Research ArticleEnhanced PID Controllers Design Based on Modified SmithPredictor Control for Unstable Process with Time Delay
Chengqiang Yin Jie Gao and Qun Sun
School of Mechanical and Automobile Engineering Liaocheng University Liaocheng 252059 China
Correspondence should be addressed to Chengqiang Yin shtjycq163com
Received 1 July 2014 Revised 30 August 2014 Accepted 1 September 2014 Published 29 September 2014
Academic Editor Hongli Dong
Copyright copy 2014 Chengqiang Yin et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A two-degree-of-freedom control structure is proposed for a class of unstable processes with time delay based on modified Smithpredictor control the superior performance of disturbance rejection and good robust stability are gained for the system The set-point tracking controller is designed using the direct synthesismethod the IMC-PID controller for disturbance rejection is designedbased on the internal mode control design principle The controller for set-point response and the controller for disturbancerejection can be adjusted and optimized independently Meanwhile the two controllers are designed in the form of PID whichis convenient for engineering application Finally simulation examples demonstrate the validity of the proposed control scheme
1 Introduction
Unstable processes are well known to be difficult to controlespecially when there exists pure time delay A time delayis introduced into the transfer function description of suchsystem due to the measurement delay or an actuator delay[1ndash5] A lot of academic research had been devoted todeveloping effective control strategies for such processesGenerally PI or PID controllers are designed using a unityfeedback control structure for these systems Two-degree-of-freedom methods based on PID control are the mostcommon methods [6ndash11] Internal mode control and Smithpredictor (SP) control are regarded as the most effectivemethods for process control andmostwidely used in industrybut cannot be used directly for unstable process with timedelay Owing to the standard SP control structure which is inessence equivalent to the internal model control structure fortime delay processes a number of control schemes based onmodified Smith predictor had been developed in recent years[12ndash18] By using the Smith predictor Rao and Chidambaram[19] proposed a two-degree-of-freedom control scheme forunstable systems with time delay in the scheme threecontrollerswere used to improve the systemperformance Liuet al [20] proposed a modified form of Smith predictor in atwo-degree-of-freedom control scheme which demonstrated
the remarkable improvement of regulatory capacity for bothof reference input tracking and load disturbance rejectionGarcıa andAlbertos [21] proposed a scheme that is equivalentto the Smith predictor but able to cope with any kind ofsystems the results showed a substantial improvement inthe performancerobustness tradeoff as well as in the tuningprocess Vijayan and Panda [22] proposed a double-feedbackloop method which was used to achieve stability and betterperformance of the process By comparison few papers [2324] developed discrete-time domain control methods foradvanced regulation of unstable processes Besides nonlinearcontrol schemes were presented to deal with integratingand unstable processes with time delay [25] Dey et al[26] proposed an autotuning proportional-derivative controlscheme the proportional and derivative gains were adjustedusing a nonlinear gain updating factor to achieve an overallimproved performance
Evaporation processes are common in the food industryIt is the stage in which the water contained in a juice iseliminated in order to obtain a juice with a higher concen-tration The dynamics of the evaporation can be representedwith integrating first order plus time delay process [27] Thecontrol of such system is difficult because of the limitationsimposed by the integrator and the time delay on the systemperformance and stability
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 521460 7 pageshttpdxdoiorg1011552014521460
2 Mathematical Problems in Engineering
r(s)F(s)
minus
minus minus
K1(s)
di(s)do(s)
y(s)
+
++
K2(s)
P0(s) eminus120579s
P(s)
Figure 1 Modified Smith control structure
Disturbance rejection is much more important than set-point tracking for many process control applications But themethods proposed previously for the disturbance rejectionhave not gained much popularity what is more it is difficultto be carried out in process industries The objective of thepresent study is to develop a practicable method to obtainenhanced disturbance rejection performance and perfect set-point tracking performance So a two-degree-of-freedomcontrol scheme based on modified Smith predictor shown inFigure 1 is proposed The set-point tracking controller 119870
1(119904)
and disturbance rejection controller 1198702(119904) are designed in
the form of PID The scheme can lead to substantial controlperformance improvement especially for the disturbancerejectionThe analysis has been carried out for the two typicaltransfer function models 119875(119904) = 119896119890
minus120579119904119904(119879119904 minus 1) and 119875(119904) =
119896119890minus120579119904
119904(119879119904 + 1)In Figure 1 119875
0(119904) is the transfer function of the process
model without the time delay that is 119875(119904) = 1198750(119904)119890minus120579119904 119870
1(119904)
is used for set-point tracking 1198702(119904) is used for disturbance
rejection 119865(119904) is the set-point filter 119903(119904) is the set point119910(119904) isthe process output and 119889
119894(119904) and 119889
119900(119904) are the disturbances
before and after process respectively As can be seen theperformance of set point and load disturbance rejectionresponse are decoupled completely and can bemonotonicallytuned to meet a good performance by controller 119870
1(119904) and
1198702(119904) respectively
2 Controller Design Procedure
21 Set-Point Tracking Controller 1198701(119904) From Figure 1 the
transfer function from 119910(119904) to 119903(119904) can be determined in theform of
In the nominal case that is 119875(119904) = 1198750(119904)119890minus120579119904 the set-point
tracking transfer function can be simplified as
119867119903(119904) =
119910 (119904)
119903 (119904)=
119875 (119904)1198701(119904)
1 + 1198750(119904) 1198701(119904)
(2)
Obviously there is no dead-time element in the characteristicequation of the nominal set-point tracking transfer function1198701(119904) can be obtained if the transfer function is determined
1198701(119904) =
119867119903119889(119904)
1 minus 119867119903119889(119904)
1
1198750(119904)
(3)
Considering the implementation and system performancethe desired set-point tracking transfer function is proposed
119867119903119889(119904) =
119910 (119904)
119903 (119904)=11988621199042+ 1198861119904 + 1
(120582119904 + 1)3
(4)
where 120582 is the adjustable parameter as for the unstableprocess type 119875(119904) = 119896119890
minus120579119904119904(119879119904 minus 1) the controller can be
derived from (3) and (4)
1198701(119904) =
119904 (119879119904 minus 1) (11988621199042+ 1198861119904 + 1)
Because of simple structure and better control performancethan the direct-action tuner the ability of PID controllersto meet most of the control objectives has led to theirwidespread acceptance in the control industry As we knowdistributed control system is widely used in process industryPID module is the basic and the most used module in thedistributed control system over 90 control points weredesigned in PID form [28] To obtain a realizable controller1198701(119904) should be realized in discrete form or approximated by
a rational transfer function So 1198701(119904) can be expressed as
1198701(119904) =
(119879119904 minus 1) (11988621199042+ 1198861119904 + 1)
119896 [12058231199042 + (31205822 minus 1198862) 119904 + (3120582 minus 119886
1)] (6)
According to the model transform method [29] order 1198861=
4120582 1198862= 61205822+ 1 A PID controller with first order lag filter
can be approximated
1198701(119904) = 119896
1(1 +
1
1205911198941119904+ 1205911198891119904)
1
120572119904 + 1 (7)
After approximate comparison we can obtain the parametersof PID 119896
1= 1198861119896 is proportional gain 120591
1198941= 1198861is integral
gain 1205911198891
= 11988621198861is derivative gain and 120572 = 120582
4119879 is the filter
parameterAs can be seen from (4) numerator of the transfer
function will result in undesired system overshoot So inorder to improve the set-point tracking performance andreduce the overshoot a set-point filer is designed 119865(119904) =
1(11988621199042+ 1198861119904 + 1)
Analogously as for the process type119875(119904) = 119896119890minus120579119904
119904(119879119904+1)1198701(119904) can be obtained from (3) and (4)
119896 [12058231199042 + (31205822 minus 1198862) 119904 + (3120582 minus 119886
1)] (8)
Using the similarmethod we obtain 1198861= 3120582 119886
2= 31205822minus1205823119879
The1198701(119904) can be derived in the form of PID as follows
1198701(119904) = 119896
1(1 +
1
1205911198941119904+ 1205911198891119904) (9)
Mathematical Problems in Engineering 3
where 1198961= 3119879119896120582
2 is proportional gain 1205911198941= 3120582 is integral
gain and 1205911198891
= 120582(1 minus 1205823119879) is derivative gain
22 Disturbance Rejection Controller 1198702(119904) In the proposed
control structure shown in Figure 1 the load disturbancetransfer functions are given by
119867119889119894(119904) =
119910 (119904)
119889119894(119904)
=119875 (119904)
1 + 1198702(119904) 119875 (119904)
119867119889119900(119904) =
119910 (119904)
119889119900(119904)
=1
1 + 1198702(119904) 119875 (119904)
(10)
At the same time we can obtain the closed-loop comple-mentary sensitivity function between the process input andoutput for the load disturbance rejection as
119879 (119904) =1198702(119904) 119875 (119904)
1 + 1198702(119904) 119875 (119904)
(11)
Here 1198702(119904) is designed using the method of unit feedback
based on internal mode control theory [30]
1198702(119904) 119875 (119904)
1 + 1198702(119904) 119875 (119904)
= 119875 (119904) 119862 (119904) (12)
where119862(119904) is the internal mode controller 119875(119904) = 119875minus(119904)119875+(119904)
119862(119904) = 119875minus1
minus(119904)119891(119904) in which 119891(119904) is the filter 119875
minus(119904) contains
the invertible portion of the model and 119875+(119904) contains all
the noninvertible portion The invertible portions are thepart of the model with stable poles and unstable poles Thenoninvertible portions are the portion of themodel with righthalf plane zeros and time delays In order to ensure that thesystem is internally stable the filter is designed as
119891 (119904) =sum119898
119894=1119887119894119904119894+ 1
(1205821015840119904 + 1)119899 (13)
where 1205821015840 is an adjustable parameter which controls the trade-off between the performance and robustness determined tocancel the unstable and integrating poles of 119875(119904) 119898 is thenumber of unstable and integrating poles 119899 is selected to belarge enough to make the internal mode controller proper 119887
