# Closed-Loop Fault Diagnosis of SDR Using Gap Metric, PCA, and Kalman’s Principle

Authors: Hongfu Wang; Qinghua Zeng; An Wang; Zongyu Zhang; Weide Liu
Journal: International Journal of Aerospace Engineering
 (2023)

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## Abstract

The measurement of the pressure in the gas generator (GG) plays a decisive role in the closed-loop regulation of the gas flow in a solid ducted rocket ramjet (SDR). Therefore, the fault detection and isolation (FDI) of the pressure sensors is particularly significant. Especially in the GG with double pressure feedback, there are issues such as less available data and sensor fault transmission, which make the FDI more difficult. In this paper, for the GG with double pressure feedback, firstly, the “Consistency Index” was constructed based on the principal component analysis (PCA) algorithm, which made it easier to evaluate the consistency of the measured values. Secondly, based on Kalman’s principle, the redundancy information of the system was used to estimate the pressure in the GG. Finally, the gap metric between the measured pressure and the estimated pressure was employed to characterize the health of the sensors. Compared with the MWPCA algorithm, it was shown that our proposed algorithm was more accurate in fault diagnosis and could avoid the problem of missing alarm when two sensors had consistent faults, which would provide strong support for the safe operation of the SDR and could further promote its application in long-endurance aircrafts.


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## Body

## 1. Introduction

The SDR gas flow regulation technology can realize the adjustable thrust of the aircraft, which is very important to realize the wide envelope and large maneuvering flight of the aircraft [1]. Adjusting the gas flow by controlling the throat area is the most practical flow adjustment scheme. Because the gas flow at high temperature and high pressure is difficult to measure directly, the gas flow is often indirectly regulated by adjusting the pressure in the GG. However, on the one hand, the pressure sensors in the GG face a high-temperature environment (when the pressure inside the GG is 1 Mpa, the temperature can reach 1000 degrees Celsius); on the other hand, the gas pipes of the sensors are easily blocked by the solid powder produced by the combustion of the propellant; there is a certain risk of failure. Therefore, the FDI of the GG can enhance the safety and reliability of the SDR, which is important to guarantee the stable flight of the aircraft.After decades of research, scholars had proposed a variety of fault diagnosis methods, which could be roughly divided into data-driven approaches, model-based approaches, and knowledge-based approaches [2–4]. Among them, the data-driven approaches usually do not rely on accurate mathematical models, but directly extract the latent information from historical data sets [5]. Therefore, the data-driven approaches were mostly used in sensors’ FDI, such as the PCA methods [6–8], time-frequency analysis methods [9, 10], machine learning-based methods [11–17], and the combination of the above methods [18–22]. Huang et al. defined the isolation index by setting each sensor in turn as a missing variable and then recalculating the corresponding fault detection statistic. The missing variable associated with the maximum isolation index was identified as the faulty sensor, and the results showed that the method is effective in detecting and isolating drift faults [6]. In addition, PCA algorithm-based fault diagnosis has been used for FDI in electric traction systems and air-conditioning systems [7, 8], while the PCA methods were usually completed under the assumption that “only one sensor fails at the same time,” which limited the application to some extent. The CIBL-RTCSA (conditionally independent Bayesian learning-based recursive transformed component statistical analysis) approach was proposed in reference [9], and the accuracy of FDI could be effectively improved compared with the naive Bayes. In reference [10], for the problem of motor fault diagnosis, the Fourier decomposition of the Hilbert modulus obtained by different measurement methods was performed to enhance the expression of its characteristic quantity, which could improve the accuracy of fault diagnosis. Obviously, the methods based on time-frequency analysis were usually used for offline FDI due to their large computational load and dependence on a large amount of sample data. Souza et al. used a predictive maintenance model with convolutional neural network (PDM-CNN) to classify bearing faults, and the validation based on a publicly available database showed that the model was more than 97% accurate in classifying faults [11]. Mousavi et al. employed fuzzy inference to estimate the severity of faults based on fuzzy rules extracted from fault patterns and used adaptive thresholds to improve the accuracy of fault estimation [12]. Toscano and Lyonnet presented a parametrization method of a fuzzy classifier from a set of examples without necessarily using a process of iterative training, and this classifier was applied to the diagnosis of a DC motor showing its effectiveness [13]. Jan et al. used a fuzzy deep neural network (FDNN) for diagnosing sensor fault types, while a support vector machine (SVM) was used for fault classification, and the fault detection performance was analysed from several aspects utilizing fault simulation and injection [14]. Saeed et al. have used the Extremely Randomized Trees method to solve the problem of fault diagnosis of wireless sensor network, and it has been shown that this method had strong robustness to signal noise [15]. Siahpour et al. adopted a deep learning-based cross-sensor domain adaptation approach for machinery fault diagnosis, which can achieve 95% diagnostic accuracy [16]. Based on a multilabel classification algorithm, Tan et al. used a single fault data for simultaneous fault diagnosis, which significantly reduced the requirements for training data in machine learning. However, he also pointed out that the algorithm is difficult to deal with the problem of data imbalance [17]. And Lei et al. also pointed out that intelligent fault diagnosis based on machine learning still relies on expert knowledge; although deep learning can automatically learn features, it requires enough labelled samples, which are impractical in engineering scenarios [23].In summary, both time-frequency analysis and machine learning-based methods rely heavily on sample data, which often leads to large and complex computations in their applications. However, the PCA methods could describe high-dimensional data with fewer principal elements and realize “data dimensionality reduction,” thereby greatly reducing the amount of data calculation, which is its significant advantage. Therefore, the PCA methods were often used in combination with the other two methods for FDI. Rezamand et al. applied a wavelet-based probability density function (PDF) to obtain the probability of maintaining health from the extracted principal components (PCs) and finally predicted the remaining useful life of the wind turbine using the Bayesian algorithm and illustrated the effectiveness of the proposed method through experimental tests [18]. For wind turbine systems, Fu et al. used the uncorrelated multilinear principal component analysis (UMPCA) to extract uncorrelated principal components from measurement data on the one hand and used the fast Fourier transform (FFT) to enhance the data classification capability of UMPCA on the other hand, illustrating the more effective fault classification capability of the proposed algorithm by comparing it with the multilinear principal component analysis (MPCA) method [19]. Kordestani et al. used the DPCA algorithm to reduce the data dimension of the vibration signal and used the support vector machine (SVM) to detect and isolate the gear faults in wind turbines. The analysis of the historical data has indicated the effectiveness of the algorithm [20]. And Li and Qu combined the convolutional neural network (CNN) and PCA methods to solve the problem of aeroengine sensors fault diagnosis, which could improve the efficiency of CNN and the accuracy of fault diagnosis [21]. A novel CNN with the function of spectrum calculation and fault diagnosis was designed, in which the spectrum calculation network and the fault diagnosis network were connected in series. Only by system input and output, the fault diagnosis could be realized, which avoided the complex process in traditional methods [22]. However, the data-driven approaches still need sufficient fault data for support and often have poor real-time performance, making it difficult to complete online FDI tasks.The model-based approaches have clear physical meanings and are important research directions for sensors’ FDI, such as the FDI of sensors based on reference models as well as residuals, but are significantly affected by the process’ disturbance model estimation [24]. Jiang et al. used the Dempster-Shafer evidence theory to establish the fault model and test model of the sensors, which could better express the uncertainty information and fit the measurement data [25]. In addition, the observer-based method can also be attributed to one of the model-based methods, for example, the adaptive neural network observer and the unknown input observer methods were used in the references [26, 27], respectively. However, in practical applications, the research objects often have many dynamic characteristics that are not easy to model, which also face many disturbances in the working process, so the mismatch of the models may lead to serious misclassification.The knowledge-based approaches are far from perfect due to limited knowledge, single repeatability of rules, and subjectivity of function definition [2], while they were usually used in combination with data-based and model-based methods [8, 26]. The above three approaches have their own advantages, for example, the model-based approaches have clear physical meaning but are affected by the uncertainty of the models. Although the data-driven approaches do not require accurate models, they rely on the historical data of the systems, while the knowledge-based approaches to fault diagnosis are highly accurate but have knowledge bottlenecks and are less versatile [3]. Therefore, it is worthwhile to focus on one of the research directions to reasonably and comprehensively apply various approaches to improve the accuracy of FDI and enhance the fault tolerance performance of aircraft [28].Some scholars had realized that when there were multiple data sources representing the same variable, a metric was needed to measure the difference between different pieces of evidence. For example, Ji et al. performed initial fault detection based on augmented the Mahalanobis distance, but this method required a priori knowledge of fault direction information [29]. Based on the classic Dempster-Shafer evidence theory, Lin et al. used the Euclidean distance to characterize the difference between different pieces of evidence, which could better analyse the conflict between them [30]. Similarly, Xiao also used the Dempster-Shafer evidence theory to characterize conflicts between pieces of evidence [31]. Compared with the classical Euclidean metric, the gap metric can be used to measure the correlation of different data in high-dimensional space, but the application of the gap metric theory to the fault diagnosis of sensors was rarely studied. Zhou proposed a new framework of fault diagnosis and fault-tolerant control based on gap metric, which has important guiding significance for the application of gap metric theory in fault diagnosis [32]. Wang et al. used the gap metric to perform fault detectability analysis for systems with model uncertainty and external disturbances and illustrated the effectiveness of the proposed method using a classical case [33]. In the reference [34], a data-driven gap metric was used to design a suitable fault cluster to improve the separability of initial faults. In addition, the gas flow regulation system of SDR is a closed-loop system. Compared with the open-loop system, the fault diagnosis performance of the closed-loop system is mainly degraded; on the one hand, the control quantity will cover up the smaller faults, and on the other hand, the feedback control may make the fault propagate in the system [35]. Zhou had pointed out that the complex impact of closed-loop feedback on fault diagnosis has not been fully analysed, and further research was needed [36]. For the GG system, due to the limited installation space of the pressure sensors and the cost of the SDR, too much sensor redundancy is usually not used. Even if multiple (more than 2) sensors are used to measure the pressure in the GG at the same time, although the faulty sensors can be isolated by the “voting method” (It is assumed that the failure of several sensors at the same time is a small probability event, so the sensor whose measured values differs from the others is considered a faulty element), however, there is still a problem, that is, when there are only two sensors left in the system working normally (assuming that the remaining sensors have been isolated), the “voting method” cannot continue to perform FDI.This paper mainly aimed at the GG system of SDR with double pressure feedback and combined gap metric, PCA, and Kalman’s principle to diagnose and isolate the fault of the pressure sensors. The main contributions of this paper are as follows:(a)
Based on PCA theory, a “consistency index” was constructed, which could ensure the monotonicity under fault conditions and the convergence under nonfault conditions, so the fault detectability was improved(b)
Based on the Kalman principle, the pressure in the GG was estimated by using the redundant information of the system, and the phase lead correction and tracking differentiator (TD) were designed to reduce the noise and delay, which could improve the accuracy and real-time performance of the estimation(c)
The gap metric between the measured pressure and the estimated pressure was employed to characterize the health of the sensors, and an FDI algorithm for GG with double pressure feedback was proposedThe rest of this paper was organized as follows: In Section2, the working principle of the GG was introduced, the failure model of the sensors was developed, and the effect of sensor failure on the closed-loop system was also analysed. The FDI algorithm for pressure sensors was presented in Section 3, which made comprehensive use of Kalman’s principle, gap metric, and PCA theory with corresponding improvements. In Section 4, several simulations were performed for the case of one pressure sensor failure or two pressure sensors failing at the same time. Finally, the conclusion and outlook were given in Section 5.


