# Active Fault Tolerance Control Based on Consistent Matrix for Multimotor Synchronous System

Authors: Haijie Mao; Wei Li; Xiaolin Feng; Haochuan Zhang
Journal: Complexity
 (2020)

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## Abstract

This paper presents an active fault-tolerant method to mitigate sensor failures in multimotor synchronous control. First, inspired by the construction of the coupling matrix in complex network synchronous output, a consistent matrix is designed based on structural redundancy in synchronous control. This consistent matrix has two advantages: one is that it can reflect different sensor output similarities and the other one is that it can detect, locate, and estimate the sensor fault. Then, the fault information is integrated into the design of tolerance control with an improved mean feedback mechanism. The proposed method is suitable for both single and multiple fault situations, and its effectiveness is finally verified by both MATLAB simulation and the ABB semiphysical experimental platform.


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## Body

## 1. Introduction

Due to great load drive capabilities and more flexible motion modes, multimotor synchronous driven systems have been widely applied in numerous industry domains including robotics, paper making, and belt conveyors [1–4]. The objective of synchronous control is to ensure synchronization of speed or displacement between different motors under different loads or disturbances [5–7]. In multimotor synchronous control systems, the sensors used to measure speed or position are easily malfunctioned due to ageing, collision, or electromagnetic interference. Once a fault occurs, the synchronization control performance will be seriously affected and disastrous consequences may happen. Therefore, identifying such faults and taking effective measures in time is of great significance for ensuring system safety, reliability, and product quality [5, 6].The development of fault detection and diagnosis (FDD) and fault-tolerant control (FTC) based on analytical redundancy provides strong support for ensuring the security of the system. Many researchers have studied FTC for sensor faults in motion control systems for decades [8–15]. They can be roughly divided into two categories, namely, mathematical model-based and data-driven-based [9]. The core of the mathematical model method is the design of various types of observers. For example, Mao et al. [10] adopted adaptive disturbance rejection control (ADRC) and an extended state observer (ESO) to estimate the speed in a current loop, which realized FTC for a speed sensor fault and improved the multimotor synchronization accuracy. Najafabadi et al. [11] solved the problem of diagnosing and isolating three sensor faults for current, voltage, and speed in induction motors through designing an adaptive current observer for rotor resistance estimation. For a speed sensor failure in an induction motor, Marino et al. [12] designed an adaptive observer to detect sensor faults online, and fault tolerance was realized based on indirect magnetic field orientation control. In recent years, aiming at the difficulty of modelling, the data-driven method has gradually become a new research hot spot, which has received extensive attention [13–15]. Based on the large amount of online and offline monitoring data existing in the system, this method can characterize the normal and fault modes in the system based on that useful information hidden in data using data mining and processing technologies. Consequently, it has been recognized as a practical diagnostic technology. As an application, by collecting line voltage and performing fast Fourier transform, a data-driven method using a two-layer Bayesian network was proposed in [13] to obtain fault characteristics. As a result, fault diagnosis of an inverter in PMSM was realized.Recalling the existing results in the literature, most results are directed to single-motor drives. Although some of them are also applicable to multimotor synchronous driving modes, they still have some limitations since the presented algorithms are complicated and difficult to implement. To overcome such limitations, in this paper, through exploring mathematical model-based and data-driven-based methods’ characters and shortages, we devise a combination method based on a complex network.The complex network model has been widely used in many fields, such as power grids and aerospace [16–19]. A typical complex network system consists of a number of subsystems that are usually coupled with each other. Typically, a multimotor synchronous control system is such a complex network. The consistency study of complex networks, that is, complex network synchronization control, is an important research topic in the field of complex networks.Redundancy is the basis of fault diagnosis and fault tolerance. In this paper, inspired by the construction of a coupling matrix in complex network systems, a consistent matrix is designed to characterize the similarities of output data of different sensors, using the structure and information redundancy in multimotor synchronous control systems. Based on online analysis and judgment of matrix elements and eigenvalues, the detection, location, and estimation for faulty sensors are realized. As a result, a novel improved mean feedback strategy is presented using structural redundancy and fault information to achieve fault tolerance. It is shown through simulation and experiment that system security and reliability are greatly improved.


## 2. Deviation-Coupled Synchronous Control Structure

The research on multimotor synchronous control strategies mainly includes two aspects, synchronous control structure and synchronous control algorithm. In terms of the control structure, there are mainly serial master-slave control [20], virtual spindle control [21], cross-coupling control [22, 23], and deviation-coupling control [24]. Among them, deviation coupling adopts the compensation control strategy and the comprehensive effect is more positive than others in aspects of starting characteristics, disturbance suppression ability, applicable range, and convenience of engineering realization. Accordingly, its practical applications have been found in a wider range of fields. In terms of control algorithms, which are mainly for load uncertainty and unknown interference, various robust control algorithms are devised [10, 24–27], such as sliding mode control, internal model control, and active disturbance rejection control. Since this paper focuses on fault diagnosis and tolerance in multimotor synchronous control systems, the deviation-coupled synchronous control structure in [10] is adopted. The control principle is shown in Figure 1, where ADRC is a kind of controller and S.C. is the synchronous compensator or controller, such as PI or PID.Figure 1
Deviation-coupling synchronization control structure.


## 3. Sensor Fault FDD and FTC Design Based on Complex Network Consistent Matrix

### 3.1. Establishment of Complex Dynamical Network Consistent Matrix

#### 3.1.1. Complex Network Coupling Matrix

Based on graph theory, a complex network system with identical dynamic systems as nodes that satisfies the dissipation conditions can be described as follows [28]:(1)x˙i=fxit+σ∑j=1NaijHxjt,where xi=xi1,xi2,…,xiNT∈Rn is the state of node i, i=1,2,…,N; f∈CRn,Rn is a known function (often a nonlinear function); σ>0 is the coupling strength of the network; A=aijN×N is the coupling matrix of the network, which satisfies ∑jaij=0,i=1,2,…,N; and H∈CRn,Rn, which is a coupling function.The coupling matrixA can be used to describe an undirected topology. If there is a connection between node i and node j, then aij=aji=k,k>0, otherwise aij=aji=0,i≠j. The diagonal elements of matrix A satisfy(2)aii=−∑j=1,j≠iNaij=−∑j=1,j≠iNaji.Definition 1.
In the dynamic network (1), the network is said to be consecutively synchronized for any initial conditions [29] if(3)limt⟶∞xit−xjt=0,i,j=1,2,…,N.
The coupling matrixA determined by the complex network structure reflects the synchronous state or consistent attribute of the network. Inspired by the nature of the coupling matrix, this paper constructs a consistent matrix with the abovementioned properties to characterize the synchronization or consistency of a complex network. When the system is in an abnormal state due to faults, the consistency will be destroyed, and the resulting abnormality will also be reflected in the coupling matrix. So, with the location and size of the matrix element that changes, the fault can be detected and located.