119894
is determined by 1 minus 119875(119904)119862(119904)|119904=1199111 119911119898
= 0 where 1199111 119911
119898
are the unstable and integrating polesAs for the unstable process type 119875(119904) = 119896119890
minus120579119904119904(119879119904 minus 1)
it can be transformed as 119875(119904) = 1198961015840119890minus120579119904
(1198791015840119904 minus 1)(119879119904 minus 1)
time constant 1198791015840 is selected to be large enough The filter isdesigned as
Since the ]1199042 term has little impact on the overall controlperformance it can be ignored Calculating by using (19) and(21) we can obtain the parameters for the controller 119870
2(119904)
1198962= 11988711198961015840(41205821015840+ 120579 minus 119887
1) is proportional gain 120591
1198942= 1198871is
4 Mathematical Problems in Engineering
integral gain and 1205911198892
= 11988721198871is derivative gain 1205721015840 = 05120579
120573 = ((11988711205792 minus 119887
2+ 21205821015840120579 + 6120582
10158402)(120579 + 4120582
1015840minus 1198871)) + 119879 +119879
1015840 120572 and120573 are parameters of the lead-lag filter
As for the process type 119875(119904) = 119896119890minus120579119904
119904(119879119904 + 1) it can betransformed as 119875(119904) = minus119896
1015840119890minus120579119904
[(1198791015840119904 minus 1)(minus119879119904 minus 1)] and the
executable controller 1198702(119904) can be obtained in the form of
PID by using (15) (18) and (20) On the basis of simulationstudy on integrating first order plus time delay processes theuse of 01120573 instead of 120573 is suitable and 120573 is about 02ndash12generally
3 System Robust Stability Analysis
A control system is robust if it is insensitive to differencesbetween the actual system and the model of the systemwhich was used to design the controllerThese differences arereferred to as model mismatch or simply model uncertainty[31] A study of robustness analysis is an important taskbecause no mathematical model of a system will be a perfectrepresentation of the actual system Small-gain theorem is119897119898119879(119904)infin
lt 1 it expresses the robustly stable condition ofa control system where 119897
119898(119904) is the bound on the process
multiplicative uncertainty and 119879(119904) is the closed-loop com-plementary sensitivity function [32]
For the process type 119875(119904) = 119896119890minus120579119904
119904(119879119904 minus 1) if there areuncertainties that exist in all three parameters that is
1003816100381610038161003816119890minus119895Δ120579119908 minus 1
1003816100381610038161003816
(25)
At the same time in order to compromise the nominalperformance with the robust stability of the closed-loop forthe load disturbance rejection the following constraint isrequired to meet [31]
where 120596(119904) is a weight function of the closed-loop sensitivityfunction 119878 = 1 minus 119879(119904) which usually can be chosen as
15
1
05
0
0 10 20 30 40 50
Time (s)
Proc
ess o
utpu
t
LiuProposed
Figure 2 Nominal system responses for Example 1
1119904 for the step change of the load disturbance It indicatesthat tuning the adjustable parameter 1205821015840 aims at the tradeoffbetween the nominal performance of the closed-loop andits robust stability That is to say decreasing 120582
1015840 improvesthe disturbance rejection performance of the closed-loopbut decays its robust stability in the presence of the pro-cess uncertainty On the contrary increasing 120582
1015840 tends tostrengthen the robust stability of the closed-loop but degradesits disturbance rejection performance
4 Simulation
Example 1 Consider the unstable process studied by Liu et al[20] 119875(119904) = 119890
minus02119904119904(119904 minus 1)
In the proposed method take 120582 = 33120579 = 066 theparameters of controllers are obtained 119896
1= 264 120591
1198941= 264
1205911198891
= 1367 120572 = 019 1198861
= 264 and 1198862
= 361 Asfor the controller 119870
2(119904) transform the process as 119875(119904) =
100119890minus02119904
(100119904 minus 1)(119904 minus 1) take 1205821015840 = 2120579 = 04 and obtain
1198702(119904) = 302 (1 +
1
179119904+ 106119904)
01119904 + 1
0008119904 + 1 (27)
The method proposed in [20] is better than others which canbe seen in this simulation effects so only compare with themethod in [20] By adding a unit step change to the set-pointinput at 119905 = 0 an inverse unit step change of load disturbanceto process output at 119905 = 20 the simulation results are obtainedas shown in Figure 2
Now suppose that there exists 30 increment for estimat-ing the process time delay and 30 reduction for the timeconstant of the process model then the perturbed systemresponses are provided in Figure 3
It can be seen from the simulation results that theproposed method gives better performances for the unstableprocess
Example 2 Consider a process studied by Liu and Gao [15]119875(119904) = 01119890
minus5119904119904(5119904 + 1)
Mathematical Problems in Engineering 5
15
1
05
0
0 10 20 30 40 50
Time (s)
Proc
ess o
utpu
t
LiuProposed
Figure 3 Perturbed system responses for Example 1
0 100 200 300 400Time (s)
12
1
08
06
04
02
0
Proc
ess o
utpu
t
LiuProposed
Figure 4 Nominal system responses for Example 2
In the proposed method as for controllers 1198701(119904) take
120582 = 08120579 = 4 and obtain 1198961= 9375 120591
1198941= 12 120591
1198891= 2933
1198861= 12 and 119886
2= 352 As for controller 119870
2(119904) transform
the process as 119875(119904) = minus10119890minus5119904
(100119904 minus 1)(minus5119904 minus 1) take1205821015840= 08120579 = 4 and obtain the controller
1198702(119904) = 22 (1 +
1
22119904+ 3864119904)
25119904 + 1
08119904 + 1 (28)
Add a unit step change to the set-point input at 119905 = 0 and aninverse step change of load disturbance to the process outputat 119905 = 200 The simulation results are obtained as shown inFigure 4 and the corresponding control action responses areshown in Figure 5 It can be observed from the figure thatthe control action response of the proposed method showssmooth variation compared to that of Liu and Gao [15]
Now suppose that there exist 10 error for estimatingthe process time delay and the time constant of the processmodel such as both of them are actually 10 larger Theperturbed system responses are provided in Figure 6 andthe corresponding control action responses are shown inFigure 7 It can be seen that the proposed control action
Con
trol s
igna
l
6
5
4
3
2
1
0
minus1
0 100 200 300 400
Time (s)
LiuProposed
Figure 5 Nominal system control signal for Example 2
15
1
05
0
0
100 200 300 400
Proc
ess o
utpu
t
Time (s)
LiuProposed
Figure 6 Perturbed system responses for Example 2
is smooth and the performance of disturbance rejection isbetter than that of Liu and Gao [15]
5 Conclusion
In order to improve the system performance of distur-bance rejection a modified Smith predictor scheme hasbeen proposed based on a two-degree-of-freedom controlstructure In the proposed control structure both of the set-point response and the load disturbance response can betuned separately by the set-point tracking controller and thedisturbance estimator respectivelyThemost advantage of themethod is that the designed system has good performanceof disturbance rejection as well as performance of set-pointtracking The two controllers are all designed in the form ofPID and they are simple and easy to be used in process indus-try Comparisons with the previous methods demonstrate aclear advantage of the proposedmethod in both nominal androbust performances in disturbance rejection
6 Mathematical Problems in Engineering
6
5
4
3
2
1
0
minus1
0 100 200 300 400
Con
trol s
igna
l
Time (s)
LiuProposed
Figure 7 Perturbed system control signal for Example 2
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Natural Science Founda-tion of Shandong Province (Grant no ZR2012CQ026) andthe Science and Technology Funds of Shandong EducationDepartment (Grant no J11LD16)
References
[1] I-L Chien S C Peng and J H Liu ldquoSimple control methodfor integrating processes with long deadtimerdquo Journal of ProcessControl vol 12 no 3 pp 391ndash404 2002
[2] M Huzmezan W A Gough G A Dumont and S KovacldquoTime delay integrating systems a challenge for process controlindustries A practical solutionrdquo Control Engineering Practicevol 10 no 10 pp 1153ndash1161 2002
[3] H Dong Z Wang and H Gao ldquoDistributed 119867infin
filteringfor a class of markovian jump nonlinear time-delay systemsover lossy sensor networksrdquo IEEE Transactions on IndustrialElectronics vol 60 no 10 pp 4665ndash4672 2013
[4] J E Normey-Rico C Bordons and E F Camacho Control ofDead-Time Processes Spinger Berlin Germany 2007
[5] Z Wang H Dong B Shen and H Gao ldquoFinite-horizon 119867infin
filtering with missing measurements and quantization effectsrdquoIEEE Transactions on Automatic Control vol 58 no 7 pp 1707ndash1718 2013
[6] A Visioli ldquoOptimal tuning of PID controllers for integraland unstable processesrdquo IEE Proceedings Control Theory andApplications vol 148 no 2 pp 180ndash184 2001
[7] C Hwang and J Hwang ldquoStabilisation of first-order plus dead-time unstable processes using PID controllersrdquo IEE ProceedingsControl Theory and Applications vol 151 no 1 pp 89ndash94 2004
[8] R C Panda ldquoSynthesis of PID controller for unstable andintegrating processesrdquoChemical Engineering Science vol 64 no12 pp 2807ndash2816 2009
[9] Y Lee J Lee and S Park ldquoPID controller tuning for integratingand unstable processes with time delayrdquo Chemical EngineeringScience vol 55 no 17 pp 3481ndash3493 2000
[10] A Seshagiri Rao V S Rao and M Chidambaram ldquoDirectsynthesis-based controller design for integrating processes withtime delayrdquo Journal of the Franklin Institute vol 346 no 1 pp38ndash56 2009
[11] D G Padhan and S Majhi ldquoA new control scheme for PIDload frequency controller of single-area and multi-area powersystemsrdquo ISA Transactions vol 52 no 2 pp 242ndash251 2013
[12] I Kaya ldquoAutotuning of a new PI-PD smith predictor based ontime domain specificationsrdquo ISA Transactions vol 42 no 4 pp559ndash575 2003
[13] S Majhi and D P Atherton ldquoObtaining controller parametersfor a new Smith predictor using autotuningrdquo Automatica vol36 no 11 pp 1651ndash1658 2000
[14] J E Normey-Rico and E F Camacho ldquoDead-time compen-sators a surveyrdquo Control Engineering Practice vol 16 no 4 pp407ndash428 2008
[15] T Liu and F R Gao ldquoEnhanced IMCdesign of load disturbancerejection for integrating and unstable processes with slowdynamicsrdquo ISA Transactions vol 50 no 2 pp 239ndash248 2011
[16] E-D Cong M-H Hu S-T Tu F-Z Xuan and H-H Shao ldquoAnovel double loop control model design for chemical unstableprocessesrdquo ISA Transactions vol 53 no 2 pp 497ndash507 2014
[17] J E Normey-Rico P Garcia and A Gonzalez ldquoRobust stabilityanalysis of filtered Smith predictor for time-varying delayprocessesrdquo Journal of Process Control vol 22 no 10 pp 1975ndash1984 2012
