1-4

ΔT1-4

Dist1-4

Ref = 0

Figure 9 Structure of the open loop system

Pertin1-16 Pertout1-16

Control1-4

ep1-4

eu1-4

ΔT1-4

Dist1-4 tension ic

Figure 10 Block diagram of open loop system

Pertin1-16 Pertout1-16

T1-4

U1-4

yc1-4

G

K

+

+

+minus

sum

sum

Dist1-4

Ref1-4

Figure 11 Structure of the closed-loop system

interconnection for open-loop and closed-loop system Thestructure of the open-loop system is represented in Figure 9In Figure 10 tension ic denotes the transfer function matrixof open loop system with 24 inputs and 28 outputs

The simulation of closed-loop system with designedcontrollers is based on the structure shown in Figure 11 Theweighting functions 119908

119901and 119908

119906are not included in the block

diagram for obvious reasons

33 119867infin

Controller The interconnection of nominal openloop system is denoted by hin ic as shown in Figure 12

The suboptimal 119867infin

controller minimizes the infinite-normof 119865119897(ℎ119894119899 119894119888 119896) overall stabilizing controller 119870 Note that

119865119897(ℎ119894119899 119894119888 119896) is the transfer function matrix of the nominal

closed-loop system from the disturbances to the errorsTo compute 119867

infinoptimal controller for LTI plant some

conditions are satisfied by plant Sometimes one of theseconditions is not satisfied and the algorithm for computingdoes not work [15 17] When this situation accrued otherrobust control methods are used although some tricks canbe used to satisfy conditions For example adding or sub-tracting epsilon may solve the problem in these conditions

Journal of Engineering 7

K

eDist hin ic

Figure 12 Closed-loop LFTs in119867infindesign

10minus1 100 101 102 103 10401

02

03

04

05

06

Frequency (rads)10minus1 100 101 102 103 104

Frequency (rads)

010

020304050607

Closed-loop complex 120583 controller number 3

Com

plex120583

Closed-loop mixed 120583 controller number 3

Mix

ed120583

Figure 13 120583 plot after third iteration

100

10minus1

10minus2

10minus4 10minus3 10minus2 10minus1 100 101 102 103 104

Robust stability

Frequency (rads)

120583-controllerReduced-order controller

Upp

er b

ound

of120583

Hinfin controller

Figure 14 Comparison of robust stability for three controllers

For this plant 119867infin

controller is numerically computed andfinally the obtained controller is of 20th orderTheminimumvalue of 120574 is 0101

34 120583 Controller The 120583 synthesis is introduced by twoiterative methods 119863-119870 and 120583-119870 iteration For this plant120583 synthesis is executed by the M-file dkit from the RobustControl Toolbox of Matlab which automates the procedureby using 119863-119870 iterations [15] After the third iteration

the maximum value of 120583 is 0567 which indicates that therobust performance has been achieved The final controllerobtained is of 104th order The 120583 plot of the closed-loopsystem is shown in Figure 13

The order of this controller is 104 which makes it difficultin implementation On the other hand high order controllerswill lead to high cost difficult commissioning poor relia-bility and potential problems in maintenance This is theweakness of 120583 controller in this design Usually the order

8 Journal of Engineering

10minus4 10minus3 10minus2 10minus1 100 101 102 103 1040

01

02

03

04

05

06

07

08

09

1Robust performance

Frequency (rads)Hinfin controller120583-controllerReduced-order controller

Upp

er b

ound

of120583

Figure 15 Comparison of robust performance for three controllers

0 10 20 30 40minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref1 (tension ST1-ST2)

Time (s)0 10 20 30 40

minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref2 (tension ST2-ST3)

Time (s)

0 10 20 30 40minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref3 (tension ST3-ST4)

Time (s)0 10 20 30 40

minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref4 (tension ST4-ST5)

Time (s)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Figure 16 Comparison of transient response to reference input

Journal of Engineering 9

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 1

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600Transient response to disturbance 2

Transient response to disturbance 4

Time (s)

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 3

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600

Time (s)

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Figure 17 Comparison of transient response to disturbance signal

can be reduced and reduction of the controller order maylead to deterioration of the closed-loop performance Variousmethods may be used to reduce the order and in this casestudy the Hankel-norm approximation is usedThe reduced-order controller is equal to 13 that leads to a controllermaintaining the robust stability and performance of theclosed-loop system (matrices for reduced-order controllerare presented in Appendix B)

4 Simulation Results

In the previous section robust controllers for tension con-trol system are introduced It is intended here to comparethe robust stability robust performance and time domainresponses of the proposed controllers Since the considereduncertainty is structured verification of the robust stabilityand performance needs the frequency response in terms of120583 values On the other hand controllers should be tested bymeans of 120583 analysis The robust stability and performancedo not maintain if the uncertainty is unstructured whichshows that the 120583 values give less conservative results if furtherinformation is known about the uncertainty

41 Robust Stability The frequency responses of the upperbound of 120583 for 119867

infincontroller and 120583 and reduced-order

controllers are shown in Figure 14Themaximum120583 values for119867infincontroller are equal to 07801 and the maximum 120583 values

for 120583 and reduced-order controllers are 0202 Thereforethe robust stability of control systems is achieved and thesystems with the 120583 controller and reduced-order controllersare characterized with better robust stability

42 Robust Performance The robust performance of controlsystems is plotted in Figure 15 The maximum 120583 valuesfor 119867

infincontroller are 099 for 120583 controller 0597 and for

reduced-order controller 069 which implies the robust 120583controller and reduced-order controller achieve better robustperformance than119867

infincontroller

43 Time Domain Simulation The transient responses of theclosed-loop system to the reference input and the disturbancesignal are shown in Figures 16 and 17 respectively The 120583 andthe reduced-order controllers give a better transient responsewith smaller overshoot than the119867

infincontroller In summary

10 Journal of Engineering

the 120583 and the reduced-order controllers lead to better trade-off between the requirements in terms of transient responsedisturbance rejection and robustness

5 Conclusion

In cold rolling the strip tension should be controlled forbetter quality of products For cold rolling with 5 standsthe state-space representation perturbed with parametricuncertainties is derived and further 119867

infincontroller and

120583 controller based on robust control theory are designedTension control is obtained by means of the proposed robustcontrollers in the presence of uncertainties and disturbancesThe order of the 120583 controller is large which requires thecontroller order reduction to preserve the closed-loop per-formance and robustness while making implementation ofthe controller easier andmore reliableThe simulation resultsshow that control systems achieve robust stability and robust

performance The comparison of the robust stability androbust performance for controllers as well as the comparisonof the corresponding transient responses shows that the 120583controller and its reduced-order are preferable although thereduced-order controller would be usually preferred dueto its lower order It should be mentioned that in suchimportant industrial process most commissioning personnelhave very little or no background in advanced control theoryespecially in the theory and application of119867

infinrobust control

techniques and are accustomed to single-input-single-output(SISO) proportional-integral (PI) type control loops Thesesimpler configurations are easy to tune at commissioningduring actual operation

Appendices

A The System Matrices

Consider

119860 =

[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

44200 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

minus44200 0 26000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 minus26800 0 18600 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 minus18400 0 12200 0

]]]]]]]]]]

]

1198612=

[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 10 0

0 0 0 0

0 0 0 10

0 0 0 0

]]]]]]]]]]]]

]

1198611=

[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 6280 003 00011 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 minus6220 0 0 0 3600 0017 00011 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 minus3600 0 0 0 2300 0011 0011 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 minus2300 0 0 0 1540 0073 0011

]]]]]]]]]]

]

1198621=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

07

147000

0

0

0

0

0

0

0

0

0

0

0

0

0

0

199000 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

0 0 075 0 0 0 0 0

minus257000 0 157000 0 0 0 0 0

minus197000 0 120000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 0 0 08 0 0 0

0 0 minus242000 0 168000 0 0 0

0 0 minus1210000 0 84000 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 0 0 08 0

0 0 0 0 minus253000 0 168000 0

0 0 0 0 minus84000 0 55000 0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

1198622=

[[[

[

0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

]]]

]

Journal of Engineering 11

11986311=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 21000 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 28300 0135 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 36700 0 0 0 21000 0 0 0 0 0 0 0 0 0 0

0 minus28200 0 0 0 16700 0077 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 minus32300 0 0 0 21000 0 0 0 0 0 0

0 0 0 0 0 minus16100 0 005 10500 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 minus31700 0 0 0 21000 0 0

0 0 0 0 0 0 0 0 0 minus10400 0 0 0 6930 0033 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986312=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 10

0 0 0 0

0 0 0 0

0 0 0 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(A1)

B Matrices for Reduced-Order Controller

Consider

ak =

[[[[[[[[[[[[[[[[[[[[

[

minus13534 9671 2455 minus1674 minus122 155 minus048 299 109 014 minus008 minus019 011