#### 3.1.2. Data-Driven Consistent Matrix Construction

Figure1 shows that multimotor synchronous control is a typical complex network system. Considering the control objectives, this paper pays more attention to judging the consistency of the complex network output. To evaluate the consistency or similarity between different nodes in a complex network, a consistent matrix, which is derived from the coupling matrix, is introduced. The process of constructing the consistent matrix can be summarized as follows.The output of theN nodes of the multimotor synchronous complex network system shown in Figure 1 is measured by N sensors and constitutes a set Y=y1,y2,…,yN. The consistency between yi and yj is represented by the coefficient aij. As aij gets larger, yj and yi become more consistent. The consistency between two different nodes is the same, which means aij=aji. Thus, the matrix A is a symmetric matrix. Constructing A is very important to effectively distinguish the consistency between yi and yj. Generally, it is set as an exponential function of the distance between yi and yj. The construction form is given as follows:(4)Ak=aij=e−yik−yjk2,i≠j,aii=−∑j=1,i≠jNaij,i=j,where yik is the output data from the sensor on node i at sampling moment k, i,j=1,2,…,N. Matrix A satisfies the conditions of the coupling matrix, that is, aij=aji>0, i≠j, rankA=N−1. It also has the same properties as the coupling matrix.Remark 1.
The matrix is completely generated by the output data of the sensor, so it can be considered as data-driven based. At the same time, the structure ofA reflects the distance relationship between two different sensor outputs. Therefore, it has the characteristics of a model. Moreover, the design of A can be considered as a combination of data-driven and model-based methods, which has the advantages of convenient data acquisition and reflecting the mechanism characteristics of the network model.Remark 2.
For the network synchronization, the abnormal or fault output of a certain node will be far away from the output of other normal nodes. It will decrease the consistency between them accordingly. This change can be directly reflected from the corresponding position in the matrix, which is the basis for follow-up fault diagnosis studies.


### 3.2. Sensor Fault Diagnosis Based on Consistent Matrix

#### 3.2.1. Consistent Matrix Analysis under Normal/Fault Condition of the Sensor

Based on the design of the consistent matrix described above, for a multimotor synchronous control system shown in Figure1, the following assumptions are made.Assumption 1.
There areN motors to be synchronized, and the output of each subsystem can be regarded as a node of a complex network.Assumption 2.
The system has good synchronous control performance under normal conditions. The system reaches steady state and has good synchronization accuracy, which means thatyi≈yj.Assumption 3.
When two or more sensors fail at the same time, the sizes or magnitudes of faults are different.
Under the previous assumptions, we can obtainaij≈1 and aii≈−N−1. The eigenvalues of A satisfy λ1=λ2=⋯=λN−1=−N and λN=0. When a sensor (taking the ith sensor as an example) fails, its output will inevitably deviate from other normal sensor outputs. The elements in the ith row and the ith column of A will show ai∗<1,a∗i<1 (except for the diagonal), and with the magnitude of the fault increases, ai∗≪1,a∗i≪1. The eigenvalues of matrix A will also change accordingly.


#### 3.2.2. Fault Detection and Location Based on Consistent Matrix Judgement

From the analysis above on different characteristics of the consistent matrixA before and after the fault, fault detection can be performed by judging the element size of A. Generally, three types of fault detection methods can be selected:(1)
Ak is compared with Ak−l. Taking the length of the data as l, generate Ak−1,…,Ak−l and compare Ak with the elements at the corresponding position of Ak−l. When a row or column of elements in the deviation matrix exceeds the predefined threshold, the fault can be detected and the fault sensor can be located according to the row or column position of the element exceeding the threshold.(2)
Elements inAk are compared with 1. With the analysis above, when the sensor is normal, aij≈1. When the ith sensor fails, the ith row and the ith column elements in Ak appear to be much smaller than 1, ai∗<1,a∗i<1. The fault sensor can be identified and located accordingly.(3)
Using the change inAk eigenvalues. The eigenvalues of the consistent matrix are also changed when a fault occurs. The normal eigenvalues are approximately −N and 0. After the fault occurs, the matrix Ak still satisfies the dissipation characteristics, although rankA=N−1, except for the eigenvalue 0, the distribution of other eigenvalues will change instead.For the abovementioned three methods, from the perspective of detection speed and reliability, there are different characteristics. Method (1) usesl sample data to test. With the sample number increasing, the reliability of the detection is improved, while a certain delay will occur. Therefore, from real-time and reliability consideration, l should not be too large. Method (2) has fast diagnostic speed and good real-time performance, but the anti-interference performance is poor, possibly leading to false alarms. Method (3) is essentially the same as Method (2). They both are based on Ak′s own elements or eigenvalues. However, the limitation of Method (3) lies in that it cannot locate the fault.Combining the characteristics analysis of the abovementioned three methods, to improve the efficiency and reliability of the diagnosis, it is a good idea to combine Methods (1) and (2). When the diagnosis program is implemented, Method (2) is mainly used. After the fault is judged, theAk comparison is made with the latter steps in combination. That is, Method (1) is used for further confirmation. Diagnosis speed, reliability, and computing resources can be simultaneously ensured.It is worth mentioning that the diagnostic method above is also applicable to multiple fault situations. It means that when multiple sensors fail simultaneously, multirow and multicolumn element values will satisfyaij<1 at the corresponding positions in the matrix.For example, whenM sensors fault simultaneously in a multimotor synchronous control system composed of N Motors (M<N) under Assumption 3, there will be CN−M2=N−MN−M−1/2 element values satisfying aij=1, while other M2N−M−1/2 elements will satisfy aij<1, except for diagonal elements. The diagnostic process is similar to the single fault scenario.However, as the number of faulty sensors increases, the types of faults diversify and the number of elements to be judged will increase greatly. Accordingly, the diagnosis time is prolonged. At the same time, the distances between two different sensors, especially between the faulty sensors, are various. All of these will decrease the overall diagnostic reliability of the method to some extent, especially when the detection threshold is set to a certain fixed value.


#### 3.2.3. Fault Estimation Based on Eigenvalues of the Consistent Matrix

In addition to detecting and locating the fault, the fault size can be estimated by further studying the performance of matrixA. The specific process is as follows.When theith sensor fails, the consistent matrix A can be approximated as (ignoring the noise and synchronization error)(5)Af≈−N−2+β1⋯β⋯11−N−2+β⋯β⋯1⋮⋮⋮⋮⋮⋮ββ⋯−N−1β⋯β⋮⋮⋮⋮⋮⋮11⋯β⋯−N−2+β,where β is the consistent coefficient between the fault sensor and other normal sensors, β<1. To estimate the fault size, the abovementioned matrix can be transformed into the following form:(6)Af∗≈−N−2+β1⋯β⋯11−N−1⋯1⋯1⋮⋮⋮⋮⋮⋮β1⋯−N−2+β⋯1⋮⋮⋮⋮⋮⋮11⋯1⋯−N−1.That is, except fora1i=ai1=β, the diagonal elements remain unchanged from the original rule and the elements in other positions are all set to 1. It is easy to obtain the eigenvalues of Af∗, which are λ1=⋯=λN−2=−N, λN−1=−N−2+2β, and λN=0. Compared with the matrix A in normal situations, the only changed eigenvalue of Af∗ is a fault-related quantity, i.e., λN−1=−N−2+2β. Therefore, the fault size can be estimated as(7)β∗=−N−2+λN−1∗2,y1−yi=ln1β∗.Considering thatAf is a singular matrix, to obtain Af∗, a generalized inverse of Af can be used to obtain a transform matrix, P=Af∗⋅Af+.Remark 3.
Compared with some traditional model-based methods, which have a complicated process of modelling, observer designing, parameter optimization, etc., the proposed method can realize the detection, location, and estimation of sensor faults only by judging the elements and eigenvalues of the consistent matrix. It is completely driven by the output data of the sensor and applicable for single and multiple fault situations. Therefore, the proposed method has some significant advantages, such as simple calculation, convenient implementation, and clear physical meaning.