[18] J Nandong and Z Zang ldquoHigh-performance multi-scale con-trol scheme for stable integrating and unstable time-delayprocessesrdquo Journal of Process Control vol 23 no 10 pp 1333ndash1343 2013
[19] A S Rao andM Chidambaram ldquoAnalytical design of modifiedSmith predictor in a two-degrees-of-freedom control schemefor second order unstable processes with time delayrdquo ISATransactions vol 47 no 4 pp 407ndash419 2008
[20] T Liu W Zhang and D Gu ldquoAnalytical design of two-degree-of-freedom control scheme for open-loop unstable processeswith time delayrdquo Journal of Process Control vol 15 no 5 pp559ndash572 2005
[21] P Garcıa and P Albertos ldquoRobust tuning of a generalizedpredictor-based controller for integrating and unstable systemswith long time-delayrdquo Journal of Process Control vol 23 no 8pp 1205ndash1216 2013
[22] VVijayan andRC Panda ldquoDesign of PID controllers in doublefeedback loops for SISO systems with set-point filtersrdquo ISATransactions vol 51 no 4 pp 514ndash521 2012
[23] P Garcıa P Albertos and T Hagglund ldquoControl of unstablenon-minimum-phase delayed systemsrdquo Journal of Process Con-trol vol 16 no 10 pp 1099ndash1111 2006
[24] P Albertos and P Garcıa ldquoRobust control design for long time-delay systemsrdquo Journal of Process Control vol 19 no 10 pp1640ndash1648 2009
[25] J C Moreno J L Guzman J E Normey-Rico A Banos andM Berenguel ldquoA combined FSP and reset control approachto improve the set-point tracking task of dead-time processesrdquoControl Engineering Practice vol 21 no 4 pp 351ndash359 2013
[26] C Dey R K Mudi and D Simhachalam ldquoA simple nonlinearPD controller for integrating processesrdquo ISA Transactions vol53 no 1 pp 162ndash172 2014
Mathematical Problems in Engineering 7
[27] J E Normey-Rico and E F Camacho ldquoUnified approachfor robust dead-time compensator designrdquo Journal of ProcessControl vol 21 no 7 pp 1080ndash1091 2011
[28] K J Astrom and H Hagglund PID Controllers Theory Designand Tuning Instrument Society of America Research TrianglePark NC USA 2nd edition 1995
[29] B A Ogunnaike and W H Ray Process Modeling Dynamicsand Control Oxford University New York NY USA 1994
[30] M Shamsuzzoha andM Lee ldquoEnhanced performance for two-degree-of-freedom control scheme for second order unstableprocesses with time delayrdquo in Proceedings of the InternationalConference on Control Automation and Systems (ICCAS rsquo07)pp 240ndash245 Seoul Republic of Korea October 2007
[31] MMorari andE ZafiriouRobust Process Control PrenticeHallEnglewood Cliffs NJ USA 1989
[32] J C Doyle B A Francis and A R Tannenbaum FeedbackControl Theory Macmillan Publishing Company New YorkNY USA 1992
Disturbance rejection is much more important than set-point tracking for many process control applications But themethods proposed previously for the disturbance rejectionhave not gained much popularity what is more it is difficultto be carried out in process industries The objective of thepresent study is to develop a practicable method to obtainenhanced disturbance rejection performance and perfect set-point tracking performance So a two-degree-of-freedomcontrol scheme based on modified Smith predictor shown inFigure 1 is proposed The set-point tracking controller 119870
1(119904)
and disturbance rejection controller 1198702(119904) are designed in
the form of PID The scheme can lead to substantial controlperformance improvement especially for the disturbancerejectionThe analysis has been carried out for the two typicaltransfer function models 119875(119904) = 119896119890
minus120579119904119904(119879119904 minus 1) and 119875(119904) =
119896119890minus120579119904
119904(119879119904 + 1)In Figure 1 119875
0(119904) is the transfer function of the process
model without the time delay that is 119875(119904) = 1198750(119904)119890minus120579119904 119870
1(119904)
is used for set-point tracking 1198702(119904) is used for disturbance
rejection 119865(119904) is the set-point filter 119903(119904) is the set point119910(119904) isthe process output and 119889
119894(119904) and 119889
119900(119904) are the disturbances
before and after process respectively As can be seen theperformance of set point and load disturbance rejectionresponse are decoupled completely and can bemonotonicallytuned to meet a good performance by controller 119870
1(119904) and
1198702(119904) respectively
2 Controller Design Procedure
21 Set-Point Tracking Controller 1198701(119904) From Figure 1 the
transfer function from 119910(119904) to 119903(119904) can be determined in theform of
In the nominal case that is 119875(119904) = 1198750(119904)119890minus120579119904 the set-point
tracking transfer function can be simplified as
119867119903(119904) =
119910 (119904)
119903 (119904)=
119875 (119904)1198701(119904)
1 + 1198750(119904) 1198701(119904)
(2)
Obviously there is no dead-time element in the characteristicequation of the nominal set-point tracking transfer function1198701(119904) can be obtained if the transfer function is determined
1198701(119904) =
119867119903119889(119904)
1 minus 119867119903119889(119904)
1
1198750(119904)
(3)
Considering the implementation and system performancethe desired set-point tracking transfer function is proposed
119867119903119889(119904) =
119910 (119904)
119903 (119904)=11988621199042+ 1198861119904 + 1
(120582119904 + 1)3
(4)
where 120582 is the adjustable parameter as for the unstableprocess type 119875(119904) = 119896119890
minus120579119904119904(119879119904 minus 1) the controller can be
derived from (3) and (4)
1198701(119904) =
119904 (119879119904 minus 1) (11988621199042+ 1198861119904 + 1)
Because of simple structure and better control performancethan the direct-action tuner the ability of PID controllersto meet most of the control objectives has led to theirwidespread acceptance in the control industry As we knowdistributed control system is widely used in process industryPID module is the basic and the most used module in thedistributed control system over 90 control points weredesigned in PID form [28] To obtain a realizable controller1198701(119904) should be realized in discrete form or approximated by
a rational transfer function So 1198701(119904) can be expressed as
1198701(119904) =
(119879119904 minus 1) (11988621199042+ 1198861119904 + 1)
119896 [12058231199042 + (31205822 minus 1198862) 119904 + (3120582 minus 119886
1)] (6)
According to the model transform method [29] order 1198861=
4120582 1198862= 61205822+ 1 A PID controller with first order lag filter
can be approximated
1198701(119904) = 119896
1(1 +
1
1205911198941119904+ 1205911198891119904)
1
120572119904 + 1 (7)
After approximate comparison we can obtain the parametersof PID 119896
1= 1198861119896 is proportional gain 120591
1198941= 1198861is integral
gain 1205911198891
= 11988621198861is derivative gain and 120572 = 120582
4119879 is the filter
parameterAs can be seen from (4) numerator of the transfer
function will result in undesired system overshoot So inorder to improve the set-point tracking performance andreduce the overshoot a set-point filer is designed 119865(119904) =
1(11988621199042+ 1198861119904 + 1)
Analogously as for the process type119875(119904) = 119896119890minus120579119904
119904(119879119904+1)1198701(119904) can be obtained from (3) and (4)
119896 [12058231199042 + (31205822 minus 1198862) 119904 + (3120582 minus 119886
1)] (8)
Using the similarmethod we obtain 1198861= 3120582 119886
2= 31205822minus1205823119879
The1198701(119904) can be derived in the form of PID as follows
1198701(119904) = 119896
1(1 +
1
1205911198941119904+ 1205911198891119904) (9)
Mathematical Problems in Engineering 3
where 1198961= 3119879119896120582
2 is proportional gain 1205911198941= 3120582 is integral
gain and 1205911198891
= 120582(1 minus 1205823119879) is derivative gain
22 Disturbance Rejection Controller 1198702(119904) In the proposed
control structure shown in Figure 1 the load disturbancetransfer functions are given by
119867119889119894(119904) =
119910 (119904)
119889119894(119904)
=119875 (119904)
1 + 1198702(119904) 119875 (119904)
119867119889119900(119904) =
119910 (119904)
119889119900(119904)
=1
1 + 1198702(119904) 119875 (119904)
(10)
At the same time we can obtain the closed-loop comple-mentary sensitivity function between the process input andoutput for the load disturbance rejection as
119879 (119904) =1198702(119904) 119875 (119904)
1 + 1198702(119904) 119875 (119904)
(11)
Here 1198702(119904) is designed using the method of unit feedback
based on internal mode control theory [30]
1198702(119904) 119875 (119904)
1 + 1198702(119904) 119875 (119904)
= 119875 (119904) 119862 (119904) (12)
where119862(119904) is the internal mode controller 119875(119904) = 119875minus(119904)119875+(119904)
119862(119904) = 119875minus1
minus(119904)119891(119904) in which 119891(119904) is the filter 119875
minus(119904) contains
the invertible portion of the model and 119875+(119904) contains all
the noninvertible portion The invertible portions are thepart of the model with stable poles and unstable poles Thenoninvertible portions are the portion of themodel with righthalf plane zeros and time delays In order to ensure that thesystem is internally stable the filter is designed as
119891 (119904) =sum119898
119894=1119887119894119904119894+ 1
(1205821015840119904 + 1)119899 (13)
where 1205821015840 is an adjustable parameter which controls the trade-off between the performance and robustness determined tocancel the unstable and integrating poles of 119875(119904) 119898 is thenumber of unstable and integrating poles 119899 is selected to belarge enough to make the internal mode controller proper 119887
119894
is determined by 1 minus 119875(119904)119862(119904)|119904=1199111 119911119898
= 0 where 1199111 119911
119898
are the unstable and integrating polesAs for the unstable process type 119875(119904) = 119896119890
minus120579119904119904(119879119904 minus 1)
it can be transformed as 119875(119904) = 1198961015840119890minus120579119904
(1198791015840119904 minus 1)(119879119904 minus 1)
time constant 1198791015840 is selected to be large enough The filter isdesigned as
Since the ]1199042 term has little impact on the overall controlperformance it can be ignored Calculating by using (19) and(21) we can obtain the parameters for the controller 119870
2(119904)
1198962= 11988711198961015840(41205821015840+ 120579 minus 119887
1) is proportional gain 120591
1198942= 1198871is
4 Mathematical Problems in Engineering
integral gain and 1205911198892
= 11988721198871is derivative gain 1205721015840 = 05120579
120573 = ((11988711205792 minus 119887
2+ 21205821015840120579 + 6120582
10158402)(120579 + 4120582
1015840minus 1198871)) + 119879 +119879
1015840 120572 and120573 are parameters of the lead-lag filter
As for the process type 119875(119904) = 119896119890minus120579119904
119904(119879119904 + 1) it can betransformed as 119875(119904) = minus119896
1015840119890minus120579119904
[(1198791015840119904 minus 1)(minus119879119904 minus 1)] and the
executable controller 1198702(119904) can be obtained in the form of
PID by using (15) (18) and (20) On the basis of simulationstudy on integrating first order plus time delay processes theuse of 01120573 instead of 120573 is suitable and 120573 is about 02ndash12generally
