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Reliability and fault diagnosis in industrial
systems
methodology summary
by Gláucio Bastos, M.B.A., Ch.E.

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abstract
 target: presentation of the advantages and
efficiency of Bayesian networks (BN) in the
formulation of reliability models for the cases of
systems with unknown structure, with common
cause failures and redundancy

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need
 different sources of quantitative and qualitative data are
incorporated into the Bayesian models, considered prior
probabilities or a priori information
 the evaluation of the reliability parameters of the Bayesian
hierarchical model method obtains a higher representation
using the Weibull distribution and probability density
function (PDF) of occurrence of failures of the generic
exponential distribution because it allows the modeling of
different regions of lifetime curve of a large number of
components
 if the probabilities a priori are not known, may be defined by
statistical sampling techniques or directed learning methods

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need
 continuous decision-making variables, which in the Bayesian
concept of fault diagnosis are the posterior probabilities of
failures are of great interest for monitoring the degradation
of components
 these variables can be used for tasks such as:
 more intelligent supervision
 preventive maintenance programs
 cost analysis of failures using nodes utilities
 risk-based reconfiguration of defective systems
controlling its overall or partial reliability (prognosis)

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need
 fault tolerant control ensures high availability and security
for current industry systems
 modern automation relates to autonomous system, the
requirements of control performance and overall system
reliability
 fault detection and isolation (FDI) techniques involve detection
in sensor readings of discrepancies or ‘residuals' in relation
to a standard, indicating the occurrence of a failure,
including its type and its location in the system

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Bayesian model for fault diagnosis
 for its structural and causal characteristics, the FDI is performed
with advantage by BN
 in system Bayesian model-based FDI the diagnosis is made by the
association between the reliability of the components in the
process being monitored and residues detected according to the
specifications of the physical model, each of these parts - the 1st
continuous and the other discrete - constituting hybrid BN
because some of the random variables are continuous and the
other discretes where continuous nodes contain a priori
probabilities of failure of components that are used by the
inference process in the discrete part to determine the posterior
probabilities of failures
 this method can be applied in large scale for all types of failure
distribution (herein was used the Weibull) of the system
components

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Bayesian model for fault diagnosis
 the system is composed by n equipments or components
C = {Ci; 1 ≤ i ≤ n} with failure distribution Weibull type
 Bayesian decision model presented in the following figure
contains random variables associated with residuals
r = {rj; 1 ≤ j ≤ p}, components Ci , as well with Bayesian
reliability model of such components
 the arc connecting node Ci to node rj indicates that rj is
sensitive to component Ci fail and it is associated with its
reliability Ri
 to a residual rj there is 02 states:
{D(Detected),ND(NotDetected)} and there is also 02
states {F(Faulty), S(Safe)} to a component Ci

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Bayesian model for fault diagnosis

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Bayesian model for fault diagnosis
 continuous part of the BN allows you to define a priori
probabilities of component failures
 then when a residual is detected at instant t, the component
Ci has a priori probabilities: P(Ci = Faulty) = Fi(t) =
1 − Ri(t), where Fi is the cumulative distribution function
(CDF)
 the discrete part has a structure that depends on the fault
signature (FSM) matrix: a standard for residuals
 when a residual rj is not sensitive to failure of a component
Ci, there is no arc between 02 nodes
 after residuals detection, the posterior probabilities of
failure p(Ci|r1, . . . , rp) can be inferred in the discrete part
of the BN

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Bayesian model for fault diagnosis
* aplication *
 the method is simulated in the system below, formed by T1
and T2 tanks, V1 and V2 valves, L1 (De1) and L2 (De2)
sensors, pump (P), proportional-integral (PI) controller and
controller 'bang-bang‘ K (On-Off)

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Bayesian model for fault diagnosis
* aplication *
 from the parameters of the failure rates, the Weibull type
PDFs of both component reliability and system are shown in
the following figure with their average and HDIs of 95%, and
the decay of its quantile to 90% with the time of operation,
especially the 1st quantile - most critical - showing that after
operating for 20,000 hrs. the overall system reliability falls
to 0.006256

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Bayesian model for fault diagnosis
* aplication *

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Bayesian model for fault diagnosis
* aplication *
 FSM of monitored system is as follows
 which was modified by a priori probabilities of false alarms
(0.05) and non-detection (0.02), considered to be identical
for all components
 it is observed that the flaws in V2 and T2 can not be isolated
because both exhibit identical patterns
 the simulation scenario presents after operating for 20,000
hrs. the following standard for residuals: [r1, r2, r3, r4, r5] =
[0, 0, 0, 0, 1], that matches the pattern of failure for the
V2 and T2 components