0 minus2734 523 1642 147 minus068 04 minus136 minus065 0005 007 01 minus0048

0 minus2872 minus277 03 minus1267 minus021 005 minus065 003 minus01 009 002 004

0 0 0 minus611 minus184 018 minus029 031 042 minus009 minus002 minus002 0018

0 0 0 145 minus662 minus008 019 minus011 minus006 minus002 0 0003 0

0 0 0 0 0 minus0008 006 minus002 minus0008 minus0001 0 0002 0001

0 0 0 0 0 minus006 minus0004 0008 0008 minus0002 0 0 0001

0 0 0 0 0 0 0 minus002 002 minus0004 minus0001 0 0

0 0 0 0 0 0 0 minus004 minus0008 0002 minus0003 0 0

0 0 0 0 0 0 0 0 0 minus0003 0 0 0

0 0 0 0 0 0 0 0 0 0 minus0002 0 0

0 0 0 0 0 0 0 0 0 0 0 minus0002 0

0 0 0 0 0 0 0 0 0 0 0 0 minus0002

]]]]]]]]]]]]]]]]]]]]

]

12 Journal of Engineering

bk =

[[[[[[[[[[[[[[[[[[[[

[

015 008 002 004

minus004 minus003 minus001 minus001

minus005 minus0002 minus002 0

minus0008 001 001 001

minus0006 0008 0003 minus001

0 0 0 0

0 0 0 0

minus0001 0 0 0

0 0 0 0

0 0 0 0

0001 0 0 0

0 0 0 minus0001

0 0 0 minus0001

]]]]]]]]]]]]]]]]]]]]

]

ck =[[[

[

minus0001 minus0006 0 0001 minus0001 0 0 0 0 0 0 0 0

006 minus002 minus0009 0001 0007 0 0 0 0 0 0 0001 0

0009 minus004 minus003 001 0 0 0 0 0 0 0001 0 0001

013 minus008 minus005 001 0 0 0001 minus0002 0 0 0001 minus0001 minus0001

]]]

]

dk =[[[

[

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

]]]

]

(B1)

Abbreviations

119881 Work roll linear velocity119880V Speed actuator input120591V Speed actuator time constant119881in Input strip velocity119881out Output strip velocityℎin Stand input thicknessℎout Stand output thickness119891 Forward slip ratio120593 Backward slip ratio119864 Youngrsquos modulus119861 Width of strip119867119894+1

Strip thickness between stand 119894 and stand 119894 + 1119871 Length between adjacent stands119879119894119894+1

Strip tension between stand 119894 and stand 119894 + 1

Conflict of Interests

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

References

[1] W S Levine Control System Applications Taylor amp FrancisBoca Raton Fla USA 2011

[2] J Pittner N S Samaras and M A Simaan ldquoA simple rollingmillmodel with linear quadratic optimal controllerrdquo inProceed-ings of the 37th IAS Annual Meeting and World Conference onIndustrial Applications of Electrical Energy pp 142ndash149 October2002

[3] H R Koofigar F Sheikholeslam and S Hosseinnia ldquoUnifiedgauge-tension control in cold rolling mills a robust regulationtechniquerdquo International Journal of Precision Engineering andManufacturing vol 12 no 3 pp 393ndash403 2011

[4] E J M Geddes and I Postlethwaite ldquoImprovements in productquality in tandem cold rolling using robust multivariablecontrolrdquo IEEE Transactions on Control Systems Technology vol6 no 2 pp 257ndash269 1998

[5] E J M Geddes and I Postlethwaite ldquoMultivariable control ofa high performance tandem cold rolling millrdquo in Proceedings ofthe International Conference on Control (CONTROL rsquo94) vol 1pp 202ndash207 Coventry UK March 1994

[6] E Arinton S Caraman and J Korbicz ldquoNeural networks formodelling and fault detection of the inter-stand strip tension ofa cold tandem millrdquo Control Engineering Practice vol 20 no 7pp 684ndash694 2012

[7] KMTakami JMahmoudi andEDahlquist ldquoAdaptive controlof cold rolling system in electrical strips production systemwithonline-offline predictorsrdquo International Journal of AdvancedManufacturing Technology vol 50 no 9ndash12 pp 917ndash930 2010

[8] G Pin V Francesconi F A Cuzzola and T Parisini ldquoAdaptivetask-space metal strip-flatness control in cold multi-roll millstandsrdquo Journal of Process Control vol 23 no 2 pp 108ndash1192013

[9] J Pittner and M A Simaan ldquoAn optimal control method forimprovement in tandem cold metal rollingrdquo in Proceedings ofthe IEEE 42nd AnnualMeeting Industry Applications Conference(IAS rsquo07) pp 382ndash389 September 2007

[10] J R Pittner Pointwise linear quadratic optimal control of atandem cold rolling mill [PhD thesis] School of EngineeringUniversity of Pittsburgh 2006

[11] L E Zarate ldquoPredictive model for the cold rolling processthrough sensitivity factors via neural networksrdquo Journal of the

Journal of Engineering 13

Brazilian Society ofMechanical Sciences and Engineering vol 28no 1 pp 111ndash117 2006

[12] S-H Song and S-K Sul ldquoA new tension controller for con-tinuous strip processing linerdquo IEEE Transactions on IndustryApplications vol 36 no 2 pp 633ndash639 2000

[13] X Zhang and Q Zhang ldquoRobust control of strip tension fortandem cold rolling millrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 2390ndash2393 Yantai China July2011

[14] J Pittner and M A Simaan Tandem Cold Metal Rolling MillControl Using Practical Advanced Methods Springer LondonUK 2011

[15] DGu PH Petkov andMKonstantinovRobust Control Designwith MATLAB Springer London UK 2005

[16] Z Xiaofeng and ZQingdong ldquoRobust control of strip thicknessfor cold rolling millrdquo in Informatics in Control Automation andRobotics vol 133 of Lecture Notes in Electrical Engineering pp777ndash785 Springer Berlin Germany 2012

[17] K Zohu and J C Doyle Essentials of Robust Control PrenticeHall 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 7

K

eDist hin ic

Figure 12 Closed-loop LFTs in119867infindesign

10minus1 100 101 102 103 10401

02

03

04

05

06

Frequency (rads)10minus1 100 101 102 103 104

Frequency (rads)

010

020304050607

Closed-loop complex 120583 controller number 3

Com

plex120583

Closed-loop mixed 120583 controller number 3

Mix

ed120583

Figure 13 120583 plot after third iteration

100

10minus1

10minus2

10minus4 10minus3 10minus2 10minus1 100 101 102 103 104

Robust stability

Frequency (rads)

120583-controllerReduced-order controller

Upp

er b

ound

of120583

Hinfin controller

Figure 14 Comparison of robust stability for three controllers

For this plant 119867infin

controller is numerically computed andfinally the obtained controller is of 20th orderTheminimumvalue of 120574 is 0101

34 120583 Controller The 120583 synthesis is introduced by twoiterative methods 119863-119870 and 120583-119870 iteration For this plant120583 synthesis is executed by the M-file dkit from the RobustControl Toolbox of Matlab which automates the procedureby using 119863-119870 iterations [15] After the third iteration

the maximum value of 120583 is 0567 which indicates that therobust performance has been achieved The final controllerobtained is of 104th order The 120583 plot of the closed-loopsystem is shown in Figure 13

The order of this controller is 104 which makes it difficultin implementation On the other hand high order controllerswill lead to high cost difficult commissioning poor relia-bility and potential problems in maintenance This is theweakness of 120583 controller in this design Usually the order

8 Journal of Engineering

10minus4 10minus3 10minus2 10minus1 100 101 102 103 1040

01

02

03

04

05

06

07

08

09

1Robust performance

Frequency (rads)Hinfin controller120583-controllerReduced-order controller

Upp

er b

ound

of120583

Figure 15 Comparison of robust performance for three controllers

0 10 20 30 40minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref1 (tension ST1-ST2)

Time (s)0 10 20 30 40

minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref2 (tension ST2-ST3)

Time (s)

0 10 20 30 40minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref3 (tension ST3-ST4)

Time (s)0 10 20 30 40

minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref4 (tension ST4-ST5)

Time (s)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Figure 16 Comparison of transient response to reference input

Journal of Engineering 9

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 1

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600Transient response to disturbance 2

Transient response to disturbance 4

Time (s)

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 3

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600

Time (s)

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Figure 17 Comparison of transient response to disturbance signal

can be reduced and reduction of the controller order maylead to deterioration of the closed-loop performance Variousmethods may be used to reduce the order and in this casestudy the Hankel-norm approximation is usedThe reduced-order controller is equal to 13 that leads to a controllermaintaining the robust stability and performance of theclosed-loop system (matrices for reduced-order controllerare presented in Appendix B)

4 Simulation Results

In the previous section robust controllers for tension con-trol system are introduced It is intended here to comparethe robust stability robust performance and time domainresponses of the proposed controllers Since the considereduncertainty is structured verification of the robust stabilityand performance needs the frequency response in terms of120583 values On the other hand controllers should be tested bymeans of 120583 analysis The robust stability and performancedo not maintain if the uncertainty is unstructured whichshows that the 120583 values give less conservative results if furtherinformation is known about the uncertainty

41 Robust Stability The frequency responses of the upperbound of 120583 for 119867

infincontroller and 120583 and reduced-order

controllers are shown in Figure 14Themaximum120583 values for119867infincontroller are equal to 07801 and the maximum 120583 values

for 120583 and reduced-order controllers are 0202 Thereforethe robust stability of control systems is achieved and thesystems with the 120583 controller and reduced-order controllersare characterized with better robust stability

