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CONTROL OF ALUMINUM ROLLING MILLSAbderrahim Abbas aa Department of Chemical Engineering College of Engineering University of Bahrain PO Box 32038 BahrainPublished online 07 Feb 2007

To cite this article Abderrahim Abbas (2001) CONTROL OF ALUMINUM ROLLING MILLS Materials and ManufacturingProcesses 165 691-700 DOI 101081AMP-100108629

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MATERIALS AND MANUFACTURING PROCESSES 16(5) 691ndash700 (2001)

CONTROL OF ALUMINUM ROLLING MILLS

Abderrahim Abbas

Department of Chemical Engineering College of EngineeringUniversity of Bahrain PO Box 32038 BahrainFax 973-684-844 E-mail arabbusenguobbh

ABSTRACT

The aluminum rolling mills suffer from the presence of large time-delay-to-time-constant ratios Conventional proportionalndashintegralndashderivative (PID)controllers and their variants do not provide adequate performance when usedto control these types of processes Advanced control strategies are neededThe Smith predictor (SP) is a relatively simple control scheme that is usedfor time delay compensation In this paper the SP has been used to controla rolling mill and its performance has been compared with that obtained froma conventional proportional plus integral (PI) controller In addition the sensi-tivity of the SP to modeling errors and changing plant conditions was investi-gated As expected it was found that for the perfect model case the SP pro-vides superior performance as compared to the classical PI The superiorityof the SP is maintained as long as the modeling errors are not too large ThePI algorithm is much more robust than the SP control scheme

Key Words Aluminum processing Control of thickness PID controlRolling mill Smith predictor Time delay

10 INTRODUCTION

Control systems of modern aluminum (metal) rolling mills have to meetstringent performance requirements and include fast response times and smallovershoots Large settling times result in significant raw material waste For exam-ple in the flattening of aluminum sheets for the beverage can industry up to 2

691

Copyright 2001 by Marcel Dekker Inc wwwdekkercom

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Figure 1 A simplified rolling mill

of the feedstock are routinely discarded in typical cold rolling mill processes dueto start-up times (1) The requirement of low overshoot limits the oscillations ofheavy rollers and hence reduces the wear on the equipment

The difficulty of controlling rolling mills arises from the fact that these pro-cesses suffer from the presence of large time delays Consider the simplified roll-ing mill shown in Figure 1 The exit thickness (h) is controlled by adjusting thegap between the rollers that squeeze the material into the desired thickness Theproblem is that h cannot be measured immediately after it leaves the roll gap but ismeasured further down the line introducing a measurement delay into the controlsystem This time delay can be 10 times larger than the process time constant(23)

Generally conventional proportionalndashintegralndashderivative (PID) controllersand their variants do not yield satisfactory performance when used to control pro-cesses having long time delays During the last four decades a number of strate-gies were proposed to control processes with significant time delays (4ndash7) Themost popular technique is the Smith predictor (SP) proposed by Smith (4) Al-though this model-based approach offers much better performance for time delaysystems than classical controllers it suffers from a number of shortcomings Themain disadvantage of the SP is the deterioration of the closed-loop performancein the face of plantmodel mismatches In certain cases however it was foundthat plantmodel mismatches could improve the performance (89)

The objective of this paper is to investigate the control of a rolling mill usingtwo different schemes (the classical proportional plus integral [PI] controller andthe SP structure) The performances of the two controllers are compared Thesensitivity of the SP to modeling errors and changing process conditions is alsoinvestigated

20 THEORY

Consider the classical feedback control loop shown in Figure 2 Gc Gp andGL are the transfer functions of the controller process and disturbance (load)

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CONTROL OF ALUMINUM ROLLING MILLS 693

Figure 2 Conventional feedback control loop

respectively R C and L are the setpoint controlled (output) variable and distur-bance respectively

For a setpoint change the closed-loop transfer function is

C

R

Gc Gp

1 Gc Gp

(1)

If the plant contains a time delay θp then its transfer function may be writtenas Gp G sdot eθps

where G is the delay-free part of the transfer function Equation (1) be-comes

C

R

Gc Geθps

1 GcGeθps(2)

The difficulty of control arises from the presence of the time delay in the denomi-nator (characteristic equation) of Eq (2)

As mentioned previously the SP overcomes the effects of the time delayFigure 3 shows the structure of the SP where Gm and θm are respectively thedelay-free and time delay of the plant model

Figure 3 Smith predictor (SP) scheme

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The closed-loop transfer function relating changes in the controlled variableto changes in the setpoint is

C

R

Gc Gp

1 GcGm Gc (Gp Gm)(3)

For a perfect model Gp Gm Eq (3) becomes

C

R

Gc Gp

1 GcG(4)

Comparison of Eqs (2) and (4) indicates that the SP eliminates the time delay fromthe characteristic equation This allows the controller to take a more aggressiveadjustment of the manipulated variable and hence yields a better performance thanthat obtained from conventional controllers

In this study the considered plant is a first-order plus time delay (FOPTD)transfer function and the controller is of the PI type

Gp Kp eθps

τps 1(5)

and

Gc Kc 1 1

τIs (6)

The plant parametersrsquo values used are those obtained by Schneider (3) for a partic-ular rolling mill Kp 1 τp 06 sec and θp 182 sec The plant model isalso assumed to have a FOPTD structure

For the conventional control loop (Fig 2) the controller was tuned usingthe internal model control (IMC)-based tuning rules proposed by Morari and Zafi-riou (10)

Kc 1

Kp

sdot2τp θp

2λ(7)

and

τI τp θp

2(8)

Following their recommendations the closed-loop time constant was chosen tobe λ 18θp leading to Kc 046 and τI 151 sec

The PI controller of the SP structure (Fig 3) was tuned using the Dahlinrsquossynthesis rules (11) Kc τp(Kpλ) and τI τp Using λ 05τp leads to Kc 2 and τI 06 sec

Note that the above two sets of tuning rules are known to yield good closed-loop performance for both setpoint and disturbance changes

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30 RESULTS

Simulation of the two control schemes yielded the setpoint and disturbanceresponses presented in Figures 4 and 5 Note that for the SP a perfect modelGm Gp was used For the disturbance case GL was taken to be equal to GpFor the mill this assumption is a good approximation to load changes in the feedthickness H (12)

An inspection of Figures 4 and 5 indicates that the SP offers much betteroverall performances for both setpoint and disturbance changes The settling times(based on 2 band) for the SP setpoint and disturbance responses are respec-tively 30 and 62 sec both of which are much shorter than those correspondingto the conventional controller (10 and 126 sec) The 2 band is based on thefact that in rolling mills it is generally desired to track setpoints so that the steady-state thickness is within 1ndash2 of the desired value (1)

One popular criterion that is usually used to measure the overall performanceof a controller is the integral of the absolute value of the error (IAE)

