The improvement of the detection and diagnosis capability for the unanticipated fault is a tendency in the research and application of fault diagnosis. In this paper, some notions and the basic principles for the unanticipated fault detection and diagnosis are given. A general process model applied to the diagnosis for the unanticipated fault is designed, by adopting a three-layer progressive structure, which is comprised of an inherent detection layer, an unanticipated isolation layer and an unanticipated recognition layer. Several key problems in the general process model are analyzed. The model and methods proposed in this paper are driven by pure data and they can detect and diagnose the unanticipated fault. The approach is evaluated by using an example of a satellite's attitude control system, and excellent results have been obtained.
At present, in the research field of fault diagnosis, a great
majority of methods proposed are based on the premise of a
perfect fault pattern database. The treatment on the fault
detection and diagnosis are carried out for anticipated fault
(AF) [
In retrospect to the existing researches, rather little
attention has been paid to research UF detection and
diagnosis. Therefore, no mature solve scheme has been shaped for
either the problem itself or the technical realization [
For actual system, some impacts such as nonlinearity,
uncertainty and external interference are inevitable in its
actual operation, which will result difficulties in setting up a
precise model for the system. Consequently, the application
of the methods for fault detection and diagnosis based on
model inference will be very limited [
This paper proposes a data-driven fault diagnosis method for UF. Combined with the fault diagnosis process, a general process model (GPM) is advanced, which is comprised of an inherent detection layer (IDL), an unanticipated isolation layer (UIL) and an unanticipated recognition layer (URL). Firstly, according to different characteristics of the monitoring data, the corresponding residual statistics are built and a detection criterion of the IDL is provided for fault detection. Secondly, the statistic of angle similarity is constructed on the basis of the fault feature direction, the isolation between the UF and the AF is realized in the UIL. Finally, in the URL, by the adoption of the contribution factor, the UF is recognized. The method, as a fault diagnosis method driven by pure data, is capable of carrying out detection, isolation and recognition for the UF.
The paper is organized as follows. In Section 2, some notions and the basic principles for UF and UFDD are discussed. A three-layer GPM for UFDD is introduced in Section 3. Sections 4 analyzes some key problems in the GPM and advances the corresponding solutions. In Section 5, performance evaluation of the proposed GPM and methods for the satellite’s attitude control system is presented. Conclusions are drawn in Section 6.
2.1
The fault can be divided into the anticipated fault (AF) and the unanticipated fault (UF).
Explanation 1: Anticipated fault (AF) is the fault which has been recognized by people, existing in the fault pattern database with the relevant monitoring data and the processing strategy.
Explanation 2: Unanticipated fault (UF) is the fault which lacks prior knowledge without any fault samples or with few fault data. UF does not exist in the fault pattern database, and the corresponding elimination strategy for it has not been detected.
A perfect fault pattern database should be a set including
all AF patterns and UF patterns. However, due to some
objective reasons, the acquisition of the perfect fault pattern
database is extremely difficult. The AF rarely occurs, and
most of faults occurs in the actual working process are UF
[
Explanation 3: UF detection is a process for judging whether UF occurs.
The tasks of UF detection and AF detection are different. The two methods apply previous normal monitoring data to train a discriminator, and then the current monitoring data is used as the testing data to be input into the discriminator to judge whether the current status is a fault. However, the UF detection is carried out after the completion of fault detection, and the fault is further judged whether to be UF. Obviously, for AF detection, all faults are always assumed to be anticipated. Consequently, if the UF occurs, it will be misjudged as a certain anticipated fault. 2.3
Explanation 4: UF diagnosis is a process of determining whether the UF occur (i.e. UF detection). In addition, the UF diagnosis further includes the isolation and the recognition of the UF after the UF detection is completed.
Compared with the AF diagnosis, due to lack of prior knowledge of the UF, the mapping relationship from fault data to fault part (essentially, the fault pattern is a function between fault data and fault part) cannot be found. Therefore, the key for UF diagnosis is to quickly establish a cognition process. The cognition comprises the recognition of superficial data characteristics or the mapping recognition from data to a physical layer. Based on a fault diagnosis method driven by pure data, this paper focuses on the recognition of superficial data characteristics. 3
By combining the notion and basic principles of the UF and the UFDD, this paper proposes a multi-layer general process model (GPM) for UF diagnosis on the basis of pure data-driven method. The structure of GPM is shown in Figure 1. The first layer is the IDL, which establishes a detection discriminator for fault detection; the second layer is the UIL, which applies the detection residual to establish a fault feature direction so as to build an isolation discriminator to realize the isolation of the AF and the UF; the third layer is the URL, which applies a contribution factor to analyze the variant which is most relevant to the current UF and to realize the fault recognition based on superficial data characteristics. The first issue that a diagnosis system faces is to carry out normal/abnormal recognition for a feature vector of the monitoring data. The task of the IDL is to determine whether the monitoring data is normal or abnormal. The detection discriminator can be used for reflecting the characteristics of the normal system. In a given threshold, the testing data is inputted to the detection discriminator for judging whether the fault exists. If a value of the discriminator is smaller than the given threshold, the system is thought to be normal; otherwise, a fault is thought to occur. Meanwhile the occurrence time (Fault time) and the feature direction of the fault (Current fault direction) should be determined, and the testing data is presented to the UIL.
