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WSEAS TRANSACTIONS on SYSTEMS and CONTROL Peyman Nazarian
Control of a radiative furnace
under precise and uncertainty conditions
Peyman Nazarian
Electrical Department
Islamic Azad University -Zanjan Branch
No. 38 - Golestan 17 - Shahrak Ansarieh – Zanjan
IRAN
pay_naz@yahoo.com
Abstract: - This paper discusses the control of a radiative furnace under precise and imprecise modeling.
Regarding nonlinearity nature of the process, in the precise case, feedback linearization technique is applied to
the furnace and under uncertainty conditions, sliding mode control is used. The conditions are completely
practical and have gained from an industrial process.
Key-Words: Feedback Linearization – Radiation – Furnace – Uncertainty – Sliding Mode Control
1. Introduction Adaptive feedback linearization control technique
Radiative furnace is the useful equipment at for chaos suppression in a chaotic system is used in
industry. It works with radiation like many other [10]. Disturbance decoupling and trajectory tracking
devices [1], [2], [3] and [4], of course by heat of nonlinear control systems using an observer-based
radiation. fuzzy feedback linearization control (FLC) is
It is assumed that we encounter with two essential developed in [11]. Application of the feedback
cases. In the first case, we have precise model and linearization to the model of a power system is
we would control it by feedback linearization investigated in [12]. A scheme for temperature
technique and in the second one, sliding mode control of a greenhouse using the feedback
control is applied due to uncertainty of the model. linearization is presented in [13].
Now we explain the basic concepts of two
aforementioned control methods and then simulate
them. 2. Feedback linearization review
Feedback linearization is an approach to nonlinear Consider a system described by
control design which has attracted a great deal of
research interest in recent years. x = f (x ) + ug (x ) (1)
Feedback linearization has been used successfully to
address some practical control problems. Design of a
nonlinear control system for a Variable Air Volume where f and g are smooth vector fields on some
Air Conditioning (VAVAC) plant through feedback open set X ⊆ R" containing 0 and f(0) = 0.
linearization is presented in [5]. For an induction There exists smooth functions q , s ∈S (X ) with
motor the sliding mode controller is combined with s ( x) ≠ 0 for all x in some neighborhood of the
an adaptive input–output feedback linearization
technique in order to preserve the system robustness origin, and a local diffeomorphism T on Rn with
with respect to stator and rotor resistances variations T(0) = 0. We define
and uncertainties [6]. In [7] a decentralized nonlinear
controller for large-scale power systems is v = q (x ) + s (x ) (2)
investigated. The proposed controller design is based
on the input–output feedback linearization z = T (x ) (3)
methodology. In reference [8] the technique of
feedback linearization is used to design controllers
The resulting variables z and v satisfy a linear
for displacement, velocity and differential pressure
differential equation of the form
control of a rotational hydraulic drive. Application of
a feedback linearization technique for control of a
distributed solar collector is described in [9]. z = Az + bv (4)
ISSN: 1991-8763 211 Issue 5, Volume 4, May 2009
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Peyman Nazarian
z = M −1T (x ) (13)
where the pair (A, b) is controllable. In this case, the
system is called feedback linearizable.
v = v − a z = q (x ) − a ' M −1 T (x ) + s ( x ) u (14)
q (x ) 1
u =− + v (5) where
s (x ) s (x )
a ' = [a0 a1 ...an −1 ] (15)
By applying a state transformation z = M −1z
such that the resulting system is in controllable
canonical form
3. Radiative furnace dynamics
−1
z = M AMz + M bv −1
(6) Fig. 1 shows a typical horizontal radiative furnace
which is used for tempered glass production.
Assuming the temperature in the whole of the glass
where and the upper and lower walls is uniformly changed
⎡ 0 1 0 0 ⎤ and the conductive heat transfer is negligible.