## 2. Types of Pressure Sensor Failures and Their Impact on the GG

### 2.1. The Working Process of the GG

As shown in Figure1, when the propellant in the GG burns to generate gas, the controller will collect the pressure data in the GG through the pressure sensors, and according to the gas flow required by the system, the control angle of the interstage valve will be calculated, so that the purpose of regulating the pressure in the GG and indirectly controlling the gas flow rate could be achieved. In the GG system studied in this paper, there are two sensor measurement points, and under normal conditions, the GG’s pressure measured at the two measurement points is approximately equal. The average of the two measurements at each moment is used as the feedback for closed-loop flow regulation.Figure 1
The structure and working principle of the GG.


### 2.2. Failure Model of Pressure Sensors

For pressure sensors, the fault types can be roughly divided into deviation faults (The faulty measurement differs from the correct measurement by a certain constant), complete failures (The measured value does not change with the actual change, and always maintains a certain value), drift faults (The degree of failure varies linearly with time), and precision degradation faults (The average error of the measured value increases). The first two of the above faults can be considered as sudden faults, which are usually easier to diagnose, while the last two can be considered as slow faults, which are relatively more difficult to diagnose. Therefore, this paper mainly aims at diagnosing and isolating the slow fault of the sensors. For drift faults, the mathematical model can be expressed by Equation (1), and the failure of precision degradation can be expressed by Equation (2) [37].
(1)P¯t=pgt+η+dt−ts,(2)P¯t=pgt+η+η∗,where P¯t is the measured value of the sensor, d is a constant, ts is the moment when the fault occurs, pgt is the true value of the pressure in GG, and η is the measurement noise, which can be assumed to obey the N0,σ12 distribution. As well as η∗ is the newly added error in the failure of precision degradation, which can be considered to obey the N0,σ22 distribution.