Essentially, the IDL is a single discriminator, which can be applied to catch the characteristics of the system in a normal pattern as well as to complete the detection and discrimination of the testing data. Two key problems are involved, the first is the residual generation and the second is the residual evaluation. The specific techniques can be seen in Section 4.1. 3.2 The task of the UIL is to finish the isolation between the UF and AF. After detected, the current fault shall be judged whether to be the AF or the UF. If it is, the current fault will be classified as some sort of AF. All AF patterns are saved in the pattern database of AF. The isolation discriminator matches the feature of the current fault pattern with all those of the AF patterns successively, so as to realize the isolation between the UF and AF. If the feature of the current fault cannot be matched with any AF pattern, it indicates that the UF occurs. The testing data is presented to the URL. The key problem of the UIL lies in the establishment of an isolator and the design of an isolation criterion. The specific techniques can be seen in Section 4.2. The task of the URL is to perform online learning and analysis for the UF data, so as to generate the fault pattern. The function of the URL is to learn and summarize the pattern found in unknown pattern. As it is different from the AF, it is difficult to find the mapping relationship from the fault data to the fault part for the UF. Therefore, the key point of recognition lies in establishing the corresponding relationship between the data and the unknown fault. Due to insufficient recognition on the UF and lack of historical information and prior knowledge, it is usually more difficult to establish the mapping relationship on the physical layer. The key point of this paper is to analyze the UF recognition based on the superficial data layer. According to contribution factor, the variant which is mostly relevant to the current UF can be found, so that the UF recognition is finished. The specific techniques can be seen in Section 4.3. 4
In the above section, a basic framework of the UF diagnosis is provided. The task of the UF diagnosis is to detect, isolate and recognize the UF. The detection is a starting point of fault diagnosis, and the target of the fault detection is to judge whether the UF occurs; the isolation is the core of fault diagnosis; and the recognition is a terminal point of fault diagnosis. Additionally, the recognition is also the starting point of fault-tolerant control (fault processing). The specific techniques on detecting, isolating and recognizing the UF can be seen below. 4.1
Just as Section 3 shows, the basic task of the IDL is to judge
whether the testing data is normal. If it is a fault,
simultaneously the occurrence time and the feature direction of the
fault shall be determined. The key point of the IDL lies in
the detection residual generation as well as the residual
evaluation. The detection statistic is established according
to the residual, and the fault detection is performed
according to the given criterion. For different monitoring data,
different residual generation approaches exist, including
simple T2 detection [
The characteristics of the monitoring system and
monitoring data can be applied to select the corresponding
detection method. The simple T2 statistic detection is applied
to a stable data [
In practical application, the characteristics of the monitoring system and the monitoring data can be applied to select the corresponding detection method.
Besides, for the three methods analyzed above, only the characteristics of data output are considered. However, for some systems (such as the satellite’s attitude control system), the object of the fault detection always comprises control input as well as measuring output, and the control input has a certain responding relationship with the measuring output. In the situation where there is no baseline training data, an input-output system identification method is needed to search a model structure for the system, and thus the fault detection both on control input and measuring output will be performed in the IDL.
If we assume that (Un−1,Yn−1 ) ∈ ( R(n−1)× p , R(n−1)×m ) are respectively as system input and system output before the nth time period, take them as the training data and make (un , yn ) ∈ ( R1× p , R1×m ) as the current testing data. The train purpose is to find the model structure of the system, usually with the rule as follows mfin Yn−1 − f (Un−1 ) (1) term, term; and (2) (3) Let ˆ Yn−1 = f (Un−1 )
is yˆn = f (un ,Un−1,Yn−1 ) Yn−1 = Yn−1 − Yˆn−1 = Yn−1 − f (Un−1 )
T is the is
tendency the
residual one-step prediction, rn = yn − yˆn is the prediction residual, then the key point for the minimum problem in (1) is to construct the function f between the system input and system output.