3 System Robust Stability Analysis
A control system is robust if it is insensitive to differencesbetween the actual system and the model of the systemwhich was used to design the controllerThese differences arereferred to as model mismatch or simply model uncertainty[31] A study of robustness analysis is an important taskbecause no mathematical model of a system will be a perfectrepresentation of the actual system Small-gain theorem is119897119898119879(119904)infin
lt 1 it expresses the robustly stable condition ofa control system where 119897
119898(119904) is the bound on the process
multiplicative uncertainty and 119879(119904) is the closed-loop com-plementary sensitivity function [32]
For the process type 119875(119904) = 119896119890minus120579119904
119904(119879119904 minus 1) if there areuncertainties that exist in all three parameters that is
1003816100381610038161003816119890minus119895Δ120579119908 minus 1
1003816100381610038161003816
(25)
At the same time in order to compromise the nominalperformance with the robust stability of the closed-loop forthe load disturbance rejection the following constraint isrequired to meet [31]
where 120596(119904) is a weight function of the closed-loop sensitivityfunction 119878 = 1 minus 119879(119904) which usually can be chosen as
15
1
05
0
0 10 20 30 40 50
Time (s)
Proc
ess o
utpu
t
LiuProposed
Figure 2 Nominal system responses for Example 1
1119904 for the step change of the load disturbance It indicatesthat tuning the adjustable parameter 1205821015840 aims at the tradeoffbetween the nominal performance of the closed-loop andits robust stability That is to say decreasing 120582
1015840 improvesthe disturbance rejection performance of the closed-loopbut decays its robust stability in the presence of the pro-cess uncertainty On the contrary increasing 120582
1015840 tends tostrengthen the robust stability of the closed-loop but degradesits disturbance rejection performance
4 Simulation
Example 1 Consider the unstable process studied by Liu et al[20] 119875(119904) = 119890
minus02119904119904(119904 minus 1)
In the proposed method take 120582 = 33120579 = 066 theparameters of controllers are obtained 119896
1= 264 120591
1198941= 264
1205911198891
= 1367 120572 = 019 1198861
= 264 and 1198862
= 361 Asfor the controller 119870
2(119904) transform the process as 119875(119904) =
100119890minus02119904
(100119904 minus 1)(119904 minus 1) take 1205821015840 = 2120579 = 04 and obtain
1198702(119904) = 302 (1 +
1
179119904+ 106119904)
01119904 + 1
0008119904 + 1 (27)
The method proposed in [20] is better than others which canbe seen in this simulation effects so only compare with themethod in [20] By adding a unit step change to the set-pointinput at 119905 = 0 an inverse unit step change of load disturbanceto process output at 119905 = 20 the simulation results are obtainedas shown in Figure 2
Now suppose that there exists 30 increment for estimat-ing the process time delay and 30 reduction for the timeconstant of the process model then the perturbed systemresponses are provided in Figure 3
It can be seen from the simulation results that theproposed method gives better performances for the unstableprocess
Example 2 Consider a process studied by Liu and Gao [15]119875(119904) = 01119890
minus5119904119904(5119904 + 1)
Mathematical Problems in Engineering 5
15
1
05
0
0 10 20 30 40 50
Time (s)
Proc
ess o
utpu
t
LiuProposed
Figure 3 Perturbed system responses for Example 1
0 100 200 300 400Time (s)
12
1
08
06
04
02
0
Proc
ess o
utpu
t
LiuProposed
Figure 4 Nominal system responses for Example 2
In the proposed method as for controllers 1198701(119904) take
120582 = 08120579 = 4 and obtain 1198961= 9375 120591
1198941= 12 120591
1198891= 2933
1198861= 12 and 119886
2= 352 As for controller 119870
2(119904) transform
the process as 119875(119904) = minus10119890minus5119904
(100119904 minus 1)(minus5119904 minus 1) take1205821015840= 08120579 = 4 and obtain the controller
1198702(119904) = 22 (1 +
1
22119904+ 3864119904)
25119904 + 1
08119904 + 1 (28)
Add a unit step change to the set-point input at 119905 = 0 and aninverse step change of load disturbance to the process outputat 119905 = 200 The simulation results are obtained as shown inFigure 4 and the corresponding control action responses areshown in Figure 5 It can be observed from the figure thatthe control action response of the proposed method showssmooth variation compared to that of Liu and Gao [15]
Now suppose that there exist 10 error for estimatingthe process time delay and the time constant of the processmodel such as both of them are actually 10 larger Theperturbed system responses are provided in Figure 6 andthe corresponding control action responses are shown inFigure 7 It can be seen that the proposed control action
Con
trol s
igna
l
6
5
4
3
2
1
0
minus1
0 100 200 300 400
Time (s)
LiuProposed
Figure 5 Nominal system control signal for Example 2
15
1
05
0
0
100 200 300 400
Proc
ess o
utpu
t
Time (s)
LiuProposed
Figure 6 Perturbed system responses for Example 2
is smooth and the performance of disturbance rejection isbetter than that of Liu and Gao [15]
5 Conclusion
In order to improve the system performance of distur-bance rejection a modified Smith predictor scheme hasbeen proposed based on a two-degree-of-freedom controlstructure In the proposed control structure both of the set-point response and the load disturbance response can betuned separately by the set-point tracking controller and thedisturbance estimator respectivelyThemost advantage of themethod is that the designed system has good performanceof disturbance rejection as well as performance of set-pointtracking The two controllers are all designed in the form ofPID and they are simple and easy to be used in process indus-try Comparisons with the previous methods demonstrate aclear advantage of the proposedmethod in both nominal androbust performances in disturbance rejection
6 Mathematical Problems in Engineering
6
5
4
3
2
1
0
minus1
0 100 200 300 400
Con
trol s
igna
l
Time (s)
LiuProposed
Figure 7 Perturbed system control signal for Example 2
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Natural Science Founda-tion of Shandong Province (Grant no ZR2012CQ026) andthe Science and Technology Funds of Shandong EducationDepartment (Grant no J11LD16)
References
[1] I-L Chien S C Peng and J H Liu ldquoSimple control methodfor integrating processes with long deadtimerdquo Journal of ProcessControl vol 12 no 3 pp 391ndash404 2002
[2] M Huzmezan W A Gough G A Dumont and S KovacldquoTime delay integrating systems a challenge for process controlindustries A practical solutionrdquo Control Engineering Practicevol 10 no 10 pp 1153ndash1161 2002
[3] H Dong Z Wang and H Gao ldquoDistributed 119867infin
filteringfor a class of markovian jump nonlinear time-delay systemsover lossy sensor networksrdquo IEEE Transactions on IndustrialElectronics vol 60 no 10 pp 4665ndash4672 2013
[4] J E Normey-Rico C Bordons and E F Camacho Control ofDead-Time Processes Spinger Berlin Germany 2007
[5] Z Wang H Dong B Shen and H Gao ldquoFinite-horizon 119867infin
filtering with missing measurements and quantization effectsrdquoIEEE Transactions on Automatic Control vol 58 no 7 pp 1707ndash1718 2013
[6] A Visioli ldquoOptimal tuning of PID controllers for integraland unstable processesrdquo IEE Proceedings Control Theory andApplications vol 148 no 2 pp 180ndash184 2001
[7] C Hwang and J Hwang ldquoStabilisation of first-order plus dead-time unstable processes using PID controllersrdquo IEE ProceedingsControl Theory and Applications vol 151 no 1 pp 89ndash94 2004
[8] R C Panda ldquoSynthesis of PID controller for unstable andintegrating processesrdquoChemical Engineering Science vol 64 no12 pp 2807ndash2816 2009
[9] Y Lee J Lee and S Park ldquoPID controller tuning for integratingand unstable processes with time delayrdquo Chemical EngineeringScience vol 55 no 17 pp 3481ndash3493 2000
[10] A Seshagiri Rao V S Rao and M Chidambaram ldquoDirectsynthesis-based controller design for integrating processes withtime delayrdquo Journal of the Franklin Institute vol 346 no 1 pp38ndash56 2009
[11] D G Padhan and S Majhi ldquoA new control scheme for PIDload frequency controller of single-area and multi-area powersystemsrdquo ISA Transactions vol 52 no 2 pp 242ndash251 2013
[12] I Kaya ldquoAutotuning of a new PI-PD smith predictor based ontime domain specificationsrdquo ISA Transactions vol 42 no 4 pp559ndash575 2003
[13] S Majhi and D P Atherton ldquoObtaining controller parametersfor a new Smith predictor using autotuningrdquo Automatica vol36 no 11 pp 1651ndash1658 2000
[14] J E Normey-Rico and E F Camacho ldquoDead-time compen-sators a surveyrdquo Control Engineering Practice vol 16 no 4 pp407ndash428 2008
[15] T Liu and F R Gao ldquoEnhanced IMCdesign of load disturbancerejection for integrating and unstable processes with slowdynamicsrdquo ISA Transactions vol 50 no 2 pp 239ndash248 2011
[16] E-D Cong M-H Hu S-T Tu F-Z Xuan and H-H Shao ldquoAnovel double loop control model design for chemical unstableprocessesrdquo ISA Transactions vol 53 no 2 pp 497ndash507 2014
[17] J E Normey-Rico P Garcia and A Gonzalez ldquoRobust stabilityanalysis of filtered Smith predictor for time-varying delayprocessesrdquo Journal of Process Control vol 22 no 10 pp 1975ndash1984 2012
[18] J Nandong and Z Zang ldquoHigh-performance multi-scale con-trol scheme for stable integrating and unstable time-delayprocessesrdquo Journal of Process Control vol 23 no 10 pp 1333ndash1343 2013
[19] A S Rao andM Chidambaram ldquoAnalytical design of modifiedSmith predictor in a two-degrees-of-freedom control schemefor second order unstable processes with time delayrdquo ISATransactions vol 47 no 4 pp 407ndash419 2008
[20] T Liu W Zhang and D Gu ldquoAnalytical design of two-degree-of-freedom control scheme for open-loop unstable processeswith time delayrdquo Journal of Process Control vol 15 no 5 pp559ndash572 2005
[21] P Garcıa and P Albertos ldquoRobust tuning of a generalizedpredictor-based controller for integrating and unstable systemswith long time-delayrdquo Journal of Process Control vol 23 no 8pp 1205ndash1216 2013
[22] VVijayan andRC Panda ldquoDesign of PID controllers in doublefeedback loops for SISO systems with set-point filtersrdquo ISATransactions vol 51 no 4 pp 514ndash521 2012
[23] P Garcıa P Albertos and T Hagglund ldquoControl of unstablenon-minimum-phase delayed systemsrdquo Journal of Process Con-trol vol 16 no 10 pp 1099ndash1111 2006
[24] P Albertos and P Garcıa ldquoRobust control design for long time-delay systemsrdquo Journal of Process Control vol 19 no 10 pp1640ndash1648 2009
[25] J C Moreno J L Guzman J E Normey-Rico A Banos andM Berenguel ldquoA combined FSP and reset control approachto improve the set-point tracking task of dead-time processesrdquoControl Engineering Practice vol 21 no 4 pp 351ndash359 2013
[26] C Dey R K Mudi and D Simhachalam ldquoA simple nonlinearPD controller for integrating processesrdquo ISA Transactions vol53 no 1 pp 162ndash172 2014
Mathematical Problems in Engineering 7
