A CERTAIN METHOD OF USING DIAGNOSTIC
INFORMATION FOR TECHNICAL RISK
ANALYSIS
Gumiński R., Radkowski S.
Warsaw University of Technology, Institute of Machine Design Fundamentals, Warsaw, Poland
Abstract: The paper presents the need for increasing the certainty and precision of probabilistic models, both in risk analysis as well as in simulation models that are helpful during regular operation. Such possibilities are presented by accounting in the analysis for the information derived during inspections or in operational diagnosis while applying proportional models or Bayes estimation. The paper also points to the need for extending the proportional models to include time-dependent models.
1. Introduction
In the case of currently used systems, which are increasingly expanded and complex and which store and process more and more goods and energy, the emergence and development of a defect may result in serious consequences for the goods, the natural environment as well as the health and life of people. As a result there is a lot of interest in the work on the safeguards, methods of detection, as well as organization of diagnostic and operational systems aimed at reducing the uncertainty of technical risk estimation . For that reason, the existing operational systems often have the task of forecasting the condition of an object, providing projections of problems and relevant responses in the future. Such systems are very extensive and the high number of the tasks they must perform makes them very costly. A system of such a type often constitutes a part of the so-called autonomous logistic structures (Fig. 1). The structure of such a logistic system requires a given object to be monitored by its own diagnostic system while the information is conveyed to the base. Once the operational requirements and forecasts are taken into account they are used in the decision-making process related to performance of subsequent measurements and potential repairs, often while accounting for the simulation of further development of a defect and forecasts of its consequences.
The essence of such an approach is the right selection of defect-oriented simulation models that reflect as precisely as possible the system’s behavior and the current condition of an object. The autonomous logistic structure also includes the procurement/supply unit, which enables maintaining of relevant stock of materials, return of the faulty goods and sending of the information on defects to manufacturers. All activities undertaken in such a system are precisely coordinated. Prognostics and Health Management (PHM) of an object plays the key role in the system. The goal of PHM is to:
increase reliability and safety of objects;
reduce the cost of maintaining of the stock and repairs; rationalize headcount;
eliminate redundant inspections;
optimum response time in the process of maintenance and delivery of spare parts; eliminate automatically errors and deficiencies/defects;
provide information on the time of maintenance operations for all levels of the logistic structure;
define the probability of undesirable events with potentially catastrophic consequences;
detect defects in their initial stage and monitor defect development; possibly reduce the number of maintenance operations.
Fig. 1. Autonomous logistic structure in operation [1]
The unit responsible for managing the condition of an object has the task of making the decisions regarding operation of the object while relying on diagnostic information, forecasts, available resources and user requirements. What is important in the case of such a task is the analysis of technical risk, with a possibility of including in it the additional information on the conditions of operation and the state of an object obtained as a result of diagnosis.
The key to effective application of risk analysis in autonomous logistic systems is its broad application at the stage of designing of products and systems. Increasingly more often it results from the need to fulfill the guidelines of the relevant EU directives. So far the issues of safety and total cost at the design stage have been included in the comparative analysis of variant profiles, which accordingly demonstrate preference for a selected scope of maintenance and repair works as a function of reliability Rn, ease of
maintenance Ro and operational susceptibility M (Fig. 2). Introduction of risk analysis
procedures at the product design stage means the need for analysis of potential types of defects, the effects of their occurrence, most often while including the FMECA – Failure Modes their Effects and Criticality Analysis. It is also worth conducting broader evaluation, already at the design stage, of the ability to diagnose the types of defects while relying on the results of Failure Mode Symptoms Analysis (FMSA) and on the relationship: defect – symptom – functional task (Fig. 3) [3].
design maintenance based small range of maintenance reliability Rn maintenance Ro large range of maintenance operational susceptibility M maintenance based object
CF(m×n) Component-failure mode matrix EC(r×m) Function-component matrix CFD Component-failure mode -diagnostic matrix EFD Function--failure mode-diagnostic matrix EF(r×n) Function-failure matrix
Fig. 3. A diagram showing the analysis of defect – symptom – functional task relationship The basis of the approach is the introduction of categorization of defects. Such an approach, which is classical in the FMEA method, has so far not found recognition in mechanical engineering. The publications devoted to defect modeling indicate that most often we assume in the adopted models such a nature of complexity of inputs, both external and internal, so as to enable the adoption of exponential distribution of durability of an object and of its components [4,5]. Also extending this method of modeling to examination of the impact of many other defects, which permanently fulfil the conditions of exponential distribution, enables us to present the final effect of the reaction/interaction as a function of systemic intensity of defects whose form is similar to the function obtained for reliable structure of a serial system which maintains the form of an exponential system. Engineering practice shows that such a model, which is correct in many cases, especially when random factors dominate, cannot serve as the basis of making operational decisions when the function of defect intensity changes in time. At the same time, adopting a model of factors leading to changes of durability with an exponential nature excludes, according to the known property of such a distribution, the need for diagnosing this property of the system [6].
