Chinese Journal of Aeronautics 25 (2012) 598-604

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Chinese Journal of Aeronautics

journal homepage: www.elsevier.com/locate/cja

JOURNAL OF

AERONAUTICS

A Fault Sample Simulation Approach for Virtual Testability

Demonstration Test

ZHANG Yong, QIU Jing*, LIU Guanjun, YANG Peng

Laboratory of Science and Technology on Integrated Logistics Support, College of Mechatronics Engineering and Automation, National University of Defense Technology, Changsha 410073, China Received 25 September 2011; revised 8 December 2011; accepted 20 December 2011

Abstract

Virtual testability demonstration test has many advantages, such as low cost, high efficiency, low risk and few restrictions. It brings new requirements to the fault sample generation. A fault sample simulation approach for virtual testability demonstration test based on stochastic process theory is proposed. First, the similarities and differences of fault sample generation between physical testability demonstration test and virtual testability demonstration test are discussed. Second, it is pointed out that the fault occurrence process subject to perfect repair is renewal process. Third, the interarrival time distribution function of the next fault event is given. Steps and flowcharts of fault sample generation are introduced. The number of faults and their occurrence time are obtained by statistical simulation. Finally, experiments are carried out on a stable tracking platform. Because a variety of types of life distributions and maintenance modes are considered and some assumptions are removed, the sample size and structure of fault sample simulation results are more similar to the actual results and more reasonable. The proposed method can effectively guide the fault injection in virtual testability demonstration test.

Keywords: fault sample; testability demonstration; virtual testability test; stochastic process; statistical simulation; Monte Carlo; maintenance

1. Introduction

It often takes a long time to obtain natural fault samples and evaluate testability indices. In order to accelerate testability demonstration, fault injection is always applied to the testability test [1-5].

However, application results indicate that testability demonstration test based on fault injection has some problems [1-7]. First, fault injection is always destructive. It is unrealistic to inject numerous faults because of the high cost. Second, some fault injection tests are forbidden because the faults are likely to cause serious accidents. Third, faults cannot be injected effectively

* Corresponding author. Tel.: +86-731-84573398. E-mail address: qiujing@nudt.edu.cn

Foundation item: National Natural Science Foundation of China (51105369)

1000-9361/$ - see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/S1000-9361(11)60424-X

because of physical location restrictions. These problems often lead to the irrational structure of fault samples. It affects the credibility of the demonstration results and brings a great uncertain risk to manufacturers and consumers.

Nowadays, virtual test technology has been gradually developed and becomes more and more mature. It can reduce the test cost and shorten the development periods. Although it is difficult to model complex equipment by current modeling technology, virtual testability test can be carried out on some subsystems or units [8-12].

In the physical testability demonstration test, fault sample selection assumes that all components have the exponential life distribution. It also assumes that the maintenance mode is breakdown maintenance and perfect maintenance [1-4]. Sample size is figured out by experience or statistical sampling formula. Then, fault samples are allocated and fault modes are sampled

based on hierarchical structure and estimated failure rates [1-5].

The fault sample selection is to determine appropriate sample size and to make fault sample structure reasonable. On the one hand, considering the limits of test cost and time cost, the fault sample size needs to be as small as possible. On the other hand, in order to improve the accuracy and precision of test demonstration results, the size needs to be as large as possible. This results in a contradiction [1-3].

Virtual testability test has many advantages, such as low cost, high efficiency, low risk, and few restrictions. The fault sample size is almost unlimited. Similarly, it requires that the fault sample structure should be reasonable. In other words, it should approximate to the actual fault occurrence sample structure. Thus, the method of fault sample generation in virtual testability demonstration test is different from that of physical testability demonstration test.

2. Problem Analysis

widely used. The formula just relates to required indices and the risk values. The fault sample sizes are the same as long as the required indices and the risk values are the same.

However, the actual fault population is associated with the work and environmental stress, equipment reliability and structure, etc. Higher stress, more complicated structure, lower technology maturity and reliability often lead to more faults. In this case, more faults should be injected to evaluate the testability indices. That is similar to allocating more faults for components of high failure rate [1]. In order to reduce test cost, the fault sample size in physical testability demonstration test is usually small. This causes the fact that many faults cannot be covered. The representativeness of fault sample is poor [1, 3-4].

