Perfect Tracking Control of Linear Sliders Using Sliding Mode Control with Uncertainty Estimation Mechanism

Abstract

This paper aims to achieve precise position control of a stage used in semiconductor exposure apparatus. The demand for smart devices, such as smartphones, is rapidly expanding, and their performance is expected to continue to improve. To manufacture these devices, it is necessary to miniaturize semiconductor devices and improve productivity. The precise control of semiconductor exposure apparatus is important for the manufacture of ultra-small semiconductor devices. The stage of semiconductor exposure apparatus uses a linear motor, and this paper performs high-precision perfect tracking control of this stage. Perfect tracking control is a control method that always follows the command value while the command value changes moment by moment, and requires high accuracy. In high-precision positioning, uncertainty in the stage model has a significant impact. Therefore, this paper proposes a method to reduce tracking errors due to the influence of uncertainty by performing uncertainty compensation using sliding mode control with the estimated value of uncertainty. The estimation of uncertainty uses a method that combines Kernel LMS with an observer. Instead of the widely used Gaussian kernel, this paper uses a generalized Gaussian kernel that allows for finer parameter settings. Furthermore, this paper proposes a method to adaptively optimize the shape parameter of the generalized Gaussian kernel. Our simulations and experiments confirm that the proposed method improves tracking performance compared to conventional sliding mode control.

1. Introduction

Currently, attention is focused on the advancement of the IoT society worldwide. Specifically, smartphones and smart home appliances are examples of such devices. In the near future, fully automated vehicles and contact lenses that perform the same functions as smartphones by simply being worn on the eyes are expected to appear. Since semiconductor devices are the foundation of these devices, it is expected that semiconductor device manufacturing technology will become increasingly important. Therefore, various studies have been conducted in different fields to meet the manufacturing requirements of semiconductor devices [1,2,3,4,5,6,7].

This paper aims to achieve high-precision positioning of the exposure stage used in semiconductor manufacturing. To achieve high-precision positioning of the stage, a precise mathematical model that accurately represents the dynamic characteristics of the stage is required. However, due to the nonlinearity and disturbance effects on the dynamic characteristics of the stage, it is difficult to construct a precise mathematical model, and its performance is degraded due to the influence of unconsidered uncertainties. Therefore, in order to achieve high-precision positioning of the stage, it is necessary to construct a control system that takes into account the influence of uncertainties. Two methods have been proposed for constructing control systems that consider uncertainties. The first method enhances the robustness of the controller against uncertainties to compensate for instability under conditions that include uncertainties [8,9,10,11,12]. The second method involves estimating uncertainties and model parameters and modifying the control input based on the estimated values [13,14,15,16]. The first method focuses on stability and does not consider improving control performance, making it unsuitable for high-precision control. Therefore, in this paper, the second method is used to identify uncertainties and modify the control input based on the identification results. In particular, sequential identification of uncertainties is performed to cope with rapidly changing uncertainties. There have been studies on sequential model identification, such as [17,18,19,20,21], and their effectiveness has been confirmed.

Previous studies have proposed a control method based on operator theory for constructing a mathematical model that represents the dynamic characteristics of the stage, focusing on the linear motor’s chain of magnetic flux [22]. Operator theory is a control theory that represents the input–output characteristics of the plant using a nonlinear mapping method called an operator. This design method does not require the state variables of the plant and is easy to implement. It has been proposed for various applications, such as wireless power transmission, processes, and nano materials, as it does not require a special form for nonlinearity [23,24,25,26]. However, previous studies did not consider the influence of uncertainties, resulting in significant discrepancies between simulation results and experimental results. Additionally, control methods using model estimation by AI have been proposed [27,28,29]. Research on stage positioning using air pressure has also been conducted [30,31,32]. In both methods, even if uncertainty compensation is performed, it is difficult to adapt to changes in uncertainties because the uncertainty model is identified offline or learning is performed at long intervals. In this paper, sequential identification of uncertainty models is performed using Kernel LMS [33] and an optimal observer, and uncertainty compensation is performed using the identified uncertainty model and sliding mode control [34,35,36,37]. Kernel LMS is a method that can adaptively identify nonlinear functions by approximating them using the kernel method, and it is widely used in the field of signal processing [38,39,40,41]. There are other methods for identifying nonlinear functions, such as Neural Networks (NNs) [42,43,44] and Support Vector Machines (SVMs) [45,46,47], but they have high learning costs and cannot easily achieve adaptive identification. On the other hand, Kernel LMS has the advantage of low learning costs and the ability to achieve adaptive identification. Research applying Kernel LMS to control has been conducted, such as in [48,49], and its ability to handle complex signals such as drones and active noise cancellation has been confirmed. Many methods using various kernels, including Gaussian kernels, have been proposed, but in this paper, identification is performed using a generalized Gaussian kernel based on generalized Gaussian distribution [50,51]. Furthermore, the shape parameter included in the kernel is sequentially optimized to identify the optimal kernel.

