Electric load simulator system control based on adaptive particle swarm optimization wavelet neural network with double sliding modes

Abstract

In this article, an adaptive particle swarm optimization wavelet neural network with double sliding modes controller is proposed to address the complex nonlinearities and uncertainties in the electric load simulator. The adaptive double sliding modes–particle swarm optimization wavelet neural network algorithm with the self-learning structures and parameters is designed as a torque tracking controller, in which a number of hidden nodes are added and pruned by the structure learning algorithm, and the parameters are online adjusted by the adaptive particle swarm optimization at the same time. Moreover, one conventional sliding mode is introduced to track the time-varying reference command, and the other complementary sliding mode is adopted to attenuate the effect of the approximation error. Furthermore, the relative parameters should comply with some estimation laws on the basis of the Lyapunov theory used to guarantee the system stability. Finally, the simulation experiments are carried out on the hardware-in-the-loop platform for the electric load simulator, the performance of the adaptive double sliding modes–particle swarm optimization wavelet neural network with structure learning is verified compared with some similar control methods. In addition, different amplitudes and frequencies of the reference commands are introduced to further evaluate the effectiveness and robustness of the proposed algorithms.

Introduction

The load simulator is designed for the guns, aircrafts, and ships,^1–3 which is also a key equipment in the hardware-in-the-loop simulation (HILS) for the test and simulation. Traditional researchers in load simulators always focus on the optimization of control parameters,^4,5 and the control strategies show poor performance when the control objectives or working conditions are changed. Due to the complex nonlinearities and uncertainties, especially like backlash, friction, and coupling between the load motor and the torque motor, many intelligent algorithms are investigated for load simulators to improve the tracking performance and robustness.^6–10 Although many of the mentioned works above are focused on the electro-hydraulic load simulators (EHLS) usually applied to the heavy loading, there are many similar characteristics between the EHLS and the electric load simulator (ELS). ¹

Among different nonlinear control techniques, methods based on the neural network (NN) have been growing into a popular research topic in recent years.^11–13 Considering the local search ability and low convergence rate, the wavelet neural network (WNN) with the merits of time domain and frequency domain search and approximation has been widely applied to the nonlinear control.^14,15 However, the methods above can only represent models with limited accuracy, so there remains approximation errors between the practical system and the constructed model.^1,7 In order to solve this problem, several stability analyses combined with the sliding mode control (SMC) are proposed to guarantee the system stability in the sense of Lyapunov theory.^2,8,15,16 Although the SMC has great fast response and transient control performance against the disturbances, it suffers from large chattering phenomena. Therefore, many compensation methods have been proposed to eliminate the external disturbance effect.^17–20 A saturated modification of traditional projection-based adaptation laws is proposed to eliminate the chattering phenomena, which also guarantees the system stability in the sense of the Lyapunov theory. ²⁰ However, many parameters cannot be exactly obtained, which are always changing in the practical working conditions, such as the friction and backlash. Since the parameters of real nonlinear systems are time variable, the structure and parameter selection has played great effects on the control performance.^15,16,21,22 In view of parameter uncertainties, chattering phenomena, and computation complexities, the structure of the WNN should be adaptive to guarantee the computation efficiency.^15,16 The structure learning (SL) algorithm can online adjust the hidden nodes to reduce the computation loading and improve the control performance. Meanwhile, the particle swarm optimization (PSO) is used to adjust the parameters of the WNN, which speeds up the convergence rate and decreases the oscillations of the SMC.^23,24

This article puts forward an adaptive particle swarm optimization wavelet neural network (PSOWNN) with double sliding modes (2S) controller, which can also vary its hidden nodes of the WNN dynamically to keep the prescribed approximation accuracy with a simple computation. Moreover, the WNN is designed to online estimate the system dynamics, the parameters of which are adjusted by the adaptive PSO. Furthermore, to speed up the convergence speed and guarantee the stability, an analytical method with 2S on the basis of Lyapunov theory is proposed to determine the dynamical learning rates and estimation laws. Finally, the simulation experiments are carried out to evaluate the performance and robustness of the proposed adaptive double sliding modes–particle swarm optimization wavelet neural network (2S-PSOWNN) with SL controller.

