EVALUATION AND REDESIGN OF SIMPLIFIED DECOUPLERS APPLIED ... · 3.2 FOPTD Model Identi¯cation of TITO pro-cess using Relay Experiment A frequency-domain identi¯cation technique

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Anais do XIX Congresso Brasileiro de Automática, CBA 2012.

ISBN: 978-85-8001-069-5

EVALUATION AND REDESIGN OF SIMPLIFIED DECOUPLERS APPLIED TO ATITO LABORATORY-SCALE THERMAL PROCESS

George Acioli Junior∗, Pericles Rezende Barros∗

∗Departamento de Engenharia EletricaUniversidade Federal de Campina Grande

Campina Grande, Paraıba, Brasil

Emails: [email protected], [email protected]

Abstract— In this paper a procedure to evaluate and redesign simplified dynamic decouplers using relay-basedexperiment is applied to a TITO laboratory-scale thermal process. This procedures was originally proposed in(Acioli Junior and Barros, 2011). The aim of the procedure is to reach effective decoupling at frequencies ofinterest for control purposes.

Keywords— Decoupling Control, Simplified Decouplers, Relay Experiment

Resumo— Neste trabalho, um procedimento de avaliacao e reprojeto de desacopladores simplificados utili-zando experimento baseado no rele e aplicado em um processo termico de escala laboratorial com duas entradase duas saıdas. Esse procedimento foi originalmente proposto em (Acioli Junior and Barros, 2011). O objetivo doprocedimento e alcancar desacoplamento efetivo nas frequencias de interesse para controle.

Palavras-chave— Controle com Desacoplamento, Desacoplador Simplificado, Experimento do Rele

1 INTRODUCTION

Most industrial processes are multi-input multi-output (MIMO) systems. Due to the interactionsbetween inputs and outputs, it is difficult to de-sign a controller for each loop independently. Asolution is to design a MIMO controller that takesthe interactions into account.

In last decades, many MIMO control ap-proaches have been proposed. However, it ismostly used on a higher-level of the control systemarchitecture to provide setpoints for regulator-level. In regulator-level, PI/PID based controlis still dominant techniques used. There are ba-sically two PI/PID based control schemes: de-centralized control and decoupling control (Xionget al., 2007).

Decentralized control consists of a set of SISO(single-input single-output) controllers which aredesigned for each loop by taking interactions intoaccount. It is a valid approach because, in somecases, the performance increase obtained by usingMIMO control structures does not justify its de-sign and maintenance. Several decentralized con-trol design methods have been proposed, includ-ing detuning methods (Luyben, 1986), sequen-tial loop closing methods (Shiu and Hwang, 1998)and equivalent transfer function methods (Huanget al., 2003). When interactions are modest it isnormally adequate. Nevertheless, it may fail togive acceptable responses if there exist severe in-teractions. To overcome this, decoupling controlscheme is used. In this scheme, a decoupler is in-troduced between the decentralized controller andthe MIMO process to eliminate the effect of inter-actions. Thus, the MIMO process can be han-dled as multiple SISO loops and a less conserva-

tive SISO control design method can be directlyapplied (Cai et al., 2008).

The theory of decoupling control has beenwell-established in literature. Most of themwere suggested for two-input two-output (TITO)processes in terms of dynamic decoupling. In(Shinskey, 1996) decoupling of TITO processes arediscussed in details. Dynamic decoupling controlmethodologies are classed in three types: ideal de-coupling, simplified decoupling and inverted de-coupling (Shinskey, 1996). The ideal decoupling israrely used in practice due to its complicated el-ements and realizability problems. Simplified de-coupling control of TITO processes are discussedin this paper.

