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Research Article
Sliding mode controllers for a tempered glass furnace
Naif B. Almutairi n, Mohamed Zribi
Electrical Engineering Department, College of Engineering and Petroleum, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
art ic l e i nf o
a b s t r a c t
Article history:
Received 2 January 2015
Received in revised form
5 October 2015
Accepted 5 November 2015
This paper investigates the design of two sliding mode controllers (SMCs) applied to a tempered glass
furnace system. The main objective of the proposed controllers is to regulate the glass plate temperature,
the upper-wall temperature and the lower-wall temperature in the furnace to a common desired temperature. The first controller is a conventional sliding mode controller. The key step in the design of this
controller is the introduction of a nonlinear transformation that maps the dynamic model of the tempered glass furnace into the generalized controller canonical form; this step facilitates the design of the
sliding mode controller. The second controller is based on a state-dependent coefficient (SDC) factorization of the tempered glass furnace dynamic model. Using an SDC factorization, a simplified sliding
mode controller is designed. The simulation results indicate that the two proposed control schemes work
very well. Moreover, the robustness of the control schemes to changes in the system's parameters as well
as to disturbances is investigated. In addition, a comparison of the proposed control schemes with a
fuzzy PID controller is performed; the results show that the proposed SDC-based sliding mode controller
gave better results.
& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords:
Sliding mode control
Temperature control
Furnace system
Tempered glass
State-dependent form
1. Introduction
This paper deals with the temperature control of a glass furnace
system which is used in the tempered glass production. Tempered
glass is a safety glass processed by a controlled thermal or chemical treatment to increase its strength compared with the
ordinary (or annealed) glass. In general, tempered glass is about
four times stronger than the normal glass. This property of tempered glass causes the broken glass to crumble into small granular
chunks rather than splintering into jagged shards. As a result of its
strength, tempered glass is used in environments where safety is
an important issue. Applications of tempered glass include the
side and the rear windows of vehicles, entrance doors, shower and
tub enclosures, racquetball courts, patio furniture, microwave
ovens and skylights, etc. [1].
Tempered glasses can be made from annealed glass via a
thermal tempering process. In the thermal tempering process of
ordinary glass, first the glass must be cut into the desired size.
Then, the glass is examined for imperfections that could cause the
breakage at any step during the tempering process. Next, the glass
begins a heat treatment process in which it travels through a
tempering furnace either in a batch or in a continuous feed. The
furnace heats the glass to a temperature above 600 [K]. After that,
n
Corresponding author. Tel.: þ 965 2498 5845; fax: þ965 2481 7451.
E-mail address: naif.ku@ku.edu.kw (N.B. Almutairi).
the glass undergoes a high-pressure cooling procedure called
quenching. During this process, which lasts few seconds, a highpressure air blasts the surface of the glass from an array of nozzles
in varying positions. The air cools the outer surfaces of the glass
quicker than the center. As the center of the glass cools, it tries to
pull back from the outer surfaces. As a result, the center remains in
tension, and the outer surfaces go into compression, which gives
the tempered glass its strength. Another approach for making
tempered glass is chemical tempering. In this tempering method,
various chemicals exchange ions on the surface of the glass in
order to create compression. But because of its higher cost when
compared to the tempering method using furnaces and quenching,
chemical tempering is not widely used [1].
The temperature control task in the tempered glass manufacturing process is the main part in the production process of
tempered glass and it has an important influence on the quality of
the products. Therefore, to produce tempered glass with a high
quality, temperature in the glass tempering furnace must be
controlled very well. Inadequate temperature control in the tempering process will lead to some product defects. Thus, in order to
successfully control the temperature in the tempered furnace, the
glass must be heated to a given temperature quickly. In addition,
the temperature difference between the various parts of the glass
surface must be very small, i.e., the glass surface must be heated
evenly. In the tempered glass furnace industry, the widely used
controller for temperature control is the conventional PI controller.
http://dx.doi.org/10.1016/j.isatra.2015.11.005
0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Almutairi NB, Zribi M. Sliding mode controllers for a tempered glass furnace. ISA Transactions (2015), http:
//dx.doi.org/10.1016/j.isatra.2015.11.005i
2
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Recent implementations of fuzzy logic controllers in temperature
control of tempered glass furnace can be found in [2]. A nonlinear
feedback linearization controller applied to a tempered glass furnace is given in [3], while an application of a sliding mode controller to the temperature control problem of a tempered furnace
is given in [4].
During the last few decades, sliding mode control has received
significant interest and has become a well-established research
area with a great potential for practical applications. Sliding mode
control can be applied to different types of systems provided that a
mathematical model of the controlled process is available. The
theoretical development aspects of SMC are well documented in
the literature (for example see [5]). The discontinuous nature of
the control action in SMC is claimed to result in outstanding
robustness features including insensitivity to parameter variations
and rejection of disturbances for both system stabilization and
system tracking problems. In addition, the sliding mode control
technique provides a systematic approach to the problem of
maintaining stability and good performance in the face of modelling uncertainty.
The following paragraph highlights the results of some applications of the sliding mode control technique to different types of
chemical processes. In [6,7], the application of dynamic sliding
mode controllers to a continuously stirred tank reactor are proposed. In [8], a sliding mode controller applied to an industrial high
temperature furnace is proposed; a linear low-order mathematical
model of the furnace is used in the design of this controller.
Applications of sliding mode controllers based on a first-order plus
dead-time model of some chemical processes are proposed in [9–
11]; the authors used a mixing tank process and a chemical rector as
examples in their works. Another sliding mode controller based on
first-order plus dead-time model of a temperature control of a
water tank system is proposed in [12]. In [13], the application of a
sliding mode controller based on a first-order plus dead-time model
is applied to an isolated chemical reactor. A sliding mode controller
based on a second-order plus dead-time model is applied to some
chemical processes in [14]. A fuzzy sliding mode controller applied
to chemical processes is proposed in [15]. A high-order sliding
mode scheme is used to control a chemical rector in [16]. Another
high-order sliding mode controller is used to regulate the temperature of a heat exchanger process in [17]. Recently, the sliding
mode control technique is applied to tubular a photo-bioreactor
[18] and to a solar power plant [19].
In the following, we survey some of the work on temperature
control of furnaces. Several researchers investigated the problem
of controlling the temperatures in different types of furnaces. A
hybrid supervisory control system is designed for an industrial
reheat furnace in [20]. This hybrid controller can replace the
experienced operator in choosing the best temperature set-point
value at each operating point of the furnace. Another hybrid
intelligent controller applied to a shaft furnace is proposed in [21].