42 Robust Performance The robust performance of controlsystems is plotted in Figure 15 The maximum 120583 valuesfor 119867

infincontroller are 099 for 120583 controller 0597 and for

reduced-order controller 069 which implies the robust 120583controller and reduced-order controller achieve better robustperformance than119867

infincontroller

43 Time Domain Simulation The transient responses of theclosed-loop system to the reference input and the disturbancesignal are shown in Figures 16 and 17 respectively The 120583 andthe reduced-order controllers give a better transient responsewith smaller overshoot than the119867

infincontroller In summary

10 Journal of Engineering

the 120583 and the reduced-order controllers lead to better trade-off between the requirements in terms of transient responsedisturbance rejection and robustness

5 Conclusion

In cold rolling the strip tension should be controlled forbetter quality of products For cold rolling with 5 standsthe state-space representation perturbed with parametricuncertainties is derived and further 119867

infincontroller and

120583 controller based on robust control theory are designedTension control is obtained by means of the proposed robustcontrollers in the presence of uncertainties and disturbancesThe order of the 120583 controller is large which requires thecontroller order reduction to preserve the closed-loop per-formance and robustness while making implementation ofthe controller easier andmore reliableThe simulation resultsshow that control systems achieve robust stability and robust

performance The comparison of the robust stability androbust performance for controllers as well as the comparisonof the corresponding transient responses shows that the 120583controller and its reduced-order are preferable although thereduced-order controller would be usually preferred dueto its lower order It should be mentioned that in suchimportant industrial process most commissioning personnelhave very little or no background in advanced control theoryespecially in the theory and application of119867

infinrobust control

techniques and are accustomed to single-input-single-output(SISO) proportional-integral (PI) type control loops Thesesimpler configurations are easy to tune at commissioningduring actual operation

Appendices

A The System Matrices

Consider

119860 =

[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

44200 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

minus44200 0 26000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 minus26800 0 18600 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 minus18400 0 12200 0

]]]]]]]]]]

]

1198612=

[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 10 0

0 0 0 0

0 0 0 10

0 0 0 0

]]]]]]]]]]]]

]

1198611=

[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 6280 003 00011 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 minus6220 0 0 0 3600 0017 00011 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 minus3600 0 0 0 2300 0011 0011 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 minus2300 0 0 0 1540 0073 0011

]]]]]]]]]]

]

1198621=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

07

147000

0

0

0

0

0

0

0

0

0

0

0

0

0

0

199000 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

0 0 075 0 0 0 0 0

minus257000 0 157000 0 0 0 0 0

minus197000 0 120000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 0 0 08 0 0 0

0 0 minus242000 0 168000 0 0 0

0 0 minus1210000 0 84000 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 0 0 08 0

0 0 0 0 minus253000 0 168000 0

0 0 0 0 minus84000 0 55000 0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

1198622=

[[[

[

0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

]]]

]

Journal of Engineering 11

11986311=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 21000 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 28300 0135 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 36700 0 0 0 21000 0 0 0 0 0 0 0 0 0 0

0 minus28200 0 0 0 16700 0077 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 minus32300 0 0 0 21000 0 0 0 0 0 0

0 0 0 0 0 minus16100 0 005 10500 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 minus31700 0 0 0 21000 0 0

0 0 0 0 0 0 0 0 0 minus10400 0 0 0 6930 0033 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986312=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 10

0 0 0 0

0 0 0 0

0 0 0 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(A1)

B Matrices for Reduced-Order Controller

Consider

ak =

[[[[[[[[[[[[[[[[[[[[

[

minus13534 9671 2455 minus1674 minus122 155 minus048 299 109 014 minus008 minus019 011

0 minus2734 523 1642 147 minus068 04 minus136 minus065 0005 007 01 minus0048

0 minus2872 minus277 03 minus1267 minus021 005 minus065 003 minus01 009 002 004

0 0 0 minus611 minus184 018 minus029 031 042 minus009 minus002 minus002 0018

0 0 0 145 minus662 minus008 019 minus011 minus006 minus002 0 0003 0

0 0 0 0 0 minus0008 006 minus002 minus0008 minus0001 0 0002 0001

0 0 0 0 0 minus006 minus0004 0008 0008 minus0002 0 0 0001

0 0 0 0 0 0 0 minus002 002 minus0004 minus0001 0 0

0 0 0 0 0 0 0 minus004 minus0008 0002 minus0003 0 0

0 0 0 0 0 0 0 0 0 minus0003 0 0 0

0 0 0 0 0 0 0 0 0 0 minus0002 0 0

0 0 0 0 0 0 0 0 0 0 0 minus0002 0

0 0 0 0 0 0 0 0 0 0 0 0 minus0002

]]]]]]]]]]]]]]]]]]]]

]

12 Journal of Engineering

bk =

[[[[[[[[[[[[[[[[[[[[

[

015 008 002 004

minus004 minus003 minus001 minus001

minus005 minus0002 minus002 0

minus0008 001 001 001

minus0006 0008 0003 minus001

0 0 0 0

0 0 0 0

minus0001 0 0 0

0 0 0 0

0 0 0 0

0001 0 0 0

0 0 0 minus0001

0 0 0 minus0001

]]]]]]]]]]]]]]]]]]]]

]

ck =[[[

[

minus0001 minus0006 0 0001 minus0001 0 0 0 0 0 0 0 0

006 minus002 minus0009 0001 0007 0 0 0 0 0 0 0001 0

0009 minus004 minus003 001 0 0 0 0 0 0 0001 0 0001

013 minus008 minus005 001 0 0 0001 minus0002 0 0 0001 minus0001 minus0001

]]]

]

dk =[[[

[

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

]]]

]

(B1)

Abbreviations

119881 Work roll linear velocity119880V Speed actuator input120591V Speed actuator time constant119881in Input strip velocity119881out Output strip velocityℎin Stand input thicknessℎout Stand output thickness119891 Forward slip ratio120593 Backward slip ratio119864 Youngrsquos modulus119861 Width of strip119867119894+1

Strip thickness between stand 119894 and stand 119894 + 1119871 Length between adjacent stands119879119894119894+1

Strip tension between stand 119894 and stand 119894 + 1

Conflict of Interests

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

References

[1] W S Levine Control System Applications Taylor amp FrancisBoca Raton Fla USA 2011

[2] J Pittner N S Samaras and M A Simaan ldquoA simple rollingmillmodel with linear quadratic optimal controllerrdquo inProceed-ings of the 37th IAS Annual Meeting and World Conference onIndustrial Applications of Electrical Energy pp 142ndash149 October2002

[3] H R Koofigar F Sheikholeslam and S Hosseinnia ldquoUnifiedgauge-tension control in cold rolling mills a robust regulationtechniquerdquo International Journal of Precision Engineering andManufacturing vol 12 no 3 pp 393ndash403 2011

[4] E J M Geddes and I Postlethwaite ldquoImprovements in productquality in tandem cold rolling using robust multivariablecontrolrdquo IEEE Transactions on Control Systems Technology vol6 no 2 pp 257ndash269 1998

[5] E J M Geddes and I Postlethwaite ldquoMultivariable control ofa high performance tandem cold rolling millrdquo in Proceedings ofthe International Conference on Control (CONTROL rsquo94) vol 1pp 202ndash207 Coventry UK March 1994

[6] E Arinton S Caraman and J Korbicz ldquoNeural networks formodelling and fault detection of the inter-stand strip tension ofa cold tandem millrdquo Control Engineering Practice vol 20 no 7pp 684ndash694 2012

[7] KMTakami JMahmoudi andEDahlquist ldquoAdaptive controlof cold rolling system in electrical strips production systemwithonline-offline predictorsrdquo International Journal of AdvancedManufacturing Technology vol 50 no 9ndash12 pp 917ndash930 2010

[8] G Pin V Francesconi F A Cuzzola and T Parisini ldquoAdaptivetask-space metal strip-flatness control in cold multi-roll millstandsrdquo Journal of Process Control vol 23 no 2 pp 108ndash1192013

[9] J Pittner and M A Simaan ldquoAn optimal control method forimprovement in tandem cold metal rollingrdquo in Proceedings ofthe IEEE 42nd AnnualMeeting Industry Applications Conference(IAS rsquo07) pp 382ndash389 September 2007

[10] J R Pittner Pointwise linear quadratic optimal control of atandem cold rolling mill [PhD thesis] School of EngineeringUniversity of Pittsburgh 2006

[11] L E Zarate ldquoPredictive model for the cold rolling processthrough sensitivity factors via neural networksrdquo Journal of the

Journal of Engineering 13

Brazilian Society ofMechanical Sciences and Engineering vol 28no 1 pp 111ndash117 2006

[12] S-H Song and S-K Sul ldquoA new tension controller for con-tinuous strip processing linerdquo IEEE Transactions on IndustryApplications vol 36 no 2 pp 633ndash639 2000

[13] X Zhang and Q Zhang ldquoRobust control of strip tension fortandem cold rolling millrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 2390ndash2393 Yantai China July2011