In (Wang et al., 2000), the simplified decou-pler was designed assuming that a FOPTD (first-order plus time-delay) TITO model was known.Because the unavoidable model mismatch, the de-coupler only effectively compensate interactionswhere the model is accurate in terms of frequency.For control purposes, it is interesting an effectivedecoupler at certain desired frequencies. Recently,a procedure to evaluate and redesign simplifieddecouplers using relay-based experiment was pro-posed in (Acioli Junior and Barros, 2011). Theaim of the proposed procedure is to design a sim-plified dynamic decoupler to be effective at fre-quencies of interest, which is achieved by firstestimating an accurate FOPTD model at thosefrequencies. A simple frequency-domain identi-fication technique with adequate relay-based ex-periment is used to estimate the FOPTD model.An initial simplified decoupler is designed usingthis model. The initial decoupler is evaluated andredesigned to achieve effective decoupling at fre-quencies of interest.

http://cba2012.dee.ufcg.edu.br/

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In this paper, the procedure proposed in(Acioli Junior and Barros, 2011) is revised. Theprocedure is applied to a TITO laboratory-scalethermal process. The paper is organized as fol-lows. In Section 2, the problem statement is pre-sented. The initial decoupler design is consideredin Section 3. In Section 4, the decoupler evalua-tion method is presented. The decoupler redesigntechnique is proposed in Section 5. The experi-mental result is discussed in Section 6 followed byconclusion in Section 7.

2 Problem Statement

Consider a TITO open-loop process G(s).

G(s) =

[G11(s) G12(s)G21(s) G22(s)

].

The aim of decoupling is to determine D(s)so that G(s)D(s) is diagonally dominant. In thispaper, a simplified decoupling method is used todesign D(s). For TITO process, it consists of:

D(s) =

[1 −G12(s)

G11(s)

−G21(s)G22(s)

1

].

To make the decoupler design simple, it isassumed that G(s) is approximated by a TITOFOPDT model as

G(s) =

[K11

1+sT11e−sL11 K12

1+sT12e−sL12

K21

1+sT21e−sL21 K22

1+sT22e−sL22

]. (1)

The problem statement is: Given G(s): 1)Design an initial simplified decoupler D0(s); 2)Evaluate the decoupled system G(s)D0(s) at fre-quencies of interest; 3) If necessary, redesignD0(s) to obtain an effective decoupling at frequen-cies of interest.

3 Initial Decoupler Design

In this section, the initial decoupler design methodis considered. Additionally, a frequency-domainidentification technique that uses a relay exper-iment to estimate a model accurate around fre-quencies of interest is presented.

3.1 Simplified Decoupler Design

The initial decoupler D0(s) is defined using a sim-plified decoupler. It is given by

D0(s) =

[1 D0

12

D021 1

], (2)

where

D012 =

−K12

K11

(T11s+ 1)

(T12s+ 1)e−v(L12−L11)s,

D021 =

−K21

K22

(T22s+ 1)

(T21s+ 1)e−v(L21−L22)s,

and

v(L) =

{1, ifL ≥ 00, ifL < 0.

(3)

Note that the decoupler design needs TITOFOPTD model parameters. In this paper, theFOPTD model in equation 1 is obtained using afrequency-domain identification method describedbelow.

3.2 FOPTD Model Identification of TITO pro-cess using Relay Experiment

A frequency-domain identification technique thatuses a relay experiment to estimate a TITOFOPTD model is detailed here.

3.2.1 Relay-Based Experiment

The standard relay feedback test is commonlyused to generate sustained oscillations of the con-trolled variable near and to get the ultimate gain(Ku) and frequency (ω180) from the experiment

data (Astrom and Hagglund, 1995). For a pureFOPTD model, the knowledge of a frequencypoint (for example, Ku and ω180) and the pro-cess static gain uniquely characterizes model. In(Astrom and Hagglund, 2006), analytical expres-sions for estimating FOPTD model parametersfrom a single relay experiment plus an estimate ofthe process static gain is presented. The same ex-pressions can be used to estimate FOPTD modelwith the knowledge of any other frequency pointinformation and the process static gain.

In this paper, a single relay plus an integra-tor is used to estimate a FOPTD model with ac-curately at frequency for which the phase is -90o

(ω90). This relay experiment is performed onG(s)as an independent single-relay, i.e. only one loopat a time is subject to the relay structure whilethe other is kept open. In order obtain the processstatic gain together with a frequency point infor-mation at ω90, the relay+integrator structure isapplied during an integer number n of sustainedoscillation periods (T90) plus a rectangular pulseof width T90/2. Here, n is usually used equal to 2or 3.

