CHEMICAL ENGINEERING TRANSACTIONS

VOL. 52, 2016

A publication of

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

Guest Editors: Petar Sabev Varbanov, Peng-Yen Liew, Jun-Yow Yong, Jiří Jaromír Klemeš, Hon Loong Lam
Copyright © 2016, AIDIC Servizi S.r.l.,

ISBN 978-88-95608-42-6; ISSN 2283-9216

Robust Controller Design for a Heat Exchanger

Anna Vasičkaninová*, Monika Bakošová, Ľuboš Čirka, Martin Kalúz

Slovak University of Technology in Bratislava, Faculty of Chemical and Food Technology, Institute of Information

Engineering, Automation, and Mathematics, Radlinského 9, 812 37 Bratislava, Slovak Republic

anna.vasickaninova@stuba.sk

The aim of the paper is to show the benefits of two advanced control strategies in heat exchanger control. Two

robust control techniques were used for controller design: H∞ control and -synthesis. H∞ optimal control is a
frequency-domain optimization and design theory that was developed in response to the need for a design

procedure that explicitly addresses questions of modelling errors. The H∞ robust controller design theory deals

with defined uncertainties, like parametric uncertainty and performance specified as weight filters. The H∞

theory has some weaknesses; for instance, it does not consider the uncertainty structure. The -synthesis

theory is a further development of the H∞ control where the uncertainty structure is considered in the design.

The designed controllers were verified in laboratory conditions on a real-time control of a laboratory heat

exchanger.

1. Introduction

Heat exchangers belong to the often used equipment in the chemical and process industry and they are

characterised by high energy demands. Many factors enter into the design of heat exchangers, including

thermal analysis, weight, size, structural strength, pressure drop and cost. The heat exchanger modelling

framework has been described and demonstrated in (Skaugen et al., 2013). The main purpose of (Daróczy et

al., 2014) is to illustrate and analyse a heat exchanger arrangement problem in its most general form. In

(Manassaldi et al., 2014) a mathematical model for the optimal design of air cooled heat exchangers is

described.

Control of heat exchangers is a complex process due to the nonlinear behaviour and complexity caused by

many phenomena such as leakage, friction, temperature dependent flow properties, contact resistance,

unknown fluid properties. Various control strategies were developed for overcoming all mentioned problems.

Simulation of a thermodynamic model of an earth-air heat exchanger was done and used along with a PID

controller to estimate savings in energy consumption in (Diaz-Mendez et al., 2014). The robust model

predictive control represents one of approaches that enable to design effective control algorithms for

optimization of the control performance as well as to take process uncertainty into account (Bakošová and

Oravec, 2013). The possibility to implement various robust model predictive control strategies for control of the

heat exchanger network with uncertainty is demonstrated in (Oravec et al., 2015). In (Shi and You, 2015), a

robust optimisation approach has been developed to handle the uncertainty in integrated planning, scheduling

and dynamic optimisation for continuous manufacturing processes. In Veselý (2013), a survey of robust

control design procedure is given. In Bell et al. (2015) , a novel and robust solution approach is presented that

can be used to predict the steady-state thermal heat transfer rate for counter flow heat exchangers. There

exist various solutions also of the standard H∞ problem. The mixed H2/H∞ performance criterion provides an

interesting measure for the controller evaluation. The theoretic motivation for the mixed H2/H∞ control problem

has been discussed in Kwakernaak (2002). The goal of Zarabadipour et al. (2011) is to design a reduced

order robust controller based on the balanced realization technique. The simulation results show that the

reduced order controller design with applied H2/H∞ has good results in frequency and time domain. In Ganji et

al. (2013), different conventional and intelligent controllers are used that are implemented with a clear

objective to control the outlet fluid temperature of the shell-and-tube heat exchanger system. For the dynamic

DOI: 10.3303/CET1652042

Please cite this article as: Vasičkaninová A., Bakošová M., Čirka L., Kalúz M., 2016, Robust controller design for a heat exchanger, Chemical
Engineering Transactions, 52, 247-252 DOI:10.3303/CET1652042

247

system with time varying characteristic and parametric uncertainties, a sliding mode controller is developed

and an optimal H∞ controller is designed based on -synthesis with DK-iteration algorithm in Moradi et

al. (2012). In De Souza et al. (2014) the problems of robust stability analysis and robust control of linear

discrete-time periodic systems is investigated. The paper of Vasičkaninová and Bakošová (2015) deals with

design and application of robust controllers for a shell-and-tube heat exchanger.

