Scholarly article on topic 'A novel testability model for health management of heading attitude system'

A novel testability model for health management of heading attitude system Academic research paper on "Materials engineering"

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{"Fault evolution" / "Functional fault analysis" / "Physics of failure model" / "Prognostics and health management" / "Quantified directed graph" / "Testability analysis"}

Abstract of research paper on Materials engineering, author of scientific article — Guanjun Liu, Shuming Yang, Jing Qiu, Peng Yang

Abstract Prognostics and health management (PHM) is very important to guarantee the reliability and safety of aerospace systems, and sensing and test are the precondition of PHM. Integrating design for testability into early design stage of system early design stage is deemed as a fundamental way to improve PHM performance, and testability model is the base of testability analysis and design. This paper discusses a hierarchical model-based approach to testability modeling and analysis for heading attitude system health management. Quantified directed graph, of which the nodes represent components and tests and the directed edges represent fault propagation paths, is used to describe fault-test dependency, and quantitative testability information is assigned to nodes and directed edges. The fault dependencies between nodes can be obtained by functional fault analysis methodology that captures the physical architecture and material flows such as energy, heat, data, and so on. By incorporating physics of failure models into component, the dynamic process of a failing or degrading component can be projected onto system behavior, i.e., system symptoms. Then, the analysis of extended failure modes, mechanisms and effects is utilized to construct fault evolution-test dependency. Using this integrated model, the designers and system analysts can assess the test suite’s fault detectability, fault isolability and fault predictability. And heading attitude system application results show that the proposed model can support testability analysis and design for PHM very well.

Academic research paper on topic "A novel testability model for health management of heading attitude system"

Chinese Journal of Aeronautics, 2013,26(1): 201-208

JOURNAL OF

AERONAUTICS

Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics

cja@buaa.edu.cn www.sciencedirect.com

A novel testability model for health management of attitude system

Liu Guanjun, Yang Shuming, Qiu Jing *, 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 14 September 2011; revised 2 November 2011; accepted 20 December 2011 Available online 16 January 2013

Abstract Prognostics and health management (PHM) is very important to guarantee the reliability and safety of aerospace systems, and sensing and test are the precondition of PHM. Integrating design for testability into early design stage of system early design stage is deemed as a fundamental way to improve PHM performance, and testability model is the base of testability analysis and design. This paper discusses a hierarchical model-based approach to testability modeling and analysis for heading attitude system health management. Quantified directed graph, of which the nodes represent components and tests and the directed edges represent fault propagation paths, is used to describe fault-test dependency, and quantitative testability information is assigned to nodes and directed edges. The fault dependencies between nodes can be obtained by functional fault analysis methodology that captures the physical architecture and material flows such as energy, heat, data, and so on. By incorporating physics of failure models into component, the dynamic process of a failing or degrading component can be projected onto system behavior, i.e., system symptoms. Then, the analysis of extended failure modes, mechanisms and effects is utilized to construct fault evolution-test dependency. Using this integrated model, the designers and system analysts can assess the test suite's fault detectability, fault isolability and fault predictability. And heading attitude system application results show that the proposed model can support testability analysis and design for PHM very well.

© 2013 CSAA & BUAA. Production and hosting by Elsevier Ltd. All rights reserved.

heading

KEYWORDS

Fault evolution; Functional fault analysis; Physics of failure model; Prognostics and health management;

Quantified directed graph; Testability analysis

1. Introduction

* Corresponding author. Tel.: +86 731 84573305. E-mail address: qiujing@nudt.edu.cn (J. Qiu). Peer review under responsibility of Editorial Committe of CJA.