Problem statement

In order to evaluate the static and dynamic performance of the gun control system (GCS) in advance, the ELS is designed to simulate the time-varying loading, especially the inertia loading and external disturbances, which is mainly composed of a host control computer, a signal conditioning center, a torque motor, a motor drive, a torque sensor, a resolver, an angular velocity sensor, an angular acceleration sensor, inertia disks and a gear reducer, and so on, as shown in Figure 1. The host control computer with the proposed algorithm to achieve torque tracking control of the ELS mainly analyzes and calculates the feedback torque of the torque sensor, the compensation torque of inertia disks, and the actual torque required. The signal conditioning center is mainly used for processing signals from the sensors and the configuration settings needed for the hardware switches. According to the output of the host control computer, the output of drive provides the appropriate current to drive the torque motor. The torque sensor is used to collect the output of the torque motor, which is fed back to the host control computer to form the closed-loop control for the loading torque. The function of the resolver is gathering the angle position of the gear reducer, which can be used to calculate the real-time command torque for the torque motor. The function of the angular velocity sensor is gathering the angular velocity of the gear reducer to provide more actual values for the identification methods and the control strategies. The angular acceleration sensor collects the angular acceleration at the end of the gear reducer to compute the inertia moment produced by the inertia disks. The number of inertia disks depends on the actual demand of the GCS, which provides various rotational inertia values to simulate the inertia torque. The position of the load motor is a strong disturbance for the torque motor. At the same time, the motion of the torque motor is also a strong disturbance for the load motor, so the ELS exists inherent coupling and interacting. In addition, the friction and backlash also play a negative impact on the ELS, which show strong nonlinearities and uncertainties at the low speed and shaft direction variation. Common control strategies cannot guarantee the system torque tracking with high control performance, and the exact model also cannot be established. ⁶ Therefore, considering the complexity of the ELS system that can be simply modeled by the following equation

$$\stackrel{··}{y}\left(t\right)=f(\mathbf{y},t)+g_{0}u\left(t\right)+d\left(t\right)$$

(1)

where $\mathbf{y}=[y(t),\stackrel{·}{y}(t)]$ and $u(t)$ are the state vector and the input of the ELS system, respectively; $f(\mathbf{y},t)$ and $g_0$ are the dynamic variables, $f(\mathbf{y},t)=-(B_m/J_m+K_T{K}_e/J_m{R}_m)\stackrel{·}{y}(t)$ , $g_0=K_T{K}_a/J_m{R}_m>0$ ; $B_m$ is the viscous frication coefficient; $J_m$ is the inertia constant of the motor; $R_m$ is the stator resistance of the motor; $K_T$ is the coefficient of the electromagnetic torque; $K_e$ is the coefficient of the counter electromotive force; $K_a$ is the amplifier gain; $d(t)$ is an unknown but bounded disturbance, and guaranteed $|d(t)|\le C_0$ , where $C_0$ is a positive constant.

Figure 1.

Structure of the electrical load simulator.

Design of adaptive 2S-PSOWNN with SL

Although the ideal controller can asymptomatically stabilize the system, some time-varying variables cannot be obtained exactly, like the external disturbance $d(t)$ . In order to address this problem, an adaptive 2S-PSOWNN with SL controller, as shown in Figure 2, and the controller output are designed as

$$U_{\mathrm{APW}}\left(t\right)=U_m\left(t\right)+U_c\left(t\right)$$

(2)

where $U_m(t)$ is a main controller, and $U_c(t)$ is a compensation controller. The former is designed to mimic the ideal controller, the latter based on a sliding surface, and bounded errors compensate for the difference between the ideal controller and the former. The inputs of the WNN controller are torque errors and the derivation of torque errors, and the torque error is the difference between the reference command and torque sensor value. In addition, adaptive PSO algorithms and variable SL are used to improve the convergence speed, and the stability analysis method with parameter estimation laws is applied in the sense of Lyapunov stability.

Figure 2.

Structure of the adaptive 2S-PSOWNN with SL controller.

WNN

The structure of the WNN controller is shown in Figure 3, which consists of an input layer, a hidden layer, and an output layer. The signal propagation and activation function in each layer are introduced as follows:

• 1. Input layer. No function is performed in this layer. Each node in this layer only transmits input variables e and $\stackrel{·}{e}$ to the hidden layer. Noted that in the WNN model application, input variables are usually normalized into the interval $[0,1]$ .

• 2. Hidden layer. In this layer, each activation function can be represented as follows ²⁵

$$\Omega \left(x\right)=2^{a/2}\Omega (2^{a}x-b)$$

(3)

where a and b are the dilation and translation parameters, respectively. The Mexican hat mother wavelet function is defined as follows

$$\phi \left(x\right)=(1-{\Vert x\Vert }^2)e^{-{\Vert x\Vert }^2/2}$$

(4)

where $||x||=x^{T}x$ . Thus, the activation function of the $j\mathrm{th}$ hidden node in connection with the input layer is described as

$${\Omega }_j\left(x_i\right)=2^{a_{i,j}/2}\left(1-{\Vert 2^{a_{i,j}}{x}_i-b_{i,j}\Vert }^2\right)e^{-{\Vert 2^{a_{i,j}}{x}_i-b_{i,j}\Vert }^2/2}i=1,2;j=1,\dots ,m$$

(5)

where ${\Omega }_j(x_i)$ is the $i\mathrm{th}$ input variable of the $j\mathrm{th}$ node in the hidden layer, and m is the number of the hidden nodes.