The proposed control scheme consists of six controllers, namely a
pre-setting controller, a feedback controller, a prediction controller, a prediction based feedforward controller, and a fault
diagnosis fault tolerant controller. A multi-mode intelligent controller is proposed in [22]; the controller is a combination of fuzzy
control, bang-bang control, feedforward control, expert control
and PID control. An adaptive controller used to control the temperature of a solar furnace is proposed in [23]. A feedback linearization scheme based on generalized predictive control for a solar
furnace is proposed in [24]; the implementation results are given
in [25]. A predictive control scheme applied to a rotary furnace is
presented in [26]. The application of a nonlinear model predictive
controller to an induction heating furnace can be found in [27].
The design and implementation of a control system applied to a
vacuum diffusion furnace is proposed in [28]. The proposed
control scheme consists of four controllers, namely a temperature
PID controller, a gas flow tracking controller, a tracking controller
and a furnace vacuum PI controller. Moreover, a cascaded PID
controller design for the heating furnace temperature control
problem is proposed in [29].
In this paper, we propose two sliding mode controllers to control
the temperatures of a glass furnace which is used in tempered glass
production. The main objectives in this work is to improve the
current performance of the controlled tempered glass furnace, and
to show that more sophisticated control schemes like SMC can be
applied to chemical processes. The paper is organized as follows.
The dynamic model of the tempered glass furnace system is presented in Section 2. The design of the first sliding mode controller is
presented in Section 3. The design of the second sliding mode
controller (the SDC-based) is developed in Section 4. The simulation
results are presented and discussed in Section 5. A comparison of
the proposed controllers with a fuzzy PID controller is presented in
Section 6. Finally the conclusion is given in Section 7.
Sometimes, the arguments of a function will be omitted in the
analysis when no confusion can arise.
2. Dynamic model of the tempered glass furnace
Fig. 1 depicts a schematic diagram of a typical horizontal
radiative furnace which is used for tempered glass production.
Assuming that the temperature in the whole of the glass and
the upper and lower walls changes uniformly and that the conductive heat transfer is negligible, the dynamic model of this
radiative furnace can be written as [4],
T_ p ðtÞ ¼ a1 ðT 4w ðtÞ T 4p ðtÞÞ
h
i
T_ w ðtÞ ¼ a2 T 4w ðtÞ F wp T 4p ðtÞ ð1 F wp ÞT 4B ðtÞ þ αPðtÞ
T_ B ðtÞ ¼ a3 ðT 4w ðtÞ T 4B ðtÞÞ
ð1Þ
where T p ðtÞ, T w ðtÞ and T B ðtÞ are the absolute temperatures of the
glass plate, the upper wall and the lower wall, respectively; the
unit of these temperatures is Kelvin [K]. It is worth mentioning
that the Kelvin scale is an absolute measurement scale for temperatures. Thus, the absolute temperature of the glass plate, the
upper wall and the lower wall are all positive quantities. The input
electric power (in Watts) is PðtÞ. The parameters of the system a1 ,
a2 and a3 are as follows,
a1 ¼ βσ F wp Aw
a2 ¼ ασ Aw
a3 ¼ ασ Aw ð1 F wp Þ
ð2Þ
where σ is the Stefan–Boltzmann constant and it is equal to
5:6697 10 8 W=m2 U K4 ; Aw is the area of the wall (in m2); F wp
is the shape factor between the upper wall and the plate; α and β
are the warming rates of the walls and the plate respectively.
Define the following state and input variables as follows,
x1 ðtÞ ¼ T p ðtÞ
x2 ðtÞ ¼ T w ðtÞ
x3 ðtÞ ¼ T B ðtÞ
uðtÞ ¼ PðtÞ
ð3Þ
Let xðtÞ ¼ x1 ðtÞ x2 ðtÞ x3 ðtÞ . Then, the dynamic model of the
tempered glass furnace system can be written as follows,
x_ 1 ðtÞ ¼ a1 ðx42 ðtÞ x41 ðtÞÞ
x_ 2 ðtÞ ¼ a2 ðx42 ðtÞ F wp x41 ðtÞ ð1 F wp Þx43 ðtÞÞ þ αuðtÞ
x_ 3 ðtÞ ¼ a3 ðx42 ðtÞ x43 ðtÞÞ
ð4Þ
Remark 1. The mathematical model of this furnace system has a
unique equilibrium point given by ðT pe ; T we ; T Be Þ ¼ ðxe ; xe ; xe Þ where
Please cite this article as: Almutairi NB, Zribi M. Sliding mode controllers for a tempered glass furnace. ISA Transactions (2015), http:
//dx.doi.org/10.1016/j.isatra.2015.11.005i
N.B. Almutairi, M. Zribi / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
xe is the equilibrium absolute temperature of the three states of
the system.
In this section, we will use a sliding mode controller to control
the tempered glass furnace. The key step in this design is the
introduction of a nonlinear transformation that maps the state
equations of the tempered glass furnace model into the generalized controller canonical form; this step facilitates the design of
the controller.
3.1. Transformation of the model of the system
Define the new state variables y1 ðtÞ, y2 ðtÞ and y3 ðtÞ such that,
x3 ðtÞ x1 ðtÞ
a3
a1
y2 ðtÞ ¼ x41 ðtÞ x43 ðtÞ
y3 ðtÞ ¼ 4a1 x31 ðtÞðx42 ðtÞ x41 ðtÞÞ þ 4a3 x33 ðtÞðx43 ðtÞ x42 ðtÞÞ
ð5Þ
Remark 2. The transformation in Eq. (5) can be written as follows,
2 3
2
3
y1
T 1 ðxÞ
6y 7
6 T ðxÞ 7
y 4 2 5 ¼ TðxÞ ¼ 4 2
5
y3
T 3 ðxÞ
This transformation was derived using a vector field method
(see [30]). The derivation of the transformation can be summarized as follows:
1) The controllability matrix of the nonlinear system is obtained
using Lie brackets.
2) Then the last row of the inverse of the controllability matrix is
used to obtain T 1 .
