Sliding Mode Control of Servo Feed System Based on Fuzzy Reaching Law

Abstract

An adaptive sliding mode control method based on the fuzzy exponential convergence law is proposed to solve the jitter problem caused by the sliding mode control of the servo-feed system and to improve the tracking performance of the system. The design of fuzzy rules for adaptive adjustment of convergence law parameters in sliding mode control improves the convergence speed of the sliding mode function, eliminates unknown disturbances in the system, and weakens the chattering of the system. The proposed method is simulated and experimentally verified by a parallel mobile platform. The results show that the sliding mode control method based on the fuzzy convergence law has strong disturbance suppression capability, high position tracking accuracy, and effective chattering suppression, and the maximum tracking error is reduced by 43.1% and 31.5%, respectively, compared with PID control and exponential sliding mode control.

1. Introduction

CNC machine tool is a kind of high precision and high efficiency machining equipment, and its servo feed system is one of the most important control systems in machine tools. The system enables precise control of parameters such as feed speed, position and force during machine tool machining by using components such as sensors, actuators and controllers, resulting in higher accuracy and stability of the machine tool machining process [1]. With the continuous development of the aviation industry, automotive industry and mold manufacturing, the performance of CNC machine tools has put forward higher requirements. Typically, the drive model of a servo feed system can be equated to a rigid model. Although the rigid body model of the system is simple, it can only reflect the dynamic characteristics of the system during low-speed operation. When the servo feed system operates at high speed, the screw drive components generate axial vibration and torsional deformation, causing structural vibration [2]. The generation of structural vibrations then limits the achievable bandwidth and is a decisive factor in the tracking and positioning accuracy in the feed system [3]. In real systems, both parameter uncertainty and external perturbations are important factors that affect high accuracy positioning and tracking performance [4]. In addition, external disturbances dominate the nonlinear friction and cutting forces [5].

Structural vibrations, external disturbances and parameter uncertainties of the servo-feed system are the main factors that lead to the inability of the classical controller to meet the requirements of high precision control [6]. In response to these challenges, a range of advanced control strategies and advanced technologies are required to develop faster and higher tracking performance servo-feed systems [7]. The literature [8,9] investigated various control issues of servo position systems, including their stability, output regulation capability, and interference immunity. To address the system uncertainty, Msukwa et al. [10] proposed a nonlinear sliding mode controller with a feedforward compensator, which led to a significant improvement in the tracking performance of the feed drive system. By means of Lyapunov stability theory, the stability of the system is analyzed and confirmed, and its convergence to the sliding surface is ensured. For the high-precision position control of electro-hydraulic servo pump control system, Song et al. [11] proposed a compensation control algorithm based on fuzzy control theory on the basis of classical PID control algorithm for reducing the influence of system by load disturbance and enhancing the anti-interference capability of the system. Chen et al. [12] proposed a composite nonlinear feedback adaptive integral sliding mode controller with convergence law (CNF-AISMRL) for fast and accurate control of a servo position control system subject to external disturbances. Zhang et al. [13] applied an autoturbation controller (ADRC) to motor position control of a servo-driven feed system to suppress vibration and disturbances, and used a simple PI controller to achieve accurate position control of the motion table. Rajabi et al. [14] proposed a sliding mode control method based on extended Kalman filter and traceless Kalman filter online state estimation to ensure the tracking accuracy at high speed feed in the presence of noise, friction and uncertainty factors in the ball screw feed system, but the isokinetic convergence law it uses leads to jitter problems.

The sliding mode control algorithm has a fast response to the system, is less affected by internal and external disturbances, and has good robustness [15], and is widely used to control servo-feed systems [16]. However, chattering is easy to occur in the controlled system because of the switching control behavior near the sliding mode surface. The presence of chattering makes it difficult to improve the performance of the control system and limits the application of sliding mode control [17]. To solve the jitter problem, domestic and foreign scholars have tried to reduce jitter by reconstructing sliding modes, constructing state observers and adjusting sliding mode parameters [18,19,20]. The main methods are the dynamic sliding mode method [21], the improved convergence law parameter method [22], the quasi-sliding mode boundary layer thickness adjustment method [23], the signal filtering method [24], and the applied interference observer method [25]. They all weaken the jitter phenomenon to some extent. Among them, the design of convergence law can not only improve the motion quality when the sliding mode function converges to the sliding mode surface, but also can effectively reduce the jitter. In the literature [26], an improved fast approximation law is proposed to shorten the time to reach the slip surface and speed up the convergence of the system. In the literature [6], a new segmentation function is applied to the power convergence law to replace the symbolic function in the traditional sliding mode control, which improves the speed of system convergence to the sliding mode surface and effectively reduces the high frequency chattering of the system. An adaptive timing fault-tolerant control scheme is proposed in the literature [27]. Using inverse trigonometric functions, a new double power convergence law is constructed to reduce the chattering phenomenon while accelerating the state stability. In the literature [28,29], an improved power exponential convergence law sliding mode velocity controller was designed to adjust the convergence law parameters of the sliding mode control using the integral term and fuzzy control, which not only improved the convergence velocity but also weakened the chattering to some extent.

