2010 American Control Conference
Marriott Waterfront, Baltimore, MD, USA
June 30-July 02, 2010
FrC21.6
Reconstruction-based Contribution for Process Monitoring with Kernel
Principal Component Analysis.
Carlos F. Alcala and S. Joe Qin
Abstract— This paper presents a new method for fault diagnosis based on kernel principal component analysis (KPCA).
The proposed method uses reconstruction-based contributions
(RBC) to diagnose simple and complex faults in nonlinear
principal component models based on KPCA. Similar to linear
PCA, a combined index, based on the weighted combination
of the Hotelling’s T 2 and SPE indices, is proposed. Control
limits for these fault detection indices are proposed using second
order moment approximation. The proposed fault detection
and diagnosis scheme is tested with a simulated CSTR process
where simple and complex faults are introduced. The simulation
results show that the proposed fault detection and diagnosis
methods are efective for KPCA.
I. I NTRODUCTION
Principal component analysis (PCA) is a multivariate
statistical tool widely used in industry for process monitoring
([11], [14], [12], [3]). It decomposes the measurement space
into a principal component space that contains the commoncause variability, and a residual space that contains the
process noise. The task of process monitoring is to detect and
diagnose faults when they happen in a process. In process
monitoring with PCA, fault detection is performed with fault
detection indices. Popular indices are the Hotelling’s T 2
statistic for monitoring the principal component subspace;
the SPE index, which monitors the residual subspace; and
a combination of both indices that monitors the whole
measurement space. Once a fault is detected, it is necessary
to diagnose its cause. A popular method for fault diagnosis is the contribution plots. They are based on the idea
that variables with large contributions to a fault detection
index are likely the cause of the fault. However, as was
demonstrated by Alcala and Qin [1], even for simple sensor
faults, contribution plots fail to guarantee correct diagnosis.
As an alternative to contribution plots, Alcala and Qin [1]
propose the use of reconstruction-based contributions (RBC)
that overcomes the aforementioned shortcoming. The RBC
method is related to fault identification by reconstruction
([6], [7]), but it does not require the knowledge of fault
directions.
The PCA-based process monitoring scheme mentioned
above is based on the assumption that the process behaves
linearly. However, when a process is nonlinear, the monitoring of a process using a linear PCA model might not
C. F. Alcala is with the Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, CA 90089,
USA, [email protected]
S. J. Qin is with the Mork Family Department of Chemical Engineering and Materials Science, and the Ming Hsieh Deparment of Electrical Engineering, University of Southern California, CA 90089, USA,
[email protected]
978-1-4244-7425-7/10/$26.00 ©2010 AACC
perform properly. In order to address this issue, several
nonlinear PCA methods have been proposed ([9], [5], [8],
[13]). One of these methods is kernel principal component
analysis (KPCA), developed by Schölkopf et al. [13], which
maps measurements from their original space to a higher
dimensional feature space where PCA is performed. KPCA
has been applied successfully for process monitoring ([10],
[4], [17], [18]). Fault detection with KPCA is done in a
similar way as with PCA using a T 2 , SPE and combined
indices. However, fault diagnosis cannot be performed with
contribution plots since there seems to be no way to calculate
them with KPCA. Choi et al. [4] propose a method for fault
diagnosis employing the reconstruction-method of Dunia et
al. [7], which looks at a fault identification index obtained
when a fault detection index has been reconstructed along the
direction of a variable. While this method applies to sensor
faults or a process fault with a known direction, it cannot be
applied to diagnosing faults with unknown directions.
In this paper, reconstruction-based contribution for kernel
PCA (RBC-KPCA) is proposed for fault diagnosis. We
present a matrix formulation for KPCA and the kernel trick
that is simpler than the standard kernel PCA description,
and is analogous to the linear PCA formulation familiar to
the process monitoring community. The T 2 and SPE fault
detection indices are expressed in terms of a kernel vector
mapped by the kernel function, rather than the intangible
vector in the feature space. Furthermore, the combined index
approach of Yue and Qin [16] is proposed for fault detection
in the feature space and new control limits for the fault
detection indices are proposed. The proposed RBC diagnosis
method is used just like the standard contribution plots for
KPCA-based fault diagnosis, although it is difficult to extend
the standard contribution plots for KPCA. It is shown that
simple and complex faults can be diagnosed correctly using
RBC-KPCA with a simulation of a CSTR process.
II. K ERNEL PCA
A. KPCA Background
In order to make the kernel PCA model of process data
with n variables, m measurements, taken under normal operT
ating conditions, form the training set X = [x1 x2 · · · xm ] .
Each of the measurements xi is an n-dimensional vector and
can be mapped into an h-dimensional space via a mapping
function φi = Φ (xi ). The h-dimensional space is called
the feature space and the n-dimensional space is called the
measurement space. An important property of the feature
space is that the dot product of two vectors φi and φj can
7022
be calculated as a function of the corresponding vectors xi
and xj ,
φTi φj = k (xi , xj )
(1)
The function k (·, ·) is called the kernel function, and there
exist several types of these functions. The gaussian kernel
function, or radial basis function, is expressed as
#
"
T
(xi − xj ) (xi − xj )
(2)
k (xi , xj ) = exp −
c
and will be used in this paper.
For process monitoring with linear PCA, the model is obtained from the training set X. In kernel PCA, the covariance
matrix of a set of m mapped vectors φi is eigen-decomposed
to obtain the principal loadings of the model. Assume that the
vectors in the feature space are scaled to zero mean and form
T
the training data as X = [φ1 φ2 · · · φm ] . Let the sample
covariance matrix of the data set in the feature space be S.
We have,
(m − 1)S = X T X =
m
X
φi φTi
(3)
i=1
KPCA in the feature space is equivalent to solving the
following eigenvector equation,
XTXν =
m
X
φi φTi ν = λν
αi . Then use Eq. 8 to find νi . In order to guarantee that
νiT νi = 1, Eqs. 6 and 4 are used to derive
αTi αi = νiT X T X νi = νiT λi νi = λi .
√
Therefore, αi needs to have a norm of λi . Let α◦i be the
unit norm eigenvector corresponding to λi ,
p
αi = λi α◦i .
The matrix with the l leading eigenvectors are the KPCA
principal loadings in the feature space, denoted as Pf =
[ν1 ν2 · · · νl ]. From Eq. 8, Pf is related to the loadings
in the measurement space as
1 T
1 T
X α1 · · ·
X αl
Pf =
λ
λl
h 1
i
1
−1
T ◦ −2
= X α1 λ1
· · · X T α◦l λl 2
1
= X T PΛ− 2
where P = [α◦1 · · · α◦l ] and Λ = diag{λ1 · · · λl }
are the l principal eigenvectors and eigenvalues of K, respectively, corresponding to the largest eigenvalues in descending
order.
For a given measurement x and its mapped vector φ =
Φ (x), the scores are calculated as t = PTf φ, which, from
Eq. 9, we can be expressed as
(4)
= Λ
X X X ν = λX ν
= X φ = φT1 φ
k(x)
=
Defining
φT1 φ1

