# An Improved Lagrange Particle Swarm Optimization Algorithm and Its Application in Multiple Fault Diagnosis

Authors: Xiaofeng Lv; Deyun Zhou; Ling Ma; Yuyuan Zhang; Yongchuan Tang
Journal: Shock and Vibration
 (2020)

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## Abstract

The fault rate in equipment increases significantly along with the service life of the equipment, especially for multiple fault. Typically, the Bayesian theory is used to construct the model of faults, and intelligent algorithm is used to solve the model. Lagrangian relaxation algorithm can be adopted to solve multiple fault diagnosis models. But the mathematical derivation process may be complex, while the updating method for Lagrangian multiplier is limited and it may fall into a local optimal solution. The particle swarm optimization (PSO) algorithm is a global search algorithm. In this paper, an improved Lagrange-particle swarm optimization algorithm is proposed. The updating of the Lagrangian multipliers is with the PSO algorithm for global searching. The difference between the upper and lower bounds is proposed to construct the fitness function of PSO. The multiple fault diagnosis model can be solved by the improved Lagrange-particle swarm optimization algorithm. Experiment on a case study of sensor data-based multiple fault diagnosis verifies the effectiveness and robustness of the proposed method.


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## Body

## 1. Introduction

The problem of fault detection and diagnosis in complex equipment is a hot topic for theory and practical research [1–5]. On one hand, if the complex electromechanical system of existing weapons or equipment fails, it is difficult to address the problems solely with traditional diagnostic methods or the experience of maintenance personnel, because the system is composed of very complicated mechanical and electronic systems [6–8]. On the other hand, studies show that the available maintenance cost of complex equipment is a key issue and may be extremely high [9, 10]. Meanwhile, because of lack of effective detection and maintenance procedure, the maintenance cost of equipment is often 3–10 times bigger than the research cost. Therefore, it is of great importance to make effective fault diagnosis for weapon equipment, especially for complex electromechanical equipment. Existing studies [11–14] play a great effort in lowering the cost of maintenance and test, simplifying the test equipment, and improving the testability of equipment.Model-based fault diagnosis and data-driven fault diagnosis are commonly used fault diagnosis methods [15, 16], among which the dependency model of Data Sciences International (DSI) company is a typical approach [6, 17, 18]. The optimization algorithm can be used to solve the objective function under the correlation model of complex system. The correlation model is transformed into a multiple fault diagnosis model in [19]. Subsequently, the multiple fault diagnosis model is solved based on the Lagrange relaxation algorithm [20, 21]. Apart from being applied in multiple fault diagnosis, the Lagrange relaxation algorithm is also applied for other applications as partitioning problem [22], scheduling problem [23], the adjacent only quadratic minimum spanning tree problem [24], the supply chain network problem [25], and so on [26–28].Lagrangian relaxation algorithm can be adopted to solve multiple fault diagnosis models. Traditionally, the method of updating Lagrange multiplier is based on the subgradient method. But the mathematical derivation process in the algorithm is complex. In addition, the updating method for Lagrangian multiplier is limited, and the traditional Lagrangian relaxation algorithm may fall into a local optimal solution. In [29], the Lagrange relaxation algorithm is improved and applied to multiple fault diagnosis problem, which brings about a better result. In this paper, the particle swarm optimization (PSO) algorithm will be adopted to update the Lagrange multiplier. The PSO algorithm is a global search algorithm [30, 31]. As a result, the PSO algorithm has a good adaptability and simplicity in practical applications [32–35]. We adopt the PSO algorithm to update the Lagrange multiplier in this paper.In this paper, we propose an improved method for multiple fault diagnosis based on the Lagrangian relaxation algorithm and PSO algorithm. The updating of Lagrange multiplier will be designed as the optimization iteration process of particle swarm. A new method for updating of speed and location is proposed based on the nonnegative characteristic of Lagrange multiplier. The fitness function in the improved optimization algorithm is designed based on the absolute value of the difference between the upper and lower bounds. An experiment based on the system state diagnosis before a launch of Apollo spacecraft is adopted for verification of the proposed method.The rest of this paper is organized as follows. The multiple fault diagnosis model is introduced in Section2. In Section 3, the Lagrangian model is briefly introduced. Then, the improved Lagrange-particle swarm optimization algorithm is proposed in Section 4. Section 5 presents the simulation and verification experiment of the proposed method. Section 6 draws the conclusion of this paper.