In the above-mentioned literature, some scholars have not conducted experimental studies on the adopted convergence law method, and the convergence law is less applied in the sliding mode control of servo-feed systems. In this paper, we address the problem of fixed convergence parameters in the traditional exponential convergence law, which leads to large sliding mode jitter and tracking errors, and use fuzzy algorithm to soften the sliding mode control signal from the perspective of combining sliding mode variable structure control with intelligent algorithm. An adaptive fuzzy sliding mode controller (FSMC) is designed for the position tracking control of a servo-feed drive system. Experiments are conducted to verify the effectiveness of the designed control strategy compared to PID controllers and sliding mode controllers.

2. Flexible Dynamics Modeling of the Feed System

The rigid body model can better describe the dynamic characteristics of the system in the low frequency band of the system, which is easy to implement the control method. However, under high-speed operating conditions, the ball screw drive sub of the servo feed system will inevitably undergo elastic deformation, making the system susceptible to shaft–torsion coupling vibration, which will have a great impact on the tracking performance of the system. Therefore, the flexible body characteristics exhibited by the system in the high frequency band must be fully considered to establish the flexible body dynamics model of the servo-feed system.

Figure 1 shows the servo feed system model: the AC servo motor drives the screw to rotate through the coupling, the nut on the screw is fixed to the table by the nut sleeve, both ends of the platform are supported by rolling linear guides and the servo motor generates motor torque through a closed-loop servo control system, which controls the position of the table.

Several two-dimensional models have been proposed in the literature [30] to simulate the rigid body model and the first vibration pattern of the ball screw drive. In these methods, the concentrated mass model can effectively capture the stiffness characteristics and the main vibration pattern on the ball screw feed drive system and be easily used for controller design [31]. According to the concentrated mass method, the servo-feed system model is equated to a two-degree-of-freedom mass-spring-damping system, as shown in Figure 2, and the physical significance of each parameter is shown in

Table 1. Servo feed system flexible body model parameters table.

Parameters	Physical Meaning
$m_1$	Equivalent mass of model rolling parts
$m_2$	Equivalent mass of model moving parts
$x_1$	Ball Screw Rotation Equivalent Displacement
$x_2$	Table linear displacement
$k$	Axial stiffness coefficient of the screw
$c$	Damping factor of ball screw to nut connection
$b_1$	Equivalent damping coefficient of motor spindle and bearing
$b_2$	Damping factor of linear guide
$d_1$	Disturbances acting on the motor
$d_2$	Perturbations acting on the moving platform
$u$	Control voltage signal for motor torque input

.

In Figure 2, the equation of motion for the concentrated mass model can be written as:

$$\begin{array}{l}{m}_1\stackrel{••}{x_1}=-b_1\stackrel{•}{x_1}+k(x_2-x_1)+c(\stackrel{•}{x_2}-\stackrel{•}{x_1})+u+d_1\\ m_2\stackrel{••}{x_2}=-b_2\stackrel{•}{x_2}+k(x_1-x_2)+c(\stackrel{•}{x_1}-\stackrel{•}{x_2})+d_2\end{array}$$

(1)

The transfer dynamics between the motor and the table can be derived from the equation of motion (1) and transformed by Laplace as:

$$\begin{array}{l}{G}_M(S)=\frac{X_1(S)}{U(S)}=\frac{m_2{s}^2+(c+b_2)s+k}{m_1{m}_2{s}^4+a_3{s}^3+a_2{s}^2+a_{1}s+a_0}\\ G_T(S)=\frac{X_2(S)}{U(S)}=\frac{cs+k}{m_1{m}_2{s}^4+a_3{s}^3+a_2{s}^2+a_{1}s+a_0}\end{array}$$

(2)

Among them:

$$\begin{array}{l}{a}_0=0\\ a_1=(b_1+b_2)k\\ a_2=b_1{b}_2+c(b_1+b_2)+k(m_1+m_2)\\ a_3=m_1{b}_2+m_2{b}_1+c(m_1+m_2)\end{array}$$

$G_M\left(S\right)$ and $G_T\left(S\right)$ are the transfer functions between motor torque to motor position and motor torque to table position, respectively.

