CHEMICAL ENGINEERING TRANSACTIONS

VOL. 76, 2019

A publication of

The Italian Association
of Chemical Engineering
Online at www.aidic.it/cet

Guest Editors: Petar S. Varbanov, Timothy G. Walmsley, Jiří J. Klemeš, Panos Seferlis
Copyright © 2019, AIDIC Servizi S.r.l.

ISBN 978-88-95608-73-0; ISSN 2283-9216

Bypass Control System Design for HENs Based on

Frequency Domain Analysis

Lin Sun*, Zheng-hua Yang, Xiong-lin Luo

Department of Automation, China University of Petroleum, Beijing, 102249, China

sunlin@cup.edu.cn

In this paper, a new approach to address bypass control system design for heat exchanger networks (HENs) is

proposed based on frequency domain analysis. The classical bypass control system design methodology relies

on the calculation of the relative gain array (RGA) which is commonly used as a measure of the interaction. The

main drawbacks of this approach are the difficulties of analyzing the unstable process, and the complexity of

measuring stability. Instead, the key idea of this work is to take stability margins into consideration when potential

bypasses and controlled variables are paired. Firstly, on the basis of dynamic model, small singular values of

the return difference matrix are presented for the simultaneous analysis of stability margin and flexibility.

Secondly, the non-square relative gain array (ns-RGA) in the frequency domain is applied to study the

interactions of potential bypasses and controlled variables, and some pairing rules are proposed. Thirdly

parameters of PID controller are designed while phase margin and gain margin are in proper region, then the

final pairing scheme is determined. Case studies illustrated the necessity to consider stability margins during

the process of the bypass control system.

1. Introduction

The application of bypasses has been demonstrated for effective control of a target temperature. However, on

one hand, as strong interaction exists in streams, HENs designed according to the chemical process often do

not have enough bypasses. On the other hand, bypasses will also affect stability and optimality of HENs.

Therefore, the BCS design is a challenging problem.

Stability margins often refer to gain margin and phase margin and are studied in many works. Structural singular

value μ is proposed to calculate stability margins (Doyle, 1982). Safonov (1982) developed a readily computable

lower bound for diagonally perturbed systems. Wang et al. (2014) pointed out that the distance between

operating point and Hopf singular point is used as a measure of stability. Lehtomaki et al. (1981) proposed the

concept of minimum singular value of return difference matrix which pays attention on gain margin and phase

margin simultaneously. This paper intends to study the application of minimum singular value of return difference

matrix in HENs.

Flexibility is a fundamental requirement of HENs and previous works on flexible analysis and designs are under

the assumption that all control variables can be adjusted and HENs is stable. However, flexible HENs may come

to be unstable with the existence of disturbance. Jiang et al. (2014) took stability into consideration when

analyzing flexibility and they have a conclusion that flexible region contains unstable parts. However, they did

not study the relationship between stability margins and flexibility.

Bypasses are widely used for optimizing the operation of HENs to maintain the control requirements. A key

issue of BCS is particularly relevant to select the appropriate potential bypasses and controlled variables.

Relative gain array (RGA) was introduced by Bristol (1966) and is commonly used as a measure of interaction.

Flexibility and controllability are taken into consideration at the process of BCS simultaneously. A computational

framework is proposed for the synthesis of HENs where flexibility and controllability are studied simultaneously

(Escobar et al., 2013). A sequential procedure for flexible HEN synthesis and control structure design has been

proposed by Braccia et al. (2018). RGA in the frequency domain is proposed (Xu et al., 2016) and can used to

analyze the unstable process. Therefore, this work plans to introduce RGA in the frequency domain to HENs.

DOI: 10.3303/CET1976128

Paper Received: 28/02/2019; Revised: 03/06/2019; Accepted: 04/06/2019
Please cite this article as: Sun L., Yang Z.-H., Luo X.-L., 2019, Bypass Control System Design for HENs Based on Frequency Domain Analysis,
Chemical Engineering Transactions, 76, 763-768 DOI:10.3303/CET1976128

763

PID controller is widely placed to ensure control performance. Nonetheless, the design of PID controller has not

been an integral part of BCS design. There are several design schemes for optimally tuning PID parameters.

PID parameters adjustment method is proposed in HENs under fouling conditions (Trafczynski et al., 2016).

Carvalho et al. (2018) introduced tune strategies for overcoming fouling effects in PID controlled heat

exchangers. Basically, they are based on the time-domain.

This paper investigates the incorporation of stability margins and BCS design. A method is presented to design

optimal bypasses while achieving proper stability margins based on frequency domain analysis. Firstly, stability

margins of HENs are analysed. Secondly, RGA in the frequency domain is applied to obtain the interaction of

potential bypasses and controlled variables, and some pairing rules are given. Thirdly PID parameters are

projected to design so that a single control system achieves proper stability margins.