This paper is structured as follows. In Section 2, the equipment used in the experiments is described. In Section 3, the problem formulation is presented. In Section 4, the derivation of the stage model is described. In Section 5, the proposed method for the controller is presented. In Section 6, the effectiveness of the proposed method is verified through simulations and experiments. In Section 7, the paper is summarized.

2. Experimental System

Figure 1 and Figure 2 depict a photograph and a schematic diagram, respectively, of the experimental apparatus employed in this study. The plant consists of a linear motor stage connected to an air slider via bolts, with the stage made to float and friction eliminated by supplying compressed air to the air slider. By applying an electric current to the linear motor, the stage can slide left and right. The position of the stage is observed using a laser sensor attached to the device, and a position signal is sent to the DSP. The DSP processes the position signal and inputs the resulting input voltage signal to the driver circuit, thereby controlling the linear motor by applying a current to it.

3. Problem Statement

The kinetic model of the linear stage P is represented by Equation (1), where $y\in Y$ denotes the stage position, $m>0$ is the mass of the slider, $f:Y\times V\to F$ is the nonlinear external force applied to the stage, and $f_u\in F$ is the manipulated variable.

$$\begin{array}{c}\hfill P:F\to Y:m\ddot{y}\left(t\right)=f(y,\dot{y})\left(t\right)+f_u\left(t\right)\end{array}$$

(1)

Let the operator f be decomposed into a known linear operator $f_0$ and an unknown bounded nonlinear uncertainty operator $∆$, as shown in Equation (2).

$$\begin{array}{c}\hfill f(y,\dot{y})=f_0(y,\dot{y})+∆(y,\dot{y})\end{array}$$

(2)

It is assumed that the following conditions hold for uncertainty $∆$.

Assumption 1. 

∆ is a bounded operator: For any bounded $y\in Y$ and $\dot{y}\in V$, $∆(y,\dot{y})<\infty$ holds. If this assumption is not satisfied, it is impossible to output a manipulated variable that eliminates the estimated uncertainty ∆.

Assumption 2. 

∆ is sufficiently small compared to $f_0$: For any bounded $y\in Y$ and $\dot{y}\in V$, $∆(y,\dot{y})\ll f_0(y,\dot{y})$ holds. Under this assumption, the influence of uncertainty can be reduced by making the estimation error sufficiently small.

Assumption 3. 

The upper limit of the second-order differential value of ∆ is known: For any bounded $y\in Y$ and $\dot{y}\in V$, $\left|{\displaystyle \frac{{\mathrm{d}}^2∆}{\mathrm{d}{t}^2}}(y,\dot{y})\right|\le {\overline{∆}}^{\left(2\right)}$ holds. Under this assumption, it is possible to stabilize the plant by suppressing the upper limit ${\overline{∆}}^{\left(2\right)}$.