3) Finally, T 2 and T 3 are obtained using the fact that: T 2 ¼ LF
T 1 and T 3 ¼ LF T 2 where LF T 1 represents the Lie derivative of
T 1 with respect to the vector field F and LF T 2 represents the Lie
derivative of T 2 with respect to the vector field F:
Using the transformation in Eq. (5), the dynamic model of the
tempered glass furnace using the new state variables is as follows,
y_ 1 ¼ y2
y_ 2 ¼ y3
þ 28a23 x63 ðx42 x43 Þ þ 16a2 x32 ða1 x31 a3 x33 Þ
ðx42 F wp x41 ð1 F wp Þx43 Þ
ð6Þ
Thus, the dynamic model of the tempered glass furnace in
Eq. (6) can be written in the following generalized controller
canonical form,
y_ 1 ¼ y2
y_ 2 ¼ y3
y_ 3 ¼ f ðxÞ þ gðxÞuðtÞ
where,
f ðxÞ ¼ 12a21 x21 x42 ðx42 x41 Þ þ 28a21 x61 ðx41 x42 Þ þ 12a23 x23 x42 ðx43 x42 Þ
þ 28a23 x63 ðx42 x43 Þ þ 16a2 x32 ða1 x31 a3 x33 Þ
ðx42 F wp x41 ð1 F wp Þx43 Þ
ð8Þ
Remark 3. At steady state, the values of the new state variables
are y1e ¼ yd , and y2e ¼ y3e ¼ 0; the constant yd is the desired value
of the new state variable y1 ðtÞ. This desired value is related to the
desired absolute temperature xd such that,
1 1
ð9Þ
x
yd ¼
a3 a1 d
Assumption 1. The term gðxÞ in Eq. (7) is different from zero.
Remark 4. It can be shown that the term gðxÞ in Eq. (7) is never
zero for the furnace model under consideration. Using Eq. (2), we
can write gðxÞ as follows,
αð1 F wp Þ 3
x3
ð10Þ
gðxÞ ¼ 16αx32 ða1 x31 a3 x33 Þ ¼ 16αx32 a3 x31 βF wp
For gðxÞ to equal zero, either x2 ¼ 0 [K], or
"sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#
βF wp
3
x3 ¼
x cx1
αð1 F wp Þ 1
ð11Þ
Recall that the absolute temperature of the upper wall, x2 ðtÞ, is
always positive. Also, given the data of the tempered glass furnace
system in [4] which is listed in Table 1, the parameter c in Eq. (11)
equals 2.17. This means that the temperature of the lower wall is
more than twice the temperature of the glass plate. This is not
physically possible. Hence, it can be concluded that gðxÞ a0 under
normal operating conditions. Therefore, Assumption 1 is valid.
3.2. Controller design
In this subsection, a sliding mode control scheme is designed.
When the system is in the generalized controller canonical form,
the simplest choice of sliding surface is a linear surface.
Let α1 , α2 and W 1 be positive scalars. Define the sliding surface
s1 such that,
s1 ¼ y3 þ α2 y2 þ α1 ðy1 yd Þ
y_ 3 ¼ 12a21 x21 x42 ðx42 x41 Þ þ 28a21 x61 ðx41 x42 Þ þ 12a23 x23 x42 ðx43 x42 Þ
þ 16αx32 ða1 x31 a3 x33 ÞuðtÞ
gðxÞ ¼ 16αx32 ða1 x31 a3 x33 Þ
The dynamical model in Eqs. (7) and (8) will be used in the
design of the first sliding mode control scheme for the furnace.
3. Design of the first sliding mode controller
y1 ðtÞ ¼
3
ð12Þ
It should be noted that the sliding surface s1 is highly nonlinear
when it is written in the original coordinates. The sliding surface s1
can be written in the original coordinates as follows,
s1 ¼ 4a1 x31 ðtÞðx42 ðtÞ x41 ðtÞÞ þ 4a3 x33 ðtÞðx43 ðtÞ x42 ðtÞÞ þ α2 ðx41 ðtÞ x43
ðtÞÞ þ α1
x3 ðtÞ x1 ðtÞ
a3 a1 yd
8
>
< þ1
0
sgnðs1 Þ ¼
>
: 1
. Also define the signum function such that,
if s1 4 0
if s1 ¼ 0
if s1 o0
The following proposition gives the first result of the paper.
ð7Þ
Proposition 1. The sliding mode controller:
1
uðtÞ ¼
ð12a21 x21 x42 ðx42 x41 Þ þ28a21 x61 ðx41 x42 Þ
16αx32 ða1 x31 a3 x33 Þ
þ12a23 x23 x42 ðx43 x42 Þ þ28a23 x63 ðx42 x43 Þ þ α2 y3 þ α1 y2
þW 1 sgnðs1 Þ þ 16a2 x32 ða1 x31 a3 x33 Þðx42 F wp x41 ð1 F wp Þx43 ÞÞ
ð13Þ
Please cite this article as: Almutairi NB, Zribi M. Sliding mode controllers for a tempered glass furnace. ISA Transactions (2015), http:
//dx.doi.org/10.1016/j.isatra.2015.11.005i
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4
Table 1
Values of the parameters of the tempered glass furnace.
Parameter
F wp
Aw
α
β
Value
0.7
1
8 10 5
35 10 5
when applied to the model of the tempered glass furnace system
given by Eq. (4), stabilizes the states x1 ðtÞ, x2 ðtÞ and x3 ðtÞ to their
common desired value xd .
Proof
Taking the time derivative of s1 in Eq. (12) and using Eq. (6),
one obtains,
s_ 1 ¼ y_ 3 þ α2 y_ 2 þ α1 y_ 1 ¼ 12a21 x21 x42 ðx42 x41 Þ þ 28a21 x61 ðx41 x42 Þ
Fig. 1. Schematic diagram of a horizontal radiative furnace [4].
Therefore,
lim y2 ðtÞ ¼ 0 ) lim ðx41 ðtÞ x43 ðtÞÞ ¼ 0
þ 12a23 x23 x42 ðx43 x42 Þ þ 28a23 x63 ðx42 x43 Þ þ 16a2 x32 ða1 x31 a3 x33 Þðx42
t-1
t-1
) lim x41 ðtÞ lim x43 ðtÞ ¼ 0
F wp x41 ð1 F wp Þx43 Þ þ 16αx32 ða1 x31 a3 x33 ÞuðtÞ þ α2 y3 þ α1 y2
ð14Þ
Using the controller given by Eq. (13) into the above equation, it
follows that,
s_ 1 ¼ W 1 sgnðs1 Þ
ð15Þ
By using the Lyapunov function V 1 ¼ 12 s21 , it can be easily
checked that the dynamics in Eq. (15) guarantees that s1 s_ 1 o0
(when s1 a0), which is the condition needed to guarantee
reaching the sliding surface s1 ¼ 0. Integrating Eq. (15) with
respect to time gives us the reaching time t r1 . The time t r1
represents the required time for the error signals to reach the
sliding surface s1 . Therefore, the reaching time for the controller
1 ð0Þj
. Clearly the choice of W 1
given by Proposition 1 is t r1 ¼ j sW
1
influence how fast the error signals reaches the sliding surface.
Thus, the trajectories associated with the unforced discontinuous
dynamics in Eq. (15) exhibit a finite time reachability to zero from
any given initial condition s1 ð0Þ, provided that the constant W 1 is
sufficiently large and positive. Therefore, it can be concluded that
s1 converges to zero in finite time.