Consider the block diagram shown in Figure1. Applying the relay structure to loop 1 (u′

1) withloop 2 open, the output response y1 = [ y11 y12 ]can be recorded. Similarly, for loop 2 (u′

2) withloop 1 open, the output response y2 = [ y21 y22 ]can be recorded.

3.2.2 The Model Gain Definition

The model gain K = |G (0) | is computed as theratio between the integral of the deviations of theoutput and input respectively which is equivalentto compute the Fourier Transform at frequency

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Figure 1: TITO System - Relay plus IntegratorExperiment

zero of the output and input signals:

K = |G (0) | =tb∫

0

∆y(t)dt/

tb∫

0

∆u(t)dt. (4)

The input-output relay experiment data u′1-y

11 , u

′1-

y12 , u′2-y

21 and u′

2-y22 is used to estimate the model

gain for G11, G21, G12 and G22, respectively.

3.2.3 Frequency-Domain Identification

Furthermore, for each input-output experimentdata u′

1-y11 , u′

1-y12 , u′

2-y21 and u′

2-y22 a fre-

quency point information (G11(jω190), G21(jω

190),

G22(jω290) and G12(jω

290), respectively) is esti-

mated using DFT. The frequency information to-gether with the gains are used to estimate themodel parameters in equation 1. The estimatorwill be detailed below.

Consider a SISO FOPTD G (s). Define|G(jωi

90)| and φ(ωi90) as the process gain and

phase at ωi90, respectively. Assume that the pro-

cess gain K is known and that G(jωi90) is esti-

mated. Define the relative gain κ(ωi90) =

|G(jωi90)|

K .Then, other the FOPTD parameters (time con-stant and delay) is computed as

T (ωi90) =

1

ωi90

√κ−2(ωi

90)− 1

L(ωi90) =

1

ωi90

(φ(ωi90)−arctan(

√κ−2(ωi

90)− 1)

Here frequency points at ω190 and ω2

90 are usedto estimate TITO FOPDT model. If the true sys-tem has a FOPTD model then the obtained modelwould be valid for all frequencies. On the otherhand, the model will be close to the true modelonly around at ωi

90, which is the frequency of in-terest for effective decoupling.

4 Relay-Based Decoupler Evaluation

In this section, the initial decoupler is evaluatedregarding frequencies of interest. These frequen-cies are ω1

90 for loop 1 and ω290 for loop 2, which

are obtained from the frequency-domain identifi-cation method. To evaluate D0(s) the decouplerevaluation excitation is applied to the decoupledsystem G(s)D0(s).

The decoupler evaluation excitation for loopi is a square wave with frequency ωi

90. The de-coupler is evaluated as follows. To evaluated D0

ij ,where i, j = 1, 2 and i 6= j, the evaluation excita-tion for loop i is applied at the opposite input uj .For loop 1 decoupler evaluation, it is illustratedin Figure 2. Therefore, for each loop i = 1, 2 theevaluation excitation is sequentially applied. Theidea is to evaluate the decoupler term at frequencyin the bandwidth of loop i.

Figure 2: Decoupled System - Decoupler Evalua-tion Excitation

The procedure is illustrated by the curvesshown in Figure 3. The decoupler evaluation ex-citation for loop 1 is applied at the input u2 (seeFigure 3). The width of this evaluation excita-tion is an integer number (N11) of time periodsT1 = 2π

ω190

plus a rectangular pulse of width T1/2.

The decoupler evaluation excitation for loop 2 issequentially applied to input u1 (see Figure 3).The width of this evaluation excitation is an inte-ger number (N12) of time periods T2 = 2π

ω290

plus

a rectangular pulse of width T2/2. Here, N11 andN12 are usually used equal to 2 or 3.