Furthermore, the ideal model $P_n$ is defined as shown in Equation (3) using the above $f_0$, and the difference from the actual model is defined as shown in Equation (4), where $y^{*}\in Y$ is the ideal position of the stage, and $f_u^{*}\in F$ is the ideal manipulated variable.

$$\begin{array}{cc}\hfill & P_n:F\to Y:m{\ddot{y}}^{*}\left(t\right)=f_0(y^{*},{\dot{y}}^{*})\left(t\right)+f_u^{*}\left(t\right)\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & e\left(t\right)=y\left(t\right)-y^{*}\left(t\right)\hfill \end{array}$$

(4)

The dynamics of e are expressed as follows:

$$\begin{array}{cc}\hfill \ddot{e}\left(t\right)& =\ddot{y}\left(t\right)-{\ddot{y}}^{*}\left(t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{m}}\left(f_0(e,\dot{e})\left(t\right)+∆(y,\dot{y})\left(t\right)+f_u\left(t\right)-f_u^{*}\left(t\right)\right)\hfill \end{array}$$

(5)

In this paper, a control system that ensures $e\to 0$ using a kernel adaptive filter to eliminate uncertainty $∆$ is proposed. Furthermore, it achieves perfect tracking to a continuous set point r by employing sliding mode control with an integral term.

In this paper, we adopt two notations for the differential quantity of a given state x, $\dot{x}$ and $x^{\left(n\right)}$, with $\ddot{x}$ representing the second-order differential and $x^{\left(n\right)}$ the higher-order cases. That is, $\ddot{x}=x^{\left(2\right)}={\displaystyle \frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}}$.

4. Model

In this paper, an improved version of the model from [22] is used as the model for the plant, which is represented below:

$$\begin{array}{cc}\hfill & m\ddot{y}\left(t\right)=-c\dot{y}\left(t\right)+\mathrm{\Phi }\left(y\right)i\left(t\right)-W^{\prime }\left(y\right)\left(t\right)\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & L\dot{i}\left(t\right)=-Ri\left(t\right)-\mathrm{\Phi }\left(y\right)\dot{y}\left(t\right)+u\left(t\right)\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & \mathrm{\Phi }\left(y\right)=\widehat{a}-\widehat{b}\left(cosh\left(\widehat{k}y\right)-1\right)\hfill \end{array}$$

(8)

$$W^{\prime }\left(y\right)=\frac{L_x{L}_z\left({\mu }_0-{\mu }_f\right)}{2{\mu }_0{\mu }_f}\left(-{\widehat{b}}^{2}sinh\left(\widehat{k}{l}_y\right)sinh\left(2\widehat{k}y\right)+4\left(\widehat{a}\widehat{b}+{\widehat{b}}^2\right)sinh\left(\frac{\widehat{k}}{2}{l}_y\right)sinh\left(\widehat{k}y\right)\right)$$

(9)

where $c>0$ is the velocity resistance of the stage, $\mathrm{\Phi }(·)$ is the operator representing the magnetic flux crossing the coil of the linear motor, $W^{\prime }(·)$ is the operator representing the force due to the magnetic potential of the linear motor, $L>0$ is the inductance of the linear motor, $R>0$ is the electrical resistance of the linear motor, $\widehat{a}>0,\widehat{b}>0$, and $\widehat{k}>0$ are the parameters of the magnetic flux, and ${\mu }_0>0$ and ${\mu }_f>0$ are the permeability of the vacuum and the coil, respectively. $L_x>0$ and $L_z>0$ are the lengths of the linear motor in each axis direction, and $l_y>0$ is the length of the coil in the y axis direction, and these parameters correspond to the structure of the linear motor shown in Figure 3.

This chapter describes the derivation of this model.

Derivation of the Model

This model considers the spring-like characteristics caused by the permanent magnet of the linear motor. Due to its structural features, the linear motor exerts a force that tries to slide the stage position towards the center based on the gradient of the magnetic potential of the permanent magnet. A conceptual diagram is shown in Figure 3.