[14] J Pittner and M A Simaan Tandem Cold Metal Rolling MillControl Using Practical Advanced Methods Springer LondonUK 2011

[15] DGu PH Petkov andMKonstantinovRobust Control Designwith MATLAB Springer London UK 2005

[16] Z Xiaofeng and ZQingdong ldquoRobust control of strip thicknessfor cold rolling millrdquo in Informatics in Control Automation andRobotics vol 133 of Lecture Notes in Electrical Engineering pp777ndash785 Springer Berlin Germany 2012

[17] K Zohu and J C Doyle Essentials of Robust Control PrenticeHall 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Engineering

10minus4 10minus3 10minus2 10minus1 100 101 102 103 1040

01

02

03

04

05

06

07

08

09

1Robust performance

Frequency (rads)Hinfin controller120583-controllerReduced-order controller

Upp

er b

ound

of120583

Figure 15 Comparison of robust performance for three controllers

0 10 20 30 40minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref1 (tension ST1-ST2)

Time (s)0 10 20 30 40

minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref2 (tension ST2-ST3)

Time (s)

0 10 20 30 40minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref3 (tension ST3-ST4)

Time (s)0 10 20 30 40

minus4000

minus2000

0

2000

4000

6000

8000Transient response to ref4 (tension ST4-ST5)

Time (s)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Figure 16 Comparison of transient response to reference input

Journal of Engineering 9

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 1

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600Transient response to disturbance 2

Transient response to disturbance 4

Time (s)

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 3

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600

Time (s)

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Figure 17 Comparison of transient response to disturbance signal

can be reduced and reduction of the controller order maylead to deterioration of the closed-loop performance Variousmethods may be used to reduce the order and in this casestudy the Hankel-norm approximation is usedThe reduced-order controller is equal to 13 that leads to a controllermaintaining the robust stability and performance of theclosed-loop system (matrices for reduced-order controllerare presented in Appendix B)

4 Simulation Results

In the previous section robust controllers for tension con-trol system are introduced It is intended here to comparethe robust stability robust performance and time domainresponses of the proposed controllers Since the considereduncertainty is structured verification of the robust stabilityand performance needs the frequency response in terms of120583 values On the other hand controllers should be tested bymeans of 120583 analysis The robust stability and performancedo not maintain if the uncertainty is unstructured whichshows that the 120583 values give less conservative results if furtherinformation is known about the uncertainty

41 Robust Stability The frequency responses of the upperbound of 120583 for 119867

infincontroller and 120583 and reduced-order

controllers are shown in Figure 14Themaximum120583 values for119867infincontroller are equal to 07801 and the maximum 120583 values

for 120583 and reduced-order controllers are 0202 Thereforethe robust stability of control systems is achieved and thesystems with the 120583 controller and reduced-order controllersare characterized with better robust stability

42 Robust Performance The robust performance of controlsystems is plotted in Figure 15 The maximum 120583 valuesfor 119867

infincontroller are 099 for 120583 controller 0597 and for

reduced-order controller 069 which implies the robust 120583controller and reduced-order controller achieve better robustperformance than119867

infincontroller

43 Time Domain Simulation The transient responses of theclosed-loop system to the reference input and the disturbancesignal are shown in Figures 16 and 17 respectively The 120583 andthe reduced-order controllers give a better transient responsewith smaller overshoot than the119867

infincontroller In summary

10 Journal of Engineering

the 120583 and the reduced-order controllers lead to better trade-off between the requirements in terms of transient responsedisturbance rejection and robustness

5 Conclusion

In cold rolling the strip tension should be controlled forbetter quality of products For cold rolling with 5 standsthe state-space representation perturbed with parametricuncertainties is derived and further 119867

infincontroller and

120583 controller based on robust control theory are designedTension control is obtained by means of the proposed robustcontrollers in the presence of uncertainties and disturbancesThe order of the 120583 controller is large which requires thecontroller order reduction to preserve the closed-loop per-formance and robustness while making implementation ofthe controller easier andmore reliableThe simulation resultsshow that control systems achieve robust stability and robust

performance The comparison of the robust stability androbust performance for controllers as well as the comparisonof the corresponding transient responses shows that the 120583controller and its reduced-order are preferable although thereduced-order controller would be usually preferred dueto its lower order It should be mentioned that in suchimportant industrial process most commissioning personnelhave very little or no background in advanced control theoryespecially in the theory and application of119867

infinrobust control

techniques and are accustomed to single-input-single-output(SISO) proportional-integral (PI) type control loops Thesesimpler configurations are easy to tune at commissioningduring actual operation

Appendices

A The System Matrices

Consider

119860 =

[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

44200 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

minus44200 0 26000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 minus26800 0 18600 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 minus18400 0 12200 0

]]]]]]]]]]

]

1198612=

[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 10 0

0 0 0 0

0 0 0 10

0 0 0 0

]]]]]]]]]]]]

]

1198611=

[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 6280 003 00011 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 minus6220 0 0 0 3600 0017 00011 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 minus3600 0 0 0 2300 0011 0011 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 minus2300 0 0 0 1540 0073 0011

]]]]]]]]]]

]

1198621=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

07

147000

0

0

0

0

0

0

0

0

0

0

0

0

0

0

199000 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

0 0 075 0 0 0 0 0

minus257000 0 157000 0 0 0 0 0

minus197000 0 120000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 0 0 08 0 0 0

0 0 minus242000 0 168000 0 0 0

0 0 minus1210000 0 84000 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 0 0 08 0

0 0 0 0 minus253000 0 168000 0

0 0 0 0 minus84000 0 55000 0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

1198622=

[[[

[

0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

]]]

]

Journal of Engineering 11

11986311=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 21000 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 28300 0135 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 36700 0 0 0 21000 0 0 0 0 0 0 0 0 0 0

0 minus28200 0 0 0 16700 0077 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 minus32300 0 0 0 21000 0 0 0 0 0 0

0 0 0 0 0 minus16100 0 005 10500 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 minus31700 0 0 0 21000 0 0

0 0 0 0 0 0 0 0 0 minus10400 0 0 0 6930 0033 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986312=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 10

0 0 0 0

0 0 0 0

0 0 0 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(A1)

B Matrices for Reduced-Order Controller

Consider

ak =

[[[[[[[[[[[[[[[[[[[[

[

minus13534 9671 2455 minus1674 minus122 155 minus048 299 109 014 minus008 minus019 011

0 minus2734 523 1642 147 minus068 04 minus136 minus065 0005 007 01 minus0048

0 minus2872 minus277 03 minus1267 minus021 005 minus065 003 minus01 009 002 004

0 0 0 minus611 minus184 018 minus029 031 042 minus009 minus002 minus002 0018

0 0 0 145 minus662 minus008 019 minus011 minus006 minus002 0 0003 0

0 0 0 0 0 minus0008 006 minus002 minus0008 minus0001 0 0002 0001

0 0 0 0 0 minus006 minus0004 0008 0008 minus0002 0 0 0001

0 0 0 0 0 0 0 minus002 002 minus0004 minus0001 0 0

0 0 0 0 0 0 0 minus004 minus0008 0002 minus0003 0 0

0 0 0 0 0 0 0 0 0 minus0003 0 0 0

0 0 0 0 0 0 0 0 0 0 minus0002 0 0

0 0 0 0 0 0 0 0 0 0 0 minus0002 0

0 0 0 0 0 0 0 0 0 0 0 0 minus0002

]]]]]]]]]]]]]]]]]]]]

]

12 Journal of Engineering

bk =

[[[[[[[[[[[[[[[[[[[[

[

015 008 002 004

minus004 minus003 minus001 minus001

minus005 minus0002 minus002 0

minus0008 001 001 001

minus0006 0008 0003 minus001

0 0 0 0

0 0 0 0

minus0001 0 0 0

0 0 0 0

0 0 0 0

0001 0 0 0

0 0 0 minus0001

0 0 0 minus0001

]]]]]]]]]]]]]]]]]]]]

]

ck =[[[

[

minus0001 minus0006 0 0001 minus0001 0 0 0 0 0 0 0 0

006 minus002 minus0009 0001 0007 0 0 0 0 0 0 0001 0

0009 minus004 minus003 001 0 0 0 0 0 0 0001 0 0001

013 minus008 minus005 001 0 0 0001 minus0002 0 0 0001 minus0001 minus0001

]]]

]

dk =[[[

[

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

]]]

]

(B1)

Abbreviations

119881 Work roll linear velocity119880V Speed actuator input120591V Speed actuator time constant119881in Input strip velocity119881out Output strip velocityℎin Stand input thicknessℎout Stand output thickness119891 Forward slip ratio120593 Backward slip ratio119864 Youngrsquos modulus119861 Width of strip119867119894+1

Strip thickness between stand 119894 and stand 119894 + 1119871 Length between adjacent stands119879119894119894+1

Strip tension between stand 119894 and stand 119894 + 1

Conflict of Interests

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

References

[1] W S Levine Control System Applications Taylor amp FrancisBoca Raton Fla USA 2011

[2] J Pittner N S Samaras and M A Simaan ldquoA simple rollingmillmodel with linear quadratic optimal controllerrdquo inProceed-ings of the 37th IAS Annual Meeting and World Conference onIndustrial Applications of Electrical Energy pp 142ndash149 October2002