### 3.3. Fault Tolerance Based on Improved Weighted Mean Feedback

After the fault is diagnosed, timely and effective isolation and fault tolerance of the faulty sensor are essential to ensure the safe and stable operation of the system. The signal collected by the sensor is used as the input signal of the mean feedback and the self-feedback. Therefore, after detecting the fault, the two signals should be isolated and reconstructed to ensure the security of the system. For the mean feedback part, it is modified from the weighted mean form as follows:(8)y¯=1∑i=1Nαi∑i=1Nαiyi,where yi is the output signal of each sensor and αi can be defined as the reliability coefficient of the ith sensor. Based on the abovementioned fault diagnosis result, when the sensor is normal, that is, completely reliable, αi=1, while when the sensor is faulty, αi=0, as seen in the following equation:(9)αi=1,Normal,0,Fault.When the sensor is normal, the feedback signal is the traditional average form1/N∑i=1Nyi, and when the sensor is faulty because the reliability coefficient is introduced, αi=0, the corresponding sensor signal is cut off and the fault sensor is automatically isolated. As a result, fault sensor isolation and system fault tolerance can be realized. It does not affect the generation of the mean signal, nor does it change the topology of the original network.For the self-feedback part, the mean signaly¯ is used to replace the faulty sensor output yi. In this situation, the system output keeps the current value unchanged.Based on the analysis above, for the sensor failure problem in a multimotor synchronous control system, the synchronous control with improved deviation-coupling structure of fault diagnosis and fault tolerance function is shown in Figure2.Figure 2
Improved structure with FDD and FTC functions.The outputs of the FDD and FTC module are the modified outputsy˜1,y˜2,…,y˜N of sensors 1 to N, the fault indicator γ, and the mean feedback y¯, respectively. When the sensor is normal, y˜i=yi, and when the sensor is faulty, the corresponding output is y˜i=y¯.Remark 4.
The original mean value is improved by introducing the reliability coefficient, and the modified mean feedback design is performed accordingly. The proposed method does not affect the mean feedback output or the system topology and can implement fault-tolerant control to assure the reliability and safety of the system.


### 3.4. Sensor Fault Detection, Isolation, and Tolerance Steps

Based on the abovementioned analysis, for a single sensor fault, the fault diagnosis, isolation, and fault tolerance processes are as follows:Step 1. Initialization: set the sampling period Ts and the diagnostic threshold th of the system operation; label the sensors in the multimotor synchronous system from 1 to N; and let αi=1, i=1,2,…,N, and the fault indication output γ=0.Step 2. Generating matrix Ak: the consistent matrix Ak is generated according to (4) by the output of each subsystem.Step 3. Fault diagnosis: judging if each element of Ak satisfies aij<th. If yes, the fault occurs and the algorithm proceeds to Step 4; otherwise, it returns to Step 2 to generate the next moment matrix Ak+1 and judges again.Step 4. Fault tolerance: according to the diagnosis result of Step 3, if the jth sensor fails, set αj=0, calculate y¯ according to (8), and set yj=y¯ and γ=j.It should be noted that to improve the efficiency of diagnosis, it only needs to judge the first row or the first column of matrixAk, not all the elements. The result of the judgment is nothing more than the following three cases. (1) If all the values of the first row or the first column satisfy a1i<th or ai1<th, the No.1 sensor fails and the fault indication output is γ=1; (2) if only one of the values satisfies a1i<th, the No. i sensor fails, the fault indication output is γ=i; and (3) if all do not satisfy a1i<th, no fault occurs, and the fault indication remains γ=0.


## 3.1. Establishment of Complex Dynamical Network Consistent Matrix

### 3.1.1. Complex Network Coupling Matrix

Based on graph theory, a complex network system with identical dynamic systems as nodes that satisfies the dissipation conditions can be described as follows [28]:(1)x˙i=fxit+σ∑j=1NaijHxjt,where xi=xi1,xi2,…,xiNT∈Rn is the state of node i, i=1,2,…,N; f∈CRn,Rn is a known function (often a nonlinear function); σ>0 is the coupling strength of the network; A=aijN×N is the coupling matrix of the network, which satisfies ∑jaij=0,i=1,2,…,N; and H∈CRn,Rn, which is a coupling function.The coupling matrixA can be used to describe an undirected topology. If there is a connection between node i and node j, then aij=aji=k,k>0, otherwise aij=aji=0,i≠j. The diagonal elements of matrix A satisfy(2)aii=−∑j=1,j≠iNaij=−∑j=1,j≠iNaji.Definition 1.
In the dynamic network (1), the network is said to be consecutively synchronized for any initial conditions [29] if(3)limt⟶∞xit−xjt=0,i,j=1,2,…,N.
The coupling matrixA determined by the complex network structure reflects the synchronous state or consistent attribute of the network. Inspired by the nature of the coupling matrix, this paper constructs a consistent matrix with the abovementioned properties to characterize the synchronization or consistency of a complex network. When the system is in an abnormal state due to faults, the consistency will be destroyed, and the resulting abnormality will also be reflected in the coupling matrix. So, with the location and size of the matrix element that changes, the fault can be detected and located.


### 3.1.2. Data-Driven Consistent Matrix Construction

Figure1 shows that multimotor synchronous control is a typical complex network system. Considering the control objectives, this paper pays more attention to judging the consistency of the complex network output. To evaluate the consistency or similarity between different nodes in a complex network, a consistent matrix, which is derived from the coupling matrix, is introduced. The process of constructing the consistent matrix can be summarized as follows.The output of theN nodes of the multimotor synchronous complex network system shown in Figure 1 is measured by N sensors and constitutes a set Y=y1,y2,…,yN. The consistency between yi and yj is represented by the coefficient aij. As aij gets larger, yj and yi become more consistent. The consistency between two different nodes is the same, which means aij=aji. Thus, the matrix A is a symmetric matrix. Constructing A is very important to effectively distinguish the consistency between yi and yj. Generally, it is set as an exponential function of the distance between yi and yj. The construction form is given as follows:(4)Ak=aij=e−yik−yjk2,i≠j,aii=−∑j=1,i≠jNaij,i=j,where yik is the output data from the sensor on node i at sampling moment k, i,j=1,2,…,N. Matrix A satisfies the conditions of the coupling matrix, that is, aij=aji>0, i≠j, rankA=N−1. It also has the same properties as the coupling matrix.Remark 1.
The matrix is completely generated by the output data of the sensor, so it can be considered as data-driven based. At the same time, the structure ofA reflects the distance relationship between two different sensor outputs. Therefore, it has the characteristics of a model. Moreover, the design of A can be considered as a combination of data-driven and model-based methods, which has the advantages of convenient data acquisition and reflecting the mechanism characteristics of the network model.Remark 2.
For the network synchronization, the abnormal or fault output of a certain node will be far away from the output of other normal nodes. It will decrease the consistency between them accordingly. This change can be directly reflected from the corresponding position in the matrix, which is the basis for follow-up fault diagnosis studies.


