Health Management Decision of Sensor System Based on Health Reliability Degree and Grey Group Decision-Making

Metal Oxide Semiconductor (MOS) gas sensor has been widely used in sensor systems for the advantages of fast response, high sensitivity, low cost, and so on. But, limited to the properties of materials, the phenomenon, such as aging, poisoning, and damage of the gas sensitive material will affect the measurement quality of MOS gas sensor array. To ensure the stability of the system, a health management decision strategy for the prognostics and health management (PHM) of a sensor system that is based on health reliability degree (HRD) and grey group decision-making (GGD) is proposed in this paper. The health management decision-making model is presented to choose the best health management strategy. Specially, GGD is utilized to provide health management suggestions for the sensor system. To evaluate the status of the sensor system, a joint HRD-GGD framework is declared as the health management decision-making. In this method, HRD of sensor system is obtained by fusing the output data of each sensor. The optimal decision-making recommendations for health management of the system is proposed by combining historical health reliability degree, maintenance probability, and overhaul rate. Experimental results on four different kinds of health levels demonstrate that the HRD-GGD method outperforms other methods in decision-making accuracy of sensor system. Particularly, the proposed HRD-GGD decision-making method achieves the best decision accuracy of 98.25%.


Introduction
Sensor systems are extensively used in many fields, such as industry, manufacturing, aviation, and aerospace. Metal Oxide Semiconductor (MOS) gas sensor has become the most common gas sensor in sensor system at present because it has the advantages of fast response to target gas, high sensitivity, simple structure, easy to operation, low cost, and so on. Limited to the properties of metal oxide gas sensitive materials, the phenomenon such as aging, poisoning and damage of the gas sensitive material will affect the measurement quality of MOS gas sensor array. As a result, the trained pattern recognition method greatly degrades the performance of odor detection and analysis to target gases [1]. The influence of the work state and measurement quality of MOS gas sensor array to the performance of sensor system cannot be ignored.
At present, the following three ways are used to improve the fallen performance of the odor detection and analysis for sensor system that is caused by the decrease of reliability of the measurement value of the MOS gas sensor array. In recent years, group decision-making technology has rapidly developed [44][45][46][47]. The main research content of group decision-making is making effective decisions when multiple decision makers make decisions simultaneously. The main problem that must be solved is how to aggregate the decision information of different experts with different preferences to obtain consistent decision results. By fusing the decision objective of each expert, the accuracy of the system can be improved. However, group decision-making technology research is undeveloped. The methods of correctly obtaining decision information, including property value, property weight, and decision maker weight information have not been established.
In this paper, a method for health management and maintenance decision based on health reliability degree and grey group decision-making (HRD-GGD) is proposed. In this method, HRD of sensor system is obtained by fusing the output data of each sensor. The optimal decision-making recommendations for health management of the system is proposed by combining historical health reliability degree, maintenance probability, and overhaul rate. The HRD-GGD is proposed to realize the maintenance of the sensor system by comprehensively considering the decision results of multiple expert sets. Not only can the system give out the system state, but also provide the maintenance suggestion for each failure mode after the system working and give the confidence degree of each maintenance proposal.
The rest of this paper is organized as follows. In Section 2, the framework of health management and maintenance decision and the corresponding methods are presented. In Section 3, the experimental setup and analytical discussion are introduced. In Section 4, two situations are presented and 400 different health status level samples are analyzed to give the results of health management decision. Finally, the conclusion is accounted in Section 5. In recent years, group decision-making technology has rapidly developed [44][45][46][47]. The main research content of group decision-making is making effective decisions when multiple decision makers make decisions simultaneously. The main problem that must be solved is how to aggregate the decision information of different experts with different preferences to obtain consistent decision results. By fusing the decision objective of each expert, the accuracy of the system can be improved. However, group decision-making technology research is undeveloped. The methods of correctly obtaining decision information, including property value, property weight, and decision maker weight information have not been established.
In this paper, a method for health management and maintenance decision based on health reliability degree and grey group decision-making (HRD-GGD) is proposed. In this method, HRD of sensor system is obtained by fusing the output data of each sensor. The optimal decision-making recommendations for health management of the system is proposed by combining historical health reliability degree, maintenance probability, and overhaul rate. The HRD-GGD is proposed to realize the maintenance of the sensor system by comprehensively considering the decision results of multiple expert sets. Not only can the system give out the system state, but also provide the maintenance suggestion for each failure mode after the system working and give the confidence degree of each maintenance proposal.
The rest of this paper is organized as follows. In Section 2, the framework of health management and maintenance decision and the corresponding methods are presented. In Section 3, the experimental setup and analytical discussion are introduced. In Section 4, two situations are presented and 400 different health status level samples are analyzed to give the results of health management decision. Finally, the conclusion is accounted in Section 5.