### 2.3. Effects of Sensor Failure on Closed-Loop Systems

Under the influence of the closed-loop control system, the “correction” ability of the control law will cover the “fault information”. As shown in Figure2, when no fault occurred, the system always maintained P¯≈pg; however, when a fault occurred, the situation became different. It was assumed that at 32.75 seconds, the pressure sensors had a slow drift fault (d1=d2=0.002); it could be seen that under the closed-loop control and the regulation of the interstage valve, although the measured value (P¯) could still follow the command (Pcmd) well, but in fact, the true value (Pg) of the system responded in the opposite direction of the fault, and the longer the drift fault occurred, the greater the deviation of P¯ from Pg, and the greater the damage to the system.Figure 2
Influence of closed-loop control on fault diagnosis.


### 2.4. Influence of Sensor Failure on the Stability of Closed-Loop System

It is assumed that multiple sensors are used to measure the pressure of GG at the same time, and the average of their measured values is used as the feedback. The mathematical model of GG can be equivalent to a first-order system [38]. For the convenience of analysis, the control law adopts the integral link; Mns represents the transfer function of the pressure sensor (As shown in Figure 3). As we have known, under ideal conditions, the transfer function of each pressure sensor is 1. When there are measurement noise and faults, the transfer function of the measurement unit can be regarded as Kt; then, the closed-loop transfer function of the system was shown in Equation (3).
(3)Gcls=KIt∙K1ts2+T1t∙s+Kt∙KIt∙K1t,where Kt=1+∑i=1nηi+ξi/n∙pg, ξt=dt−ts.Figure 3
Multisensor redundant closed-loop control system for GG’s pressure.For this system, it is clear thatK1t, T1t, and KIt are all positive values (It has been described in reference [38]), so the root’s polarity of the characteristic equation can be judged according to Veda’s theorem (When the sum of the two roots is negative and the product of the two roots is positive, both roots of the system will be negative). Then, according to the Hurwitz criterion, the stability condition of the system can be obtained. When ∑i=1nηi+ξi>−n∙pg, the system is stable.


## 2.1. The Working Process of the GG

As shown in Figure1, when the propellant in the GG burns to generate gas, the controller will collect the pressure data in the GG through the pressure sensors, and according to the gas flow required by the system, the control angle of the interstage valve will be calculated, so that the purpose of regulating the pressure in the GG and indirectly controlling the gas flow rate could be achieved. In the GG system studied in this paper, there are two sensor measurement points, and under normal conditions, the GG’s pressure measured at the two measurement points is approximately equal. The average of the two measurements at each moment is used as the feedback for closed-loop flow regulation.Figure 1
The structure and working principle of the GG.


## 2.2. Failure Model of Pressure Sensors

For pressure sensors, the fault types can be roughly divided into deviation faults (The faulty measurement differs from the correct measurement by a certain constant), complete failures (The measured value does not change with the actual change, and always maintains a certain value), drift faults (The degree of failure varies linearly with time), and precision degradation faults (The average error of the measured value increases). The first two of the above faults can be considered as sudden faults, which are usually easier to diagnose, while the last two can be considered as slow faults, which are relatively more difficult to diagnose. Therefore, this paper mainly aims at diagnosing and isolating the slow fault of the sensors. For drift faults, the mathematical model can be expressed by Equation (1), and the failure of precision degradation can be expressed by Equation (2) [37].
(1)P¯t=pgt+η+dt−ts,(2)P¯t=pgt+η+η∗,where P¯t is the measured value of the sensor, d is a constant, ts is the moment when the fault occurs, pgt is the true value of the pressure in GG, and η is the measurement noise, which can be assumed to obey the N0,σ12 distribution. As well as η∗ is the newly added error in the failure of precision degradation, which can be considered to obey the N0,σ22 distribution.


## 2.3. Effects of Sensor Failure on Closed-Loop Systems

Under the influence of the closed-loop control system, the “correction” ability of the control law will cover the “fault information”. As shown in Figure2, when no fault occurred, the system always maintained P¯≈pg; however, when a fault occurred, the situation became different. It was assumed that at 32.75 seconds, the pressure sensors had a slow drift fault (d1=d2=0.002); it could be seen that under the closed-loop control and the regulation of the interstage valve, although the measured value (P¯) could still follow the command (Pcmd) well, but in fact, the true value (Pg) of the system responded in the opposite direction of the fault, and the longer the drift fault occurred, the greater the deviation of P¯ from Pg, and the greater the damage to the system.Figure 2
Influence of closed-loop control on fault diagnosis.


## 2.4. Influence of Sensor Failure on the Stability of Closed-Loop System

It is assumed that multiple sensors are used to measure the pressure of GG at the same time, and the average of their measured values is used as the feedback. The mathematical model of GG can be equivalent to a first-order system [38]. For the convenience of analysis, the control law adopts the integral link; Mns represents the transfer function of the pressure sensor (As shown in Figure 3). As we have known, under ideal conditions, the transfer function of each pressure sensor is 1. When there are measurement noise and faults, the transfer function of the measurement unit can be regarded as Kt; then, the closed-loop transfer function of the system was shown in Equation (3).
(3)Gcls=KIt∙K1ts2+T1t∙s+Kt∙KIt∙K1t,where Kt=1+∑i=1nηi+ξi/n∙pg, ξt=dt−ts.Figure 3
Multisensor redundant closed-loop control system for GG’s pressure.For this system, it is clear thatK1t, T1t, and KIt are all positive values (It has been described in reference [38]), so the root’s polarity of the characteristic equation can be judged according to Veda’s theorem (When the sum of the two roots is negative and the product of the two roots is positive, both roots of the system will be negative). Then, according to the Hurwitz criterion, the stability condition of the system can be obtained. When ∑i=1nηi+ξi>−n∙pg, the system is stable.