Next, we will examine the dynamics of the system on the
sliding surface. From Eq. (12), and when s1 ¼ 0, one obtains,
y3 ¼ α2 y2 α1 ðy1 yd Þ
ð16Þ
Using Eq. (6), the dynamic model of the reduced-order system
on the sliding surface is governed by,
t-1
t-1
ð20Þ
which gives,
lim x1 ðtÞ ¼ lim x3 ðtÞ
t-1
ð21Þ
t-1
since both x1 ðtÞ and x3 ðtÞ are positive quantities.
Following the same approach, and using the result of Eq. (21), it
follows that,
lim y3 ðtÞ ¼ 0 ) lim ð4a1 x31 ðx42 x41 Þ þ 4a3 x33 ðx43 x42 ÞÞ
t-1
t-1
¼ 0 ) 4ð a1 þ a3 Þ lim ðx33 ðx43 x42 ÞÞ ¼ 0
t-1
ð22Þ
Recall that x3 ðtÞ, a1 , and a3 are all positive quantities, this
implies that,
lim ðx43 x42 Þ ¼ 0
ð23Þ
t-1
which gives,
lim x2 ðtÞ ¼ lim x3 ðtÞ
t-1
ð24Þ
t-1
since both x2 ðtÞ and x3 ðtÞ are positive quantities.
Finally, since y1 ðtÞ converges to yd asymptotically, we have,
x3 ðtÞ x1 ðtÞ
lim y1 ðtÞ ¼ yd ) lim
t-1
t-1
a3
a1
x1 ðtÞ
x1 ðtÞ
lim
t-1 a1
a3
1 1
¼ yd )
lim x1 ðtÞ ¼ yd
a3 a1 t-1
¼ yd ) lim
t-1
y_ 1 ¼ y2
y_ 2 ¼ α2 y2 α1 ðy yd Þ
ð17Þ
Let e1 ðtÞ ¼ y1 ðtÞ yd and e2 ðtÞ ¼ y2 ðtÞ y2f ¼ y2 ðtÞ. Then, Eq. (17)
can be written as,
" # "
#" #
0
1
e1
e_ 1
¼
ð18Þ
e2
α1 α2
e_ 2
The characteristic equation of the linear system in Eq. (18) is
given by,
Δ1 ð λ Þ ¼ λ þ α 2 λ þ α 1 ¼ 0
2
which gives,
lim x1 ðtÞ ¼ h
t-1
The choice of the zeroes of the characteristic equation in Eq.
(19) (from which we can obtain the values of the gains α1 and α2 )
affects the speed at which the error signals e1 ðtÞ and e2 ðtÞ converge
to zero. Thus, by choosing α1 and α2 to be positive constants, we
are guaranteed that the errors e1 ðtÞ and e2 ðtÞ converge to zero
asymptotically. Hence, y1 ðtÞ yd and y2 converge to zero asymptotically. Moreover, using Eq. (16), we are guaranteed that y3 ðtÞ will
also converge to zero asymptotically.
yd
1
1
a3 a1
i xd
ð26Þ
Thus, from Eqs. (21), (24) and (26), we conclude that,
lim x1 ðtÞ ¼ lim x2 ðtÞ ¼ lim x3 ðtÞ ¼ xd
t-1
ð19Þ
ð25Þ
t-1
t-1
ð27Þ
This completes the proof.
A Simulink model showing the control structure of the proposed controller is given in Fig. 2.
Although it is numerically feasible to implement the proposed
sliding mode controller given by Eq. (13), it is of interest to
develop a control scheme which is less computationally intensive
and at the same time has the features of the SMC technique. In the
following section, a second sliding mode controller which is based
on the SDC factorization is proposed.
Please cite this article as: Almutairi NB, Zribi M. Sliding mode controllers for a tempered glass furnace. ISA Transactions (2015), http:
//dx.doi.org/10.1016/j.isatra.2015.11.005i
N.B. Almutairi, M. Zribi / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
5
the nonlinear system [31]. Factorizing the mathematical model of
the tempered glass furnace given by Eq. (4), we obtain the following state-dependent model of the furnace system,
_ ¼ AðxÞxðtÞ þ BuðtÞ
xðtÞ
ð28Þ
where,
2
2
3
x1 ðtÞ
6 x ðtÞ 7
xðtÞ ¼ 4 2 5;
x3 ðtÞ
a1 x31
6 a F
3
AðxÞ ¼ 4
2 wp x1
0
a1 x32
0
a2 x32
a3 x32
a2 ð1 F wp Þx32
a3 x33
2 3
0
6 7
7
5; B ¼ 4 α 5
0
3
ð29Þ
3
33
31
Here x A ℜ is the state vector, AðxÞ A ℜ
and B A ℜ
are the
SDC matrices. It should be mentioned that the SDC factorization in
Eqs. (28) and (29) is not unique.
Motivated by the work done in [32], the SMC design method for
linear time invariant (LTI) systems is applied to the SDC factorized
model of the furnace system. The SMC design for LTI systems was
studied thoroughly in the literature, for example the reader can
refer to the work done in [33]. In order to design a sliding mode
control for an LTI system, the system is transformed so that it can
be separated into two parts such that one of the parts is so-called
reduced-order form in which the control inputs are absent. In
order to do so, the change of variables z1 ðtÞ ¼ x1 ðtÞ, z2 ðtÞ ¼ x3 ðtÞ,
z3 ðtÞ ¼ x2 ðtÞ will be used. Then, the new SDC form of the furnace
system is given by,
z_ ¼ FðzÞz þGu
ð30Þ
where,
2
3
z1
6z 7
z ¼ 4 2 5;
z3
2
6
FðzÞ ¼ 4
a1 z31
0
a2 F wp z31
0
a3 z32
a2 ð1 F wp Þz32
3
a1 z33
37
a3 z3 5;
a2 z33
2
3
0
607
G¼4 5
α
ð31Þ
The SDC factorized model given by Eqs. (30) and (31) will be
used in the design the second sliding mode control scheme.
4.2. Controller design
This subsection proposes a design of an SDC-based sliding
mode control scheme to the tempered glass furnace system. The
control scheme is designed based on the SDC factorization of the
system given in Eqs. (30) and (31).
Since the model of the system is written in a linearly structured
mathematical form with state-dependent coefficients, then the
simplest choice of sliding surface is a linear surface. Let c1 , c2 , and
W 2 be positive scalars. Also, let the sliding surface s2 be such that,
s2 ¼ ðz3 xd Þ þ c1 ðz1 xd Þ þ c2 ðz2 xd Þ
¼ ðx2 xd Þ þ c1 ðx1 xd Þ þ c2 ðx3 xd Þ
Fig. 2. A Simulink model showing the control structure of the first SMC.