With the increase of the function, structure and technology complexity of aerospace systems, equipment maintenance and support mode is gradually converting to predictive maintenance, autonomic logistics from corrective maintenance, preventive maintenance and condition-based maintenance.1-3 Health monitoring and prognostics of complex equipments is a basic requirement to the new logistics modes in many application domains where safety, reliability, and availability of the systems are considered mission critical. As a key complement

1000-9361 © 2013 CSAA & BUAA. Production and hosting by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cja.2012.12.011

to the new logistics modes, prognostics and health management (PHM) is an approach to system life-cycle support that seeks to reduce/eliminate inspections and time-based maintenance through accurate monitoring, incipient fault detection, fault evolution track and remaining useful life prediction.4 PHM is significant to improve aerospace system safety, reliability, maintainability and affordability to reduce life cycle cost and realize autonomic logistics.5'6 With the rapid development of information sensing, condition monitoring and fault prognostics, etc., PHM has been an important part in aerospace systems such as helicopter, aircraft engine, missile, etc.

Generally speaking, PHM mainly consists of condition monitoring, fault diagnostics, fault prognostics, health evaluation and decision control.7,8 In order to conduct these functions very well, a much broader range of asset health related test information should be collected. There is no doubt that test information forms the foundation upon which all the PHM systems are based.9-11

Testability is a design characteristic which allows the status (operable, inoperable, or degraded) of an item to be determined and the isolation of faults within the item to be performed in a timely manner.12 In order to reduce development cycle and cost and improve integration effects, design for testability (DFT) should be integrated into early design stage of system and developed concurrently with system design. In a word, information sensing and test are the precondition of PHM, and the traditional attached test design methodology should give way to the systematic concurrent DFT techniques. At present, model based DFT is very popular.13,14 One aspect is that the model-based way is convenient to amend the design scheme according to feedback testability analysis results and hence reduces system development cycle and cost; the other is that the knowledge reusability of models enables different engineers at different design stages to have consistent understandings and hence makes DFT be developed concurrently and consistently.

Obviously, testability model is the base of testability analysis, design and validation. The existing testability models such as information flow graph15 and multi-signal flow graph,16 which are effective for fault detection and isolation, only describe fault-test dependency qualitatively and lack quantitative PHM-related testability information. However, DFT for PHM should pay more attention to the requirements of fault predictability rather than only to fault detection and isolation needs. The fault-test dependency only describes the true/false detect-ability of test to fault, and it is insufficient to analyze the support level of testability for fault prognostics. As we know that a fault is detectable doesn't mean it is predictable. Whether a fault is predictable or not depends on two basic conditions: one is the fault should be gradual in nature; the other is the fault should be a key fault. Besides, the predictability of a fault is also related to timely detection and evolution track. If a fault is detected when or after the fault leads to a failure, fault prognostics becomes insignificant; further, if the evolution process of a fault cannot be tracked by tests, (data driven-based) fault prognostics may not be realized. So testability model for PHM, besides describing fault-test dependency, should be able to describe detectability of test to fault early state and trackability of test to fault evolution process which is called fault evolution-test dependency in the paper. It is necessary to study a novel testability modeling approach for PHM of aerospace systems.

2. Hierarchical modeling methodology

To address these problems, a hierarchical modeling methodology is proposed in the paper. The approach to multi-level modeling of complex aerospace systems combines system-level and component-level modeling. At the system level, functional fault analysis (FFA) and quantified directed graph (QDG) are used to describe fault-test dependency and PHM-related testability information; while at the component level, physics of failure (PoF) models and extended failure modes, mechanisms and effects analysis (FMMEA) are combined to analyze fault evolution-test dependency.

2.1. Fault-test dependency modeling

Inputs that are needed to build the system model include, but are not limited to, system schematics (component connectivity topology), failure modes, test resource, system functional behavior, information flow (energy, material, data) and expert experience. Many approaches such as Petri net, fault tree can be used to model the knowledge. Considering the limited knowledge at early design stage of system and in order to make the system model highly consistent with system constituents, we propose QDG. QDG is able to describe system cause-effect dependency easily and can be developed from partial information such as system structure and function. In QDG, nodes represent components and tests and the directed edges represent fault effect propagation paths. Quantitative testability information is assigned to nodes and directed edges in the forms of probability, fuzziness and uncertainty.