If a mathematical model can be obtained for the system
equation by the physical mechanism, the estimation of f can
be converted into the parameter estimation (Gray-Box
Model); and if there is no physical background, f can be
estimated only according to the experiment and the system
identification (Black-Box Model). Common linear black
box models comprise an autoregression model (AR Model)
with external input, an autoregressive moving average
model (ARMA Model) with external input, an output error
model (OE Model), a Box-Jenkins model (BJ Model) and a
prediction error minimized model (PEM Model); and
common nonlinear black box models comprise a nonlinear
autoregression moving average model (NLARMA Model)
and a nonlinear Hammerstein-Wiener model (NLHW
Model) [
After obtaining the prediction residual, the detection statistics are as below:
T 2 ( yn ) = rnT cov (Y )-1 rn where cov (Y ) is the covariance of the residual term Y , and a judging threshold is set to be
m ( n)( n − 2) Tα2 = ( n − 1)( n − 1 - m ) F(1−α ) ( m, n − 1 − m) current directions from the same pattern. ξ2 is another true direction, corresponding to another fault pattern. The origin of the coordinates can be regarded as the true direction for the normal pattern.
ξ1 ξ2 ξ1 ξ2 where F(1−α ) ( m, n −1 − m) indicates a quantile of F distribution function when a significance level is α , the degree of freedom is ( m, n −1 − m) .
If T 2 ( yn ) > Tα2 , yn−1 is considered as the fault point. However, a false alarm is inevitable because of noise, thus we need a more reliable criterion for detection as follows.
Criterion 1: If T 2 ( yn ) > Tα2 holds continuously for W times, then the fault has really happened, where W is called time threshold. The W-th alarm time is considered as the fault time (tf) (i.e. the occurrence time of the fault) and the residual r of the fault time is called the current fault direction or current direction (i.e. the feature direction of the fault).
The detection statistic threshold is decided by Equation (3). The time threshold should not be too large (usually 2 to 4) to avoid any false alarms. A larger time threshold makes a more reliable decision, but it will cause some detection delay which will cause harm to the system. Current fault direction is the key information of each fault, and it is the base for the isolation fault. According to Criterion 1, the current fault is detectable if and only if | rn ||> Tα2 ( rnT cov(Y )−1rn ) −1 (4)
In the IDL, the fault detection is realized by the adoption of the input-output system identification method. Moreover, the occurrence time and feature direction of the fault can also be obtained.
Obviously, the input-output system identification method is provided with all the advantages of the time-series modeling prediction method. It is particularly suitable for the system with discontinuous input and discontinuous output at the same time, its defect is that the calculation amount is large, and the iteration process is relatively difficult. 4.2
The basic task of the UIL is to utilize the feature direction of the fault obtained in the IDL to establish the isolation discriminator, and then to realize the isolation between the AF and the UF. The key point lies in the isolator establishment. Here the concept of direction similarity is induced, and a fault isolation criterion is given. In Criterion 1, the definition of current fault direction or current direction (i.e. the feature direction of a fault) is given. We adopt the true fault feature direction as defined below to be the fault’s pattern characteristics on superficial data layer.
Explanation 5: True (fault) direction of a fault pattern is defined as the unified mean of all possible current fault directions from the same pattern.
The relationship between the current directions and the true direction is just like that between discrete random variable and its expectation. It is easy to understand that 1 n ξ = lim ∑ ri / n→∞ n i=1 r = r ξ + ε 1 n
∑ ri n i=1 2 (5) (6) where {ri}in=1 are all possible current directions from the same pattern, and ε is the noise and r is the magnitude of the current direction.
It is shown in Figure 2 that there are two opposite true directions for each fault pattern, e.g. the true direction , ξ1 , is in the center of a symmetric cone, around which are the
Denote θ ( r, ξ ) is the angle between the current direction and the true direction, Ddisc ( r, ξ ) = 1 − cos (θ ( r, ξ )) is called the directional discrepancy between them. We can find that if they are from the same pattern, Ddisc (r, ξ ) will be small, otherwise, it will be large.
Suppose that ε ∼ N (0, Ω) , the current direction is r = ε + r ξ , and {ξi}iq=1 is all anticipated true directions, and q , then the isolation statistic is ξi0 = argξmin{1 − cos(r, ξi ) }i=1 given as follows
Iso(r) = r (1 − cos(r, ξi0 ) )
ξiT0Ωξi0
Iso(r) ∼ N (0,1) Theorem 1: If Iso(r ) is defined in Equation (7), then Proof: Suppose that the current direction is r = ε + r ξ , where ξ is the true direction and ε is the observation noise, and ε ∼ N (0, Ω ) . According to Explanation 5 we have ξ = 1 . If cos(r, ξ ) ≥ 0 , we can approximately obtain that cos(r, ξ ) = ξ Tr = ξ Tε + 1 ∼ N (1, r −2 ξ TΩξ ) ξ r r i.e. cos(r, ξ ) satisfies truncated normal distribution.