2. Stability margin analysis of HENs

Stability is a prerequisite for a control system and stability criterion of open-loop system can be obtained

according to Argument Principle,

0=+= ZRP (1)

Where P is the number of open-loop system poles in the right half-plane, Z is the number of open-loop system

zeros in the right half-plane, and R is the number of turns of Nyquist curve surrounding the origin. When P is

equal to 0, the open-loop system is stable.

This paper will study stability margins of HENs based on the inverse return difference matrix (Li et al., 2014).

2
arcsin2

)1lg(20

m
PM

mGM

(2)

Where m is minimum singular value of the inverse return difference matrix of HENs, PM is phase margin and

GM is gain margin.

Stability and flexibility have been analyzed by Jiang et al. (2014) simultaneously. However, they did not take a

deep insight into different stable points with different flexibility as shown in Figure 1.

θ1
N +FΔθ1

θ1

θ2

θc

θ2
N +F1Δθ2

θ1
N +FΔθ2

θ2
N +FΔθ2

θ2
N +F1Δθ2

θ1
N +FΔθ1

-θ1
N +F1Δθ1

-
θ1

N +F1Δθ1
+

θ1
c

θ2
N

θ1
N

GM1

Figure 1: Flexible region with different stability margins

In Figure 1, flexibility index, gain margin and phase margin of HENs are F, GM and PM respectively when the

design variable is d. Flexibility index, gain margin and phase margin of HENs are F1, GM1 and PM1 respectively

when design variable is d1 .

A simple HENs provided by Sun et al. (2011) will be used to illustrate the relationship between stability margins

and flexibility index. The nominal data is listed in Table 1 and the structure is showed in Figure 2.

764

H1 1 2 c1

h1 C1

h2 C2

190℃
100℃

80℃

80℃
140℃

20℃

70℃ 30℃

160℃

130℃

Figure 2: The structure of HENs

Table 1: The parameters of the hot and cold fluid

Stream

number
Input temperature/℃ Output temperature/℃ Heat capacity/(MW/℃)

Hot(H1) 190 30 0.10

Cold1(C1) 80 160 0.15

Cold2(C2) 20 130 0.05

When the area of heat exchanger 1 and heat exchanger 2 is 1258.71 m2 and 428.57m2, the Nyquist curve is

shown in Figure 3a.

0 0.1 0.2-0.1

Real axis

Imaginary axis

-2

-4

-6

-8

×10-4

×10-5

19.4 19.6 19.8 20.0 20.2 20.4 20.6

0.6

0.7

0.8

0.9

1.0

1.1

1.2

F
I

GM/db

Figure 3a: Nyquist curve of HENs Figure 3b: Relationship between flexibility index and gain margin

As shown in Figure 3a, the number of turns of Nyquist curve enveloping (0,0j) is 0, and the number of open-

loop system zeros in the right half-plane is 0, so the HENs is stable.

In order to have better dynamic characteristics, gain margin of control systems is generally taken as Am=[3, 6],

and phase margin is taken as φm= [30°, 60°]. It can be known from Figure 3b that when the FI is more than 1,

gain margin is more than 6db. So the HENs is over stable and PID controller should be placed to improve

stability margins at the process of BCS design.

3. HEN control system design

3.1 HEN frequency domain RGA

Non-square RGA (ns-RGA) is used to measure the interactions of different variables for the number of controlled

and manipulated variables is usually unequal in HENs. Compared with conventional ns-RGA in the time domain,

in this paper s is replaced by jω, then G(s) in the time domain becomes G(jω) in the frequency domain. Assumed

that there are q controlled variables and n potential bypasses in HENs. Ns-RGA in the frequency domain can

be calculated by Eq(3) for q times.

T
jGjGΛ )]([)()( 


= (3)

The symbol  means an element-by-element multiplication (Schur product). Gϕ(jω) is the pseudo-inverse matrix
of G(jω).

765

Each element in Eq(3) is plural number and cannot quantitatively analyse the impact of each potential bypass

on the controlled variable. Then interaction measurement index (IMI) and the unit circle diagram in complex

plane are proposed (Xu et al., 2016).

Steps of calculating IMI in HENs is as following,

Step 1: Choosing proper controlled variables and potential bypasses of HENs, then calculate the Λ(ω).

Step 2: Assumed that the element Λip(ω) in Λ(ω) is described as a(ω)+ b(ω)j, and then the distance between
Λip(ω) and (1,0j) is formulated as λip(ω). Then IMI is calculated in Eq(4),


−

=
n

dD
n






 0
)(

1
ip

ip
(4)

The method based on the unit circle diagram in complex plane is as following,

Step 1: Choosing the controlled variables and potential bypasses of HENs, then calculates the Λ(ω).

Step 2: Since Λ(ω) is a function of ω, the integral mean of Λ(ω) to ω is calculated in Eq(5). And Λ0 is defined as

average ns-RGA in the frequency domain.


−

=
n

dΛΛ
n






 0
)(

0
(5)

In this paper the ω0 is 0, and the ωn is the bandwidth.