Figure 3: Decoupler Evaluation Excitation

Consider the TITO process with initial simpli-fied decoupler shown in Figure 2. Applying the de-coupler evaluation excitation at input ui, i = 1, 2,

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an input/output relation can be defined as

Hi(s) =Yi(s)

Uj(s)= Gij(s) +Gii(s)D

0ij(s), (5)

where i, j = 1, 2 and i 6= j.Perfect decoupling should make equation 5

zero. Values different from zero characterizes de-coupling errors. Thus, the decoupling error indexfor loop i at frequency ωi

90 is defined as

Hi(jωi90) = Gij(jω

i90) +Gii(jω

i90)D

0ij(jω

i90), (6)

Therefore, if the decoupling error index arenot close to zero, it is desirable to redesign D0(s)for effective decoupling at frequencies ωi

90. Thedecoupler redesign is presented below.

5 Decoupler Redesign

For loop i effective decoupling at frequencies ωi90

(ω190 and ω2

90), the decoupler terms (D012 and D0

21)must be redesigned to achieve corresponding de-coupling error close to zero.

Lemma 1 The decoupler redesign equation forloop i is given by

[D1ij(jω

i90)−D0

ij(jωi90)] =

−Hi(jωi90)

Gii(jωi90)

. (7)

where i, j = 1, 2 and i 6= j, and D1ij(jω

i90) is the

redesigned version of D0ij(jω

i90).

Proof: See (Acioli Junior and Barros, 2011). 2

The decoupler term D0,1ij has a simple form

D0,1ij (jωi

90) = K0,1

i

(jωi90T

0,1ii + 1)e−jωi

90L0,1

(jωi90T

0,1ij + 1)

. (8)

Thus, for loop i effective decoupling the corre-sponding decoupler term D1

ij must meet equation

7. For this, parameters K1

i ,T1ii and T 1

ij should be

modified in respect to K0

i ,T0ii and T 0

ij . So, thereare three possibilities to redesign the decouplerterms. Three redesign cases are outlined below.

5.1 The Decoupler Redesign - Case 1

In case 1, only the gain K1

i (see equation 8) ismodified for each loop i corresponding decouplerterm.

Lemma 2 The decoupler term gain for loop i ismodified as

∆Ki = Re

−Hi(jωi90)

Gii(jωi90)

(jωi90T

0ii+1)e−jωi

90θ0

(jωi90T

0ij+1)

, (9)

where, ∆Ki = (K1

i −K0

i ).



In case 2, the gain K1

i and time constant T 1ij are

modified for each loop i.

Lemma 3 K1

i and T 1ij are modified as

T 1ij =

−Im(Γi(jωi90))

ωi90Re(Γi(jωi

90))

K1

i = Re(Γi(jω

i90)

)[1 + (ωi

90T1ij)

2],(10)

where

Γi(jωi90) =

−Hi(jωi90)

Gii(jωi90)

+D0ij(jω

i90)

(jωi90T

0ii + 1)e−jωi

90θ0.



In case 3, the gain K1

i and time constant T 1ii are

modified for each loop i.

Lemma 4 K1

i and T 1ii are modified as

K1

i = < (Ψi(jω

i90)

)

T 1ii =

=(Ψi(jωi90))

ωi90K

1i

.(11)

where

Ψi(jωi90) = Hi(jω

i90)(jω

i90T

0ij + 1)e+jωi

90θ0


6 Experimental Results

In this section the procedure to evaluate andredesign simplified decoupler is applied to alaboratory-scale temperature system.

6.1 Experimental Setup description

The experimental setup is a laboratory-scale ther-moelectric system. This consists of two peltiermodules, two LM35 temperature sensors, a metalplate, two heat exchangers, two fans, a PLC(Programmable Logic Controller) and a PC withSCADA (Supervisory Control And Data Acquisi-tion). The peltier modules act as heat pumperson two sections of a flat metal plate heat load.The heat exchangers and fans are used to transferheat from the opposite faces of each peltier mod-ule. The process works as a coupled two-inputtwo-output process. Power is applied using PWMactuators while the temperatures are measured us-ing LM35 sensors. In (Barros et al., 2008) theexperimental setup is described in details.