⎢ 0 0 1. 0 ⎥
M AM = ⎢
−1 ⎥ (7)
⎢ .. .. . . ⎥
⎢ ⎥
⎣ −a0 −a1 −a2 .. −an −1 ⎦
⎡0⎤
⎢0⎥
M −1b = ⎢ ⎥ (8)
⎢:⎥
⎢ ⎥
⎣1 ⎦
Figure 1. Horizontal radiative furnace
And the ai 's are the coefficients of the
characteristic polynomial From radiation heat transfer we have
n −1
| sI − A |= s n + ∑ a j s j (9) ⎧ T p = a1 (Tw4 −T p4 )
⎪
⎨Tw = a2 [Tw − Fw − pT p − (1 − Fw − p )T B ] + α P (16)
i =0 4 4 4
⎪ T B = a3 (Tw4 −T B4 )
A further state feedback form ⎩
v = v + [a0a1...an −1 ] z (10) where TP, Tw and TB are glass plate, upper wall and
B
lower wall temperatures respectively. P is the input
electric power and a1, a2 and a3 are as below
results in the closed loop system
a 1 = βσFw −p A
z = Az + bv (11)
a 2 = −ασA
where a 3 = σαA(1 − Fw −p ) (17)
⎡0 1 0 0⎤ ⎡0⎤ where the σ, the Stefan-Boltzmann constant is
⎢0 0 1.. 0 ⎥⎥ ⎢0⎥ 5.6697*10-8 W/m2.K4, A is the area of the wall, Fw-p
A =⎢ ,b =⎢ ⎥ (12) is the shape factor between upper wall and plate and
⎢. .. .. .. ⎥ ⎢:⎥ α, β are the warming rate of the walls and plate
⎢ ⎥ ⎢ ⎥ respectively.
⎣0 0 0 0⎦ ⎣1 ⎦
In simulation following practical data have been
used.
ISSN: 1991-8763 212 Issue 5, Volume 4, May 2009
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Peyman Nazarian
y(2) = v → x1(2) = v (27)
Fw-p = 0.7
α = 8*10 -5 u = m ( x ) + n ( x )v (28)
β = 35*10 -5
A=1 (18)
Lf 2 h
m (x ) = − (29)
Now apply feedback linearization to furnace system. L g Lf h
We define state variables as follows
1
Tp = x1 n (x ) = (30)
Tw = x2 L g Lf h
TB = x3 (19)
From MATLAB symbolic toolbox we obtain
Then arrange equations in companion form
m (x ) = 4 β 2σ 2 Fw2− p x 13 (x 2 − x 14 ) +
4
x = f ( x ) + g ( x )u (20)
4αβσ 2 Fw − p x 2 [x 2 − Fw − p x 14 − (1 − Fw − p )x 34 ] (31)
3 4
h (x ) = y = x 1 (21) n(x) = 4σαβFw-px24 (32)
⎛ a1 (x 2 − x 14 )
4
⎞ For tracking control task define desire path x1d and
⎜ 4 ⎟ tracking error is
⎜ a2 [x 2 − Fw − p x 1 − (1 − Fw − p )x 3 ] ⎟
4 4
f (x ) = (22)
⎜ a3 (x 2 − x 34 )
4 ⎟ e = x1- x1d (33)
⎜
⎜ ⎟
⎟
⎝ ⎠ and
⎛0 ⎞ v = x1d(2) - k1e(1) – k0e (34)
⎜ ⎟
g (x ) = ⎜ α ⎟ (23)
⎜0 ⎟ meanwhile, when Fw − p is one, the parameter a3 is
⎝ ⎠
set to zero and since the relative degree remains 2
and system order is decreased to 2, the I.O.L.
Lg h (x ) = 0 converts to I.S.L. and there is no internal dynamics
L g Lf h ( x ) ≠ 0 (24) appears in this case. This case occurs when the plate
and the wall below have the same size and therefore
it is a feasible subject.
Thus we have relative degree 2 and since the system We use this situation for simplifying the necessary
order is 3, the order of the internal dynamics is 1. equations under uncertainty condition and applying
Now to investigate stability of zero dynamics, define sliding mode control as below. In section 5 we
would explain more the sliding mode control in
μ1 = h(x) = x1 details.
μ2 = Lf h(x)
φ(x) = [ μ1 μ2 ψ ] (24) x ( 2 ) = f (x ) + b(x ).u
φ(x) is diffeomorphism so we have
f ( x ) = 4 ( x + x )3 4a a [ x + (1 − F ) x 4 ] − 4a x 3 x
a 1 2 a w−p 1
Lg ψ(x) = 0 → ψ(x) = x3 (25) 1 1
μ1 = μ2 = 0 → ψ ( x ) = −a 3 ψ 4 ( x ) (26)
b( x ) = 4 ( x + x )3 4a α (35)
a 1
a3 is a positive value, so zero dynamics is stable and
1
thus we could apply the input-output linearization.
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WSEAS TRANSACTIONS on SYSTEMS and CONTROL Peyman Nazarian
4. Simulation results with feedback Figure 4. Internal dynamics behavior with two poles
at s = -0.2 and step input
linearization
Suppose electrical system of the furnace power could Feedback linearization of radiation furnace
vary from zero to 16kW linearly. Also initial 16000
conditions for plate and walls are 300oK and 500oK 14000
respectively. Set point changes from 700oK to 12000
900oK.
