New Optimization Design Method for a Double Secondary Linear Motor Based on R-DNN Modeling Method and MCS Optimization Algorithm

: Traditional linear motor optimization methods typically use analytical models combined with intelligent optimization algorithms. However, this approach has disadvantages, e.g., the analytical model might not be accurate enough, and the intelligent optimization algorithm can easily fall into local optimization. A new linear motor optimization strategy combining an R-deep neural network (R-DNN) and modified cuckoo search (MCS) is proposed; additionally, the thrust lifting and thrust fluctuation reductions are regarded as optimization objectives. The R-DNN is a deep neural network modeling method using the rectified linear unit (RELU) activation function, and the MCS provides a faster convergence speed and stronger data search capability as compared with genetic algorithms, particle swarm optimization, and standard CS algorithms. Finally, the validity and accuracy of this work are proven based on prototype experiments.


Introduction
Linear motors can convert electrical energy into mechanical energy for linear motion. They have advantages in regards to their small size, rapid dynamic response, and high positioning accuracy. Therefore, they are widely used in industrial automation [1][2][3][4][5] .
Linear motor parameter optimization is important in the research of linear motors. The traditional parameter optimization method for a linear motor is usually an analytical model combined with an intelligent optimization algorithm. In Ref. [6], an analytical model is established based on the equivalent circuit method, and then a genetic algorithm (GA) is used to search for the optimal design parameters of the motor. The experimental results show that the optimized parameters correspond to an increase of Manuscript received June 17, 2020; revised July 14, 2020; accepted July 21, 2020. Date of publication September 30, 2020; date of current version August 6, 2020. * Corresponding Author, E-mail: 06009@ahu.edu.cn * Supported by the National Natural Science Foundation of China (51837001, 51907001, 51707002). Digital Object Identifier: 10.23919/CJEE.2020.000023 thrust and a decrease of thrust fluctuation. In Ref.
[7], the thrust and efficiency are specified as objective functions, and then a pareto-based multi-objective differential evolutionary (DE) algorithm is adopted to search for the optimal size parameters of the motor, ultimately obtaining satisfactory thrust performance improvement results. In Ref. [8], a performance analysis model for a linear motor is established based on an equivalent magnetization method. Then, a particle swarm optimization (PSO) algorithm is used for an iterative calculation of the objective functions, to determine the optimal combination of structural parameters under multiple working conditions. These parameter optimization methods (based on analytical models and optimization algorithms) have important reference value for the study of linear motor. Analytical model can help researchers understand the mutual restriction and connection between the internal parameters of motor, and has important reference value for the qualitative analysis of motor, intelligent optimization algorithm can accurately locate the optimal structural parameters of the motor [9] . However, there are many idealized processes in the analytical model, resulting in a slightly lower analytical accuracy, and the general optimization algorithms have difficulty escaping from local optimal values [10] .
In this study, a deep neural network modelling method using a rectified linear unit (RELU) activation function is investigated, to build a more accurate agent model for a double secondary linear motor (DSLM). To determine the optimal structure parameters of the linear motor, we introduce a novel and efficient global cuckoo search (CS) optimization algorithm, and improve it. Compared with the standard CS algorithm and two other common optimization algorithms (PSO and GA), the modified CS (MCS) algorithm searches for the global optimal value more quickly and accurately. Finally, the effectiveness of the proposed optimization design method is verified, based on experiments.

