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

-e 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. -e particle swarm optimization (PSO) algorithm is a global search algorithm. In this paper, an improved Lagrange-particle swarm optimization algorithm is proposed. -e updating of the Lagrangian multipliers is with the PSO algorithm for global searching. -e difference between the upper and lower bounds is proposed to construct the fitness function of PSO. -e 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.


Introduction
e problem of fault detection and diagnosis in complex equipment is a hot topic for theory and practical research [1][2][3][4][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][7][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. erefore, it is of great importance to make effective fault diagnosis for weapon equipment, especially for complex electromechanical equipment. Existing studies [11][12][13][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]. e optimization algorithm can be used to solve the objective function under the correlation model of complex system. e 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][27][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. e 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][33][34][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. e 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. e 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. e rest of this paper is organized as follows. e multiple fault diagnosis model is introduced in Section 2. In Section 3, the Lagrangian model is briefly introduced. en, 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.

Multiple Fault Diagnosis Model
e multiple fault diagnosis problem consists of the following elements. e fault source vector of the system can be denoted as F � (f 1 , f 2 , · · · , f m ), where f i represents the ith fault state of the system. e set of test points in the system is denoted as T � (t 1 , t 2 , · · · , t n ), where t j represents the jth test point. e fault source-test point dependency matrix is D � [d ij ], where the element d ij represents the dependency between the fault state f i and the test point t j . In addition, if the fault state and the test point are dependent on each other, then d ij � 1; otherwise, d ij � 0. e prior probability vector of the fault is P � (p 1 , p 2 , · · · , p m ), where p i 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 sources X ⊆ F can be calculated to maximize the posterior probability. e posterior probability can be converted to a prior probability with Bayesian formula [19]: where T p is the test set that has passed the test. T f is the test set that has not passed the test. e vector x of m dimension is defined to represent the fault state of the component.
When the fault state f i occurs, x i � 1; otherwise, x i � 0. P(T p , T f ) is a constant that is independent of X. us, we can get the following function: where where P i is the prior probability of the failure in component i, calculating the probability of the occurrence P(X) regarding the fault source set X. If the fault state f i happens, P e values of P(T p | X) and P(T f | X) are relative to the information in the matrix D [19]: where t j and t k denote the test points j and k respectively. Obtaining the maximum value of P(T p | X)P(T f | X), since they can be only 0 or 1, thus, both P(T p | X) and P(T f | X) must be 1. e following four cases exist: According to the aforementioned analysis, a lemma can be summarized as follows [19].

Lemma 1.
e maximum value of P(T p | X)P(T f | X) can be calculated on the condition that the following conditions are satisfied: Shock and Vibration To fulfil the aforementioned condition 1, delete the fault set that has been confirmed not to happen from the fault set C and get the fault set C − . In other words, the value of the confirmed fault x i 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 t k ∈ T f , Ax ≥ e, where A can be calculated as the transposed matrix of D in which the test column without failure has been deleted. e size of A is |T f | × 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]:

Lagrangian Relaxation Model
e Lagrangian relaxation algorithm can be adopted to solve the aforementioned multiple fault diagnosis problem. e original multiple fault diagnosis model is shown as follows [19]: 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. c j depends on the prior probability p(s j ), and c j can be presented as follows: Lagrangian relaxation algorithm is applied to the model. e improved optimization model is proposed as follows [19]: is the Lagrangian multiplier. Subsequently, the Lagrangian dual model can be represented as follows: Finally, the fault source x can be obtained by solving the Lagrangian dual model Z LD .