• 3. Output layer. According to a sum method, the output is obtained as follows

$$\begin{array}{c}{o}_i=\sum _{j=1}^m{\Omega }_j\left(x_i\right)w_{j,i}\hfill \\ i=1,2;j=1,\dots ,m\hfill \end{array}$$

(6)

where $w_{j,i}$ is the connection weight between the $j\mathrm{th}$ hidden node and $i\mathrm{th}$ output node.

Figure 3.

Structure of the WNN.

The final output vector of the WNN is defined as $[\begin{array}{cc}\hfill o_1\hfill & \hfill o_2\hfill \end{array}]=[\begin{array}{cc}\hfill \hat{P}\hfill & \hfill \hat{Q}\hfill \end{array}]$ , which represents the signals which are transmitted to the control center of the torque motor, which are main components of the main controller and are rewritten as

$$o_1=\hat{P}\left(\frac{e}{\hat{\mathbf{\alpha }}}\right)={\hat{\mathbf{\alpha }}}^T\mathbf{\Omega }$$

(7)

$$o_2=\hat{Q}\left(\frac{e}{\hat{\mathbf{\beta }}}\right)={\hat{\mathbf{\beta }}}^T\mathbf{\Omega }$$

(8)

where $\mathbf{\Omega }=[\begin{array}{cccc}\hfill {\Omega }_1\hfill & \hfill {\Omega }_2\hfill & \hfill \cdots \hfill & \hfill {\Omega }_m\hfill \end{array}{]}^T$ is the activation function vector, and $\hat{\mathbf{\alpha }}=[\begin{array}{cccc}\hfill w_{1,1}\hfill & \hfill w_{2,1}\hfill & \hfill \cdots \hfill & \hfill w_{m,1}\hfill \end{array}{]}^T$ and $\hat{\mathbf{\beta }}=[\begin{array}{cccc}\hfill w_{1,2}\hfill & \hfill w_{2,2}\hfill & \hfill \cdots \hfill & \hfill w_{m,2}\hfill \end{array}{]}^T$ are the weight vectors. In order to further study, ideal WNN estimators are designed as follows

$$P=P^{*}\left(\frac{e}{{\mathbf{\alpha }}^{*}}\right)+{\Delta }_p={\mathbf{\alpha }}^{*T}\mathbf{\Omega }+{\Delta }_p$$

(9)

$$Q=Q^{*}\left(\frac{e}{{\mathbf{\beta }}^{*}}\right)+{\Delta }_q={\mathbf{\beta }}^{*T}\mathbf{\Omega }+{\Delta }_q$$

(10)

where ${\mathbf{\alpha }}^{*}$ and ${\mathbf{\beta }}^{*}$ are the ideal weight vectors that achieve the perfect approximation, and ${\Delta }_p$ and ${\Delta }_q$ are the minimum approximation errors. In addition, $|{\Delta }_p|<{\delta }_p$ and $|{\Delta }_q|<{\delta }_q$ need to be guaranteed, where both ${\delta }_p$ and ${\delta }_q$ are positive constants.

Adaptive PSO

In order to improve the computation efficiency, the adaptive PSO algorithm is designed to adjust the parameters of the WNN. The position and speed of particles are always updated by tracking the individual and global extremums in the solution space, and initial values of the particle position and speed are usually given in the start stage. The parameters of the WNN needed to be adjusted are the optimized objects of the adaptive PSO. A relative equation is defined as follows

$$\{\begin{array}{c}\hfill v_i(t+1)=\varpi v_i(t)+c_1{r}_1(h_i-s_i(t))+c_2{r}_2(h_g-s_i(t))\hfill \\ \multicolumn{1}{c}{s_i(t+1)=s_i(t)+v_i(t+1)i=1,2,\dots ,L}\end{array}$$

(11)

where $v_i(t)$ is the particle search speed, $s_i(t)$ is the particle search position, t is the iteration, $\varpi$ is the inertia weight, $c_1$ and $c_2$ are learning factors, $r_1$ and $r_2$ are random numbers distributed in the $[0,1]$ uniformly, $h_i$ and $h_g$ are the individual and global extremums, respectively, and L is the number of particles. During the process of the particle swarm evolution, the inertia weight greatly influences the global and local optimization abilities.^26–28 The bigger the weight is, the stronger the global search ability is, and vice versa. The conventional inertia weight is linear attenuation in standard PSO algorithm, which greatly increases the optimization difficulty. In this article, the S-type function is proposed to make the search speed vary from high to low at different stages. Finally, the search speed keeps constant. The improved inertia weight computation is formatted as follows

$$\varpi =\frac{{\varpi }_{max}-{\varpi }_{min}}{1+{\vartheta }^{(2\vartheta \times \mathrm{Iter}/{\mathrm{Iter}}_{max}-\vartheta )}}+{\varpi }_{min}$$

