Universidad Nacional de Chimborazo
NOVASINERGIA 2019, Vol. 2, No. 1, diciembre-mayo (80-87)
ISSN: 2631-2654
https://doi.org/10.37135/unach.ns.001.03.07
Research Article
http://novasinergia.unach.edu.ec
Discontinuous feedback control strategies for a heat exchanger
Estrategias de control de retroalimentación discontinuas para un
intercambiador de calor
Maribel Pérez Pirela
1
*, Juan Paulo García Sandoval
1
, Oscar Camacho
2
1
Chemical Engineering Department, University of Guadalajara, Guadalajara, México, 44430
2
Automation and Industrial Control Department, Escuela Politécnica Nacional, Quito, Ecuador, 170525;
paulo.garcia@cucei.udg.mx; oscar.camacho@epn.edu.ec
* Correspondence: maribel.pirela@alumno.udg.mx
Recibido 15 abril 2019; Aceptado 17 mayo 2019; Publicado 06 junio 2019
Abstract:
Sliding mode Control (SMC) is one of the robust and nonlinear control methods.
The SMC has several advantages, such as robustness against external
disturbances and uncertainties in parameters. On the other hand, the chattering
effect is a common problem for the method. In the literature, some approaches
have been proposed to overcome the problem of chattering. In this document,
an evaluation of simulation of conventional techniques of (first-order) sliding
mode control is investigated. Simulations applications are made using a heat
exchanger system for the control of temperature monitoring and regulation of
interference problems. A qualitative performance analysis is done through radio
charts. The graphical results are illustrated and performance measurements are
tabulated based on the time domain analysis. The results of simulations indicate
that the sliding mode control is applicable to practical control systems at the cost
of some disadvantages.
Keywords:
Sliding-mode control, heat exchange system, radio charts.
Resumen:
El control de modo deslizante (SMC) es uno de los métodos de control robustos
y no lineales. El SMC tiene varias ventajas, como la robustez frente a
perturbaciones externas e incertidumbres en los parámetros. Por otro lado, el
efecto de charlando es un problema común para el método. En la literatura, se
han propuesto algunos enfoques para superar el problema del chattering. En
este documento, se investiga una evaluación de la simulación de las técnicas
convencionales de control de modo deslizante (de primer orden). Las
aplicaciones de simulaciones se realizan utilizando un sistema de
intercambiador de calor para el control del monitoreo de temperatura y la
regulación de problemas de interferencia. Un análisis de rendimiento
cualitativo se realiza a través de gráficos radiales. Los resultados gráficos se
ilustran y las mediciones de rendimiento se tabulan en función del análisis del
dominio de tiempo. Los resultados de las simulaciones indican que el control de
modo deslizante es aplicable a los sistemas de control prácticos a costa de
algunas desventajas.
Palabras clave:
Control por modos deslizantes, Sistema de intercambio de calor, Control de
temperatura
http://novasinergia.unach.edu.ec 81
1 Introduction
In recent years, technological advances have
generated a huge variety of new problems and non-
linear applications that are commonly seen in major
modern engineering applications (Yu & Kaynak,
2009). In this sense, it is well known that the process
industries are an integral part of the economic
development of a nation and chemical processes use
non-linear systems such as distillation columns,
boilers, chemical reactors, heat exchangers, among
others. These processes are complex, have time
delays and different types of non-linearity, higher
order, slow dynamic behavior, time delay and
external disturbances (Stephanopoulos, 1984). It is
not always possible to control them with classic
control schemes, such as the feedback control
scheme and conventional controllers such as
proportional (P), proportional-integral (PI),
proportional-integral-derivative (PID), etc.
Thus, general practice of controller design for
process control systems requires a mathematical
model, however determining an accurate model is
almost impossible. Hence, a working model of the
plant is obtained using techniques of system
identification.
Therefore, to control this type of systems, robust
control schemes are required being a concrete
approximation to the robust control design the so-
called sliding mode control (SMC) method, which
constitutes a particular type of control by variable
structure. In general, the SMC procedure produces a
complex controller, which could contain four or
more parameters resulting in a difficult tuning job.
Therefore, the use of SMC's traditional procedures
presents disadvantages in its application to chemical
processes.
There are several papers where successfully
designed and applied SMCs for regulation and
tracking of systems. Camacho and Smith, (2000)
proposed SMC for chemical processes designed
from a PID sliding surface and a reduced First Order
Plus Delay Time (FOPDT) model of plant with
tuning parameters as a function of the characteristic
parameters of the FOPDT. Eker (2006) presented a
sliding mode control system with a PID sliding
surface adopted to control the speed of an
electromechanical plant. Herrera et al. (2015),
designed and applied a SMC to a Quadrotor, Báez
et al. (2018), presented a real implementation of a
SMC applied to a cooling tower in an Arduino Mega
microcontroller.