## 2. Multiple Fault Diagnosis Model

The multiple fault diagnosis problem consists of the following elements. The fault source vector of the system can be denoted asF=f1,f2,⋯,fm, where fi represents the ith fault state of the system. The set of test points in the system is denoted as T=t1,t2,⋯,tn, where tj represents the jth test point. The fault source-test point dependency matrix is D=dij, where the element dij represents the dependency between the fault state fi and the test point tj. In addition, if the fault state and the test point are dependent on each other, then dij = 1; otherwise, dij = 0. The prior probability vector of the fault is P=p1,p2,⋯,pm, where pi represents the prior probability of the ith fault state.Basing on the aforementioned factors, the process of multiple fault diagnosis can be described as follows. Basing on the test results, a set of possible fault sourcesX⊆F can be calculated to maximize the posterior probability. The posterior probability can be converted to a prior probability with Bayesian formula [19]:(1)maxPXTp,Tf=maxPTp,TfXPXPTp,Tf,where Tp is the test set that has passed the test. Tf is the test set that has not passed the test. The vector x of m dimension is defined to represent the fault state of the component. When the fault state fi occurs, xi = 1; otherwise, xi = 0. PTp,Tf is a constant that is independent of X. Thus, we can get the following function:(2)maxPXTp,Tf=maxPTp,TfXPX=maxPTpXPTfXPX,where(3)PX=∏i=1mPsixi1−Psi1−xi,where Pi is the prior probability of the failure in component i, calculating the probability of the occurrence PX regarding the fault source set X. If the fault state fi happens, Pixi=Pi, 1−Pi1−xi=1, then, do multiplication of the prior probability regarding the fault state fi. Otherwise, the fault state fi does not happen; do multiplication of 1−Pi, which is the prior probability of the fault state fi that does not happen. The values of PTpX and PTfX are relative to the information in the matrix D [19]:(4)PTpX=∏tj∈TpPtj∈TpX=∏tj∈Tp∏i=1m1−dijxi,PTfX=∏tk∈TfPtk∈TfX=∏tk∈Tf1−∏i=1m1−dikxi,where tj and tk denote the test points j and k respectively.Obtaining the maximum value ofPTpXPTfX, since they can be only 0 or 1, thus, both PTpX and PTfX must be 1. The following four cases exist:(i)
If the test pointtj passes the test, for the faulty component xi = 1, the test tj must be independent with the component xi (dij = 0)(ii)
If the test pointtj passes the test, for the component without failure xi = 0, the test tj must be independent with the component xi (dij = 0 or dij = 1)(iii)
If the test pointtk fails to pass the test, for the component without failure xi = 0, the test tk must be independent with the component xi (dik = 0 or dik = 1)(iv)
If the test pointtk fails to pass the test, there is at least one faulty component xi = 1 which makes the test tk dependent on the fault component xi (dik = 1)According to the aforementioned analysis, a lemma can be summarized as follows [19].Lemma 1.
The maximum value ofPTpXPTfX can be calculated on the condition that the following conditions are satisfied:(1)
Iftj∈Tp and xi = 1, then dij = 0(2)
Iftk∈Tf, there is at least one faulty component (xi = 1) which makes the test tk dependent on the fault component xi, subsequently, dik = 1To fulfil the aforementioned condition 1, delete the fault set that has been confirmed not to happen from the fault setC and get the fault set C−. In other words, the value of the confirmed fault xi that will not happen is 0, and the value of others is still unknown. Subsequently, condition 1) can be converted into the following constraints: for tk∈Tf, Ax≥e, where A can be calculated as the transposed matrix of D in which the test column without failure has been deleted. The size of A is Tf×m. e is a unit vector of column.Taking the negative natural logarithm of formula(3) and ignoring its constant, the problem can be converted as follows [19]:(5)ZIP=minX∈C−∑ci∈C−bixis.t.Ax≥e,xi∈0,1,i=1,⋯,m,where bi=−lnpi/1−pi,i=1,⋯,m.