(12)

where ${\varpi }_{max}$ and ${\varpi }_{min}$ are the maximum and minimum inertia weights, respectively; Iter and ${\mathrm{Iter}}_{max}$ are the current and maximum iterations, here ${\mathrm{Iter}}_{max}=100$ ; and $\vartheta$ is the speed variation coefficient, here $\vartheta =10$ . The learning factor represents the self-recognitive and self-learning characteristics, and the particle swarms take advantages of the local learning ability when the learning factors are too big. However, all the particle swarms are easily trapped in the local minimum. Therefore, the self-adaptive and high-efficient adjustments of inertia weights and learning factors guarantee the improvement of the global optimum convergence. The evolution factor is adopted to evaluate the particle swarm state, and relative executive steps are as follows:

• 1. Compute the Euclidean distance between every particle and other particles

$$d_i=\frac{1}{L-1}\sqrt{\sum _{i=1,i\ne j}^L{(s_i-s_j)}^2}$$

(13)

• 2. Define the evolutionary factor when the distance of the global optimum particle is named as $d_g$ ²⁹

$$f=\frac{d_g-d_{min}}{d_{max}-d_{min}}$$

(14)

The states of particle swarms are effectively evaluated with the constructed evolutionary factors, and the learning factors are adjusted in the range $[0.5,2.5]$

$$c_1=\frac{1}{0.2+1.5e^{-2.6f}}$$

(15)

$$c_2=\frac{1}{0.1+0.2e^{-1.85f}}$$

(16)

If $c_1+c_2$ is greater than 4, the value needs to be normalized as follows

$$c_i=\frac{4c_i}{\sum _{i=1}^2{c}_i}$$

(17)

SL algorithm

Determining the number of hidden nodes is an important issue because of the trade-off between the computation loading and learning performance. In order to attack this problem, a simple SL algorithm is introduced. The activation intensity and output errors are used as the judgment rules. If the maximum output of the activation function is less than specific threshold, then a number of hidden nodes need to be added. The relative norm is defined as

$${\Omega }_{max}=\underset{1\le j\le m}{max}\left({\Omega }_j(a_{\mathrm{ij}},b_{\mathrm{ij}},x_i)\right)$$

(18)

$${\Omega }_{max}\le G_{\mathrm{th}}$$

(19)

where $G_{\mathrm{th}}$ is a positive constant. Meanwhile, when the output errors are too big, the approximation ability is also enhanced. However, an average error in the sliding window is designed to avoid the abnormal data, and the other addition rule is defined as follows

$$\mathrm{error}=\frac{\sum _{k=t-M+1}^{t}e\left(k\right)}{M}$$

(20)

$$\left|\mathrm{error}\right|\ge E_{\mathrm{th}}$$

(21)

where $e(k)=y_d(k)-y(k)$ means the input of the controller at the time k, M is the number of data in the sliding window, and $E_{\mathrm{th}}$ is a positive constant. If the addition condition is reached, a hidden node is added, and the parameters are as follows

$$\{\begin{array}{c}\hfill {\mathbf{a}}_{m+1}=\mathbf{a}\hfill \\ \multicolumn{1}{c}{{\mathbf{b}}_{m+1}=\mathbf{b}}\\ \multicolumn{1}{c}{{\mathbf{w}}_{m+1}=\mathbf{0}}\\ \multicolumn{1}{c}{m(t+1)=m\left(t\right)+1}\end{array}$$

(22)

To avoid the number of hidden nodes growing unboundedly, an algorithm for pruning the hidden nodes is necessary to prevent the heavy computation loading, and a significant index based on the activation function is determined for the importance of the $j\mathrm{th}$ hidden node ³⁰

$$\begin{array}{c}{I}_j(k+1)=\{\begin{array}{c}\hfill I_j\left(k\right)\exp (-\tau )\mathrm{if}{\Omega }_j>\rho \hfill \\ \multicolumn{1}{c}{I_j\left(k\right)\mathrm{if}{\Omega }_j\le \rho }\end{array}\hfill \\ j=1,2,\dots ,m\hfill \end{array}$$

(23)

where $I_j(k)$ is the significant index of the $j\mathrm{th}$ hidden node whose initial value is 1, $\rho$ is the elimination threshold value, and $\tau$ is the elimination speed constant. If the $I_j(k)\le I_{\mathrm{th}}$ is satisfied, where $I_{\mathrm{th}}$ is a pre-given threshold, then the $j\mathrm{th}$ node is canceled.