[3] H R Koofigar F Sheikholeslam and S Hosseinnia ldquoUnifiedgauge-tension control in cold rolling mills a robust regulationtechniquerdquo International Journal of Precision Engineering andManufacturing vol 12 no 3 pp 393ndash403 2011

[4] E J M Geddes and I Postlethwaite ldquoImprovements in productquality in tandem cold rolling using robust multivariablecontrolrdquo IEEE Transactions on Control Systems Technology vol6 no 2 pp 257ndash269 1998

[5] E J M Geddes and I Postlethwaite ldquoMultivariable control ofa high performance tandem cold rolling millrdquo in Proceedings ofthe International Conference on Control (CONTROL rsquo94) vol 1pp 202ndash207 Coventry UK March 1994

[6] E Arinton S Caraman and J Korbicz ldquoNeural networks formodelling and fault detection of the inter-stand strip tension ofa cold tandem millrdquo Control Engineering Practice vol 20 no 7pp 684ndash694 2012

[7] KMTakami JMahmoudi andEDahlquist ldquoAdaptive controlof cold rolling system in electrical strips production systemwithonline-offline predictorsrdquo International Journal of AdvancedManufacturing Technology vol 50 no 9ndash12 pp 917ndash930 2010

[8] G Pin V Francesconi F A Cuzzola and T Parisini ldquoAdaptivetask-space metal strip-flatness control in cold multi-roll millstandsrdquo Journal of Process Control vol 23 no 2 pp 108ndash1192013

[9] J Pittner and M A Simaan ldquoAn optimal control method forimprovement in tandem cold metal rollingrdquo in Proceedings ofthe IEEE 42nd AnnualMeeting Industry Applications Conference(IAS rsquo07) pp 382ndash389 September 2007

[10] J R Pittner Pointwise linear quadratic optimal control of atandem cold rolling mill [PhD thesis] School of EngineeringUniversity of Pittsburgh 2006

[11] L E Zarate ldquoPredictive model for the cold rolling processthrough sensitivity factors via neural networksrdquo Journal of the

Journal of Engineering 13

Brazilian Society ofMechanical Sciences and Engineering vol 28no 1 pp 111ndash117 2006

[12] S-H Song and S-K Sul ldquoA new tension controller for con-tinuous strip processing linerdquo IEEE Transactions on IndustryApplications vol 36 no 2 pp 633ndash639 2000

[13] X Zhang and Q Zhang ldquoRobust control of strip tension fortandem cold rolling millrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 2390ndash2393 Yantai China July2011

[14] J Pittner and M A Simaan Tandem Cold Metal Rolling MillControl Using Practical Advanced Methods Springer LondonUK 2011

[15] DGu PH Petkov andMKonstantinovRobust Control Designwith MATLAB Springer London UK 2005

[16] Z Xiaofeng and ZQingdong ldquoRobust control of strip thicknessfor cold rolling millrdquo in Informatics in Control Automation andRobotics vol 133 of Lecture Notes in Electrical Engineering pp777ndash785 Springer Berlin Germany 2012

[17] K Zohu and J C Doyle Essentials of Robust Control PrenticeHall 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 9

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 1

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600Transient response to disturbance 2

Transient response to disturbance 4

Time (s)

0 10 20 30 40minus600

minus400

minus200

0

200

400

600Transient response to disturbance 3

Time (s)0 10 20 30 40

minus600

minus400

minus200

0

200

400

600

Time (s)

Tens

ion1

-2(N

)

Tens

ion2

-3(N

)

Tens

ion3

-4(N

)

Tens

ion4

-5(N

)

Hinfin controller120583-controller

Reduced-order controllerRef

Hinfin controller120583-controller

Reduced-order controllerRef

Figure 17 Comparison of transient response to disturbance signal

can be reduced and reduction of the controller order maylead to deterioration of the closed-loop performance Variousmethods may be used to reduce the order and in this casestudy the Hankel-norm approximation is usedThe reduced-order controller is equal to 13 that leads to a controllermaintaining the robust stability and performance of theclosed-loop system (matrices for reduced-order controllerare presented in Appendix B)

4 Simulation Results

In the previous section robust controllers for tension con-trol system are introduced It is intended here to comparethe robust stability robust performance and time domainresponses of the proposed controllers Since the considereduncertainty is structured verification of the robust stabilityand performance needs the frequency response in terms of120583 values On the other hand controllers should be tested bymeans of 120583 analysis The robust stability and performancedo not maintain if the uncertainty is unstructured whichshows that the 120583 values give less conservative results if furtherinformation is known about the uncertainty

41 Robust Stability The frequency responses of the upperbound of 120583 for 119867

infincontroller and 120583 and reduced-order

controllers are shown in Figure 14Themaximum120583 values for119867infincontroller are equal to 07801 and the maximum 120583 values

for 120583 and reduced-order controllers are 0202 Thereforethe robust stability of control systems is achieved and thesystems with the 120583 controller and reduced-order controllersare characterized with better robust stability

42 Robust Performance The robust performance of controlsystems is plotted in Figure 15 The maximum 120583 valuesfor 119867

infincontroller are 099 for 120583 controller 0597 and for

reduced-order controller 069 which implies the robust 120583controller and reduced-order controller achieve better robustperformance than119867

infincontroller

43 Time Domain Simulation The transient responses of theclosed-loop system to the reference input and the disturbancesignal are shown in Figures 16 and 17 respectively The 120583 andthe reduced-order controllers give a better transient responsewith smaller overshoot than the119867

infincontroller In summary

10 Journal of Engineering

the 120583 and the reduced-order controllers lead to better trade-off between the requirements in terms of transient responsedisturbance rejection and robustness

5 Conclusion

In cold rolling the strip tension should be controlled forbetter quality of products For cold rolling with 5 standsthe state-space representation perturbed with parametricuncertainties is derived and further 119867

infincontroller and

120583 controller based on robust control theory are designedTension control is obtained by means of the proposed robustcontrollers in the presence of uncertainties and disturbancesThe order of the 120583 controller is large which requires thecontroller order reduction to preserve the closed-loop per-formance and robustness while making implementation ofthe controller easier andmore reliableThe simulation resultsshow that control systems achieve robust stability and robust

performance The comparison of the robust stability androbust performance for controllers as well as the comparisonof the corresponding transient responses shows that the 120583controller and its reduced-order are preferable although thereduced-order controller would be usually preferred dueto its lower order It should be mentioned that in suchimportant industrial process most commissioning personnelhave very little or no background in advanced control theoryespecially in the theory and application of119867

infinrobust control

techniques and are accustomed to single-input-single-output(SISO) proportional-integral (PI) type control loops Thesesimpler configurations are easy to tune at commissioningduring actual operation

Appendices

A The System Matrices

Consider

119860 =

[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

44200 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

minus44200 0 26000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 minus26800 0 18600 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 minus18400 0 12200 0

]]]]]]]]]]

]

1198612=

[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 10 0

0 0 0 0

0 0 0 10

0 0 0 0

]]]]]]]]]]]]

]

1198611=

[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 6280 003 00011 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 minus6220 0 0 0 3600 0017 00011 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 minus3600 0 0 0 2300 0011 0011 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 minus2300 0 0 0 1540 0073 0011

]]]]]]]]]]

]

1198621=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

07

147000

0

0

0

0

0

0

0

0

0

0

0

0

0

0

199000 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

0 0 075 0 0 0 0 0

minus257000 0 157000 0 0 0 0 0

minus197000 0 120000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 0 0 08 0 0 0

0 0 minus242000 0 168000 0 0 0

0 0 minus1210000 0 84000 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 0 0 08 0

0 0 0 0 minus253000 0 168000 0

0 0 0 0 minus84000 0 55000 0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

1198622=

[[[

[

0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

]]]

]

Journal of Engineering 11

11986311=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 21000 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 28300 0135 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 36700 0 0 0 21000 0 0 0 0 0 0 0 0 0 0

0 minus28200 0 0 0 16700 0077 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 minus32300 0 0 0 21000 0 0 0 0 0 0

0 0 0 0 0 minus16100 0 005 10500 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 minus31700 0 0 0 21000 0 0

0 0 0 0 0 0 0 0 0 minus10400 0 0 0 6930 0033 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986312=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 10

0 0 0 0

0 0 0 0

0 0 0 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(A1)

B Matrices for Reduced-Order Controller

Consider

ak =

[[[[[[[[[[[[[[[[[[[[

[

minus13534 9671 2455 minus1674 minus122 155 minus048 299 109 014 minus008 minus019 011

0 minus2734 523 1642 147 minus068 04 minus136 minus065 0005 007 01 minus0048

0 minus2872 minus277 03 minus1267 minus021 005 minus065 003 minus01 009 002 004

0 0 0 minus611 minus184 018 minus029 031 042 minus009 minus002 minus002 0018

0 0 0 145 minus662 minus008 019 minus011 minus006 minus002 0 0003 0

0 0 0 0 0 minus0008 006 minus002 minus0008 minus0001 0 0002 0001

0 0 0 0 0 minus006 minus0004 0008 0008 minus0002 0 0 0001

0 0 0 0 0 0 0 minus002 002 minus0004 minus0001 0 0

0 0 0 0 0 0 0 minus004 minus0008 0002 minus0003 0 0

0 0 0 0 0 0 0 0 0 minus0003 0 0 0

0 0 0 0 0 0 0 0 0 0 minus0002 0 0

0 0 0 0 0 0 0 0 0 0 0 minus0002 0

0 0 0 0 0 0 0 0 0 0 0 0 minus0002

]]]]]]]]]]]]]]]]]]]]