In Pérez-Pirela and García-Sandoval (2018) a
dynamic model was developed and validated to
describe the behavior of a heat exchanger and the
proposed SMC for chemical processes was based in
this non-linear dynamic model.
The contribution of this paper is that the SMC
techniques presented in Camacho and Smith, (2000)
and Pérez-Pirela et al. (2018) are simulated for a
heat exchanger system to demonstrate applicability
of the techniques to practical systems, with integral-
differential sliding surface, whose control law is the
sum of the switching signal and the equivalent
control signal. The results are presented graphically
and comparison measures based on time-domain
analysis are tabulated. It also presents the potential
application in control systems of the representation
of radial graphics, because they are an easy way to
see how effective the controllers are when the
performance of both approaches are compared.
2 Fundamental Sliding-Mode
Control
Robustness and systematic design procedures are
well-known sliding mode controllers’ advantages
(Slotine, 1984). Traditionally, conventional SMC
has been designed for systems with relative grade
one. If the control input appears on the first
derivative of the sliding surface, its relative degree
with respect to the control is one. Under these
features, the control method is called the first-order
SMC. Then, in order to control an output with a
relative degree greater than one, it will have to add
as many outputs as necessary to display the control
input.
The SMC control law is composed of two parts: the
control law of sliding mode and the control law of
reach mode. The first is responsible for keeping the
dynamic system controlled on a sliding surface,
which represents the desired closed-loop behavior.
The second control law is designed to reach the
desired surface. System trajectories are sensitive to
parameter variations and disturbances during
trajectory range mode, but are insensitive in slide
mode (Sira-Ramírez, 2015). The first step in SMC is
the choice of the sliding surface or sliding function
that is usually formulated as a linear function of the
system states, expressed as a function of the tracking
error
, which is the difference between the
measured output and the reference value. In this
sense, Slotine (1984) defined an integral-differential
sliding surface of order n that applies the complete
error of follow-up of the form:
(1)
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where, is the process order model, and
is
an adjustment parameter. The aim of the control is
to ensure that the controlled variable is equal to its
reference value at all times, which means that
and its derivatives must be null. Once the reference
value is reached, it indicates that
reaches a
constant value. Keeping
at this constant value
means that
is zero at all times; that is:
Once the sliding surface is selected, attention must
be paid to the design of the control law which drives
the controlled variable to its reference value and
satisfies equation 2. Thus, the homogeneous
differential equation that has a single solution is
obtained by fixing
. Therefore, the error will
come asymptotically to zero with a proper control
law that keeps the trajectory on the sliding surface.
It is only necessary and sufficient to derive to
Equation 1 once, so that the input
appears. This
becomes a first-order stabilization problem based on
. The direct method of Lyapunov can be used to
obtain the control law that maintains
at zero
and a function of Lyapunov candidate is:
with
,
(Khalil,
2002). A sufficient condition for the stability of the
system is:
where is a real constant, strictly positive, which
determines the speed of convergence of the
trajectory to the sliding surface (Slotine, et al.,1991).
The inequality of equation 4 ensures that the distance
to the sliding surface decreases along all the
trajectories and consequently, the system is stable.
Therefore, equation 4 is called the attainability
condition for the sliding surface. Substituting the
sliding surface in equation 4 you get:
Thus, a control input that satisfies the attainability
condition can be chosen as:
where
is the estimate of the equation of state , it
is the gain of the discontinuous control which is a
strictly positive real constant, with a lower limit that
depends on the estimations of the system parameters
and external disturbances. The function sign ()
denotes the sign function defined as:
The sliding surface design is a powerful tool for
improving system performance. It is also possible to
shorten the time of reach and thus decrease the effect
of the disturbances by increasing the amplitude of
the gain of discontinuous control in equation 6.
However, increasing gain has negative effects
such as high sensitivity to the dynamics of
unmodeled systems, unchattering of amplitude and
saturation of the actuator. Therefore, the increase in
discontinuous control gain is generally undesirable
for physical systems and is not a viable alternative to
sliding surface design. A good interchange between
the time of reach and the speed of response is
obtained by changing the parameters of the sliding
surface (Yu & Efe, 2015).
The control input in equation 6 consists of two parts.
The first part is a continuous term known as the
equivalent control, which is based on the estimated
system parameters and compensates for the
estimated undesirable dynamics of the system. The
second part with the function sign is the law of
discontinuous control, which requires an infinite
switching by the control signal and the actuator at
the intersection of the error trajectory of the state and
the sliding surface. Thus, the trajectory is forced to
move always towards the sliding surface (Utkin,
1992).