In the virtual testability demonstration test, the fault sample size can be very large and almost without cost constraints, which is the significant difference between physical testability demonstration test and virtual testability demonstration test.

In testability demonstration test, the main indices are fault detection rate, fault isolation rate, false alarm rate. For example, fault detection rate yFD is defined by

rFD = — x 100%

where ND is the number of detected faults, and N the total number of occurring faults.

The definitions of fault isolation rate and false alarm rate are similar to fault detection rate. They are related to actual fault occurrence, fault detection and isolation results. Within a specified time period, both fault occurrence process and fault number are random due to a variety of random factors, such as working stress, environmental stress and parts quality. N and ND are uncertain until the complete statistics are acquired. Hence, fault samples are often used to infer the testability indices.

In physical testability demonstration test, the fault sample size n and fault modes are solved by failure sample selection (n<N). Then the fault modes in fault sample are injected into equipment to infer its testability indices. The larger the number n and the more reasonable the fault sample structure, the more accurate and credible the testability demonstration results.

There are two traditional types of methods to determine the fault sample size. One is empirical method and the other is statistical method. Empirical method usually requires the minimum fault sample size 30, or more than 30 according to the engineering experience and the reliability of equipment. This method is convenient, simple, but arbitrary and subjective.

Statistical method is based on statistics theory. The fault sample size n is figured out by statistical formula when the manufacturer risk, consumer risk and required indices are determined. This approach has uniform standards and formulas. It is also simple and

3. Mathematical Description of Fault Occurrence Process

Let N(t) be the total number of faults up to time t, then N(t) has the following properties:

1) N(t) > 0.

2) N(t) is integer valued.

3) If g< t, then N(g) <N(t).

4) For g<t, N(t)-N(g) equals the number of faults that occur in the interval (g,t).

According to the definition of counting process,

fault occurrence process {N(t), t > 0} is a counting process [13-14].

For t1<t2 < t3<t4, [t1, t2) and (t3, t4] are the two disjoint time intervals. In [t1, t2) and (t3, t4], the numbers of faults are N(t2)-N(t1) and N(t4)-N(t3), respectively. So {N(t), t > 0} is an independent increment process.

Generally, single fault assumption is widely used in reliability, maintainability and testability engineering. The possibility of two or more faults occurring at the same time is negligible. Then {N(t), t > 0} has the following properties:

1) N(0)=0.

2) The process has independent increments.

3) lim P(N(t + h) - N(t) = 1) = A (t)h + o(h).

4) lim P(N(t + h) - N(t) > 2) = o(h).

Fault occurrence process {N(t), t > 0} is said to be a homogeneous Poisson process or nonhomogeneous Poisson process with rate !p(t), h describes the time plot, and o(h) is the dimensionless. If !p(t) is a constant, {N(t), t > 0} is a homogeneous Poisson process. Otherwise, it is a nonhomogeneous Poisson process [14].

Consider a fault occurrence process and denote the time of the first fault event by s1. Furthermore, for n > 1, let sn denote the elapsed time between the (n-1)th and the nth fault events. The sequence {sn, n=1, 2, 3, •••}

is called the sequence of fault interarrival times.

Suppose {s1, s2, s3, •••} is the fault interarrival time sequence of a homogeneous Poisson process having rate Ap=A0 , where k0 is the parameter of homogeneous Poisson process. Variable sn(n=1, 2, 3, •••) is the mutually independent random variable. The event {s1 > t} takes place if and only if no event of the Poisson process occurs in the interval [0, t) and thus,

P(s1 > t) = P(N(t) = 0) = e-v Then, for t > 0, distribution function of s1 is Fs (t) = P(s1 < t) = 1 - P(s1 > t) = 1 - e"v The probability density function of s1 is f (t) = F' (t) = V-V

Hence, s1 has an exponential distribution with parameter A0. The mean value of s1 is

.S ^ __1 + ¿00 .-Aot ¿0

E ( si) _ ft 0 dt _--^ e

Similarly,

P(s2 > t ) _ e

_ „_¿01

FS2 (t) = P(s2 < t) = 1 - P(s2 > t) = 1 - e

fs2(t ) = Fs2(t ) = ^0e - ^

Repeating the same argument yields FSn (t) = 1 - e- v

fs (t) = FS (t) = V-V

(9) (10)

Hence, if fault occurrence process is a homogeneous Poisson process with rate 10, fault interarrival time sn(n=1, 2, 3, •••) has an exponential distribution with parameter 10.