]

12 Journal of Engineering

bk =

[[[[[[[[[[[[[[[[[[[[

[

015 008 002 004

minus004 minus003 minus001 minus001

minus005 minus0002 minus002 0

minus0008 001 001 001

minus0006 0008 0003 minus001

0 0 0 0

0 0 0 0

minus0001 0 0 0

0 0 0 0

0 0 0 0

0001 0 0 0

0 0 0 minus0001

0 0 0 minus0001

]]]]]]]]]]]]]]]]]]]]

]

ck =[[[

[

minus0001 minus0006 0 0001 minus0001 0 0 0 0 0 0 0 0

006 minus002 minus0009 0001 0007 0 0 0 0 0 0 0001 0

0009 minus004 minus003 001 0 0 0 0 0 0 0001 0 0001

013 minus008 minus005 001 0 0 0001 minus0002 0 0 0001 minus0001 minus0001

]]]

]

dk =[[[

[

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

]]]

]

(B1)

Abbreviations

119881 Work roll linear velocity119880V Speed actuator input120591V Speed actuator time constant119881in Input strip velocity119881out Output strip velocityℎin Stand input thicknessℎout Stand output thickness119891 Forward slip ratio120593 Backward slip ratio119864 Youngrsquos modulus119861 Width of strip119867119894+1

Strip thickness between stand 119894 and stand 119894 + 1119871 Length between adjacent stands119879119894119894+1

Strip tension between stand 119894 and stand 119894 + 1

Conflict of Interests

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

References

[1] W S Levine Control System Applications Taylor amp FrancisBoca Raton Fla USA 2011

[2] J Pittner N S Samaras and M A Simaan ldquoA simple rollingmillmodel with linear quadratic optimal controllerrdquo inProceed-ings of the 37th IAS Annual Meeting and World Conference onIndustrial Applications of Electrical Energy pp 142ndash149 October2002

[3] H R Koofigar F Sheikholeslam and S Hosseinnia ldquoUnifiedgauge-tension control in cold rolling mills a robust regulationtechniquerdquo International Journal of Precision Engineering andManufacturing vol 12 no 3 pp 393ndash403 2011

[4] E J M Geddes and I Postlethwaite ldquoImprovements in productquality in tandem cold rolling using robust multivariablecontrolrdquo IEEE Transactions on Control Systems Technology vol6 no 2 pp 257ndash269 1998

[5] E J M Geddes and I Postlethwaite ldquoMultivariable control ofa high performance tandem cold rolling millrdquo in Proceedings ofthe International Conference on Control (CONTROL rsquo94) vol 1pp 202ndash207 Coventry UK March 1994

[6] E Arinton S Caraman and J Korbicz ldquoNeural networks formodelling and fault detection of the inter-stand strip tension ofa cold tandem millrdquo Control Engineering Practice vol 20 no 7pp 684ndash694 2012

[7] KMTakami JMahmoudi andEDahlquist ldquoAdaptive controlof cold rolling system in electrical strips production systemwithonline-offline predictorsrdquo International Journal of AdvancedManufacturing Technology vol 50 no 9ndash12 pp 917ndash930 2010

[8] G Pin V Francesconi F A Cuzzola and T Parisini ldquoAdaptivetask-space metal strip-flatness control in cold multi-roll millstandsrdquo Journal of Process Control vol 23 no 2 pp 108ndash1192013

[9] J Pittner and M A Simaan ldquoAn optimal control method forimprovement in tandem cold metal rollingrdquo in Proceedings ofthe IEEE 42nd AnnualMeeting Industry Applications Conference(IAS rsquo07) pp 382ndash389 September 2007

[10] J R Pittner Pointwise linear quadratic optimal control of atandem cold rolling mill [PhD thesis] School of EngineeringUniversity of Pittsburgh 2006

[11] L E Zarate ldquoPredictive model for the cold rolling processthrough sensitivity factors via neural networksrdquo Journal of the

Journal of Engineering 13

Brazilian Society ofMechanical Sciences and Engineering vol 28no 1 pp 111ndash117 2006

[12] S-H Song and S-K Sul ldquoA new tension controller for con-tinuous strip processing linerdquo IEEE Transactions on IndustryApplications vol 36 no 2 pp 633ndash639 2000

[13] X Zhang and Q Zhang ldquoRobust control of strip tension fortandem cold rolling millrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 2390ndash2393 Yantai China July2011

[14] J Pittner and M A Simaan Tandem Cold Metal Rolling MillControl Using Practical Advanced Methods Springer LondonUK 2011

[15] DGu PH Petkov andMKonstantinovRobust Control Designwith MATLAB Springer London UK 2005

[16] Z Xiaofeng and ZQingdong ldquoRobust control of strip thicknessfor cold rolling millrdquo in Informatics in Control Automation andRobotics vol 133 of Lecture Notes in Electrical Engineering pp777ndash785 Springer Berlin Germany 2012

[17] K Zohu and J C Doyle Essentials of Robust Control PrenticeHall 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Journal of Engineering

the 120583 and the reduced-order controllers lead to better trade-off between the requirements in terms of transient responsedisturbance rejection and robustness

5 Conclusion

In cold rolling the strip tension should be controlled forbetter quality of products For cold rolling with 5 standsthe state-space representation perturbed with parametricuncertainties is derived and further 119867

infincontroller and

120583 controller based on robust control theory are designedTension control is obtained by means of the proposed robustcontrollers in the presence of uncertainties and disturbancesThe order of the 120583 controller is large which requires thecontroller order reduction to preserve the closed-loop per-formance and robustness while making implementation ofthe controller easier andmore reliableThe simulation resultsshow that control systems achieve robust stability and robust

performance The comparison of the robust stability androbust performance for controllers as well as the comparisonof the corresponding transient responses shows that the 120583controller and its reduced-order are preferable although thereduced-order controller would be usually preferred dueto its lower order It should be mentioned that in suchimportant industrial process most commissioning personnelhave very little or no background in advanced control theoryespecially in the theory and application of119867

infinrobust control

techniques and are accustomed to single-input-single-output(SISO) proportional-integral (PI) type control loops Thesesimpler configurations are easy to tune at commissioningduring actual operation

Appendices

A The System Matrices

Consider

119860 =

[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

44200 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

minus44200 0 26000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 minus26800 0 18600 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 minus18400 0 12200 0

]]]]]]]]]]

]

1198612=

[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 10 0

0 0 0 0

0 0 0 10

0 0 0 0

]]]]]]]]]]]]

]

1198611=

[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 6280 003 00011 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 minus6220 0 0 0 3600 0017 00011 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 minus3600 0 0 0 2300 0011 0011 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 minus2300 0 0 0 1540 0073 0011

]]]]]]]]]]

]

1198621=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus10 0 0 0 0 0 0 0

07

147000

0

0

0

0

0

0

0

0

0

0

0

0

0

0

199000 0 0 0 0 0 0 0

0 0 minus10 0 0 0 0 0

0 0 075 0 0 0 0 0

minus257000 0 157000 0 0 0 0 0

minus197000 0 120000 0 0 0 0 0

0 0 0 0 minus10 0 0 0

0 0 0 0 08 0 0 0

0 0 minus242000 0 168000 0 0 0

0 0 minus1210000 0 84000 0 0 0

0 0 0 0 0 0 minus10 0

0 0 0 0 0 0 08 0

0 0 0 0 minus253000 0 168000 0

0 0 0 0 minus84000 0 55000 0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

1198622=

[[[

[

0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

]]]

]

Journal of Engineering 11

11986311=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 21000 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 28300 0135 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 36700 0 0 0 21000 0 0 0 0 0 0 0 0 0 0

0 minus28200 0 0 0 16700 0077 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 minus32300 0 0 0 21000 0 0 0 0 0 0

0 0 0 0 0 minus16100 0 005 10500 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 minus31700 0 0 0 21000 0 0

0 0 0 0 0 0 0 0 0 minus10400 0 0 0 6930 0033 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986312=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 10

0 0 0 0

0 0 0 0

0 0 0 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(A1)

B Matrices for Reduced-Order Controller

Consider

ak =

[[[[[[[[[[[[[[[[[[[[

[

minus13534 9671 2455 minus1674 minus122 155 minus048 299 109 014 minus008 minus019 011