IAE infin

0

|e(t) |dt (9)

where e(t) is the error (difference between the setpoint and controlled variable)The IAE values for the setpoint and disturbance responses of the SP structure

are 0316 and 211 whereas those corresponding to the conventional PI are 146and 328 respectively Note that the IAE values for the setpoint changes excludethe areas of the responses corresponding to the inevitable delay (182 sec)

Figure 4 Response to a unit step change in the setpoint

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Figure 5 Response to a unit step change in the disturbance

Figure 6 IAE versus percent error in the model gain for the SP scheme (setpoint input) Thedotted line represents the IAE value for the conventional loop

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CONTROL OF ALUMINUM ROLLING MILLS 697

Figure 7 IAE versus percent error in the time delay for the SP scheme (setpoint input)

Figure 8 IAE versus percent error in the model time constant for the SP scheme (setpoint input)

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To investigate the sensitivity of the SP to plantmodel mismatches errorswere introduced in the three model parameters and the closed-loop responses tounit step changes in the setpoint were simulated The results are shown in Figures6ndash8 From these figures it can be concluded that in general the performancedeteriorates as the error in each parameter increases and that the SP controlscheme is more sensitive to modeling errors in the time delay than in the othertwo parameters (steady state gain and time constant) Figure 7 shows that theperformance of the SP as measured by IAE is better than that of the conventionalPI as long as the error in the time delay is within the range 176 to 197The SP system is unstable if the error in θp is outside the range 23 to 264

Figures 6 and 8 show that in certain cases modeling errors may lead toimprovements rather than deterioration of the system performance For example5 error in the model time constant results in about 4 reduction in the IAE

The effect of modeling errors on the performance of the SP structure forload disturbances exhibited similar trends to those corresponding to setpointchanges (see Fig 9)

To compare the sensitivities (robustness) of the classical PI and SP systemto changing plant conditions (not modeling errors) the two control systems weresimulated with errors introduced in the most sensitive parameter namely the timedelay The results are presented in Figure 10 from which it can be clearly seenthat the PI is more robust than the SP The former is quite insensitive to negativechanges in θp

Figure 9 IAE versus percent error in the model delay for the SP scheme (load change)

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CONTROL OF ALUMINUM ROLLING MILLS 699

Figure 10 IAE versus percent error in the plant delay

40 CONCLUSIONS

The SP was used to control a rolling mill having a high time-delay-to-time-constant ratio (about 3) The SP resulted in a much superior performance com-pared to the classical PI controller As expected the SP was found to be moresensitive to modeling errors in time delay than to changes in the other plant param-eters ie the plant time constant and gain The superiority of the SP over the PIwas maintained as long as the modeling errors or changes in the processing condi-tions were not very large On the other hand the PI exhibited a much better ro-bustness than the SP

50 NOMENCLATURE

C output variablee(t) errorFOPTD first-order plus time delayG transfer functionG delay-free part of the transfer functionh exit thicknessH feed thicknessIAE integral of the absolute errorK gain

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L disturbancePI proportional plus integralPID proportionalndashintegralndashderivativeR setpoints Laplace operatorSP Smith predictort time sec

Greek Symbols

λ closed-loop time constant secτ time constant secτI integral time secθ time delay sec

Subscripts

L load (disturbance)m modelp process

60 REFERENCES

1 Khan BZ Lehman B IEEE Trans Control Syst Technol 1996 4 (4) 459ndash4662 Bissell C Control Engineering Chapman amp Hall New York 19943 Schneider DM IEEE Trans Ind Appl 1988 24 (2) 186ndash1914 Smith OJM Chem Eng Prog 1957 53 217ndash2195 Wang QG Bi Q Zhang Y ISA Trans 2000 39 79ndash926 Lee D Lee M Sung S Lee I J Process Control 1999 9 79ndash857 Donoghue JF ISA Trans 1977 16 27ndash348 Abbas A Marshall JE Walton K IEE Proc D 1986 133 (6) 313ndash3149 Marshall JE Control of Time-Delay Systems Peter Peregrinus Ltd London 1979

10 Morari M Zafiriou E Robust Process Control Prentice Hall NJ 1989 12111 Chidambaram M Applied Process Control Allied Publishers Ltd New Delhi

1998 3012 Goodwin GC Graebe SF Salgado ME Control System Design Prentice-Hall

NJ 2001 226

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Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081AMP100108629

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

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692 ABBAS

Figure 1 A simplified rolling mill

of the feedstock are routinely discarded in typical cold rolling mill processes dueto start-up times (1) The requirement of low overshoot limits the oscillations ofheavy rollers and hence reduces the wear on the equipment

The difficulty of controlling rolling mills arises from the fact that these pro-cesses suffer from the presence of large time delays Consider the simplified roll-ing mill shown in Figure 1 The exit thickness (h) is controlled by adjusting thegap between the rollers that squeeze the material into the desired thickness Theproblem is that h cannot be measured immediately after it leaves the roll gap but ismeasured further down the line introducing a measurement delay into the controlsystem This time delay can be 10 times larger than the process time constant(23)

Generally conventional proportionalndashintegralndashderivative (PID) controllersand their variants do not yield satisfactory performance when used to control pro-cesses having long time delays During the last four decades a number of strate-gies were proposed to control processes with significant time delays (4ndash7) Themost popular technique is the Smith predictor (SP) proposed by Smith (4) Al-though this model-based approach offers much better performance for time delaysystems than classical controllers it suffers from a number of shortcomings Themain disadvantage of the SP is the deterioration of the closed-loop performancein the face of plantmodel mismatches In certain cases however it was foundthat plantmodel mismatches could improve the performance (89)

The objective of this paper is to investigate the control of a rolling mill usingtwo different schemes (the classical proportional plus integral [PI] controller andthe SP structure) The performances of the two controllers are compared Thesensitivity of the SP to modeling errors and changing process conditions is alsoinvestigated

20 THEORY

Consider the classical feedback control loop shown in Figure 2 Gc Gp andGL are the transfer functions of the controller process and disturbance (load)

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CONTROL OF ALUMINUM ROLLING MILLS 693

Figure 2 Conventional feedback control loop

respectively R C and L are the setpoint controlled (output) variable and distur-bance respectively

For a setpoint change the closed-loop transfer function is

C

R

Gc Gp

1 Gc Gp

(1)

If the plant contains a time delay θp then its transfer function may be writtenas Gp G sdot eθps

where G is the delay-free part of the transfer function Equation (1) be-comes

C

R

Gc Geθps

1 GcGeθps(2)

The difficulty of control arises from the presence of the time delay in the denomi-nator (characteristic equation) of Eq (2)

As mentioned previously the SP overcomes the effects of the time delayFigure 3 shows the structure of the SP where Gm and θm are respectively thedelay-free and time delay of the plant model

Figure 3 Smith predictor (SP) scheme

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The closed-loop transfer function relating changes in the controlled variableto changes in the setpoint is

C

R

Gc Gp

1 GcGm Gc (Gp Gm)(3)

For a perfect model Gp Gm Eq (3) becomes

C

R

Gc Gp

1 GcG(4)

Comparison of Eqs (2) and (4) indicates that the SP eliminates the time delay fromthe characteristic equation This allows the controller to take a more aggressiveadjustment of the manipulated variable and hence yields a better performance thanthat obtained from conventional controllers