Formally, a QDG consists of four sections.

(1) A finite set of components making up of a system, C = {c1, c2,..., cL}, and component attribute denotes whether a component is a key and/or important one or not.

(2) A finite set of m available failure modes, F = f1, f2,..., fm}. Each component is associated with a set of failure modes, Fi = {fl1, fl2,..., fp}, 1 6 16 L. Obviously, F = U Fl. Failure mode attributes related closely to testability analysis are fault type (F_Type), failure prior probability (F_Prob) and failure criticality (F_Crit). Generally, F_Type can be gradual fault (F_Type = 1) or abrupt fault (F_Type = 0); F_Prob can be calculated according to history data, reliability design documents and expert experience; F_Crit can be type I (catastrophic), type II (fatal), type III (critical) or type IV (light).

(3) A finite set of n available tests T = {t1, t2,..., tn}. Each test may be of the following quantitative attributes: test time (T_Time), test cost (T_Cost), test signal to noise rate (T_Stnr) and test failure rate (T_Rate). Test is of general meanings, usually including build in test (BIT), automatic test equipment (ATE), a variety of sensors and even virtue test methods.

(4) A finite set of directed edges, E = {e1, e2,..., eo}. The directed edges denote fault effect propagation paths, which may be a physical link in the system or a virtue path. A directed edge is uniquely characterized by fault propagation probability (E_Prob), fault propagation time (E_Time) and fault propagation gain (E_Gain).

E_Time and E_Gain can be defined based on the step response, i.e., the propagation time is defined as the rise time when the response reaches 10% of the steady gain, while the fault propagation gain is defined as the steady-state gain.17

In the multi-signal flow graphs or information flow graphs, the directed edges specify the system structural connectivity which can be called physical paths in the paper. Generally, physical paths do not mean fault propagation paths, but fault effect propagation must be through the physical paths, so physical paths are necessary but not the sufficient condition of fault effect propagation. However, in the QDG, the directed edges represent fault propagation paths. The propagation paths may be physical links which might be fluid, thermal, electrical, etc., or virtual paths which may not be a physical link but represent certain fault effect propagation. So QDG has great advantages to describe fault-fault dependency and fault-test dependency, but may deviate from system structure topology to some extent.

In QDG, fault dependency analysis is very important. For digital systems, fault dependencies can be obtained by failure modes, effects and criticality analysis (FMECA) due to its good modularity and fault propagation certainty. However, for complex aerospace systems consisting of mechanics, electronics and hydraulics, the fault dependencies obtained only by FMECA are unilateral or even wrong. For these systems, fault propagation is usually accompanied by flow of energy, signal or data, so fault dependencies can be analyzed by system dynamic physical behavior analysis combined with system structure and function. FFA provides a feasible way to the idea.

FFA is a high-level, functional model of the system that captures the physical architecture, including the physical connectivity of energy, material, and data flows. The methodology can be used to analyze the fault effect propagation paths.18 As a simple example, considering tank external leak failure mode, the failure mode effects propagate along the liquid (material) flow to the next component "valve" and along the pressure signal (signal) flow to the test node "pressure sensor'', so fault (external leak)-test (pressure sensor) dependency can be obtained.

2.2. Fault evolution-test dependency modeling

The intrinsic difference of testability for PHM from the traditional testability (for fault detection and isolation) is that the former pays much attention to the support of DFT for PHM, especially fault prognostics besides fault detection and isolation requirements. Therefore, gradual key faults are especially taken into account. As stated previously, it is necessary to consider the detectability of test to fault early state and trackability of test to fault evolution process, i.e., fault evolution-test dependencies. Component level model is mainly used to analyze fault evolution-test dependencies.