Equation (2) describes the overall transfer dynamics of the servo-feed system. Since the characteristic equations are of fourth order, they are converted into state-space form by selecting four state quantities. In servo feed system installations, the position and speed of the motor and table are measured by rotary encoder and scale systems, respectively [32]. Select the state vector as $x=[\begin{array}{cccc}{x}_1& \stackrel{•}{x_1}& x_2& \stackrel{•}{x_2}\end{array}{]}^T$, then Equation (2) can be rewritten as the equation of state for the two-degree-of-freedom flexible body model:

$$\stackrel{•}{x}=A_{M}v+B_{M}u$$

(3)

Among them:

$$A_M=\left[\begin{array}{cccc}0& 1& 0& 0\\ -\frac{k}{m_1}& -\frac{(c+b_1)}{m_1}& \frac{k}{m_1}& \frac{c}{m_1}\\ 0& 0& 0& 1\\ \frac{k}{m_2}& \frac{c}{m_2}& -\frac{k}{m_2}& -\frac{(c+b_2)}{m_2}\end{array}\right]$$

$$B_M=[\begin{array}{cccc}0& \frac{1}{m_1}& 0& 0\end{array}{]}^T$$

$$x={\left[\begin{array}{cccc}{x}_1& \stackrel{•}{x_1}& x_2& \stackrel{•}{x_2}\end{array}\right]}^T$$

3. Design of Fuzzy Convergence Law Sliding Mode Controller

3.1. Sliding Mode Controller Design

Sliding mode variable structure control means that the controlled system constantly changes its own state through the designed control law, so that the trajectory of the system moves towards the switching surface. The design of the switching surface does not need to consider the parameter changes of the controlled system and external disturbances, so its robust performance is better. However, in the process of following the switching surface, the system keeps crossing the switching surface due to the constant change of its own state, and the control mode shows discontinuity, thus triggering the chattering phenomenon.

Before designing the sliding mode controller, define the position tracking error of the system as follows:

$$e=x_r-x$$

(4)

where, $e={\left[e_1\hspace{1em}{e}_2\hspace{1em}{e}_3\hspace{1em}{e}_4\right]}^T$, $x={\left[x_1\hspace{1em}\stackrel{•}{x_1}\hspace{1em}{x}_2\hspace{1em}\stackrel{•}{x_2}\right]}^T$, $x_r={\left[x_{1r}\hspace{1em}\stackrel{•}{x_{1r}}\hspace{1em}{x}_{2r}\hspace{1em}\stackrel{•}{x_{2r}}\right]}^T$, $x_r$ is the reference value of the state variable. From Equations (1) and (4), the dynamic error of the system can be introduced as:

$$\left\{\begin{array}{l}\stackrel{•}{e_1}=\stackrel{•}{x_{1r}}-\stackrel{•}{x_1}=e_2\\ \stackrel{•}{e_2}=\stackrel{••}{x_{1r}}-\stackrel{••}{x_1}=\stackrel{••}{x_{1r}}+f_1-\frac{1}{m_1}u-\frac{1}{m_1}{d}_1\\ \stackrel{•}{e_3}=\stackrel{•}{x_{2r}}-\stackrel{•}{x_2}=e_4\\ \stackrel{•}{e_4}=\stackrel{••}{x_{2r}}-\stackrel{••}{x_2}=\stackrel{••}{x_{2r}}+f_2-\frac{1}{m_2}{d}_2\end{array}\right.$$

(5)

where:

$$\left\{\begin{array}{l}{f}_1=\frac{k}{m_1}{x}_1-\frac{k}{m_1}{x}_2+\frac{b_1+c}{m_1}\stackrel{•}{x_1}-\frac{c}{m_1}\stackrel{•}{x_2}\\ f_2=-\frac{k}{m_2}{x}_1+\frac{k}{m_2}{x}_2-\frac{c}{m_2}\stackrel{•}{x_1}+\frac{b_2+c}{m_2}\stackrel{•}{x_2}\end{array}\right.$$

To achieve position tracking control of the moving platform, the sliding mode function must be associated with the error of the state variable to ensure that the system can effectively track the reference trajectory, defining the sliding function as follows:

$$\sigma ={\lambda }_1{e}_1+{\lambda }_2{e}_2+{\lambda }_3{e}_3+{\lambda }_4{e}_4$$

