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Virtual testability demonstration test brings new requirements to the fault sample generation. First, fault occurrence process is described by stochastic process theory. It is discussed that fault occurrence process subject to minimal repair is nonhomogeneous Poisson process (NHPP). Second, the interarrival time distribution function of the next fault event is proposed and three typical kinds of parameterized NHPP are discussed. Third, the procedure of fault sample generation is put forward with the assumptions of minimal maintenance and scheduled replacement. The fault modes and their occurrence time subject to specified conditions and time period can be obtained. Finally, an antenna driving subsystem in automatic pointing and tracking platform is taken as a case to illustrate the proposed method. Results indicate that both the size and structure of the fault samples generated by the proposed method are reasonable and effective. The proposed method can be applied to virtual testability demonstration test well.

Recently, testability test has two basic methods, including fault injection test and field test. Both of them are physical tests. It often takes long time to get enough original fault samples in field test. In order to accelerate testability demonstration, fault injection is always applied in the testability test [

However, application results indicate that testability demonstration test based on fault injection has two unavoidable problems [

The fault sample selection is to determine appropriate sample size and to make fault sample structure reasonable, that is, to select representative fault samples [

Nowadays, many researches attach importance to virtual test. Virtual test can simulate the process of a real test and obtain test results in an efficient way. It means that virtual test can effectively decrease the test cost and risk and shorten the test period compared with physical test. According to recent studies, large-scale system modeling and simulation are difficult while small-scale system modeling and simulation can be performed in the present technical conditions [

As mentioned above, virtual test has many advantages, such as high efficiency, short test period, and low cost. As the fault sample size of virtual testability test is almost unlimited, it overcomes some deficiencies of physical testability test. Thus, the fault sample generation in virtual testability test is different from fault sample selection in physical testability test.

The combination of minimal maintenance and scheduled replacement is the main maintenance mode for many systems. The occurrences of faults are nonhomogeneous because faults occur randomly and are repairable. On the basis of Monte Carlo method, Zhao et al. proposed a fault sample generation method which was subject to exponential distribution [

The nonhomogeneous Poisson process has clear physical meanings and theoretical basis. It is widely applied to system reliability analysis, reliability indices calculation, and reliability growth test.

This paper discusses the occurrence process of faults and describes it by NHPP. A suitable fault sample simulation method for virtual testability demonstration is proposed. The main idea of the method proposed in this paper is obtaining the value and composition of fault sample based on fault statistical model and statistical simulation. The purpose is obtaining an implementation of fault occurrence within the specified time and conditions, which is called fault sample simulation in this paper.

Let

the process has independent increments;

According to the definition of counting process, fault occurrence process

Fault occurrence process is an independent increment process in disjoint time intervals. That is, for

Let

For a nonhomogeneous Poisson process with fault occurrence intensity function

Consider a repairable system that is put into operation at time

Fault occurrence process.

If fault occurrence process is a HPP having rate

It is important to note that some fault occurrence processes do not have stationary increments. The rate of occurrence of faults varies with time rather than being a constant. This means that failures may be more or less likely to occur at certain time than others, and hence the interarrival time is generally neither independent nor identically distributed [

The NHPP is generalization of HPP having the HPP as a special case. It is often used to model repairable systems that are subject to a minimal repair strategy with negligible repair time. Minimal repair means that a failed system is restored just back to functioning state. After a minimal repair, the system continues as if nothing had happened. The likelihood of a system fault is the same immediately before and after a fault.

Consider a system consisting of many components. Suppose that a component fails and causes a system failure and this component is immediately replaced by a component of the same type, thus causing a negligible system downtime. Since only a small fraction of the system is replaced, it seems natural to assume that the system’s reliability after the repair essentially is the same as immediately before the failure. In other words, the assumption of minimal repair is a realistic approximation. The minimal repair assumption is therefore often applicable and the NHPP may be accepted as a realistic model [

Schematic diagram of fault detection process is shown in Figure

Schematic diagram of fault detection process.