Implementation Framework of Health Management and Maintenance Decision
The main purpose of the health management decision is to obtain the working state of sensor system quantitatively and to provide the maintenance decision for the system at this status. In order to evaluate the state of the sensor system, the historical failure information, historical maintenance records, and trends of historical health status for the system is used in order to model the health management mode. As the state of the system is clearer, it is easy to reduce the proportion of unscheduled maintenance in the maintenance plan and change the unscheduled maintenance to predictive maintenance (scheduled maintenance).
The establishment of framework is the core of health management decision theory. The health management suggestion is dynamically obtained by collecting fault information, health status, failure prediction conclusion, and historical maintenance situation. The framework is shown in Figure 2. The system input vector is composed of three parameters: historical health trend, maintenance probability, and overhaul rate. The historical health trend indicates the working state of the system during the last period of time. The parameter is acquired by fusing the historical HRD during this period. Maintenance probability is obtained from historical maintenance records. The value is equal to history maintenance times/total test times. The more frequent maintenance of the system, the greater the probability of failure. The value of maintenance probability is larger at this condition. Overhaul rate is the parameter of unpredictable maintenance task. The value is equal to the next inspection time/overhaul cycle. The longer the overhaul time, the greater the uncertainty of the system. The system is inclined to failure in this way.

Implementation Framework of Health Management and Maintenance Decision
The main purpose of the health management decision is to obtain the working state of sensor system quantitatively and to provide the maintenance decision for the system at this status. In order to evaluate the state of the sensor system, the historical failure information, historical maintenance records, and trends of historical health status for the system is used in order to model the health management mode. As the state of the system is clearer, it is easy to reduce the proportion of unscheduled maintenance in the maintenance plan and change the unscheduled maintenance to predictive maintenance (scheduled maintenance).
The establishment of framework is the core of health management decision theory. The health management suggestion is dynamically obtained by collecting fault information, health status, failure prediction conclusion, and historical maintenance situation. The framework is shown in Figure 2. The system input vector is composed of three parameters: historical health trend, maintenance probability, and overhaul rate. The historical health trend indicates the working state of the system during the last period of time. The parameter is acquired by fusing the historical HRD during this period. Maintenance probability is obtained from historical maintenance records. The value is equal to history maintenance times/total test times. The more frequent maintenance of the system, the greater the probability of failure. The value of maintenance probability is larger at this condition. Overhaul rate is the parameter of unpredictable maintenance task. The value is equal to the next inspection time/overhaul cycle. The longer the overhaul time, the greater the uncertainty of the system. The system is inclined to failure in this way.  There are three experts in the expert set. The experts can be changed when facing different problems. In the system, three algorithms are used as the experts, namely D-S evidence theory, Bayes theory, and fuzzy set theory. Three experts give their suggestion, respectively, according to the above three parameters. It can be found from the experiment that these three methods have their limitations, respectively, which will be discussed in Section 4.
The second part is the group decision-making. This part is responsible for data fusion of the decisions of the expert set to obtain the final decision result. The decision information is recorded as part of the next decision. The decision information is recorded and used as the basis for the next decision. The solution set of grey group decision-making is 1   There are three experts in the expert set. The experts can be changed when facing different problems. In the system, three algorithms are used as the experts, namely D-S evidence theory, Bayes theory, and fuzzy set theory. Three experts give their suggestion, respectively, according to the above three parameters. It can be found from the experiment that these three methods have their limitations, respectively, which will be discussed in Section 4.
The second part is the group decision-making. This part is responsible for data fusion of the decisions of the expert set to obtain the final decision result. The decision information is recorded as part of the next decision. The decision information is recorded and used as the basis for the next decision. The solution set of grey group decision-making is {A 1 , A 2 , A 3 , A 4 }, the corresponding decision frameworks are A 1 {no maintenance}, A 2 {preventive maintenance}, A 3 {corrective maintenance}, and A 4 {immediate maintenance}. The decision expert set is {e 1 , e 2 , e 3 }, which represent three experts, respectively. The decision index set is {u 1 , u 2 , u 3 }. The corresponding evidences are u 1 {historical health reliability degree}, u 2 {unpredictable maintenance task}, and u 3 {historical maintenance record}. The evidence property weight vector for three evidences is ω = {ω 1 , ω 2 , ω 3 }. The size of decision framework is four. The decision framework and its corresponding health status levels and maintenance levels are shown in Table 1.
The state description and corresponding maintenance suggestion are shown, as follows: 1.
Health: The whole system is very healthy. All of the sensors are also healthy. Their measurements are close to the expected value. There is no need to repair the system.