Lagrange-Particle Swarm
Optimization Algorithm e Z LR (λ) 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 Z LD . e 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. us, 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.
at is to say, the subgradient method is replaced by the PSO algorithm for calculating the Lagrangian multiplier λ.
e problem of defining a fitness function is involved in PSO algorithm. Here, the absolute value of the difference between upper and lower bounds J is chosen as the reference standard. e upper bound of the original multiple fault model (Z IP ) is introduced as a parameter Z UP ; it can be replaced by the value of a feasible solution. Z DW is a lower bound that is the nearest value to Z IP for the function Z LR (λ). e absolute value of the difference between upper and lower bounds is J � |Z UP − Z DW |. e 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 Z UP and lower bound Z DW 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]. e 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. e flow chart of the algorithm is shown in Figure 1. 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.
Defining Z DQ as the optimal value for function Z LR (λ) for the current situation of λ. Above all, the Lagrangeparticle swarm optimization algorithm can be summarized as follows: (i) Step 1. Initialization of parameters. e initial population is λ(0) � λ 0 1 , λ 0 2 , · · · , λ 0 g , the size of the population is g, the best position for initial particle is P b (0) � λ(0) � λ 0 1 , λ 0 2 , · · · , λ 0 g and Shock and Vibration 3 the initial velocity vector is v(0) � v 0 1 , v 0 2 , · · · , v 0 g . e initial velocity is required to be between v max and v min . e optimal fitness function value for each initial particle is set as F 0 b � 0, 0, · · · , 0 { }, where the dimension is g. e difference of the absolute value between the minimum upper and lower bounds for each initial particle is J b (0) � ∞, ∞, · · · , ∞ { }, where the dimension is g. e difference of the absolute value between the optimal upper and lower bounds for the overall initial situation is J t b1 � ∞.
Step 2. Searching the global optimal particle.
Step 2.1. Do the following five steps for each element λ t i , (i � 1, 2, · · · , g) in the tth iteration of the particle denoted as λ(t) � λ t 1 , λ t 2 , · · · , λ t m . (1) Solving equation (8) with λ t i . e optimal value calculated by equation (8) is strategy is used for (1) removing unnecessary fault sources in the following application, (2) reducing the gap between Z t UP,i , and Z t DW,i , and (3) terminating the algorithm as soon as possible. to determine whether to update the optimal position of the particle P t b,i . If F t b,i < F t i , update the position as P t b,i � λ t i , the value of fitness function as F t b,i � F t i , and the difference between upper and lower bounds as J t b,i � J t i ; otherwise, update no parameter.
Step 2.2. Basing on the value of the fitness Determine whether the number of iterations and the difference between upper and lower bounds meets the requirements

Population initialization. Input initial parameter
Calculate the absolute value of difference between upper and lower bounds J and fitness function F Update the historical optimal position of each particle P b and global optimal particle λ b , and find its difference between upper and lower bounds J b1 Update the velocity v and position λ to produce next generation of particle and increase the number of iterations t = t + 1 Output the optimal fault source X = (x 1 , …, x m ) with the global optimal solution Update the inertia weight w Step1: initialize parameter Step 2: search for global optimal particle Step 3: update the particles Step 4: output the faults Yes No Figure 1: e flow chart of the Lagrange-particle swarm optimization algorithm. the global optimal particle λ t b in the optimal position P b (t) � P t b,1 , P t b,2 , · · · , P t b,g } among all the particles. In addition, calculating the corresponding difference between the upper and lower bounds as J t b1 , which is also the global smallest difference value between the upper and lower bounds.
Step 3.1. Updating the inertia weight w t � w max − t(w max − w min )/N max .
Step 3.2. Updating the velocity and position of the particles. e velocity of the ith particle in the d dimension for the t + 1 times iteration is . If the updated velocity is negative, defining v � − K 1 e − (v+v max ) to slow down the movement speed in the negative direction.
is is because the Lagrangian multiplier is required to be a nonnegative value. e updated position is λ t+1 Updating the time of the iteration as t � t + 1.
Step 3.3. Determining whether the maximum number of the iterations N max has been reached. If t > N max or the absolute value (J t b1 ) of the difference between the upper and lower bounds obtained by calculating the global optimal particle λ t b 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 λ t b . Outputting the fault source X � (x 1 , · · · , x m ). e 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 multiplier λ, 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. is 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.

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. e dependency matrix of the system is shown in Table 1.
Establishing the corresponding Lagrangian dual model, the proposed Lagrange-particle swarm optimization algorithm is adopted to solve the model. e solving of the model is by programming in MATLAB R2014a. e parameters in the algorithm are chosen as follows. e maximum iteration number N max � 200, population size popsize � 50, acceleration factor c 1 � 1.3 and c 2 � 1.5, maximum of inertia weight w max � 1.2, minimum inertia of weight w min � 0.4, maximum of velocity v max � 2, minimum of velocity v min � − 2, proportional coefficient K � 5 and K 1 � 0.2, and the absolute value of the minimum difference between upper and lower bounds J min � 0.01. J min affects the quality of solution and the calculating time significantly. If J min 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 J min 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, J min is finally to be chosen as 0.01.
Take the double-fault diagnosis problem as an example, the results are shown in Table 2. e corresponding fault isolation rate is 91.11%. e 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.
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. e experiment results with the original Lagrange relaxation algorithm and the improved algorithm are shown in Table 3.
e proposed method has a higher accuracy in fault diagnosis.
is 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.

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. e fitness function is constructed based on the absolute value of the difference between the upper and lower bounds. e 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. e 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. e 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.

Data Availability
All relevant data are provided within the paper. Shock and Vibration 7