If the life distribution of a component is exponential distribution and maintenance mode is breakdown and perfect repair, the interval time between two adjacent faults is its lifetime. Then its fault occurrence process is a homogeneous Poisson process.

Exponential distribution is a simple, single-parameter distribution, which is a common type of life distribution and easy to use [14-15]. However, other distributions are playing an important role in life distribution, such as normal distribution, lognormal distribution, Weibull distribution and gamma distribution. If we assume that life distributions of all components are exponential distribution, the fault sample simulation results will have errors.

One possible generalization is to consider a fault occurrence process that the times between successive fault events are independent and identically distributed with an arbitrary distribution, such as normal distribution, lognormal distribution, Weibull distribution and

gamma distribution. Let {N(t), t > 0} be a counting process and let sn denote the time between the (n-1)th and the nth fault event of this process. If the sequence of nonnegative random variables {s1, s2, s3, •••} is independent and identically distributed, then the counting process {N(t), t> 0} is said to be a renewal process.

Renewal process is a generalization of homogeneous Poisson process. It is applied to and developed in machine maintenance, counters, traffic flow and many other fields [13-18]. Consider a component under breakdown maintenance. It can be replaced by new one immediately when it fails, and the repair time can be negligible. Failures of the components are independent. Their lifetime is random and has the same probability distribution at the same environmental and working stress level. Fault occurrence process {N(t), t > 0} is a renewal process. It is indicated in Fig. 1, where vn(n=1, 2, 3, •••) denotes the fault events, tn(n=1, 2, 3, •••) the occurrence time of faults.

Fig. 1 Fault occurrence process.

Schematic diagram of fault detection process is shown in Fig. 2. Let xq(q=1, 2, 3, •••) be the interval time of adjacent fault detection. Variable xq(q=1, 2, 3, •••) are influenced by fault occurrence process and testability plan. t0 is the initial time, and tF is the final time.

t0 Detection(l) Detection(2) Detection Detection(y') Detection^/) 'v

Fig. 2 Schematic diagram of fault detection process.

The number of fault detection and the interval time of fault detection are random. The number of detected faults is random, too. Generally, the observed values of fault detection rate always change in the specified time period (0,t]. The formula is

r (t) _ x 100%

/FDW N (t )

where ND(t) is the number of detected faults up to time t. As discussed above, ND(t) and N(t) are random. Thus, the observed value of fault detection rate fFDis random, too.

4. Fault Sample Simulation

Let M(t) be the mathematical expression of N(t), and

M(t) is called mean-value or renewal function of the renewal process. It can be shown that M(t) uniquely determines the renewal process [14].

M (t) = E (N (t)) = £ nP( N (t) = n) =

£ n[P(N(t) > n) - P(N(t) > n +1)] =

£ n[P(tn < t) - P(tn+1 < t)] (12)

It can be seen from the definition of renewal process

that tn =£^^ . If these random variables sn(n=1, 2,

3, •••) have the same distribution function F(t), the distribution of tn is n-fold convolution of F(t). It is denoted as Fn(t). So Eq. (12) can be written as

M (t) = £ n (Fn (t) - Fn+1(t) ) =

№ №

£ [ nFn (t) - (n -1) Fn (t)] = £ Fn (t) (13)

n=1 n =1

As Fn(t) and F(t) are uniquely determined by each other, Fn(t) and M(t) are uniquely identified by each other, too. If F(t) or its probability density function f(t) is known, the renewal function M(t) can be solved and the renewal process can be determined mathematically. Under the assumptions of excluding the repair time and perfect repair, the renewal process of fault occurrence and repair can be uniquely identified by cumulative fault distribution function or its density function. The cumulative fault distribution function F(t) or its density function f(t) can be estimated according to historical experience data and reliability test data. Environmental coefficients can also be estimated according to statistical test data with different environmental and working stresses. Then, F(t) or f(t) at different levels of environment and working stress can be solved.

Monte Carlo (MC) method is also known as random simulation method, random sampling method or statistical test method. It can effectively solve uncertainty problems and complex computing problems. It is widely applied to financial engineering, statistical physics, computational mathematics, reliability engineering and other fields [19-22]. In this paper, MC method is used to sample the fault occurrence process and generate the fault samples.