Stability analysis

The control objective is to find a control method so that the state trajectory $y(t)$ can track the reference command $y_d(t)$ in the presence of nonlinearities and uncertainties. By including the control variables with the tracking error, a conventional sliding mode is defined^31,32

$$s_m\left(t\right)=\stackrel{·}{e}\left(t\right)+2\lambda e\left(t\right)+{\lambda }^2\int e\left(t\right)\mathrm{dt}$$

(24)

where $\lambda$ is a positive constant. Taking the derivative of the above formula yields

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{s}}_m\left(t\right)}& =\stackrel{··}{e}\left(t\right)+2\lambda \stackrel{·}{e}\left(t\right)+{\lambda }^{2}e\left(t\right)\hfill \\ \multicolumn{1}{c}{}& ={\stackrel{··}{y}}_d\left(t\right)-\stackrel{··}{y}\left(t\right)+2\lambda \stackrel{·}{e}\left(t\right)+{\lambda }^{2}e\left(t\right)\hfill \\ \multicolumn{1}{c}{}& ={\stackrel{··}{y}}_d\left(t\right)-f(\mathbf{y},t)-g_{0}u\left(t\right)-d\left(t\right)+2\lambda \stackrel{·}{e}\left(t\right)+{\lambda }^{2}e\left(t\right)\hfill \end{array}$$

(25)

If the system parameters in equation (1) are known, there exists an ideal feedback controller as follows^14,30

$$U_{\mathrm{SM}}^{*}\left(t\right)=g_0^{-1}[{\stackrel{··}{y}}_d\left(t\right)-f(\mathbf{y},t)-d\left(t\right)+{\lambda }^{2}e\left(t\right)+2\lambda \stackrel{·}{e}\left(t\right)]$$

(26)

Considering the existing errors caused by the external disturbances, the other complementary sliding mode is defined as follows ¹⁵

$$s_c\left(t\right)=\stackrel{·}{e}\left(t\right)-{\lambda }^2\int e\left(t\right)\mathrm{dt}$$

(27)

Therefore, the sum $\sigma (t)$ of $s_m(t)$ and $s_c(t)$ can be rewritten as

$${\stackrel{·}{s}}_c\left(t\right)+\lambda \sigma \left(t\right)={\stackrel{·}{s}}_m\left(t\right)$$

(28)

Main theorem

Considering the ELS system represented by equation (1) and the WNN designed as equations (9) and (10), where $U_m(t)$ and $U_c(t)$ are given in equations (29) and (30) with the online parameter estimation laws given in equations (31)–(34), then the closed-loop stability for $\underset{t\to \infty }{lim}e(t)=0$ can be guaranteed

$$U_m\left(t\right)=\hat{P}\left(\frac{e}{\hat{\mathbf{\alpha }}}\right)+\lambda \hat{Q}\left(\frac{e}{\hat{\mathbf{\beta }}}\right)$$

(29)

$$U_c\left(t\right)={\hat{\Delta }}_p+\lambda [{\hat{\Delta }}_q+s_m\left(t\right)]$$

(30)

$$\stackrel{·}{\hat{\mathbf{\alpha }}}={\eta }_{\alpha }\sigma \left(t\right)\mathbf{\Omega }$$

(31)

$$\stackrel{·}{\hat{\mathbf{\beta }}}={\eta }_{\beta }\lambda \sigma \left(t\right)\mathbf{\Omega }$$

(32)

$${\stackrel{·}{\hat{\Delta }}}_p={\eta }_p\sigma \left(t\right)$$

(33)

$${\stackrel{·}{\hat{\Delta }}}_q={\eta }_q\lambda \sigma \left(t\right)$$

(34)

Proof

The Lyapunov function is given by

$$V\left(t\right)=\frac{s_m^2\left(t\right)}{2}+\frac{s_c^2\left(t\right)}{2}+\frac{g_0}{2{\eta }_{\alpha }}{\tilde{\mathbf{\alpha }}}^T\tilde{\mathbf{\alpha }}+\frac{g_0}{2{\eta }_{\beta }}{\tilde{\mathbf{\beta }}}^T\tilde{\mathbf{\beta }}+\frac{g_0}{2{\eta }_p}{\tilde{\Delta }}_p^2+\frac{g_0}{2{\eta }_q}{\tilde{\Delta }}_q^2$$

(35)

where $\tilde{\mathbf{\alpha }}={\mathbf{\alpha }}^{*}-\hat{\mathbf{\alpha }}$ , $\tilde{\mathbf{\beta }}={\mathbf{\beta }}^{*}-\hat{\mathbf{\beta }}$ , ${\tilde{\Delta }}_p={\Delta }_p-{\hat{\Delta }}_p$ , and ${\tilde{\Delta }}_q={\Delta }_q-{\hat{\Delta }}_q$ ; ${\eta }_{\alpha }$ , ${\eta }_{\beta }$ , ${\eta }_p$ , and ${\eta }_q$ are positive constants. Taking the derivative of the Lyapunov function and using equations (24)–(28) yields