Essentially: 1 t ) 2 t 1 t ( 1 2 1 1 1 2 1 1 2 1
e
e
)
t
T
(
P
)
t
t
T
(
P
)
t
T
(
P
)
t
T
t
t
T
(
P
)
t
T
/
t
t
T
(
P
(1)Thus: 2 t 2 2
)
R
(
t
)
e
t
T
(
P
(2)In this case, irrespectively of whether the element was operational at time t1, the
probability of defect occurrence is resolved only by the total time of use t2 and the value
of defect intensity, which is constant in time. Thus the only theoretically justified type of diagnosis is the diagnosis involving inspection of the quality of manufacturing and assembly.
Such diagnosis, which is closer to the analysis of processes of degradation and wear and tear, especially the non-linear relationship between defect development and the time of operation, inclines many researchers to make attempts of including the diagnostic information in reliability evaluation. Selected issues related to this scope were presented by the authors of this paper during the most recent Reliability School in Szczyrk [7]. The presented review indicates that the issue of categorization of defects is strictly correlated with the selection of the methods of defect development prediction and the relevant algorithms for detection of diagnostic information. The issue attracting the biggest interest from the point of view of value of diagnostic information is the use of the results of a diagnostic experiment for explaining the mechanism of influence of random factors. While solving the task formulated in such a way we usually assume that we know the classes of probability distribution and the experiment itself can be restricted to estimation of the distribution's parameters. In a situation when there exist grounds for examining the degree of influence of various system variables, the function of defect intensity can be assumed to be a function with two independent variables: the base function of defect intensity, which is constant in time or which is time dependent, and intensity of defects as a function of systemic variables. Most often the function of defect intensity has the form of a product. Let us note that generally this function can be analyzed as a conditional random variable. The conditions can be imposed by both, the relevantly defined external and internal factors, maintenance and repair activities, as well as the phase of defect development. Adoption of the last model results in use of Bayes’ formulas, used to determine the influence of a posteriori information in the examination of probability distribution parameter.
Let us consider the problem of estimation of the probability density function while using the conditional probability model f(x/W) for a given set of data D={x1,...,xN}. Since the
proper value of parameter W is uncertain, thus in the case of Bayes’ model we assume that the analyzed figure is a random variable with a defined distribution as per Jaynes’ rule [8]: if there are no reasons for adopting the a priori distribution, then monotonous distribution should be assumed. When observing the D data set, we can see the updating of the data for conditional a posteriori distribution f(W/D) as per Bayes formula:
) D ( f ) W ( f ) W / D ( f ) D / W ( f (3)
A good illustration of the discussed method of using Bayes formula to evaluate the parameters of distribution based on diagnostic information is demonstrated by Cruse in [9] where Bayes theorem is used to determine the value of the parameters describing the growth of fatigue-related defect while accounting for the observation of crack development.
The essence of this approach involves updating the estimated parameters of a probabilistic model in order to achieve higher compliance of results of modeling and observation. In accordance with the assumptions presented above, we assume that the parameters that are either unknown or uncertain are random variables. The uncertainty of estimation of parameters can be associated with the changes of random variables with the use of Bayes theorem.
Then, while assuming that we will estimate the parameters of the a priori distribution of parameter a-f(a) and that D is an observation set, the parameters of the a posteriori distribution will have the following notation:
) D ( f ) a ( f ) a / D ( f ) D / a ( f (4) where: f(D/a)f(a)da ) D ( f
Additionally it is assumed that the denominator which contains the integral of the function of the a posteriori probability density is constant and equals 1 and that f(D/a) is the probability of observation that can be expressed by a credibility function. Equation (4) can be expressed in the following form:
) a ( f ] a / D [ L K ) D / a ( f B (5) where: B
K - the normalizing constant
] a / D [ L - credibility function
In a situation when the observations enable both the detection of a defect or the defection of lack of such a defect, then the credibility function for a probabilistic model of a crack developing due to fatigue presents the probability of defect for a defined number of cycles
N)
P(Nj . Accounting for the fact that the distribution of the values of the crack is
exponential, the obtained number of cycles until a defect appears depends on the function of intensity of defects. The credibility function has the following form in such a case:
n 1 i m 1 j sj i f/
)
1
P
(
N
/
)
N
(
p
/
B
L
(6)where:
n – denotes the set of detected defects,
m – denotes the set of events determining non-existence of a defect.