If a fault event occurs at time z, it is independent of the fault events occurring before time z. {N(t)-N(t-h)=1} denotes that a fault event occurs at time z (h^0). This event is recorded as Az. Let tA be the interval time between Az and the next fault event. Then the event {tA<x} is equal to a fault event occurring in (z, z+x), that is{N(z+x)-N(z)=1}. The fault interarrival time distribution function Fz)(x) of the next fault event after time

• 601 • z is

F(z}(x) = P(tA< x | Az) = P(N(z + x) - N(z) = 1| Az) (14)

According to the independent assumption of fault event, Eq. (14) can be simplified as

F(z} (x) = P(N(z + x) - N(z) = 1) (15)

According to the definition and character of the renewal process, (x) has the same distribution function. The fault interarrival time distribution function is the cumulative fault distribution function.

The flowchart of fault sample simulation under breakdown maintenance is showed in Fig. 3. We use the cumulative fault distribution function to determine the renewal process. The basic steps of fault sample simulation are as follows:

Step 1 Determine the parameters of the renewal process.

Step 2 Initialize the specified statistical time T* and initialize i=0, t,=0.

Step 3 Solve the inverse function F ~l(U) of the cumulative fault distribution function F(t).

Step 4 Generate the random number ui.

Step 5 Generate the time ti of the ith fault event based on direct sampling method, si=F _1(ui), ti=ti-1+si. The variable si is the interarrival time between the (i-1)th fault event and the ith fault event, and th the ith fault occurrence time.

Step 6 If ti > T*, stop.

Step 7 Obtain fault samples based on probability sampling method according to the proportion of each fault mode or random sampling method.

Determine the renewal process +

Initialize T*,i=0, t= 0

Solve the inverse function

1—► i=i+1

Generate random number il

Calculate ,v;

t = ti_+si

Fault mode sampling

Fig. 3 Flowchart of fault sample simulation under breakdown maintenance.

Scheduled maintenance is a scheduled service carried out by a competent and suitable agent. If a fault event occurs before the interval time Tw, the part is processed by breakdown maintenance. If no fault event occurs before Tw, the part should be replaced by new one at time Tw regardless of its health condition.

The flowchart of fault sample simulation under scheduled maintenance is showed in Fig. 4. The basic steps are as follows:

Step 1 Determine the parameters of the renewal process and set the interval time Tw of scheduled maintenance.

Step 2 Initialize the specified statistical time T* and initialize i=0, j=0, S=0, tj=0.

Step 3 Solve the inverse function F _1(U) of the cumulative fault distribution function F(t).

Step 4 Generate the random number ui.

Step 5 Generate the lifetime si of the ith new component based on the direct sampling method, si= F ~\ui).

Step 6 If s/<Tw, it shows that the component has

Fig. 4 Flowchart of fault sample simulation under scheduled maintenance.

broken down before the scheduled maintenance, and the cumulative working time S=S+si. The fault number j=j+l. If si > Tw, it shows that the component is good until the scheduled maintenance time. Then, set S=S+Tw.

Step 7 If S> T*, stop.

Step 8 Obtain fault samples based on probability sampling method according to the proportion of each fault mode or random sampling method.

5. Examples

A stable tracking platform has the ability to isolate the movement of moving vehicle, such as car, ship, aircraft, etc. It can automatically track the target and maintain stable communication. We take stepping motors and storage batteries as examples to carry out experiments. One stable tracking platform contains two stepping motors named pitching stepping motor and rotating stepping motor.

We get some credible lifetime statistics and maintenance strategy from the manufacturer and consumer. In the case of rated work intensity and environment on car, the lifetime of the two types of stepping motor is Weibull distribution and the lifetime of storage battery is exponential distribution. The lifetime distribution functions, fault modes and maintenance modes of the two types of component are shown in Table l. In the table, m, 7, y are the three parameters of Weibull distribution, and A is the parameter of exponential distribution.