According to Equation (10) and Equation (11), we obtain r (1 − cos(ξi0 , r)) ∼ N (0,ξiT0 Ωξi0 ) r (1 + cos(ξ , r)) ∼ N (0,ξ T Ωξ ) r (1 − cos(ξ , r) ) ∼ N (0,ξ T Ωξ ) Iso(r) = r (1 − cos (r, ξi0 ) )
∼ N (0,1) ξiT0 Ωξi0 and thus the theorem is proved. Therefore, the threshold for Iso(r ) is Φ(1−α ) , where α is the significance level, and Φ is the inverse of the normal cumulative distribution function. We provide the isolation criterion as follows. (7) (8) (9) (10) (11) (12) (13) Criterion 2: If Iso(r ) > Φ1−α holds true, the current fault is unanticipated; otherwise, it is anticipated.
Criterion 2 indicates that UF with too small a magnitude cannot be isolated. If the current fault is unanticipated, a new fault pattern is found and the unified current direction is regarded as its true direction. If the current fault is anticipated, then the current direction should be added to the corresponding AF direction database in UIL of the GPM, and the true direction shall be updated. 4.3
The basic task of the URL is to carry out online learning and analysis for UF data. The key point of recognition or identification is to establish the corresponding relationship from the monitoring data to the unknown fault or the characteristics of the unknown fault. The UF diagnosis discussed in this paper is an approach driven by pure data, thus the characteristic recognition on the data layer is more focused. According to the contribution factor, the variant which is most relevant to the current UF can be found, and then the UF recognition is completed.
Known from Criterion 1 that after the residual detection statistic is established, if T 2 ( yn ) > Tα2 , it is thought that a fault occurs at time period n-1. For the system with the control input and measure output, firstly a residual covariance matrix R (i.e. cov(Y ) in Equation (2)) is subjected to the singular value decomposition, which is
R = P T diag ( λ)P (14) where λ = (λ1,…,λm ) , P = ( p1,…, pm ) , pi indicates the ith column of P , and p ji indicates the jth component of pi . Let ti = r T pi , and rj indicates the jth component of the current fault feature direction r, where 1 ≤ j ≤ m .
Explanation 6: The contribution factor of the jth variant to the current fault feature direction r is
m Cont ( j ) = ∑ (ti rj p ji / λi ) (15)
i=1 From the aspect of characteristic recognition in the data layer, the variant with the largest contribution factor is the fault variant. If it is a sensor fault, the sensor corresponding to the variant with the largest contribution factor is the sensor hardware with the fault. 5
The effectiveness of the proposed GPM and the corresponding UF fault detection, isolation and recognition method are demonstrated in this section through a satellite’s attitude control system model. 5.1
The satellite’s attitude control system is a main part of a
satellite, which consists of four main parts: a satellite body,
a controller, an execution mechanism and a measuring
mechanism [
As the complexity of the satellite’s attitude control system, faults particularly for the measuring mechanism and the execution mechanism occur rather frequently.
Here on consideration of the monitoring data for the satellite’s attitude control system. The monitoring data are provided by China Aerospace Science and Technology Corporation (CASA).
Variable subscript 1 2 3 4 1 2 3 4 5
The monitoring data comprises of not only the output data of the measuring mechanism, but also the control input of the execution mechanism. The dimension of the data output by the measuring mechanism is m = 7 , The dimension of the data input by the execution mechanism is p = 4 , which can be seen in Table 1. There are altogether 10 batches of monitoring data, which can be seen in Table 2. The first batch is the normal data, and the normal pattern data is discontinuous and unstable (Figure 3). The subsequent 9 batches are used for testing, and different fault patterns (a sudden-change fault, a gradual-change fault and so on) are given. In Figure 3, the comparison of the monitoring data in the fault with drift-increasing of gyro at roll axis and the normal pattern is given. The time of each batch of data is 45000s-48000s; each piece data is collected per second, and the data length n = 3000 .
Additionally, the public parameters used in the simulation are assigned as follows: The significance level α = 0.01 and the time threshold defined in Criterion 1 is W=3. The monitoring data are relatively more complex, comprising of the output data of the measuring mechanism and the control input of the execution mechanism (seen in Table 1). The normal pattern data is discontinuous and unstable (seen in Figure 3), and the fault pattern is diversified (with sudden-change fault, gradual-change fault and so on). Therefore, the normal pattern data is difficult to be discriminated from the fault pattern data (seen from Figure 3).