Generally, fault degradation process always accompanies system behavior variation such as performance parameters (e.g., speed, pressure, strain, resistance, current and voltage); environmental parameters (e.g., temperature, vibration, acoustics and humidity); operational parameters (e.g., usage frequency, usage severity, usage time, power, and heat dissipation). Based on the understandings, physics of failure models can be incorporated into component to map the dynamic process of a failing or degrading component to system

behavior, i.e., system symptoms. And system symptoms are closely related to fault mechanisms and test configuration, so the extended FMMEA is further utilized to analyze fault evolution-test dependency.

The traditional FMECA analyzes all the possible failure modes and the corresponding causes and effects statically in order to determine the proper detection methods, and is very suitable for the traditional testability analysis and design. However, FMECA lacks fault mechanism description which is very important for fault prognostics and fault evolution-test dependency analysis. FMMEA is a PoF based methodology for assessing the root cause failure mechanisms of a given sys-tem.9'10 In fact, fault occurrence usually accompanies a series of responses, including fault effects, fault symptoms, fault models, monitoring parameters and test configuration. So the extended FMMEA proposed by the paper is shown in Fig. 1.

The turbine pump vane is introduced as an example to illustrate the component level modeling process.19 Due to corrosion/erosion, the main failure mode of vane is cross-sectional area loss, which may change the surface and reduce the total area of vanes available for moving fluids and further cause turbulence in the pump and reduce the efficiency in the fluid flow. The schematic of fault degradation model is shown in Fig. 2, where q is corrosion rate, Da total vane area loss, At time interval, r pitting growth radius.

It is simply assumed that the corrosion rate q of vane material is following Balbaud-Celerier and Barbier rule.20

q = K(cs - a) (1)

where K is the mass transfer coefficient dependent on the flow velocity, cs the corrosion product concentration at the

Identity life cycie profile i

Define system and identify elements and functions to be analyzed

Identify potential failure modes

Failure signatures Failure causes

m2 J m5J

Monitoring parameters Failure mechanisms

m, } m<\

Test configurations Failure models

Failure effects

Document the process

Fig. 1 Flowchart and relationships of extended FMMEA.

Fig. 2 Schematic of fault degradation model.

liquid-solid interface dependent on the local temperature, and cb the concentration in the bulk flow often set to zero.20 q, K and cs are constants when constant flow velocity and temperature, and no change in fluid concentration are assumed. With Fig. 2, the area loss at one pitting location can be expressed as

Dai = qprDt = qpqtDt (2)

The total vane area loss Da can be expressed as

Aa = £ "—a = 1 pn2t2

{ n = (cs - cb)2ELK

According to the conservation of power and momentum

Pout = (ah - b/out)h (4)

where pout is the pump pressure, /out the corresponding mass flow rate, h angular velocity of the pump rotor, a the total area of its vanes and b the effective loss in moved mass due to the curvature of the vanes. Based on the first order perturbation theory, we get

Da(t) —

bD/(t)0 - Dp(t) h2

The fault degradation process is projected onto system behaviors by Eq. (5), and fault evolution process Da can be detected and tracked by the continuous mass flow and pump pressure monitoring.

The FMMEA of the vane cross-sectional area loss is shown in Table 1. One can see clearly that the fault evolution process can be detected and tracked by pressure test and/or speed test.

3. Testability analysis based on hierarchical model

Once the system hierarchical model is built, the designers and system analysts can assess the capability of the test suite to detect the occurrence of faults, isolate the location of faults and predicate the trend of faults. Further, the model can be used to analyze redundant tests, back loops and ambiguity groups, and so on.

3.1. Preliminary theory

Generally speaking, testability analysis for PHM mainly includes inherent testability analysis and achieved testability analysis.16 The inherent testability only depends on system hardware design and is not affected by test stimulation and response data. Inherent testability analysis usually includes undetectable faults (UFs), detectable faults (DFs), isolable faults (IFs), predictable faults (PFs), ambiguity groups

(AGs), redundant tests (RTs) and feedback loops (FLs), etc. Achieved testability, which is a system design attribute, can be defined as the ability to observe system behavior under special test stimulation. Achieved testability analysis usually includes fault detectable rate (FDR), fault isolable rate (FIR) and fault predictable rate (FPR). Most of the mentioned concepts are the same as those in traditional testability theory16'21 except predictable faults and fault predictable rate.