In this study the considered plant is a first-order plus time delay (FOPTD)transfer function and the controller is of the PI type

Gp Kp eθps

τps 1(5)

and

Gc Kc 1 1

τIs (6)

The plant parametersrsquo values used are those obtained by Schneider (3) for a partic-ular rolling mill Kp 1 τp 06 sec and θp 182 sec The plant model isalso assumed to have a FOPTD structure

For the conventional control loop (Fig 2) the controller was tuned usingthe internal model control (IMC)-based tuning rules proposed by Morari and Zafi-riou (10)

Kc 1

Kp

sdot2τp θp

2λ(7)

and

τI τp θp

2(8)

Following their recommendations the closed-loop time constant was chosen tobe λ 18θp leading to Kc 046 and τI 151 sec

The PI controller of the SP structure (Fig 3) was tuned using the Dahlinrsquossynthesis rules (11) Kc τp(Kpλ) and τI τp Using λ 05τp leads to Kc 2 and τI 06 sec

Note that the above two sets of tuning rules are known to yield good closed-loop performance for both setpoint and disturbance changes

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CONTROL OF ALUMINUM ROLLING MILLS 695

30 RESULTS

Simulation of the two control schemes yielded the setpoint and disturbanceresponses presented in Figures 4 and 5 Note that for the SP a perfect modelGm Gp was used For the disturbance case GL was taken to be equal to GpFor the mill this assumption is a good approximation to load changes in the feedthickness H (12)

An inspection of Figures 4 and 5 indicates that the SP offers much betteroverall performances for both setpoint and disturbance changes The settling times(based on 2 band) for the SP setpoint and disturbance responses are respec-tively 30 and 62 sec both of which are much shorter than those correspondingto the conventional controller (10 and 126 sec) The 2 band is based on thefact that in rolling mills it is generally desired to track setpoints so that the steady-state thickness is within 1ndash2 of the desired value (1)

One popular criterion that is usually used to measure the overall performanceof a controller is the integral of the absolute value of the error (IAE)

IAE infin

0

|e(t) |dt (9)

where e(t) is the error (difference between the setpoint and controlled variable)The IAE values for the setpoint and disturbance responses of the SP structure

are 0316 and 211 whereas those corresponding to the conventional PI are 146and 328 respectively Note that the IAE values for the setpoint changes excludethe areas of the responses corresponding to the inevitable delay (182 sec)

Figure 4 Response to a unit step change in the setpoint

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Figure 5 Response to a unit step change in the disturbance

Figure 6 IAE versus percent error in the model gain for the SP scheme (setpoint input) Thedotted line represents the IAE value for the conventional loop

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CONTROL OF ALUMINUM ROLLING MILLS 697

Figure 7 IAE versus percent error in the time delay for the SP scheme (setpoint input)

Figure 8 IAE versus percent error in the model time constant for the SP scheme (setpoint input)

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To investigate the sensitivity of the SP to plantmodel mismatches errorswere introduced in the three model parameters and the closed-loop responses tounit step changes in the setpoint were simulated The results are shown in Figures6ndash8 From these figures it can be concluded that in general the performancedeteriorates as the error in each parameter increases and that the SP controlscheme is more sensitive to modeling errors in the time delay than in the othertwo parameters (steady state gain and time constant) Figure 7 shows that theperformance of the SP as measured by IAE is better than that of the conventionalPI as long as the error in the time delay is within the range 176 to 197The SP system is unstable if the error in θp is outside the range 23 to 264

Figures 6 and 8 show that in certain cases modeling errors may lead toimprovements rather than deterioration of the system performance For example5 error in the model time constant results in about 4 reduction in the IAE

The effect of modeling errors on the performance of the SP structure forload disturbances exhibited similar trends to those corresponding to setpointchanges (see Fig 9)

To compare the sensitivities (robustness) of the classical PI and SP systemto changing plant conditions (not modeling errors) the two control systems weresimulated with errors introduced in the most sensitive parameter namely the timedelay The results are presented in Figure 10 from which it can be clearly seenthat the PI is more robust than the SP The former is quite insensitive to negativechanges in θp

Figure 9 IAE versus percent error in the model delay for the SP scheme (load change)

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CONTROL OF ALUMINUM ROLLING MILLS 699

Figure 10 IAE versus percent error in the plant delay

40 CONCLUSIONS

The SP was used to control a rolling mill having a high time-delay-to-time-constant ratio (about 3) The SP resulted in a much superior performance com-pared to the classical PI controller As expected the SP was found to be moresensitive to modeling errors in time delay than to changes in the other plant param-eters ie the plant time constant and gain The superiority of the SP over the PIwas maintained as long as the modeling errors or changes in the processing condi-tions were not very large On the other hand the PI exhibited a much better ro-bustness than the SP

50 NOMENCLATURE

C output variablee(t) errorFOPTD first-order plus time delayG transfer functionG delay-free part of the transfer functionh exit thicknessH feed thicknessIAE integral of the absolute errorK gain

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700 ABBAS

L disturbancePI proportional plus integralPID proportionalndashintegralndashderivativeR setpoints Laplace operatorSP Smith predictort time sec

Greek Symbols

λ closed-loop time constant secτ time constant secτI integral time secθ time delay sec

Subscripts

L load (disturbance)m modelp process

60 REFERENCES

1 Khan BZ Lehman B IEEE Trans Control Syst Technol 1996 4 (4) 459ndash4662 Bissell C Control Engineering Chapman amp Hall New York 19943 Schneider DM IEEE Trans Ind Appl 1988 24 (2) 186ndash1914 Smith OJM Chem Eng Prog 1957 53 217ndash2195 Wang QG Bi Q Zhang Y ISA Trans 2000 39 79ndash926 Lee D Lee M Sung S Lee I J Process Control 1999 9 79ndash857 Donoghue JF ISA Trans 1977 16 27ndash348 Abbas A Marshall JE Walton K IEE Proc D 1986 133 (6) 313ndash3149 Marshall JE Control of Time-Delay Systems Peter Peregrinus Ltd London 1979

10 Morari M Zafiriou E Robust Process Control Prentice Hall NJ 1989 12111 Chidambaram M Applied Process Control Allied Publishers Ltd New Delhi

1998 3012 Goodwin GC Graebe SF Salgado ME Control System Design Prentice-Hall

NJ 2001 226

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Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081AMP100108629

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

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ORDER REPRINTS

CONTROL OF ALUMINUM ROLLING MILLS 693

Figure 2 Conventional feedback control loop

respectively R C and L are the setpoint controlled (output) variable and distur-bance respectively

For a setpoint change the closed-loop transfer function is

C

R

Gc Gp

1 Gc Gp

(1)