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{·}{V}\left(t\right)}& =\sigma \left(t\right)[{\stackrel{·}{s}}_m\left(t\right)-\lambda s_c\left(t\right)]-\frac{g_0}{{\eta }_{\alpha }}{\tilde{\mathbf{\alpha }}}^T\stackrel{·}{\hat{\mathbf{\alpha }}}-\frac{g_0}{{\eta }_{\beta }}{\tilde{\mathbf{\beta }}}^T\stackrel{·}{\hat{\mathbf{\beta }}}-\frac{g_0}{{\eta }_p}{\tilde{\Delta }}_p{\stackrel{·}{\hat{\Delta }}}_p-\frac{g_0}{{\eta }_q}{\tilde{\Delta }}_q{\stackrel{·}{\hat{\Delta }}}_q\hfill \\ \multicolumn{1}{c}{}& =\sigma \left(t\right)[{\stackrel{··}{y}}_d\left(t\right)-f(\mathbf{y},t)-g_{0}u\left(t\right)-d\left(t\right)+2\lambda \stackrel{·}{e}\left(t\right)+{\lambda }^{2}e\left(t\right)-\lambda s_c\left(t\right)]-\frac{g_0}{{\eta }_{\alpha }}{\tilde{\mathbf{\alpha }}}^T\stackrel{·}{\hat{\mathbf{\alpha }}}-\frac{g_0}{{\eta }_{\beta }}{\tilde{\mathbf{\beta }}}^T\stackrel{·}{\hat{\mathbf{\beta }}}-\frac{g_0}{{\eta }_p}{\tilde{\Delta }}_p{\stackrel{·}{\hat{\Delta }}}_p-\frac{g_0}{{\eta }_q}{\tilde{\Delta }}_q{\stackrel{·}{\hat{\Delta }}}_q\hfill \\ \multicolumn{1}{c}{}& =\sigma \left(t\right)g_0[g_0^{-1}H(\mathbf{y},t)-u\left(t\right)+\lambda g_0^{-1}O(\mathbf{y},t)-\lambda s_c\left(t\right)]-\frac{g_0}{{\eta }_{\alpha }}{\tilde{\mathbf{\alpha }}}^T\stackrel{·}{\hat{\mathbf{\alpha }}}-\frac{g_0}{{\eta }_{\beta }}{\tilde{\mathbf{\beta }}}^T\stackrel{·}{\hat{\mathbf{\beta }}}-\frac{g_0}{{\eta }_p}{\tilde{\Delta }}_p{\stackrel{·}{\hat{\Delta }}}_p-\frac{g_0}{{\eta }_q}{\tilde{\Delta }}_q{\stackrel{·}{\hat{\Delta }}}_q\hfill \\ \multicolumn{1}{c}{}& =\sigma \left(t\right)g_0[P\left(t\right)-u\left(t\right)+\lambda Q\left(t\right)-\lambda s_c\left(t\right)]-\frac{g_0}{{\eta }_{\alpha }}{\tilde{\mathbf{\alpha }}}^T\stackrel{·}{\hat{\mathbf{\alpha }}}-\frac{g_0}{{\eta }_{\beta }}{\tilde{\mathbf{\beta }}}^T\stackrel{·}{\hat{\mathbf{\beta }}}-\frac{g_0}{{\eta }_p}{\tilde{\Delta }}_p{\stackrel{·}{\hat{\Delta }}}_p-\frac{g_0}{{\eta }_q}{\tilde{\Delta }}_q{\stackrel{·}{\hat{\Delta }}}_q\hfill \\ \multicolumn{1}{c}{}& =\sigma \left(t\right)g_0[P^{*}\left(t\right)+{\Delta }_p+\lambda (Q^{*}\left(t\right)+{\Delta }_q)-\lambda s_c\left(t\right)-u\left(t\right)]-\frac{g_0}{{\eta }_{\alpha }}{\tilde{\mathbf{\alpha }}}^T\stackrel{·}{\hat{\mathbf{\alpha }}}-\frac{g_0}{{\eta }_{\beta }}{\tilde{\mathbf{\beta }}}^T\stackrel{·}{\hat{\mathbf{\beta }}}-\frac{g_0}{{\eta }_p}{\tilde{\Delta }}_p{\stackrel{·}{\hat{\Delta }}}_p-\frac{g_0}{{\eta }_q}{\tilde{\Delta }}_q{\stackrel{·}{\hat{\Delta }}}_q\hfill \end{array}$$

(36)

where $H(\mathbf{y},t)={\stackrel{··}{y}}_d(t)-f(\mathbf{y},t)-d(t)$ , $O(\mathbf{y},t)=2\stackrel{·}{e}(t)+\lambda e(t)-s_c(t)+g_0{s}_c(t)$ , $P(t)=g_0^{-1}H(\mathbf{y},t)$ , and $Q(t)=g_0^{-1}O(\mathbf{y},t)$ . The adaptive 2S-PSOWNN with SL controller is proposed to estimate and replace the two complex dynamic functions $P(t)$ and $Q(t)$ including the external disturbance $d(t)$ . Using equations (29) and (30), the formula above can be rewritten as

