Instrumentation & Measurement Magazine 26-5 - 4

Kalgren et al. also provide a definition of PHM. They
say PHM is " a health management approach utilizing measurements,
models, and software to perform incipient fault
detection, condition assessment, and failure progression prediction "
[7]. Their view includes incipient fault detection and
condition assessment, which ties back to the current health
state of the system. However, their views related to failure progression
prediction largely depend upon physics-of-failure
models, which are neither generalizable nor scalable in complex
systems.
Li et al. pose PHM more literally, as we do. " Prognostic and
Health Management (PHM) systems support aircraft maintenance
through the provision of diagnostic and prognostic
capabilities, leveraging the increased availability of sensor
data on modern aircraft. Diagnostics provide the functionalities
of failure detection and isolation, whereas prognostics
can predict the remaining useful life (RUL) of the system " [8].
In this definition, diagnostics are limited to on-board systems,
and prognostics are focused on RUL. We adapt this idea to consider
off-board diagnostics and time-to-failure.
We also consider the ideas expressed in the recently
approved IEEE Standard 1856, which divides the definition
of PHM into two parts [9]. First, the standard defines
prognostics to be " the process of predicting an object system's
RUL by predicting the progression of a fault given the
current degree of degradation, the load history, and the anticipated
future operational and environmental conditions
to estimate the time at which the object system will no longer
perform its intended function within the desired specifications. "
Once again, the focus is on remaining useful life and
on failure progression, which would largely be from a point
of failure perspective. Second, the standard defines health
management as " The process of decision-making and implementation
of actions based on the estimate of the state of
health derived from health monitoring and expected future
use of the system. " This is good in the sense that the dependence
is on state of health, but the definition excludes the
health assessment itself.
We have previously asserted that all aspects of health assessment,
including fault detection, localization, isolation,
and even determining there are no faults, are diagnostic processes
[10]. We assert that PHM begins with diagnosis and then
proceeds to determine when future failures might occur (prognosis).
We like to refer to prognostics as predictive diagnostics
in that we also want to know what faults are occurring when.
This sets up a pipeline process whereby PHM consists of a
sequence of five steps: monitoring; health state assessment (diagnosis);
prediction (prognosis); assessment; and action. This
results in an evidence-based decision-making process that
leads to the overall support of the system.
Risk-based PHM
Motivated by Vichare and Pecht, who draw on reliability information,
we employ a " risk-based " approach to PHM (rPHM).
We seek to introduce a framework that includes both diagnostics
and prognostics and incorporates effects or hazards
4
using the same model semantics. By building hazards into
the model, predictions can be made about the risks associated
with likely faults and downstream results of those faults.
Our approach incorporates user-specified performance functions
(i.e., utility functions) that place value on various system
states, which allows one to assess potential impact on mission
outcomes should hazards be realized or averted. Hence,
the framework also allows modeling of risk mitigation strategies
to be employed directly into the decision-making process.
Our approach combines two types of models, one focused on
diagnostics and another on prognostics. We use BNs for diagnostics,
allowing us to estimate (with uncertainty) the current
health state of the system. Once health state is determined, we
use this as " virtual evidence " in a companion CTBN model to
reason through time.
Bayesian Networks
Here, we provide a brief introduction to BNs. A BN is a
graph-based representation of a joint probability distribution.
Given a set of random variables XX Xn
= ...{}1,, , the
BN provides a compact representation of joint distribution
PX PX Xn
() = ...()1,,
by applying the product rule of probabilities
and properties of conditional independence among the
variables. A BN can be regarded as a " factored " representation
of the joint distribution corresponding to:
P X X
()... = ∏ ()P X Pa X
XXi∈
1,, n
ii
| ()
Representing conditional probabilities PX Xij|
(1)
() in a directed
acyclic graph, the vertex for Xj is connected by an
outward directed edge to the vertex for Xi, in which case we
say Xj is a parent of Xi (i.e., X Pa Xji∈ ( )). The graph structure,
combined with a parameterization of the local distributions
for each random variable Xi, corresponds to the specification
of a BN.
Continuous-Time Bayesian Networks
For predictive modeling, we use CTBNs. At the heart of a
CTBN is a Continuous-Time Markov Process (CTMP). A
CTMP is a model over continuous-time random process X ,
consisting of two parts: an initial distribution PX 0() and a transition
intensity matrix QX defined over the states of X. The
entries qij,
in QX govern the rate of transition from state xi to
state xj as a function of time. The i th diagonal entry, denoted
qi, is constrained to be the negative sum of the rest of the row
(i.e., qqi
= −∑ ≠ , ). The distribution indicating if the process
j i ij
remains in state i is exponential with rate qi:
f q
qi qti
i = −
e ()xp .
(2)
Conditional on a transition out of state i occurring at time
tX, transitions from state xi to state xj according to a multinomial
distribution with probabilities:
Px x t|, .
()= −
ji
IEEE Instrumentation & Measurement Magazine
q
q
j
i
(3)
August 2023
Instrumentation & Measurement Magazine 26-5

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