If the plant contains a time delay θp then its transfer function may be writtenas Gp G sdot eθps

where G is the delay-free part of the transfer function Equation (1) be-comes

C

R

Gc Geθps

1 GcGeθps(2)

The difficulty of control arises from the presence of the time delay in the denomi-nator (characteristic equation) of Eq (2)

As mentioned previously the SP overcomes the effects of the time delayFigure 3 shows the structure of the SP where Gm and θm are respectively thedelay-free and time delay of the plant model

Figure 3 Smith predictor (SP) scheme

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694 ABBAS

The closed-loop transfer function relating changes in the controlled variableto changes in the setpoint is

C

R

Gc Gp

1 GcGm Gc (Gp Gm)(3)

For a perfect model Gp Gm Eq (3) becomes

C

R

Gc Gp

1 GcG(4)

Comparison of Eqs (2) and (4) indicates that the SP eliminates the time delay fromthe characteristic equation This allows the controller to take a more aggressiveadjustment of the manipulated variable and hence yields a better performance thanthat obtained from conventional controllers

In this study the considered plant is a first-order plus time delay (FOPTD)transfer function and the controller is of the PI type

Gp Kp eθps

τps 1(5)

and

Gc Kc 1 1

τIs (6)

The plant parametersrsquo values used are those obtained by Schneider (3) for a partic-ular rolling mill Kp 1 τp 06 sec and θp 182 sec The plant model isalso assumed to have a FOPTD structure

For the conventional control loop (Fig 2) the controller was tuned usingthe internal model control (IMC)-based tuning rules proposed by Morari and Zafi-riou (10)

Kc 1

Kp

sdot2τp θp

2λ(7)

and

τI τp θp

2(8)

Following their recommendations the closed-loop time constant was chosen tobe λ 18θp leading to Kc 046 and τI 151 sec

The PI controller of the SP structure (Fig 3) was tuned using the Dahlinrsquossynthesis rules (11) Kc τp(Kpλ) and τI τp Using λ 05τp leads to Kc 2 and τI 06 sec

Note that the above two sets of tuning rules are known to yield good closed-loop performance for both setpoint and disturbance changes

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

CONTROL OF ALUMINUM ROLLING MILLS 695

30 RESULTS

Simulation of the two control schemes yielded the setpoint and disturbanceresponses presented in Figures 4 and 5 Note that for the SP a perfect modelGm Gp was used For the disturbance case GL was taken to be equal to GpFor the mill this assumption is a good approximation to load changes in the feedthickness H (12)

An inspection of Figures 4 and 5 indicates that the SP offers much betteroverall performances for both setpoint and disturbance changes The settling times(based on 2 band) for the SP setpoint and disturbance responses are respec-tively 30 and 62 sec both of which are much shorter than those correspondingto the conventional controller (10 and 126 sec) The 2 band is based on thefact that in rolling mills it is generally desired to track setpoints so that the steady-state thickness is within 1ndash2 of the desired value (1)

One popular criterion that is usually used to measure the overall performanceof a controller is the integral of the absolute value of the error (IAE)

IAE infin

0

|e(t) |dt (9)

where e(t) is the error (difference between the setpoint and controlled variable)The IAE values for the setpoint and disturbance responses of the SP structure

are 0316 and 211 whereas those corresponding to the conventional PI are 146and 328 respectively Note that the IAE values for the setpoint changes excludethe areas of the responses corresponding to the inevitable delay (182 sec)

Figure 4 Response to a unit step change in the setpoint

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by [

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rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

696 ABBAS

Figure 5 Response to a unit step change in the disturbance

Figure 6 IAE versus percent error in the model gain for the SP scheme (setpoint input) Thedotted line represents the IAE value for the conventional loop

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ded

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nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

CONTROL OF ALUMINUM ROLLING MILLS 697

Figure 7 IAE versus percent error in the time delay for the SP scheme (setpoint input)

Figure 8 IAE versus percent error in the model time constant for the SP scheme (setpoint input)

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by [

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ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

698 ABBAS

To investigate the sensitivity of the SP to plantmodel mismatches errorswere introduced in the three model parameters and the closed-loop responses tounit step changes in the setpoint were simulated The results are shown in Figures6ndash8 From these figures it can be concluded that in general the performancedeteriorates as the error in each parameter increases and that the SP controlscheme is more sensitive to modeling errors in the time delay than in the othertwo parameters (steady state gain and time constant) Figure 7 shows that theperformance of the SP as measured by IAE is better than that of the conventionalPI as long as the error in the time delay is within the range 176 to 197The SP system is unstable if the error in θp is outside the range 23 to 264

Figures 6 and 8 show that in certain cases modeling errors may lead toimprovements rather than deterioration of the system performance For example5 error in the model time constant results in about 4 reduction in the IAE

The effect of modeling errors on the performance of the SP structure forload disturbances exhibited similar trends to those corresponding to setpointchanges (see Fig 9)

To compare the sensitivities (robustness) of the classical PI and SP systemto changing plant conditions (not modeling errors) the two control systems weresimulated with errors introduced in the most sensitive parameter namely the timedelay The results are presented in Figure 10 from which it can be clearly seenthat the PI is more robust than the SP The former is quite insensitive to negativechanges in θp

Figure 9 IAE versus percent error in the model delay for the SP scheme (load change)

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ded

by [

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ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

CONTROL OF ALUMINUM ROLLING MILLS 699

Figure 10 IAE versus percent error in the plant delay

40 CONCLUSIONS

The SP was used to control a rolling mill having a high time-delay-to-time-constant ratio (about 3) The SP resulted in a much superior performance com-pared to the classical PI controller As expected the SP was found to be moresensitive to modeling errors in time delay than to changes in the other plant param-eters ie the plant time constant and gain The superiority of the SP over the PIwas maintained as long as the modeling errors or changes in the processing condi-tions were not very large On the other hand the PI exhibited a much better ro-bustness than the SP

50 NOMENCLATURE

C output variablee(t) errorFOPTD first-order plus time delayG transfer functionG delay-free part of the transfer functionh exit thicknessH feed thicknessIAE integral of the absolute errorK gain

Dow

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ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

700 ABBAS

L disturbancePI proportional plus integralPID proportionalndashintegralndashderivativeR setpoints Laplace operatorSP Smith predictort time sec

Greek Symbols

λ closed-loop time constant secτ time constant secτI integral time secθ time delay sec

Subscripts

L load (disturbance)m modelp process

60 REFERENCES

1 Khan BZ Lehman B IEEE Trans Control Syst Technol 1996 4 (4) 459ndash4662 Bissell C Control Engineering Chapman amp Hall New York 19943 Schneider DM IEEE Trans Ind Appl 1988 24 (2) 186ndash1914 Smith OJM Chem Eng Prog 1957 53 217ndash2195 Wang QG Bi Q Zhang Y ISA Trans 2000 39 79ndash926 Lee D Lee M Sung S Lee I J Process Control 1999 9 79ndash857 Donoghue JF ISA Trans 1977 16 27ndash348 Abbas A Marshall JE Walton K IEE Proc D 1986 133 (6) 313ndash3149 Marshall JE Control of Time-Delay Systems Peter Peregrinus Ltd London 1979