## 3. Fault Diagnosis Principle and Algorithm

### 3.1. Construction of the “Consistency Index”

The essence of PCA is to find a set of “bases”, so that the projection of the original data on this set of “bases” is as centrally as possible. For this system, the two sensors measure the same static variable (GG’s pressure), and the measured values of the two sensors can maintain good consistency during normal operation. However, when one of them fails, the consistency will be broken. Since the measurements were updated in real time, we used a moving window to perform principal component analysis (MWPCA) on the two sets of dynamic measurement data. In this way, the “consistency” of them could be judged and evaluated. The general process of PCA was shown in Figure4.Figure 4
General process of PCA.For this system, the data acquisition matrix of the two sensors in the moving window lengthk may be expressed by Equation (4), and the moving step size of the window is d, which can be expressed by Equation (5). For the Step 4 in Figure 4, the traditional calculation method was shown by Equation (6).
(4)Xt=P¯1t−k+1P¯1t−k+2P¯2t−k+1P¯2t−k+2⋯P¯1t⋯P¯2tk×2,t>k,(5)Xt+d=P¯1t−k+1+dP¯1t−k+2+dP¯2t−k+1+dP¯2t−k+2+d⋯P¯1t+d⋯P¯2t+dk×2,(6)PC1tPC2t=Xt∙PtP~t.The traditional method detects faults throughSPE statistics and the relationship between SPE and clα. When SPE≤clα, it was judged that a fault occurs [39]. However, it is insensitive to minor sensor faults, which are relatively difficult to detect [6]. For the double pressure feedback system, the effect of the closed-loop control makes Err tend to 0; then, P¯1t and P¯2t are approximately symmetric about Pcmd; therefore, when one of the sensors drifts, the actual pressure of the system must respond in the opposite direction. The value of PC2 can well reflect the consistency of the two sets of data. The better the consistency, the closer its value to 0. However, the characteristics of PC2 will be changed under the influence of the closed-loop system.
(7)Err=Pcmd−12∙P¯1t+P¯2t.Therefore, we use Equation (8) to characterize the consistency of the two sensors and refer to Pe as the “ Consistency Index “.
(8)Pet=minXtk1,Xtk2maxXtk1,Xtk2T∙P~t11P~t12,where the subscript “ij” means row i, column j of the matrix, for example, the Xtk1 means the element at row k and column 1 of Xt, the superscript “T” represents the transpose of the matrix, and ∗ is the absolute value.For ak×2 matrix, when the eigenvalues of its covariance matrix satisfy λ1>λ2≥0, it is easy to obtain P~t11<P~t12, so Equation (8) can guarantee the divergence characteristics of Pe when a fault occurs. When there is no fault, P~t11≈P~t12, Xtk1≈Ytk1, so it does not affect its convergence (See Part A in the Supplementary Material for more analysis, where Figure S1 is a visual illustration of Part A). As shown in Figure 5, when d2=0.001,d1=0, the second principal element calculated by the two methods was symmetric about x=0, and there was no difference after taking the absolute value. However, when d2=−0.001, the PC2 and Pe were quite different, while the Pe still maintained the good monotonicity. (As shown in Table 1)Figure 5
ThePC2 and Pe under different fault conditions.Table 1
Under closed-loop control, the characteristics ofPC2 and Pe are compared.
d1d2η1η2PC2tPet0.00100.2%0.2%NonmonotonicityMonotonicity-0.00100.2%0.2%MonotonicityMonotonicity00.0010.2%0.2%MonotonicityMonotonicity0-0.0010.2%0.2%NonmonotonicityMonotonicity000.2%2%NonmonotonicityNonmonotonicity


### 3.2. Gap Metric and the Sensor’s Health

The geometric meaning of the gap metric between two real numbers is their chord distance after projection on the unit of Riemann’s sphere. Compared with the Euclidean metric, it can highlight the correlation between different variables, as shown in Equation (9). For example, when P¯1t1=1MPa, P¯2t1=1.5MPa, then ∆P¯t1=P¯2t1−P¯1t1=0.5MPa, δP¯1t1,P¯2t1≈0.196, and when P¯1t2=4MPa, P¯2t2=4.5MPa, then ∆P¯t2=P¯2t2−P¯1t2=0.5MPa, δP¯1t2,P¯2t2≈0.026, obviously ∆P¯t1=∆P¯t2, but δP¯1t1,P¯2t1>δP¯1t2,P¯2t2. In fact, the gap metric could cover both the amount and the rate of change under the Euclidean metric.
(9)δp1,p2=p1−p21+p12∙1+p22.For this system, we used the gap metric between the measurement (P¯) and the estimate pressure (P^) to measure the health of the sensor. And the difference in the health of the two sensors can better reflect the relative health status between the two sensors, which are conducive to fault detection, as shown in Equation (10). When the sensors work normally, P^≈P¯, H1, or H2 is close to 0, while when there is a fault, P^≠P¯, the health status deteriorates, and the worse the health status is, the closer H1 or H2 is to 1. Hm is used to indicate the cumulative amount of health difference between two sensors, and the larger Hm indicates the greater health difference between two sensors, Hm∈−∞,+∞, while H1&H2∈0,1.
(10)Hm=∫0tH1−H2∙dt,H1=δP^P^+P¯1,P¯1P^+P¯1,H2=δP^P^+P¯2,P¯2P^+P¯2.As shown in Figure6 (The double vertical axis legend was used here, where H1 and H2 correspond to the vertical axis on the right side of the figure, and Hm corresponds to the vertical axis on the left side of the figure), at the 30th second, P¯1 has a drift fault, d1=−0.001, d2=0, H1 and H2 are approximately equal before the fault occurs (H1≈H2≈0). After the fault occurred, H1 and H2 gradually differed, the H1 increased with time, reaching about 0.32 at the 300th second, and correspondingly, the Hm also rose rapidly and eventually reached around 32. Similarly, if a drift fault occurs in P¯2, Hm will drop rapidly, and when the fault is isolated, Hm will be less than zero.Figure 6
Variation trends ofH and Hm when P¯1 had a drift fault.