..
=
.
φTm φ1
K =
XX


= 
T

φT1 φm

..

.
T
φm φm

k (x1 , xm )

..

.
···
..
.
···
···
..
.
k (xm , x1 ) · · ·
k (x1 , x1 )
..
.
PT k(x).
(10)
φT2 φ
···
[k(x1 , x) k(x2 , x)
···
φTm φ
T
T
k(xm , x)] (11)
B. Scaling
(5)
The calculation of the covariance matrix in Eq. 3 holds if
the vector φ in the feature space has zero mean. If this is not
the case, the vectors in the feature space have to be scaled to
zero mean using the sample mean of the training data. The
scaled vector φ̄ is
k (xm , xm )
m
φ̄ = φ −
and denoting
α = Xν
(6)
Kα = λα
(7)
we have
Eq. 7 shows that α and λ are an eigenvector and eigenvalue of K, respectively. In order to solve ν from Eq. 6, we
premultiply it by X T and use Eq. 4,
1 X
φi = φ − [φ1 · · · φm ] 1m
m i
where 1m is an m-dimensional vector whose elements are
1/m. The kernel function of two scaled vectors φ̄i and φ¯j
is
k̄ (xi , xj )
=
φ̄Ti φ̄j
=
k (xi , xj ) − k (xi ) 1m − k (xj ) 1m · · ·
T
T
+1Tm K1m
X T α = X T X ν = λν
(12)
Similarly, the scaling of the kernel vector k (x) is
T
k̄ (x) = φ̄1 φ̄2 · · · φ̄m φ̄
which shows that ν is given by
ν = λ−1 X T α.
− 21
where
T