10 Morari M Zafiriou E Robust Process Control Prentice Hall NJ 1989 12111 Chidambaram M Applied Process Control Allied Publishers Ltd New Delhi

1998 3012 Goodwin GC Graebe SF Salgado ME Control System Design Prentice-Hall

NJ 2001 226

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081AMP100108629

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

Dow

nloa

ded

by [

Tuf

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nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

694 ABBAS

The closed-loop transfer function relating changes in the controlled variableto changes in the setpoint is

C

R

Gc Gp

1 GcGm Gc (Gp Gm)(3)

For a perfect model Gp Gm Eq (3) becomes

C

R

Gc Gp

1 GcG(4)

Comparison of Eqs (2) and (4) indicates that the SP eliminates the time delay fromthe characteristic equation This allows the controller to take a more aggressiveadjustment of the manipulated variable and hence yields a better performance thanthat obtained from conventional controllers

In this study the considered plant is a first-order plus time delay (FOPTD)transfer function and the controller is of the PI type

Gp Kp eθps

τps 1(5)

and

Gc Kc 1 1

τIs (6)

The plant parametersrsquo values used are those obtained by Schneider (3) for a partic-ular rolling mill Kp 1 τp 06 sec and θp 182 sec The plant model isalso assumed to have a FOPTD structure

For the conventional control loop (Fig 2) the controller was tuned usingthe internal model control (IMC)-based tuning rules proposed by Morari and Zafi-riou (10)

Kc 1

Kp

sdot2τp θp

2λ(7)

and

τI τp θp

2(8)

Following their recommendations the closed-loop time constant was chosen tobe λ 18θp leading to Kc 046 and τI 151 sec

The PI controller of the SP structure (Fig 3) was tuned using the Dahlinrsquossynthesis rules (11) Kc τp(Kpλ) and τI τp Using λ 05τp leads to Kc 2 and τI 06 sec

Note that the above two sets of tuning rules are known to yield good closed-loop performance for both setpoint and disturbance changes

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

CONTROL OF ALUMINUM ROLLING MILLS 695

30 RESULTS

Simulation of the two control schemes yielded the setpoint and disturbanceresponses presented in Figures 4 and 5 Note that for the SP a perfect modelGm Gp was used For the disturbance case GL was taken to be equal to GpFor the mill this assumption is a good approximation to load changes in the feedthickness H (12)

An inspection of Figures 4 and 5 indicates that the SP offers much betteroverall performances for both setpoint and disturbance changes The settling times(based on 2 band) for the SP setpoint and disturbance responses are respec-tively 30 and 62 sec both of which are much shorter than those correspondingto the conventional controller (10 and 126 sec) The 2 band is based on thefact that in rolling mills it is generally desired to track setpoints so that the steady-state thickness is within 1ndash2 of the desired value (1)

One popular criterion that is usually used to measure the overall performanceof a controller is the integral of the absolute value of the error (IAE)

IAE infin

0

|e(t) |dt (9)

where e(t) is the error (difference between the setpoint and controlled variable)The IAE values for the setpoint and disturbance responses of the SP structure

are 0316 and 211 whereas those corresponding to the conventional PI are 146and 328 respectively Note that the IAE values for the setpoint changes excludethe areas of the responses corresponding to the inevitable delay (182 sec)

Figure 4 Response to a unit step change in the setpoint

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rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

696 ABBAS

Figure 5 Response to a unit step change in the disturbance

Figure 6 IAE versus percent error in the model gain for the SP scheme (setpoint input) Thedotted line represents the IAE value for the conventional loop

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nloa

ded

by [

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ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

CONTROL OF ALUMINUM ROLLING MILLS 697

Figure 7 IAE versus percent error in the time delay for the SP scheme (setpoint input)

Figure 8 IAE versus percent error in the model time constant for the SP scheme (setpoint input)

Dow

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ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

698 ABBAS

To investigate the sensitivity of the SP to plantmodel mismatches errorswere introduced in the three model parameters and the closed-loop responses tounit step changes in the setpoint were simulated The results are shown in Figures6ndash8 From these figures it can be concluded that in general the performancedeteriorates as the error in each parameter increases and that the SP controlscheme is more sensitive to modeling errors in the time delay than in the othertwo parameters (steady state gain and time constant) Figure 7 shows that theperformance of the SP as measured by IAE is better than that of the conventionalPI as long as the error in the time delay is within the range 176 to 197The SP system is unstable if the error in θp is outside the range 23 to 264

Figures 6 and 8 show that in certain cases modeling errors may lead toimprovements rather than deterioration of the system performance For example5 error in the model time constant results in about 4 reduction in the IAE

The effect of modeling errors on the performance of the SP structure forload disturbances exhibited similar trends to those corresponding to setpointchanges (see Fig 9)

To compare the sensitivities (robustness) of the classical PI and SP systemto changing plant conditions (not modeling errors) the two control systems weresimulated with errors introduced in the most sensitive parameter namely the timedelay The results are presented in Figure 10 from which it can be clearly seenthat the PI is more robust than the SP The former is quite insensitive to negativechanges in θp

Figure 9 IAE versus percent error in the model delay for the SP scheme (load change)

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

CONTROL OF ALUMINUM ROLLING MILLS 699

Figure 10 IAE versus percent error in the plant delay

40 CONCLUSIONS

The SP was used to control a rolling mill having a high time-delay-to-time-constant ratio (about 3) The SP resulted in a much superior performance com-pared to the classical PI controller As expected the SP was found to be moresensitive to modeling errors in time delay than to changes in the other plant param-eters ie the plant time constant and gain The superiority of the SP over the PIwas maintained as long as the modeling errors or changes in the processing condi-tions were not very large On the other hand the PI exhibited a much better ro-bustness than the SP

50 NOMENCLATURE

C output variablee(t) errorFOPTD first-order plus time delayG transfer functionG delay-free part of the transfer functionh exit thicknessH feed thicknessIAE integral of the absolute errorK gain

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

700 ABBAS

L disturbancePI proportional plus integralPID proportionalndashintegralndashderivativeR setpoints Laplace operatorSP Smith predictort time sec

Greek Symbols

λ closed-loop time constant secτ time constant secτI integral time secθ time delay sec

Subscripts

L load (disturbance)m modelp process

60 REFERENCES

1 Khan BZ Lehman B IEEE Trans Control Syst Technol 1996 4 (4) 459ndash4662 Bissell C Control Engineering Chapman amp Hall New York 19943 Schneider DM IEEE Trans Ind Appl 1988 24 (2) 186ndash1914 Smith OJM Chem Eng Prog 1957 53 217ndash2195 Wang QG Bi Q Zhang Y ISA Trans 2000 39 79ndash926 Lee D Lee M Sung S Lee I J Process Control 1999 9 79ndash857 Donoghue JF ISA Trans 1977 16 27ndash348 Abbas A Marshall JE Walton K IEE Proc D 1986 133 (6) 313ndash3149 Marshall JE Control of Time-Delay Systems Peter Peregrinus Ltd London 1979