## 3.1.1. Complex Network Coupling Matrix

Based on graph theory, a complex network system with identical dynamic systems as nodes that satisfies the dissipation conditions can be described as follows [28]:(1)x˙i=fxit+σ∑j=1NaijHxjt,where xi=xi1,xi2,…,xiNT∈Rn is the state of node i, i=1,2,…,N; f∈CRn,Rn is a known function (often a nonlinear function); σ>0 is the coupling strength of the network; A=aijN×N is the coupling matrix of the network, which satisfies ∑jaij=0,i=1,2,…,N; and H∈CRn,Rn, which is a coupling function.The coupling matrixA can be used to describe an undirected topology. If there is a connection between node i and node j, then aij=aji=k,k>0, otherwise aij=aji=0,i≠j. The diagonal elements of matrix A satisfy(2)aii=−∑j=1,j≠iNaij=−∑j=1,j≠iNaji.Definition 1.
In the dynamic network (1), the network is said to be consecutively synchronized for any initial conditions [29] if(3)limt⟶∞xit−xjt=0,i,j=1,2,…,N.
The coupling matrixA determined by the complex network structure reflects the synchronous state or consistent attribute of the network. Inspired by the nature of the coupling matrix, this paper constructs a consistent matrix with the abovementioned properties to characterize the synchronization or consistency of a complex network. When the system is in an abnormal state due to faults, the consistency will be destroyed, and the resulting abnormality will also be reflected in the coupling matrix. So, with the location and size of the matrix element that changes, the fault can be detected and located.


## 3.1.2. Data-Driven Consistent Matrix Construction

Figure1 shows that multimotor synchronous control is a typical complex network system. Considering the control objectives, this paper pays more attention to judging the consistency of the complex network output. To evaluate the consistency or similarity between different nodes in a complex network, a consistent matrix, which is derived from the coupling matrix, is introduced. The process of constructing the consistent matrix can be summarized as follows.The output of theN nodes of the multimotor synchronous complex network system shown in Figure 1 is measured by N sensors and constitutes a set Y=y1,y2,…,yN. The consistency between yi and yj is represented by the coefficient aij. As aij gets larger, yj and yi become more consistent. The consistency between two different nodes is the same, which means aij=aji. Thus, the matrix A is a symmetric matrix. Constructing A is very important to effectively distinguish the consistency between yi and yj. Generally, it is set as an exponential function of the distance between yi and yj. The construction form is given as follows:(4)Ak=aij=e−yik−yjk2,i≠j,aii=−∑j=1,i≠jNaij,i=j,where yik is the output data from the sensor on node i at sampling moment k, i,j=1,2,…,N. Matrix A satisfies the conditions of the coupling matrix, that is, aij=aji>0, i≠j, rankA=N−1. It also has the same properties as the coupling matrix.Remark 1.
The matrix is completely generated by the output data of the sensor, so it can be considered as data-driven based. At the same time, the structure ofA reflects the distance relationship between two different sensor outputs. Therefore, it has the characteristics of a model. Moreover, the design of A can be considered as a combination of data-driven and model-based methods, which has the advantages of convenient data acquisition and reflecting the mechanism characteristics of the network model.Remark 2.
For the network synchronization, the abnormal or fault output of a certain node will be far away from the output of other normal nodes. It will decrease the consistency between them accordingly. This change can be directly reflected from the corresponding position in the matrix, which is the basis for follow-up fault diagnosis studies.


## 3.2. Sensor Fault Diagnosis Based on Consistent Matrix

### 3.2.1. Consistent Matrix Analysis under Normal/Fault Condition of the Sensor

Based on the design of the consistent matrix described above, for a multimotor synchronous control system shown in Figure1, the following assumptions are made.Assumption 1.
There areN motors to be synchronized, and the output of each subsystem can be regarded as a node of a complex network.Assumption 2.
The system has good synchronous control performance under normal conditions. The system reaches steady state and has good synchronization accuracy, which means thatyi≈yj.Assumption 3.
When two or more sensors fail at the same time, the sizes or magnitudes of faults are different.
Under the previous assumptions, we can obtainaij≈1 and aii≈−N−1. The eigenvalues of A satisfy λ1=λ2=⋯=λN−1=−N and λN=0. When a sensor (taking the ith sensor as an example) fails, its output will inevitably deviate from other normal sensor outputs. The elements in the ith row and the ith column of A will show ai∗<1,a∗i<1 (except for the diagonal), and with the magnitude of the fault increases, ai∗≪1,a∗i≪1. The eigenvalues of matrix A will also change accordingly.


### 3.2.2. Fault Detection and Location Based on Consistent Matrix Judgement

From the analysis above on different characteristics of the consistent matrixA before and after the fault, fault detection can be performed by judging the element size of A. Generally, three types of fault detection methods can be selected:(1)
Ak is compared with Ak−l. Taking the length of the data as l, generate Ak−1,…,Ak−l and compare Ak with the elements at the corresponding position of Ak−l. When a row or column of elements in the deviation matrix exceeds the predefined threshold, the fault can be detected and the fault sensor can be located according to the row or column position of the element exceeding the threshold.(2)
Elements inAk are compared with 1. With the analysis above, when the sensor is normal, aij≈1. When the ith sensor fails, the ith row and the ith column elements in Ak appear to be much smaller than 1, ai∗<1,a∗i<1. The fault sensor can be identified and located accordingly.(3)
Using the change inAk eigenvalues. The eigenvalues of the consistent matrix are also changed when a fault occurs. The normal eigenvalues are approximately −N and 0. After the fault occurs, the matrix Ak still satisfies the dissipation characteristics, although rankA=N−1, except for the eigenvalue 0, the distribution of other eigenvalues will change instead.For the abovementioned three methods, from the perspective of detection speed and reliability, there are different characteristics. Method (1) usesl sample data to test. With the sample number increasing, the reliability of the detection is improved, while a certain delay will occur. Therefore, from real-time and reliability consideration, l should not be too large. Method (2) has fast diagnostic speed and good real-time performance, but the anti-interference performance is poor, possibly leading to false alarms. Method (3) is essentially the same as Method (2). They both are based on Ak′s own elements or eigenvalues. However, the limitation of Method (3) lies in that it cannot locate the fault.Combining the characteristics analysis of the abovementioned three methods, to improve the efficiency and reliability of the diagnosis, it is a good idea to combine Methods (1) and (2). When the diagnosis program is implemented, Method (2) is mainly used. After the fault is judged, theAk comparison is made with the latter steps in combination. That is, Method (1) is used for further confirmation. Diagnosis speed, reliability, and computing resources can be simultaneously ensured.It is worth mentioning that the diagnostic method above is also applicable to multiple fault situations. It means that when multiple sensors fail simultaneously, multirow and multicolumn element values will satisfyaij<1 at the corresponding positions in the matrix.For example, whenM sensors fault simultaneously in a multimotor synchronous control system composed of N Motors (M<N) under Assumption 3, there will be CN−M2=N−MN−M−1/2 element values satisfying aij=1, while other M2N−M−1/2 elements will satisfy aij<1, except for diagonal elements. The diagnostic process is similar to the single fault scenario.However, as the number of faulty sensors increases, the types of faults diversify and the number of elements to be judged will increase greatly. Accordingly, the diagnosis time is prolonged. At the same time, the distances between two different sensors, especially between the faulty sensors, are various. All of these will decrease the overall diagnostic reliability of the method to some extent, especially when the detection threshold is set to a certain fixed value.