4. Design of an SDC-based sliding mode controller
In this section, an SDC-based sliding mode controller applied to
the tempered glass furnace is proposed. The proposed controller is
based on the state-dependent factorization of the dynamical
model of the tempered glass furnace.
4.1. SDC Factorization of the tempered glass furnace system
ð32Þ
The following proposition gives the second result of the paper.
Proposition 2. The sliding mode controller,
1
uðtÞ ¼ ða2 F wp þ c1 a1 Þx41 ða2 þ c1 a1 þ c2 a3 Þx42
α
þ ðc2 a3 þa2 ð1 F wp ÞÞx43 W 2 sgnðs2 Þ
ð33Þ
when applied to the model of the furnace system given by Eq. (4),
stabilizes the states x1 ðtÞ, x2 ðtÞ and x3 ðtÞ to their common
desired value.
Proof
Taking the time derivative of s2 in Eq. (32) and using Eq. (4),
one obtains,
s_ 2 ¼ x_ 2 þ c1 x_ 1 þ c2 x_ 3
Rearrangement of a set of nonlinear differential equations
describing a system in a linearly structured mathematical model
with state-dependent coefficients is called an SDC factorization of
¼ a2 ðx42 F wp x41 ð1 F wp Þx43 Þ þ αu þc1 a1 ðx42 x41 Þ þc2 a3 ðx42 x43 Þ
ð34Þ
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Using the controller given by Eq. (33) into the above equation,
its follows that,
s_ 2 ¼ W 2 sgnðs2 Þ
ð35Þ
Using the Lyapunov function V 2 ¼ 12 s22 , it can be easily checked
that the dynamics in Eq. (35) guarantees that s2 s_ 2 o 0 (when s2 a 0),
which is the condition needed to guarantee reaching the sliding
surface s2 ¼ 0. Integrating Eq. (35) with respect to time gives us the
reaching time t r2 . The time t r2 represents the required time for the
error signals to reach the sliding surface s2 . Therefore, the reaching
2 ð0Þj
time for the controller given by Proposition 2 is t r2 ¼ j sW
. Clearly
2
the choice of W 2 influence how fast the error signals reaches the
sliding surface. Thus, the trajectories associated with the unforced
discontinuous dynamics in Eq. (35) exhibit a finite time reachability
to zero from any given initial condition s2 ð0Þ, provided that the
constant W 2 is sufficiently large and positive. Therefore, it can be
concluded that s2 converges to zero in finite time.
Next, we will examine the dynamics of s2 on the sliding surface.
Define the following error variables,
e1 ðtÞ ¼ x1 ðtÞ xd
ð36Þ
e2 ðtÞ ¼ x3 ðtÞ xd
ð37Þ
Then, the dynamics of the system on the sliding surface s2 ¼ 0
in Eq. (32) is given by,
x2 ¼ xd ðc1 e1 þ c2 e2 Þ
ð38Þ
Hence, the dynamic model of the system on the sliding surface
s2 ¼ 0 is given by,
e_ 1 ¼ a1 ðxd ðc1 e1 þc2 e2 ÞÞ4 a1 ðe1 þ xd Þ4
ð39Þ
e_ 2 ¼ a3 ðxd ðc1 e1 þc2 e2 ÞÞ4 a3 ðe2 þ xd Þ4
ð40Þ
The system of ordinary differential equations given by Eqs. (39)
and (40) can be written in a compact form as,
e_ ¼ f ðeÞ
where,
"
f ðeÞ ¼
ð41Þ
a1 ðxd ðc1 e1 þ c2 e2 ÞÞ4 a1 ðe1 þ xd Þ4
a3 ðxd ðc1 e1 þ c2 e2 ÞÞ4 a3 ðe2 þ xd Þ4
#
"
;
e¼
e1
e2
#
ð42Þ
We want to study the local stability of the nonlinear system of
ordinary differential equations given by Eq. (41). It can be shown
that the equilibrium point of this autonomous system is the origin
e ¼ 0. When this system is linearized around this equilibrium
point, we obtain the following linearized system,
e_ ¼ Ae
where,
A¼
∂f
j
¼
∂e e ¼ 0
ð43Þ
"
4a1 ð1 þ c1 Þx3d
4a1 c2 x3d
4a3 c1 x3d
4a3 ð1 þ c2 Þx3d
#
ð44Þ
The characteristic equation of the linearized system in Eq. (43)
is,
Δ2 ðλÞ ¼ λ2 þ 4x3d ða1 þ a3 þ a1 c1 þ a3 c2 Þλ þ 16a1 a3 x6d ð1 þ c1 þ c2 Þ ¼ 0
ð45Þ
The necessary and sufficient condition for the local stability of
the error system given by Eq. (43) is,
4x3d ða1 þ a3 þ a1 c1 þ a3 c2 Þ 4 0
16a1 a3 x6d ð1 þ c1 þ c2 Þ 4 0
ð46Þ
These conditions are satisfied because that the parameters of the
system a1 , a2 , a3 and the design parameters, c1 and c2 , are all
positive; also the desired state xd is positive. Therefore it can be
concluded that e1 ðtÞ and e2 ðtÞ converge to zero. Thus, the linearized
Fig. 3. A Simulink model showing the control structure of the SDC-based SMC.
system is stable. Therefore x1 ðtÞ and x3 ðtÞ converge to the desired
value xd . In addition, since e1 ðtÞ and e2 ðtÞ converge to zero as t tends
to infinity, Eq. (38) implies that x2 ðtÞ converges to xd as t tends to
infinity. Therefore, it can be concluded that all the states of the
system converge to their desired values as t tends to infinity. This
completes the proof.
It should be mentioned that stability of the closed loop system
when using the SDC-based SMC controller is local. The control
structure of the proposed SDC-based SMC is shown in Fig. 3.
Remark 5 The essential difference between the two sliding
mode controllers originates from the selected sliding surfaces. The
first selected sliding surface is highly nonlinear, whereas the second selected sliding surface is linear. The structures of the two
controllers are different. The first controller is based on the usage
of a transformation that reduces the system to the generalized
controller canonical form. The second sliding mode controller is
designed based on writing the model of the system using a statedependent coefficient factorization. It is clear from equations Eqs.
(12) and (13) and Eqs. (32) and (33) that the second sliding mode
controller is simpler and less computationally intensive than the
first sliding mode controller. Hence the SDC-based controller is
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Fig. 4. The profile of the desired temperature versus time.
Fig. 5. The temperatures versus time when using the first SMC.
easier to implement than the first controller. However, it should be
mentioned that the stability of the closed loop system when using
the SDC-based sliding mode controller is local.
5. Simulation results
The controllers designed in Sections 3 and 4 are simulated
using the MATLAB software [34]. The parameters of the tempered
glass furnace system are listed in Table 1.