Thus by using formula (5), the a posterior intensity of defects will expressed by the following relationship: ) ( f ] / D [ L K ) D / ( f B (7)
As has been indicated in the quoted paper, such an approach allows reduction of uncertainty of evaluation conducted on the basis of a small number of results obtained in comparable conditions.
Let us note that Bayes formula (4) can be written as a ratio of a posteriori and a priori distribution: ) a ( f ) a / D ( f ) D ( f ) a ( f ) a / D ( f ) D / a ( f (8)
In addition let us note that according to Jeffreys’s law [10] the a priori density of probability is proportionate to the square root of the determinant of the Fisher information matrix: 2 1 )) a ( (det ) a ( f (9) where:
2 2a
a
/
D
f
ln
E
)
a
(
- is calculated as the matrix of average second derivatives of the logarithm of credibility function determined on the basis of the experiment’s results.Thus ultimately formula (7) has the following form:
2 1 )) ( )(det / D ( L ) D / ( f (10)
While analyzing the experience from applying Bayes’ formula in medical diagnosis [11], we can extend the presented method of using the a posteriori information to the task of similar evaluation of results obtained in a diagnostic experiment.
First let us assume that the infinite set of potential states X of the diagnosed object, determined as a result diagnosis during inspections, can be divided into two disjoint subsets X’, X”, with the subset X’ including all the states in which the object fulfils its functional tasks, meaning that we cannot detect any defect (undamaged object), while the subset X” will include all the states in which the object is unable to fulfill its functions, meaning that we detect the emergence and development of a defect (a damaged object). Accordingly the probability of diagnosing an object from the defective subset is denoted as: ) U D ( P 1 (11)
) NU D ( P 2 (12)
Let us assume, in accordance with the Cox model [12] which the authors analyzed, that the decision-making process should account for the influence of systemic variables expressed in the form of vector X. In addition, let us assume that probability p(X) denotes an object which is influenced by the vector of systemic variables. Let probability p0(X)=P(U/D) denote a defective object, diagnosed while accounting for vector X of
systemic variables. Hence, by applying Bayes formula we will obtain the following: ) D ( P ) U ( P ) U D ( P ) X ( p ) D / U ( P 0 (13)
By including formula (11) and notation p(X)= P(U) and while bearing in mind that sets U and NU are disjoint, we will obtain:
) D NU ( P ) D U ( P ) D ( P (14) where ) X ( p ) U ( P ) U D ( P ) D U ( P 1 (15) while )) X ( p 1 ( ) NU ( P ) NU D ( P ) NU D ( P 2 (16) hence, )) X ( p 1 ( ) X ( p ) D ( P 1 2 (17)
Finally, while taking into account relationship (17), formula (13) will have the following form: ) U ( P 1 ) U ( P ) ) D U (( P 1 ) D U ( P 2 1 (18)
Let us note that upon finding the logarithm for both sides of the equation we can obtain expressions compliant with the definition of logit:
)) U ( P ( logit ln ) ) D U ( P ( logit 2 1 (19)
At the same time, in accordance with the definition, logit is a linear function of systemic variables [13]:
n 1 i i i 0X
)
)
D
U
(
P
(
logit
(20) Thus:
n 1 i i i 0 n 1 i i i 0X
exp
1
X
exp
)
D
U
(
P
(21)On the assumption of proportionate model of defect intensity and by conducting the diagnostic experiment in such a way so that it explains the dependence of probability of an object’s defect on systemic variables, formula (21) can be noted in the following way:
) x , t ( ) x , t ( ) x , t ( ) D U ( P 0 (22)
3. Conclusions
The quality of manufacturing and assembly as well as the occurring degradation processes can on the one hand lead to changes of intensity of defects and on the other be the reason of change of probability distribution parameters in the function of time and change of systemic variables. Use of vibroacoustic diagnosis methods offers a possibility for accounting for the influence of defect development phases, treated as systemic variables by means of vibroacoustic diagnosis, on the intensity of defects in the examined elements of kinematic nodes. Use of proportionate models is possible in diagnosis conducted during inspections while operational diagnosis requires use of models in which the intensity of defects depends on two values: the systemic variable and the time of operation. Use of a posteriori information obtained as a result of observation or active diagnostic experiment and application of Bayes model leads to reduction of uncertainty while estimating the impact of the degradation phase on an element’s reliability and while estimating the level of technical risk.
The research is financed from the funds of the Ministry of Science and Development of Information Technologies for years 2005-2008 as a requested research project
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