2.
Subhealth: The system is working at subhealthy status. The output of the system is within a normal range. All of the parameters may fluctuate near their expected value. It is essential to execute preventive maintenance regularly. Failure detection and failure isolation methods should be used in this situation.

3.
Failure Edge: The system is nearly failure. Their actual measurements have deviated from the expected value, but they have not deviated completely. In this status some sensors may be faulty, but the system can work effectively when fault recovery is performed. Corrective maintenance is needed after experiment [25]. Failure recovery method will be applied in this status to improve the work status sometimes. 4. Failure: The system is failure. Most of sensors are failure. The actual output has completely deviated from its expected results. Immediate repaired the failure components or replacement failure components immediately may be the best choice.
Maintenance decision method can provide the maintenance suggestion for each failure mode and give the confidence degree of each maintenance proposal. HRD is a novel conception to define a quantitative health level. HRD represents the health level of whole system. The HRD of the system is fused by the health level of all the sensitive elements in the system. The value ranges from 0 to 1. When the value is 0, the system works at a severe failure state. When the value is 1, the system works at 100% healthy state. The larger the HRD, the higher the health level. The relationship between HRD and health level is defined as Table 2. When to evaluate the health status of the system, the four health status levels, healthy status (HS), subhealthy status (SHS), failure edge status (FES), and failure status (FS). The specific values vary according to different application objects [26]. Table 2. Health status level.

Solution Set
Range of Health Reliability Degree Health Status Level HRD is fused of four belonging relationship degree (brd) of sensor system by applying grey theory. The four parameters brd are the keys to computation HRD. The values can be expressed in a simplified way, as shown in Figure 3. If the brd is equal to 1, then the current working status is completely belonged to its corresponding status completely. If the brd is equal to 0, then the current status is completely not belonged to its corresponding status. The brd is changed with the fluctuation of HRD. In order to map the relationship between HRD and brd, Relevance Vector Machine (RVM) is used to fuse four brd to HRD.

Solution Set Range of Health Reliability Degree Health Status Level
The relationship of brds and output parameters are shown in Figure 4. In summary, the whitening function of four grey sets are obtained by Equations (1)-(4). The relationship of brds and output parameters are shown in Figure 4. In summary, the whitening function of four grey sets are obtained by Equations (1)-(4).
For the all of the components in sensor system, the grey sample evaluation (GSE) matrix at the single time point can be expressed as GSE j = (gse ijk ) m×n (i = 1, 2, · · · , m; k = 1, 2, · · · , n), which is shown in (5).
where j represents the time point, S 1 indicates all the elements in sensor system. I k is the evaluating criterion. For the all of the components in sensor system, the grey sample evaluation (GSE) matrix at the single time point can be expressed as where j represents the time point, S1 indicates all the elements in sensor system. Ik is the evaluating criterion.
The decision weight vector of different elements in sensor system is obtained by using information entropy method. The probabilistic proportion of kth assessment criterion of ith elements is shown in (6).
Then compute the information the information entropy of the ith elements by (7).
The weight vector After obtaining the decision weight, the comprehensive grey assessment values (CGAV) are calculated by (9).
where [ ] The decision weight vector of different elements in sensor system is obtained by using information entropy method. The probabilistic proportion of kth assessment criterion of ith elements is shown in (6).
Then compute the information the information entropy of the ith elements by (7).
The weight vector W j = w 1j , w 2j , · · · , u mj is determined by After obtaining the decision weight, the comprehensive grey assessment values (CGAV) are calculated by (9).
where CGAV = [ brd SH brd SHS brd FES brd FS ]. The flowchart of the HRD methodology is shown in Figure 5 and the detail steps are shown in Table 3. The correlation among multiple parameters has been fully considered for the weight distribution of different sensors. The flowchart of the HRD methodology is shown in Figure 5 and the detail steps are shown in Table 3. The correlation among multiple parameters has been fully considered for the weight distribution of different sensors.

Input:
Output of the sensor system Output: Health Reliability Degree Procedure: Step 1: Establish the grey evaluating criterions, which is shown as (HS, SHS, FES, FS).
Step 2: Determine the whitening function of the grey model according to Equations (1)-(4).
Step 3: Compute decision weights by using information entropy method.
Step 5: Calculate the CGAV under evaluating criterion sets.