[27] J E Normey-Rico and E F Camacho ldquoUnified approachfor robust dead-time compensator designrdquo Journal of ProcessControl vol 21 no 7 pp 1080ndash1091 2011
[28] K J Astrom and H Hagglund PID Controllers Theory Designand Tuning Instrument Society of America Research TrianglePark NC USA 2nd edition 1995
[29] B A Ogunnaike and W H Ray Process Modeling Dynamicsand Control Oxford University New York NY USA 1994
[30] M Shamsuzzoha andM Lee ldquoEnhanced performance for two-degree-of-freedom control scheme for second order unstableprocesses with time delayrdquo in Proceedings of the InternationalConference on Control Automation and Systems (ICCAS rsquo07)pp 240ndash245 Seoul Republic of Korea October 2007
[31] MMorari andE ZafiriouRobust Process Control PrenticeHallEnglewood Cliffs NJ USA 1989
[32] J C Doyle B A Francis and A R Tannenbaum FeedbackControl Theory Macmillan Publishing Company New YorkNY USA 1992
2 is proportional gain 1205911198941= 3120582 is integral
gain and 1205911198891
= 120582(1 minus 1205823119879) is derivative gain
22 Disturbance Rejection Controller 1198702(119904) In the proposed
control structure shown in Figure 1 the load disturbancetransfer functions are given by
119867119889119894(119904) =
119910 (119904)
119889119894(119904)
=119875 (119904)
1 + 1198702(119904) 119875 (119904)
119867119889119900(119904) =
119910 (119904)
119889119900(119904)
=1
1 + 1198702(119904) 119875 (119904)
(10)
At the same time we can obtain the closed-loop comple-mentary sensitivity function between the process input andoutput for the load disturbance rejection as
119879 (119904) =1198702(119904) 119875 (119904)
1 + 1198702(119904) 119875 (119904)
(11)
Here 1198702(119904) is designed using the method of unit feedback
based on internal mode control theory [30]
1198702(119904) 119875 (119904)
1 + 1198702(119904) 119875 (119904)
= 119875 (119904) 119862 (119904) (12)
where119862(119904) is the internal mode controller 119875(119904) = 119875minus(119904)119875+(119904)
119862(119904) = 119875minus1
minus(119904)119891(119904) in which 119891(119904) is the filter 119875
minus(119904) contains
the invertible portion of the model and 119875+(119904) contains all
the noninvertible portion The invertible portions are thepart of the model with stable poles and unstable poles Thenoninvertible portions are the portion of themodel with righthalf plane zeros and time delays In order to ensure that thesystem is internally stable the filter is designed as
119891 (119904) =sum119898
119894=1119887119894119904119894+ 1
(1205821015840119904 + 1)119899 (13)
where 1205821015840 is an adjustable parameter which controls the trade-off between the performance and robustness determined tocancel the unstable and integrating poles of 119875(119904) 119898 is thenumber of unstable and integrating poles 119899 is selected to belarge enough to make the internal mode controller proper 119887
119894
is determined by 1 minus 119875(119904)119862(119904)|119904=1199111 119911119898
= 0 where 1199111 119911
119898
are the unstable and integrating polesAs for the unstable process type 119875(119904) = 119896119890
minus120579119904119904(119879119904 minus 1)
it can be transformed as 119875(119904) = 1198961015840119890minus120579119904
(1198791015840119904 minus 1)(119879119904 minus 1)
time constant 1198791015840 is selected to be large enough The filter isdesigned as
Since the ]1199042 term has little impact on the overall controlperformance it can be ignored Calculating by using (19) and(21) we can obtain the parameters for the controller 119870
2(119904)
1198962= 11988711198961015840(41205821015840+ 120579 minus 119887
1) is proportional gain 120591
1198942= 1198871is
4 Mathematical Problems in Engineering
integral gain and 1205911198892
= 11988721198871is derivative gain 1205721015840 = 05120579
120573 = ((11988711205792 minus 119887
2+ 21205821015840120579 + 6120582
10158402)(120579 + 4120582
1015840minus 1198871)) + 119879 +119879
1015840 120572 and120573 are parameters of the lead-lag filter
As for the process type 119875(119904) = 119896119890minus120579119904
119904(119879119904 + 1) it can betransformed as 119875(119904) = minus119896
1015840119890minus120579119904
[(1198791015840119904 minus 1)(minus119879119904 minus 1)] and the
executable controller 1198702(119904) can be obtained in the form of
PID by using (15) (18) and (20) On the basis of simulationstudy on integrating first order plus time delay processes theuse of 01120573 instead of 120573 is suitable and 120573 is about 02ndash12generally
3 System Robust Stability Analysis
A control system is robust if it is insensitive to differencesbetween the actual system and the model of the systemwhich was used to design the controllerThese differences arereferred to as model mismatch or simply model uncertainty[31] A study of robustness analysis is an important taskbecause no mathematical model of a system will be a perfectrepresentation of the actual system Small-gain theorem is119897119898119879(119904)infin
lt 1 it expresses the robustly stable condition ofa control system where 119897
119898(119904) is the bound on the process
multiplicative uncertainty and 119879(119904) is the closed-loop com-plementary sensitivity function [32]
For the process type 119875(119904) = 119896119890minus120579119904
119904(119879119904 minus 1) if there areuncertainties that exist in all three parameters that is
1003816100381610038161003816119890minus119895Δ120579119908 minus 1
1003816100381610038161003816
(25)
At the same time in order to compromise the nominalperformance with the robust stability of the closed-loop forthe load disturbance rejection the following constraint isrequired to meet [31]
where 120596(119904) is a weight function of the closed-loop sensitivityfunction 119878 = 1 minus 119879(119904) which usually can be chosen as
15
1
05
0
0 10 20 30 40 50
Time (s)
Proc
ess o
utpu
t
LiuProposed
Figure 2 Nominal system responses for Example 1
1119904 for the step change of the load disturbance It indicatesthat tuning the adjustable parameter 1205821015840 aims at the tradeoffbetween the nominal performance of the closed-loop andits robust stability That is to say decreasing 120582
1015840 improvesthe disturbance rejection performance of the closed-loopbut decays its robust stability in the presence of the pro-cess uncertainty On the contrary increasing 120582
1015840 tends tostrengthen the robust stability of the closed-loop but degradesits disturbance rejection performance
4 Simulation
Example 1 Consider the unstable process studied by Liu et al[20] 119875(119904) = 119890
minus02119904119904(119904 minus 1)
In the proposed method take 120582 = 33120579 = 066 theparameters of controllers are obtained 119896
1= 264 120591
1198941= 264
1205911198891
= 1367 120572 = 019 1198861
= 264 and 1198862
= 361 Asfor the controller 119870
2(119904) transform the process as 119875(119904) =
100119890minus02119904
(100119904 minus 1)(119904 minus 1) take 1205821015840 = 2120579 = 04 and obtain
1198702(119904) = 302 (1 +
1
179119904+ 106119904)
01119904 + 1
0008119904 + 1 (27)
The method proposed in [20] is better than others which canbe seen in this simulation effects so only compare with themethod in [20] By adding a unit step change to the set-pointinput at 119905 = 0 an inverse unit step change of load disturbanceto process output at 119905 = 20 the simulation results are obtainedas shown in Figure 2
Now suppose that there exists 30 increment for estimat-ing the process time delay and 30 reduction for the timeconstant of the process model then the perturbed systemresponses are provided in Figure 3
It can be seen from the simulation results that theproposed method gives better performances for the unstableprocess
Example 2 Consider a process studied by Liu and Gao [15]119875(119904) = 01119890
minus5119904119904(5119904 + 1)
Mathematical Problems in Engineering 5
15
1
05
0
0 10 20 30 40 50
Time (s)
Proc
ess o
utpu
t
LiuProposed
Figure 3 Perturbed system responses for Example 1
0 100 200 300 400Time (s)
12
1
08
06
04
02
0
Proc
ess o
utpu
t
LiuProposed
Figure 4 Nominal system responses for Example 2
In the proposed method as for controllers 1198701(119904) take
120582 = 08120579 = 4 and obtain 1198961= 9375 120591
1198941= 12 120591
1198891= 2933
1198861= 12 and 119886
2= 352 As for controller 119870
2(119904) transform
the process as 119875(119904) = minus10119890minus5119904
(100119904 minus 1)(minus5119904 minus 1) take1205821015840= 08120579 = 4 and obtain the controller
1198702(119904) = 22 (1 +
1
22119904+ 3864119904)
25119904 + 1
08119904 + 1 (28)
Add a unit step change to the set-point input at 119905 = 0 and aninverse step change of load disturbance to the process outputat 119905 = 200 The simulation results are obtained as shown inFigure 4 and the corresponding control action responses areshown in Figure 5 It can be observed from the figure thatthe control action response of the proposed method showssmooth variation compared to that of Liu and Gao [15]
Now suppose that there exist 10 error for estimatingthe process time delay and the time constant of the processmodel such as both of them are actually 10 larger Theperturbed system responses are provided in Figure 6 andthe corresponding control action responses are shown inFigure 7 It can be seen that the proposed control action
Con
trol s
igna
l
6
5
4
3
2
1
0
minus1
0 100 200 300 400
Time (s)
LiuProposed
Figure 5 Nominal system control signal for Example 2
15
1
05
0
0
100 200 300 400
Proc
ess o
utpu
t
Time (s)
LiuProposed
Figure 6 Perturbed system responses for Example 2
is smooth and the performance of disturbance rejection isbetter than that of Liu and Gao [15]
5 Conclusion
In order to improve the system performance of distur-bance rejection a modified Smith predictor scheme hasbeen proposed based on a two-degree-of-freedom controlstructure In the proposed control structure both of the set-point response and the load disturbance response can betuned separately by the set-point tracking controller and thedisturbance estimator respectivelyThemost advantage of themethod is that the designed system has good performanceof disturbance rejection as well as performance of set-pointtracking The two controllers are all designed in the form ofPID and they are simple and easy to be used in process indus-try Comparisons with the previous methods demonstrate aclear advantage of the proposedmethod in both nominal androbust performances in disturbance rejection
6 Mathematical Problems in Engineering
6
5
4
3
2
1
0
minus1
0 100 200 300 400
Con
trol s
igna
l
Time (s)
LiuProposed
Figure 7 Perturbed system control signal for Example 2
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Natural Science Founda-tion of Shandong Province (Grant no ZR2012CQ026) andthe Science and Technology Funds of Shandong EducationDepartment (Grant no J11LD16)
References
[1] I-L Chien S C Peng and J H Liu ldquoSimple control methodfor integrating processes with long deadtimerdquo Journal of ProcessControl vol 12 no 3 pp 391ndash404 2002
[2] M Huzmezan W A Gough G A Dumont and S KovacldquoTime delay integrating systems a challenge for process controlindustries A practical solutionrdquo Control Engineering Practicevol 10 no 10 pp 1153ndash1161 2002