Generally, the observed values of fault detection rate always change in the specified time period

Let

Note that if a fault event occurs at time

According to (

We can now simulate the fault event time

The key of fault events simulation is the distribution function and its inverse function. If fault occurrence process is described by NHPP, it can be uniquely determined by the rate of occurrence of faults

In the linear model, the ROCOF of the NHPP is defined as

The interarrival time distribution function

The inverse functionof

A repairable system modeled by the linear model is deteriorating if

In the power law model, the ROCOF of the NHPP is defined as

Thus,

A repairable system modeled by the power law model is seen to be improving if

In the log-linear model, the ROCOF of the NHPP is defined by

A repairable system modeled by the log-linear model is improving if

The cumulative number of faults and occurrence time of each fault subject to specified conditions and time period can be obtained according to reliability test and other trials. Appropriate parametric NHPP model will be selected according to failure statistics. We do not intend to discuss the model selection and parameter estimation method in this paper.

In this paper, we assume that the repair or maintenance time is negligible and the corrective maintenance is minimal maintenance or repair, that is, the maintenance action which restores the part to the failure rate it had when it failed. The part after repair is as bad as old.

If fault events occur before the scheduled replacement, the part will be processed by breakdown maintenance. If no fault event occurs before scheduled replacement, the part should be replaced by a brand new one regardless of its health condition when it meets the replacement requirement.

Statistical simulation 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. For example, Monte Carlo method is widely applied in financial engineering, statistical physics, computational mathematics, reliability engineering, and other fields [

The flow chart of fault sample simulation is showed in Figure

The flow chart of fault sample simulation.

The basic steps of fault sample simulation are as follows.

Determine the parameters of the NHPP and set the interval replacement time

Initialize the specified statistical time

Solve the interarrival time distribution function

Solve the inverse function

Generate the random number

Calculate the interarrival time

If

If

Obtain fault samples based on probability proportional to size (PPS) sampling method. Each fault mode is set to be proportional to its occurrence percentage ratio (OPR).

An automatic pointing and tracking platform has the ability to isolate the movement of moving vehicles, such as car, ship, and aircraft. It can automatically track the target and maintain stable communication. The stable tracking platform consists of multiple subsystems. We take antenna driving subsystem as example to carry out experiments. The lifetime of the automatic pointing and tracking platform is 15 years. The average working time is 1500 hours per year. The lifetime of the antenna drive subsystem is 7500 working hours. The antenna driving subsystems are replaced by new ones every 5 years. In the subsystem’s life cycle, breakdown maintenance and the assumption of minimal repair with negligible repair times are taken when it fails. The subsystems are replaced by new ones with the assumption of perfect repair after the end of their life cycle.

As the platform is new equipment, the failure statistics in full life cycle are poor. The same antenna driving subsystems have been tried out for 5 years in advance. We collected some credible and valuable failure and maintenance statistics of the subsystem in their single life cycle. The statistics contain 56 complete sets of trial data. Fault modes and their occurrence percentage ratio of the antenna driving subsystem are shown in Table

Fault modes and their occurrence percentage ratio.

Code | Fault mode | Fault unit | OPR |
---|---|---|---|

A1 | No signal output | Antenna | 7.1% |

A2 | Unstable signal output | Antenna | 3.7% |

A3 | No rotation output | Pitching motor | 10.7% |

A4 | Angle output tolerance | Pitching motor | 7.1% |

A5 | No turning signal output | Pitching motor driver | 12.5% |

A6 | Turning signal output tolerance | Pitching motor driver | 5.4% |

A7 | No rotation output | Azimuth motor | 8.9% |

A8 | Angle output tolerance | Azimuth motor | 5.4% |

A9 | No turning signal output | Azimuth motor driver | 10.7% |

A10 | Turning signal output tolerance | Azimuth motor driver | 7.1% |

A11 | No control signal output | Motion control card | 8.9% |

A12 | Control signal output error | Motion control card | 12.5% |

The interarrival time distribution function

The inverse function of

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. The fault modes and their occurrence time are obtained. A simulation result of fault sample is shown in Table

A simulation result of fault sample.