### 3.3. Kalman’s Filter Algorithm for Measurement Delay

For the problem of fault diagnosis, we should focus on the redundant information in the system and use them as the evidence of fault diagnosis. From the Equation (11) of the GG, (See Part B in the Supplementary Material for a detailed derivation process) the steady-state pressure in the GG is independent of the free-volume but only related to the throat area (Assuming no change in propellant parameters). And because the valve angle directly reflects the change of the throat area, the valve angle is redundant information that can be used. In GG system of the SDR, the valve angle can be measured in real-time, we can roughly calculate the ∆Pg using the ∆θ.
(11)dPgdt=Rg·TgV·ρb·Ab·a·Pgn−Pg·AtCr,where Pg is the gas pressure in GG, Rg is the gas constant, Tg is the gas temperature, and V represents the free-volume, which means the volume between the propellant end face and the throat. ρb represents the propellant density, Ab represents the burning area of the propellant, a represents the propellant combustion rate coefficient, n represents the pressure index, At represents the throat area, Cr represents the characteristic velocity of the gas, and θ represents the swing angle of the interstage valve.In addition, the pressure of the GG ultimately affects the pressure of the ram combustor (RC), and the pressure of the RC and the flight conditions (height, Mach, attack angle, sideslip angle) jointly determines the thrust and the axial overload. Therefore, the measured value of the overload sensor is another piece of redundant information that can be used. However, there is usually a certain time delay between the axial overload and the pressure of the GG. Taking a ground test of our research group as an example, by observing the time delay between the pressure response of the GG and the RC, the time delay between the pressure of the GG and the axial overload of the aircraft was indirectly equivalent (Ignore the time delay between the pressure response of the RC and axial overload). As shown in Figure7, the moment of pressure rise was selected as the observation point, so there was a delay of about 0.8 seconds between the pressure response of GG and RC.Figure 7
The time delay between the pressure response of the GG and the RC.For the time delay problem, a tracking differentiator with lead correction was introduced here. On the one hand, the time delay could be reduced, and on the other hand, the noise caused by the overload measurement could also be reduced. The corresponding algorithm was shown in following equations:(12)fh=fhanv1t−1−fNx,H,Ma,alpha,beta,v2t−1,r0,h0,v1t=v1t−1+h·v2t−1,v2t=v2t−1+h·fh,Pxt=v1t+h∙α∙v2t,(13)d=r0·h02,a0=h0·x2,y=x1+a0,a1=d·d+8·y,a2=a0+signy·a1−d2,sy=signy+d−signy−d2,a=a0+y−a2·sy+a2,sa=signa+d−signa−d2,fhan=−r·ad+signa·sa−r0·signa,where fNx,H,Ma,alpha,beta is the function of calculating the pressure of the GG from the overload and flight conditions, the Nx is the axial overload of the aircraft, the v1 and v2 are the tracking and differential tracking signals of the input signal, respectively, the α is the correction coefficient, and the Px is the output of the transition process.Then, we could use the Kalman information fusion algorithm to obtain the “estimated pressure” according to the valve angle and axial overload and make further preparations for FDI (Equations (14)–(18)). It was assumed that there would be a large error in calculating the pressure of GG (Pg) from the overload information (Px). Therefore, using the Kalman principle to fuse Px and the valve angle to optimally estimate the pressure (P^) would not be particularly ideal but would contain large errors. Figure 8 could reflect the relationship among P^, Px, and Pg.
(14)Pkt=P^t−1+f∆θ,(15)PA−t=PAt−1+QA,(16)Kt=PA−tPA−t+R,(17)P^t=Pkt+Kt∙Pxt−Pkt,(18)PAt=1−Kt∙PA−t,where Δθ is the variation of the valve angle, QA is the variance when the pressure is calculated from the valve angle, and R is the variance when the pressure is calculated from the axial overload.Figure 8
Pressure estimation of GG based on Kalman’s principle.


### 3.4. Fault Isolation Algorithm

How the FDI works in the control system had been represented in Figure9, and the detailed principle inside the red box would be illustrated in Figure 10. The blue box represented the fault information of the system. In this paper, the fault was simulated based on the mathematical model. Firstly, based on the redundancy of Nx and θ, the Kalman principle was used to optimally estimate the pressure in the GG, which was P^ in Figure 10. Secondly, the Pe was used to evaluate the consistency of the two pressure sensors’ measured values (In order to reduce the probability of misdiagnosis, Hm was also used to assess the difference in health between the two sensors, that is, the consistency of the measured values). And when it was poor, the Hm would be used to judge which sensor should be isolated. Since it was possible to maintain good consistency when both sensors failed, the HR could be used to judge whether to isolate the two sensors at the same time. The definition of HR was shown in Equation (19).Figure 9
Control schematic with fault diagnosis and isolation algorithms.Figure 10
Principles of fault diagnosis and isolation algorithms.In addition, the polarity ofHm and its relationship with the corresponding threshold could be used to determine which sensor had lower accuracy. When Hm or its absolute value exceeds the threshold value, it is determined that a precision degradation fault occurs. When “Flag=1,” it means that the precision degradation occurs in P¯1, and when “Flag=2,” it means that the precision degradation occurs in P¯2. Since such faults are relatively less harmful to the closed-loop system, only the monitoring function was performed here. In addition, when two kinds of faults (drift fault and precision degradation fault) occur simultaneously, only the drift fault is recognized, and the accuracy monitoring function is performed when there is no drift fault.
(19)HR=δP^P^+PR,PRP^+PR.From Figure11(a), it could be seen that when the fault was isolated, the pressure offset due to different εP (when d1=d2=0.003). If the εP was too large, the fault diagnosis process will be too long, and if the εP was too small, the diagnosis algorithm will be too sensitive and cause misdiagnosis. If εP=0.1, when the fault was isolated, the deviation between the measured pressure and the real pressure would be within 0.04MPa. From Figure 11(b), it could be seen that if εH was too small, it was too easy to isolate both sensors. For example, when the value was 0.05, the two sensors have been isolated at the beginning of the simulation. But if εH was too large, the fault identification will be too slow, which will increase the damage to the system. After comprehensive consideration, we choose εH as 0.1; when the pressure command was in the range of 0.6 MPa and 4.5 MPa, the pressure offset was between 0.12 and 1.06 MPa. According to a similar method, εHm could be taken as 0.01, and the process would not be repeated.Figure 11
Selection of thresholds for sensor fault isolation.
(a)
Selection ofεP(b)
Selection ofεH


## 3.1. Construction of the “Consistency Index”

The essence of PCA is to find a set of “bases”, so that the projection of the original data on this set of “bases” is as centrally as possible. For this system, the two sensors measure the same static variable (GG’s pressure), and the measured values of the two sensors can maintain good consistency during normal operation. However, when one of them fails, the consistency will be broken. Since the measurements were updated in real time, we used a moving window to perform principal component analysis (MWPCA) on the two sets of dynamic measurement data. In this way, the “consistency” of them could be judged and evaluated. The general process of PCA was shown in Figure4.Figure 4
General process of PCA.For this system, the data acquisition matrix of the two sensors in the moving window lengthk may be expressed by Equation (4), and the moving step size of the window is d, which can be expressed by Equation (5). For the Step 4 in Figure 4, the traditional calculation method was shown by Equation (6).
(4)Xt=P¯1t−k+1P¯1t−k+2P¯2t−k+1P¯2t−k+2⋯P¯1t⋯P¯2tk×2,t>k,(5)Xt+d=P¯1t−k+1+dP¯1t−k+2+dP¯2t−k+1+dP¯2t−k+2+d⋯P¯1t+d⋯P¯2t+dk×2,(6)PC1tPC2t=Xt∙PtP~t.The traditional method detects faults throughSPE statistics and the relationship between SPE and clα. When SPE≤clα, it was judged that a fault occurs [39]. However, it is insensitive to minor sensor faults, which are relatively difficult to detect [6]. For the double pressure feedback system, the effect of the closed-loop control makes Err tend to 0; then, P¯1t and P¯2t are approximately symmetric about Pcmd; therefore, when one of the sensors drifts, the actual pressure of the system must respond in the opposite direction. The value of PC2 can well reflect the consistency of the two sets of data. The better the consistency, the closer its value to 0. However, the characteristics of PC2 will be changed under the influence of the closed-loop system.
(7)Err=Pcmd−12∙P¯1t+P¯2t.Therefore, we use Equation (8) to characterize the consistency of the two sensors and refer to Pe as the “ Consistency Index “.
(8)Pet=minXtk1,Xtk2maxXtk1,Xtk2T∙P~t11P~t12,where the subscript “ij” means row i, column j of the matrix, for example, the Xtk1 means the element at row k and column 1 of Xt, the superscript “T” represents the transpose of the matrix, and ∗ is the absolute value.For ak×2 matrix, when the eigenvalues of its covariance matrix satisfy λ1>λ2≥0, it is easy to obtain P~t11<P~t12, so Equation (8) can guarantee the divergence characteristics of Pe when a fault occurs. When there is no fault, P~t11≈P~t12, Xtk1≈Ytk1, so it does not affect its convergence (See Part A in the Supplementary Material for more analysis, where Figure S1 is a visual illustration of Part A). As shown in Figure 5, when d2=0.001,d1=0, the second principal element calculated by the two methods was symmetric about x=0, and there was no difference after taking the absolute value. However, when d2=−0.001, the PC2 and Pe were quite different, while the Pe still maintained the good monotonicity. (As shown in Table 1)Figure 5
ThePC2 and Pe under different fault conditions.Table 1
Under closed-loop control, the characteristics ofPC2 and Pe are compared.
d1d2η1η2PC2tPet0.00100.2%0.2%NonmonotonicityMonotonicity-0.00100.2%0.2%MonotonicityMonotonicity00.0010.2%0.2%MonotonicityMonotonicity0-0.0010.2%0.2%NonmonotonicityMonotonicity000.2%2%NonmonotonicityNonmonotonicity