Grey Risk Decision-Making
The scheme-set of grey risk group decision-making problem is (A 1 , A 2 , . . . , A n ), the decision indicator set is (u 1 , u 2 , . . . , u m ), and the decision group set is (e 1 , e 2 , . . . , e q ) (q ≥ 2), where e s represents the s-th decision maker. For each decision indicator u j , there are l possible states θ = {θ 1 , θ 2 , . . . , θ l }. The probability of the state θ t occurring is p s tj (1 ≤ t ≤ l) for the decision maker e s under the decision indicator u j , which fits (0 ≤ p s tj ≤ l), [48]. The expert decision-making is divided into f -levels. The grey expert attribute set is denoted as For the decision expert attribute set M of level f, the decision expert fuzzy attribute value is m s = k(1 ≤ k ≤ f ). Define the deviation coefficient a s , which indicates the difference between m s and the s-th expert actual importance of the expert and a s ∈ [−0.5, 0.5].
The index ω = (ω 1 , ω 2 , . . . , ω m ) weight is obtained while using the reciprocal of entropy weight method to increase the experts weight with higher accuracy. If the information entropy index is smaller, the more information it provides. The greater the role that it plays in the comprehensive evaluation, the greater the weight of the index. In order to reduce the effluence, the reciprocal was used in this way. The weight of the decision-maker s is where p s tj is the input data. Deviation factor is where a s ∈ [−0.5, 0.5]. Let H s = minH s , a s = 0.5. If H s = maxH s , a s = −0.5. Therefore, the decision-making expert attributes after recuperation are m s = m s + a s . The expert weight is obtained, as follows: As H s increases, more right information included in the decision maker s increases the effectiveness and increases the weight. Thus, it is more realistic to correct the importance of the decision maker using entropy to obtain the decision maker weights.

Grey Group Decision Model for Decision-Making
The grey group decision-making theory is used to make the effective choice of the three decision algorithms and the final comprehensive decision is made by fusing the three types of algorithms. The specific process is shown in Figure 6. The comprehensive step of the grey group decision-making theory is shown in Table 4. In summary, the maintenance decision results and the comprehensive confidence of the system failure modes can be obtained while using the grey group decision-making method, and the dynamic maintenance plan can be formulated on this basis. Table 4. The comprehensive decision-making step of grey group decision-making theory.

Input:
Historical Health Reliability Degree (HHRD): The parameter is composed of the last n HRDs. Maintenance Probability (MP): Maintenance probability is equal to history maintenance times/total test times. Overhaul Rate (OR): Overhaul rate is equal to the next inspection time/overhaul cycle. Output: Decision Result: The parameter is the level of the maintenance decision-making. The size of the framework is four: {no maintenance, preventive maintenance, corrective maintenance, immediate maintenance} Confidence: The parameter is output vector of the maintenance decision-making confidence.

Procedure:
Step 1: Calculate the interval grey numbers of each decision result for each evidence in the decision framework under each decision method. The interval grey number is expressed as a s ij (⊗), a s ij (⊗) ∈ [a s ij , a s ij ]. a s ij represents the grey number lower limit and a s ij is the grey number upper limit (i = 1, 2, 3, 4, j = 1, 2, 3, s = 1, 2, 3).
Step 2: According to the upper-lower limit [a s ij , a s ij ] in the interval grey number a s ij (⊗) in Step 1, establish the comprehensive decision matrix (CDM) of each decision method as shown in Table 5.
Step 3: By utilizing the interval grey number weakening transformation, the decision matrix of three decision methods is initialized and transformed to obtain the standardized decision matrix.
Step 4: Calculate the weight of each decision method. First, determine the effect vector of the three decision methods for all decision frameworks according to the standardized decision matrix in Step 3. The matrix elements are the effect vectors of the three decision methods for each decision result in the decision framework. Then, according to the interval grey number vector distance formula and the weight of each evidence attribute ω j (j = 1, 2, 3), the space projector distance of each effect vector is calculated. Finally, the ratio between the vector distance of a decision method and the sum of effect vector distance for the other decision methods is the weight coefficient λ s (s = 1, 2, 3, 4) of the decision method.
Step 5: Calculate the comprehensive decision results. According to the maintenance decision result of the experts, the confidence of each decision result corresponding to the four decision frames can be obtained. Finally, according to the weight coefficients of each decision method that were obtained in Step 4, the final decision result is obtained by applying weighted averaging to the confidence.