[3] H Dong Z Wang and H Gao ldquoDistributed 119867infin
filteringfor a class of markovian jump nonlinear time-delay systemsover lossy sensor networksrdquo IEEE Transactions on IndustrialElectronics vol 60 no 10 pp 4665ndash4672 2013
[4] J E Normey-Rico C Bordons and E F Camacho Control ofDead-Time Processes Spinger Berlin Germany 2007
[5] Z Wang H Dong B Shen and H Gao ldquoFinite-horizon 119867infin
filtering with missing measurements and quantization effectsrdquoIEEE Transactions on Automatic Control vol 58 no 7 pp 1707ndash1718 2013
[6] A Visioli ldquoOptimal tuning of PID controllers for integraland unstable processesrdquo IEE Proceedings Control Theory andApplications vol 148 no 2 pp 180ndash184 2001
[7] C Hwang and J Hwang ldquoStabilisation of first-order plus dead-time unstable processes using PID controllersrdquo IEE ProceedingsControl Theory and Applications vol 151 no 1 pp 89ndash94 2004
[8] R C Panda ldquoSynthesis of PID controller for unstable andintegrating processesrdquoChemical Engineering Science vol 64 no12 pp 2807ndash2816 2009
[9] Y Lee J Lee and S Park ldquoPID controller tuning for integratingand unstable processes with time delayrdquo Chemical EngineeringScience vol 55 no 17 pp 3481ndash3493 2000
[10] A Seshagiri Rao V S Rao and M Chidambaram ldquoDirectsynthesis-based controller design for integrating processes withtime delayrdquo Journal of the Franklin Institute vol 346 no 1 pp38ndash56 2009
[11] D G Padhan and S Majhi ldquoA new control scheme for PIDload frequency controller of single-area and multi-area powersystemsrdquo ISA Transactions vol 52 no 2 pp 242ndash251 2013
[12] I Kaya ldquoAutotuning of a new PI-PD smith predictor based ontime domain specificationsrdquo ISA Transactions vol 42 no 4 pp559ndash575 2003
[13] S Majhi and D P Atherton ldquoObtaining controller parametersfor a new Smith predictor using autotuningrdquo Automatica vol36 no 11 pp 1651ndash1658 2000
[14] J E Normey-Rico and E F Camacho ldquoDead-time compen-sators a surveyrdquo Control Engineering Practice vol 16 no 4 pp407ndash428 2008
[15] T Liu and F R Gao ldquoEnhanced IMCdesign of load disturbancerejection for integrating and unstable processes with slowdynamicsrdquo ISA Transactions vol 50 no 2 pp 239ndash248 2011
[16] E-D Cong M-H Hu S-T Tu F-Z Xuan and H-H Shao ldquoAnovel double loop control model design for chemical unstableprocessesrdquo ISA Transactions vol 53 no 2 pp 497ndash507 2014
[17] J E Normey-Rico P Garcia and A Gonzalez ldquoRobust stabilityanalysis of filtered Smith predictor for time-varying delayprocessesrdquo Journal of Process Control vol 22 no 10 pp 1975ndash1984 2012
[18] J Nandong and Z Zang ldquoHigh-performance multi-scale con-trol scheme for stable integrating and unstable time-delayprocessesrdquo Journal of Process Control vol 23 no 10 pp 1333ndash1343 2013
[19] A S Rao andM Chidambaram ldquoAnalytical design of modifiedSmith predictor in a two-degrees-of-freedom control schemefor second order unstable processes with time delayrdquo ISATransactions vol 47 no 4 pp 407ndash419 2008
[20] T Liu W Zhang and D Gu ldquoAnalytical design of two-degree-of-freedom control scheme for open-loop unstable processeswith time delayrdquo Journal of Process Control vol 15 no 5 pp559ndash572 2005
[21] P Garcıa and P Albertos ldquoRobust tuning of a generalizedpredictor-based controller for integrating and unstable systemswith long time-delayrdquo Journal of Process Control vol 23 no 8pp 1205ndash1216 2013
[22] VVijayan andRC Panda ldquoDesign of PID controllers in doublefeedback loops for SISO systems with set-point filtersrdquo ISATransactions vol 51 no 4 pp 514ndash521 2012
[23] P Garcıa P Albertos and T Hagglund ldquoControl of unstablenon-minimum-phase delayed systemsrdquo Journal of Process Con-trol vol 16 no 10 pp 1099ndash1111 2006
[24] P Albertos and P Garcıa ldquoRobust control design for long time-delay systemsrdquo Journal of Process Control vol 19 no 10 pp1640ndash1648 2009
[25] J C Moreno J L Guzman J E Normey-Rico A Banos andM Berenguel ldquoA combined FSP and reset control approachto improve the set-point tracking task of dead-time processesrdquoControl Engineering Practice vol 21 no 4 pp 351ndash359 2013
[26] C Dey R K Mudi and D Simhachalam ldquoA simple nonlinearPD controller for integrating processesrdquo ISA Transactions vol53 no 1 pp 162ndash172 2014
Mathematical Problems in Engineering 7
[27] J E Normey-Rico and E F Camacho ldquoUnified approachfor robust dead-time compensator designrdquo Journal of ProcessControl vol 21 no 7 pp 1080ndash1091 2011
[28] K J Astrom and H Hagglund PID Controllers Theory Designand Tuning Instrument Society of America Research TrianglePark NC USA 2nd edition 1995
[29] B A Ogunnaike and W H Ray Process Modeling Dynamicsand Control Oxford University New York NY USA 1994
[30] M Shamsuzzoha andM Lee ldquoEnhanced performance for two-degree-of-freedom control scheme for second order unstableprocesses with time delayrdquo in Proceedings of the InternationalConference on Control Automation and Systems (ICCAS rsquo07)pp 240ndash245 Seoul Republic of Korea October 2007
[31] MMorari andE ZafiriouRobust Process Control PrenticeHallEnglewood Cliffs NJ USA 1989
[32] J C Doyle B A Francis and A R Tannenbaum FeedbackControl Theory Macmillan Publishing Company New YorkNY USA 1992
= 11988721198871is derivative gain 1205721015840 = 05120579
120573 = ((11988711205792 minus 119887
2+ 21205821015840120579 + 6120582
10158402)(120579 + 4120582
1015840minus 1198871)) + 119879 +119879
1015840 120572 and120573 are parameters of the lead-lag filter
As for the process type 119875(119904) = 119896119890minus120579119904
119904(119879119904 + 1) it can betransformed as 119875(119904) = minus119896
1015840119890minus120579119904
[(1198791015840119904 minus 1)(minus119879119904 minus 1)] and the
executable controller 1198702(119904) can be obtained in the form of
PID by using (15) (18) and (20) On the basis of simulationstudy on integrating first order plus time delay processes theuse of 01120573 instead of 120573 is suitable and 120573 is about 02ndash12generally
3 System Robust Stability Analysis
A control system is robust if it is insensitive to differencesbetween the actual system and the model of the systemwhich was used to design the controllerThese differences arereferred to as model mismatch or simply model uncertainty[31] A study of robustness analysis is an important taskbecause no mathematical model of a system will be a perfectrepresentation of the actual system Small-gain theorem is119897119898119879(119904)infin
lt 1 it expresses the robustly stable condition ofa control system where 119897
119898(119904) is the bound on the process
multiplicative uncertainty and 119879(119904) is the closed-loop com-plementary sensitivity function [32]
For the process type 119875(119904) = 119896119890minus120579119904
119904(119879119904 minus 1) if there areuncertainties that exist in all three parameters that is
1003816100381610038161003816119890minus119895Δ120579119908 minus 1
1003816100381610038161003816
(25)
At the same time in order to compromise the nominalperformance with the robust stability of the closed-loop forthe load disturbance rejection the following constraint isrequired to meet [31]
where 120596(119904) is a weight function of the closed-loop sensitivityfunction 119878 = 1 minus 119879(119904) which usually can be chosen as
15
1
05
0
0 10 20 30 40 50
Time (s)
Proc
ess o
utpu
t
LiuProposed
Figure 2 Nominal system responses for Example 1
1119904 for the step change of the load disturbance It indicatesthat tuning the adjustable parameter 1205821015840 aims at the tradeoffbetween the nominal performance of the closed-loop andits robust stability That is to say decreasing 120582
1015840 improvesthe disturbance rejection performance of the closed-loopbut decays its robust stability in the presence of the pro-cess uncertainty On the contrary increasing 120582
1015840 tends tostrengthen the robust stability of the closed-loop but degradesits disturbance rejection performance
4 Simulation
Example 1 Consider the unstable process studied by Liu et al[20] 119875(119904) = 119890
minus02119904119904(119904 minus 1)
In the proposed method take 120582 = 33120579 = 066 theparameters of controllers are obtained 119896
1= 264 120591
1198941= 264
1205911198891
= 1367 120572 = 019 1198861
= 264 and 1198862
= 361 Asfor the controller 119870
2(119904) transform the process as 119875(119904) =
100119890minus02119904
(100119904 minus 1)(119904 minus 1) take 1205821015840 = 2120579 = 04 and obtain
1198702(119904) = 302 (1 +
1
179119904+ 106119904)
01119904 + 1
0008119904 + 1 (27)
The method proposed in [20] is better than others which canbe seen in this simulation effects so only compare with themethod in [20] By adding a unit step change to the set-pointinput at 119905 = 0 an inverse unit step change of load disturbanceto process output at 119905 = 20 the simulation results are obtainedas shown in Figure 2
Now suppose that there exists 30 increment for estimat-ing the process time delay and 30 reduction for the timeconstant of the process model then the perturbed systemresponses are provided in Figure 3
It can be seen from the simulation results that theproposed method gives better performances for the unstableprocess
Example 2 Consider a process studied by Liu and Gao [15]119875(119904) = 01119890
minus5119904119904(5119904 + 1)
Mathematical Problems in Engineering 5
15
1
05
0
0 10 20 30 40 50
Time (s)
Proc
ess o
utpu
t
LiuProposed
Figure 3 Perturbed system responses for Example 1
0 100 200 300 400Time (s)
12
1
08
06
04
02
0
Proc
ess o
utpu
t
LiuProposed
Figure 4 Nominal system responses for Example 2
In the proposed method as for controllers 1198701(119904) take
120582 = 08120579 = 4 and obtain 1198961= 9375 120591
1198941= 12 120591
1198891= 2933
1198861= 12 and 119886
2= 352 As for controller 119870
2(119904) transform
the process as 119875(119904) = minus10119890minus5119904
(100119904 minus 1)(minus5119904 minus 1) take1205821015840= 08120579 = 4 and obtain the controller
1198702(119904) = 22 (1 +
1
22119904+ 3864119904)
25119904 + 1
08119904 + 1 (28)
Add a unit step change to the set-point input at 119905 = 0 and aninverse step change of load disturbance to the process outputat 119905 = 200 The simulation results are obtained as shown inFigure 4 and the corresponding control action responses areshown in Figure 5 It can be observed from the figure thatthe control action response of the proposed method showssmooth variation compared to that of Liu and Gao [15]
Now suppose that there exist 10 error for estimatingthe process time delay and the time constant of the processmodel such as both of them are actually 10 larger Theperturbed system responses are provided in Figure 6 andthe corresponding control action responses are shown inFigure 7 It can be seen that the proposed control action
Con
trol s
igna
l
6
5
4
3
2
1
0
minus1
0 100 200 300 400
Time (s)
LiuProposed
Figure 5 Nominal system control signal for Example 2
15
1
05
0
0
100 200 300 400
Proc
ess o
utpu
t
Time (s)