## 3.2. Gap Metric and the Sensor’s Health

The geometric meaning of the gap metric between two real numbers is their chord distance after projection on the unit of Riemann’s sphere. Compared with the Euclidean metric, it can highlight the correlation between different variables, as shown in Equation (9). For example, when P¯1t1=1MPa, P¯2t1=1.5MPa, then ∆P¯t1=P¯2t1−P¯1t1=0.5MPa, δP¯1t1,P¯2t1≈0.196, and when P¯1t2=4MPa, P¯2t2=4.5MPa, then ∆P¯t2=P¯2t2−P¯1t2=0.5MPa, δP¯1t2,P¯2t2≈0.026, obviously ∆P¯t1=∆P¯t2, but δP¯1t1,P¯2t1>δP¯1t2,P¯2t2. In fact, the gap metric could cover both the amount and the rate of change under the Euclidean metric.
(9)δp1,p2=p1−p21+p12∙1+p22.For this system, we used the gap metric between the measurement (P¯) and the estimate pressure (P^) to measure the health of the sensor. And the difference in the health of the two sensors can better reflect the relative health status between the two sensors, which are conducive to fault detection, as shown in Equation (10). When the sensors work normally, P^≈P¯, H1, or H2 is close to 0, while when there is a fault, P^≠P¯, the health status deteriorates, and the worse the health status is, the closer H1 or H2 is to 1. Hm is used to indicate the cumulative amount of health difference between two sensors, and the larger Hm indicates the greater health difference between two sensors, Hm∈−∞,+∞, while H1&H2∈0,1.
(10)Hm=∫0tH1−H2∙dt,H1=δP^P^+P¯1,P¯1P^+P¯1,H2=δP^P^+P¯2,P¯2P^+P¯2.As shown in Figure6 (The double vertical axis legend was used here, where H1 and H2 correspond to the vertical axis on the right side of the figure, and Hm corresponds to the vertical axis on the left side of the figure), at the 30th second, P¯1 has a drift fault, d1=−0.001, d2=0, H1 and H2 are approximately equal before the fault occurs (H1≈H2≈0). After the fault occurred, H1 and H2 gradually differed, the H1 increased with time, reaching about 0.32 at the 300th second, and correspondingly, the Hm also rose rapidly and eventually reached around 32. Similarly, if a drift fault occurs in P¯2, Hm will drop rapidly, and when the fault is isolated, Hm will be less than zero.Figure 6
Variation trends ofH and Hm when P¯1 had a drift fault.


## 3.3. Kalman’s Filter Algorithm for Measurement Delay

For the problem of fault diagnosis, we should focus on the redundant information in the system and use them as the evidence of fault diagnosis. From the Equation (11) of the GG, (See Part B in the Supplementary Material for a detailed derivation process) the steady-state pressure in the GG is independent of the free-volume but only related to the throat area (Assuming no change in propellant parameters). And because the valve angle directly reflects the change of the throat area, the valve angle is redundant information that can be used. In GG system of the SDR, the valve angle can be measured in real-time, we can roughly calculate the ∆Pg using the ∆θ.
(11)dPgdt=Rg·TgV·ρb·Ab·a·Pgn−Pg·AtCr,where Pg is the gas pressure in GG, Rg is the gas constant, Tg is the gas temperature, and V represents the free-volume, which means the volume between the propellant end face and the throat. ρb represents the propellant density, Ab represents the burning area of the propellant, a represents the propellant combustion rate coefficient, n represents the pressure index, At represents the throat area, Cr represents the characteristic velocity of the gas, and θ represents the swing angle of the interstage valve.In addition, the pressure of the GG ultimately affects the pressure of the ram combustor (RC), and the pressure of the RC and the flight conditions (height, Mach, attack angle, sideslip angle) jointly determines the thrust and the axial overload. Therefore, the measured value of the overload sensor is another piece of redundant information that can be used. However, there is usually a certain time delay between the axial overload and the pressure of the GG. Taking a ground test of our research group as an example, by observing the time delay between the pressure response of the GG and the RC, the time delay between the pressure of the GG and the axial overload of the aircraft was indirectly equivalent (Ignore the time delay between the pressure response of the RC and axial overload). As shown in Figure7, the moment of pressure rise was selected as the observation point, so there was a delay of about 0.8 seconds between the pressure response of GG and RC.Figure 7
The time delay between the pressure response of the GG and the RC.For the time delay problem, a tracking differentiator with lead correction was introduced here. On the one hand, the time delay could be reduced, and on the other hand, the noise caused by the overload measurement could also be reduced. The corresponding algorithm was shown in following equations:(12)fh=fhanv1t−1−fNx,H,Ma,alpha,beta,v2t−1,r0,h0,v1t=v1t−1+h·v2t−1,v2t=v2t−1+h·fh,Pxt=v1t+h∙α∙v2t,(13)d=r0·h02,a0=h0·x2,y=x1+a0,a1=d·d+8·y,a2=a0+signy·a1−d2,sy=signy+d−signy−d2,a=a0+y−a2·sy+a2,sa=signa+d−signa−d2,fhan=−r·ad+signa·sa−r0·signa,where fNx,H,Ma,alpha,beta is the function of calculating the pressure of the GG from the overload and flight conditions, the Nx is the axial overload of the aircraft, the v1 and v2 are the tracking and differential tracking signals of the input signal, respectively, the α is the correction coefficient, and the Px is the output of the transition process.Then, we could use the Kalman information fusion algorithm to obtain the “estimated pressure” according to the valve angle and axial overload and make further preparations for FDI (Equations (14)–(18)). It was assumed that there would be a large error in calculating the pressure of GG (Pg) from the overload information (Px). Therefore, using the Kalman principle to fuse Px and the valve angle to optimally estimate the pressure (P^) would not be particularly ideal but would contain large errors. Figure 8 could reflect the relationship among P^, Px, and Pg.
(14)Pkt=P^t−1+f∆θ,(15)PA−t=PAt−1+QA,(16)Kt=PA−tPA−t+R,(17)P^t=Pkt+Kt∙Pxt−Pkt,(18)PAt=1−Kt∙PA−t,where Δθ is the variation of the valve angle, QA is the variance when the pressure is calculated from the valve angle, and R is the variance when the pressure is calculated from the axial overload.Figure 8
Pressure estimation of GG based on Kalman’s principle.