Based on the analysis of fault predictability, possible predictable (PPF) fault can be defined as follows:

Definition 1. Possible predictable fault is a gradual key fault or key components' fault.

Further, predictable fault (PF) can be defined:

Definition 2. Predictable fault is a possible predictable fault of which the early state is detectable and the evolution process is trackable.

Definition 2 describes fault predictability through fault early state detectability and fault evolution process trackabili-ty. Predictable faults can be obtained by fault mechanism analysis and test detectability analysis. In applications, we usually suppose that if a test can detect the early state of a fault, it also means the test can track the fault evolution process.

Based on Definition 1 and Definition 2, fault predictable rate can be defined as follows:

Definition 3. During the stated time span, the ratio of the number of predictable faults is determined correctly by tests to the total number of possible predictable faults in a system.

Fault/fault evolution-test dependency described by the integrated model can be represented by a binary dependency matrix based on which testability analysis can be realized.

3.2. Binary dependency matrix generation

Given the system fault set is F = fi, f2,..., fm} and the available test set is T = {t1, t2,..., tn}. A binary dependency matrix B = [bij]mxn is defined to represent the fault/fault evolution-test dependencies. The rows of B correspond to faults and the columns correspond to tests. Element by is a two-tuple, by = (u, v). If test tj can detect fault f and its early state, by = (1, 1). If test tj can detect fault fi but cannot detect its early state, by = (1, 0); the reasons for this scenario may be that the test has no ability to detect early state of the fault or the fault is abrupt. If test ty cannot detect fault fi and its early state, by = (0, 0) (by = 0 for short). Generally, if a test can detect

Table 1 Extended FMMEA of vane cross-sectional area loss.

Item Function Failure mode Failure Failure Failure effect Monitoring Test

mechanism symptom Immediate Downstream End effect effect effect parameter configuration

Pump vane Pumping mass flow Cross sectional Corrosion/ area loss erosion Flow mass loss Pressure difference variation None Decrease pump output efficiency Mission loss Flow speed signal Pressure signal Speed sensor Pressure sensor

early state of a fault, it also means the test can detect the fault, so the case by = (0, 1) will not exist.

3.3. Testability analysis based on matrix B

Given U denotes Boolean variable OR operation, © denotes set XOR operation. by(k) denotes the kth item of the two-tuple, by = (u, v), k = 1,2. Tfi denotes the test set which can detect fault f, i.e., Tfi = (t,|by(1) = 1,Vty}, Tfi c T. Ftj denotes the fault set which can be detected by test ty, i.e., Ftj = (f|bj(1) = 1, Vfi}, Fty c F. FPP c F denotes possible predictable faults. Fault prior probability vector is k = [k1 k2.. .km], and test failure rate vector is R = [r1 r2.. .rn]. Test design result vector is X = [x1 x2.. .xn], where xy (1 6 j 6 n) denotes the number of the designed test ty, and the vector Q = [qy] denotes the upper limit of X, i.e., Vxy 6 qy, xy 2 Z+.

detection probability greatly depends on test practical attributes such as test reliability, test signal to noise rate (SNR), sensitivity, timely detection and symptom duration. Obviously, it is more rational and applicable to take test attributes into account when analyzing achieved testability level in aerospace systems. Generally, test reliability can be featured by test failure rate ry (1 6 y 6 n), test reliability impacts on detectability and predictability of fault fi can be formulated respectively by

(R\ = 1 - n^«

| R2 = 1 - (6)

I ty2T

Test SNR, sensitivity, timely detection and symptom duration can be named as sensing probability which can be featured by parameter py17:

-10(V,-0.5n

, e-(SNR,-0.5)^ 0

TTDj,\0.5/SyDi,\0.2 .f l^1 TTFy-y VTTJ if

TTDj < TTFj

if TTDtJ p TTF¡j

(1) Inherent testability analysis

Undetectable faults FUD = (f|f 2 F, Tfi = 0}; detectable

faults, Fd = [f lf- 2 F; U by(1) = 1}; isolable faults Fi = (f|-

f 2 Fd, Tfi © Ty = 1, Vf 2 F, f „f}; ambiguity groups Fag = (F C F|Tfi = Ty Vf, f 2 F0, i„y}; the number of faults in an ambiguity group is called ambiguity group size, denoted by |F|. Generally, suppose the given ambiguity group size is L, then, Fi = ff 2 Fd, £ Tfi © Ty 6 L, Vf 2 F, f „ f}, and the opterator © satifies that Tfi © Tfy = 1 if Tfi © Tfy = 0;

predicatable faults FP = {ftf 2 FPP nFD, U by (2) = 1};

redundant tests TR = (Tc T^ti = Fty, Vtt, ty 2 T, i „y}. Denote all the gross feedback loops as FB, and is the kth gross feedback loop. The gross feedback loop searching method based on matrix B can be referred to Ref.14

(2) Achieved testability analysis

According to Ref. 21, fault detectable rate and fault isolable rate are defined as follows.

Definition 4. During the stated time span, the ratio of the number of faults is detected correctly by tests to the total number of system faults.

Definition 5. During the stated time span, the ratio of the number of faults is isolated correctly to no more than the stated replaceable units by tests to the number of the detected faults during the same time span.

For digital systems, a test ty relating to a fault f also means the fault fi can be detected by the test ty with probability 1 when the fault fi occurs. So, we can calculate the FDR, FIR and FPR directly according to Definitions 3-5. However, for complex aerospace systems, a test relating to a fault may not mean the fault can be detected by the test with probability 1. Fault

where Vy denotes detection sensitivity of test ty to fault f, SNRy denotes SNR of test ty, TTDiy denotes the time span between the initiation of fault fi (potential failure) and the detection of the fault by the test ty, TTF y refers to the duration between the initiation of the fault fi and the time when the failure occurs. SyDy denotes symptom duration time span of test ty to fault fi. Time to detection, time to failure and symptom duration time span can be obtained by fault simulation or fault propagation timing analysis method.22

Sensing probability impacts on detectability and predictability of fault fi can be formulated respectively by

P = £pj-*A(i) / Z*A(1)

tj 2T / tjeT

p2 = £ Pyxj) 5>A(2)

According to Eqs. (6) and (8), the total detectable and predictable probability of fault f can be formulated respectively by

FD,1 = R1 x P1

FD? = r2 x P2

So FDR, FIR and FPR can be formulated respectively by

FDR = £ k,FD1 / ^

/¡2Fd / fi2F

FIR = £k¡FD1 / 5>FD1

fi 2Fd

FPR = £k¡FD2 /

ft 2Fp / fi2Fpp

4. Application for heading attitude system

The heading attitude system of certain helicopter mainly consists of aviation horizon, combined compass, magnetic

compass and course position indicator. Aviation horizon, which is usually used to measure the pitch angle and inclination angel of helicopter, is one of the important components in the system. Besides, aviation horizon is also the key component resulting in the reduction of reliability and availability of helicopter, so it is of great significance to develop PHM for aviation horizon. The schematic diagram of certain aviation horizon is shown in Fig. 3.

Fig. 3 Structure diagram of horizon system.

The aviation horizon includes gyro, static converter, circuit board, corrective mechanism, quick righting mechanism, syn-chronic generator, indicative mechanism, etc. The function schematic is shown in Fig. 4.

Fig. 4 Function schematic of horizon system.

The QDG of the horizon system modeled through testability analysis, design and evaluation system (TADES©) developed by our team is shown in Fig. 5.

The quantitative failure mode attributes are shown in Table 2. Assume all the test failure rates are 0.001, the quantitative test attributes are shown in Table 3.

Static converter is a key component and testability should make the component predictable. We build component level model based on PoF in order to obtain fault evolution-test dependencies.