## 3. Lagrangian Relaxation Model

The Lagrangian relaxation algorithm can be adopted to solve the aforementioned multiple fault diagnosis problem. The original multiple fault diagnosis model is shown as follows [19]:(6)ZIP=minX∈S−∑Sj∈S−cjxjs.t.Ax≥e,xj∈0,1,j=1,⋯,m,where S− is the set of fault sources in which the fault from the set S that has been confirmed to never occur is removed. A is the matrix that is transposed from matrix D where the test column corresponding to the fault that did not occur has been removed. cj depends on the prior probability psj, and cj can be presented as follows:(7)cj=−lnpsj1−psj,j=1,⋯,m.Lagrangian relaxation algorithm is applied to the model. The improved optimization model is proposed as follows [19]:(8)ZLRλ=min∑j=1mljxj+∑i=1Tfλis.t.xj∈0,1,j=1,⋯,m,where lj=cj−∑i=1Tfλiaij, λi≥0,i=1,⋯,Tf is the Lagrangian multiplier. Subsequently, the Lagrangian dual model can be represented as follows:(9)ZLD=maxZLRλ.Finally, the fault sourcex can be obtained by solving the Lagrangian dual model ZLD.


## 4. Lagrange-Particle Swarm Optimization Algorithm

TheZLRλ gradually approaches its optimal solution in the original problem by improving the value of Lagrangian multiplier λ during the process of solving the Lagrangian dual model ZLD. The optimal solution of the problem can be calculated once the optimized value of Lagrangian multiplier λ is found out. In general, the method of updating multiplier λ in Lagrange relaxation algorithm is with the subgradient method, which means complex mathematical tool and theoretical analysis. Thus, the PSO algorithm is adopted to update the multiplier λ because it is not that complex and can solve the problem of optimization solution. Consequently, the PSO algorithm is combined with the Lagrangian relaxation model. In Lagrange-particle swarm optimization algorithm, the PSO algorithm is adopted to find the optimized value of Lagrangian multiplier λ. That is to say, the subgradient method is replaced by the PSO algorithm for calculating the Lagrangian multiplier λ.The problem of defining a fitness function is involved in PSO algorithm. Here, the absolute value of the difference between upper and lower boundsJ is chosen as the reference standard. The upper bound of the original multiple fault model (ZIP) is introduced as a parameter ZUP; it can be replaced by the value of a feasible solution. ZDW is a lower bound that is the nearest value to ZIP for the function ZLRλ. The absolute value of the difference between upper and lower bounds is J=ZUP−ZDW. The fitness function is F=1/1+KJ, where K is a scaling coefficient. If the particle λ reaches the optimal solution, then, the distance calculated by the particle between the upper bound ZUP and lower bound ZDW decreases. A smaller difference value between the upper and lower bounds means a bigger value of the fitness function, which shows the rationality of the designed fitness function F. In addition, the absolute value of the difference between upper and lower bounds J is a positive value; thus, the interval for the value of the fitness function F is [0, 1]. The normalization method is beneficial for the position updating of the particle λ and the searching for the global optimal particle. Consequently, the fitness function meets the requirement of the algorithm.The flow chart of the algorithm is shown in Figure1. It should be noted that the particle λ should be a nonnegative value for updating [19]. Consequently, the speed and position of the particle shall be limited during the updating process to slow down the speed of moving to the negative direction.Figure 1
The flow chart of the Lagrange-particle swarm optimization algorithm.DefiningZDQ as the optimal value for function ZLRλ for the current situation of λ. Above all, the Lagrange-particle swarm optimization algorithm can be summarized as follows:(i)
Step 1. Initialization of parameters.The initial population isλ0=λ10,λ20,⋯,λg0, the size of the population is g, the best position for initial particle is Pb0=λ0=λ10,λ20,⋯,λg0 and the initial velocity vector is v0=v10,v20,⋯,vg0. The initial velocity is required to be between vmax and vmin. The optimal fitness function value for each initial particle is set as Fb0=0,0,⋯,0, where the dimension is g. The difference of the absolute value between the minimum upper and lower bounds for each initial particle is Jb0=∞,∞,⋯,∞, where the dimension is g. The difference of the absolute value between the optimal upper and lower bounds for the overall initial situation is Jb1t=∞. Define ZUP0=∞,∞,⋯,∞ and ZDW0=−∞,−∞,⋯,−∞, where the dimension is g.(iii)