### 3.2.3. Fault Estimation Based on Eigenvalues of the Consistent Matrix

In addition to detecting and locating the fault, the fault size can be estimated by further studying the performance of matrixA. The specific process is as follows.When theith sensor fails, the consistent matrix A can be approximated as (ignoring the noise and synchronization error)(5)Af≈−N−2+β1⋯β⋯11−N−2+β⋯β⋯1⋮⋮⋮⋮⋮⋮ββ⋯−N−1β⋯β⋮⋮⋮⋮⋮⋮11⋯β⋯−N−2+β,where β is the consistent coefficient between the fault sensor and other normal sensors, β<1. To estimate the fault size, the abovementioned matrix can be transformed into the following form:(6)Af∗≈−N−2+β1⋯β⋯11−N−1⋯1⋯1⋮⋮⋮⋮⋮⋮β1⋯−N−2+β⋯1⋮⋮⋮⋮⋮⋮11⋯1⋯−N−1.That is, except fora1i=ai1=β, the diagonal elements remain unchanged from the original rule and the elements in other positions are all set to 1. It is easy to obtain the eigenvalues of Af∗, which are λ1=⋯=λN−2=−N, λN−1=−N−2+2β, and λN=0. Compared with the matrix A in normal situations, the only changed eigenvalue of Af∗ is a fault-related quantity, i.e., λN−1=−N−2+2β. Therefore, the fault size can be estimated as(7)β∗=−N−2+λN−1∗2,y1−yi=ln1β∗.Considering thatAf is a singular matrix, to obtain Af∗, a generalized inverse of Af can be used to obtain a transform matrix, P=Af∗⋅Af+.Remark 3.
Compared with some traditional model-based methods, which have a complicated process of modelling, observer designing, parameter optimization, etc., the proposed method can realize the detection, location, and estimation of sensor faults only by judging the elements and eigenvalues of the consistent matrix. It is completely driven by the output data of the sensor and applicable for single and multiple fault situations. Therefore, the proposed method has some significant advantages, such as simple calculation, convenient implementation, and clear physical meaning.


## 3.2.1. Consistent Matrix Analysis under Normal/Fault Condition of the Sensor

Based on the design of the consistent matrix described above, for a multimotor synchronous control system shown in Figure1, the following assumptions are made.Assumption 1.
There areN motors to be synchronized, and the output of each subsystem can be regarded as a node of a complex network.Assumption 2.
The system has good synchronous control performance under normal conditions. The system reaches steady state and has good synchronization accuracy, which means thatyi≈yj.Assumption 3.
When two or more sensors fail at the same time, the sizes or magnitudes of faults are different.
Under the previous assumptions, we can obtainaij≈1 and aii≈−N−1. The eigenvalues of A satisfy λ1=λ2=⋯=λN−1=−N and λN=0. When a sensor (taking the ith sensor as an example) fails, its output will inevitably deviate from other normal sensor outputs. The elements in the ith row and the ith column of A will show ai∗<1,a∗i<1 (except for the diagonal), and with the magnitude of the fault increases, ai∗≪1,a∗i≪1. The eigenvalues of matrix A will also change accordingly.


## 3.2.2. Fault Detection and Location Based on Consistent Matrix Judgement

From the analysis above on different characteristics of the consistent matrixA before and after the fault, fault detection can be performed by judging the element size of A. Generally, three types of fault detection methods can be selected:(1)
Ak is compared with Ak−l. Taking the length of the data as l, generate Ak−1,…,Ak−l and compare Ak with the elements at the corresponding position of Ak−l. When a row or column of elements in the deviation matrix exceeds the predefined threshold, the fault can be detected and the fault sensor can be located according to the row or column position of the element exceeding the threshold.(2)
Elements inAk are compared with 1. With the analysis above, when the sensor is normal, aij≈1. When the ith sensor fails, the ith row and the ith column elements in Ak appear to be much smaller than 1, ai∗<1,a∗i<1. The fault sensor can be identified and located accordingly.(3)
Using the change inAk eigenvalues. The eigenvalues of the consistent matrix are also changed when a fault occurs. The normal eigenvalues are approximately −N and 0. After the fault occurs, the matrix Ak still satisfies the dissipation characteristics, although rankA=N−1, except for the eigenvalue 0, the distribution of other eigenvalues will change instead.For the abovementioned three methods, from the perspective of detection speed and reliability, there are different characteristics. Method (1) usesl sample data to test. With the sample number increasing, the reliability of the detection is improved, while a certain delay will occur. Therefore, from real-time and reliability consideration, l should not be too large. Method (2) has fast diagnostic speed and good real-time performance, but the anti-interference performance is poor, possibly leading to false alarms. Method (3) is essentially the same as Method (2). They both are based on Ak′s own elements or eigenvalues. However, the limitation of Method (3) lies in that it cannot locate the fault.Combining the characteristics analysis of the abovementioned three methods, to improve the efficiency and reliability of the diagnosis, it is a good idea to combine Methods (1) and (2). When the diagnosis program is implemented, Method (2) is mainly used. After the fault is judged, theAk comparison is made with the latter steps in combination. That is, Method (1) is used for further confirmation. Diagnosis speed, reliability, and computing resources can be simultaneously ensured.It is worth mentioning that the diagnostic method above is also applicable to multiple fault situations. It means that when multiple sensors fail simultaneously, multirow and multicolumn element values will satisfyaij<1 at the corresponding positions in the matrix.For example, whenM sensors fault simultaneously in a multimotor synchronous control system composed of N Motors (M<N) under Assumption 3, there will be CN−M2=N−MN−M−1/2 element values satisfying aij=1, while other M2N−M−1/2 elements will satisfy aij<1, except for diagonal elements. The diagnostic process is similar to the single fault scenario.However, as the number of faulty sensors increases, the types of faults diversify and the number of elements to be judged will increase greatly. Accordingly, the diagnosis time is prolonged. At the same time, the distances between two different sensors, especially between the faulty sensors, are various. All of these will decrease the overall diagnostic reliability of the method to some extent, especially when the detection threshold is set to a certain fixed value.