Connecting piece and MOSFET power transistor are the main constituents of the static converter. The main failure mechanism of connecting piece is temperature stress and mechanical stress. Generally, the impact of temperature stress on connecting piece fatigue life can be modeled by23

Nf — -

The impact of mechanical stress on connecting piece fatigue life can be modeled by24

SmP(S)dS

E(P) — m4/m2 mn — lfG(f)df

We can see that the connecting piece degradation process can be detected and tracked by vibration sensors and temperature sensors.

For MOSFET power transistor, the degradation model can be described as25

DD — Kta

K — C[(Ids/W)ex?(-/J(qkEm))T

Obviously, the power transistor degradation process can be detected and tracked by temperature sensors and electrical stress sensors.

Based on the system and component level models, the binary matrix for the aviation horizon system is shown in Table 4.

By fault mechanism analysis, f4 and f5 are the key gradual faults. According to Definition 1, FPP = {f4,f5}. Based on Table 4, results of inherent testability analysis are FUD = 0, Fd = {fi,\i = 1, 2, ..., 9}, Fi = {fu\i =1,2, ..., 9}. If ambigu-

Fig. 5 System level model.

Table 2 Failure mode attributes.

Failure mode Belongs to component F_Type F_Prob F_Crit

Voltage output error (f1) 28 V DC power (C1) 0 6.1 x 10~6 II

Voltage and frequency output error (f2) 26 V AC power (C2) 0 6.3 x 10~6 II

Wear or jamming (f3) Quick righting mechanism (C3) 0 1.7 x 10~6 III

Connecting piece fatigue (f4) Static converter (C4) 1 8.0 x 10~6 I

MOSFET degradation (f5) Static converter (C4) 1 8.0 x 10~6 I

Gyro output drift f6) Gyro (C5) 0 6.6 x 10~6 II

Low corrective speed or no correction (f7) Corrective mechanism (C6) 0 2.6 x 10~6 III

Heading output error f8) Synchronic transmitter (C7) 0 1.3 x 10~6 III

Display error f9) Indicative mechanism (C8) 0 1.1 x 10~6 III

Table 3 Test attributes.

Test T Time T_Cost T Stnr T Rate

(s) ($) (dB)

Voltage (tj) 50 6.0 10 0.001

Voltage (t2) 50 6.0 8 0.001

Voltage and frequency (t3) 60 6.5 2 0.001

Current (t4) 50 15.2 5 0.001

Voltage and frequency (t5) 60 9.4 4 0.001

Voltage and frequency (t6) 60 7.2 7 0.001

Voltage and frequency (t7) 45 6.0 6 0.001

Voltage (t8) 40 7.0 8 0.001

Vibration (t9) 45 6.6 10 0.001

Temperature (tj0) 55 8.2 12 0.001

Electrical stress (tu) 40 8.6 15 0.001

Table 6 Related calculation results.

Failure mode Parameter

R1 P1 FD1 R2i P2¡ fd2

/1 1.000 0.5812 0.5812 0 0 0

/2 1.000 0.5971 0.5971 0 0 0

/3 1.000 0.5958 0.5958 0 0 0

/4 1.000 0.6422 0.6422 1.000 0.6934 0.6934

/5 1.000 0.6468 0.6468 1.000 0.7097 0.7097

/6 1.000 0.6675 0.6675 0 0 0

/7 1.000 0.6290 0.6290 0 0 0

/8 0.999 0.6029 0.6023 0 0 0

/9 0.999 0.7602 0.7594 0 0 0

ity group size equals 1, then FAG = 0. FP = f4, f5}, TR = 0. There are no feedback loops in the system.

We can see that the system's inherent testability is very good. In engineering applications, if inherent testability analysis cannot satisfy the system's requirements, system design

scheme should be changed such as adding tests, adjusting component layout or reducing feedback loops, etc. When the scheme is amended, testability model and dependency matrix should be regenerated for further inherent testability analysis until system's requirements are satisfied.