With the input-output system identification method, the Hammerstein-Wiener model (NLHW) is adopted. Equation (1) is optimized, and the responding function f between the input and output is estimated. Similarly, for the same data (Drift-increasing fault data of gyro at roll axis (the batch number is 8) in Table 2), the detection result of the IDL is given in Figure 4, which can be seen that the fault detection is timely, the detection effect is remarkable, and 4s detection is delayed caused by the time threshold, W = 3 . 4.6
4.7 es-x ture direction and the direction similarity is valid, and the isolation between the UF and the AF can be truly realized.
By adopting the input-output system identification method, the detection results in the IDL for the data in Table 2 are shown in Table 3. The fault detection is timely, and the detection effect is more obvious (both of the FAP (false alarm probability) and the MAP (missing alarm probability) are much lower).
In the IDL, the fault detection can be realized, and the fault time and the current fault direction are also determined. In the UIL, Criterion 2 is adopted to realize the isolation between the UF and the AF. In the initial stage, the AF pattern is assumed to be empty, therefore, when the second batch of data in Table 2 is filled into the UIL, the detected fault must be the UF, and then the isolation result is transferred into the URL. When the third batch of data in Table 2 is filled into the IDL, the fault time is that t = 1001s , the statistic of the directional similarity is r (1− cos(r,ξ1) ) / ξ1TRξ1 = 7.3179, and the isolation threshold of the UF is also Φ0.99 = 2.3263 . Obviously r (1− cos(r,ξ1) ) / ξ1TRξ1 > Φ0.99 , the current fault pattern is different from the first fault pattern, and an UF occurs. Then the UF is transferred into the URL. The fault isolation result for all the tested data in Table 2 can be seen in Table 4. From Table 4, we know that the isolator with the fault fea
In the IDL, the fault detection can be realized, and the fault time and the current fault direction are also determined. In the UIL, Criterion 2 is adopted to realize the isolation between the UF and the AF. In the initial stage, the AF pattern is assumed to be empty, therefore, when the second batch of data in Table 2 is filled into the UIL, the detected fault must be the UF, and then the isolation result is transferred into the URL. When the third batch of data in Table 2 is filled into the IDL, the fault time is that t = 1001s , the statistic of the directional similarity is r (1− cos(r,ξ1) ) / ξ1TRξ1 = 7.3179, and the isolation threshold of the UF is also Φ0.99 = 2.3263 . Obviously r (1− cos(r,ξ1) ) / ξ1TRξ1 > Φ0.99 , the current fault pattern is different from the first fault pattern, and an UF occurs. Then the UF is transferred into the URL. The fault isolation result for all the tested data in Table 2 can be seen in Table 4. From Table 4, we know that the isolator with the fault feature direction and the direction similarity is valid, and the isolation between the UF and the AF can be truly realized.
After isolating the UF, the recognition of the UF should be carried out on the data layer. For the data in Table 2, the recognition result is that: the fault feature direction is (0.9876,-0.0042,0.041,-0.053,0.0453, -0.1342, 0.0678 )T . The variance with the largest contribution factor is the first dimension. According to Explanation 6, the contribution factor reaches 97 percent, and it indicates that the fault occurs for the earth sensor at the roll axis. Similarly, the result of the UF recognition in the URL for other batches of data is shown in Table 5. From Table 5, the recognition of the UF corresponding to the fault variance is correct, and the UF recognition of the data layer is reached.
The paper firstly takes the UF as a main diagnosis object. The detection and diagnosis method based on data driven for the UFs has been researched. The GPM for the UF diagnosis has been designed. The GPM is comprised of the IDL, the UIL and the URL. This GPM has provided a framework support for the UF diagnosis. According to the system both with the control input and the measure output, the system identification detection method corresponding to the IDL has been provided. The current fault feature direction and the feature direction of the AF pattern have been used to establish the statistic of directional similarity. The isolation between the AF and the UF has been realized in the UIL. According to the singular value decomposition, the fault contribution factor of each variance has been obtained, and the fault recognition in data layer has been completed. The application to fault diagnosis of the satellite’s control system has demonstrated its validity.
Our research shall be furthered in two directions. Firstly, based on the framework of the GPM, the fault detection, isolation and recognition method on the foundation of model inference shall be researched. Secondly, the GPM and methods shall be applied to the diagnosis of other complex system for both military and civil use.
This work was supported in part by National Natural Science Foundation of China (NSFC) under Grant No. 61304119. Besides, we would like to especially thank China Aerospace Science and Technology Corporation (CASA) for providing the satellite control system data.