(6)

where, ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4$ are normal numbers. From Equation (6), the state variable $\stackrel{•}{x_1}$ are directly related to the control law $u$, for the sake of mathematical derivation, take ${\lambda }_2=1$. Then the derivative of the sliding mode function sliding mode surface is:

$$\begin{array}{ll}\stackrel{•}{\sigma }& ={\lambda }_1\stackrel{•}{e_1}+\stackrel{•}{e_2}+{\lambda }_3\stackrel{•}{e_3}+{\lambda }_4\stackrel{•}{e_4}\hfill \\ & ={\lambda }_1{e}_2+\stackrel{••}{x_{1r}}+f_1+{\lambda }_3{e}_4+{\lambda }_4(\stackrel{••}{x_{2r}}+f_2)-\frac{1}{m_1}u+D\hfill \end{array}$$

(7)

where, $D$ is the total disturbance to the system, $D=-\frac{1}{m_1}{d}_1-{\lambda }_4\frac{1}{m_2}{d}_2$. To confirm that the control system is stable, a control law needs to be selected to drive the system to follow the reference trajectory. The chosen control law must ensure that the error decays exponentially with time. Due to the presence of unknown disturbances in the system, the following robust terms are designed:

$$\stackrel{•}{\sigma }=-h\sigma -\epsilon \mathrm{sgn}(\sigma )$$

(8)

where, $h>0$, $\epsilon >0$, $\mathrm{sgn}\left(·\right)$ is the symbolic function, $-h\sigma$ is an exponential convergence law, $-\epsilon \mathrm{sgn}\left(\sigma \right)$ is an isokinetic convergence law. The control law u of the system is obtained by associating (7) and (8):

$$u=m_1\left[{\lambda }_1{e}_2+{\lambda }_3{e}_4+\stackrel{••}{x_{1r}}+f_1+{\lambda }_4(\stackrel{••}{x_{2r}}+f_2)+D+h\sigma +\epsilon \mathrm{sgn}(\sigma )\right]$$

(9)

If the control law in Equation (9) is used and the upper bound of the disturbance is known, then the controlled system is asymptotically stabilized over time and the tracking error of the system state variables eventually converges to zero.

Proof. 

Define the Lyapunov function:

$$V_1=\frac{1}{2}{\sigma }^2$$

(10)

Combining Equations (9) and (10), we can see that:

$$\begin{array}{cc}{\stackrel{•}{V}}_1& =\sigma \stackrel{•}{\sigma }\hfill \\ & =\sigma [{\lambda }_1{e}_2+{\lambda }_3{e}_4+\stackrel{••}{x_{1r}}+f_1+{\lambda }_4(\stackrel{••}{x_{2r}}+f_2)-\frac{1}{m_1}u+D]\hfill \\ & =\sigma [-D_{\sup }\mathrm{sgn}(\sigma )+D-h\sigma -\epsilon \mathrm{sgn}(\sigma )]\hfill \\ & =-D_{\sup }\left|\sigma \right|+D\sigma -h{\sigma }^2-\epsilon \left|\sigma \right|\hfill \\ & \le \left(\left|D\right|-D_{\sup }\right)\left|\sigma \right|-h{\sigma }^2-\epsilon \left|\sigma \right|\hfill \\ & \le 0\hfill \end{array}$$

(11)

In summary, $V_1\ge 0$, ${\stackrel{•}{V}}_1\le 0$, Lyapunov stability condition is satisfied, the stability of the system is ensured. □

3.2. Fuzzy Convergence Law Sliding Mode Controller Design

3.2.1. Design of a Novel Fuzzy Exponential Convergence Law

In the traditional exponential convergence law sliding mode controller, the convergence law parameters $\epsilon$ and $h$ are scalars, and the values are fixed. The different values of the convergence law parameters affect the arrival time of the system at the sliding mode surface and the state after arrival, specifically $\epsilon$ affects the arrival time and $h$ affects the convergence speed. To enhance system robustness and anti-interference capability, $\epsilon$ and $h$ should be adaptively adjusted according to system changes to meet the requirements of high-performance controllers. The expression of the traditional exponential convergence law is:

$$\stackrel{•}{\sigma }=-\epsilon \mathrm{sgn}(\sigma )-h\sigma \hspace{1em}\hspace{1em}(\epsilon ,h>0)$$

(12)

From the expressions, the values of $\epsilon$ and $h$ can be dynamically adjusted to weaken the system chattering, with $\epsilon$ ensuring a finite arrival time and $h$ ensuring a fast arrival. Therefore, to ensure the rapidity of the convergence motion and to suppress the chattering, the parameters $\epsilon$ and $h$ need to be selected appropriately. With $\sigma$ and $\stackrel{•}{\sigma }$ as the input fuzzy variables of the system and $\epsilon$ and $h$ as the fuzzy outputs, define $\epsilon =fuzzy{\epsilon }_n(\sigma ,\stackrel{•}{\sigma }),\hspace{1em}h=fuzzy{h}_n(\sigma ,\stackrel{•}{\sigma })$, such that ε and h can be adaptively adjusted according to the system changes.

