Mathematical Problems in Engineering

Volume 2015, Article ID 645047, 8 pages

http://dx.doi.org/10.1155/2015/645047

## Fault Sample Generation for Virtual Testability Demonstration Test Subject to Minimal Maintenance and Scheduled Replacement

^{1}Science and Technology on Integrated Logistics Support Laboratory, National University of Defense Technology, Changsha, Hunan 410073, China^{2}College of Mechatronic Engineering and Automation, National University of Defense Technology, Changsha, Hunan 410073, China

Received 30 October 2014; Revised 19 January 2015; Accepted 19 January 2015

Academic Editor: Gang Li

Copyright © 2015 Yong Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1–4].

However, application results indicate that testability demonstration test based on fault injection has two unavoidable problems [1–5]. First, large numbers of fault injection tests lead to high cost. Second, some faults cannot be allowed to be injected because of destroyable influence and some faults cannot be effectively injected because of restricted fault injection means. These two shortcomings lead to unreasonable fault sample structure and low confidence.

The fault sample selection is to determine appropriate sample size and to make fault sample structure reasonable, that is, to select representative fault samples [2–6]. On 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 fault sample size needs to be as large as possible. It results in a difficult contradiction [1–4].

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 [1, 7–9].

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 [4]. Considering various types of life distribution and assuming perfect maintenance, Zhang et al. proposed a fault sample generation method based on renewal process [1].

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.

#### 2. Description of Fault Occurrence Process

Let be the total number of faults up to time . Faults more than two at time would be ignored under single fault assumption. So has the following properties:(1);(2);(3) is integer valued;(4)the process has independent increments;(5);(6).

According to the definition of counting process, fault occurrence process is a Poisson process with the parameter [10, 11]. The parameter is also defined as the intensity function, which describes the intensity level of fault occurrence. If is a constant, is a homogeneous Poisson process (HPP). Otherwise, it is a nonhomogeneous Poisson process. The nonhomogeneous Poisson process is the generalized form of homogeneous Poisson process. Equally, the homogeneous Poisson process is the special case of a nonhomogeneous Poisson process [11].

Fault occurrence process is an independent increment process in disjoint time intervals. That is, for , [) and (] are the two disjoint time intervals. In [) and (], the number of faults is () and , respectively. So is an independent increment process.

Let denote the mean number of faults in the interval ;

is also called cumulative number of occurrence of failures. Thus,where is called the* rate of occurrence of faults* (ROCOF) at time . It can be regarded as the mean number of faults per time unit at time . If is a constant, is a HPP. Otherwise, it is a NHPP. It has been verified that the mean number of faults in the interval is Poisson distributed [11].

For a nonhomogeneous Poisson process with fault occurrence intensity function , is a Poisson distribution with parameter , where , . can be called a Poisson process with mean value . That is,

Consider a repairable system that is put into operation at time . The first fault event of the system will occur at time . The second fault will occur at time and so on. We thus get a sequence of fault time . Let be the time between the th fault event and the th fault event for , where is taken to be zero. is called the interarrival time . is called the sequence of fault interarrival time. Fault occurrence process is indicated in Figure 1.