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Bayesian model for fault diagnosis
* aplication *
 the figure below shows the result of analysis for isolation of
faulty components, where the a priori probabilities are
determined in the continuous part of the BN and posteriors in
the discrete part
0
0.2
0.4
0.6
0.8
1
L1 L2 P K V1 V2 T1 T2 PI
Prior
Posterior
Classic
Probababilities of failures for the simulated scenario

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Bayesian model for fault diagnosis
* aplication *
 the figure boasts the explicit advantage of application of
hybrid BN in FDI:
 although the probabilities of failure calculated by the
conventional method does not allow to isolate the faulty
component between V2 and T2, they are statistically
identical for each one
 comparing the posterior probabilities defined from the
standard residuals, the highest probability of failure (0.74)
for component V2 relative to that one for the other
component T2 (0.51) indicates the malfunction of V2 as the
most likely cause of failure of the overall system under this
scenario simulation

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diagnosis with dynamic BN
 temporal dependencies between components to reliability
calculations can be modeled by dynamic BN (DBN) with 02
partitions of time, called 2-temporal BN (2-TBN), where the
same model describes the BN to the next partition of the
sample with 02 networks interconnected by arcs
 DBN model has Markovian properties and is therefore only
applicable to Markov processes (MC)
 besides MC other stochastic models like Input-Output Hidden
Markov Model (IOHMM) and, in general, Conditional Markov
Process (CMP) - conditional probability distribution (CPD) in
BN - can be represented by interconnections of a DBN

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diagnosis with dynamic BN
 therefore various types of degradation in dynamic systems
can be modeled by DBNs, which represent this way more
complex types of faults considering the influence of time, as
well as exogenous variables (abrupt changes in operation)
and environmental conditions (eg. humidity, temperature )
 as the DBN is a graphical description of a system evolving in
time, allows monitoring and updating of the system over
time and also predict the subsequent behavior of the
system, hence its application in the field of decision and
fault diagnosis in supervisory activities

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diagnosis with dynamic BN
 in the case of a type 2-TBN, for any variable their transition
probabilities are completely determined by the values ​​of the
variable in the current time phase and in the previous one -
what is the 1st. order Markov property
 for systems with 1st order stationary exponential PDF of
faults this is not guaranteed for the lifetime of a component
with PDF Weibull but this can be bypassed considering there
stationarity for a given time sequence i, which is feasible in
diagnosis in real time, where the sampling period is
extremely small to display the dynamics of residuals

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diagnosis with dynamic BN
* aplication *
 for the diagnosis of 02 tanks system whose static form was
presented in the previous example, is applied the concept of
IOHMM modeled by a DBN, as shown below
 it is a distribution over the states of the external observable
exogenous variable of input U(i)
t−1 that influences the behavior of
the hidden (unobservable) X(i)
t−1 variables which result is observed
through the outputs Y(i)
t−1 which are modes of component failure,
 therefore applies the formalism of
Hidden Markov Model (with
unobservable state) of
Input-Output – IOHMM

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diagnosis with dynamic BN
* aplication *
 the inputs U(i)
t−1 are not considered in DBN out of being the
model hybrid, result of inference in continuous part and
represent the reliabilities of the components assumed
constant throughout the sequence of partitions T of time
investigated
 the states of components X(i)
t−1 are determined by CPD
p(X(i)
t−1|U(i)
t−1)
 the states Y(i)
t−1 resulting from evaluation of residuals rj are
associated to CPD p(Y(i)
t−1|X(i)
t−1)

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diagnosis with dynamic BN
* aplication *
 the current states X(i)
t re calculated using the following
conditional probabilities:
p(X(i)
t = Faulty | X(i)
t−1 = Faulty) = 1,
p(X(i)
t = Faulty | X(i)
t−1 = Safe) = 1 − R(i)
C(T),
p(X(i)
t = Safe | X(i)
t−1 = Faulty) = 0 ,
p(X(i)
t = Safe | X(i)
t−1 = Safe) = R(i)
C(T),
where R(i)
C are components reliabilities estimated
during the sequence T

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diagnosis with dynamic BN
* aplication *
 the DBN model in compact form is shown below:
 the simulated scenario displays an active residue r5 for only a time
partition (02) and then suffers new activation after the time
partition 05, which persists until the end of the sequence

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diagnosis with dynamic BN
* aplication *
 as can be seen in the following figure, there is no diagnostic action for
partition 02, featuring the DBN robustness against false alarms
 when residual remains after the partition 05, the simulation shows
posterior probabilities of component V2 slightly larger than those of the
component T2