10 Morari M Zafiriou E Robust Process Control Prentice Hall NJ 1989 12111 Chidambaram M Applied Process Control Allied Publishers Ltd New Delhi

1998 3012 Goodwin GC Graebe SF Salgado ME Control System Design Prentice-Hall

NJ 2001 226

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081AMP100108629

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

CONTROL OF ALUMINUM ROLLING MILLS 695

30 RESULTS

Simulation of the two control schemes yielded the setpoint and disturbanceresponses presented in Figures 4 and 5 Note that for the SP a perfect modelGm Gp was used For the disturbance case GL was taken to be equal to GpFor the mill this assumption is a good approximation to load changes in the feedthickness H (12)

An inspection of Figures 4 and 5 indicates that the SP offers much betteroverall performances for both setpoint and disturbance changes The settling times(based on 2 band) for the SP setpoint and disturbance responses are respec-tively 30 and 62 sec both of which are much shorter than those correspondingto the conventional controller (10 and 126 sec) The 2 band is based on thefact that in rolling mills it is generally desired to track setpoints so that the steady-state thickness is within 1ndash2 of the desired value (1)

One popular criterion that is usually used to measure the overall performanceof a controller is the integral of the absolute value of the error (IAE)

IAE infin

0

|e(t) |dt (9)

where e(t) is the error (difference between the setpoint and controlled variable)The IAE values for the setpoint and disturbance responses of the SP structure

are 0316 and 211 whereas those corresponding to the conventional PI are 146and 328 respectively Note that the IAE values for the setpoint changes excludethe areas of the responses corresponding to the inevitable delay (182 sec)

Figure 4 Response to a unit step change in the setpoint

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rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

696 ABBAS

Figure 5 Response to a unit step change in the disturbance

Figure 6 IAE versus percent error in the model gain for the SP scheme (setpoint input) Thedotted line represents the IAE value for the conventional loop

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

CONTROL OF ALUMINUM ROLLING MILLS 697

Figure 7 IAE versus percent error in the time delay for the SP scheme (setpoint input)

Figure 8 IAE versus percent error in the model time constant for the SP scheme (setpoint input)

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

698 ABBAS

To investigate the sensitivity of the SP to plantmodel mismatches errorswere introduced in the three model parameters and the closed-loop responses tounit step changes in the setpoint were simulated The results are shown in Figures6ndash8 From these figures it can be concluded that in general the performancedeteriorates as the error in each parameter increases and that the SP controlscheme is more sensitive to modeling errors in the time delay than in the othertwo parameters (steady state gain and time constant) Figure 7 shows that theperformance of the SP as measured by IAE is better than that of the conventionalPI as long as the error in the time delay is within the range 176 to 197The SP system is unstable if the error in θp is outside the range 23 to 264

Figures 6 and 8 show that in certain cases modeling errors may lead toimprovements rather than deterioration of the system performance For example5 error in the model time constant results in about 4 reduction in the IAE

The effect of modeling errors on the performance of the SP structure forload disturbances exhibited similar trends to those corresponding to setpointchanges (see Fig 9)

To compare the sensitivities (robustness) of the classical PI and SP systemto changing plant conditions (not modeling errors) the two control systems weresimulated with errors introduced in the most sensitive parameter namely the timedelay The results are presented in Figure 10 from which it can be clearly seenthat the PI is more robust than the SP The former is quite insensitive to negativechanges in θp

Figure 9 IAE versus percent error in the model delay for the SP scheme (load change)

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

CONTROL OF ALUMINUM ROLLING MILLS 699

Figure 10 IAE versus percent error in the plant delay

40 CONCLUSIONS

The SP was used to control a rolling mill having a high time-delay-to-time-constant ratio (about 3) The SP resulted in a much superior performance com-pared to the classical PI controller As expected the SP was found to be moresensitive to modeling errors in time delay than to changes in the other plant param-eters ie the plant time constant and gain The superiority of the SP over the PIwas maintained as long as the modeling errors or changes in the processing condi-tions were not very large On the other hand the PI exhibited a much better ro-bustness than the SP

50 NOMENCLATURE

C output variablee(t) errorFOPTD first-order plus time delayG transfer functionG delay-free part of the transfer functionh exit thicknessH feed thicknessIAE integral of the absolute errorK gain

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

700 ABBAS

L disturbancePI proportional plus integralPID proportionalndashintegralndashderivativeR setpoints Laplace operatorSP Smith predictort time sec

Greek Symbols

λ closed-loop time constant secτ time constant secτI integral time secθ time delay sec

Subscripts

L load (disturbance)m modelp process

60 REFERENCES

1 Khan BZ Lehman B IEEE Trans Control Syst Technol 1996 4 (4) 459ndash4662 Bissell C Control Engineering Chapman amp Hall New York 19943 Schneider DM IEEE Trans Ind Appl 1988 24 (2) 186ndash1914 Smith OJM Chem Eng Prog 1957 53 217ndash2195 Wang QG Bi Q Zhang Y ISA Trans 2000 39 79ndash926 Lee D Lee M Sung S Lee I J Process Control 1999 9 79ndash857 Donoghue JF ISA Trans 1977 16 27ndash348 Abbas A Marshall JE Walton K IEE Proc D 1986 133 (6) 313ndash3149 Marshall JE Control of Time-Delay Systems Peter Peregrinus Ltd London 1979

10 Morari M Zafiriou E Robust Process Control Prentice Hall NJ 1989 12111 Chidambaram M Applied Process Control Allied Publishers Ltd New Delhi

1998 3012 Goodwin GC Graebe SF Salgado ME Control System Design Prentice-Hall

NJ 2001 226

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081AMP100108629

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

696 ABBAS

Figure 5 Response to a unit step change in the disturbance

Figure 6 IAE versus percent error in the model gain for the SP scheme (setpoint input) Thedotted line represents the IAE value for the conventional loop

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

CONTROL OF ALUMINUM ROLLING MILLS 697

Figure 7 IAE versus percent error in the time delay for the SP scheme (setpoint input)

Figure 8 IAE versus percent error in the model time constant for the SP scheme (setpoint input)

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

698 ABBAS

To investigate the sensitivity of the SP to plantmodel mismatches errorswere introduced in the three model parameters and the closed-loop responses tounit step changes in the setpoint were simulated The results are shown in Figures6ndash8 From these figures it can be concluded that in general the performancedeteriorates as the error in each parameter increases and that the SP controlscheme is more sensitive to modeling errors in the time delay than in the othertwo parameters (steady state gain and time constant) Figure 7 shows that theperformance of the SP as measured by IAE is better than that of the conventionalPI as long as the error in the time delay is within the range 176 to 197The SP system is unstable if the error in θp is outside the range 23 to 264

Figures 6 and 8 show that in certain cases modeling errors may lead toimprovements rather than deterioration of the system performance For example5 error in the model time constant results in about 4 reduction in the IAE