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{·}{V}\left(t\right)}& =\sigma \left(t\right)g_0\hfill \\ \multicolumn{1}{c}{}& [P^{*}\left(t\right)+\lambda Q^{*}\left(t\right)-U_m\left(t\right)+{\Delta }_p+\lambda {\Delta }_q-U_c\left(t\right)-\lambda s_c\left(t\right)]\hfill \\ \multicolumn{1}{c}{}& -\frac{g_0}{{\eta }_{\alpha }}{\tilde{\mathbf{\alpha }}}^T\stackrel{·}{\hat{\mathbf{\alpha }}}-\frac{g_0}{{\eta }_{\beta }}{\tilde{\mathbf{\beta }}}^T\stackrel{·}{\hat{\mathbf{\beta }}}-\frac{g_0}{{\eta }_p}{\tilde{\Delta }}_p{\stackrel{·}{\hat{\Delta }}}_p-\frac{g_0}{{\eta }_q}{\tilde{\Delta }}_q{\stackrel{·}{\hat{\Delta }}}_q\hfill \\ \multicolumn{1}{c}{}& =-\lambda g_0{\sigma }^2\left(t\right)+\sigma \left(t\right)g_0[{\tilde{\mathbf{\alpha }}}^T\mathbf{\Omega }+\lambda {\tilde{\mathbf{\beta }}}^T\mathbf{\Omega }+{\tilde{\Delta }}_p+\lambda {\tilde{\Delta }}_q]\hfill \\ \multicolumn{1}{c}{}& -\frac{g_0}{{\eta }_{\alpha }}{\tilde{\mathbf{\alpha }}}^T\stackrel{·}{\hat{\mathbf{\alpha }}}-\frac{g_0}{{\eta }_{\beta }}{\tilde{\mathbf{\beta }}}^T\stackrel{·}{\hat{\mathbf{\beta }}}-\frac{g_0}{{\eta }_p}{\tilde{\Delta }}_p{\stackrel{·}{\hat{\Delta }}}_p-\frac{g_0}{{\eta }_q}{\tilde{\Delta }}_q{\stackrel{·}{\hat{\Delta }}}_q\hfill \end{array}$$

(37)

Substituting the parameter estimation laws (equations (31)–(34)) into equation (37) yields

$$\stackrel{·}{V}\left(t\right)=-\lambda g_0{\sigma }^2\left(t\right)$$

(38)

Since $\stackrel{·}{V}(t)$ is negative semidefinite, that is, $V(t)\le V(0)$ , it implies that $s_m(t)$ , $s_c(t)$ , $\tilde{\mathbf{\alpha }}$ , $\tilde{\mathbf{\beta }}$ , ${\tilde{\Delta }}_p$ , and ${\tilde{\Delta }}_q$ are bounded. As $V(0)$ is bounded, $V(t)$ is nonlinear increasing and bounded, the conclusion can be obtained: $e(t)$ will converge to 0 as $t\to \infty$ , the stability of the ELS system can be guaranteed.

Simulation

In order to demonstrate the applicability and effectiveness of the adaptive 2S-PSOWNN with SL controller, the reference commands with different algorithms are applied to the simulation platform for the ELS, as shown in Figure 1, which are evaluated by the root mean square error (RMSE) of the output torque. The following simulations are carried out in an Intel Core i5 CPU with 3.2 GHz rate, 4 GB RAM and 64-bit operating system. The parameters of the actual gun control system are as follows: the rotary inertia of the turret is 7000 kg m², the total friction torque $M_f$ is less than 1200 N m, and the center position of gravity is located in 0.35 m away from the rotation center of the turret; the reduction ratio is 360. The relative parameters are selected as $\lambda =1.6$ , $L=40$ , $E_{\mathrm{th}}=0.1$ , $I_{\mathrm{th}}=0.01$ , $G_{\mathrm{th}}=0.6$ , $\rho =0.3$ , $\tau =0.01$ , ${\eta }_{\alpha }=0.004$ , $d_g=0.25$ , and ${\eta }_{\beta }={\eta }_p={\eta }_q=0.0012$ , the control period is 5 ms, and the sampling frequency is 10 kHz. The important parameters of the torque motor are shown in Table 1.

Table 1.

Parameters of the torque motor.

Parameters	Values
Rated power, $p_r$	2.8 kW
Biggest locked-rotor torque, $T_{max}$	19.8 N
Continuous locked-rotor torque, $T_{\mathrm{cont}.}$	12 N
Continuous current, $I_{\mathrm{cont}.}$	80 A
Rated speed, $v_r$	1200 r/min
Coefficient of electromagnetic torque, $K_t$	2.6 N m/A
Coefficient of counter electromotive force, $K_e$	9.6 V/k r/min
Equivalent inductance of armature circuit, $L_m$	0.0036 H
Equivalent resistance of armature circuit, $R_m$	1.2 Ω
Rotational inertia, $J_m$	0.038 kg m2
Viscous friction coefficient, $B_m$	0.22 N m s/rad

Convergence analysis

Taking the real-time characteristics of the ELS system into consideration, the high computation efficiency and the great convergence rate must be guaranteed. Figure 4 depicts the RMSE profiles for all the five different methods (PSO, backpropagation (BP), double sliding modes–particle swarm optimization with backpropagation (2S-PSOBP), 2S-PSOWNN, and 2S-PSOWNN with SL). ³³ It is clear that the adaptive 2S-PSOWNN with SL meets the control precision about 22 iterations quickly when the required RMSE is set 0.001 N m. The worst convergence iterations are even 68, which is relatively not suitable for the ELS system.