Step 2. Searching the global optimal particle.Step 2.1. Do the following five steps for each elementλit,i=1,2,⋯,g in the tth iteration of the particle denoted as λt=λ1t,λ2t,⋯,λmt.(1)
Solving equation (8) with λit. The optimal value calculated by equation (8) is ZDQ,it. Then, updating ZDQ,it=maxZDW,it,ZDQ,it.(2)
Constructing a feasible solution vectorsCit for the original problem in equation (6) with the following four steps:(a)
DefiningCit=jxj=1,j=1,⋯,m, where the vector x is the calculation result of equation (8) by using λit. The value of Cit is chosen as the subscript j of the element in x corresponding to xj=1.(b)
For the rowi uncovered in Cit (namely, ∑j=1maijxj=0), add the corresponding value of j to Cit regarding minjaij=1,cj<∞,j=1,⋯,m.(c)
Checking all the subscriptj satisfying j∈Cit. If Cit−j is still the feasible solution of ZIP, define Cit=Cit−j. This strategy is used for (1) removing unnecessary fault sources in the following application, (2) reducing the gap between ZUP,it, and ZDW,it, and (3) terminating the algorithm as soon as possible.(d)
Updating theZUP,it=minZUP,it,∑j∈Citcj.(3)
Calculating the absolute value of the difference between the upper and lower bounds denoted asJit=ZUP,it−ZDW,it.(4)
Calculating the value of the fitness function denoted asFit=1/1+KJit.(5)
Basing on the fitness value of the global optimal valueFb,it and the current generation’s value Fit to determine whether to update the optimal position of the particle Pb,it. If Fb,it<Fit, update the position as Pb,it=λit, the value of fitness function as Fb,it=Fit, and the difference between upper and lower bounds as Jb,it=Jit; otherwise, update no parameter.Step 2.2. Basing on the value of the fitness functionFbt=Fb,1t,Fb,2t,⋯,Fb,gt, selecting the global optimal particle λbt in the optimal position Pbt=Pb,1t,Pb,2t,⋯,Pb,gt among all the particles. In addition, calculating the corresponding difference between the upper and lower bounds as Jb1t, which is also the global smallest difference value between the upper and lower bounds.(iv)
Step 3. Updating the particles.Step 3.1. Updating the inertia weightwt=wmax−twmax−wmin/Nmax.Step 3.2. Updating the velocity and position of the particles. The velocity of theith particle in the d dimension for the t+1 times iteration is vi,dt+1=wtvi,dt+c1r1tPb_i,dt−λi,dt+c2r2tλb,dt−λi,dt. If the updated velocity is negative, defining v=−K1e−v+vmax to slow down the movement speed in the negative direction. This is because the Lagrangian multiplier is required to be a nonnegative value. The updated position is λi,dt+1=λi,dt+vi,dt+1. If λi,dt+1<0, setting λi,dt+1=0. Updating the time of the iteration as t=t+1.Step 3.3. Determining whether the maximum number of the iterationsNmax has been reached. If t>Nmax or the absolute value (Jb1t) of the difference between the upper and lower bounds obtained by calculating the global optimal particle λbt is less than the preset threshold, then, continue and turn to Step 4; else, turn to Step 2.(v)
Step 4. Solving equation (8) with the global optimal solution λbt. Outputting the fault source X = x1,⋯,xm.The proposed Lagrange-particle swarm updating algorithm inherits the advantages in PSO algorithm and Lagrangian relaxation model. On one hand, the objective function of the proposed algorithm integrates the complex constraint conditions, which makes it easier for solving the multiple fault diagnosis. On the other hand, the subgradient method is replaced by the PSO algorithm for calculating the Lagrangian multiplierλ, which simplifies the theoretical analysis and solving process. It should be noted that the proposed algorithm has more than one feasible solution, which shows priority over the original Lagrange relaxation algorithm. While using PSO to update the Lagrangian multiplier λ, there are many alternative particles except for the optimal ones. This contributes to more alternative fault diagnosis strategies in solving multiple fault diagnosis problem, which makes the proposed method more suitable for locating the multiple fault types in fault diagnosis.