Based on ns-RGA in the frequency domain, some principles are obtained when potential bypasses and

controlled variables in a control loop are paired.

(1) Removing the potential bypasses with interaction measurement indexes of 1 or close to 1.

(2) Drawing the remaining bypasses on unit circle of complex plane and removing the bypasses in the III and

IV regions.

(3) Selecting the point close to the (1, 0j) point on unit circle of complex plane as the potential optimal bypass.

3.2 Controller parameters design

The interaction of potential bypasses and controlled variables is analyzed in section 3.1. However, controller

parameters are neglected making it hard to meet stability margins.

Assume that HENs is a second-order system, and its frequency characteristics are describe as Eq(6),






jcd
jG

+
=

2
-b

)( (6)

The frequency characteristics of PI controller is shown in Eq(7),

j
K

KjG I
Pc


 −=)( (7)

The frequency characteristics of control system is shown in Eq(8),

)(
)-b(

)()(
2

j
K

K
ja

jcd
jGjG I

pc



 −

+
=

(8)

Controller parameters KP and KI are non-negative. Assumed that gain margin and phase margin are Am and φm

respectively. The crossover frequency and cut-off frequency are ωx and ωc.













−+
=

+−
−=

)(

dcA

bcadc
K

dcA

acdbd
K









(9)













+−
+

+−
=

+−
−

+−
=

cccc

222

))(cos())(sin(

cbcad

acdbd
K

acdbd

cbcad
K

mm
I

mm
P
















(10)

Therefore, a proper region of KP and KI can be calculated by Eq(9) and Eq(10) to meet the requirements of

stability margins.

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A control loop should meet the stability requirements. Figure 4 shows the steps of BCS design in HENs based

on frequency domain analysis.

Begin

End

The one with positive KP and KI is set as

the optimal bypass of controlled

temperature i

Yes
KP or KI is

negetive

Calculating KP and KI according to Eq(9)

and Eq(10)

Dip and Λ are obtained according to Eq(4)

and Eq(5)

Removing the potential bypasses with IMI

of 1 or close to 1

Determining the optimal bypass according

to pairing rules (2) and (3)

Figure 4: Control system process design of HENs

4. Case study

A simple HENs provided by Sun et al. (2011). The controlled target is the output temperature of hot stream and

there are four potential bypasses in this HEN. The IMI in the frequency domain are shown in Table 2. According

to pairing rules, Ns-RGA in the frequency domain described on the unit circle diagram are shown in Figure 5a.

From the Figure 5a, the hot stream of heat exchanger 2 is the potential optimal bypass.

Table 2: IMI and ns-RGA in the frequency domain

Potential bypasses
Hot stream of heat

exchanger 1

Cold stream of heat

exchanger 1

Hot stream of heat

exchanger 2

Cold stream of heat

exchanger 2

IMI 0.6356 0.9623 0.4805 0.8886

III

IIIV

(1,0)
(-1,0)

（0.1174-0.0258i)
（0.0589-0.0752i)

（0.2373+0.011i)

0 200 400 600 800 1000 1200
69.60

69.65

69.70

69.75

69.80

69.85

69.90

69.95

70.00

70.05

o
u

tp
u

t
te

m
p

e
ra

tu
re

time /s

in this paper

Hou et al. (2011)

Figure 5a: Average ns-RGA in the frequency domain Figure 5b: Output temperature of hot stream

767

KP and KI are designed according to Eq(9) and Eq(10). When KP and KI is 0.5334 and 0.942, respectively, the

gain margin and phase margin are 52db and 36°. KP and KI are non-negative. So the hot stream of heat

exchanger 2 is the optimal bypass.

Hou et al. (2011) have studied structural controllability to design optimal bypass location and the result of optimal

bypass is hot stream of heat exchanger 1 based on structural controllability. Therefore, when the input

temperature of hot stream is changed from 190℃ to 185℃, the output temperature of hot stream under the

control of Hou et al. (2011) and this paper is shown in Figure 5b. It can be concluded from Figure 5b that output

temperature in this paper has faster response speed and smaller overshoot than the method in Hou et al. (2011);

hence the control performance is better.

5. Conclusions

In this work bypass control system design based on frequency domain analysis considering stability margins

has been addressed because previous studies only explored ns-RGA in the time domain using steady-state

HENs. Then ns-RGA in the frequency domain is proposed to optimize potential bypasses. The main conclusions

of this paper are as follows:

(1) It is proved that the flexible HENs is over stable, and stability margins should be considered when designing

the control system.

(2) The IMI and ns-RGA in the frequency domain is used to analyze the optimal bypass combined the dynamic

characteristics. On the basis of optimal bypasses and tuning strategies in PI controlled HENs, the control system

will have a proper stability margins with gain margin of 3db-6db and phase margin of 30°-60°.

(3) This method will have a better control performance compared with other method based on ns-RGA in the

time domain.

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