The Process of Health Management Decision Method Based on Grey Group Decision
In order to improve the accuracy of the evaluation. The health management decision method combines grey group decision making and HRD theory to implement health management decision to sensor system. The detail steps of HRD-GGD are shown, as follows.
Step 1: To obtain the system measurement point parameters of the sensor or network interface, the data acquisition device is used to collect data.
Step 2: Pre-process the parameters collected by each measuring point in the system. Pre-processing includes abnormal value elimination, filtering noise reduction, calculating average value, and 3σ standard deviation.
Step 3: Failure detection, isolation, and diagnostic algorithms should be applied to determine the location of and the type of failure [21,22].
Step 4: If there is no failure in the array, give out the best estimation value of each sensor, according to the correlation of sensor array. If failure occurs in the sensor array, the best estimation value of failure sensor can be obtained, according to the normal data and the value of normal sensor can be got from the best estimation value of failure sensor and the other normal sensors.
Step 5: Establish the health reliability degree evaluation mode. The results of HRD are used as the inputs of the historical health status trend. The calculation steps of HRD are thoroughly introduced in Section 2.2.
Step 6: Establish a health management decision model based on grey group decision-making theory. Make maintenance decision on various failure modes and give out the corresponding maintenance suggestion.
Firstly, obtain the evidences for each expert. The number of evidence is three, expressed as u 1 , u 2 , u 3 , known, as follows: u 1 : Analysis of health reliability degree and historical health reliability degree; u 2 : Historical failure information and corresponding maintenance records; and, u 3 : Prepared system maintenance program. Evidence u 1 is fused according to three decision methods. Evidence u 2 and u 3 is obtained by using the whitening function as shown in Figure 7. brd is grey parameters. HS represents Health Status. SHS represents Subhealth Status. FES represents Failure Edge Status and FS represents Failure Status Level.
The size of the decision framework is 4. The framework is expressed with a 4-bit binary number: A 1 : no maintenance, 0001, A 2 : preventive maintenance, 0010, A 3 : corrective maintenance, 0100, and A 4 : immediate maintenance, 1000. u3: Prepared system maintenance program. Evidence u1 is fused according to three decision methods. Evidence u2 and u3 is obtained by using the whitening function as shown in Figure 7. brd is grey parameters. HS represents Health Status. SHS represents Subhealth Status. FES represents Failure Edge Status and FS represents Failure Status Level.
The size of the decision framework is 4. The framework is expressed with a 4-bit binary number: A1: no maintenance, 0001, A2: preventive maintenance, 0010, A3: corrective maintenance, 0100, and A4: immediate maintenance, 1000.  Utilizing the historical experiment data in Step 4. Analysis and prediction of the trend of the historical health parameters is based on the set of data. Determine the weight of each evidence attribute ω = (ω 1 , ω 2 , ω 3 ) based on historical failure information and maintenance records combined with the prepared maintenance program. ω 2 indicates that maintenance probability is equal to history maintenance times/total test times. ω 3 is the overhaul rate, which is equal to the next inspection time/overhaul cycle. ω 1 = 1 − ω 2 − ω 3 .
Secondly, to process the input data and turn the data to grey numbers. Calculate the confidence intervals of each evidence with a significance level of 0.05.
Thirdly, establish the comprehensive decision matrix (CDM) of each decision method. The CDM of the three experts is calculated for each failure model. Then, the decision matrix is standardized to calculate the standardization comprehensive decision matrix. Finally, the final comprehensive decision results are obtained. The matrix includes upper bound evaluation matrices, lower bound evaluation matrices, and whitening evaluation matrices. The upper bound evaluation matrices are CDM (1+) , CDM (2+) , and CDM (3+) . The lower bound evaluation matrices are CDM (1−) , CDM (2−) , and CDM (3−) . The whitening evaluation matrices are CDM (1) , CDM (2) , and CDM (3) . The upper bound evaluation matrices represent the maximum confidence of three evidences in the system. The lower bound evaluation matrices represent the minimum confidence of three evidences in the system. Whitening evaluation matrixes represent the confidence of three evidences for whitening degree in the system. The matrices are shown as (14): where k represents the expert (k = 1, e 1 ; k = 2, e 2 ; k = 3, e 3 ). σ ij represents the confidence of each evidence. i is the health level of the decision framework (i = 1, A 1 ; i = 2, A 2 ; i = 3, A 3 ; i = 4, A 4 ). j is the evidence (j = 1, historical health trends; j = 2, maintenance probability; j = 3, overhaul rate). The steps to calculate the health status level using CDM are shown in Table 6.