## 5. Experiment and Discussion

To test the performance of Lagrange-particle swarm updating algorithm in multiple fault diagnosis problem, an example of system state diagnosis before the launch of Apollo spacecraft was adopted from [36, 37]. In the case study, 10 fault states and 15 sensor test points are known and the prior probabilities among all the fault states are assumed to be equal. The dependency matrix of the system is shown in Table 1.Table 1
The dependency matrix for fault status and test in the system of Apollo spacecraft.
Fault no.Test pointPrior probability12345678910111213141510001000101111000.120010100000110100.130000111011110010.140100011000000110.150101011111001100.160001100111011110.171001100000010110.181110011011100100.191001100000010110.1101111001000101100.1Establishing the corresponding Lagrangian dual model, the proposed Lagrange-particle swarm optimization algorithm is adopted to solve the model. The solving of the model is by programming in MATLAB R2014a. The parameters in the algorithm are chosen as follows. The maximum iteration numberNmax = 200, population size popsize = 50, acceleration factor c1=1.3 and c2=1.5, maximum of inertia weight wmax=1.2, minimum inertia of weight wmin=0.4, maximum of velocity vmax=2, minimum of velocity vmin=−2, proportional coefficient K=5 and K1=0.2, and the absolute value of the minimum difference between upper and lower bounds Jmin=0.01. Jmin affects the quality of solution and the calculating time significantly. If Jmin is too small, it may lead to a failure of the solution to be less than the threshold even after the solution reached the maximum number of iterations, which means a high but unnecessary requirement on the quality of solution. On the contrary, if Jmin is too big, the number of iterations may be too less, which may lead to a bad quality of solution. After many times of trial, Jmin is finally to be chosen as 0.01.Take the double-fault diagnosis problem as an example, the results are shown in Table2. The corresponding fault isolation rate is 91.11%. The proposed method has a right diagnosis result on almost all the double-fault types. As for the fault No. (4 and 6), the proposed method gives a solution as (4, 5, and 6), which includes all the potential fault elements 4, 5, and/or 6. In this case, the fault diagnosis with the proposed method eliminates all possible faults without the missed one. Similar situation also exists in the fault No. (5 and 6), (5 and 8), and (5 and 10) according to Table 2.Table 2
The results of double-fault diagnosis with Lagrange relaxation-particle swarm optimization algorithm.
Fault no.Sensor report no.Diagnosis results1, 23, 4, 5, 8, 10, 11, 12, 13, 141, 21, 34, 5, 6, 7, 8, 9, 10, 11, 12, 13, 151, 31, 42, 4, 6, 7, 8, 10, 11, 12, 13, 14, 151, 41, 52, 4, 6, 7, 8, 9, 10, 11, 12, 13, 141, 51, 64, 5, 8, 9, 10, 11, 12, 13, 14, 151, 61, 71, 4, 5, 8, 10, 11, 12, 13, 151, 71, 81, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 141, 81, 91, 4, 5, 8, 11, 12, 13, 14, 151, 91, 101, 2, 3, 4, 7, 8, 10, 11, 12, 13, 141, 102, 33, 5, 6, 7, 9, 10, 11, 12, 14, 152, 32, 42, 3, 5, 6, 7, 11, 12, 14, 152, 42, 52, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 142, 52, 63, 4, 5, 8, 9, 10, 11, 12, 13, 14, 152, 62, 71, 3, 4, 5, 8, 11, 12, 13, 14, 152, 72, 81, 2, 3, 5, 6, 7, 9, 10, 11, 12, 142, 82, 91, 3, 4, 5, 11, 12, 14, 152, 92, 101, 2, 3, 4, 5, 7, 11, 12, 13, 142, 103, 42, 5, 6, 7, 9, 10, 11, 12, 14, 153, 43, 52, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 153, 53, 64, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 153, 63, 71, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 153, 73, 81, 2, 3, 5, 6, 7, 9, 10, 11, 12, 14, 153, 83, 91, 4, 5, 6, 7, 9, 10, 11, 12, 14, 153, 93, 101, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 153, 104, 52, 4, 6, 7, 8, 9, 10, 13, 14, 154, 54, 62, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 154, 5, 64, 71, 2, 4, 5, 6, 7, 8, 11, 13, 14, 154, 74, 81, 2, 3, 6, 7, 9, 10, 11, 14, 154, 84, 91, 2, 4, 5, 6, 7, 12, 14, 154, 94, 101, 2, 3, 4, 6, 7, 11, 13, 14, 154, 105, 62, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 154, 5, 65, 71, 2, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 155, 75, 81, 2, 3, 4, 6, 7, 8, 9, 10, 11, 13, 145, 8, 105, 91, 2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 155, 95, 101, 2, 3, 4, 6, 7, 8, 9, 10, 11, 13, 145, 8, 106, 71, 4, 5, 8, 9, 10, 11, 12, 13, 14, 156, 76, 81, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 156, 86, 91, 4, 5, 8, 9, 10, 12, 13, 14, 156, 96, 101, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 156, 107, 81, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 157, 87, 91, 4, 5, 8, 11, 12, 13, 14, 157, 97, 101, 2, 3, 4, 5, 7, 8, 11, 13, 14, 157, 108, 91, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 158, 98, 101, 2, 3, 4, 6, 7, 9, 10, 11, 13, 148, 109, 101, 2, 3, 4, 5, 7, 11, 12, 13, 14, 159, 10Fault isolation rate91.11%Adopting the case study in [19] where Lagrange relaxation algorithm is proposed for the diagnosis of double-fault types, the failure states are not recognized with the fault No.: (1) 3 and 5, (2) 6 and 8, (3) 8 and 9, (4) 4 and 6, (5) 5 and 6, (6) 5 and 10, and (7) 5 and 8. The experiment results with the original Lagrange relaxation algorithm and the improved algorithm are shown in Table 3.Table 3