1
= Λ− 2 PT X φ
t
i=1
Note that φi is not explicitly defined nor is φi φTj . The socalled kernel trick pre-multiplies Eq. 4 by X :
(9)
= F [k (x) − K1m ]
(8)
Therefore, to calculate the PCA model (λi and νi ), we
first perform eigen-decomposition of Eq. 7 to obtain λi and
(13)
where
7023
F=I−E
(14)
In this equation, I is the identity matrix, and E is an m×m
matrix with elements 1/m. Finally, the scaled kernel matrix
K, K̄, is calculated as
T K̄ = φ̄1 φ̄2 · · · φ̄m
φ̄1 φ̄2 · · · φ̄m
= FKF
(15)
Therefore, it is straightforward to transform vectors in the
feature space to zero mean before KPCA. Without loss of
generality, we assume the vectors in the feature space are
zero mean for the rest of the paper.
III. FAULT D ETECTION WITH KPCA
A. Fault Detection Indices
Since the key idea of KPCA is to map the measurement
space into the feature space, so that data in the feature space
are linearly distributed, it is natural to perform fault detection
by defining statistics in the feature space.
The Hotelling’s T 2 index is calculated as T 2 (x) =
T −1
t Λ t, where Λ = diag (λ1 , · · · , λl ) is the covariance
of the scores t in the feature space. From Eq. 10, T 2 is
calculated using kernel functions as
T
T 2 = k (x) Dk (x)
(16)
with D = PΛ−2 PT .
The SPE index is defined as the norm of the residual vector
in the feature space and is calculated as φT C̃f φ, where
C̃f is the projection matrix of the residual space, which
is orthogonal to the principal component space. Let t̃ be
the residual components and P̃f the corresponding loading
matrix,
t̃ = P̃Tf φ = [νl+1
T
···] φ
νl+2
and residual spaces in the feature space. A combined index
for fault detection in the feature space has been proposed
by Choi et al. [4]; however, their definition is different to
the one proposed here as they use an energy approximation
concept to calculate the index. The combined index proposed
for the feature space is defined as
ϕ (x) =
SP E (x) T 2 (x)
+
δ2
τ2
where δ 2 and τ 2 are the control limits for the SPE and T 2
indices, respectively. The calculation of these control limits
is provided in the next subsection. The combined index can
be calculated with kernel functions as
k (x, x)
T
ϕ (x) =
+ k (x) Ωk (x)
(19)
δ2
where
Ω=
PΛ−2 PT
PΛ−1 PT
D
C
−
= 2 − 2.
τ2
δ2
τ
δ
B. Control limits
In the feature space, the fault detection indices have the
quadratic form
J = xT Ax
(21)
with A ≥ 0. If it is considered that x is zero mean and
normally distributed, the results of Box [2] can be applied
to calculate control limits for the fault detection indices.
These limits can be calculated as g Index χ2α hIndex , with
a confidence level of (1 − α) × 100%, and the parameters
g Index and hIndex calculated as
g Index
=
tr{SA}2
tr{SA}
hIndex
=
[tr{SA}]
tr{SA}2
SP E = t̃T t̃ = φT P̃f P̃Tf φ.
SP E = φ
I−
T
φ=φ φ−φ
T
IV. R ECONSTRUCTION - BASED C ONTRIBUTION FOR
FAULT D IAGNOSIS
Pf PTf φ.
From Eqs. 1, 9 and 11 we have
T
SP E = k (x, x) − k (x) Ck (x)
(23)
where tr{A} is the trace of matrix A. We use Index to
represent either of the SPE, T 2 or ϕ indices. The parameters
g Index and hIndex for the detection indices are shown in
Table I. In the table, the parameter λi is the ith eigenvalue
of the kernel matrix K.
and leads to
(22)
2
Since we do not know the dimension of the feature space,
it is not possible to know the number of residual components
there. Thus, we cannot calculate explicitly the loading matrix
P̃f . However, we can calculate the product P̃f P̃Tf as the
projection orthogonal to the principal component space,
which is
C̃f = P̃f P̃Tf = I − Pf PTf
Pf PTf
(20)
The combined index has a control limit ζ 2 and its calculation is provided in the next section.
The SPE index is calculated as the squared norm of the
residual components
T
(18)
(17)
where C = PΛ−1 PT .
Yue and Qin [16] proposed the use of a combined index for
monitoring the principal and residual space simultaneously.