Table 4 Fault/fault evolution-test matrix.

Failure mode Time

<1 <2 <3 <4 <5 <6 <7 <8 <9 <10 <11

/1 (1,0) 0 (1,0) (1,0) (1,0) 0 (1,0) (1,0) 0 0 0

/2 0 (1,0) 0 0 (1,0) 0 0 (1,0) 0 0 0

/3 0 0 0 (1,0) (1,0) (1,0) (1,0) (1,0) 0 0 0

/4 0 0 (1,0) (1,0) (1,0) 0 (1,0) (1,0) (1,1) (1,1) 0

/5 0 0 (1,0) (1,0) (1,0) 0 (1,0) (1,0) 0 (1,1) (1,1)

/6 0 0 0 (1,0) (1,0) 0 (1,0) (1,0) 0 0 0

/7 0 0 0 (1,0) (1,0) 0 (1,0) (1,0) 0 0 0

/8 0 0 0 0 (1,0) 0 0 0 0 0 0

/9 0 0 0 0 0 0 0 (1,0) 0 0 0

Table 5 p¡j results.

Failure mode Time

<1 <2 <3 <4 <5 <6 <7 <8 <9 <10 <11

/1 0.6943 0 0.5078 0.5320 0.4766 0 0.6561 0.6207 0 0 0

/2 0 0.6940 0 0 0.4766 0 0 0.6207 0 0 0

/3 0 0 0 0.5320 0.4766 0.6201 0.6561 0.6940 0 0 0

/4 0 0 0.6219 0.6143 0.5221 0 0.6916 0.6584 0.7282 0.6587 0

/5 0 0 0.6219 0.6143 0.5221 0 0.6916 0.6584 0 0.6587 0.7607

/6 0 0 0 0.6868 0.5639 0 0.6916 0.7279 0 0 0

/7 0 0 0 0.5746 0.5221 0 0.7253 0.6940 0 0 0

/8 0 0 0 0 0.6029 0 0 0 0 0 0

/9 0 0 0 0 0 0 0 0.7602 0 0 0

Assume the test symptom duration is equivalent to the time-to-failure, which is 100 time units. The fault detection sensitivity of all the tests to all the faults is 0.9, and the number of each type test is 1. Accroding to Eq. (7), the py results are shown in Table 5.

According to Eqs. 6, 8, and 9, the related calculation results are shown in Table 6.

According to Eq. (10), the achieved testability analysis results are FDR = 0.6992, FIR = 1.0000 and FPR = 0.7016. One can see that the fault detectable level and fault predictable level are not good, so test reliability and confidence degree should be enhanced in practical applications.

5. Conclusions

To address the problems that the traditional testability models do not include any quantitative PHM-related testability information and cannot describe fault evolution-test dependency, an integrated testability model for aerospace system PHM is proposed.

(1) Based on QDG and FFA, a system level modeling approach is introduced, which can describe fault-test dependency and quantitative testability information effectively.

(2) By incorporating PoF into components and combining the extended FMMEA, component level model can be constructed, which can be used to analyze fault evolution-test dependency.

(3) Testability analysis for PHM is presented in great detail. Compared to the traditional analysis, the analysis process adds some fault prognostics-related information such as possible predictable faults, predictable faults and fault predictable rate. Further, test actual attributes are considered in the analysis process.

(4) A case is given in detail to demonstrate the proposed hierarchical modeling methodology for a heading attitude system. The application results show the proposed approach is feasible and effective, and this approach can be used for testability analysis and design for PHM of any system.

Acknowledgement

This study was supported by National Natural Science Foundation of China (No. 51175502).

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Qiu Jing received B.S. degree from Beihang University in 1985, M.S.

and Ph.D. degrees from National University of Defense Technology

(NUDT) in 1988 and 1998 respectively, and now he is a professor in

NUDT. His main research interests are condition monitoring and fault

diagnostics, design for testability, prognostics and health management.