The experiment results with the Lagrange relaxation algorithm and the proposed method.
FaultSensor report no.Lagrange algorithmProposed method3, 52, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 151, 3, 43, 56, 81, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 152, 3, 4, 76, 88, 91, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 152, 3, 4, 98, 94, 62, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 154, 64, 5, 65, 62, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 154, 64, 5, 65, 101, 2, 3, 4, 6, 7, 8, 9, 10, 11, 13, 145, 85, 8, 105, 81, 2, 3, 4, 6, 7, 8, 9, 10, 11, 13, 145, 85, 8, 10According to Table3, the fault status exists in fault No. (3 and 5), and the diagnosis result with the original Lagrange relaxation algorithm is fault No. (1, 3 and 4), while the result with the proposed method is fault No. 3 and 5. The proposed method can successfully find the right fault types. In addition, if the fault status exists in fault No. (4 and 6), (5 and 6), the improved method can locate all the potential faults denoted as No. (4, 5 and 6), which can ensure that all potential fault statuses can be located during the fault diagnosis process. For the fault No. (3 and 5), (6 and 8), and (8 and 9), the proposed method gets accurate and right solution, while the method with only Lagrange relaxation algorithm gets wrong diagnosis results where some elements with no fault are also included in the fault diagnosis reports (e.g., element No. 1 and 2). As for the fault No. (4 and 6), (5 and 6), (5 and 8), and (5 and 10), the original Lagrange relaxation algorithm cannot distinguish the fault status with the same alarm sensor report numbers, while the proposed method reports the diagnosis results including all the fault elements without missing fault type or including normal element. The fault isolation rate of the improved algorithm for double-fault diagnosis reaches 91.11% which is higher than that of the literature [19] (where the fault isolation rate is 88.89%). In addition, the troubleshooting rate with the proposed method reaches 100% in this case. Above all, the proposed method has a better performance on the fault isolation rate and the troubleshooting rate which is helpful for solving the problem of multiple fault diagnosis in complex equipment.Simulation experiment result shows that the fault isolation rate of the proposed method is higher than that of the original Lagrange relaxation algorithm. The proposed method has a higher accuracy in fault diagnosis. This is because the Lagrangian multiplier in the original Lagrange relaxation algorithm has been improved. In the proposed method, (1) the PSO algorithm is adopted to update the Lagrangian multiplier, and (2) the fitness function is constructed by using the difference between upper and lower bounds. With PSO, the initial particle and its iterative updating process have the feature of a random process; consequently, a global search can be achieved and contribute to more choices for selecting particles.


## 6. Conclusions

In this paper, the Lagrange-particle swarm optimization algorithm is proposed to solve the multiple fault diagnosis model. Being different from the original subgradient ascending method, the Lagrangian multiplier in the original Lagrangian relaxation algorithm is updated by the PSO algorithm, which makes it easier for theoretical analysis and algorithm implementation. The fitness function is constructed based on the absolute value of the difference between the upper and lower bounds. The new fitness function can be a main adjusted index that is responsible for the effectiveness of the multiplier. By controlling the speed of updating speed to the negative direction, we can improve the search performance in the proposed approach. The effectiveness of the new method is verified according to the simulation experiment, and the superiority is also discussed in comparison with the original algorithm.In the following work, the reliability of the test can be taken into consideration. The multiple fault diagnosis model may include the occurrence of failure in sensors. Currently, it is assumed that the test of fault detection is reliable. If a fault occurs, the fault can be detected by the test safely and reliably. However, in practical system, the fault detection rate may not reach 100%, because the test is composed of several sensors, while the sensor itself may fail.



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