The effect of modeling errors on the performance of the SP structure forload disturbances exhibited similar trends to those corresponding to setpointchanges (see Fig 9)

To compare the sensitivities (robustness) of the classical PI and SP systemto changing plant conditions (not modeling errors) the two control systems weresimulated with errors introduced in the most sensitive parameter namely the timedelay The results are presented in Figure 10 from which it can be clearly seenthat the PI is more robust than the SP The former is quite insensitive to negativechanges in θp

Figure 9 IAE versus percent error in the model delay for the SP scheme (load change)

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

CONTROL OF ALUMINUM ROLLING MILLS 699

Figure 10 IAE versus percent error in the plant delay

40 CONCLUSIONS

The SP was used to control a rolling mill having a high time-delay-to-time-constant ratio (about 3) The SP resulted in a much superior performance com-pared to the classical PI controller As expected the SP was found to be moresensitive to modeling errors in time delay than to changes in the other plant param-eters ie the plant time constant and gain The superiority of the SP over the PIwas maintained as long as the modeling errors or changes in the processing condi-tions were not very large On the other hand the PI exhibited a much better ro-bustness than the SP

50 NOMENCLATURE

C output variablee(t) errorFOPTD first-order plus time delayG transfer functionG delay-free part of the transfer functionh exit thicknessH feed thicknessIAE integral of the absolute errorK gain

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

700 ABBAS

L disturbancePI proportional plus integralPID proportionalndashintegralndashderivativeR setpoints Laplace operatorSP Smith predictort time sec

Greek Symbols

λ closed-loop time constant secτ time constant secτI integral time secθ time delay sec

Subscripts

L load (disturbance)m modelp process

60 REFERENCES

1 Khan BZ Lehman B IEEE Trans Control Syst Technol 1996 4 (4) 459ndash4662 Bissell C Control Engineering Chapman amp Hall New York 19943 Schneider DM IEEE Trans Ind Appl 1988 24 (2) 186ndash1914 Smith OJM Chem Eng Prog 1957 53 217ndash2195 Wang QG Bi Q Zhang Y ISA Trans 2000 39 79ndash926 Lee D Lee M Sung S Lee I J Process Control 1999 9 79ndash857 Donoghue JF ISA Trans 1977 16 27ndash348 Abbas A Marshall JE Walton K IEE Proc D 1986 133 (6) 313ndash3149 Marshall JE Control of Time-Delay Systems Peter Peregrinus Ltd London 1979

10 Morari M Zafiriou E Robust Process Control Prentice Hall NJ 1989 12111 Chidambaram M Applied Process Control Allied Publishers Ltd New Delhi

1998 3012 Goodwin GC Graebe SF Salgado ME Control System Design Prentice-Hall

NJ 2001 226

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081AMP100108629

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Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

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rsity

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ber

2014

ORDER REPRINTS

CONTROL OF ALUMINUM ROLLING MILLS 697

Figure 7 IAE versus percent error in the time delay for the SP scheme (setpoint input)

Figure 8 IAE versus percent error in the model time constant for the SP scheme (setpoint input)

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09

19 1

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ovem

ber

2014

ORDER REPRINTS

698 ABBAS

To investigate the sensitivity of the SP to plantmodel mismatches errorswere introduced in the three model parameters and the closed-loop responses tounit step changes in the setpoint were simulated The results are shown in Figures6ndash8 From these figures it can be concluded that in general the performancedeteriorates as the error in each parameter increases and that the SP controlscheme is more sensitive to modeling errors in the time delay than in the othertwo parameters (steady state gain and time constant) Figure 7 shows that theperformance of the SP as measured by IAE is better than that of the conventionalPI as long as the error in the time delay is within the range 176 to 197The SP system is unstable if the error in θp is outside the range 23 to 264

Figures 6 and 8 show that in certain cases modeling errors may lead toimprovements rather than deterioration of the system performance For example5 error in the model time constant results in about 4 reduction in the IAE

The effect of modeling errors on the performance of the SP structure forload disturbances exhibited similar trends to those corresponding to setpointchanges (see Fig 9)

To compare the sensitivities (robustness) of the classical PI and SP systemto changing plant conditions (not modeling errors) the two control systems weresimulated with errors introduced in the most sensitive parameter namely the timedelay The results are presented in Figure 10 from which it can be clearly seenthat the PI is more robust than the SP The former is quite insensitive to negativechanges in θp

Figure 9 IAE versus percent error in the model delay for the SP scheme (load change)

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

CONTROL OF ALUMINUM ROLLING MILLS 699

Figure 10 IAE versus percent error in the plant delay

40 CONCLUSIONS

The SP was used to control a rolling mill having a high time-delay-to-time-constant ratio (about 3) The SP resulted in a much superior performance com-pared to the classical PI controller As expected the SP was found to be moresensitive to modeling errors in time delay than to changes in the other plant param-eters ie the plant time constant and gain The superiority of the SP over the PIwas maintained as long as the modeling errors or changes in the processing condi-tions were not very large On the other hand the PI exhibited a much better ro-bustness than the SP

50 NOMENCLATURE

C output variablee(t) errorFOPTD first-order plus time delayG transfer functionG delay-free part of the transfer functionh exit thicknessH feed thicknessIAE integral of the absolute errorK gain

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

700 ABBAS

L disturbancePI proportional plus integralPID proportionalndashintegralndashderivativeR setpoints Laplace operatorSP Smith predictort time sec

Greek Symbols

λ closed-loop time constant secτ time constant secτI integral time secθ time delay sec

Subscripts

L load (disturbance)m modelp process

60 REFERENCES

1 Khan BZ Lehman B IEEE Trans Control Syst Technol 1996 4 (4) 459ndash4662 Bissell C Control Engineering Chapman amp Hall New York 19943 Schneider DM IEEE Trans Ind Appl 1988 24 (2) 186ndash1914 Smith OJM Chem Eng Prog 1957 53 217ndash2195 Wang QG Bi Q Zhang Y ISA Trans 2000 39 79ndash926 Lee D Lee M Sung S Lee I J Process Control 1999 9 79ndash857 Donoghue JF ISA Trans 1977 16 27ndash348 Abbas A Marshall JE Walton K IEE Proc D 1986 133 (6) 313ndash3149 Marshall JE Control of Time-Delay Systems Peter Peregrinus Ltd London 1979

10 Morari M Zafiriou E Robust Process Control Prentice Hall NJ 1989 12111 Chidambaram M Applied Process Control Allied Publishers Ltd New Delhi

1998 3012 Goodwin GC Graebe SF Salgado ME Control System Design Prentice-Hall

NJ 2001 226

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081AMP100108629

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

698 ABBAS

To investigate the sensitivity of the SP to plantmodel mismatches errorswere introduced in the three model parameters and the closed-loop responses tounit step changes in the setpoint were simulated The results are shown in Figures6ndash8 From these figures it can be concluded that in general the performancedeteriorates as the error in each parameter increases and that the SP controlscheme is more sensitive to modeling errors in the time delay than in the othertwo parameters (steady state gain and time constant) Figure 7 shows that theperformance of the SP as measured by IAE is better than that of the conventionalPI as long as the error in the time delay is within the range 176 to 197The SP system is unstable if the error in θp is outside the range 23 to 264