The dynamic behavior of a thermoelectric sys-tem results in a complex model that is highly non-linear. For control purposes, a model reductioncan be made.

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6.2 Results

The relay+integrator experiment is applied to theexperimental setup. The input-output relay ex-periment data are shown in Figs. 4 and 5. Thesedata are used to estimate model gain and a fre-quency point information. The frequency pointinformation that are used to estimate the TITOFOPDT model is shown in Table 1.

0 50 100 150 200 250 300 350 40050

60

70

80

90

time(seconds)

Relay 90 − y1

u´1y1

0 50 100 150 200 250 300 350 40050

60

70

80

90

time(seconds)

Relay 90 − y2

u´1y2

Figure 4: Relay plus Integrator Experiment - u′1

0 50 100 150 200 250 300 350 400

50

60

70

80

90

time(seconds)

Relay 90 − y2

0 50 100 150 200 250 300 350 400

55

60

65

70

75

80

85

time(seconds)

Relay 90 − y1

u´2y2

u´2y1

Figure 5: Relay plus Integrator Experiment - u′2

Table 1: Estimated Frequency Information

in-out jωi90 |G(jωi

90)| φ(ωi90)(rad/s)

G11 u′1 − y11 0.123 0.24 -1.57

G21 u′1 − y12 0.123 0.02 -2.69

G12 u′2 − y21 0.074 0.005 -2.97

G22 u′2 − y22 0.074 0.18 -1.54

The TITO FOPDT model is estimated usingthe frequency-domain identification method as

Gexp (s) =

[0.0342e−6.45s

8.0332s+10.0278e−21.51s

69.2767s+10.0517e−12.6693s

17.3451s+10.0955e−14.7591s

11.5545s+1

].

The initial decoupler designed is given by

D0exp (s)=

[1 −0.81(8.03s+1)e−15.06s

(69.28s+1)−0.54(11.55s+1)

(17.34s+1) 1

]

To evaluateD0exp (s), the decoupler evaluation

excitation is applied to the decoupled experimen-tal setup. The evaluation excitation response isshown in Figs. 6 and 7.

0 100 200 300 400 500 600

60

80

time(seconds)Man

ipul

ated

Var

(%)

Decoupler Evaluation Excitation − u2

0 50 100 150 200 250 300 350 400

10

15

20

time(seconds)

tem

pera

ture

(ºC

) Decoupler Evaluation Excitation − y1

0 50 100 150 200 250 300 350 40012

14

16

18

time(seconds)

tem

pera

ture

(ºC


u1

y1

y2

Figure 6: Decoupler Evaluation Excitation - u1

0 50 100 150 200 250 300 350 400

60

80

time(seconds)Man

ipul

ated

Var

(%)


0 50 100 150 200 250 300 350 400

16

18

20

time(seconds)

tem

pera

ture

(ºC


0 50 100 150 200 250 300 350 40010

15

time(seconds)

tem

pera

ture

(ºC


u2

y2

y1

Figure 7: Decoupler Evaluation Excitation - u2

In this example, only the decoupler termD0

21exp (s) is redesigned for each redesign cases.The modified parameters for D21exp (s) in eachredesign case are shown in Table 2.

Table 2: D21exp (s) Modified Parameters

K T21 T22

Initial -0.5413 17.3451 11.5545Case 1 -0.2966 - -Case 2 -0.3805 24.3874 -Case 3 -0.3364 - 8.3078

For each case, the obtained redesigned decou-pler D1

exp is evaluated. The evaluation excitationresponse for case 1 and 2 are shown in Figs. 8 and9. The evaluation excitation response for case 3 isvery similar to the case 2 and was suppressed.

The frequency-domain decoupler error in-dexes evaluated for initial decoupler and for eachredesign case are shown in Table 3.