One of the problems generated by the infinity
switching or oscillation of the discontinuous control
is the chattering effect. This effect produces that in
practice the control law cannot be implemented in its
natural form, since its direct application will cause
the actuators to deteriorate. The main cause of this
problem is due to the discontinuous function
which evaluates to the sliding surface.
The solution to this problem is to try to make the
signal
have a smooth level transition
while trying to keep your property. To do this,
(Slotine et al., 1991) raises the saturation function as
follows:
where
is the setting parameter responsible for
the reaching mode and is a positive constant that
helps reduce the chattering. If is too small, its
behavior will resemble the
, so when the
controllers are implemented by slider mode, this
constant will be chosen so that the control signal
prevents the chattering and generates a soft control
signal to ensure control objectives.
(7)
(6)
(8)
(2)
(3)
(4)
(5)
http://novasinergia.unach.edu.ec 83
3 Methodology
3.1 Experimental Setup
A laboratory-scale heat exchanger was used for the
implementation and testing of control strategies
based on the sliding mode control method, as shown
in figure 1. The heat exchange system was
composed of a stainless steel electric heater with a
length of 0.29 m, which contains an electrical
resistance of 1000 W, the internal and external
diameters are d1 =2" y d2 = 1", respectively, and the
fluid enters with a temperature T2,i (t) and passes
through the heater with a volumetric flow (F) of 2
L/min.
Figure 1: Experimental heat Exchange system.
The control objective was to regulate the output
temperature of the fluid,
, manipulating the
power supplied by the electrical resistance, while the
initial temperatures (
,
) and
inlet temperature, T2,i (t) ϵ C
2
, are considered as
disturbances. Inlet and outlet flow temperatures
were measured with J-type thermocouples. The
power of the electrical resistance was regulated with
a coil relay connected to a PWM device. Fluid flow
was controlled by a Asco® Posiflow® proportional
solenoid valve model SD8202G086V with a PWM
control unit Asco® model 8908A001 using an
auxiliary control loop that measures the volumetric
flow rate with an FLS® Flow sensor model
ULF03.H.0. All signals were read and manipulated
with a national Instruments®, Compact Field Point
device, operated by the user through a virtual
interface developed in LabView, which runs on a
desktop PC that communicates with the controller
via Ethernet.
3.2 Heat Exchanger Models
Non-Linear Model
Based on a distributed parameter model for the heat
exchanger described in the previous section, (Pérez-
Pirela et al., 2018) developed a simplified
mathematical model for this system, which
describes the dynamic behavior of the temperature
at the output ( ), by means of an ordinary
differential equation of second order with delay in
the input (u), and the disturbances ( y ):
where
and
is an auxiliary variable whose dynamic is
with initial conditions
y
.
Here,
, for ; are the
characteristic times of heat transport for each
material (resistance y fluid ),
is the overall characteristic time,
is the time of fluid residency
within the exchanger,
and
is the control variable (for
more details consult (Pérez-Pirela et al., 2018). By
observing equation 9, it is clear that the relative
degree between the and the
output is two.
First Order Plus Dead Time (FOPDT) Model
The reaction curve of the process, figure 2, is a
commonly used method for the identification of
dynamic models (Smith & Corripio, 1997). This
method is simple to perform and provides suitable
models for many applications; thus, the first-order
model with delay is used to approximate the model
of the heat exchanger system. For this purpose, the
curve is obtained by introducing a series of step
changes in the output of the controller through the
power of the electrical resistance as shown in table
1 and recording the output of the transmitter with the
output temperature of the fluid.
Table 1: Step changes in the power of the electrical
resistance.
u ( W )
u (%energy)
478
50
956
100
574
60
287
30
0
0
(9)
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In this way, when performing the step tests, the
following reaction curves are obtained for the
system as shown in figure 2:
Figure 2: Reaction curve Process.
In this way it is able to provide a reduced suitable
model for the application of the heat exchanger.
From the process curve shown in figure 2, and the
procedure presented in (Smith & Corripio,1997), the
numerical values of the terms of the FOPDT model
given in equation 10 are obtained:
where is the static gain, is the time constant and
is the delay time. Using the input/output data for
the system, the average coefficients of the plant are
, y
.
The dynamic behavior of the non-linear model and
the reduced order model are shown in figure 3; it
can be seen an acceptable deviation in both models.
Figure 3: Real Output measurement and output model.
(10)
http://novasinergia.unach.edu.ec 85
3.3 Sliding Mode Control Algorithms
The results are presented to demonstrate the
operation of two selected SMC techniques which
mainly characterize the classic SMC for the purpose
of regulating error
. Controller
parameters are tuned during experiments, avoiding
complicated calculations that can cause large
chattering that is dangerous to the actuator.
