Saint-Petersburg State
Technological Institute
FAULT DIAGNOSIS IN CHEMICAL
PROCESSES AND EQUIPMENT
WITH FEEDBACKS
L.A.Rusinov
N.V.Vorobjev
V.V.Kurkina
I.V.Rudakova
FAULT DIAGNOSIS IN OBJECTS
WITH FEEDBACKS
The subject-matter of my report concerns the
application of chemometrics approach to solving
industrial problems relevant in particular to
developing diagnostic systems for fault diagnosing
in difficult cases.
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Why is the diagnostics
necessary?
•
•
•
•
•
The majority of chemical technological processes
refer to the class of potentially dangerous (PDTP) or
hazardous.
As a rule PDTP are characterized by:
high level of uncertainty,
large uncontrollable disturbances,
essential internal nonlinearity,
bad observability.
lack of mathematical descriptions (often).
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Why is the diagnostics
necessary?
The operation of protection systems obligatory
in the PDTP is usually accompanied
• by emergency dumping of a reactionary mass,
• irreversible repressing of a reaction and
• other operations resulting in essential losses.
The early diagnostics can define the faults at their
incipient stages and thus allows undertaking
necessary acts to avoid the protection systems
actuating.
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What is necessary for
diagnostics?
Continious monitoring and fault diagnostics are
carried out on the basis of a diagnostic models (DM)
connecting faults in the process under control
(abnormal situations) with their observable
symptoms.
For this reason, mathematical descriptions of
PDP cannot often be used as DM because they are
usually valid only in PDP working zones and
unsuitable for abnormal situations.
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Classification of diagnostic
models
Diagnostic Models
Quantitative
Models
Observers
Parity
Space
EKF
Qualitative
Models
Causal
Models
Digraphs
Process
History Data
Models
Abstraction
Hierarchy Qualitative
Quantitative
Expert Fuzzy
Systems expert
systems
Statistical
Functional
Structural
Fault
Trees
PCA/PLS
Neural
Networks
Statistical
Classifiers
V. Venkatasubramanian, R. Rengaswamy, K Yin, S.N. Kavuri, Computers and Chemical
Engineering 27 (2003) 293-311)
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THE MASKING EFFECT FROM THE FEEDBACK
Control action
100%
0%
Regulator has
exhausted its
resource
The object of
diagnostics
The sensor signal
PROCESS
Recycle
or
Regulator
Regulator has
exhausted its
resource
Emergency situation
ABNORMAL SITUATION
Fault development
Normal regime
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THE WAY OF SOLVING THE PROBLEM
For solving the problem, the diagnostic model
(DM) should be built for the process section or
equipment in the closed loop formed by
feedback (further - object).
Deviations from this model can be used in
detecting the fact that an abnormal situation
has arisen.
However, it is impossible to identify a cause of
the fault while using deviations from the
model.
For this purpose, the models describing all
possible abnormal situations are required.
As a result, it is necessary to have a bank of
models
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THE STRUCTURE FLOW CHART
OF THE OBJECT DIAGNOSTIC SYSTEM
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THE FLOW-CHART OF THE
DIAGNOSTICS’ PROCEDURE
1. THE OBJECT MONITORING
FAULT DETECTION
THE FAULT DETECTION CRITERION:
i>;
-
threshold
2. INITIATION OF THE MODELS OF THE BANK DESCRIBING OBJECT FAULTS
3. THE FAULT REASON (FR) IDENTIFICATION:
Restrictions of the method:
1. The necessity of the presence of diagnostic models,
easy-in-use in real time.
2. The necessity to obtain the knowledge of all possible
faults of the object under control.
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THE FUZZY DIAGNOSTIC MODEL
Each fault Fi is described by fuzzy rule Ri of Takagi-Sugeno
type:
Ri : IF x1 == Ai1 AND... AND xn = = Ain,
THEN yi = ai1x1 +... + ain xn + bi,
where Ri – i-th rule of fuzzy model, i [1,k]; k – the number of
rules;
X = {xkj} - a matrix of input variables samples [Nxn]; k 
[1, N]; N – number of samples; j[1,n]; n - number of inputs;
Аi = {Aij} - the fuzzy terms-sets that enter into the
conditional part of each i-th rule;
yi - an output of i th fuzzy rule;
ai = [ai1, …, ai n] and bi - parameters of fuzzy model:
Ti=[aTi, bi].
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FLOW-CHART OF COMPUTING THE RESULT OF
FUZZY MODEL
The result of fuzzy model computing is determined by
combination of contributions of all rules in a common
k
inference:
y 
 