Such an index is a combination of the T 2 and SPE indices
weighted by their control limits. The same concept is used
here to define a fault detection index to monitor the principal
Reconstruction-based contribution (RBC), proposed by
Alcala and Qin [1], defines the reconstruction of a fault
detection index along the direction of a variable as the variables’ contribution for fault diagnosis. The variable with the
largest amount of reconstruction is likely a major contributor
to the fault. This method is useful for it can not only obtain
contributions along the direction of a variable, but along
an arbitrary direction. Therefore, it is possible to obtain
the contribution of simple faults, as well as complex faults
provided that their directions are available in a fault library.
7024
TABLE I
C ONTROL LIMITS FOR THE FAULT DETECTION INDICES .
Limit = g Index χα hIndex .
Index
Limit
FS
11
TC
CA
1
T0
CAS
2
7
10
Parameters
Solvent
Flow
2
T2
M
gT =
τ2
2
1
m−1
hT = l
FA
SP
FC
SP
9
TC
4
3
Cooling
Water Flow
CAA
SP E
δ2
Pm
λ2
Pm i
g SP E = (m−1)i=l+1
λi
i=l+1
2
P
m
λ
i
i=l+1
hSP E = Pm
λ2
i=l+1 i
ζ2
P
2
4
l/τ 4 + m
i=l+1 λi /δ
gϕ =
P
(m−1) l/τ 2 + m
λ
/δ 2
i
i=l+1
2 Pm
2 2
l/τ + i=l+1 λi /δ
hϕ = l/τ 4 +Pm
λ2 /δ 4
i=l+1 i
8
Pure A
Solute Flow
Fig. 1.
ϕ
For SPE, via Eq. 17, the derived expression for fi is
ξ T BT 1m + FCk̄ (zi )
fi = Ti
k (zi ) 1m + FCk̄ (zi )
The objective of RBC is to find the magnitude fi of a
vector with direction ξi such that the fault detection index
of the reconstructed measurement
zi = x − ξ i f i
(24)
is minimized. This is, we want to find fi
=
arg min Index (x − ξi fi ). The same concept can be
applied to KPCA and find fi such that
fi = arg min Index (k (x − ξi fi ))
(25)
The RBC value for the direction ξi is fi2 . If we want to do
reconstruction along the direction of a variable, the direction
can be written as ξi = [0 0 · · · 1 · · · 0], where 1 is placed at
the ith position. When complex faults are reconstructed, the
direction of the fault can be any unitary vector ξi .
In order to find the RBC along a direction ξi for a fault
detection index, we have to perform a nonlinear search of
the value of fi that minimizes Index (k (x − ξi f )). This is
done by obtaining the first derivative of the fault detection
index with respect to fi , then equate to zero and solve for fi .
However, the resulting expression is not an explicit solution
for fi and it has to be iterated until fi has converged.
Employing the kernel function in Eq. 2, and Eq. 16, the
derived expression of fi for the T 2 index is
fi =
ξiT BT FDk̄ (zi )
kT (zi ) FDk̄ (zi )
Diagram of the CSTR process [15]
For the index ϕ, using Eq. 19, fi is calculated as
ξiT BT 1δm
2 − FΩk̄ (zi )
fi = T
k (zi ) 1δm
2 − FΩk̄ (zi )
(29)
In the next section a simulation is performed to test the
fault detection and diagnosis methods discussed so far.
V. CSTR S IMULATION
In this section, a nonisothermal continuous stirred tank
reactor (CSTR) is simulated. The process model and simulation conditions are similar to the ones provided by Yoon
and MacGregor [15], and they are given in Appendix A; a
diagram of the proces is shown in Figure 1. The simulation
is performed according to the following model
E
dCA
F
F
= CA0 − CA − k0 e− RT CA
dt
V
V
V ρcp dT
dt
= ρcp F (T0 − T ) −
(30)
aFCb+1
FC + aFCb / (2ρC cpc )
E
(T − TC,in ) + (−∆Hrxn ) V k0 e− RT CA (31)
The process is monitored measuring the cooling water
temperature TC , the inlet temperature T0 , the inlet concentrations CAA and CAS , the solvent flow FS , the cooling
water flow FC , the outlet concentration CA , the temperature
T , and the reactant flow FA . These nine variables form the
measurement vector
(26)
x = [TC T0 CAA CAS FS FC CA T FA ]
In this equation, F is the scaling matrix defined in Eq. 14,
k (zi ) is the vector with the unscaled values of k (zi , xi ),
k̄ (zi ) contains the scaled values and is calculated with Eq.
13. The matrix B is calculated as