Structure of DSLM
The topological structure of an air-core DSLM as applied to the servo system of a microsecond laser engraving machine is shown in Fig. 1. It comprises two secondary back-irons, six rectangular coils, and periodic permanent magnets. The slot-pole combination of the DSLM is 7P/12S. The DSLM has no cogging effect or core loss, owing to its coreless and symmetrical structure. The optimization design problem for the motor should include the optimization goal, i.e., the variable to be optimized, and the boundary of the variable [11] . A high output thrust and low thrust fluctuations are important factors affecting the application of a DSLM in microsecond laser engraving machines. Therefore, this study selects thrust and thrust fluctuations as optimization goals. Considering the cost constraints, processing technology, previous experience, and dimensional constraints, the permanent magnet width, permanent magnet thickness, polar distance, air gap, and coil width are selected as the key parameters to be optimized, and their boundaries are shown in Tab. 1. The layout of the key parameters in the DSLM topology is shown in Fig. 2. When analyzing the magnetic field of the DSLM, the motor's magnetic field is often simplified as a two-dimensional magnetic field, lateral end effects are often not considered, the demagnetization curve of the magnet is assumed to be a straight line, and the permeability of the secondary core is assumed to be infinite [12][13] . Based on these assumptions and idealizations, the magnetic field of the motor can be calculated using Eq. (1). where B r is the residual magnetization, and m n is calculated using Eq. (2), as follows where n is a positive integer.
The back electromotive force (EMF) of phase a can be calculated using Eq. (3).
where E A is the EMF of coil A; E X is the EMF of coil X; L is the effective length of the conductor; N is the number of conductors contained in each coil; f is the frequency of the current; and w is the distance between two adjacent coils.   The comparison results in Fig. 4 show that some assumptions and idealizations reduce the prediction accuracy of the analytical models. As a result, the analytical models are more suitable for qualitative analysis than for precise quantitative analysis [14] . Thus, it is crucial to obtain a more accurate motor model. In this study, a modeling algorithm called R-deep neural network (DNN) is used to replace the analytical model.

R-DNN modeling
To meet the requirements of the optimization design for model accuracy, the R-DNN (a deep neural network modeling method using the RELU activation function) is studied in depth, to establish an accurate agent model for the DSLM. Fig. 5 shows a diagram of R-DNN modeling.

Create sample library
The selected design variables can form 4 5 =1 024 combinations, according to an orthogonal data combination method. The FEA method is used to calculate the thrust performance corresponding to each parameter combination. Then, a sample library can be obtained, as shown in Tab

R-DNN regression modeling
A DNN is a powerful neural network model [15] . It is composed of multi-layer adaptive nonlinear elements, and has been effectively used in wind speed prediction [16] , fault diagnosis [17] and image processing [18] .
In this study, an enhanced DNN algorithm, the R-DNN is introduced to train a DSLM model with satisfactory accuracy. The main steps for establishing the DSLM model with the R-DNN can be summarized as follows.
(1) Two sets of data are divided from the sample library. One is used to train the R-DNN model, the other is used to test its accuracy.
(2) The number of hidden layers and number of neurons in each layer are set. RELU is selected as the activation function, and the output of i-th layer can be expressed as follows (4) where W i and b i are the weight and bias between the i and i-1 layers, respectively, and σ(x) is the activation function, which is defined as follows (3) The R-DNN model is trained by an error back propagation algorithm (BP algorithm), W i and b i are updated by the Stochastic gradient descent method.
where ε is the learning rate, and D is the cost function, usually the mean squared error (MSE).
where i Y is the predicted value, and Y i is the real value.
(4) The model is regularized by dropout to avoid over-fitting.

Model accuracy test
Another well-known machine learning modeling method called k-nearest neighbors (KNN) is used in comparative experiments, to prove the superiority of R-DNN modeling method. The accuracy test results for thrust and thrust fluctuation are shown in Fig. 6 and Fig. 7, respectively. This test is performed in Matlab. An evaluation index called R 2 [19] is often used to evaluate the fitting accuracy of machine learning algorithms. The closer R 2 is to 1, the higher the accuracy of the corresponding model. The calculation formula of R 2 is as follows where i y is the average of the real value, n is the number of test samples, ŷ i is the predicted value, E sst is the sum of the total dispersion squares, and E sse is the residual sum of squares. Fig. 8 shows an accuracy comparison between the R-DNN and KNN models. The results show that the fitting accuracy of the R-DDN model is higher than that of the (famous) KNN, and can meet the requirements for subsequent optimization design.