i
yi
i 1
k
 
i
i 1
where βi - the degree of activation of i-th rule that is
determined by max-product composition:
n
i  П A (xj),
j 1
ij
 A ( x j )  [0,1] - the contribution of each term-set Aij to the
conditional part of the rule Ri.
ij
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FLOW-CHART OF COMPUTING THE RESULT OF
FUZZY MODEL
Coefficients of a right part of rules are determined by solution
of the system equations by means of weighed МLS:
The solution is:
y=Xch θ ; Xch=[X,1]
1
 i  [ X ch W i X ch ] X ch W i y
T
T
 A ( x j ) - the diagonal matrix with the degrees of activation β
ij
on its diagonal
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THE DETERMINE OF MEMBERSHIP
FUNCTIONS.
By means of fuzzy clustering of the data object array,
for example, by the algorithm of Gustafson-Kessel, the
number of clusters с and the matrix of their fuzzy
separation U are defined.
Membership functions of fuzzy sets in the conditional
part of rules are extracted from the matrix U which
(g, s)-th member mgs[0,1] characterizes the value of
membership of an input-output combination in s-th
column in the cluster g.
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THE DETERMINE OF MEMBERSHIP
FUNCTIONS.
To obtain one-dimensional fuzzy set Ggj, the
multidimensional fuzzy sets, defined pointwise in g-th
row of the separation matrix U, are projected into input
variables space Xj.
Resulting fuzzy sets Ggj are usually nonconvex. To
obtain the convex (unimodal) fuzzy sets, approximating
by appropriate forms of membership functions (for
example, Gaussian) is needed.
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MEMBERSHIP FUNCTIONS OBTAINED BY
CLUSTERISATION
6 clusters – by threes clusters for a forward and reverse valve strokes
M
E
M
B
E
R
S
H
I
P
1
2
3
4
5
6
F
U
N
C
TI
O
N
S
THE NORMALIZED VALUES OF INPUT VARIABLES
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THE DIAGNOSTIC MODEL BASED ON
THE KALMAN FILTER
In this case DM is developed in the space of object states.
The Kalman filter is actually searching for an optimum
estimate with the least-squares method.
The linear object model for Kalman filter is of the form:
x ( k  1)  A  x ( k )  B  u ( k )  n
y ( k  1)  C  x ( k )  w
where x(k) is the state vector,
y(k) - the vector of filter output variables at the kth step,
x(k-+1) - the predicted (extrapolated) state vector value at
the (k+1)th step;
A, B, C - are known prediction, control and observation
matrices;
n,w – noises.
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FLOW-CHART OF COMPUTING THE
RESULT OF KALMAN MODEL
The matrix of filter gain factors is given as:

K ( k )  P ( k )  H  [ S ( k )]
where: P  ( k )


T
1
, S(k) – correlation matrixes
P ( k )  А  P ( k  1)  А  w k
T
S (k )  C  P  (k )  C T  v k
The specified estimation for the system state vector
is:
x  (k )  x  (k )  K  Re sidual(k); Re sidual(k)  y(k )  C  x  (k )
And finally, the specified covariance matrix of
estimation of the system state vector is given in the


form:
P ( k )  I  K ( k )  C   P ( k )
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THE DIAGNOSTIC MODEL BASED ON
THE EXTENDED KALMAN FILTER
For nonlinear objects the model is of the form:
x ( k )  f ( x ( k  1), u ( k  1))
y ( k )  h ( x ( k  1), u ( k  1))
In this case, the filter does not use fixed matrices A(x) and
C(x), but linearizes them recursively based on the previous
state estimate with the use of matrices of first partial
derivatives of the state equations:
Аi , j 
C i, j 
 f i ( x ( k  1), u ( k  1))
 x j ( k  1)