## 3.4. Fault Isolation Algorithm

How the FDI works in the control system had been represented in Figure9, and the detailed principle inside the red box would be illustrated in Figure 10. The blue box represented the fault information of the system. In this paper, the fault was simulated based on the mathematical model. Firstly, based on the redundancy of Nx and θ, the Kalman principle was used to optimally estimate the pressure in the GG, which was P^ in Figure 10. Secondly, the Pe was used to evaluate the consistency of the two pressure sensors’ measured values (In order to reduce the probability of misdiagnosis, Hm was also used to assess the difference in health between the two sensors, that is, the consistency of the measured values). And when it was poor, the Hm would be used to judge which sensor should be isolated. Since it was possible to maintain good consistency when both sensors failed, the HR could be used to judge whether to isolate the two sensors at the same time. The definition of HR was shown in Equation (19).Figure 9
Control schematic with fault diagnosis and isolation algorithms.Figure 10
Principles of fault diagnosis and isolation algorithms.In addition, the polarity ofHm and its relationship with the corresponding threshold could be used to determine which sensor had lower accuracy. When Hm or its absolute value exceeds the threshold value, it is determined that a precision degradation fault occurs. When “Flag=1,” it means that the precision degradation occurs in P¯1, and when “Flag=2,” it means that the precision degradation occurs in P¯2. Since such faults are relatively less harmful to the closed-loop system, only the monitoring function was performed here. In addition, when two kinds of faults (drift fault and precision degradation fault) occur simultaneously, only the drift fault is recognized, and the accuracy monitoring function is performed when there is no drift fault.
(19)HR=δP^P^+PR,PRP^+PR.From Figure11(a), it could be seen that when the fault was isolated, the pressure offset due to different εP (when d1=d2=0.003). If the εP was too large, the fault diagnosis process will be too long, and if the εP was too small, the diagnosis algorithm will be too sensitive and cause misdiagnosis. If εP=0.1, when the fault was isolated, the deviation between the measured pressure and the real pressure would be within 0.04MPa. From Figure 11(b), it could be seen that if εH was too small, it was too easy to isolate both sensors. For example, when the value was 0.05, the two sensors have been isolated at the beginning of the simulation. But if εH was too large, the fault identification will be too slow, which will increase the damage to the system. After comprehensive consideration, we choose εH as 0.1; when the pressure command was in the range of 0.6 MPa and 4.5 MPa, the pressure offset was between 0.12 and 1.06 MPa. According to a similar method, εHm could be taken as 0.01, and the process would not be repeated.Figure 11
Selection of thresholds for sensor fault isolation.
(a)
Selection ofεP(b)
Selection ofεH


## 4. Mathematical Simulation and Analysis

In this section, simulations were performed for the cases of one of sensors failure and both sensors failure at the same time, so that the effectiveness of the FDI algorithm in the paper could be fully judged and analysed. Among them, the simultaneous failure of the two sensors includes a failure with the same drift trend and a failure with an inconsistent drift trend. Matlab/Simulink was used as the simulation software, and the simulation conditions were given in Table2.Table 2
Simulation conditions.
ParametersPhysical meaningValueUnitParametersPhysical meaningValueUnitr0Parameters in thefhan function0.3/QAThe variance when estimating pressure fromθ0.01MPa2αCorrection coefficient100/RThe variance when estimating pressure fromNx0.01MPa2τDelay in estimating pressure fromNx100∗hshThe sampling period0.01sεHmPrecision degradation monitoring threshold0.01/h0Parameters in thefhan function0.01/εPIndividual sensor isolation thresholds0.1/εHBoth sensors isolation thresholds0.1/ηThe measurement noise0.2%/η∗The newly added error in the failure of precision degradation0.8%/


### 4.1. Only One of the Pressure Sensors Is Faulty

We could makeP¯2 drift fault from the 30th second, d2=0.003, d1=0, and the pressure command was a trapezoidal wave command with a slope of 0.25 MPa/s from 0.7 to 1.5 MPa. As could be seen in Figure 12, before the fault was isolated, P¯A entered the control closed loop as feedback. At 53.62 seconds, P¯2 was diagnosed as a faulty sensor and was isolated; after that, P¯1 alone entered the control closed loop as feedback. And P¯1 reflected the real situation of the system.Figure 12
Only one of the pressure sensors was faulty.


### 4.2. Both Pressure Sensors Are Faulty

It was set that both sensors had drift faults from the 30th second,d1=d2=0.003, and the pressure command was a trapezoidal wave command with a slope of 0.25 MPa/s from 0.7 to 1.5 MPa. From Figure 13, it could be seen that before the failure, the combined health of the sensors (HR) varies within a certain range, until the health threshold was exceeded at 61.32 seconds, and both sensors were isolated. After that, closed-loop control was performed using the P^ as the feedback.Figure 13
Both sensors fail with the same drift trend.It was set that both sensors had drift faults from the 30th second,d1=−0.002, d2=0.003, and the pressure command was a trapezoidal wave command with a slope of 0.25 MPa/s from 0.7 to 1.5 MPa. As shown in Figure 14, at 44.25 seconds, Pe exceeded the threshold firstly, and P¯2 was isolated firstly due to its greater degree of failure. At the 107.4 seconds, the HR exceeds the threshold, P¯1 was also isolated; after that, closed-loop control was performed using the P^ as the feedback.Figure 14
Drift failure with inconsistent trend between two sensors.