Modified cuckoo search algorithm
To achieve the optimization goal of this study, i.e., to improve the thrust and reduce the thrust fluctuations of a DSLM, a powerful global optimization algorithm (the MCS algorithm) is proposed for optimizing the motor's structural parameters.
A CS is a new heuristic optimization algorithm that imitates the parasitic breeding strategy of a cuckoo population as enhanced by Levy flight, rather than using a simple isotropic random walk. Some research work has been conducted to show that the CS algorithm is more effective than DE algorithms, GAs, and others [20][21] . The CS is simple and efficient, and has an excellent random search path. It has been successfully applied in engineering optimization, image processing, artificial intelligence, and other fields [22][23][24][25] .
However, as a bionic algorithm based on population behavior, the CS algorithm still has the disadvantages in regards to a slow search rate, and difficulty in jumping out of a local optimum [26] . Considering these shortcomings, we modified the standard CS algorithm to purpose the MCS. In the MCS, the exploration and mining ability of CS algorithm is enhanced by introducing a non-linear decreasing inertia weight to modify the population renewal formula. The convergence speed of CS algorithm is improved by introducing Gauss disturbances to increase the change vitality of nest positions.
In the MSC algorithm, there is one egg per nest, and the cuckoo's egg represents a new solution. The goal is to use new and potentially better solutions to replace worse solutions. Three rules are specified in the MCS algorithm, as follows [27] .
(1) The cuckoo lays only one egg at a time, representing a solution to the problem, and randomly places the eggs in a nest for incubation.
(2) A part of the nests contains good-quality eggs, i.e., good solutions to problems. These nests will be retained for the next generation.
(3) The total number of nests is certain. The probability of an alien egg being found is P a (P a ∈[0,1]).
The flow chart of the MCS algorithm is shown in Fig. 9.
The main steps are as follows.
Step Step 2: Retain the best nest position x (t-1) b from the previous generation. Then, according to Eq. (10), the positions of the other nests are updated. Compare the nest position after being updated with that before, and retain the nest position with the better fitness value, so that a set of better nest positions g t =[x (t) 1 , i represents the position of the i-th nest in the t-th generation, α is the step size, L(λ) is the Levy random search path, and L~u=t -λ ,(1<λ≤3), w is a non-linear decreasing inertia weight.
where iter represents the number of iterations.
Step 3: A random number r∈[0,1] which obeys a uniform distribution is used as the probability of alien eggs being found. Compared with P a , the nest position with a low probability of being found is retained, whereas a nest position with high probability of being found is changed according to Eq. (12). Compare the nest position after the change with that before the change, and retain the nest position with the better fitness value, so that a new and better group of nest where r is the scaling factor. x (t-1) j and x (t-1) k represent two random different solutions.
Step 4: According to Eq. (13), a Gaussian perturbation is applied to the nest position k t to obtain a new set of nest positions, t p′ . Each nest in t p′ is compared with each corresponding nest in k t to retain the nest position with the better fitness value. Finally, a better nest position p t is obtained.
In the above, ε is a random matrix of the same order as k t , and a is a constant.
Step 5: The optimal nest position x (t) b and its fitness value f b are selected. If the iteration stop condition is reached, the optimal fitness value f b and the corresponding optimal position x (t) b are output. Otherwise, the process returns to Step 2 to continue with the iterative update.  The results show that the MCS algorithm has the fastest search speed and highest fitness value corresponding to the solution. It further shows that the MCS algorithm is more suitable for solving the optimal structural parameters of complex motors. The optimal structural parameters found by MCS algorithm are shown in Tab. 3.

Experiment
To prove the feasibility and effectiveness of the motor optimization method used in this study, an optimized prototype is manufactured and tested on an experimental platform, as shown in Fig. 11. The thrust curve comparison test results of the initial motor and optimized motor are presented in Fig. 12. where F i is the thrust of each sampling point, and n is the number of sampling points.
The calculated results show that the average thrust of the initial motor is 58.28 N, and the thrust fluctuation is 3.54%. The average thrust of the optimized motor is 63.19 N, and the thrust fluctuation is 1.95%. Therefore, by applying the optimization strategy proposed in this study, the average thrust of the motor is increased by 8.42%, and the thrust fluctuation is reduced by 44.92%.

Conclusions
Given that an analytical model is not accurate enough to meet the needs of a motor's optimal design, we proposed an effective alternative method, R-DNN. Compared with a traditional machine learning modeling method (KNN), R-DNN has stronger modeling capability. As general optimization algorithms have slow convergence rates and it is difficult to jump out of local optimal solutions, an efficient global optimization algorithm, i.e., the MCS, is proposed. Compared with GAs, PSOs, and standard CSs, the MCS algorithm has the strongest search capability and fastest search speed. Finally, the experimental results from the optimized prototype are evidently better than those of the original motor are, proving the effectiveness of the optimization design work performed in this study.     [15] S Doi, H Sasaki, H Igarashi. Multi-objective topology