2. Controller design

2.1 H∞ control
Various techniques are available for the design of H∞-controllers. Mixed sensitivity is the name given to the

transfer function shaping problems in which the sensitivity function S = (I +GK)-1 is shaped along with one or

more other closed-loop transfer functions such as KS or the complementary sensitivity function T = I - S in a

typical one degree-of-freedom configuration, where G is the plant transfer function and K is the transfer

function of the (sub-) optimal controller to be found (Skogestad and Postlethwaite, 2005). The shaping of

multivariable transfer functions is based on the idea that a satisfactory definition of gain (range of gain) for

a matrix transfer function is given by the singular values  of the transfer function. Hence, the classical loop-

shaping ideas of feedback design can be generalized to multivariable systems. In addition to the requirement

that K stabilizes G, the closed-loop objectives are as follows:

 For disturbance rejection make )(S small,

 For noise attenuation make )(T small,

 For reference tracking make 1)()(  TT  ,

 For input usage (control energy) reduction make )(KS small,

 For robust stability in the presence of an additive perturbation  GGP , make )(KS small,

 For robust stability in the presence of a multiplicative output perturbation GIGP )(  , make )(T

small,

where )(A is the maximum singular value of A and )(A is the minimum singular value of A. It is known that

the robust controller is designed to minimize the H-norm of the plant. Three weight functions are added to the

control system for loop shaping (Bansal and Sharma, 2013). The classical feedback control system structure

with weighting is shown in Figure 1.

Figure 1: Control system for the synthesis of H∞ controller

The objective is to find the controller K(s) in the form of a rational function and to make the closed loop system

stable satisfying expression 


),( KGN for a given   0.






















KSW

KGN

),(

(1)

The user-defined weighting functions W1, W2, W3 bound the largest singular values of the closed-loop transfer

functions S (for performance), KS (to penalize large inputs) and T (for robustness and to avoid sensitivity to

noise), respectively (Skogestad and Postlethwaite, 2005).

2.2 μ-synthesis with DK-iteration
There is no analytical method to calculate a μ-optimal controller. However, a numerical method for complex

perturbations known as DK-iteration can be used (Balas et al., 1998). The generalized open-loop

representation of the configuration in Figure 2 can be given by Eq(2) (Griffin and Fleming, 2003):

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http://www.kirp.chtf.stuba.sk/~vasickanin/?show_id=3&show_pub=all





































































 

GGG

GWGWGW

pdpp

u00

(2)

where the input and output vectors for this configuration contain the inputs and outputs relating to the input

and output perturbations u and y. Uncertainty at the plant input and performance requirements at the system

output are described by weighting functions Wu and Wp, respectively, K is the transfer function of the controller

and Gd is disturbance model.

The closed-loop interconnection structure N, which incorporates P and K is given











 


SGWGSW

SKGWKGSW
N

dpp

duu
(3)

Figure 2: Closed-loop system with uncertainty and performance weighting

μ-synthesis with DK-iteration is based on the upper bound:

  1min)(  DNDN
D

 (4)

A scaling matrix D is chosen such that it commutes with , the plant perturbation, i.e. D=D. The synthesis

problem is then to find the controller such that

  















1
)(minmin DKDN

(5)

by alternating the minimization with respect to K and D (keeping the other fixed). The iteration goes as follows

(Skogestad and Postlethwaite, 1996):

 K-step. Synthesize an H∞controller for the scaled problem with fixed D(s).