The fuzzy convergence law can effectively weaken the jitter of the system, but the traditional exponential convergence law uses the discontinuous symbolic function $\mathrm{sgn}()$ to make the system state variables switch back and forth under the sliding mode surface, and as long as the control contains the symbolic function of $\mathrm{sgn}()$, chattering is inevitably present in the control output. The saturation function can effectively weaken the chattering and make the output smooth and bounded. In this paper, the convergence law is designed using hyperbolic tangent function (a saturation function) instead of the sign function, and the hyperbolic tangent function is as follows:

$$\tanh (s)=\frac{\exp (\sigma )-\exp (-\sigma )}{\exp (\sigma )+\exp (-\sigma )}$$

(13)

A comparison of the symbolic and hyperbolic tangent functions is shown in Figure 3. The hyperbolic tangent function can make the switching process continuous and smooth, which is very important to reduce chattering.

Then the new fuzzy exponential convergence law is designed as follows:

$$\stackrel{•}{\sigma }=-fuzzy{\epsilon }_n(\sigma ,\stackrel{•}{\sigma })\hspace{0.17em}\tanh (\sigma )-fuzzy{h}_n(\sigma ,\stackrel{•}{\sigma })\hspace{0.17em}\sigma$$

(14)

3.2.2. Design of a New Fuzzy Convergence Law Sliding Mode Controller

A new fuzzy convergence law is introduced for the sliding mode controller designed in Section 3.1, and the expression of the controller is obtained by associating (7) and Equation (14) as:

$$u=m_1\left[\begin{array}{l}{\lambda }_1(\stackrel{•}{x_{1r}}-\stackrel{•}{x_1})+{\lambda }_3(\stackrel{•}{x_{2r}}-\stackrel{•}{x_2})+{\lambda }_2(\stackrel{••}{x_{1r}}+\frac{k}{m_2}{x}_1-\frac{k}{m_2}{x}_2+\frac{b_2+c}{m_2}\stackrel{•}{x_1}-\frac{c}{m_2}\stackrel{•}{x_2})\\ +(\stackrel{••}{x_{2r}}-\frac{k}{m_1}{x}_1+\frac{k}{m_1}{x}_2+\frac{b_1+c}{m_1}\stackrel{•}{x_2}-\frac{c}{m_1}\stackrel{•}{x_1})+D+fuzzy{h}_n(\sigma ,\stackrel{•}{\sigma })\sigma \\ +fuzzy{\epsilon }_n(\sigma ,\stackrel{•}{\sigma })\tanh (\sigma )\end{array}\right]$$

(15)

Proof. 

Define the Lyapunov function

$$V_2=\frac{1}{2}{\sigma }^2$$

(16)

Combining (7) and Equation (14), we can see that

$$\begin{array}{cc}\hfill {\stackrel{•}{V}}_2=& \sigma \stackrel{•}{\sigma }\hfill \\ \hfill =& \sigma [{\lambda }_1({\stackrel{•}{x}}_{1r}-{\stackrel{•}{x}}_1)+{\lambda }_2(\stackrel{••}{x_{1r}}+\frac{k}{m_2}{x}_1-\frac{k}{m_2}{x}_2+\frac{b_2+c}{m_2}{\stackrel{•}{x}}_1-\frac{c}{m_2}\stackrel{•}{x_2}-\frac{1}{m_2}{d}_2)\hfill \\ & +{\lambda }_3(\stackrel{•}{x_{2r}}-\stackrel{•}{x_2})+(\stackrel{••}{x_{2r}}-\frac{k}{m_1}{x}_1+\frac{k}{m_1}{x}_2+\frac{b_1+c}{m_1}{\stackrel{•}{x}}_2-\frac{c}{m_1}{\stackrel{•}{x}}_1-\frac{1}{m_1}{d}_1-\frac{1}{m_1}u)]\hfill \\ \hfill =& \sigma [-D_{\sup }tanh\left(\sigma \right)+D-fuzzy{h}_n(\sigma ,\stackrel{•}{\sigma })\sigma -fuzzy{\epsilon }_n(\sigma ,\stackrel{•}{\sigma })tanh\left(\sigma \right)]\hfill \\ \hfill =& -D_{\sup \hspace{0.17em}}\left|\sigma \right|+D\sigma -fuzzy{h}_n(\sigma ,\stackrel{•}{\sigma }){\sigma }^2-fuzzy{\epsilon }_n(\sigma ,\stackrel{•}{\sigma })\left|\sigma \right|\hfill \\ \hfill \le & (\left|D\right|-D_{\sup \hspace{0.17em}})\left|\sigma \right|-fuzzy{h}_n(\sigma ,\stackrel{•}{\sigma }){\sigma }^2-fuzzy{\epsilon }_n(\sigma ,\stackrel{•}{\sigma })\left|\sigma \right|\hfill \\ \hfill \le & 0\hfill \end{array}$$