0 minus2734 523 1642 147 minus068 04 minus136 minus065 0005 007 01 minus0048

0 minus2872 minus277 03 minus1267 minus021 005 minus065 003 minus01 009 002 004

0 0 0 minus611 minus184 018 minus029 031 042 minus009 minus002 minus002 0018

0 0 0 145 minus662 minus008 019 minus011 minus006 minus002 0 0003 0

0 0 0 0 0 minus0008 006 minus002 minus0008 minus0001 0 0002 0001

0 0 0 0 0 minus006 minus0004 0008 0008 minus0002 0 0 0001

0 0 0 0 0 0 0 minus002 002 minus0004 minus0001 0 0

0 0 0 0 0 0 0 minus004 minus0008 0002 minus0003 0 0

0 0 0 0 0 0 0 0 0 minus0003 0 0 0

0 0 0 0 0 0 0 0 0 0 minus0002 0 0

0 0 0 0 0 0 0 0 0 0 0 minus0002 0

0 0 0 0 0 0 0 0 0 0 0 0 minus0002

]]]]]]]]]]]]]]]]]]]]

]

12 Journal of Engineering

bk =

[[[[[[[[[[[[[[[[[[[[

[

015 008 002 004

minus004 minus003 minus001 minus001

minus005 minus0002 minus002 0

minus0008 001 001 001

minus0006 0008 0003 minus001

0 0 0 0

0 0 0 0

minus0001 0 0 0

0 0 0 0

0 0 0 0

0001 0 0 0

0 0 0 minus0001

0 0 0 minus0001

]]]]]]]]]]]]]]]]]]]]

]

ck =[[[

[

minus0001 minus0006 0 0001 minus0001 0 0 0 0 0 0 0 0

006 minus002 minus0009 0001 0007 0 0 0 0 0 0 0001 0

0009 minus004 minus003 001 0 0 0 0 0 0 0001 0 0001

013 minus008 minus005 001 0 0 0001 minus0002 0 0 0001 minus0001 minus0001

]]]

]

dk =[[[

[

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

]]]

]

(B1)

Abbreviations

119881 Work roll linear velocity119880V Speed actuator input120591V Speed actuator time constant119881in Input strip velocity119881out Output strip velocityℎin Stand input thicknessℎout Stand output thickness119891 Forward slip ratio120593 Backward slip ratio119864 Youngrsquos modulus119861 Width of strip119867119894+1

Strip thickness between stand 119894 and stand 119894 + 1119871 Length between adjacent stands119879119894119894+1

Strip tension between stand 119894 and stand 119894 + 1

Conflict of Interests

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

References

[1] W S Levine Control System Applications Taylor amp FrancisBoca Raton Fla USA 2011

[2] J Pittner N S Samaras and M A Simaan ldquoA simple rollingmillmodel with linear quadratic optimal controllerrdquo inProceed-ings of the 37th IAS Annual Meeting and World Conference onIndustrial Applications of Electrical Energy pp 142ndash149 October2002

[3] H R Koofigar F Sheikholeslam and S Hosseinnia ldquoUnifiedgauge-tension control in cold rolling mills a robust regulationtechniquerdquo International Journal of Precision Engineering andManufacturing vol 12 no 3 pp 393ndash403 2011

[4] E J M Geddes and I Postlethwaite ldquoImprovements in productquality in tandem cold rolling using robust multivariablecontrolrdquo IEEE Transactions on Control Systems Technology vol6 no 2 pp 257ndash269 1998

[5] E J M Geddes and I Postlethwaite ldquoMultivariable control ofa high performance tandem cold rolling millrdquo in Proceedings ofthe International Conference on Control (CONTROL rsquo94) vol 1pp 202ndash207 Coventry UK March 1994

[6] E Arinton S Caraman and J Korbicz ldquoNeural networks formodelling and fault detection of the inter-stand strip tension ofa cold tandem millrdquo Control Engineering Practice vol 20 no 7pp 684ndash694 2012

[7] KMTakami JMahmoudi andEDahlquist ldquoAdaptive controlof cold rolling system in electrical strips production systemwithonline-offline predictorsrdquo International Journal of AdvancedManufacturing Technology vol 50 no 9ndash12 pp 917ndash930 2010

[8] G Pin V Francesconi F A Cuzzola and T Parisini ldquoAdaptivetask-space metal strip-flatness control in cold multi-roll millstandsrdquo Journal of Process Control vol 23 no 2 pp 108ndash1192013

[9] J Pittner and M A Simaan ldquoAn optimal control method forimprovement in tandem cold metal rollingrdquo in Proceedings ofthe IEEE 42nd AnnualMeeting Industry Applications Conference(IAS rsquo07) pp 382ndash389 September 2007

[10] J R Pittner Pointwise linear quadratic optimal control of atandem cold rolling mill [PhD thesis] School of EngineeringUniversity of Pittsburgh 2006

[11] L E Zarate ldquoPredictive model for the cold rolling processthrough sensitivity factors via neural networksrdquo Journal of the

Journal of Engineering 13

Brazilian Society ofMechanical Sciences and Engineering vol 28no 1 pp 111ndash117 2006

[12] S-H Song and S-K Sul ldquoA new tension controller for con-tinuous strip processing linerdquo IEEE Transactions on IndustryApplications vol 36 no 2 pp 633ndash639 2000

[13] X Zhang and Q Zhang ldquoRobust control of strip tension fortandem cold rolling millrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 2390ndash2393 Yantai China July2011

[14] J Pittner and M A Simaan Tandem Cold Metal Rolling MillControl Using Practical Advanced Methods Springer LondonUK 2011

[15] DGu PH Petkov andMKonstantinovRobust Control Designwith MATLAB Springer London UK 2005

[16] Z Xiaofeng and ZQingdong ldquoRobust control of strip thicknessfor cold rolling millrdquo in Informatics in Control Automation andRobotics vol 133 of Lecture Notes in Electrical Engineering pp777ndash785 Springer Berlin Germany 2012

[17] K Zohu and J C Doyle Essentials of Robust Control PrenticeHall 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 11

11986311=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 21000 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 28300 0135 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 minus02 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 36700 0 0 0 21000 0 0 0 0 0 0 0 0 0 0

0 minus28200 0 0 0 16700 0077 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 minus02 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 minus32300 0 0 0 21000 0 0 0 0 0 0

0 0 0 0 0 minus16100 0 005 10500 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 minus02 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 minus31700 0 0 0 21000 0 0

0 0 0 0 0 0 0 0 0 minus10400 0 0 0 6930 0033 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986312=

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

10 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 10 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 10

0 0 0 0

0 0 0 0

0 0 0 0

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(A1)

B Matrices for Reduced-Order Controller

Consider

ak =

[[[[[[[[[[[[[[[[[[[[

[

minus13534 9671 2455 minus1674 minus122 155 minus048 299 109 014 minus008 minus019 011

0 minus2734 523 1642 147 minus068 04 minus136 minus065 0005 007 01 minus0048

0 minus2872 minus277 03 minus1267 minus021 005 minus065 003 minus01 009 002 004

0 0 0 minus611 minus184 018 minus029 031 042 minus009 minus002 minus002 0018

0 0 0 145 minus662 minus008 019 minus011 minus006 minus002 0 0003 0

0 0 0 0 0 minus0008 006 minus002 minus0008 minus0001 0 0002 0001

0 0 0 0 0 minus006 minus0004 0008 0008 minus0002 0 0 0001

0 0 0 0 0 0 0 minus002 002 minus0004 minus0001 0 0

0 0 0 0 0 0 0 minus004 minus0008 0002 minus0003 0 0

0 0 0 0 0 0 0 0 0 minus0003 0 0 0

0 0 0 0 0 0 0 0 0 0 minus0002 0 0

0 0 0 0 0 0 0 0 0 0 0 minus0002 0

0 0 0 0 0 0 0 0 0 0 0 0 minus0002

]]]]]]]]]]]]]]]]]]]]

]

12 Journal of Engineering

bk =

[[[[[[[[[[[[[[[[[[[[

[

015 008 002 004

minus004 minus003 minus001 minus001

minus005 minus0002 minus002 0

minus0008 001 001 001

minus0006 0008 0003 minus001

0 0 0 0

0 0 0 0

minus0001 0 0 0

0 0 0 0

0 0 0 0

0001 0 0 0

0 0 0 minus0001

0 0 0 minus0001

]]]]]]]]]]]]]]]]]]]]

]

ck =[[[

[

minus0001 minus0006 0 0001 minus0001 0 0 0 0 0 0 0 0

006 minus002 minus0009 0001 0007 0 0 0 0 0 0 0001 0

0009 minus004 minus003 001 0 0 0 0 0 0 0001 0 0001

013 minus008 minus005 001 0 0 0001 minus0002 0 0 0001 minus0001 minus0001

]]]

]

dk =[[[

[

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

]]]

]

(B1)

Abbreviations

119881 Work roll linear velocity119880V Speed actuator input120591V Speed actuator time constant119881in Input strip velocity119881out Output strip velocityℎin Stand input thicknessℎout Stand output thickness119891 Forward slip ratio120593 Backward slip ratio119864 Youngrsquos modulus119861 Width of strip119867119894+1

Strip thickness between stand 119894 and stand 119894 + 1119871 Length between adjacent stands119879119894119894+1

Strip tension between stand 119894 and stand 119894 + 1

Conflict of Interests

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

References

[1] W S Levine Control System Applications Taylor amp FrancisBoca Raton Fla USA 2011