T 
k (zi , x1 ) (x − x1 )
 k (zi , x2 ) (x − x2 )T 


B=
(27)

..


.
T
k (zi , xm ) (x − xm )
(28)
T
(32)
The variables are sampled every minute and 100 samples,
taken under normal conditions, are used as the training set.
Four different faults are introduced into the system. The first
3 faults are sensor faults similar to the ones shown in Choi et
al. [4]. The first simulated fault, Fault 1, is a bias in the sensor
of the output temperature T ; the bias magnitude is 1(K).
Since T is a controlled variable, the effect of the fault will
be removed by the PI controller and its effect will propagate
to other variables. This fault is considered a complex fault
7025
100
1
0.5
0
0
50
Measurement
ϕ
100
ϕ value
40
20
0
0
Fig. 2.
50
Measurement
100
2
T value
SPE value
SPE
1
0
50
Measurement
2
T
100
1
0.5
0
0
50
Measurement
ϕ
100
40
20
0
0
Fig. 3.
0
1 2 3 4 5 6 7 8 9 10
Direction
2
T
0.2
0.1
0
1 2 3 4 5 6 7 8 9 10
Direction
ϕ
4
2
0
1 2 3 4 5 6 7 8 9 10
Direction
Fault detection indices and RBC values for Fault 1
2
0
Average RBC
50
Measurement
2
T
2
50
Measurement
100
SPE
Average RBC
0
SPE
4
0.5
Average RBC
0
Average RBC
1
Average RBC
2
0.1
1
0
1
2
3 4 5 6 7 8 9
Direction
2
T
1
2
3 4 5 6 7 8 9
Direction
ϕ
1
2
3 4 5 6 7 8 9
Direction
0.2
Average RBC
2
T value
SPE value
SPE
ϕ value
since it affects several variables. The second and third
faults, Fault 2 and Fault 3, are biases in the sensors of the
inlet temperature To and inlet reactant concentration CAA ,
respectively; the bias magnitude for T0 is 1.5(K) and for CAA
is 1.0 (kmole/m3 ). These faults are considered simple faults
since they only affect one variable. The last fault, Fault 4, is
a slow drift in the reaction kinetics; the fault has the form
of an exponential degradation of the reaction rate caused by
catalyst poisoning; in this case, the reaction rate coefficient
will change with time as k0 (t + 1) = 0.996 ∗ k0 (t). This
process fault is a complex fault that affects several variables
such as the output temperature T , concentration CA , and the
cooling water flow FC .
For each of the fault scenarios mentioned above, 100
measurements are simulated and the fault is introduced at
the 51st measurement. The KPCA model is built using 4
principal components and the parameter c in the kernel
equation 2 is set to 0.65.
Figures 2 to 5 show the fault detection and diagnosis results for the faults. The first column in each set of plots shows
the SPE, T 2 and ϕ indices for the simulated measurements.
The second colum shows the results of fault diagnosis given
by RBC with KPCA for the three fault detection indices. In
Figure 2, it is shown that the SP E and combined indices
detect the fault as soon as it happens, not doing the same
the T 2 index. The first nine bars of the RBC plots, for the
three fault detection indices, show the RBC values of all
variables. Since the bias fault happens in the temperature
sensor, Variable 8, this variable is supposed to have the
largest RBC; however, due to the feedback control, the effect
of the fault is transferred to the cooling water flow, Variable
6, as indicated by the RBCs of SPE, T 2 and ϕ. In this case,
more information is needed to determine the cause of the
fault. Once the cause of the fault has been determined and
stored in a library of known faults, its direction can be used to
calculate the RBC value along this fault direction. The tenth
bar in the right column of Figure 2 shows the RBC value
for the direction of Fault 1. We can see that this fault has
the largest RBC value with the three fault detection indices.
The detection and diagnosis results for Fault 2 are shown
in Figure 3. Since this is a bias fault in T0 , Variable 2, it is
expected that this variable has the largest RBC value, which
actually happens for the SP E and ϕ indices. Again, the T 2
index does not detect nor diagnose this fault correctly. Figure
4 shows the results for the fault in the sensor CAA , Variable
3. In this case all detection indices show CAA as the cause of
the problem, however the SP E and combined indices make