## 3.2.3. Fault Estimation Based on Eigenvalues of the Consistent Matrix

In addition to detecting and locating the fault, the fault size can be estimated by further studying the performance of matrixA. The specific process is as follows.When theith sensor fails, the consistent matrix A can be approximated as (ignoring the noise and synchronization error)(5)Af≈−N−2+β1⋯β⋯11−N−2+β⋯β⋯1⋮⋮⋮⋮⋮⋮ββ⋯−N−1β⋯β⋮⋮⋮⋮⋮⋮11⋯β⋯−N−2+β,where β is the consistent coefficient between the fault sensor and other normal sensors, β<1. To estimate the fault size, the abovementioned matrix can be transformed into the following form:(6)Af∗≈−N−2+β1⋯β⋯11−N−1⋯1⋯1⋮⋮⋮⋮⋮⋮β1⋯−N−2+β⋯1⋮⋮⋮⋮⋮⋮11⋯1⋯−N−1.That is, except fora1i=ai1=β, the diagonal elements remain unchanged from the original rule and the elements in other positions are all set to 1. It is easy to obtain the eigenvalues of Af∗, which are λ1=⋯=λN−2=−N, λN−1=−N−2+2β, and λN=0. Compared with the matrix A in normal situations, the only changed eigenvalue of Af∗ is a fault-related quantity, i.e., λN−1=−N−2+2β. Therefore, the fault size can be estimated as(7)β∗=−N−2+λN−1∗2,y1−yi=ln1β∗.Considering thatAf is a singular matrix, to obtain Af∗, a generalized inverse of Af can be used to obtain a transform matrix, P=Af∗⋅Af+.Remark 3.
Compared with some traditional model-based methods, which have a complicated process of modelling, observer designing, parameter optimization, etc., the proposed method can realize the detection, location, and estimation of sensor faults only by judging the elements and eigenvalues of the consistent matrix. It is completely driven by the output data of the sensor and applicable for single and multiple fault situations. Therefore, the proposed method has some significant advantages, such as simple calculation, convenient implementation, and clear physical meaning.


## 3.3. Fault Tolerance Based on Improved Weighted Mean Feedback

After the fault is diagnosed, timely and effective isolation and fault tolerance of the faulty sensor are essential to ensure the safe and stable operation of the system. The signal collected by the sensor is used as the input signal of the mean feedback and the self-feedback. Therefore, after detecting the fault, the two signals should be isolated and reconstructed to ensure the security of the system. For the mean feedback part, it is modified from the weighted mean form as follows:(8)y¯=1∑i=1Nαi∑i=1Nαiyi,where yi is the output signal of each sensor and αi can be defined as the reliability coefficient of the ith sensor. Based on the abovementioned fault diagnosis result, when the sensor is normal, that is, completely reliable, αi=1, while when the sensor is faulty, αi=0, as seen in the following equation:(9)αi=1,Normal,0,Fault.When the sensor is normal, the feedback signal is the traditional average form1/N∑i=1Nyi, and when the sensor is faulty because the reliability coefficient is introduced, αi=0, the corresponding sensor signal is cut off and the fault sensor is automatically isolated. As a result, fault sensor isolation and system fault tolerance can be realized. It does not affect the generation of the mean signal, nor does it change the topology of the original network.For the self-feedback part, the mean signaly¯ is used to replace the faulty sensor output yi. In this situation, the system output keeps the current value unchanged.Based on the analysis above, for the sensor failure problem in a multimotor synchronous control system, the synchronous control with improved deviation-coupling structure of fault diagnosis and fault tolerance function is shown in Figure2.Figure 2
Improved structure with FDD and FTC functions.The outputs of the FDD and FTC module are the modified outputsy˜1,y˜2,…,y˜N of sensors 1 to N, the fault indicator γ, and the mean feedback y¯, respectively. When the sensor is normal, y˜i=yi, and when the sensor is faulty, the corresponding output is y˜i=y¯.Remark 4.
The original mean value is improved by introducing the reliability coefficient, and the modified mean feedback design is performed accordingly. The proposed method does not affect the mean feedback output or the system topology and can implement fault-tolerant control to assure the reliability and safety of the system.


## 3.4. Sensor Fault Detection, Isolation, and Tolerance Steps

Based on the abovementioned analysis, for a single sensor fault, the fault diagnosis, isolation, and fault tolerance processes are as follows:Step 1. Initialization: set the sampling period Ts and the diagnostic threshold th of the system operation; label the sensors in the multimotor synchronous system from 1 to N; and let αi=1, i=1,2,…,N, and the fault indication output γ=0.Step 2. Generating matrix Ak: the consistent matrix Ak is generated according to (4) by the output of each subsystem.Step 3. Fault diagnosis: judging if each element of Ak satisfies aij<th. If yes, the fault occurs and the algorithm proceeds to Step 4; otherwise, it returns to Step 2 to generate the next moment matrix Ak+1 and judges again.Step 4. Fault tolerance: according to the diagnosis result of Step 3, if the jth sensor fails, set αj=0, calculate y¯ according to (8), and set yj=y¯ and γ=j.It should be noted that to improve the efficiency of diagnosis, it only needs to judge the first row or the first column of matrixAk, not all the elements. The result of the judgment is nothing more than the following three cases. (1) If all the values of the first row or the first column satisfy a1i<th or ai1<th, the No.1 sensor fails and the fault indication output is γ=1; (2) if only one of the values satisfies a1i<th, the No. i sensor fails, the fault indication output is γ=i; and (3) if all do not satisfy a1i<th, no fault occurs, and the fault indication remains γ=0.


## 4. Simulation Results and Analysis

To verify the effectiveness of the fault diagnosis and fault-tolerant method proposed above, a simulation is carried out for both a single sensor fault and multisensor simultaneous fault in the MATLAB/Simulink environment based on the control structure shown in Figure2.


### 4.1. Single Sensor Fault Simulation

For a single sensor fault situation, the synchronous control of three permanent magnet synchronous motors (PMSM) is taken as an example. To verify the advantages of the proposed method, under different speed inputs, the sensors in subsystems 1–3 have three different types of faults such as constant deviation, stuck, and constant gain at different times. The PMSM models are shown in (10)–(13), and the parameters of each motor are given in Table1. The simulation time is 35 s, the simulation step size is Ts=0.001s, and threshold th=0.1. The simulation results are shown in Figures 3–6:(10)id⋅=1Ldud−Rsid+ωLqiq,(11)iq⋅=1Lquq−Rsiq−ωLdid−ωϕf,(12)ω⋅=1JPnTe−PnTl−Bω,(13)Te=1.5Pnϕfiq.Table 1
Three-motor mechanical parameters.
Variable
Symbol
Motor 1
Motor 2
Motor 3
Rated power (kW)
P
1.5
1.5
1.5
Rated speed (rpm)
N
1500
1500
1500
Rated torque (N·m)
T
3
3
3
D-axis self-inductance (mH)
L
d
3.5
4.5
3
Q-axis self-inductance (mH)
L
q
3.5
4.5
3
Stator resistance (ῼ)
R
s
0.94
0.89
1.32
Moment of inertiakg⋅m2
J
0.01
0.007
0.005
Number of pole pairs
p
2
2
2Figure 3
Fault indicator output of the method.Figure 4
Fault estimation of the method in this paper.Figure 5
Tolerance synchronous output after the fault.Figure 6
Tolerance synchronous output error.First, to reflect the impact of the fault, three types of fault shown in Table2 are first applied to the No. 1 sensor only. Figure 7 shows the result of no action taken after the fault occurs. A fault of the No. 1 sensor seriously affects the outputs of subsystems 2 and 3 at this moment. Clearly, the whole system cannot be synchronized for a period from the occurrence to the end of a fault.Table 2
Sensor failure information.
Sensor
Fault type
Mathematical description
Time
1
Constant deviation
y
˜
k
=
y
k
+
c
1
5–6 s
2
Stuck
y
˜
k
=
c
2
15–16 s
3
Constant gain
y
˜
k
=
c
3
y
k
25–26 sFigure 7
No FDD synchronous output after failure.Second, to verify the effect of the proposed fault diagnosis method, different types of faults, as shown in Table2, are applied to three sensors at different times. Figure 3 shows the fault indicator results. When different types of sensor fault occur at 5, 15, and 25 s, the fault indicator output is γ=1,2,and3, respectively. It is clear to see that the proposed method can detect and locate the fault quickly and accurately. Figure 4 shows the results of fault estimation. The maximum estimation error shown in the detail figure is only about 3 rad/s. The estimation is accurate.Third, with the abovementioned diagnosis results, the improved weighted mean feedback based on the reliability coefficient was used to isolate the faulty sensor and achieve fault-tolerant control. The results are shown in Figure5 and 6.It can be seen from Figure5 that the system still maintains a satisfactory synchronization accuracy after the fault occurs, in addition to the slight fluctuations caused by the adjustment of the controller in a short time. Furthermore, from the synchronous output error of Figure 6, one can see that the error caused by the fault tolerance is much smaller than the error of control when the desired rotational speed is changed. The maximum tolerance error is about 0.1 rad/s.