Technique I: This technique is presented to regulate
chemical processes by Camacho y Smith (2000),
applying a reduced FODPT model; and the control
signal is the sum of the switching signal and the
equivalent control signal.
Technique II: This technique was proposed by
(Pérez-Pirela et al., 2018), where conventional SMC
techniques applied to an experimental non-linear
heat exchanger model are designed, validated and
compared. The control law is the sum of the
switching signal and the equivalent control signal.
In both techniques a sliding surface is used based on
the integral of the error, for a heat exchange system,
the sliding surface related to is the one shown
below
The control signal is the sum of the switching signal
and the equivalent control signal.
Simulations were performed for the heat exchanger
coupled with each controller. The initial conditions
of the system, Begin with the heat exchanger being
in is in permanent mode at a temperature of thermal
equilibrium of , with a volumetric flow of
, an inlet temperature, ; variable as
load disturbance. At the controller
started with a reference temperature of , then
at the reference temperature was
changed to, then to to decrease
the reference temperature to , thus it is
intended to measure the characteristics of tracking
to a system reference. Subsequently, the volumetric
flow was decreased to and the
previously commented tests were performed again,
in order to observe the system behavior before the
variation in one of its nominal parameters.
The performance of the techniques was evaluated
with the following performance indices:
The integral of the absolute error (IAE) =
The integral of the control input Absolute (IACI) =
.
4 Results and Discussion
The controllers of the techniques were implemented
in the MATLAB environment and the sampling time
was selected to be 100 s. To test the system's
regulatory properties, the reference temperature
changes mentioned in subsection 2.3 were applied
and the responses are shown in figure 4. As shown
in this figure, both techniques have a very similar
performance, so it was also compared to the
performance indices presented in table 2. These
results show that the system with the technique II
sliding mode controller has better performance than
the system with technique I.
Table 2: Performance Index results to changes in the
reference.
Figure 4: System response to changes in the reference.
To test the robustness of the system in the face of
parameter variation, the volumetric flow was
decreased to 1.3 L/min and the results are shown in
figure 5. The performance of the techniques was also
compared to performance indices and presented in
table 3. The system with conventional SMC
Techniques SMC
IAE
IACI
I
619.29
4.8905× 10
4
II
611.05
4.8854× 10
4
(11)
http://novasinergia.unach.edu.ec 86
oscillates during the recovery of the disturbance,
showing that technique II again has better
performance than technique I.
Table 3: Performance Index results to parameter variation.
In this work, we also wanted to show the results in
the representation of radial charts, because radial
charts are the most effective when you are
comparing target vs achieved performance to a
standard or a group's performance. Figure 6 shows
the controllers’ performance to the reference
temperature changes for both techniques; thus, they
can be easily compared along their own axis, and the
global similarities are evident by the size and shape
of the polygons that are generated. Similarly, figure
7 shows the performance of the controllers by
decreasing the flow from 2 L/min to 1.3 L/min, and
overall differences are apparent by the size and
shape of the polygons.
Figure 5: System response to parameter variance.
(a)
(b)
Figure 6: Radial Chart representation for tracking. (a)
Output Process. (b) Control law.
(a)
(b)
Figure 7: Radial Chart representation for regulation. (a)
Output Process. (b) Output Controller.
5 Conclusion
In this study, two conventional sliding mode control
techniques selected for an experimental heat
exchanger system have been evaluated to investigate
the applicability of the proposed techniques. A first-
rate model with delay approaches its use in
Techniques SMC
IAE
IACI
I
968.03
3.6030 × 10
4
II
694.16
3.5860× 10
4
http://novasinergia.unach.edu.ec 87
experiments, as most real systems can be represented
by a reduced first-order model with delay. The step
response, the control signal and the variations of the
switching signal, error versus failure derived from
the error were obtained to compare the performances
of the techniques. During the experiments, the
parameters were tuned manually, as the presence of
chattering can cause a detrimental effect on the
system components.
According to the results and analysis in the time
domain tabulated in table 2 and table 3, it is clear that
the techniques presented in (Pérez-Pirela et al.,
2018), have produced better results than the
technique presented in (Camacho et al., 2000). Both
techniques have less chattering in the control signal
than can be acceptable for the actual systems. Since
the tracking error converges exponentially to zero
under uncertainties, the SMC techniques presented
in (Camacho et al., 2000) and (Pérez Pirela et al.,
2018) can be candidates for their use in industrial
applications as an alternative to the PID controller
commonly used. In addition, the first-order slider-
mode control algorithm has always systematic
solution. So, it is easy to understand and apply to real
systems.
Conflict of interest
No potential conflict of interest was reported by the
authors.
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