(17)

In summary, $V_2\ge 0$, $\stackrel{•}{V_2}\le 0$, Lyapunov stability condition is satisfied, thus proving that the new fuzzy convergence law sliding mode controller designed in this paper is stable. □

3.2.3. Parameter Fuzzification and Fuzzy Rule Design

The sliding mode control system of the servo feed system is built in Matlab/simulink, and the range of variation in the basic theoretical domain of each fuzzy variable is obtained. According to the range of variation, the fuzzy domains of input $\sigma$, $\stackrel{•}{\sigma }$ and output $\epsilon$, $h$ are divided into the following classes: $\left\{\sigma ,\stackrel{•}{\sigma }\right\}\in \left\{-3,-2,-1,0,1,2,3\right\}$, $\left\{\epsilon \right\}\in \left\{0,1,2,3\right\}$, $\left\{h\right\}\in \left\{0,1,2,3,4,5,6,7,8,9,10,11,12\right\}$. The ranges of the basic and fuzzy domains are shown in

Table 2. Scope of thesis.

Variables	Basic Domain	Fuzzy Domain
$\sigma$	[−1.5 × 10−3, 1.5 × 10−3]	[−3, 3]
$\stackrel{•}{\sigma }$	[−1.5, 1.5]	[−3, 3]
$\epsilon$	[0, 100]	[0, 3]
$h$	[0, 100]	[0, 12]

.

From the exponential convergence law, it can be solved that:

$$\sigma (t)=\frac{\epsilon }{h}+({\sigma }_0-\frac{\epsilon }{h})e^{-ht}$$

(18)

From Equation (18), it can be seen that decreasing $\epsilon$ and increasing $h$ can accelerate the convergence process and reduce the chattering. When $\left|\sigma \right|$ is large, increasing $h$ will have a large control effect. However, when $\left|\sigma \right|$ is small and $\left|\sigma \right|$ becomes small, a large $h$ should be chosen to ensure the convergence speed. Similarly, when $\left|\sigma \right|$ is large, a larger value of $\epsilon$ ensures the arrival time; When $\left|\sigma \right|$ is small, the value of $\epsilon$ is smaller to avoid the increase in chattering. In summary, the following conclusions can be drawn: When $\left|\sigma \right|$ is large, a large value of $\epsilon$ and a small value of $h$ are selected, and when $\left|\sigma \right|$ is small, a small value of $\epsilon$ and a large value of $h$ are selected. Based on the above findings, a fuzzy control rule table for the two output parameters was developed, as in

Table 3. Table of fuzzy control rules for $h$.

$\mathit{h}$		$\mathit{\sigma }$
		NB	NM	NS	Z	PS	PM	PB
$\stackrel{•}{\sigma }$	NB	NB	NM	NS	Z	PS	PM	PB
	NM	PB	PB	NS	NM	NB	PM	PS
	NS	PB	PB	NM	NM	NB	PM	PS
	Z	PB	PB	NM	NB	NB	PM	PS
	PS	PM	PB	NM	NB	NM	PB	PM
	PM	PS	PM	NB	NB	NM	PB	PB
	PB	PS	PM	NB	NS	NM	PB	PB

and

Table 4. Table of fuzzy control rules for $\epsilon$.

$\mathit{\epsilon }$		$\mathit{\sigma }$
		NB	NM	NS	Z	PS	PM	PB
$\stackrel{•}{\sigma }$	NB	NS	NS	PS	PM	PS	NM	NB
	NM	NM	NS	PM	PM	PS	NM	NB
	NS	NM	NS	PM	PM	Z	NM	NB
	Z	NM	NS	PM	PS	NS	NS	NM
	PS	NB	NM	PB	PS	NS	NS	NM
	PM	NB	NM	PB	PS	NM	NS	NM
	PB	NB	NM	PB	PS	NM	NS	NS

.

