Fuzzy Wavelet Neural Network with the Improved Levenberg–Marquardt Algorithm for the AC Servo System

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Introduction
In recent years, varies studies show that the AC servo system exhibits good dynamical property [1,2], but the stability still needs to improve. For the AC servo system, the dynamic mathematical model is a complex system with characteristics of large load, which can lead to nonlinearity and uncertain disturbance. In practical applications, an AC servo system performance may be affected due to unmodeled dynamics changed [3,4].
After referring to many references, a lot of studies have shown the neural network is an important component of a complicated nonlinear system control policy under the circumstance of the lack of full model details [5][6][7][8]. e most prominent advantage of the neural network is approximate capability, and it can approximate function with any precision. However, it is hard to avoid local minimization for the BP neural network. If we use a sigmoid function as the stimulation function, it also causes slow convergence speed. Moreover, it cannot realize the mapping rules in time [9]. Fuzzy logic has become a hot topic research of neural networks in many studies. Dong et al. [10,11] provide theoretical basis to modelling and controlling the nonlinear system. Consider there are many uncertainties existing in the fuzzy control process. Wang [12] provides a fuzzy neural network along with utilization to improve system robustness without accurate control; however, the parametric learning algorithm is presupposed for the topology of fuzzy systems.
As an alternative, multiple research studies concentrate on the use of the wavelet neural network (WNN) [13][14][15][16]. Compared with the usual sigmoid function neural network, the wavelet function possesses a better learning capacity in aspects of system identification. In recent years, Zekri et al. [17][18][19] studied the combination of wavelet theory and the fuzzy neural network (FNN). In the FWNN, fuzzy rules are corresponding to the sub-WNN, respectively, and the wavelet and fuzzy sets parameters learning can improve the FWNN approximation accuracy [20][21][22][23]. However, the main drawback of the WNN is that due to its feed-forward network structure, its application area is limited to static issues. e Levenberg-Marquardt (LM) method has a remarkable characteristic of local learning and a fast convergence performance at the same time [24]. However, the LM algorithm increases memory demands with the method of calculating some problems that come from the error function with the Jacobian matrix [25]. Moreover, another disadvantage is that the LM algorithm is still a local optimization method. e particle swarm algorithm (PSO) is a global optimization algorithm, through collaboration and competition between individuals to find the optimal solution, and the particle swarm optimization search process is started from the entire group, with the implicit parallel search features to improve the performance of the algorithm [22]. However, the PSO algorithm has some disadvantages such as slow convergence speed.
Based on the above analysis, in this study, an adaptive fuzzy wavelet neural network controller with LM is proposed to control the rotor position of the AC servo system for tracing reference trajectory with robustness. In the proposed control structure, the FWNN is a controller, and the LMPSO algorithm is employed for the online training of all weights of the FWNN. Moreover, potentiality of fuzzy rules (PFR) with using error reduction ratio (ERR) is developed to adjust the parameters and organize the structure of the FWNN. e stability of the system can be proved by using Lyapunov theory [26]. Finally, studies demonstrate promising results of a prototype AC servo system that can verify the feasibility and effectiveness by using the proposed algorithm. e contents of this study can be listed as follows: the second section analyzes the servo system. After briefly introducing the FWNN in the third section, the following section four develops the FWNN-LM which has been proposed at great length. Afterwards, the convergence of the algorithm is analyzed in section five. And then, the simulation outcomes are discussed in section six. Last, a conclusion has been mentioned in the last section.

AC Servo System
Analysis e AC servo system control structure is shown in Figure 1. In the stationary (d-q) frame of reference, the mathematical models of the permanent magnet synchronous motor can be expressed as follows: where i d , i q , u d , u q , and L d , L q represent the electric currents, voltages, and inductance coefficient of the motor d and q axes, respectively; R is the motor stator resistor (Ohm), ψ f represents the motor permanent magnet flux, p represents the motor pair of poles, J represents the motor inertia constant, B represents the viscous friction coefficient, ω r stands for the motor angular velocity, T e stands for motor electromagnetic torque, and T L stands for the load torque. e system is applied to a three-closed-loop control system. It uses the magnetic field-oriented control technology to complete the motor position and achieve high performance. Additionally, the simplification of the motor control system uses the i d � 0 vector control approach.
When i d � 0, the motor mechanical equation can be expressed as where ω b is the mechanical angular velocity, and T e can be written as where K t is a moment constant that needs to be adjusted. Generally, compared with the mechanical time constant, the motor current time constant has a much smaller numerical value; thus, the delay time of the current responding can be neglected. e state variables can be set as x 1 � θ and x 2 � _ θ; substitute equation (2) into equation (3), and the AC servo system can be rewritten as where − (B/J), K t /J, and − (1/J)T L represent the nonlinear dynamic equations and the external disturbance, respectively.