### 4.2. Multisensor Fault Simulation

As mentioned above, the proposed method is also suitable for multifault situation. So, another simulated synchronous system composed of four motors is presented, considering the simultaneous failure of two sensors. The parameters of the four motors are shown in Table3. The simulation time is 35 s, th=0.1, and the simulation step size Ts=0.001s.Table 3
4 motor mechanical parameters.
Variable
Motor 1
Motor 2
Motor 3
Motor 4
Rated speed (rpm)
1500
1500
1500
1500
Rated torque (N·m)
3
3.3
4.5
4
D-axis self-inductance (mH)
3.5
3.7
4.1
3.8
Q-axis self-inductance (mH)
3.2
3.6
4.0
3.6
Stator resistance (ῼ)
0.9
0.94
1.32
1
Moment of inertiakg⋅m2
0.01
0.007
0.005
0.006
Number of pole pairs
2
2
2
2For the synchronous control system consisting of four motors, when two sensors fail simultaneously, there are six combinations. Three combinations of them can be taken into consideration. Three types of sensor failures (constant gain, stuck, and constant deviation) are considered. The fault information is shown in Table4.Table 4
Sensor fault information.
Sensor
Fault type
Mathematical description
Fault amplitude
Time (s)
No. 1, No. 2
Constant deviation
y
˜
k
=
y
k
+
c
1
c
1
=
−
40
,
c
1
′
=
−
20
5–6
No. 3, No. 4
Stuck
y
˜
k
=
c
2
c
2
=
22
,
c
2
′
=
18
15–16
No. 2, No. 4
Constant gain
y
˜
k
=
c
3
y
k
c
3
=
100
,
c
3
′
=
200
25–26Figure8 shows the results of the system synchronization output after 2 sensors fail simultaneously without taking any measures. Similar to the single failure scenario, the synchronization performance of the system is degraded for a period of time. Figures 9‐11 show the results of FDD and FTC using the proposed method.Figure 8
Synchronous output without taking any action when two sensors fail simultaneously.Figure 9
Fault indication output when two sensors fail simultaneously.Figure 10
Synchronous output of FTC.Figure 11
Synchronization error of FTC.It can be seen from the time indicated by the fault shown in Figure9 that the method can detect the fault quickly and exactly in 5–6, 15–16, and 25–26 s. Fault indicator amplitudes 12, 34, and 24 indicated that No. 1–No. 2, No. 3–No. 4, and No. 2–No. 4 failed simultaneously, respectively. The fault location is also satisfied for multisensor faults.After the fault is diagnosed, Figure10 shows the fault-tolerant results using the improved weighted mean feedback. When two sensors fail at the same time, only the outputs of the remaining two normal sensors are used, and a satisfactory synchronization performance is achieved. Furthermore, from Figure 11, the synchronization error at t=25s is relatively large (about 2 rad/s) due to the slow change characteristics of constant gain fault. The synchronization error of tolerance is far less than the control error of the expected speed change. The fault tolerance is satisfactory overall.It should be pointed out that, when two sensors have different amplitude faults, the distances between the faulty sensor and a normal sensor, as well as between two faulty sensors, become complicated and diverse. It is difficult to estimate the fault size. In addition, considering the existence of the synchronization error and the randomness of the distance between different sensors outputs, the threshold selection is more sensitive.


## 4.1. Single Sensor Fault Simulation

For a single sensor fault situation, the synchronous control of three permanent magnet synchronous motors (PMSM) is taken as an example. To verify the advantages of the proposed method, under different speed inputs, the sensors in subsystems 1–3 have three different types of faults such as constant deviation, stuck, and constant gain at different times. The PMSM models are shown in (10)–(13), and the parameters of each motor are given in Table1. The simulation time is 35 s, the simulation step size is Ts=0.001s, and threshold th=0.1. The simulation results are shown in Figures 3–6:(10)id⋅=1Ldud−Rsid+ωLqiq,(11)iq⋅=1Lquq−Rsiq−ωLdid−ωϕf,(12)ω⋅=1JPnTe−PnTl−Bω,(13)Te=1.5Pnϕfiq.Table 1
Three-motor mechanical parameters.
Variable
Symbol
Motor 1
Motor 2
Motor 3
Rated power (kW)
P
1.5
1.5
1.5
Rated speed (rpm)
N
1500
1500
1500
Rated torque (N·m)
T
3
3
3
D-axis self-inductance (mH)
L
d
3.5
4.5
3
Q-axis self-inductance (mH)
L
q
3.5
4.5
3
Stator resistance (ῼ)
R
s
0.94
0.89
1.32
Moment of inertiakg⋅m2
J
0.01
0.007
0.005
Number of pole pairs
p
2
2
2Figure 3
Fault indicator output of the method.Figure 4
Fault estimation of the method in this paper.Figure 5
Tolerance synchronous output after the fault.Figure 6
Tolerance synchronous output error.First, to reflect the impact of the fault, three types of fault shown in Table2 are first applied to the No. 1 sensor only. Figure 7 shows the result of no action taken after the fault occurs. A fault of the No. 1 sensor seriously affects the outputs of subsystems 2 and 3 at this moment. Clearly, the whole system cannot be synchronized for a period from the occurrence to the end of a fault.Table 2
Sensor failure information.
Sensor
Fault type
Mathematical description
Time
1
Constant deviation
y
˜
k
=
y
k
+
c
1
5–6 s
2
Stuck
y
˜
k
=
c
2
15–16 s
3
Constant gain
y
˜
k
=
c
3
y
k
25–26 sFigure 7
No FDD synchronous output after failure.Second, to verify the effect of the proposed fault diagnosis method, different types of faults, as shown in Table2, are applied to three sensors at different times. Figure 3 shows the fault indicator results. When different types of sensor fault occur at 5, 15, and 25 s, the fault indicator output is γ=1,2,and3, respectively. It is clear to see that the proposed method can detect and locate the fault quickly and accurately. Figure 4 shows the results of fault estimation. The maximum estimation error shown in the detail figure is only about 3 rad/s. The estimation is accurate.Third, with the abovementioned diagnosis results, the improved weighted mean feedback based on the reliability coefficient was used to isolate the faulty sensor and achieve fault-tolerant control. The results are shown in Figure5 and 6.It can be seen from Figure5 that the system still maintains a satisfactory synchronization accuracy after the fault occurs, in addition to the slight fluctuations caused by the adjustment of the controller in a short time. Furthermore, from the synchronous output error of Figure 6, one can see that the error caused by the fault tolerance is much smaller than the error of control when the desired rotational speed is changed. The maximum tolerance error is about 0.1 rad/s.