By inputting the above empirical rules into the software, a fuzzy 3D rule map can be obtained, as shown in Figure 4.

4. Simulation Analysis

In order to verify the control performance of the fuzzy convergence law sliding mode control method proposed in this paper, a flexible body model of the servo feed system is established and model simulation comparison experiments are conducted. The structure of the fuzzy convergence law sliding mode control system is shown in Figure 5.

The main parameters of the flexible body model of the servo-feed system are shown in

Table 5. Main parameters of the flexible body model.

Parameter	Numeric Value
$m_1(\mathrm{kg})$	50
$m_2(\mathrm{kg})$	80
$k(\mathrm{N}/\mathrm{m})$	5.994 × 107
$c(\mathrm{N}/\mathrm{m}·{\mathrm{s}}^{-1})$	4.596 × 103
$b_1(\mathrm{N}/\mathrm{m}·{\mathrm{s}}^{-1})$	0.0694
$b_2(\mathrm{N}/\mathrm{m}·{\mathrm{s}}^{-1})$	1.382 × 103

.

The smooth reference trajectory helps to suppress the vibration, and the cubic acceleration trajectory is chosen as the reference command for the table position. The maximum displacement is 0.065 m, the maximum velocity is 0.1 m/s, and the maximum acceleration is 1 m/s², as shown in Figure 6.

The flexible body model of the servo feed system is added to the controller designed in this paper, and the given reference trajectory is tracked and compared with the sliding mode control and PID control to verify the feasibility and control effect of the fuzzy convergence law sliding mode control algorithm. The simulation model built by MATLAB /Simulink simulation module is shown in Figure 7.

The fuzzy convergence law sliding mode controller is implemented by programming the S-function. The system inputs are the reference command xr, the ball screw rotational equivalent displacement x₁, the table linear displacement x₂, and the convergence law parameters $\epsilon$ and $h$. The outputs are the slide surface function σ, the slide surface derivative dσ, and the voltage signal ut for controlling the torque. σ and dσ are used as inputs to the fuzzy controller to adjust the parameters $\epsilon$ and $h$ in the exponential convergence law adaptively through fuzzy rules to reduce the chattering phenomenon of the system.

Take the coefficients λ1, λ2, λ3 and λ4 of the sliding mode function as 30, 1, 30 and 0.1, respectively, without considering the interference term, so that d1 = d2 = 0, then the simulation results of dynamic regulation of parameters ε and h are shown in Figure 8.

The fuzzy controller feeds the adjusted parameters $h$ and $\epsilon$ back to the fuzzy convergence law sliding mode controller in real time for the voltage signal adjustment. The voltage signal is passed through the AC servo motor output to control the torque signal of the load. The voltage signal outputs the torque signal of the control load through the AC servo motor, so as to accurately locate the table. The simulation results of the position trajectory tracking of the table are shown in Figure 9.

As can be seen from Figure 9, the table generates significant tracking errors when it is in the speed rise and fall phase. In order to compare the performance of the controller more intuitively, the tracking error and control voltage signal of the platform are simulated, and the simulation results are shown in Figure 10 and Figure 11.

From Figure 10 and Figure 11, it can be seen that the maximum tracking error of PID control is 0.541 mm, and there is a steady-state error of 0.046 mm in the stage of uniform travel of the table, and the response to the control voltage signal is also slow. The maximum tracking error of the sliding mode control is 0.101 mm, and there is obvious chattering in the control voltage signal during the table acceleration and deceleration phase. With the fuzzy convergence law sliding mode controller designed in this paper, the voltage signal is significantly smoother and less chattering, and the maximum tracking error is reduced to 0.051 mm, the comparison results are shown in

Table 6. Comparison of simulation results for PID, sliding mode, and fuzzy convergence law sliding mode controllers.

Control Method	Maximum Tracking Error	Reduction/% Compared to PID, Sliding Mode
PID control	0.541 mm	90.6%
Sliding mode control	0.101 mm	49.5%
Fuzzy convergence law Sliding mode control	0.051 mm	-

. Compared with the previous two, the fuzzy convergence law sliding mode controller has better speed and stability, and the position tracking performance of the table is significantly improved.

Considering the influence of the interference term on the precise positioning of the table, adding d₁ = d₂ = sin(πt) in the simulation, the simulation results are shown in Figure 12 and Figure 13.