Wavelet Neural Network
Structure. e structure of wavelet neural network is shown in Figure 2. As Figure 2 illustrates, K is the master nodes of the input layer, the hidden layer number is M, ω km is the connection weighing between node k of the input layer and node m of the hidden layer, ω m is the connection weighing between node m and the output layer, b m is the translation parameter of wavelet function, and a m is the scale variable of the wavelet function. e output can be written as [21] where

Fuzzy Wavelet Neural Network Structure.
In the fuzzy wavelet network, each fuzzy rule corresponds to a given wavelet scale values of the wavelet neural network [27]. In order to describe FWNN-LMPSO clearly, a simple structure of the FWNN is shown in Figure 3.
e N F fuzzy IF-THEN rules can be expressed as follows: .., and x m is A mn , then

Complexity
where R n is the fuzzy rule (1 ≤ n ≤ N F ); A mn is the membership function for the fuzzy set of Gaussian function, which can be expressed as where x m is the input of m � 1: N in , N in represents the number of input neurons; n � 1: N F . e canter c mn and width σ mn can be used to define as a subordinate function.
e output of the entire FWNN structure by using product rules and defuzzification is shown as e output mean square error of the online learning is Input layer Hidden layer Output layer Figure 2: e structure of the wavelet neural network. Figure 3: e structure of the FWNN.
Complexity 3 where O d is the expected output of the training data. According to the descent algorithm, the FWNN parameters adjustment formulas are shown as where c l (l � 1: 6) is the learning rate, and the arguments of the FWNN controller can be expressed as

FWNN with LM Algorithm-Based PSO
In the neural network, a sigmoid function is used as the activation function of the BP neural network, which leads to the result that the BP neural network is easy to get into a local minimum, slow convergence speed. us, to improve the performance of FWNN, the training process is required to adjust both the structure size and the parameters. An LMPSO method is used for adjusting the parameters; and a PFR is developed to design the structure of FWNNs. In the following, the LMPSO method and PFR method are described in detail.

LM Algorithm-Modified FWNN.
e LM algorithm is an approximate Newton algorithm, which proves that the LMbased BPNN algorithm converges quick and accurate performance [28]. In this study, the total mean square error of P is given as e LM algorithm is written as where the Jacobian matrix J(h)is e structure of the fuzzy wavelet neural network (FWNN) based on the LMPSO controller is shown in Figure 4.

LM Algorithm-Based PSO.
PSO is a population-based heuristic global optimization technique. In this algorithm, the population is called a swarm, and the trajectory of each particle in the search space is adjusted by dynamically altering its velocity, according to its own flying experience and swarm experience in the search space. In the PSO algorithm, a group of particles represent a candidate solution. e velocity and position updating formulas of the PSO are illustrated as WFNN Controller Eq (5)-Eq (11) AC servo System Eq (1)-Eq (4) On-line Learning Algorithm Eq (12)   Complexity where v i (k) represents the current rate of particle i th during the iteration k; x i (k) represents the current position of the i th particle; P i (k) is on behalf of the optimum position of the i th particle previously appeared; P g (k) denotes the best previous position among all the particles; c 1 and c 2 are on behalf of the acceleration factors; r 1 and r 2 uniformed random number in the interval [0, 1]; wstands for the inertia weight in the interval [0.4, 0.9]. An appropriate fitness function to calculate the appropriate value is where F(t) is the fitness value; e MAE represents the mean absolute error, e max is the maximum absolute error, and connected with LM-based FWNN, there is In this study, the PSO-LM algorithm is used to make adjustments of FWNN which can make e MAE and e max more appropriate in actual conditions. A few particulars about the PSO procedure are shown as follows: (1) Initialize the PSO parameters (2) Determine P g (k) and P i (k) (3) Refresh the particle speed as well as the position by taking advantage of equation (29) (4) Get the current fitness value FT and updateP i (k), P g (k); if FT i < P i (k), FT i < P g (k). (5) Ifi < N, set i � i + 1 and then go to step (3); otherwise, proceed to the next step. (6) If k < maxgen, set k � k + 1 and then go to step (2) or output P g (k) to LM-based FWNN.