Health Status Level
Step 1: to calculate the health level of the decision framework and the confidence of every evidence.
Step 2: In grey group decision, the weight vector of each evidence attribute is ω = (ω 1 , ω 2 , ω 3 ) Step 3: calculate the decision confidence for the expert set by confidence i = ω·CDM.
Step 5: calculate the confidence of the final health status level by confidence = Finally, ranking all of the alternative health status levels in accordance with the confidence and choosing the optimal health management suggestion with largest confidence.

Experimental Setup and Analytical Discussion
The detailed HRD-GGD process is given in Section 2.4. In this part, the problem of health management decision for an atmosphere pollution gas sensor system is taken as an example to verify the HRD-GGD method.

Sensor System Experimental System
The sensor system used for testing atmosphere pollution gas was mainly composed of gas source, MFCs, gas chamber, sensor array, heater driver circuit, signal conditional circuit, data acquisition circuit, power supply, and laptop PC. The sensor array that consisted of four different types of gas sensors (CO, NO 2 , O 3 , SO 2 ) and temperature, humidity, and pressure sensors, was fixed in the gas chamber. The number of each type gas sensor is three. The gas chamber temperature was maintained at 30 • C by constant temperature control. The structure of sensor system is shown in Figure 8 and the physical picture of sensor system is shown in Figure 9. The normal working ranges and units for the 15 sensors (three CO sensors, three NO 2 sensors, three O 3 sensors, three SO 2 sensors, one temperature, one humidity, and one pressure sensor) are given in the Table 7. The health management decision platform is applied to the system. In the experiment, QT is used as the experimental software platform combined with SQL server software to realize database storage function. The HRD-GGD algorithm is implemented using Visual Studio.

Experiment Data
The data of each sensor are the voltage values collected by the 15 sensors in Table 8. Both the test and training samples include normal samples and fault samples. The normal samples are historical experimental data, i.e., system history experimental samples. The major frequent failure part in the sensor system is the sensor array. So, the failure type is mainly aimed at sensor array and heater drive circuit. The system HRD is the HRD of sensor array. The fault samples are analysed using fault simulation software, according to the fault modes. Figure 10 illustrates the sensor response process when exposed to 50 ppm CO in experiment. Each set of data includes 15 measurement points, with a sampling time of 1 s. The experiment involved 2 min for sensor to response completely and 2 min for the sensor to recover. When the system works for a long time, the performance of the sensor system will decrease with the increasing of running time. The historical HRD of the sensor system in 200 h is shown in Figure 11. F1-F5 are the failure of sensors, F6 is the failure of heater circuit. Because of the correlation among components, the output tends to be abnormal when the system fails. F1 to F6 are both sensors or circuit faults for single sensor. Every sensor fault can be diagnosed as a kind of failure. Due to the same form of expression and different location, it can be classified as a kind of failure. When multiple failures occur, which is to say that different sensors have different failures at the same time. This situation can be understood as multiple fault superposition, not as new failure. 200 sets of historical The system is used in the laboratory. The acquisition device utilized USB-bus cards (USB-2089, Art Technology Development Co. Ltd., Beijing, China) with 16 analog inputs at up to 400 KHz and a 14-bit A/D conversion accuracy. The sampling period is once per second. The failure data is generated by the failure simulating software. By analysis of the feature of historical failure data, the failure forms of different failures are summarized in the software. The maintenance suggestions are given according to practical experience. The failure information and maintenance suggestions are given in Table 7.

Experiment Data
The data of each sensor are the voltage values collected by the 15 sensors in Table 8. Both the test and training samples include normal samples and fault samples. The normal samples are historical experimental data, i.e., system history experimental samples. The major frequent failure part in the sensor system is the sensor array. So, the failure type is mainly aimed at sensor array and heater drive circuit. The system HRD is the HRD of sensor array. The fault samples are analysed using fault simulation software, according to the fault modes. Figure 10 illustrates the sensor response process when exposed to 50 ppm CO in experiment. Each set of data includes 15 measurement points, with a sampling time of 1 s. The experiment involved 2 min for sensor to response completely and 2 min for the sensor to recover. When the system works for a long time, the performance of the sensor system will decrease with the increasing of running time. The historical HRD of the sensor system in 200 h is shown in Figure 11. F1-F5 are the failure of sensors, F6 is the failure of heater circuit. Because of the correlation among components, the output tends to be abnormal when the system fails. F1 to F6 are both sensors or circuit faults for single sensor. Every sensor fault can be diagnosed as a kind of failure. Due to the same form of expression and different location, it can be classified as a kind of failure. When multiple failures occur, which is to say that different sensors have different failures at the same time. This situation can be understood as multiple fault superposition, not as new failure. 200 sets of historical data are used for obtaining the best work state. 400 groups data of different health status level (each type of health status level contains 100 groups data) are used for testing.