Figure 4.

Number of iterations.

Step response

The step response with the external disturbance of the adaptive 2S-PSOWNN and 2S-PSOWNN with SL is evaluated in Figure 5, which further shows the rapidity and stability of the proposed algorithms in the ELS. From the simulation results, it is clear that the adaptive PSOWNN with SL controller meets the requirements in 63 ms when the amplitude of the reference command is 10 N m. However, the adaptive 2S-PSOWNN controller achieves the goal which is 25 ms later than the former above. At the time of 145 ms, an external disturbance is exerted to the ELS system, and the latter performs worser than the former, which proves that the addition of SL algorithm improves the robustness of the control system to some extent.

Figure 5.

Comparison of step response with the external disturbance.

Sinusoidal tracking

In order to simulate the complexities existing in the practical GCS, the sinusoidal motions of the torque motor are taken as the reference command when the load motor is working in the different working styles. First, the output torque of the adaptive 2S-PSOWNN and 2S-PSOWNN with SL controller is shown in Figure 6 with the reference command $x_{d1}(t)=5\sin 2\pi t$ when the load motor is set in the static state. The amplitude errors are $4.61\%$ and $1.86\%$ , and the phase errors are $-2.3844\circ$ and $-0.9318\circ$ , which indicate that the latter performs better than the former.

Figure 6.

Sinusoidal tracking at the static loading.

Then, considering the time-varying position of the load motor and further analyzing the stability and precision of the desired outputs with different frequencies and amplitudes, the reference commands of the torque motor are selected as $x_{d2}(t)=5\sin 20\pi t$ , $x_{d3}(t)=5\sin 10\pi t$ , $x_{d4}(t)=\sin 2\pi t$ , $x_{d5}(t)=5\sin 2\pi t$ , and $x_{d6}(t)=10\sin 2\pi t$ , and the load motor works in the sinusoidal motion $\Delta \mathrm{dit}=\sin (5\pi t)\mathrm{rad}$ based on the practical GCS angle variation, and the practical load motor position is collected by the resolver and is shown in Figure 7. The experimental results are presented in Figures 8–12, and the dynamic experimental results of the sinusoidal tracking are shown in Table 2.

Figure 7.

Sinusoidal motion of the load motor.

Figure 8.

Sinusoidal tracking with $x_{d2}(t)=5\sin 20\pi t$ .

Figure 9.

Sinusoidal tracking with $x_{d3}(t)=5\sin 10\pi t$ .

Figure 10.

Sinusoidal tracking with $x_{d4}(t)=\sin 2\pi t$ .

Figure 11.

Sinusoidal tracking with $x_{d5}(t)=5\sin 2\pi t$ .

Figure 12.

Sinusoidal tracking with $x_{d6}(t)=10\sin 2\pi t$ .

Table 2.

Dynamic experiment results of the sinusoidal tracking.

Reference command	Amplitude error (%)	Phase error (°)
$5\sin 20\pi t$	6.74	−8.9154
$5\sin 10\pi t$	5.81	−6.5217
$\sin 2\pi t$	3.49	−4.2476
$5\sin 2\pi t$	2.73	−2.9712
$10\sin 2\pi t$	2.35	−2.1173

From Table 2, the experiment results indicate that the amplitudes and frequencies of the reference commands play great effects on the performance of the ELS system, but the proposed algorithm wholly meets the control requirements. In combination with the time-varying position of the load motor, the controller shows better characteristics when the load motor is working in the static state. When the load motor is set in the sinusoidal motion, the bigger amplitude and the lower frequency of the reference commands are superior to the smaller amplitude and the higher frequency. However, what we need to remind is that the control effectiveness is relatively poorer when the direction of the torque motor is changed, which may be influenced by the friction and backlash and needs to be further studied in the future.

Conclusion

In this article, an adaptive 2S-PSOWNN with SL algorithm is proposed and applied to the ELS system. The parameters of the WNN are online tuned by the adaptive PSO, and the SL algorithm is adopted to add and prune the number of hidden nodes to improve the computation efficiency. 2S not only enhance the approximation accuracy but also guarantee the stability in the sense of Lyapunov theory. The simulation results show that the proposed controller has superior performance and great robustness in many ways.