10000
U(Electric Power)
Feedback linearization of radiation furnace 8000
1000
6000
900
4000
800 2000
Glass Temperature
0
700 0 500 1000 1500 2000 2500
time
600 Figure 5. Electric power behavior with two poles at
500
s = -0.2 and step input
400 Feedback linearization of radiation furnace
1000
300
0 500 1000 1500 2000 2500 900
time
Figure 2. Step response of plate temperature with 800
two pole at s = -0.2
Glass Temperature
700
Feedback linearization of radiation furnace 600
950
900 500
850
400
800
Up Temperature
300
750 0 500 1000 1500 2000 2500
time
700
650
Figure 6. Ramp response of plate temperature with
two pole at s = -0.2
600
550
Feedback linearization of radiation furnace
1000
500
0 500 1000 1500 2000 2500
time 950
Figure 3. Step response of upper wall temperature 900
with two poles at s = -0.2 850
Up Temperature
800
Feedback linearization of radiation furnace
900 750
700
850
650
800
600
Down Temperature
750
550
700
500
0 500 1000 1500 2000 2500
650 time
600 Figure 7. Ramp response of upper wall temperature
550
with two poles at s = -0.2
500
0 500 1000 1500 2000 2500
time
ISSN: 1991-8763 214 Issue 5, Volume 4, May 2009
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Peyman Nazarian
950
Feedback linearization of radiation furnace Now we change the one pair poles of the controller
form s = −0.2 to s = −0.03 and investigate
900
responses. We expect the response be slow that the
850
following simulations display this.
800
Down Temperature
Feedback linearization of radiation furnace
750 900
700
800
650
600
700
Glass Temperature
550
600
500
0 500 1000 1500 2000 2500
time
500
Figure 8. Internal dynamics behavior with two poles
at s = -0.2 and ramp input 400
Feedback linearization of radiation furnace 300
16000 0 500 1000 1500 2000 2500
time
14000
Figure 11. Step response of plate temperature with
12000 two pole at s = -0.03
10000
U(Electric Power)
Feedback linearization of radiation furnace
16000
8000
14000
6000
12000
4000
10000
U(Electric Power)
2000
8000
0
0 500 1000 1500 2000 2500
time 6000
Figure 9. Electric power behavior with two poles at 4000
s = -0.2 and ramp input
2000
Now we simulate ramp input response of the furnace 0
0 500 1000 1500 2000 2500
system with a step disturbance. time
Figure 12. Electric power behavior with two poles at
Feedback linearization of radiation furnace
1000 s = -0.03 and step input
900
800 5. Sliding Mode Control review
Consider the single input dynamic system as the
Glass Temperature
700
following dynamics
600
x ( n ) = f ( X ) + b ( X ) .u (36)
500
400 Where the scalar x is the output of interest, the scalar
300
u is the control input and X is the state vector as
0 500 1000
time
1500 2000 2500 below
Figure 10. Ramp response of plate temperature with
two pole at s = -0.2 and step disturbance
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WSEAS TRANSACTIONS on SYSTEMS and CONTROL Peyman Nazarian
⎡ x ⎤ s = x + λx (43)
⎢ x ⎥
X =⎢ ⎥ (37) Satisfying condition or sliding condition for
⎢ ⎥ remaining system trajectories on the surface is
⎢ ( n −1) ⎥
⎣x ⎦
1d 2
s ≤ −η s (44)
In equation (36) the function f(X) (in general 2 dt
nonlinear) is not exactly known, but the extent of the
imprecision on f(X) is upper bounded by a known Where η is a strictly positive constant. Essentially,
continuous function of X ; similarly, the control gain equation (44) states that the squared "distance" to the
b(X) is not exactly known, but is of known sign and
surface, as measured by s 2 , decreases along all
is bounded by known, continuous function of X.
system trajectories.