## 4.2. Multisensor Fault Simulation

As mentioned above, the proposed method is also suitable for multifault situation. So, another simulated synchronous system composed of four motors is presented, considering the simultaneous failure of two sensors. The parameters of the four motors are shown in Table3. The simulation time is 35 s, th=0.1, and the simulation step size Ts=0.001s.Table 3
4 motor mechanical parameters.
Variable
Motor 1
Motor 2
Motor 3
Motor 4
Rated speed (rpm)
1500
1500
1500
1500
Rated torque (N·m)
3
3.3
4.5
4
D-axis self-inductance (mH)
3.5
3.7
4.1
3.8
Q-axis self-inductance (mH)
3.2
3.6
4.0
3.6
Stator resistance (ῼ)
0.9
0.94
1.32
1
Moment of inertiakg⋅m2
0.01
0.007
0.005
0.006
Number of pole pairs
2
2
2
2For the synchronous control system consisting of four motors, when two sensors fail simultaneously, there are six combinations. Three combinations of them can be taken into consideration. Three types of sensor failures (constant gain, stuck, and constant deviation) are considered. The fault information is shown in Table4.Table 4
Sensor fault information.
Sensor
Fault type
Mathematical description
Fault amplitude
Time (s)
No. 1, No. 2
Constant deviation
y
˜
k
=
y
k
+
c
1
c
1
=
−
40
,
c
1
′
=
−
20
5–6
No. 3, No. 4
Stuck
y
˜
k
=
c
2
c
2
=
22
,
c
2
′
=
18
15–16
No. 2, No. 4
Constant gain
y
˜
k
=
c
3
y
k
c
3
=
100
,
c
3
′
=
200
25–26Figure8 shows the results of the system synchronization output after 2 sensors fail simultaneously without taking any measures. Similar to the single failure scenario, the synchronization performance of the system is degraded for a period of time. Figures 9‐11 show the results of FDD and FTC using the proposed method.Figure 8
Synchronous output without taking any action when two sensors fail simultaneously.Figure 9
Fault indication output when two sensors fail simultaneously.Figure 10
Synchronous output of FTC.Figure 11
Synchronization error of FTC.It can be seen from the time indicated by the fault shown in Figure9 that the method can detect the fault quickly and exactly in 5–6, 15–16, and 25–26 s. Fault indicator amplitudes 12, 34, and 24 indicated that No. 1–No. 2, No. 3–No. 4, and No. 2–No. 4 failed simultaneously, respectively. The fault location is also satisfied for multisensor faults.After the fault is diagnosed, Figure10 shows the fault-tolerant results using the improved weighted mean feedback. When two sensors fail at the same time, only the outputs of the remaining two normal sensors are used, and a satisfactory synchronization performance is achieved. Furthermore, from Figure 11, the synchronization error at t=25s is relatively large (about 2 rad/s) due to the slow change characteristics of constant gain fault. The synchronization error of tolerance is far less than the control error of the expected speed change. The fault tolerance is satisfactory overall.It should be pointed out that, when two sensors have different amplitude faults, the distances between the faulty sensor and a normal sensor, as well as between two faulty sensors, become complicated and diverse. It is difficult to estimate the fault size. In addition, considering the existence of the synchronization error and the randomness of the distance between different sensors outputs, the threshold selection is more sensitive.


## 5. Experimental Results and Analysis

To test the engineering applicability of the proposed method, a multimotor synchronous control experimental platform composed of 4 motors is established. The hardware mainly includes ABB’s AC500-eCo PLC; input and output modules DX561, DC562, and AX561; ACS355 frequency converter and permanent magnet synchronous motor (see Table5 for parameter information); and 360 line photoelectric encoder. The software mainly includes Control Builder Plus (referred to as CBP, integrated with CoDeSys), OPC configurator, and MATLAB/Simulink. Because the AC500-eCo PLC contains only two high-speed counting channels, the four motors are controlled by two PLCs through four frequency converters. The established experimental platform is shown in Figure 12.Table 5
Experimental motor mechanical parameters.
Variable
Symbol
Motor
Power (kW)
P
0.4
Rated voltage (V)
U
220
Rated current (A)
I
1.8
Rated speed (rpm)
N
3000
Rated torque (N·m)
T
1.2
Number of pole pairs
p
1Figure 12
Semiphysical experiment platform.Setting sensor No. 2–4 to have three kinds of fault, constant deviation, constant gain, and stuck, the corresponding fault occurrence and duration time is1−1.5s, 2.5−3s, and 4−4.5s, respectively. The expected speed is 2000 rpm. The results of fault diagnosis and fault tolerance are shown in Figures 13 and 14.Figure 13
Fault indication output.Figure 14
Synchronous output after fault tolerance.The fault diagnosis is timely and reliable. The design of the improved weighted average feedback mechanism based on the diagnosis results realized the fault-tolerant control. The synchronous precision of the system after fault tolerance is satisfactory in the semiphysical experiment. The effectiveness of the method is verified once again in the platform. It also demonstrates that the fault diagnosis method based on complex network consistent matrix and the tolerance design of improved weighted mean feedback have good engineering applicability.


## 6. Conclusion

A fault diagnosis and fault-tolerant control method based on complex network synchronization has been presented to mitigate sensor fault problems in multimotor synchronous control. Based on the concept of distance, inspired by the idea of the coupling matrix in complex network synchronization, a consistent matrix is devised, which can reflect the similarity of different sensor output data. From the online correlation of elements and features in the matrix, the sensor fault can be detected, located, and estimated by the element value judgments in the matrix. Based on the fault diagnosis information, a fault-tolerant mechanism has been designed by introducing an improved weighted mean feedback to achieve effective isolation of the faulty sensor. Compared with the existing theories and techniques, the proposed method has several advantages, such as simple principles, small calculation amount, no need to change the topology structure of the original network, easy engineering realization, and suitability for single and multiple failure.From simulations and experiments, in order to implement satisfactory synchronous control, by considering load interference, noise, and uncertainty, a strong robustness control algorithm is necessary when the sensor is normal. In the meantime, the proposed method can be superimposed on the system as a further synchronous output guarantee. The reason is that, due to interference and other factors, when the synchronization error is large at a certain time, the method has the function of automatically rejecting the deviation from the larger output, and adopting other closer output averages as the feedback function, which makes the mean feedback closer to the true value. Therefore, this method can achieve better performance than conventional mean feedback, that is, it can realize fault tolerance when there is a fault and improve synchronization accuracy when there is no fault.In our future work, the focus will be on the robust and adaptive threshold selection study for multisensor fault situation.



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*Source: 1426193-2020-04-21.xml*