28 having intensity function
t n
, then t
N is a Poisson random variable having mean
t Z
, where t
Z is the cumulative intensity function.
t
du u
n t
Z
2-6
Consider that the system has an increasing intensity function, which tells us that failures will be likely to occur more frequently and at some time t it will become more
economical to replace the system. To find the optimum value of t ,
t
, we minimize the cost model,
1 ]
[ t
Z C
C t
t C
E
f p
2-7
where
p
C
is the cost of replacingpreventive the system.
f
C
is the cost of a failure.
Such applications can be found in Aven 1996, Ascher et al. 1995, Percy et al. 1998 and Watson et al. 2002. PIM is just an extension of PHM Kumar, 1996, which also
assumes that the covariates will influence the intensity, hence it experiences the same shortcomings as those of PHM discussed above.
2.8.3 Markov Models
Another approach that is widely used for modelling CBM uses Markov models. In all cases of condition monitoring, the condition of a technical unit is changing continuously.
To use a Markov model, the deterioration process is normally assumed to be described in terms of a limited number of condition states, such as ‗normal‘, ‗defective‘ and
‗failure‘.
Figure 2-6
illustrated the concept of Markov model with a transition probabilities link the current state with a maintenance action to a future state.
29
Figure 2-6: Illustration of 3 states Markov model
A transition probability
13 21
12
,... ,
P P
P
is defined as the probability that a component will move from one state to another, depending on the action taken. The Markov
property assumes that the probability of deteriorating to another state does not depend on the history of the process, which implies that the future lifetime of a
componentsystem does not depend on how long it has already operated. Applications of this approach in CBM can be found in Coolen and Dekker 1995 and Hontelez et al.
1996 . However, the ‗memoryless‘ property of this approach, which considers the
future condition as dependent solely on the current information, without using the whole history of observation, limits the capability of this technique.
As an extension of the Markov model, a general model of the deterioration process pioneered by Thorstensen and Rasmussen 1999, can also be used to estimate the
deterioration function of an asset and then incorporate a stochastic component to represent the failure of the system. Thorstensen and Rasmussen 1999 present the
deterioration process at time t,
t
as
U t
b t
g
t
1
, ~ N
U 2-8
where t
g represents the deterministic deterioration function of an asset and
U t
b
represents the stochastic nature of the deterioration. The deterioration function is then divided into
M
condition levels in which condition level 1 represents the new condition of the component, while condition level M represents a total breakdown. Here, the
concept of the Markov model is used and the boundary of each condition level is introduced. This model looks interesting, but as future deterioration is independent of
the deterioration in current time units, it has similar drawbacks to the original Markov model. It is noted that the Markov model assumes that the observed information
30 completely describes the system state while in most CBM cases this assumption does
not hold.
2.8.4 Stochastic Filtering