The control problem is to get the state X to track a
specific time-varying state Xd in the presence of
model imprecision on f(X) and b(X) as the following
definition 6. Simulation results with sliding mode
control
⎡ xd ⎤ Uncertainty in variable α causes an uncertainty in
⎢ x ⎥ f(X) and g(X). The practical range for α as the
Xd =⎢ d ⎥ (38) following
⎢ ⎥
⎢ ( n −1) ⎥ 6 ×10 −5 ≤ α ≤ 9 ×10 −5 (45)
⎣x d ⎦
For the tracking task to be achievable using a finite ~ ⎡~ ⎤
x
control u, the initial desired state Xd(0) must be such X = X − X d = ⎢~ ⎥ (46)
that ⎣x ⎦
Xd(0) = X(0) (39) d
S( x, t ) = ( + λ )~
x (47)
Let x be the tracking error in the variable x as dt
below
S = ~ + λ~ = 0 ⇒ u = −f + x d − λ~
x x ˆ ˆ x (48)
x = x −xd (40)
fˆ = −3 ×10−4 a1 4 ( x + x )3 ×
And let a
1 (49)
⎡ x ⎤ ×[ x + (1 − F )x 4 ] − 4a x 3x
⎢ x ⎥ a w −p 1
1
X = X −X d = ⎢ ⎥ (41)
⎢ ⎥
⎢ ( n −1) ⎥ ˆ
f −f ≤ F (50)
⎣x ⎦
Be the tracking error vector. Furthermore, let us F = 6 ×10−5 a1 4 ( x + x )3 ×
define a time-varying surface S(t) in the state-space a
1
R(n) by the scalar equation s (X ; t ) = 0 , where (51)
× x + (1 − F )x 4
d a w −p
s (X ; t ) = ( + λ ) n −1 x (42) 1
dt
For security of the sliding condition, we add a
And λ is a strictly positive constant, whose choice discontinuous around surface s=0 as below
we shall interpret later. For instance, if n = 2 ,
ISSN: 1991-8763 216 Issue 5, Volume 4, May 2009
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Peyman Nazarian
u = b −1 [u − k ⋅ sgn (s )]
ˆ ˆ 4
Sliding Control for vertical furnace
(52) 10
x 10
9.5
Where
9
Sliding condition check
8.5
b = 4 54 × 10 −5 a 1 4 ( x + x )3
ˆ (53)
a 8
1 7.5
k ≥ β ( F + η ) + ( β − 1) u
7
ˆ (54)
6.5
6
1
β = ⎛ max ⎞
b 2
⎜ = 1.5
b min ⎟
(55) 5.5
0 500 1000 1500 2000 2500
⎝ ⎠ Time
Figure 13. Sliding condition check
To choose an appropriate value for k , one may be
use a constant value of k that satisfies equation (54) 900
Sliding Control for vertical furnace
at all times. In case of absence of this fulfillment, it
should be select a greater k for satisfaction. 800
Also to decrease input chattering with high
⎛s ⎞ 700
frequency switches, one can uses sat ⎜ ⎟ instead
⎝ϕ ⎠
Plate Temp.
600
of sgn(s ) in equation (52). sat is saturation
element and sgn means sign function or element in 500
MATLAB.
In addition, if x d (0 ) ≠ x (0 ) , then reach time to the
400
sliding surface is related to the following inequality 300
0 500 1000 1500 2000 2500
Time
s (t = 0 ) Figure 14. Step input response with SMC
t ≤ (56)
reach
η Sliding Control for vertical furnace
16000
In reach time, state path reaches to the sliding 14000
surface and then tends to x d with time constant 12000
n −1
that for our problem with n = 2 is 1 . 10000
U(Electric Power
λ λ
8000
Now regarding the above mentioned equations, we
can simulate responses in MATLAB environment. 6000
First we consider the saturation element between 0 4000
and 16000 according to power limitations. Also
2000
initial conditions are assumed 300 for plate and 500
for the wall. We want to examine different cases in 0
0 500 1000 1500 2000 2500
simulation. Time
First suppose η = 1 , λ = 1000 , ϕ = 50 and Figure 15. Electric power behavior with SMC and
k = 100000 . Then check satisfaction of the sliding step input
condition for simulation times as shown in Fig. 13.
Positive values of the plotted curve displays the
fulfillment of sliding condition and therefore the
simulation results are valid.
ISSN: 1991-8763 217 Issue 5, Volume 4, May 2009
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Peyman Nazarian
Sliding Control for vertical furnace Sliding Control for vertical furnace
800 16000
750
14000
700
12000
650
10000
U(Electric Power
600
Plate Temp.
550 8000
500
6000
450
4000
400
350 2000
300 0
0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500
Time Time
Figure 16. Ramp input response with SMC Figure 19. Input signal chattering occurs with
ϕ = 0.00001
Sliding Control for vertical furnace
16000
14000
Now examine the case k = 40000 . In this case the
sliding condition is contravened as shown in Fig. 20.