### 4.3. The Accuracy of the FDI

In this section, the accuracy of the FDI algorithm described in this paper was calculated for drift faults and measurement precision degradation faults. For the GG system with double pressure feedback studied in this paper, its FDI algorithm was rarely studied, so this paper adopted the classic MWPCA algorithm and SPE detection method as the control group. Among them,Pcmd was taken as a fixed value between 0.6 and 4.5 MPa. For drift faults, the drift rate d was taken as a value between -0.002 and -0.020. For the failure of measurement accuracy decline, η was taken as 0.2%, and the newly added measurement error of the faulty sensor (η∗) was taken as a value between 0.8% and 2.3%. The simulation results were shown in Table 3 (Detailed simulation results were provided in Tables S1-S5 in Part C of the Supplementary Material). For the detection of precision degradation faults, the fault would be missed because the fault was too small, and the Pe and Hm would exceed the thresholds because the fault was too serious, so the faulty sensor was isolated. The situation was classified as a missing alarm in this paper, that is, the precision degradation fault was falsely reported as a drift fault. The confusion matrix of the two algorithms for fault diagnosis was shown in Figure 15. It could be seen that the algorithm proposed in this paper was better than the MWPCA-SPE algorithm, especially for the diagnosis of B1 type faults.Table 3
Statistics of simulation results.
Fault typeThe total number of simulationsThe algorithm proposed in this paperMWPCA-SPEMissing alarmFalse alarmAccuracyMissing alarmFalse alarmAccuracyA40000100%6.1%093.9%B140000100%100%00B240000100%7%093%C14001.25%3%95.8%8.2%091.8%C24001.25%3%95.8%8.2%091.8%Remark: the “Fault Type A” means only one of the pressure sensors was drift faulty, the “Fault Type B1” means both the pressure sensors were drift faulty, andd1=d2=d, the “Fault Type B2” means d2=d, d1=d/2. The “Fault Type C1” means precision degradation faults, and η1=η2=η=0.2%, η1∗=0, η2∗=η∗, the “Fault Type C2” means η1=η2=η=0.2%, η1∗=η∗, η2∗=0.Figure 15
Confusion matrix for fault diagnosis of two algorithms.
(a)
The confusion matrix of the algorithm proposed in this paper(b)
Confusion matrix of MWPCA-SPE


## 4.1. Only One of the Pressure Sensors Is Faulty

We could makeP¯2 drift fault from the 30th second, d2=0.003, d1=0, and the pressure command was a trapezoidal wave command with a slope of 0.25 MPa/s from 0.7 to 1.5 MPa. As could be seen in Figure 12, before the fault was isolated, P¯A entered the control closed loop as feedback. At 53.62 seconds, P¯2 was diagnosed as a faulty sensor and was isolated; after that, P¯1 alone entered the control closed loop as feedback. And P¯1 reflected the real situation of the system.Figure 12
Only one of the pressure sensors was faulty.


## 4.2. Both Pressure Sensors Are Faulty

It was set that both sensors had drift faults from the 30th second,d1=d2=0.003, and the pressure command was a trapezoidal wave command with a slope of 0.25 MPa/s from 0.7 to 1.5 MPa. From Figure 13, it could be seen that before the failure, the combined health of the sensors (HR) varies within a certain range, until the health threshold was exceeded at 61.32 seconds, and both sensors were isolated. After that, closed-loop control was performed using the P^ as the feedback.Figure 13
Both sensors fail with the same drift trend.It was set that both sensors had drift faults from the 30th second,d1=−0.002, d2=0.003, and the pressure command was a trapezoidal wave command with a slope of 0.25 MPa/s from 0.7 to 1.5 MPa. As shown in Figure 14, at 44.25 seconds, Pe exceeded the threshold firstly, and P¯2 was isolated firstly due to its greater degree of failure. At the 107.4 seconds, the HR exceeds the threshold, P¯1 was also isolated; after that, closed-loop control was performed using the P^ as the feedback.Figure 14
Drift failure with inconsistent trend between two sensors.


## 4.3. The Accuracy of the FDI

In this section, the accuracy of the FDI algorithm described in this paper was calculated for drift faults and measurement precision degradation faults. For the GG system with double pressure feedback studied in this paper, its FDI algorithm was rarely studied, so this paper adopted the classic MWPCA algorithm and SPE detection method as the control group. Among them,Pcmd was taken as a fixed value between 0.6 and 4.5 MPa. For drift faults, the drift rate d was taken as a value between -0.002 and -0.020. For the failure of measurement accuracy decline, η was taken as 0.2%, and the newly added measurement error of the faulty sensor (η∗) was taken as a value between 0.8% and 2.3%. The simulation results were shown in Table 3 (Detailed simulation results were provided in Tables S1-S5 in Part C of the Supplementary Material). For the detection of precision degradation faults, the fault would be missed because the fault was too small, and the Pe and Hm would exceed the thresholds because the fault was too serious, so the faulty sensor was isolated. The situation was classified as a missing alarm in this paper, that is, the precision degradation fault was falsely reported as a drift fault. The confusion matrix of the two algorithms for fault diagnosis was shown in Figure 15. It could be seen that the algorithm proposed in this paper was better than the MWPCA-SPE algorithm, especially for the diagnosis of B1 type faults.Table 3
Statistics of simulation results.
Fault typeThe total number of simulationsThe algorithm proposed in this paperMWPCA-SPEMissing alarmFalse alarmAccuracyMissing alarmFalse alarmAccuracyA40000100%6.1%093.9%B140000100%100%00B240000100%7%093%C14001.25%3%95.8%8.2%091.8%C24001.25%3%95.8%8.2%091.8%Remark: the “Fault Type A” means only one of the pressure sensors was drift faulty, the “Fault Type B1” means both the pressure sensors were drift faulty, andd1=d2=d, the “Fault Type B2” means d2=d, d1=d/2. The “Fault Type C1” means precision degradation faults, and η1=η2=η=0.2%, η1∗=0, η2∗=η∗, the “Fault Type C2” means η1=η2=η=0.2%, η1∗=η∗, η2∗=0.Figure 15
Confusion matrix for fault diagnosis of two algorithms.
(a)
The confusion matrix of the algorithm proposed in this paper(b)
Confusion matrix of MWPCA-SPE


## 5. Conclusion

Hardware redundancy is often used in engineering to enhance the fault tolerance of the system. In a redundant system with sensors, there will be problems of fault diagnosis and isolation between two sensors. In this paper, for the GG system with double pressure feedback, a novel fault diagnosis algorithm was proposed by combining the Kalman principle (knowledge-based approach), principal component analysis principle (data-driven approach), and gap metric theory (model-based approach). The simulations have shown that the method could cope well with drift failures and precision degradation failures of two sensors. This could help increase the flight safety of long-endurance aircrafts and could also provide a useful reference for other similar systems.However, in the actual engineering system, there are some small and self-recoverable faults. The recoverability of failures and the response oscillations that fault-tolerant control may face are the next important issues to be studied.



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*Source: 9994093-2023-01-05.xml*