Sensor
Range Unit Sensor Range Unit

Results and Analysis
The health management decision of sensor system means that the maintenance suggestion is implemented in a global way and it refers to all of the sensors and components. In this section, two situations that represent different health levels are introduced to interpret the proposed strategy.

Situation 1: all the sensor and components are fault free
The collected data is processed using the grey data fusion method to obtain the historical health

Results and Analysis
The health management decision of sensor system means that the maintenance suggestion is implemented in a global way and it refers to all of the sensors and components. In this section, two situations that represent different health levels are introduced to interpret the proposed strategy.
Situation 1: all the sensor and components are fault free Figure 11. The historical HRD of the sensor system.

Results and Analysis
The health management decision of sensor system means that the maintenance suggestion is implemented in a global way and it refers to all of the sensors and components. In this section, two situations that represent different health levels are introduced to interpret the proposed strategy.

Situation 1: all the sensor and components are fault free
The collected data is processed using the grey data fusion method to obtain the historical health reliability degree, the details are shown in Section 2.2. The last seven health reliability degrees are used as the reference historical health parameter. The analysis and prediction of the historical health parameters trend are performed based on a data set. Historical failure information, maintenance records, and the established maintenance program are also considered.
Consider a random experiment as an example. The size of decision framework is 4. The framework is expressed as: {no maintenance}: A 1 , {preventive maintenance}: A 2 , {corrective maintenance}: A 3 , {immediate maintenance}: A 4 . Regard a series of historical health reliability degree as evidence. First, obtain the series of experimental data and the previous seven series. The historical health parameters are shown in Table 9 and are fused by three experts' decision methods. Deal with data by grey processing to obtain the grey interval and calculate separately. Deal with data by grey processing to obtain the grey interval and calculate the confidence of each level separately. The decision result of evidence 1: historical health trends are shown as (15). According to detection conditions, the historical maintenance number is selected as five and the total experiment time is 100 times. The next inspection time is set as 300 days and the overhaul cycle is 365 days. Attribute weights are determined as ω = (ω 1 , ω 2 , ω 3 ) = (0.7719, 0.05, 0.1781). ω 2 = 0.05 is the maintenance probability, which is equal to historical maintenance times/total test times. ω 3 = 0.1781 is the overhaul rate, which is equal to the next inspection time/overhaul cycle. In the example, the historical maintenance times are five and total test times are 100. The next inspection time is 300 days and the overhaul cycle are 365 days. The decision matrix that was established by the fusion of each decision method is shown from Tables 10-12. The attribute weights of these three decision methods are (0.4158, 0.3467, 0.2375), which is attained by (10)~ (12). The whitening degree is the mathematical expression of whitening rules in the grey set under the existing information. Whitening degree is obtained by whitening function, which is shown in Figure 7. The comprehensive decision matrixes that were obtained using the three decision methods by using (14).  Since the results of the matrix are floating from 0 to 1, normalization is not required. The final grey group decision-making result can be obtained by directly fusing with the weight. For another decision-making method, such as D-S evidence theory, Bayes theory, and fuzzy set theory are used as the off-the-shelf maintenance decision-making method. The decision results are shown in Figure 12. According to descending order of the maintenance confidence, the ranks and maintenance suggestions that are based on grey group decision and another three comparative decision methods are shown in Table 13, the rank of four methods are shown as A 1 > A 2 > A 3 > A 4 , the final maintenance suggestion of four methods are all A 1 : no maintenance, which is suitable to the status description. In order to verify the evaluation ability of maintenance decision method for health status, 100 groups of health status data at different times were used to test the accuracy of evaluation. The confidence of the 100 groups health state data to four health status level are shown in Figure 13. False alarm occurred when using Bayes theory. The accuracy for the four methods is shown in Table 14. Grey group decision ignores the disadvantage of Bayes theory for this situation to improve the decision accuracy.  In order to verify the evaluation ability of maintenance decision method for health status, 100 groups of health status data at different times were used to test the accuracy of evaluation. The confidence of the 100 groups health state data to four health status level are shown in Figure 13. False alarm occurred when using Bayes theory. The accuracy for the four methods is shown in Table 14. Grey group decision ignores the disadvantage of Bayes theory for this situation to improve the decision accuracy. Table 13. The rank and maintenance suggestion of Situation 1.