a better detection of the fault.
In Figure 5 we can see the detection and diagnosis results
for the slow drift on the reaction kinetics. In this case the
SPE and combined indices start growing up slowly until they
become very large compared to their control limits; this is
not the case of T 2 since it does not change much. Looking
only at the first nine bars of the diagnosis results, they would
point to the cooling water flow, Variable 6, as the origin of the
fault. However, if we have information of this fault from past
experience, we can calculate RBC values along its direction;
0
2
1
0
Fault detection indices and RBC values for Fault 2
the average of these values is shown in the last bar of the
plot. It can be seen that the RBC values are the largest ones
along this direction; thus, we can diagnose this fault as a
drift in the kinetics.
VI. C ONCLUSIONS
It is shown in this paper that reconstruction-based contribution can be used with kernel PCA for the diagnosis of
faults in nonlinear processes. The proposed RBC method
works like the standard contribution plots in linear PCA,
which does not require the fault direction to be known
beforehand. Further, contributions along sensor directions
as well as known process fault directions can be obtained
with RBC. The combined index is proposed to monitor the
whole feature space and new control limits are proposed for
the three fault detection indices in the feature space. The
detection and diagnosis power of the proposed methods has
7026
2
T value
1
0.5
0
0
50
Measurement
ϕ
100
ϕ value
40
20
0
0
Fig. 4.
50
Measurement
100
SPE value
SPE
1
0
50
Measurement
2
T
100
T value
1
2
0.5
0
0
50
Measurement
ϕ
100
ϕ value
40
20
0
0
Fig. 5.
0.5
0
1
2
3 4 5 6 7 8 9
Direction
2
T
1
2
3 4 5 6 7 8 9
Direction
ϕ
1
2
3 4 5 6 7 8 9
Direction
1
0
1
0.5
0
Fault detection indices and RBC values for Fault 3
2
0
Average RBC
100
Average RBC
50
Measurement
2
T
50
Measurement
100
Average RBC
0
Average RBC
0
0.5
SPE
0.4
0.2
0
1
2 3 4 5 6 7 8 9
Direction
2
T
1
2 3 4 5 6 7 8 9
Direction
ϕ
1
2 3 4 5 6 7 8 9
Direction
0.1
0.05
Average RBC
SPE value
1
R EFERENCES
SPE
1
Average RBC
SPE
2
0
0.2
0.1
0
Fault detection indices and RBC values for Fault 4
been tested with the simulation of a CSTR process with
sensor and process faults. It was showed that, while the
SPE and T 2 indices sometimes disagreed on their results, the
combined index was effective to correctly diagnose simple
and complex faults.
Several open issues deserve further study. One is the high
dimensionality of the feature space when the number of
samples is very large. Another issue is the selection of the
radius of the kernel function. This parameter, while being
determined by trial and error at the moment, affects the
nonlinearity of the mapping from the data space to the feature
space.
VII. ACKNOWLEDGEMENTS
We appreciate the financial support from the Roberto
Rocca Education Program and from the Texas-WisconsinCalifornia Control Consortium.
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of analysis of variance problems, I. effect of inequality of variance in
the one-way classification. Ann. Math. Statistics, 25:290–302, 1954.
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VIII. A PPENDIX
A. CSTR Simulation Parameters and Initial Conditions
The CSTR simulation parameters are V = 1(m3 ), ρ =
106 (g/m3 ), ρC = 106 (g/m3 ), E/R = 8333.1 (K), cp = 1
(cal/g K), cpc = 1(cal/gK), b = 0.5, k0 = 1010 (m3 /kmole
× min), a = 1.678 × 106 (cal/min K), ∆Hrxn = −1.3 × 107
(cal/kmole). The parameters of the temperature PI controller
are KC = −1.5 and TI = 5.0.
The initial conditions for the simulation are T0 = 370.0
(K), TC = 365.0 (K), FC = 15 (m3 /min), T = 368.25
(K), Fs = 0.9 (m3 /min), FA = 0.1 (m3 /min), CA =
0.8 (kmole/m3 ), CA S = 0.1 (kmole/m3 ), CAA = 19.1
(kmole/m3 ).
Gaussian noise with small variance was added to the
measurements and disturbances. For more details of the
simulation check [15].
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