x ( k 1 )  x ( k )
 h i ( x ( k  1), u ( k  1))
 x j ( k  1)

x ( k 1 )  x ( k )
These matrices are calculated at every step and then
inserted into the standard Kalman filter formulas.
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CASE STUDY
Case study was carried out on two types of
objects:
1. The object in control loop electropneumatic valve with the positioner
2. The object in recycle circuit - Tennessee
Eastman process
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THE STRUCTURE FLOW CHART
OF THE ELECTROPNEUMATIC VALVE WITH
THE POSITIONER
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THE STRUCTURE FLOW CHART
OF THE TENNESSEE EASTMAN PROCESS
*) Applicability of the method to processes with recycles is presented in the
Vorobiev’s poster presentation.
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THE ELECTROPNEUMATIC VALVE WITH THE
POSITIONER
The POSITIONER has the mathematical model
developed in the European Interuniversity Project
DAMADICS. 19 various positioner faults have been
considered by the model.
But it does not fulfill to the first restriction for
diagnostic models of objects of the class under
study: it is difficult and not easy-to-use in real
time.
So, models on the basis of fuzzy logic and Kalman
filtering are developed and the DAMADICS model
was used for their training.
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MODELED POSITIONER FAULTS
F2
ABRUPT
FAULT
«FLUID BOILING UP IN THE
VALVE CAVITY AT THE
EXTREME FLOW RATE»
F1
INCIPIENT
FAULT
«SEDIMENTATION»
Input
variables
CV1(k), CV1(k-1), CV1(k-2) –
Control signal values entered
from controller at kth, (k-1)th
and (k-2)th steps
ZT – position of the valve
plunger at the kth step
Output
variable
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FT – The flow rate through
the valve
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OUTPUT RESIDUALS OF FUZZY MODELS
(INCIPIENT FAULT F1)
DETECTION
RESIDUALS ,%
(MODEL OF
NORMAL REGIME)
time
RESIDUALS ,%
(MODEL OF
FAULT F1
time
RESIDUALS ,%
(MODEL OF
FAULT F2
IDENTIFICATION
time
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OUTPUT RESIDUALS OF FUZZY MODELS
(ABRUPT FAULT F2)
DETECTION
Forward
plunger
stroke
RESIDUALS ,%
(MODEL OF
NORMAL REGIME)
time
Reverse
plunger
stroke
RESIDUALS ,%
(MODEL OF
FAULT F1
time
RESIDUALS ,%
(MODEL OF
FAULT F2
IDENTIFICATION
time
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OUTPUT RESIDUALS OF MODELS BASED ON
THE KALMAN FILTERS (INCIPIENT FAULT F1)
DETECTION
RESIDUALS ,%
(MODEL OF
NORMAL REGIME)
time
RESIDUALS ,%
(MODEL OF
FAULT F1
time
IDENTIFICATION
RESIDUALS ,%
(MODEL OF
FAULT F2
time
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OUTPUT RESIDUALS OF MODELS BASED ON
THE KALMAN FILTERS (ABRUPT FAULT F2)
RESIDUALS ,%
(MODEL OF
NORMAL REGIME)
DETECTION
time
RESIDUALS ,%
(MODEL OF
FAULT F1
time
IDENTIFICATION
RESIDUALS ,%
(MODEL OF
FAULT F2
time
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CONCLUSIONS
Сhemometrics methods are very effective for execution
the monitoring and diagnostics of technological processes
in chemical and related industries, even in difficult cases
at diagnostics of the objects in circuits with feedbacks
because of feedback masking effects
Statistical methods allow constructing diagnostic
models on the base of the history process data, not
demanding the knowledge of process chemism and the
presence of its mathematical descriptions.
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CONCLUSIONS
For diagnosing such faults, it is suggested to use the
bank of diagnostic models describing normal operation of
the objects under control and their operation when faults
are available.
Applicability of the method is illustrated by the example
of system development with two types of diagnostic
models: the model with fuzzy rules of Takagi-Sugeno type
and on the basis of extended Kalman filters. Both models
have demonstrated approximately equal results when
diagnosing both incipient and abrupt faults.
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