The tolerance handling in analogue fault diagnosis is a challenging problem. Although lots of methods are effective for fault diagnosis, it is hard to apply them to the case with tolerance influence. In this paper, a robust statisticsbased approach is introduced for toleranceinfluencing fault diagnosis. The advantage of this proposed method is that it can accurately locate the data fusion among fault states. In addition, the results in analogue benchmark (e.g., linear voltage divider circuit) indicate that it is effective in fault diagnosis in accordance with given fault diagnostic requirements (e.g., fault diagnosis error, fault detection rate).
The tolerance is one of the most common problems in the production of analogue circuits [
In the past, some scientists often accept an unauthenticated assumption of nominal distribution to circuit response (voltage, current, and other fault signatures) according to the statement in [
(a) The inaccurate distribution
To solve these questions as above, this paper presents a robust statisticsbased method for the fault diagnosis. The robustness means the statistical fault modelling is established according to the strong support of theories. Thus, the capabilities of fault diagnosis for an analogue circuit can be estimated in a trustable manner. Furthermore, there are at least 2 more advantages:
The rest of paper is organized as follows. At first, in the last part of this section, all of critical problems in fault diagnosis, including the fault detection, identification, and diagnosis error control, are integrated in the working flow of circuit diagnosis as shown in Figure
A scheme on the fault diagnosis with tolerance handling: in the diagnostic design of an analogue circuit, the fault diagnosis error limit control assures that the fault diagnosis can be successful with specific requirements. Actually the fault detection or identification is accomplished with the measurement on the optimum testnodes set, in order to reduce the data fusion for a best diagnostics.
Without loss of generality, assume that the circuit under test (CUT) in Figure
A representative linearanalogue circuit under test (CUT) in practice.
A linear circuit in Figure
At first, suppose that there is a vertex
Equation (
Lemma
When the linearanalogue circuits are designed with tolerance, the tolerance influence can be modelled as disturbance on
Suppose that a matrix of
In accordance with theTheorem
Given a linear CUT network composed of an independent voltage source
Furthermore, the testnodes set (accessible nodes to voltage measurement) is set as
Based on the substitution theorem in [
The linear analogue circuit network with fault on component
With the superposition theorem, a nodal voltage is the algebraic sum of the following two components: the first component is caused by
Here,
If the parameter of
Eliminating
Setting
The FSV was proposed in [
Fault slope value (FSV) is a random variable
Furthermore,
In practice, there exist lots of numerical algorithms for estimation of the ratio of two normal random variables. For instance, the computation in [
The derivation (
In a circuit under test (CUT), the estimated range of FSV is represented in Table
Range of FSV for all fault states on the testnode pairs.




 































The determination of
A fault can be located with a fault diagnosis error <100(2
(a) Circuit fault (
Without loss of generality, in Figure
Once the required fault diagnosis error is satisfied with a given tolerance, such tolerance is less than the tolerance limit value, which is defined as
Given a fault diagnosis error
According to Definition
Then based on the corresponding value of FSV in Table
To be convenient in the following statement, we define
Then the result of fault diagnosis is measured through fault detection rate (FDR), fault isolation rate (FIR) as follows.
Fault detection rate (FDR): the ratio of fault states that can be detected in a given fault states set:
where the number of fault states is
Fault isolation rate (FIR): the ratio of fault states pair can be isolated based on a given fault states pair set:
where the maximum number of isolated fault states pair is
The basic idea of statistics approach in fault diagnosis can also help us identify fault parameter. In this paper, the fault parameter identification needs the determination of the diagnostic fault parameter border, which decide how small the alteration of component parameter could be distinguished from faultfree state. As for the diagnostic fault parameter border, Theorems
In the single fault case, there is a parameter range
In the single fault case, the variation in corresponding element of
Assume that there are potential
Assume that
Furthermore, Theorem
Assume that the value of
According to the linear expression of
The optimum testnodes set selection is also a critical problem in fault diagnosis that can be solved on the basis of the result of accurate statistics. Here, according to the statistical result, a bipartite decision network including fault states nodes and accessible testnodes pair vertices is established, in which the connections from a testnodes pair node
A simple example of bipartite decision network is shown in its matrix form of (
Once the bipartite decision network is established, the proposed process of testnodes selection is as follows.
The target testnodes set
The degree
All the connections from the selected testnode pair vertexes to the fault states pair vertexes should be eliminated. Then, the corresponding fault states pair is set be diagnosed.
If there are fault state pairs having not been diagnosed, update the values of degree
The algorithm ends up with the condition that there are no connections in the bipartite network.
In this section, there are two examples to validate the proposed fault diagnosis based on the accurate statistical analysis of FSV. At first, the simulations are implemented with PSPICE and R language programming on the respective software platform (Orcad 10.5 and RStudio) to determinate the proper testnodes set for fault diagnosis. Both of them are executed with a given requirement of
Since the hard faults (catastrophic ones) can be thought of as a case of soft faults and the single fault in circuit is the most probable occurrence, the discussions in these examples focus on the case of single soft fault.
The first example is shown in Figure
(a) The linear analogue circuit with test nodes
Based on Theorem
The FSV range estimated through the proposed normal ratio distribution for the circuit in Figure


 


—  —  — 

(3.97, 4.12)  (7.89, 8.38)  (15.7, 16.8) 

(−1.99, −1.78)  (−3.96, −3.48)  (−7.89, −7.04) 

(−0.49, −0.42)  (−0.98, −0.82)  (−1.96, −1.61) 

(0.24, 0.25)  (−0.25, −0.22)  (−0.48, −0.42) 

(0.24, 0.25)  (0.12, 0.13)  (−0.13, −0.12) 