Considering stepping motor, the inverse function of cumulative fault distribution function is

_L _5_

t = (-n ln(1 - z))m = (-16 000 ln(1 - z))26

Considering storage battery, the inverse function of cumulative fault distribution function is

1 W1 . 500 000, .

t = -J-( " z) =--ln(1 - z)

It is assumed that the specified statistical time of testability demonstration is 15 years. The fault samples are generated by fault occurrence process simulation based on the proposed method. Thle fault samples of pitching stepping motor, rotating stepping motor and storage battery are respectively shown in Tables 2-4.

Table 1 Lifetime distribution functions, fault modes and maintenance modes

Module Stress level

Distribution type

Fault mode

Fault mode sampling method

Maintenance mode

Repair effect

^ . Rated work Weibull distribution m ™ m No output (A1) ,

Steppmg mtens>ty ami ^=4.2, 16 000, 1 - exp(- i-) mtm-1exp(-i-) Reverse output(A2) Rand°m sam" motor environment ,=0) n n n Tblerance(A3) ' P''"g

environment

on car Rated work

Storage intensity and Exponential distribution battery environment on car

1 , . N . , . N No output (B1) Random sam-1-exp(-lt) -^texp(-^t> Unstiible ouitput (B2) pling

Breakdown maintenance

Scheduled maintenance (Tw =2 years)

Perfect repair

Perfect repair

Table 2 Fault sample simulation results of pitching stepping motor

Sequence number Time/h Fault mode

1 12 574 A1

2 30 440 A3

3 48 012 A2

4 56 525 A1

5 68 502 A1

6 88 645 A2

7 109 336 A1

8 128 892 A3

Table 3 Fault sample simulation results of rotating stepping motor

Sequence number Time/h Fault mode

1 13 690 A2

2 25 513 A1

3 34 763 A1

4 46 937 A2

5 61 440 A1

6 81 331 A3

7 90 839 A1

8 111 078 A3

9 125 555 A1

Table 4 Fault sample simulation results of storage battery

Sequence number Time/h Fault mode

1 20 614 B2

2 25 667 B1

3 72 257 B2

4 112 989 B2

The cumulative number of pitching stepping motor fault, rotating stepping motor fault and storage battery fault are shown in Fig. 5. Abscissa represents the cumulative time; ordinate represents the cumulative number of faults. The cumulative number of faults increases by one when a fault occurs. In the case of other different working intensities and working environments, we can also obtain the fault samples by the proposed simulation method.

0 2 4 6 S 10 12 (/( 10J h)

(a) Pitching stepping motor fault

0 2 4 6 8 10 12

//(104h)

(b) Rotating stepping motor fault 4 I-.-.-.-.-.-.-.-.-.-.-rf-r-

0 2 4 6 8 10 12

//(104 h) (c) Storage battery fault

Fig. 5 Cumulative number of three faults.

Not only exponential distribution but also other distributions are considered in the proposed fault sample simulation method. Fault samples occur under breakdown maintenance and scheduled maintenance can be simulated. Because some assumptions are eliminated, the size and structure of the fault sample are more similar to the actual results and more reasonable.

6. Conclusions

1) Not only exponential distribution but also other life distributions are considered in the proposed method. Both breakdown maintenance and scheduled maintenance are taken into account in fault sample simulation.

2) Because some assumptions are eliminated, the size and structure of the fault sample are more similar to the actual results and more reasonable. The random faults sample simulated by proposed method can be applied to virtual testability demonstration test and used to guide the fault injection.

3) Only the perfect repair is taken into account in this paper. In the case of imperfect repair and condition-based maintenance, the fault sample simulation needs further study.

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Biographies:

ZHANG Yong is a Ph.D. student at laboratory of science and technology on integrated logistics support, National University of Defense Technology. He received the B.S. degree from Wuhan University in 2004. His area of research includes testability engineering and fault diagnosis. E-mail: zhangy21cn@126.com

QIU Jing is a professor and Ph.D. supervisor at National University of Defense Technology. He received the Ph.D. degree from the same university in 1998. His current research interests are testability engineering, fault diagnosis and prognostics.

E-mail: qiujing@nudt.edu.cn

LIU Guanjun is a professor at National University of Defense Technology. He received the Ph.D. degree from the same university in 2000. His current research interests are testability engineering and embedded test. E-mail: gjliu342@sina.com

YANG Peng is a teaching assistant at National University of Defense Technology. He received the Ph.D. degree from the same university in 2008. His current research interests are testability engineering and fault diagnosis. E-mail: yp7894@163.com