Potentiality of Fuzzy Rules (PFR).
e PFR values can be used to calculate the potentiality of fuzzy rules and extract the contributions of the normalized neuron.
e FWNN model is expressed as follows: where + 2), . . . , w(t)] T is the weight between the output layer and normalized layer, and Φ(t) is given by e matrix Φ(t) can be transformed into a set of orthogonal basis vectors by QR decomposition as where R(t) is an upper triangular matrix, and Q(t) � [q 1 (t), q 1 (t), . . . , q n (t)] have the same dimension as Φ(t). en, the ERR is given by [29] , l � 1, 2, . . . , N F . (21) e PFR value of the l th normalized neuron can be expressed as follows: where PFR l (t) ∈ (0, 1) is the potentiality of fuzzy rule in the l th normalized neuron, and where 0 < η < 1 is a constant.

Stability Analysis
Lyapunov function is used to assess the system stability, and it can be defined as where e(k) � (y d (k) − y(k)).y d is the desired output, and y(k) is the actual output.

Simulation Test and Discussion
In order to test and verify the effectiveness of the FWNN-LMPSO control, it will be compared with FWNN-LM.

Simulation Experiment.
In addition, all simulation programs are conducted in Matlab/Simulink, and a clock rate of 2.6 GHz and 4 GB of RAM on a PC running in a Microsoft 7.0 environment are selected. e main parameters of the AC servo system are given in Table 1. Figures 5-9 show the simulation results.
As shown in Figure 5, the moment of inertia changes from the initial value to 1.5 times. FWNN-LM generates an overshoot; it takes 4.15 s to reach the stable condition. Using FWNN-LMPSO control, the system responds quickly, and only needs 1.6 s to reach the steady state without overshoot. Figure 6 shows step response when a 360 nm disturbance added at 3 s.
As Figure 6 shows, when the load added, it gets more deviation results on account of the response of the algorithm of FWNN-LM control. It also has a 5.15°delay in tracking the reference position. However, when using the FWNN-LMIPSO control algorithm, the offset can decrease to 1.25°. It costs 0.35 s to reach the target position. Above all, the system can perform better in the aspect of load disturbance suppressing.
Experimental result of tracking step signal with random disturbances is shown in Figure 7. It can be seen from Figure 7 that when adding random disturbance to the response signal, there is no offset occurring by using FWNN-LMPSO control. Moreover, random disturbance is also added in sinusoidal tracking experiment. e maximum error of FWNN-LM and FWNN-LMPSO is 0.089°and 0.057°, respectively. e sinusoidal tracking error curves with a frequency of 1.67 rad/s and amplitude of 30 degree is shown in Figure 8.
In Figure 9, the number of FWNN-LM iterations is about 220 steps, the training error is 0.128, and the training error of FWNN-LMPSO is 0.035 when the number of iterations is about 95 steps. erefore, the convergence rate of FWNN-LMPSO is better than the FWNN-LM method.

Semiphysical Experiment.
e semiphysical experiment platform structure is shown in Figure 10. A step response with FWNN-LM control and FWNN-LMPSO control are

Conclusions
is study offers a new fuzzy wavelet neural network method in the AC servo system. Compared with the FWNN-LM controller, the proposed FWNN-LMPSO controller can be designed more accurately, more meaningful, and simpler. e main advantages of the existing method based on FWNN-LMPSO are as follows: first, in the FWNN-LMPSO based on the PFR method, fuzzy rules can be added to and removed from the structure learning method with the method of using the ERR value. Second, the LM algorithm improves the control accuracy through the adjustment of parameters, and the PSO learning algorithm is used to improve the learning speed. Lyapunov theory is also introduced to analysis system stability. Last, experimental results show the method has strong robustness and better dynamic performance.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.