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0 100 200 300 400 500 600

60

80

time(seconds)Man

ipul

ated

Var

(%)


0 50 100 150 200 250 300 350 400

10

15

20

time(seconds)

tem

pera

ture

(ºC


0 50 100 150 200 250 300 350 400

18

20

22

time(seconds)

tem

pera

ture

(ºC


u1

y1

y2

Figure 8: Decoupler Evaluation Excitation - Ex-perimental - Case 1

0 100 200 300 400 500 600

60

80

time(seconds)Man

ipul

ated

Var

(%)


0 50 100 150 200 250 300 350 400

10

15

20

time(seconds)

tem

pera

ture

(ºC


0 50 100 150 200 250 300 350 400

18

20

22

time(seconds)

tem

pera

ture

(ºC


u1

y1

y2

Figure 9: Decoupler Evaluation Excitation - Ex-perimental - Case 2

From the frequency-domain decoupler errorindexes values (Table 3) it can be seen that theprocedure to evaluate and redesign decouplershave improved the decoupling at frequency of in-terest. In this example, the redesign decouplercases 2 and 3 presented similar improvement overthe case 1.

7 Conclusions

In this paper the procedure to evaluate and re-design simplified decouplers using relay exper-iment presented in (Acioli Junior and Barros,2011) was revised. This procedure is applied to aTITO laboratory-scale thermal process. The pro-cedure improved the decoupling at frequency ofinterest by the evaluation and redesign an initialsimplified decoupler. The decoupler redesign case2 and 3 provides similar results.

References

Acioli Junior, G. and Barros, P. R. (2011). Evalu-ation and redesign of decouplers for tito pro-

Table 3: Decoupler Error Indexes - Experimental

|H1| 6 H1(rad/s) |H2| 6 H2(rad/s)Initial 0.007 -0.910 0.015 1.33Case 1 - - 0.024 1.37Case 2 - - 0.011 1.27Case 3 - - 0.013 1.26

cesses using relay experiment, Proceedings ofthe 2011 IEEE International Conference onControl Applications (CCA) - Part of 2011IEEE Multi-Conference on Systems and Con-trol, Denver, CO, USA, pp. 1145 – 1150.

Astrom, K. J. and Hagglund, T. (1995). PIDControllers: Theory, Design and Tuning,2nd edn, Instrument Society of America, Re-search Triangle Park, North Carolina.

Astrom, K. J. and Hagglund, T. (2006). AdvancedPID Control, Instrument Society of America,Research Triangle Park, North Carolina.

Barros, P. R., Junior, G. A. and dos San-tos, J. B. M. (2008). Two-input two-output laboratory-scale temperature systembased on peltier modules, Proceedings of the17th IFAC World Congress, Seoul, Korea,pp. 9737–9772.

Cai, W.-J., Ni, W., He, M.-J. and Ni, C.-Y.(2008). Normalized decoupling - a new ap-proach for mimo process control system de-sign, Ind. Eng. Chem. Res. 47: 7347 – 7356.

Huang, H. P., Jeng, J. C., Chiang, C. H. and Pan,W. A. (2003). A direct method for multi-looppi/pid controller design, Journal of ProcessControl 13: 769 – 786.

Luyben, W. (1986). Simple method for tuningsiso controllers in multivariable systems, Ind.Eng. Chem. Process Des. Dev. 25: 654–660.

Shinskey, F. G. (1996). Process control systems:Application, design and tuning, McGraw-Hill, New York.

Shiu, S.-J. and Hwang, S.-H. (1998). Sequen-tial design method for multivariable decou-pling and multiloop pid controllers, Ind. Eng.Chem. Res 37(1): 107–119.

Wang, Q. G., Hwang, B. and Guo, X. (2000).Auto-tuning of tito decoupling controllersfrom step tests, ISA Transactions 39: 407–418.

Xiong, Q., Cai, W.-J. and He, M.-J. (2007).Equivalent transfer function method forpi/pid controller design of mimo processes,Journal of Process Control 17: 665 – 673.

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EVALUATION AND REDESIGN OF SIMPLIFIED DECOUPLERS APPLIED ... · 3.2 FOPTD Model Identi¯cation of TITO pro-cess using Relay Experiment A frequency-domain identi¯cation technique