 










1
)(min DKDN

(6)

 D-step. Find D(j) to minimize at each frequency

 )()( 1  jNDjD  (7)

 Fit the magnitude of each element of D(j) to a stable and minimum-phase transfer function D(s). Go

to K-step.

Continue iteration until 1
1





DND or until the norm no longer decreases.

3. Description of the multifunction process control teaching system

The described approach was verified in a real-time control of the laboratory heat exchanger realized on the

Multifunction Process Control Teaching System (MPCTS) the Armfield PCT 40 MPCTS (2007). The MPCTS is

designed especially for teaching of a wide range of technological and chemical processes. The PCT40 is

connected to the computer through data acquisition and control card, and each actuator (pumps, valves,

heater, etc.) can be controlled directly from MATLAB software. The computer also displays the readings from

various measurement sensors. The through-flow heat exchanger is a part of the MPCTS (Figure 3).

249

Figure 3: PCT 40 - heat exchanger

Figure 4: Schematic diagram of the heat exchanger

The heat exchanger (Figure 4) consists of a cylindrical vessel with fluid inlet and outlet, the heating element,

internal cooling system and a set of sensors. The inputs are the heater power, handled by Solid State Relay,

and the water flow rate qc in the cooling coil. The controlled variable is the temperature measured by the

sensor T1. A constant amount of inlet and outlet water, which ensures a constant water level in the tank, is

realized by equal adjustment of flow rate q by means of two peristaltic pumps.

The objective of the work was to identify the controlled process, to determine the controller parameters and to

evaluate the controller performance using these parameters. For the identification, a series of the positive and

negative step changes of the input variable (opening of the valve) were generated, and the step responses of

the outlet temperature were measured. According to these step changes, the nominal model was identified

using the Strejc method (Mikleš and Fikar, 2007) in the form of the first order transfer function with the gain

Z = -0.53 oC %-1 and the time constant  = 258 s.

The PI controller parameters were tuned for the model with the nominal values of the identified parameters

using pole placement method. The enumerated PI controller parameters are kp = -8.8935, ti = 164.9245 s for

poles s1, s2 = -0.01 (Mikleš and Fikar, 2007). In Figure 5 the control input opening of the valve and the PI

control of the output temperature are shown.

Figure 5: PI control of the heat exchanger

In Figure 6 the control input and the H∞-control of the output temperature are shown.

The H∞-controller for the heat exchanger was calculated in the form

0.00020.2489s

0.04468 - s -12.73
)(

2



s

sK (8)

250

Figure 6: H∞ control of the heat exchanger

The μ-synthesis with DK-iteration controller for the heat exchanger was calculated in the form

0.002735 + s 1.695 + s 261.8 + s 179.9 + s

1.56 - s 891.6 - s 127720 - 87560s-
)(

234

sK

(9)

In Figure 7 the control input and the μ-synthesis control of the output temperature are shown.

Figure 7: μ-synthesis control of the heat exchanger

The comparison of the proposed controllers was made using IAE integral performance index. The IAE values
are given in Table 1. The smaller value of IAE is assured using the μ-synthesis control, achievement of
setpoint value is fast but the disadvantage is the higher vibration of the input and output variables.

Table 1: Values of IAE

Controller IAE

PI control 2.8148 × 105

H∞ control 2.8091 × 105

μ-synthesis control 2.8019 × 105

4. Conclusion

Two methods were used for robust controller design, H∞-controller and μ-synthesis with DK-iteration controller.

The presented approach was verified on the real laboratory system MPCTS - the Armfield PCT 40. The

251

control tests executed on the laboratory model gave satisfactory results. As the laboratory heat exchanger

simulates an industrial process, it was proved that the investigated advanced control design methods could be

successfully used to control industrial heat exchangers.

Acknowledgment

The authors gratefully acknowledge the contribution of the Scientific Grant Agency of the Slovak Republic

under the grant 1/0112/16.

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