Number | Working time (h) | Fault |
---|---|---|

1 | 1287 | A12 |

2 | 4422 | A10 |

3 | 5953 | A3 |

4 | 6900 | A11 |

5 | 7166 | A5 |

6 | 7261 | A2 |

7 | 7482 | A1 |

8 | 9833 | A7 |

9 | 10061 | A11 |

10 | 10963 | A3 |

11 | 12487 | A9 |

12 | 13369 | A1 |

13 | 13932 | A5 |

14 | 14392 | A12 |

15 | 14554 | A10 |

16 | 14740 | A4 |

17 | 14815 | A9 |

18 | 14870 | A12 |

19 | 16008 | A6 |

20 | 17212 | A3 |

21 | 18181 | A10 |

22 | 19384 | A6 |

23 | 21639 | A11 |

24 | 21784 | A5 |

25 | 21903 | A7 |

26 | 22032 | A12 |

27 | 22208 | A5 |

28 | 22316 | A8 |

29 | 22354 | A4 |

The cumulative number of the subsystem faults is shown in Figure

Cumulative number of faults.

We implement the statistical simulation 1000 times. The specified statistical simulation time is single life cycle of the antenna driving subsystem. 1000 groups of fault samples are generated automatically by simulation. The numbers of faults in fault sample are random variables. We compare some statistics of the actual samples and the simulation samples to examine the effectiveness of the proposed method. The comparison is shown in Table

The comparison of sample statistics.

Statistic | The actual samples | The simulation samples |
---|---|---|

Sample mean | 10.46 | 10.57 |

Sample variance | 2.35 | 2.29 |

Two-order origin moment | 118.26 | 121.64 |

Sample median | 10.5 | 11 |

Percentage ratio of A1 | 7.1% | 6.9% |

Percentage ratio of A2 | 3.7% | 4.0% |

Percentage ratio of A3 | 10.7% | 10.5% |

Percentage ratio of A4 | 7.1% | 7.1% |

Percentage ratio of A5 | 12.5% | 12.4% |

Percentage ratio of A6 | 5.4% | 5.6% |

Percentage ratio of A7 | 8.9% | 8.7% |

Percentage ratio of A8 | 5.4% | 5.6% |

Percentage ratio of A9 | 10.7% | 10.5% |

Percentage ratio of A10 | 7.1% | 7.5% |

Percentage ratio of A11 | 8.9% | 8.8% |

Percentage ratio of A12 | 12.5% | 12.4% |

The sample variance is

The sample values are arranged in increasing order so as to meet

The sample median is

We can get that the statistics of the simulation results are nearly consistent with the actual fault samples according to the comparison. The composition of the simulation samples is rational. The results show that the proposed method is feasible and effective. The random fault samples generated by statistical simulation can be applied to virtual testability demonstration test.

(1) It is analyzed and pointed out that the fault sample generation in virtual testability test is different from fault sample selection in physical testability test.

(2) In the case of minimal repair and scheduled replacement, the fault occurrence process can be described by NHPP theory. A fault sample generation approach for virtual testability demonstration test is proposed.

(3) As some assumptions are eliminated, the size and structure of the fault samples simulated by proposed method are reasonable. Experiment results show that the proposed method is feasible and effective. It can also be applied to virtual maintainability test and integrated logistics support scheme design.

The number of detected faults up to time

The number of detected faults up to time

Fault detection rate

Fault sample size

Mean number of faults in the interval

The rate of occurrence of faults at time

The interval time of adjacent fault detection

The event denoting one occurring fault

The interval time between

Interval time distribution function of the next fault event after time

The coefficient determines the shape of the

The scheduled interval replacement time

Random variables having the uniform distribution in

The time

The cumulative working time of the

The cumulative number of fault events

The occurrence time of the

The cumulative working time of the parts

The specified statistical time.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported by Research Fund for Shanghai Astronautics Science and Technology and National Natural Science Foundation of China. The authors would like to thank Peng Yang, Kehong Lv, and Chenxu Zhao for enlightening discussions. They would also like to thank Chao Wang and Chao Wu for correction.