[2] J Pittner N S Samaras and M A Simaan ldquoA simple rollingmillmodel with linear quadratic optimal controllerrdquo inProceed-ings of the 37th IAS Annual Meeting and World Conference onIndustrial Applications of Electrical Energy pp 142ndash149 October2002

[3] H R Koofigar F Sheikholeslam and S Hosseinnia ldquoUnifiedgauge-tension control in cold rolling mills a robust regulationtechniquerdquo International Journal of Precision Engineering andManufacturing vol 12 no 3 pp 393ndash403 2011

[4] E J M Geddes and I Postlethwaite ldquoImprovements in productquality in tandem cold rolling using robust multivariablecontrolrdquo IEEE Transactions on Control Systems Technology vol6 no 2 pp 257ndash269 1998

[5] E J M Geddes and I Postlethwaite ldquoMultivariable control ofa high performance tandem cold rolling millrdquo in Proceedings ofthe International Conference on Control (CONTROL rsquo94) vol 1pp 202ndash207 Coventry UK March 1994

[6] E Arinton S Caraman and J Korbicz ldquoNeural networks formodelling and fault detection of the inter-stand strip tension ofa cold tandem millrdquo Control Engineering Practice vol 20 no 7pp 684ndash694 2012

[7] KMTakami JMahmoudi andEDahlquist ldquoAdaptive controlof cold rolling system in electrical strips production systemwithonline-offline predictorsrdquo International Journal of AdvancedManufacturing Technology vol 50 no 9ndash12 pp 917ndash930 2010

[8] G Pin V Francesconi F A Cuzzola and T Parisini ldquoAdaptivetask-space metal strip-flatness control in cold multi-roll millstandsrdquo Journal of Process Control vol 23 no 2 pp 108ndash1192013

[9] J Pittner and M A Simaan ldquoAn optimal control method forimprovement in tandem cold metal rollingrdquo in Proceedings ofthe IEEE 42nd AnnualMeeting Industry Applications Conference(IAS rsquo07) pp 382ndash389 September 2007

[10] J R Pittner Pointwise linear quadratic optimal control of atandem cold rolling mill [PhD thesis] School of EngineeringUniversity of Pittsburgh 2006

[11] L E Zarate ldquoPredictive model for the cold rolling processthrough sensitivity factors via neural networksrdquo Journal of the

Journal of Engineering 13

Brazilian Society ofMechanical Sciences and Engineering vol 28no 1 pp 111ndash117 2006

[12] S-H Song and S-K Sul ldquoA new tension controller for con-tinuous strip processing linerdquo IEEE Transactions on IndustryApplications vol 36 no 2 pp 633ndash639 2000

[13] X Zhang and Q Zhang ldquoRobust control of strip tension fortandem cold rolling millrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 2390ndash2393 Yantai China July2011

[14] J Pittner and M A Simaan Tandem Cold Metal Rolling MillControl Using Practical Advanced Methods Springer LondonUK 2011

[15] DGu PH Petkov andMKonstantinovRobust Control Designwith MATLAB Springer London UK 2005

[16] Z Xiaofeng and ZQingdong ldquoRobust control of strip thicknessfor cold rolling millrdquo in Informatics in Control Automation andRobotics vol 133 of Lecture Notes in Electrical Engineering pp777ndash785 Springer Berlin Germany 2012

[17] K Zohu and J C Doyle Essentials of Robust Control PrenticeHall 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Journal of Engineering

bk =

[[[[[[[[[[[[[[[[[[[[

[

015 008 002 004

minus004 minus003 minus001 minus001

minus005 minus0002 minus002 0

minus0008 001 001 001

minus0006 0008 0003 minus001

0 0 0 0

0 0 0 0

minus0001 0 0 0

0 0 0 0

0 0 0 0

0001 0 0 0

0 0 0 minus0001

0 0 0 minus0001

]]]]]]]]]]]]]]]]]]]]

]

ck =[[[

[

minus0001 minus0006 0 0001 minus0001 0 0 0 0 0 0 0 0

006 minus002 minus0009 0001 0007 0 0 0 0 0 0 0001 0

0009 minus004 minus003 001 0 0 0 0 0 0 0001 0 0001

013 minus008 minus005 001 0 0 0001 minus0002 0 0 0001 minus0001 minus0001

]]]

]

dk =[[[

[

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

]]]

]

(B1)

Abbreviations

119881 Work roll linear velocity119880V Speed actuator input120591V Speed actuator time constant119881in Input strip velocity119881out Output strip velocityℎin Stand input thicknessℎout Stand output thickness119891 Forward slip ratio120593 Backward slip ratio119864 Youngrsquos modulus119861 Width of strip119867119894+1

Strip thickness between stand 119894 and stand 119894 + 1119871 Length between adjacent stands119879119894119894+1

Strip tension between stand 119894 and stand 119894 + 1

Conflict of Interests

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

References

[1] W S Levine Control System Applications Taylor amp FrancisBoca Raton Fla USA 2011

[2] J Pittner N S Samaras and M A Simaan ldquoA simple rollingmillmodel with linear quadratic optimal controllerrdquo inProceed-ings of the 37th IAS Annual Meeting and World Conference onIndustrial Applications of Electrical Energy pp 142ndash149 October2002

[3] H R Koofigar F Sheikholeslam and S Hosseinnia ldquoUnifiedgauge-tension control in cold rolling mills a robust regulationtechniquerdquo International Journal of Precision Engineering andManufacturing vol 12 no 3 pp 393ndash403 2011

[4] E J M Geddes and I Postlethwaite ldquoImprovements in productquality in tandem cold rolling using robust multivariablecontrolrdquo IEEE Transactions on Control Systems Technology vol6 no 2 pp 257ndash269 1998

[5] E J M Geddes and I Postlethwaite ldquoMultivariable control ofa high performance tandem cold rolling millrdquo in Proceedings ofthe International Conference on Control (CONTROL rsquo94) vol 1pp 202ndash207 Coventry UK March 1994

[6] E Arinton S Caraman and J Korbicz ldquoNeural networks formodelling and fault detection of the inter-stand strip tension ofa cold tandem millrdquo Control Engineering Practice vol 20 no 7pp 684ndash694 2012

[7] KMTakami JMahmoudi andEDahlquist ldquoAdaptive controlof cold rolling system in electrical strips production systemwithonline-offline predictorsrdquo International Journal of AdvancedManufacturing Technology vol 50 no 9ndash12 pp 917ndash930 2010

[8] G Pin V Francesconi F A Cuzzola and T Parisini ldquoAdaptivetask-space metal strip-flatness control in cold multi-roll millstandsrdquo Journal of Process Control vol 23 no 2 pp 108ndash1192013

[9] J Pittner and M A Simaan ldquoAn optimal control method forimprovement in tandem cold metal rollingrdquo in Proceedings ofthe IEEE 42nd AnnualMeeting Industry Applications Conference(IAS rsquo07) pp 382ndash389 September 2007

[10] J R Pittner Pointwise linear quadratic optimal control of atandem cold rolling mill [PhD thesis] School of EngineeringUniversity of Pittsburgh 2006

[11] L E Zarate ldquoPredictive model for the cold rolling processthrough sensitivity factors via neural networksrdquo Journal of the

Journal of Engineering 13

Brazilian Society ofMechanical Sciences and Engineering vol 28no 1 pp 111ndash117 2006

[12] S-H Song and S-K Sul ldquoA new tension controller for con-tinuous strip processing linerdquo IEEE Transactions on IndustryApplications vol 36 no 2 pp 633ndash639 2000

[13] X Zhang and Q Zhang ldquoRobust control of strip tension fortandem cold rolling millrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 2390ndash2393 Yantai China July2011

[14] J Pittner and M A Simaan Tandem Cold Metal Rolling MillControl Using Practical Advanced Methods Springer LondonUK 2011

[15] DGu PH Petkov andMKonstantinovRobust Control Designwith MATLAB Springer London UK 2005

[16] Z Xiaofeng and ZQingdong ldquoRobust control of strip thicknessfor cold rolling millrdquo in Informatics in Control Automation andRobotics vol 133 of Lecture Notes in Electrical Engineering pp777ndash785 Springer Berlin Germany 2012

[17] K Zohu and J C Doyle Essentials of Robust Control PrenticeHall 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 13

Brazilian Society ofMechanical Sciences and Engineering vol 28no 1 pp 111ndash117 2006

[12] S-H Song and S-K Sul ldquoA new tension controller for con-tinuous strip processing linerdquo IEEE Transactions on IndustryApplications vol 36 no 2 pp 633ndash639 2000

[13] X Zhang and Q Zhang ldquoRobust control of strip tension fortandem cold rolling millrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 2390ndash2393 Yantai China July2011

[14] J Pittner and M A Simaan Tandem Cold Metal Rolling MillControl Using Practical Advanced Methods Springer LondonUK 2011

[15] DGu PH Petkov andMKonstantinovRobust Control Designwith MATLAB Springer London UK 2005

[16] Z Xiaofeng and ZQingdong ldquoRobust control of strip thicknessfor cold rolling millrdquo in Informatics in Control Automation andRobotics vol 133 of Lecture Notes in Electrical Engineering pp777ndash785 Springer Berlin Germany 2012

[17] K Zohu and J C Doyle Essentials of Robust Control PrenticeHall 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of