12000
4
10000 x 10 Sliding Control for vertical furnace
U(Electric Power
4
8000
3
6000
Sliding condition check
4000 2
2000
1
0
0 500 1000 1500 2000 2500
Time 0
Figure 17. Electric power behavior with SMC and
-1
ramp input
-2
Now we change the ϕ = 50 to ϕ = 0.00001 for 0 500 1000
Time
1500 2000 2500
investigating the chattering of the input. Figure 20. Sliding condition contravening
Sliding Control for vertical furnace
900 We can see that in some simulation times, the plotted
curve is negative and this means contravening the
800 sliding condition.
700 Sliding Control for vertical furnace
900
Plate Temp.
600
800
500
700
Plate Temp.
400
600
300
0 500 1000 1500 2000 2500
500
Time
Figure 18. Step input response with SMC and 400
selecting ϕ = 0.00001
300
0 500 1000 1500 2000 2500
Time
ISSN: 1991-8763 218 Issue 5, Volume 4, May 2009
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Peyman Nazarian
Figure 21. Step input response with SMC and 7. Conclusion
selecting ϕ = 0.00002 and λ = 1 In this paper an industrial furnace as a nonlinear
plant investigated and two different controller
16000
Sliding Control for vertical furnace designed for it, as well as a linear system by
feedback linearization technique in precise modeling
14000 case and sliding mode control in uncertainty of
12000 modeling parameters.
In feedback linearization control, stability and
10000
response speed adjusted by changing k0 and k1 in
U(Electric Power
8000 spite of nonlinearity and large dynamic range of the
6000
furnace temperature and totally two pairs of poles is
selected and the various outputs behaviors were
4000 investigated.
2000
In sliding mode control, response speed adjusted by
changing the combination of λ , ϕ . Also for
0
0 500 1000
Time
1500 2000 2500
softening the control signal and avoiding chattering
used an saturation element with large value of ϕ .
Figure 22. Input signal chattering occurs with
Parameter η used to adjust the time to reaching
ϕ = 0.00002 and λ = 1
sliding surface. Increasing the parameter k also
Sliding Control for vertical furnace
helps for retaining satisfying condition.
900
In both of the aforementioned controllers, a step
disturbance signal applied and the behavior of
800
various parts of the furnace studied.
700
Plate Temp.
600
References:
[1] E. Mohseni Languri, D.D. Ganji, N. Jamshidi,
500
Variational iteration and Homotopy perturbation
methods for fin efficiency of convective straight fins
400 with temperature-dependent thermal conductivity,
5th WSEAS Int. Conf. on FLUID MECHANICS
300
0 500 1000 1500 2000 2500
(FLUIDS'08) Acapulco, Mexico, January 25-27,
Time 2008.
Figure 22. Step response of plate temperature with
step disturbance and ϕ = 0.00002 and λ = 1 [2] Dr.A. Alizadeh-Attar, H.R. Ghoohestani, I. Nasr
Isfahani, Reducing Flare Emissions from Chemical
Sliding Control for vertical furnace
Plants and Refineries Through the Application of
16000 Fuzzy Control System, Proceedings of the 6th
14000
WSEAS International Conference on Applied
Computer Science, Hangzhou, China, April 15-17,
12000
2007.
10000
U(Electric Power
[3] HIMANSHU DEHRA, The Electrodynamics of a
8000
Pair of PV Modules with Connected Building
6000 Resistance, Proc. of the 3rd IASME/WSEAS Int.
4000
Conf. on Energy, Environment, Ecosystems and
Sustainable Development, Agios Nikolaos, Greece,
2000
July 24-26, 2007.
0
0 500 1000 1500 2000 2500
Time [4] Khajornsak Sopajaree, and Apisit Sancom,
Figure 23. Input signal behavior with step CHEMICAL AND PHYSICAL PROPERTY OF
RICE STRAW WASTE AND HOSPITAL
disturbance and ϕ = 0.00002 and λ = 1
SEWAGE SLUDGE IN TURNED WINDROW
AERATION SYSTEM, 3rd IASME/WSEAS Int.
ISSN: 1991-8763 219 Issue 5, Volume 4, May 2009
WSEAS TRANSACTIONS on SYSTEMS and CONTROL Peyman Nazarian
Conf. on Energy & Environment, University of Systems, Volume 29, Issue 4, May 2007, Pages 322-
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[7] Enrico De Tuglie, Silvio Marcello Iannone,
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[8] Jaho Seo, Ravinder Venugopal, Jean-Pierre
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ISSN: 1991-8763 220 Issue 5, Volume 4, May 2009
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