(3.97, 4.14)  (7.89, 8.40)  (15.7, 16.8) 

(0.93, 1.10)  (1.84, 2.01)  (3.63, 3.99) 

(0.24, 0.25)  (0.46, 0.49)  (0.93, 0.99) 

(0.23, 0.25)  (0.11, 0.13)  (0.23, 0.25) 

(0.24, 0.25)  (0.12, 0.13)  (0.05, 0.06) 
These fault isolation or detection result can be coded into a bipartite network. To be convenient, it can be expressed as a matrix shown in (
The selected testnodes pair according to the maximum value of




 














FIR = 98%  FDR = 100% 
In this paper, the simulation of the circuit in Figure
The estimated FSV range through the normal distribution (NDS) or proposed statisticsbased method (NQS) for fault state
1%  5%  10%  15%  


NQS  (1.99, 2.02)  (1.98, 2.10)  (1.96, 2.23)  (1.97, 2.14) 
NDS  (1.99, 2.00)  (1.98, 1.99)  (1.96, 1.98)  (1.97, 1.98)  

NQS  (3.99, 4.02)  (3.97, 4.14)  (3.95, 4.28)  (3.99, 4.52) 
NDS  (3.99, 3.99)  (3.97, 3.98)  (3.94, 3.98)  (3.95, 4.05)  

NQS  (7.98, 8.07)  (7.89, 8.40)  (7.79, 8.89)  (8.06, 9.71) 
NDS  (7.98, 7.98)  (7.89, 7.94)  (7.88, 8.04)  (8.60, 8.88)  

NQS  (15.9, 16.2)  (15.7, 16.8)  (15.4, 17.9)  (16.2, 22.3) 
NDS  (15.9, 15.9)  (15.4, 15.9)  (14.2, 16.8)  (16.8, 21.0)  
MMFR  66.7%  91.7%  92.5%  94.1% 
Up to now, the simulationbased illustrations in the linear circuits of Figure
The fault diagnosis in the actual circuit of Figure



Diagnosis?  


5.92 V  0.38 V  —  — 

4.00 V  0.26 V  16.7  N 

6.95 V  0.25 V  −7.69  Y 

6.20 V  0.24 V  −2.00  Y 

5.99 V  0.24 V  −0.5  Y 

5.94 V  0.25 V  −0.12  Y 

5.48 V  0.35 V  14.7  N 

6.05 V  0.41 V  3.45  Y 

5.96 V  0.41 V  0.8  Y 

5.93 V  0.41 V  0.2  Y 

5.94 V  0.64 V  0.06  Y 
According to Table
The proposed fault diagnosis method is also effective in the analogue circuit composed of linear amplifiers. In this case, the tolerance is
With the proposed statisticsbased method, the estimations of FSV value have been shown in Table
The FSV range estimated through normal quantile distribution in the circuit of Figure





Diagnosis?  


—  —  —  —  —  

(0.64, 0.86)  (0.72, 1.06)  (0.74, 1.11)  (0.71, 1.34)  (0.76, 1.46)  Y 

(−0.84, −0.56)  (−0.90, −0.54)  (−0.93, −0.61)  (−1.37, −0.55)  (−1.33, −0.74)  Y 

(−0.47, −0.16)  (−1.03, −0.23)  (−0.10, −0.24)  (0.20, 0.95)  (0.26, 1.06)  Y 

(0.14, 0.60)  (1.63, 4.00)  (1.68, 4.31)  (1.61, 4.32)  (2.12, 4.95)  Y 

(−1.59, −0.25)  (−1.61, −0.37)  (−2.76, −1.87)  (−3.55, −2.06)  (−4.08, −2.06)  Y 

(−3.03, −1.39)  (−3.81, −1.58)  (−0.79, −0.58)  (−1.17, −0.49)  (2.99, 4.67)  Y 
Fault diagnosis result in the actual circuit of Figure





Diagnosis?  


0.55 V  0.43 V  0.46 V  —  —  

0.72 V  0.57 V  0.61 V  1.21  1.13  Y 

0.74 V  0.29 V  0.31 V  −1.36  −1.13  Y 

0.50 V  0.26 V  0.28 V  0.29  0.28  Y 

0.61 V  0.50 V  0.76 V  2.00  4.66  Y 

0.42 V  0.49 V  0.52 V  −2.16  −2.16  Y 

0.41 V  0.55 V  0.42 V  −1.17  3.50  Y 
(a) The test nodes in this circuit are
This paper builds a statisticsbased viewpoint in order to solve the tolerance problem in accordance with given fault diagnostic requirements. Based on this point of view, the relationship between tolerance limit and fault diagnosis error limit can be discovered in this paper. And in the process of fault diagnosis, the accurate statistical feature discussion let us know more accurate response varying range, which assure that the measurement reduction (testnodes selection) and fault parameter identification can avoid incorrectness in fault diagnosis applications. As a matter of fact, all of these advantages have been analysed and tested in the experiment of linearcircuit benchmarks.
This is an acute view to handle the tolerance problem in analogue circuit diagnostic design and fault diagnosis. Furthermore, it can be generalized in the following cases.
First and foremost, one of the authors would like to show my sincere thanks to these colleagues in my lab, such as Weimin Xian, JinYu Zhou, and Wei Li, who give me sound pieces of advice in amendments of words and pictures as well as corresponding data. Then he shall extend his thanks to Vice Professor Long for all his kindness, which has also helped him to develop the fundamental academic competence in the programming of R code and Spice simulation. At last, this work was supported in part by the National Natural Science Foundation of China under Grants nos. 61071029, 60934002, 61271035, and 61201009, and by Ministry Level PreResearch Foundation of China under Grant no. 9140A17060411DZ0205.