It should be mentioned that the power of the electrical system
of the furnace could vary from zero to 16 KW. Hence, to obtain
realistic results, the simulations are carried out using the following
input electric power constraint,
0 r uðtÞ r 16 KW
ð47Þ
The initial temperatures used in all simulations are T p ð0Þ ¼ 300
[K], and T w ð0Þ ¼ T B ð0Þ ¼ 500 [K]. To show the effectiveness of the
proposed control schemes, the desired temperature profile, xd , has
three different set point values. First the desired temperature is set
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Fig. 6. The control signal versus time when using the first SMC.
Table 2
Settling times (in minutes) when using the first SMC.
Set point value
xd ¼ 600 [K]
xd ¼ 800 [K]
xd ¼ 1000 [K]
Glass plate temperature, T p ðtÞ
Upper wall temperature, T w ðtÞ
Lower wall temperature, T B ðtÞ
Ts ¼18.63
Ts ¼17.20
Ts ¼19.85
Ts¼ 8.55
Ts¼ 7.73
Ts¼ 24.47
Ts¼ 19.12
Ts¼ 18.78
Ts¼ 22.98
to 600 [K], then it changes from 600 [K] to 800 [K], and finally
from 800 [K] to 1000 [K] as shown in Fig. 4.
system is such that,
5.1. Simulation results when using the first SMC
which is negative definite (for s1 a 0). Thus, the trajectories associated
with the unforced discontinuous dynamics in Eq. (48) exhibit a finite
time reachability to zero from any given initial condition s1 ð0Þ, provided that the constants K 1 and W 1 are positive. Therefore, it can be
concluded that s1 converges to zero in finite time.
Using the reaching law in Eq. (48), a term K 1 s1 will be added to
the first sliding mode controller given by Eq. (13). To show that
using the reaching law in Eq. (48) reduces the chattering in the
control signal while keeping the value of W 1 small, we simulate
the system using W 1 ¼ 10, and K 1 ¼ 1. The simulation results are
shown in Figs. 7 and 8.
Comparing the results in Figs. 5 and 6 and the results in
Figs. 7 and 8, it is noticed that the chattering is greatly reduced and
the time response of the system is almost unchanged as seen in
Table 3.
The simulations of the first sliding mode controller when it is
applied to the tempered glass furnace are carried out using
MATLAB. The controller parameters, α1 and α2 , are selected such
that the roots of the characteristic equation in Eq. (19) are at
0.01 and 0.002. The weighting parameter used is taken to be
W 1 ¼ 107 . The simulation results are presented in Figs. 5 and 6.
The choice of W 1 influences how fast the sliding surface s1 ¼ 0
is reached. The choice of the parameters α1 and α2 influence how
fast the error signals converge to zero. After some tuning of the
values of the control parameters W 1 and α1 , α2 , we obtained
responses of error signals which converge to zero in a reasonably
fast manner. This means that the sliding surface s1 ¼ 0 is reached
quickly (the attraction of the sliding surface is quite fast) and after
reaching the sliding surface s1 ¼ 0, the error signals converge to
zero in a well behaved manner. Note that the parameters α1 and
α2 are taken to be 2 10 5 and 0.012 respectively. These parameters result a linearized system with a settling time of about
40 (min).
It can be seen from Fig. 5 that the glass plate absolute temperature,
the upper wall absolute temperature and the lower wall absolute
temperature reached their desired values in a reasonable time for the
three different values of xd . The settling times are summarized in
Table 2.
As shown in Fig. 6, the proposed control scheme suffers from the
chattering problem. The chattering is due to the assumption that the
control signal can be switched from one value to another at any
moment and with almost zero time delay. Also, it is noticed that the
value of W 1 is quite large. Therefore, to reduce the chattering in the
control signal we need to reduce the value of W 1 . This can be achieved
by replacing the reaching law given by Eq. (15) with the following
reaching law,
s_ 1 ¼ K 1 s1 W 1 sgnðs1 Þ
ð48Þ
where K 1 is a positive design parameter. To show that this reaching
law exhibit a finite time reachability to zero from any given initial
condition s1 ð0Þ, we again use the Lyapunov function V 1 ¼ 12 s21 ; the
time derivative of V 1 with respect to the trajectories of the furnace
V_ 1 ¼ s1 s_ 1 ¼ K 1 s21 W 1 js1 j
ð49Þ
5.2. Simulation results of the SDC-based sliding mode controller
The SDC-based sliding mode controller given by Eq. (33) is
applied to the tempered glass furnace system in Eq. (4). The
parameters of the sliding surface in Eq. (32) are taken to be c1 ¼
0:25 and c2 ¼ 0:15; while the constant W 2 is 1.0. The simulation
results are presented in Figs. 9 and 10.
The choice of W 2 influences how fast the sliding surface s2 ¼ 0 is
reached. The choice of the parameters c1 and c2 influence how fast the
error signals converge to zero. After some tuning of the values of the
control parameters W 2 and c1 , c2 , we obtained responses of error
signals which converge to zero in a reasonably fast manner. This
means that the sliding surface s2 ¼ 0 is reached quickly. After reaching
the sliding surface s2 ¼ 0, the error signals converge to zero in a well
behaved manner. Note that the parameters c1 and c2 are taken to be
0.25 and 0.15 respectively. These parameters result a linearized system
with a settling time of about 40 (min).
It can be seen from Fig. 9 that the three absolute temperatures
reached their desired values in a reasonable time. The results are
summarized in Table 4.
Clearly, the simulation results show that the proposed SDCbased SMC suffers slightly from chattering. In order to eliminate
this undesirable chattering, we have used the same reaching law
Please cite this article as: Almutairi NB, Zribi M. Sliding mode controllers for a tempered glass furnace. ISA Transactions (2015), http:
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Fig. 7. Simulation reults when using the first SMC with a different reaching law.
Fig. 8. Control signal of the first SMC with a different reaching law.
used in eliminating the chattering in the first sliding mode controller. Figs. 11 and 12 depict the simulation results when the SDCbased SMC with the reaching law given by Eq. (48) is used.
Fig. 11 and Table 5 indicate that the performance of the absolute temperatures of this controller is similar to the performance
of the controller in Eq. (33). However, a comparison between
Figs. 10 and 12 indicates a great reduction in the chattering of the
control signal (Table 5).
For comparison purposes, on the same graph, we plotted the
plate glass absolute temperatures of the controlled tempered glass
furnace for the two applications of the proposed control schemes.
The results are shown in Fig. 13.
Please cite this article as: Almutairi NB, Zribi M. Sliding mode controllers for a tempered glass furnace. ISA Transactions (2015), http:
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Table 3
Settling times (in minutes) when using the first SMC with a different reaching law.