LiuProposed
Figure 6 Perturbed system responses for Example 2
is smooth and the performance of disturbance rejection isbetter than that of Liu and Gao [15]
5 Conclusion
In order to improve the system performance of distur-bance rejection a modified Smith predictor scheme hasbeen proposed based on a two-degree-of-freedom controlstructure In the proposed control structure both of the set-point response and the load disturbance response can betuned separately by the set-point tracking controller and thedisturbance estimator respectivelyThemost advantage of themethod is that the designed system has good performanceof disturbance rejection as well as performance of set-pointtracking The two controllers are all designed in the form ofPID and they are simple and easy to be used in process indus-try Comparisons with the previous methods demonstrate aclear advantage of the proposedmethod in both nominal androbust performances in disturbance rejection
6 Mathematical Problems in Engineering
6
5
4
3
2
1
0
minus1
0 100 200 300 400
Con
trol s
igna
l
Time (s)
LiuProposed
Figure 7 Perturbed system control signal for Example 2
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Natural Science Founda-tion of Shandong Province (Grant no ZR2012CQ026) andthe Science and Technology Funds of Shandong EducationDepartment (Grant no J11LD16)
References
[1] I-L Chien S C Peng and J H Liu ldquoSimple control methodfor integrating processes with long deadtimerdquo Journal of ProcessControl vol 12 no 3 pp 391ndash404 2002
[2] M Huzmezan W A Gough G A Dumont and S KovacldquoTime delay integrating systems a challenge for process controlindustries A practical solutionrdquo Control Engineering Practicevol 10 no 10 pp 1153ndash1161 2002
[3] H Dong Z Wang and H Gao ldquoDistributed 119867infin
filteringfor a class of markovian jump nonlinear time-delay systemsover lossy sensor networksrdquo IEEE Transactions on IndustrialElectronics vol 60 no 10 pp 4665ndash4672 2013
[4] J E Normey-Rico C Bordons and E F Camacho Control ofDead-Time Processes Spinger Berlin Germany 2007
[5] Z Wang H Dong B Shen and H Gao ldquoFinite-horizon 119867infin
filtering with missing measurements and quantization effectsrdquoIEEE Transactions on Automatic Control vol 58 no 7 pp 1707ndash1718 2013
[6] A Visioli ldquoOptimal tuning of PID controllers for integraland unstable processesrdquo IEE Proceedings Control Theory andApplications vol 148 no 2 pp 180ndash184 2001
[7] C Hwang and J Hwang ldquoStabilisation of first-order plus dead-time unstable processes using PID controllersrdquo IEE ProceedingsControl Theory and Applications vol 151 no 1 pp 89ndash94 2004
[8] R C Panda ldquoSynthesis of PID controller for unstable andintegrating processesrdquoChemical Engineering Science vol 64 no12 pp 2807ndash2816 2009
[9] Y Lee J Lee and S Park ldquoPID controller tuning for integratingand unstable processes with time delayrdquo Chemical EngineeringScience vol 55 no 17 pp 3481ndash3493 2000
[10] A Seshagiri Rao V S Rao and M Chidambaram ldquoDirectsynthesis-based controller design for integrating processes withtime delayrdquo Journal of the Franklin Institute vol 346 no 1 pp38ndash56 2009
[11] D G Padhan and S Majhi ldquoA new control scheme for PIDload frequency controller of single-area and multi-area powersystemsrdquo ISA Transactions vol 52 no 2 pp 242ndash251 2013
[12] I Kaya ldquoAutotuning of a new PI-PD smith predictor based ontime domain specificationsrdquo ISA Transactions vol 42 no 4 pp559ndash575 2003
[13] S Majhi and D P Atherton ldquoObtaining controller parametersfor a new Smith predictor using autotuningrdquo Automatica vol36 no 11 pp 1651ndash1658 2000
[14] J E Normey-Rico and E F Camacho ldquoDead-time compen-sators a surveyrdquo Control Engineering Practice vol 16 no 4 pp407ndash428 2008
[15] T Liu and F R Gao ldquoEnhanced IMCdesign of load disturbancerejection for integrating and unstable processes with slowdynamicsrdquo ISA Transactions vol 50 no 2 pp 239ndash248 2011
[16] E-D Cong M-H Hu S-T Tu F-Z Xuan and H-H Shao ldquoAnovel double loop control model design for chemical unstableprocessesrdquo ISA Transactions vol 53 no 2 pp 497ndash507 2014
[17] J E Normey-Rico P Garcia and A Gonzalez ldquoRobust stabilityanalysis of filtered Smith predictor for time-varying delayprocessesrdquo Journal of Process Control vol 22 no 10 pp 1975ndash1984 2012
[18] J Nandong and Z Zang ldquoHigh-performance multi-scale con-trol scheme for stable integrating and unstable time-delayprocessesrdquo Journal of Process Control vol 23 no 10 pp 1333ndash1343 2013
[19] A S Rao andM Chidambaram ldquoAnalytical design of modifiedSmith predictor in a two-degrees-of-freedom control schemefor second order unstable processes with time delayrdquo ISATransactions vol 47 no 4 pp 407ndash419 2008
[20] T Liu W Zhang and D Gu ldquoAnalytical design of two-degree-of-freedom control scheme for open-loop unstable processeswith time delayrdquo Journal of Process Control vol 15 no 5 pp559ndash572 2005
[21] P Garcıa and P Albertos ldquoRobust tuning of a generalizedpredictor-based controller for integrating and unstable systemswith long time-delayrdquo Journal of Process Control vol 23 no 8pp 1205ndash1216 2013
[22] VVijayan andRC Panda ldquoDesign of PID controllers in doublefeedback loops for SISO systems with set-point filtersrdquo ISATransactions vol 51 no 4 pp 514ndash521 2012
[23] P Garcıa P Albertos and T Hagglund ldquoControl of unstablenon-minimum-phase delayed systemsrdquo Journal of Process Con-trol vol 16 no 10 pp 1099ndash1111 2006
[24] P Albertos and P Garcıa ldquoRobust control design for long time-delay systemsrdquo Journal of Process Control vol 19 no 10 pp1640ndash1648 2009
[25] J C Moreno J L Guzman J E Normey-Rico A Banos andM Berenguel ldquoA combined FSP and reset control approachto improve the set-point tracking task of dead-time processesrdquoControl Engineering Practice vol 21 no 4 pp 351ndash359 2013
[26] C Dey R K Mudi and D Simhachalam ldquoA simple nonlinearPD controller for integrating processesrdquo ISA Transactions vol53 no 1 pp 162ndash172 2014
Mathematical Problems in Engineering 7
[27] J E Normey-Rico and E F Camacho ldquoUnified approachfor robust dead-time compensator designrdquo Journal of ProcessControl vol 21 no 7 pp 1080ndash1091 2011
[28] K J Astrom and H Hagglund PID Controllers Theory Designand Tuning Instrument Society of America Research TrianglePark NC USA 2nd edition 1995
[29] B A Ogunnaike and W H Ray Process Modeling Dynamicsand Control Oxford University New York NY USA 1994
[30] M Shamsuzzoha andM Lee ldquoEnhanced performance for two-degree-of-freedom control scheme for second order unstableprocesses with time delayrdquo in Proceedings of the InternationalConference on Control Automation and Systems (ICCAS rsquo07)pp 240ndash245 Seoul Republic of Korea October 2007
[31] MMorari andE ZafiriouRobust Process Control PrenticeHallEnglewood Cliffs NJ USA 1989
[32] J C Doyle B A Francis and A R Tannenbaum FeedbackControl Theory Macmillan Publishing Company New YorkNY USA 1992
In the proposed method as for controllers 1198701(119904) take
120582 = 08120579 = 4 and obtain 1198961= 9375 120591
1198941= 12 120591
1198891= 2933
1198861= 12 and 119886
2= 352 As for controller 119870
2(119904) transform
the process as 119875(119904) = minus10119890minus5119904
(100119904 minus 1)(minus5119904 minus 1) take1205821015840= 08120579 = 4 and obtain the controller
1198702(119904) = 22 (1 +
1
22119904+ 3864119904)
25119904 + 1
08119904 + 1 (28)
Add a unit step change to the set-point input at 119905 = 0 and aninverse step change of load disturbance to the process outputat 119905 = 200 The simulation results are obtained as shown inFigure 4 and the corresponding control action responses areshown in Figure 5 It can be observed from the figure thatthe control action response of the proposed method showssmooth variation compared to that of Liu and Gao [15]
Now suppose that there exist 10 error for estimatingthe process time delay and the time constant of the processmodel such as both of them are actually 10 larger Theperturbed system responses are provided in Figure 6 andthe corresponding control action responses are shown inFigure 7 It can be seen that the proposed control action
Con
trol s
igna
l
6
5
4
3
2
1
0
minus1
0 100 200 300 400
Time (s)
LiuProposed
Figure 5 Nominal system control signal for Example 2
15
1
05
0
0
100 200 300 400
Proc
ess o
utpu
t
Time (s)
LiuProposed
Figure 6 Perturbed system responses for Example 2
is smooth and the performance of disturbance rejection isbetter than that of Liu and Gao [15]
5 Conclusion
In order to improve the system performance of distur-bance rejection a modified Smith predictor scheme hasbeen proposed based on a two-degree-of-freedom controlstructure In the proposed control structure both of the set-point response and the load disturbance response can betuned separately by the set-point tracking controller and thedisturbance estimator respectivelyThemost advantage of themethod is that the designed system has good performanceof disturbance rejection as well as performance of set-pointtracking The two controllers are all designed in the form ofPID and they are simple and easy to be used in process indus-try Comparisons with the previous methods demonstrate aclear advantage of the proposedmethod in both nominal androbust performances in disturbance rejection
6 Mathematical Problems in Engineering
6
5
4
3
2
1
0
minus1
0 100 200 300 400
Con
trol s
igna
l
Time (s)
LiuProposed
Figure 7 Perturbed system control signal for Example 2
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Natural Science Founda-tion of Shandong Province (Grant no ZR2012CQ026) andthe Science and Technology Funds of Shandong EducationDepartment (Grant no J11LD16)
References
[1] I-L Chien S C Peng and J H Liu ldquoSimple control methodfor integrating processes with long deadtimerdquo Journal of ProcessControl vol 12 no 3 pp 391ndash404 2002
[2] M Huzmezan W A Gough G A Dumont and S KovacldquoTime delay integrating systems a challenge for process controlindustries A practical solutionrdquo Control Engineering Practicevol 10 no 10 pp 1153ndash1161 2002
[3] H Dong Z Wang and H Gao ldquoDistributed 119867infin
filteringfor a class of markovian jump nonlinear time-delay systemsover lossy sensor networksrdquo IEEE Transactions on IndustrialElectronics vol 60 no 10 pp 4665ndash4672 2013
[4] J E Normey-Rico C Bordons and E F Camacho Control ofDead-Time Processes Spinger Berlin Germany 2007
[5] Z Wang H Dong B Shen and H Gao ldquoFinite-horizon 119867infin
filtering with missing measurements and quantization effectsrdquoIEEE Transactions on Automatic Control vol 58 no 7 pp 1707ndash1718 2013
[6] A Visioli ldquoOptimal tuning of PID controllers for integraland unstable processesrdquo IEE Proceedings Control Theory andApplications vol 148 no 2 pp 180ndash184 2001
[7] C Hwang and J Hwang ldquoStabilisation of first-order plus dead-time unstable processes using PID controllersrdquo IEE ProceedingsControl Theory and Applications vol 151 no 1 pp 89ndash94 2004
[8] R C Panda ldquoSynthesis of PID controller for unstable andintegrating processesrdquoChemical Engineering Science vol 64 no12 pp 2807ndash2816 2009