Rank Maintenance Suggestion
Grey Group Decision In order to verify the evaluation ability of maintenance decision method for health status, 100 groups of health status data at different times were used to test the accuracy of evaluation. The confidence of the 100 groups health state data to four health status level are shown in Figure 13. False alarm occurred when using Bayes theory. The accuracy for the four methods is shown in Table 14. Grey group decision ignores the disadvantage of Bayes theory for this situation to improve the decision accuracy.  Figure 13. The confidence of 100 groups health state data to different health status level. Figure 13. The confidence of 100 groups health state data to different health status level.

Situation 2: single sensor is failure
To verify the effectiveness of the grey group maintenance decision method, situation 2 is used to verify the versatility and correctness of the method. In situation 2, the data of each group are all kinds of failures simulated by fault simulation software. The attribute weights are ω = (ω 1 , ω 2 , ω 3 ) = (0.0932, 0.4, 0.5068). In this example, the historical maintenance times are 40 and the total test times are 100. The next inspection time is 250 days and the overhaul cycle is 365 days.
The decision results of four methods are shown in Figure 14. All of the decision results are immediate maintenance except for the fuzzy set. The ranks and maintenance suggestions based on grey group decision and other three comparative decision methods are shown in Table 15. To verify the effectiveness of the grey group maintenance decision method for single failure. 100 groups of failure status data are used for testing. The decision results are shown in Figure 15. The decision accuracies are shown in Table 14. The result of grey group decision is the same as Bayes theory and it is superior to the other algorithm. The result of D-S evidence theory is fluctuated and it is hard to distinguish the optimal maintenance result. The confidences of four health status level of fuzzy set theory are almost at the same level and it is difficult to realize the optimal maintenance result Grey group decision utilizes the advantage of Bayes theory for this situation to ignore the inaccuracy result of the other experts. To verify the effectiveness of the grey group maintenance decision method for single failure. 100 groups of failure status data are used for testing. The decision results are shown in Figure 15. The decision accuracies are shown in Table 14. The result of grey group decision is the same as Bayes theory and it is superior to the other algorithm. The result of D-S evidence theory is fluctuated and it is hard to distinguish the optimal maintenance result. The confidences of four health status level of fuzzy set theory are almost at the same level and it is difficult to realize the optimal maintenance result Grey group decision utilizes the advantage of Bayes theory for this situation to ignore the inaccuracy result of the other experts. The accuracy of maintenance decision-making which is verified by 400 groups different health status level samples (100 groups for each health status level) are shown in Table 14. It is not difficult to find out from the above test that D-S evidence theory has good detection results at health state. However, it cannot be detected effectively at the failure state because of the large range data fluctuation. Bayes theory has a good decision-making result for the failure status, but false alarm will appear at health state. The fuzzy set theory is very good in the health state. But, in the case of failure, the health statuses often appear with a similar confidence, and cannot be effectively evaluated. According to the group decision method, the grey group decision exerts the advantage of the D-S evidence theory and fuzzy set theory in the health state and it reduces the missing alarm rate. In the The accuracy of maintenance decision-making which is verified by 400 groups different health status level samples (100 groups for each health status level) are shown in Table 14. It is not difficult to find out from the above test that D-S evidence theory has good detection results at health state. However, it cannot be detected effectively at the failure state because of the large range data fluctuation.
Bayes theory has a good decision-making result for the failure status, but false alarm will appear at health state. The fuzzy set theory is very good in the health state. But, in the case of failure, the health statuses often appear with a similar confidence, and cannot be effectively evaluated. According to the group decision method, the grey group decision exerts the advantage of the D-S evidence theory and fuzzy set theory in the health state and it reduces the missing alarm rate. In the case of failure, it plays the advantage of the Bayes theory and it reduces the probability of false alarm.

Conclusions
In this paper, a method of health management decision strategy of a sensor system is proposed by utilizing HRD-GGD theory. Health reliability degree strategy is utilized to quantify system state and to provide support for decision making. The system can provide the maintenance suggestion after the system runs and give the confidence degree of each maintenance proposal. The experimental results show that this method can evaluate the system state effectively, and the accuracy rate of maintenance recommendation is 98.25%. The result proves that the accuracy is improved over 2% when compared with the other methods and the decision results are optimal under all health status levels.
In the future, we will investigate the remain life of sensor system and failure prediction by analysis of the historical trend of MOS sensor degradation in the sensor system.