Figures 6 and 8 show that in certain cases modeling errors may lead toimprovements rather than deterioration of the system performance For example5 error in the model time constant results in about 4 reduction in the IAE

The effect of modeling errors on the performance of the SP structure forload disturbances exhibited similar trends to those corresponding to setpointchanges (see Fig 9)

To compare the sensitivities (robustness) of the classical PI and SP systemto changing plant conditions (not modeling errors) the two control systems weresimulated with errors introduced in the most sensitive parameter namely the timedelay The results are presented in Figure 10 from which it can be clearly seenthat the PI is more robust than the SP The former is quite insensitive to negativechanges in θp

Figure 9 IAE versus percent error in the model delay for the SP scheme (load change)

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

CONTROL OF ALUMINUM ROLLING MILLS 699

Figure 10 IAE versus percent error in the plant delay

40 CONCLUSIONS

The SP was used to control a rolling mill having a high time-delay-to-time-constant ratio (about 3) The SP resulted in a much superior performance com-pared to the classical PI controller As expected the SP was found to be moresensitive to modeling errors in time delay than to changes in the other plant param-eters ie the plant time constant and gain The superiority of the SP over the PIwas maintained as long as the modeling errors or changes in the processing condi-tions were not very large On the other hand the PI exhibited a much better ro-bustness than the SP

50 NOMENCLATURE

C output variablee(t) errorFOPTD first-order plus time delayG transfer functionG delay-free part of the transfer functionh exit thicknessH feed thicknessIAE integral of the absolute errorK gain

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

700 ABBAS

L disturbancePI proportional plus integralPID proportionalndashintegralndashderivativeR setpoints Laplace operatorSP Smith predictort time sec

Greek Symbols

λ closed-loop time constant secτ time constant secτI integral time secθ time delay sec

Subscripts

L load (disturbance)m modelp process

60 REFERENCES

1 Khan BZ Lehman B IEEE Trans Control Syst Technol 1996 4 (4) 459ndash4662 Bissell C Control Engineering Chapman amp Hall New York 19943 Schneider DM IEEE Trans Ind Appl 1988 24 (2) 186ndash1914 Smith OJM Chem Eng Prog 1957 53 217ndash2195 Wang QG Bi Q Zhang Y ISA Trans 2000 39 79ndash926 Lee D Lee M Sung S Lee I J Process Control 1999 9 79ndash857 Donoghue JF ISA Trans 1977 16 27ndash348 Abbas A Marshall JE Walton K IEE Proc D 1986 133 (6) 313ndash3149 Marshall JE Control of Time-Delay Systems Peter Peregrinus Ltd London 1979

10 Morari M Zafiriou E Robust Process Control Prentice Hall NJ 1989 12111 Chidambaram M Applied Process Control Allied Publishers Ltd New Delhi

1998 3012 Goodwin GC Graebe SF Salgado ME Control System Design Prentice-Hall

NJ 2001 226

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081AMP100108629

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

CONTROL OF ALUMINUM ROLLING MILLS 699

Figure 10 IAE versus percent error in the plant delay

40 CONCLUSIONS

The SP was used to control a rolling mill having a high time-delay-to-time-constant ratio (about 3) The SP resulted in a much superior performance com-pared to the classical PI controller As expected the SP was found to be moresensitive to modeling errors in time delay than to changes in the other plant param-eters ie the plant time constant and gain The superiority of the SP over the PIwas maintained as long as the modeling errors or changes in the processing condi-tions were not very large On the other hand the PI exhibited a much better ro-bustness than the SP

50 NOMENCLATURE

C output variablee(t) errorFOPTD first-order plus time delayG transfer functionG delay-free part of the transfer functionh exit thicknessH feed thicknessIAE integral of the absolute errorK gain

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

700 ABBAS

L disturbancePI proportional plus integralPID proportionalndashintegralndashderivativeR setpoints Laplace operatorSP Smith predictort time sec

Greek Symbols

λ closed-loop time constant secτ time constant secτI integral time secθ time delay sec

Subscripts

L load (disturbance)m modelp process

60 REFERENCES

1 Khan BZ Lehman B IEEE Trans Control Syst Technol 1996 4 (4) 459ndash4662 Bissell C Control Engineering Chapman amp Hall New York 19943 Schneider DM IEEE Trans Ind Appl 1988 24 (2) 186ndash1914 Smith OJM Chem Eng Prog 1957 53 217ndash2195 Wang QG Bi Q Zhang Y ISA Trans 2000 39 79ndash926 Lee D Lee M Sung S Lee I J Process Control 1999 9 79ndash857 Donoghue JF ISA Trans 1977 16 27ndash348 Abbas A Marshall JE Walton K IEE Proc D 1986 133 (6) 313ndash3149 Marshall JE Control of Time-Delay Systems Peter Peregrinus Ltd London 1979

10 Morari M Zafiriou E Robust Process Control Prentice Hall NJ 1989 12111 Chidambaram M Applied Process Control Allied Publishers Ltd New Delhi

1998 3012 Goodwin GC Graebe SF Salgado ME Control System Design Prentice-Hall

NJ 2001 226

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081AMP100108629

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

ORDER REPRINTS

700 ABBAS

L disturbancePI proportional plus integralPID proportionalndashintegralndashderivativeR setpoints Laplace operatorSP Smith predictort time sec

Greek Symbols

λ closed-loop time constant secτ time constant secτI integral time secθ time delay sec

Subscripts

L load (disturbance)m modelp process

60 REFERENCES

1 Khan BZ Lehman B IEEE Trans Control Syst Technol 1996 4 (4) 459ndash4662 Bissell C Control Engineering Chapman amp Hall New York 19943 Schneider DM IEEE Trans Ind Appl 1988 24 (2) 186ndash1914 Smith OJM Chem Eng Prog 1957 53 217ndash2195 Wang QG Bi Q Zhang Y ISA Trans 2000 39 79ndash926 Lee D Lee M Sung S Lee I J Process Control 1999 9 79ndash857 Donoghue JF ISA Trans 1977 16 27ndash348 Abbas A Marshall JE Walton K IEE Proc D 1986 133 (6) 313ndash3149 Marshall JE Control of Time-Delay Systems Peter Peregrinus Ltd London 1979

10 Morari M Zafiriou E Robust Process Control Prentice Hall NJ 1989 12111 Chidambaram M Applied Process Control Allied Publishers Ltd New Delhi

1998 3012 Goodwin GC Graebe SF Salgado ME Control System Design Prentice-Hall

NJ 2001 226

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081AMP100108629

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014

Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081AMP100108629

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

09

19 1

4 N

ovem

ber

2014