Set point value
xd ¼ 600 [K]
xd ¼800 [K]
xd ¼ 1000 [K]
Glass plate temperature, T p ðtÞ
Upper wall temperature, T w ðtÞ
Lower wall temperature, T B ðtÞ
Ts¼ 18.30
Ts¼ 16.90
Ts¼ 19.65
Ts ¼7.28
Ts ¼6.45
Ts ¼23.63
Ts¼ 17.23
Ts¼ 16.90
Ts¼ 21.10
Fig. 9. The temperatures versus time when using the SDC-based SMC.
Therefore, the simulation results indicate that the proposed
SDC-based sliding mode controller gave slightly better results
when compared to the first sliding mode controller.
5.3. Uncertainty in the warming rates
The performance of the tempered glass furnace is simulated
when some of the parameters of the system are assumed not to be
known exactly. The nominal values of the warming rates of the
walls and the plate are 8 10 5 and 35 10 5 , respectively. We
will repeat the simulations assuming that α is changed by þ 200%
from its nominal value while β is changed by 50% of its
nominal value.
The response of T p ðtÞ when the furnace system is controlled
using the first SMC (when α used in the mathematical model of
the system is increased by 200% and β is decreased by 50%) is
shown in Fig. 14 (solid line). The response of T p ðtÞ when the furnace system is controlled using the SDC-based SMC is shown in
Fig. 15 (solid line). In both figures, the response of the controlled
furnace system with the nominal values of α and β is plotted using
the dotted lines. It can be seen from these figures that the glass
plate absolute temperature converges to its desired value.
It should be mentioned that similar results were obtained for
the other two absolute temperatures of the furnace, namely the
absolute temperatures of the upper wall and the lower wall; these
figures are not included in the article because of space limitations.
The simulation results are summarized in Tables 6 and 7.
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Fig. 10. The control signal versus time when using the SDC-based SMC.
Table 4
Settling times (in minutes) when using the SDC-based SMC.
Set point value
xd ¼ 600 [K]
xd ¼ 800 [K]
xd ¼ 1000 [K]
Glass plate temperature, T p ðtÞ
Upper wall temperature, T w ðtÞ
Lower wall temperature, T B ðtÞ
Ts¼ 4.92
Ts¼ 5.42
Ts¼ 29.15
Ts¼ 4.43
Ts¼ 5.72
Ts¼ 17.87
Ts ¼4.27
Ts ¼3.93
Ts ¼9.77
Fig. 11. The temperatures versus time when using the SDC-based SMC with a different reaching law.
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Fig. 12. The control signal versus time when using the SDC-based SMC with a different reaching law.
Table 5
Settling times (in minutes) when using the SDC-based SMC with a different reaching law.
Set point value
xd ¼ 600 [K]
xd ¼800 [K]
xd ¼ 1000 [K]
Glass plate temperature, T p ðtÞ
Upper wall temperature, T w ðtÞ
Lower wall temperature, T B ðtÞ
Ts¼ 4.57
Ts¼ 5.07
Ts¼ 28.70
Ts ¼3.90
Ts ¼5.15
Ts ¼17.47
Ts¼ 3.98
Ts¼ 3.65
Ts¼ 9.50
Fig. 13. The plate glass absolute temperature of the tempered glass furnace using the proposed SMC control schemes.
Fig. 14. (a) The glass plate absolute temperature (solid line) using the first SMC when α is changed by þ 200% and β is changed by 50%; the dotted line represents the
nominal case. (b) The control signal versus time; the dotted line represents the nominal case.
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Therefore, it can be concluded that the proposed two sliding
mode control schemes are robust to changes in the warming rates
of the walls and the plate in the furnace system. It should be
mentioned that this is an expected result as it is well established
that sliding mode controllers are robust to uncertainties in some of
the system’s parameters.
5.4. Load disturbance rejection
In order to show the effectiveness of the developed sliding
mode controllers in the presence of an input disturbance in
addition to the previous uncertainties in the warming rates, a
reference signal of magnitude xd ¼600 [K] is used while a load step
disturbance of magnitude 4 KW is also applied to the system.
This load step disturbance is applied at t¼80 min (i.e., when the
system has reached the equilibrium).
Table 6
Settling times (in minutes) using the first SMC when α changes by þ 200% and β is
changed by 50%.
Set point value
xd ¼ 600 [K]
Glass plate temperature, T p ðtÞ
Upper wall temperature, T w ðtÞ
Lower wall temperature, T B ðtÞ
Ts¼ 9.65
Ts¼ 9.08
Ts¼ 16.92
Table 7
Settling times (in minutes) using the SDC-based SMC when α changes by þ 200%
and β is changed by 50%.
Set point value
xd ¼ 600 [K]
Glass plate temperature, T p ðtÞ
Upper wall temperature, T w ðtÞ
Lower wall temperature, T B ðtÞ
Ts¼ 8.3
Ts¼ 6.43
Ts¼ 11.43
13
The simulations of the controlled tempered glass furnace are
carried out when the two proposed sliding mode control schemes
are used. Fig. 16(a) shows the response of the furnace absolute
temperatures when the first sliding mode controller is used; it
shows that the proposed control scheme forces all absolute temperatures to reach their common desired value even with the
application of a load step disturbance. The control signal uðtÞ ¼ PðtÞ
(the input electric power to the furnace) is depicted in Fig. 16(b).
Note that the control signal adjusts its value to compensate for the
step load disturbance.
The response of the controlled furnace when using the SDCbased sliding mode controller is shown in Fig. 17. It is clear from
the responses of the absolute temperatures of the tempered glass
furnace that this control scheme is also robust against a load step
disturbance.
Comparing the responses of the proposed sliding mode control
schemes, we can see that both proposed control schemes are
robust against a load step disturbance as well as against uncertainties in the warming rates.
6. Comparison of the proposed controllers with a fuzzy PID
controller
Fuzzy control and conventional control have similarities and
differences. They are similar in that they try to solve the same type
of problems, and the mathematical tools used to analyze the
designed control systems are similar. However, the fundamental
difference between them is that the conventional control starts
with a mathematical model of the process and a controller is
designed for that model, while a fuzzy controller starts with
heuristics and human expertize (in terms of fuzzy IF-THEN rules)
and a controller is designed by synthesizing these fuzzy rules.
Due to the existence of nonlinearities in thermal tempering
processes, and the availability of human expertize for adjusting the
control process, many fuzzy logic controllers that improved the
Fig. 15. (a) The glass plate absolute temperature (solid line) using the SDC-based SMC when α is changed by þ 200% and β is changed by 50%; the dotted line represents the
nominal case. (b) The control signal versus time; the dotted line represents the nominal case.