From Figure 12 and Figure 13, it can be seen that the maximum tracking error of all three control methods increases after adding the perturbation term to the simulation model. The maximum tracking error of PID control is 1.10 mm, which is 1.054 mm higher than that without disturbance. The maximum tracking error of sliding mode control is 0.615 mm, which is 0.514 mm higher than that without disturbance. However, after the table stops moving, the steady-state errors of 0.304 mm and 0.234 mm exist for the PID control and sliding mode control, respectively, and the tracking accuracy is significantly reduced. Compared with the previous two, the maximum tracking error of the fuzzy convergence law sliding mode control is 0.268 mm, which only increases 0.217 mm. The tracking error curve does not change much from before the disturbance was added, and it can still track the reference trajectory accurately.

Combining the above simulation results and analysis, the fuzzy convergence law sliding mode controller designed in this paper can effectively improve the position tracking accuracy of the servo feed system and suppress the chattering phenomenon of the control voltage signal. Its good robustness can effectively eliminate the steady-state error of the system and achieve high accuracy position tracking of the table when there is disturbance in the system.

5. Experimental Validation

In order to verify the actual control effect of the sliding mode control method based on the fuzzy convergence law on the servo feed system, experiments are conducted under the parallel moving platform [33] test system shown in Figure 14.

The system consists of table, upper computer, linear scale, servo motor, motion controller, AC servo driver, and power supply system. In order to meet the requirements of precise positioning of the platform, the transmission amplitude grating ruler WTB0.5-0300 is used for real-time position feedback of the table. The GTS-400-PV-PCI-G motion controller supports calling dynamic link libraries and compiling the controller code on the PC side using Visual Studio. The motion controller and servo driver are controlled by analog control method. The DAC of the motion controller outputs analog voltage signals to the servo motor driver, and then controls the position of the table.

The trajectory tracking experiments of the table were conducted using PID, sliding mode and fuzzy convergence law sliding mode control methods, respectively, and the motion process of the table was planned according to the curve in Figure 6. The trajectory tracking error and control voltage signal are recorded, and the experimentally collected results are shown in Figure 15, Figure 16 and Figure 17.

Figure 15 shows the experimental results of the PID controller, the tracking error of the table in the acceleration and deceleration stages varies greatly, the maximum tracking error is 0.065 mm, and there is obvious chattering in the control signal; Figure 16 shows the experimental results of the sliding mode controller, the maximum tracking error is 0.054 mm, using the sliding mode control method compared with the PID control method to reduce the tracking error, but the chattering phenomenon of the control signal is not significantly improved. Figure 17 shows the experimental results of the sliding mode controller with fuzzy convergence law, the maximum tracking error is reduced to 0.037 mm, and the chattering of the control signal is significantly reduced.

In order to further compare the control effects of PID controller, sliding mode controller and the fuzzy convergence law sliding mode controller designed in this paper, the experimental results of the three methods are compared, as shown in Figure 18.

After calculation, the maximum tracking error of the table under the fuzzy convergence law sliding mode control is reduced by about 43.1% compared with the PID control method and about 31.5% compared with the sliding mode control method. This control method can further reduce the maximum tracking error of the table, improve the control accuracy, weaken the chattering in the control signal, and have better control effect and better controller performance. The comparison results of the experimental data of the maximum tracking error are shown in

Table 7. Comparison of experimental results of PID, sliding mode, and fuzzy convergence law sliding mode controllers.

Control Method	Maximum Tracking Error	Reduction/% Compared to PID, Sliding Mode
PID control	0.065	43.1%
Sliding mode control	0.054	31.5%
Fuzzy convergence law sliding mode control	0.037	-

.

6. Conclusions

(1)

A two-degree-of-freedom flexible body model reflecting the dynamics of the servo-feed system was developed, and a conventional exponential sliding mode controller was designed based on this model. For the problems of jitter and poor tracking performance caused by fixed exponential convergence parameters, an adaptive sliding mode control method based on the fuzzy convergence law is proposed and the stability of the method is demonstrated.

(2)

The overall simulation model of the system is built by Simulink, and the fuzzy convergence law sliding mode controller is simulated and compared with the traditional PID control and sliding mode control. From the simulation results, it can be seen that the optimized controller effectively suppresses the jitter phenomenon, improves the tracking accuracy of the control system, and eliminates the steady-state error in the uniform speed section.

(3)

The trajectory tracking test of servo feed system with different control algorithms was completed with the help of parallel moving platform. The experimental results show that the sliding mode control algorithm based on the fuzzy convergence law can meet the dynamic and steady-state performance of the servo-feed system better than the PID algorithm and the sliding mode control algorithm, which effectively weaken the jitter in the control signal and reduce the trajectory tracking error of the table.