[9] Y Lee J Lee and S Park ldquoPID controller tuning for integratingand unstable processes with time delayrdquo Chemical EngineeringScience vol 55 no 17 pp 3481ndash3493 2000
[10] A Seshagiri Rao V S Rao and M Chidambaram ldquoDirectsynthesis-based controller design for integrating processes withtime delayrdquo Journal of the Franklin Institute vol 346 no 1 pp38ndash56 2009
[11] D G Padhan and S Majhi ldquoA new control scheme for PIDload frequency controller of single-area and multi-area powersystemsrdquo ISA Transactions vol 52 no 2 pp 242ndash251 2013
[12] I Kaya ldquoAutotuning of a new PI-PD smith predictor based ontime domain specificationsrdquo ISA Transactions vol 42 no 4 pp559ndash575 2003
[13] S Majhi and D P Atherton ldquoObtaining controller parametersfor a new Smith predictor using autotuningrdquo Automatica vol36 no 11 pp 1651ndash1658 2000
[14] J E Normey-Rico and E F Camacho ldquoDead-time compen-sators a surveyrdquo Control Engineering Practice vol 16 no 4 pp407ndash428 2008
[15] T Liu and F R Gao ldquoEnhanced IMCdesign of load disturbancerejection for integrating and unstable processes with slowdynamicsrdquo ISA Transactions vol 50 no 2 pp 239ndash248 2011
[16] E-D Cong M-H Hu S-T Tu F-Z Xuan and H-H Shao ldquoAnovel double loop control model design for chemical unstableprocessesrdquo ISA Transactions vol 53 no 2 pp 497ndash507 2014
[17] J E Normey-Rico P Garcia and A Gonzalez ldquoRobust stabilityanalysis of filtered Smith predictor for time-varying delayprocessesrdquo Journal of Process Control vol 22 no 10 pp 1975ndash1984 2012
[18] J Nandong and Z Zang ldquoHigh-performance multi-scale con-trol scheme for stable integrating and unstable time-delayprocessesrdquo Journal of Process Control vol 23 no 10 pp 1333ndash1343 2013
[19] A S Rao andM Chidambaram ldquoAnalytical design of modifiedSmith predictor in a two-degrees-of-freedom control schemefor second order unstable processes with time delayrdquo ISATransactions vol 47 no 4 pp 407ndash419 2008
[20] T Liu W Zhang and D Gu ldquoAnalytical design of two-degree-of-freedom control scheme for open-loop unstable processeswith time delayrdquo Journal of Process Control vol 15 no 5 pp559ndash572 2005
[21] P Garcıa and P Albertos ldquoRobust tuning of a generalizedpredictor-based controller for integrating and unstable systemswith long time-delayrdquo Journal of Process Control vol 23 no 8pp 1205ndash1216 2013
[22] VVijayan andRC Panda ldquoDesign of PID controllers in doublefeedback loops for SISO systems with set-point filtersrdquo ISATransactions vol 51 no 4 pp 514ndash521 2012
[23] P Garcıa P Albertos and T Hagglund ldquoControl of unstablenon-minimum-phase delayed systemsrdquo Journal of Process Con-trol vol 16 no 10 pp 1099ndash1111 2006
[24] P Albertos and P Garcıa ldquoRobust control design for long time-delay systemsrdquo Journal of Process Control vol 19 no 10 pp1640ndash1648 2009
[25] J C Moreno J L Guzman J E Normey-Rico A Banos andM Berenguel ldquoA combined FSP and reset control approachto improve the set-point tracking task of dead-time processesrdquoControl Engineering Practice vol 21 no 4 pp 351ndash359 2013
[26] C Dey R K Mudi and D Simhachalam ldquoA simple nonlinearPD controller for integrating processesrdquo ISA Transactions vol53 no 1 pp 162ndash172 2014
Mathematical Problems in Engineering 7
[27] J E Normey-Rico and E F Camacho ldquoUnified approachfor robust dead-time compensator designrdquo Journal of ProcessControl vol 21 no 7 pp 1080ndash1091 2011
[28] K J Astrom and H Hagglund PID Controllers Theory Designand Tuning Instrument Society of America Research TrianglePark NC USA 2nd edition 1995
[29] B A Ogunnaike and W H Ray Process Modeling Dynamicsand Control Oxford University New York NY USA 1994
[30] M Shamsuzzoha andM Lee ldquoEnhanced performance for two-degree-of-freedom control scheme for second order unstableprocesses with time delayrdquo in Proceedings of the InternationalConference on Control Automation and Systems (ICCAS rsquo07)pp 240ndash245 Seoul Republic of Korea October 2007
[31] MMorari andE ZafiriouRobust Process Control PrenticeHallEnglewood Cliffs NJ USA 1989
[32] J C Doyle B A Francis and A R Tannenbaum FeedbackControl Theory Macmillan Publishing Company New YorkNY USA 1992
Figure 7 Perturbed system control signal for Example 2
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Natural Science Founda-tion of Shandong Province (Grant no ZR2012CQ026) andthe Science and Technology Funds of Shandong EducationDepartment (Grant no J11LD16)
References
[1] I-L Chien S C Peng and J H Liu ldquoSimple control methodfor integrating processes with long deadtimerdquo Journal of ProcessControl vol 12 no 3 pp 391ndash404 2002
[2] M Huzmezan W A Gough G A Dumont and S KovacldquoTime delay integrating systems a challenge for process controlindustries A practical solutionrdquo Control Engineering Practicevol 10 no 10 pp 1153ndash1161 2002
[3] H Dong Z Wang and H Gao ldquoDistributed 119867infin
filteringfor a class of markovian jump nonlinear time-delay systemsover lossy sensor networksrdquo IEEE Transactions on IndustrialElectronics vol 60 no 10 pp 4665ndash4672 2013
[4] J E Normey-Rico C Bordons and E F Camacho Control ofDead-Time Processes Spinger Berlin Germany 2007
[5] Z Wang H Dong B Shen and H Gao ldquoFinite-horizon 119867infin
filtering with missing measurements and quantization effectsrdquoIEEE Transactions on Automatic Control vol 58 no 7 pp 1707ndash1718 2013
[6] A Visioli ldquoOptimal tuning of PID controllers for integraland unstable processesrdquo IEE Proceedings Control Theory andApplications vol 148 no 2 pp 180ndash184 2001
[7] C Hwang and J Hwang ldquoStabilisation of first-order plus dead-time unstable processes using PID controllersrdquo IEE ProceedingsControl Theory and Applications vol 151 no 1 pp 89ndash94 2004
[8] R C Panda ldquoSynthesis of PID controller for unstable andintegrating processesrdquoChemical Engineering Science vol 64 no12 pp 2807ndash2816 2009
[9] Y Lee J Lee and S Park ldquoPID controller tuning for integratingand unstable processes with time delayrdquo Chemical EngineeringScience vol 55 no 17 pp 3481ndash3493 2000
[10] A Seshagiri Rao V S Rao and M Chidambaram ldquoDirectsynthesis-based controller design for integrating processes withtime delayrdquo Journal of the Franklin Institute vol 346 no 1 pp38ndash56 2009
[11] D G Padhan and S Majhi ldquoA new control scheme for PIDload frequency controller of single-area and multi-area powersystemsrdquo ISA Transactions vol 52 no 2 pp 242ndash251 2013
[12] I Kaya ldquoAutotuning of a new PI-PD smith predictor based ontime domain specificationsrdquo ISA Transactions vol 42 no 4 pp559ndash575 2003
[13] S Majhi and D P Atherton ldquoObtaining controller parametersfor a new Smith predictor using autotuningrdquo Automatica vol36 no 11 pp 1651ndash1658 2000
[14] J E Normey-Rico and E F Camacho ldquoDead-time compen-sators a surveyrdquo Control Engineering Practice vol 16 no 4 pp407ndash428 2008
[15] T Liu and F R Gao ldquoEnhanced IMCdesign of load disturbancerejection for integrating and unstable processes with slowdynamicsrdquo ISA Transactions vol 50 no 2 pp 239ndash248 2011
[16] E-D Cong M-H Hu S-T Tu F-Z Xuan and H-H Shao ldquoAnovel double loop control model design for chemical unstableprocessesrdquo ISA Transactions vol 53 no 2 pp 497ndash507 2014
[17] J E Normey-Rico P Garcia and A Gonzalez ldquoRobust stabilityanalysis of filtered Smith predictor for time-varying delayprocessesrdquo Journal of Process Control vol 22 no 10 pp 1975ndash1984 2012
[18] J Nandong and Z Zang ldquoHigh-performance multi-scale con-trol scheme for stable integrating and unstable time-delayprocessesrdquo Journal of Process Control vol 23 no 10 pp 1333ndash1343 2013
[19] A S Rao andM Chidambaram ldquoAnalytical design of modifiedSmith predictor in a two-degrees-of-freedom control schemefor second order unstable processes with time delayrdquo ISATransactions vol 47 no 4 pp 407ndash419 2008
[20] T Liu W Zhang and D Gu ldquoAnalytical design of two-degree-of-freedom control scheme for open-loop unstable processeswith time delayrdquo Journal of Process Control vol 15 no 5 pp559ndash572 2005
[21] P Garcıa and P Albertos ldquoRobust tuning of a generalizedpredictor-based controller for integrating and unstable systemswith long time-delayrdquo Journal of Process Control vol 23 no 8pp 1205ndash1216 2013
[22] VVijayan andRC Panda ldquoDesign of PID controllers in doublefeedback loops for SISO systems with set-point filtersrdquo ISATransactions vol 51 no 4 pp 514ndash521 2012
[23] P Garcıa P Albertos and T Hagglund ldquoControl of unstablenon-minimum-phase delayed systemsrdquo Journal of Process Con-trol vol 16 no 10 pp 1099ndash1111 2006
[24] P Albertos and P Garcıa ldquoRobust control design for long time-delay systemsrdquo Journal of Process Control vol 19 no 10 pp1640ndash1648 2009
[25] J C Moreno J L Guzman J E Normey-Rico A Banos andM Berenguel ldquoA combined FSP and reset control approachto improve the set-point tracking task of dead-time processesrdquoControl Engineering Practice vol 21 no 4 pp 351ndash359 2013
[26] C Dey R K Mudi and D Simhachalam ldquoA simple nonlinearPD controller for integrating processesrdquo ISA Transactions vol53 no 1 pp 162ndash172 2014
Mathematical Problems in Engineering 7
[27] J E Normey-Rico and E F Camacho ldquoUnified approachfor robust dead-time compensator designrdquo Journal of ProcessControl vol 21 no 7 pp 1080ndash1091 2011
[28] K J Astrom and H Hagglund PID Controllers Theory Designand Tuning Instrument Society of America Research TrianglePark NC USA 2nd edition 1995
[29] B A Ogunnaike and W H Ray Process Modeling Dynamicsand Control Oxford University New York NY USA 1994
[30] M Shamsuzzoha andM Lee ldquoEnhanced performance for two-degree-of-freedom control scheme for second order unstableprocesses with time delayrdquo in Proceedings of the InternationalConference on Control Automation and Systems (ICCAS rsquo07)pp 240ndash245 Seoul Republic of Korea October 2007
[31] MMorari andE ZafiriouRobust Process Control PrenticeHallEnglewood Cliffs NJ USA 1989
[32] J C Doyle B A Francis and A R Tannenbaum FeedbackControl Theory Macmillan Publishing Company New YorkNY USA 1992
[27] J E Normey-Rico and E F Camacho ldquoUnified approachfor robust dead-time compensator designrdquo Journal of ProcessControl vol 21 no 7 pp 1080ndash1091 2011
[28] K J Astrom and H Hagglund PID Controllers Theory Designand Tuning Instrument Society of America Research TrianglePark NC USA 2nd edition 1995
[29] B A Ogunnaike and W H Ray Process Modeling Dynamicsand Control Oxford University New York NY USA 1994
[30] M Shamsuzzoha andM Lee ldquoEnhanced performance for two-degree-of-freedom control scheme for second order unstableprocesses with time delayrdquo in Proceedings of the InternationalConference on Control Automation and Systems (ICCAS rsquo07)pp 240ndash245 Seoul Republic of Korea October 2007
[31] MMorari andE ZafiriouRobust Process Control PrenticeHallEnglewood Cliffs NJ USA 1989
[32] J C Doyle B A Francis and A R Tannenbaum FeedbackControl Theory Macmillan Publishing Company New YorkNY USA 1992