Please cite this article as: Almutairi NB, Zribi M. Sliding mode controllers for a tempered glass furnace. ISA Transactions (2015), http:
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Fig. 16. (a) The response of the absolute temperatures of the controlled tempered glass furnace when the first SMC is used and a load step disturbance is applied in addition
to uncertainties in the warming rates. (b) The control signal versus time.
Fig. 17. (a) The response of the absolute temperatures of the controlled tempered glass furnace when the SDC-based SMC is used and a load step disturbance is applied in
addition to uncertainties in the warming rates. (b) The control signal versus time.
temperature control performance were designed and applied to
different types of furnaces. A fuzzy controller combined with a
conventional PI controller was used in [35] to control the temperature in a glass-melting furnace where a simple first-order-
plus-dead-time model of the system was used. Other fuzzy controllers applied to high temperature furnaces were proposed in
[36,37]. In addition, fuzzy PID controllers applied to heating furnaces can be found in [38,39]. A two-stage fuzzy neural network
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Fig. 18. The temperature T p ðtÞ versus time when using the proposed two controllers and the fuzzy PID controller.
Table 8
The setting times (in minutes) of the proposed two controllers and the fuzzy PID controller for the glass plate absolute temperature.
Desired absolute temperature
First SMC
SDC-based SMC
Fuzzy PID controller
xd ¼ 600 [K]
xd ¼ 800 [K]
xd ¼ 1000 [K]
Ts ¼18.30
Ts ¼7.28
Ts ¼17.23
Ts¼ 4.57
Ts¼ 3.90
Ts¼ 3.98
Ts¼ 5.72
Ts¼ 21.88
Ts¼ 17.43
Table 9
The computational times (in seconds) of the proposed two controllers and the
fuzzy PID controller.
Computational Time
First SMC
SDC-based SMC
Fuzzy PID controller
4.14
1.90
19.41
temperature controller for an industrial heating furnace was
developed in [40].
In this section, we will compare the results of the proposed
controllers with the results obtained when using a fuzzy PID
controller. Recall that the standard PID controller can be written
as,
Z t
deðtÞ
uðtÞ ¼ K p eðtÞ þK i
ð50Þ
eðτÞdτ þK d
dt
0
where eðtÞ ¼ xd T p ðtÞ represent the error signal; K p is the proportional gain, K i is the integral gain and K d is the derivative gain
of the controller.
Motivated by the design methodology of the fuzzy PID proposed in [41], we use an online gain scheduling scheme to design
the parameters ðK p ; K i ; K d Þ based on fuzzy systems. Three different
fuzzy systems, each with a different fuzzy rule base, were
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designed. Each data base consists of 49 fuzzy rules. An educated
guess is used to tune the fuzzy member functions parameters. We
used a product inference engine, a singleton fuzzifier, and a center
average defuzzifier to combine the 49 rules in each set.
Fig. 18(a) shows the glass plate absolute temperature when the
first sliding mode controller is used; the trajectory of the glass
plate absolute temperature when using the SDC-based sliding
mode controller is shown in Fig. 18(b). Fig. 18(c) shows the glass
plate absolute temperature when using the fuzzy PID controller. It
can be seen that when using the three different control schemes,
the glass plate absolute temperature converges to its desired
values in a reasonable time.
It should be mentioned that we obtained similar responses for
the absolute temperatures T w ðtÞ and T B ðtÞ versus time; those plots
are not shown because of space limitations.
To quantify the comparison of the different control schemes,
we investigated the settling times as well as the computational
times of each of the controllers. The settling times of the glass
plate absolute temperature for different desired temperatures are
listed in Table 8; while the computational times needed to
implement each control scheme are given in Table 9. It is noted
that these computational times are obtained using the MATLAB
commands tic and toc.
It can be seen from Tables 8 and 9 that the SDC-based sliding
mode controller gave the fastest settling times and the lowest
computational time when compared to the other two controllers.
Remark 6. In the proposed controllers, all states of the furnace
system, namely the absolute temperatures of the glass plate, the
upper wall and the lower wall, are supposed to be available for
measurement in order to compute the control laws. High temperatures (up to 1200 [K]) of industrial furnaces can be measured
using noble metal thermocouples. Also, Platinum resistance thermometers (PRTs) can be used to measure temperatures in the
range of 13 [K] to 1200 [K] [42]. PRTs are more accurate than noble
metal thermocouples. In addition, thermal radiation thermometers are also used to measure the temperatures of furnaces and
for calibration and control purposes of these furnaces. Radiation
thermometers can be used to measure temperatures without
contact with the object whose temperature is being measured. The
intensity of the infrared energy emitted by the object is proportional to its temperature; therefore the temperature of an object
can be determined by using the amount of infrared energy emitted
by the object and its emissivity. Radiation thermometers are used
in cases when noble metal thermocouples and PRTs cannot be
used or when these thermocouples do not yield accurate temperature measurements for some reasons.
To show that the proposed control schemes work very well in
the presence of measurement noise in the values of the absolute
temperatures, a white Gaussian noise of zero mean and unity
variance is used to simulate this type of noise. The simulation
results of the controlled tempered glass furnace using the proposed sliding mode controllers are shown in Fig. 19. This figure
indicates that the proposed control schemes work well even in the
presence of bounded measurement noise.
7. Conclusion
The temperature control of a tempered glass furnace system is
addressed in this paper. Two sliding mode control schemes are
proposed for this system. The designed controllers are conventional sliding mode controller and an SDC-based sliding mode
controller. The latter is proposed because it is much simplified and
requires less computation than the first one. The performances of
the controlled system are simulated using MATLAB. Also, the
performances are studied under variations in two of the parameters of the system as well as in the presence of an external load
disturbance. The simulation results indicate that the proposed
control schemes work very well and are robust to change in the
parameters of the system as well as to disturbances acting on the
system. Moreover, the results of the proposed controllers are
compared to the results of the controlled furnace system using a
Fig. 19. (a) Absolute temperatures of the controlled tempered glass furnace with a measurement noise (a) when using the first SMC. (b) when using the SDC-based SMC.
Please cite this article as: Almutairi NB, Zribi M. Sliding mode controllers for a tempered glass furnace. ISA Transactions (2015), http:
//dx.doi.org/10.1016/j.isatra.2015.11.005i
N.B. Almutairi, M. Zribi / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
fuzzy PID controller. The simulation results indicate that the SDCbased SMC gave better results when compared to the results of the
first SMC and the results of the fuzzy PID controller.
Future work will address the application of higher-order sliding
mode controllers to the temperature control problem of the
tempered glass furnace.
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Please cite this article as: Almutairi NB, Zribi M. Sliding mode controllers for a tempered glass furnace. ISA Transactions (2015), http:
//dx.doi.org/10.1016/j.isatra.2015.11.005i