The magnetic energy per unit volume $\varphi$ is represented by Equation (10) using the magnetic flux density $\mathrm{\Phi }$:

$$\begin{array}{c}\hfill \varphi =\frac{1}{2\mu }{\mathrm{\Phi }}^2\end{array}$$

(10)

where $\mu$ is the magnetic permeability. Assuming that the magnetic flux density in the regions with and without coils in Figure 3 is constant in the x and z directions, the total magnetic energy W of the linear motor is represented by Equation (11):

$$\begin{array}{c}\hfill W\left(y\right)=L_x{L}_z\frac{1}{2{\mu }_0}{\int }_{-\frac{L_y}{2}}^{\frac{L_y}{2}}{\mathrm{\Phi }}^2\left(y\right)\mathrm{d}y+l_x{L}_z\left(\frac{1}{2{\mu }_f}-\frac{1}{2{\mu }_0}\right){\int }_{-\frac{l_y}{2}+y}^{\frac{l_y}{2}+y}{\mathrm{\Phi }}^2\left(y\right)\mathrm{d}y\end{array}$$

(11)

where $L_x,L_y$, and $L_z$ are the lengths of the linear motor in the $x,y$, and z directions, and $l_x$ and $l_y$ are the lengths of the coil in the x and y directions, respectively. The first term of this equation represents the total magnetic energy when the coil is not present, and the second term represents the change in magnetic energy when the coil is added. When the coil is not present, the permeability of the entire region is ${\mu }_0$, so the entire region is integrated as in the first term. When the coil is present, the region of $l_x\times l_y$ is replaced by the coil, so the region of the coil $l_x\times l_y$ is integrated as in the second term. By substituting Equation (8) into Equations (11) and (12), we obtain

$$\begin{array}{cc}\hfill W\left(y\right)& =\frac{L_x{L}_z}{2{\mu }_0}\left({\widehat{a}}^2+2\widehat{a}\widehat{b}+\frac{3}{2}{\widehat{b}}^2\right)\left(L_y-l_y+\frac{{\mu }_0}{{\mu }_f}{l}_y\right)\hfill \\ \hfill & +\frac{L_x{L}_z}{4{\mu }_0\widehat{k}}\left({\widehat{b}}^{2}sinh\left(\widehat{k}{L}_y\right)-8\left(\widehat{a}\widehat{b}+{\widehat{b}}^2\right)sinh\left(\frac{\widehat{k}}{2}{L}_y\right)\right)\hfill \\ \hfill & +\frac{L_x{L}_z\left({\mu }_0-{\mu }_f\right)}{4{\mu }_0{\mu }_f\widehat{k}}\left({\widehat{b}}^{2}sinh\left(\widehat{k}{l}_y\right)cosh\left(2\widehat{k}y\right)-8\left(\widehat{a}\widehat{b}+{\widehat{b}}^2\right)sinh\left(\frac{\widehat{k}}{2}{l}_y\right)cosh\left(\widehat{k}y\right)\right)\hfill \\ \hfill \end{array}$$

(12)

By differentiating Equation (12) with respect to y, the force due to the total magnetic energy of the linear motor $W^{\prime }$ in Equation (9) is obtained.

5. Proposed Control Method

This chapter describes a method for estimating uncertainty using a kernel adaptive filter and perfect tracking control with sliding mode control, which is proposed in this paper.

5.1. Uncertainty Estimation

The estimated value of uncertainty $\widehat{∆}$ is expressed by using the model with kernel regression, as shown in Equations (13)–(16):

$$\begin{array}{cc}\hfill & \widehat{∆}\left(t\right)=\widehat{\mathit{w}}{\left(t\right)}^T\mathit{k}\left(\mathit{x}\right(t\left)\right)\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & \mathit{k}\left(\mathit{x}\right(t\left)\right)={\left(\begin{array}{ccc}\kappa (\mathit{x}\left(t\right),{\tilde{\mathit{x}}}_1)& \cdots & \kappa (\mathit{x}\left(t\right),{\tilde{\mathit{x}}}_N)\end{array}\right)}^T\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & \kappa (\mathit{x},\tilde{\mathit{x}})=exp\left(-{\left({\displaystyle \frac{g\left(\gamma \right)\parallel \mathit{x}-\tilde{\mathit{x}}\parallel }{\sigma }}\right)}^{\gamma }\right)\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & g\left(\gamma \right)=\sqrt{{\displaystyle \frac{\mathrm{\Gamma }(3/\gamma )}{\mathrm{\Gamma }(1/\gamma )}}}\hfill \end{array}$$

(16)

where $\mathit{x}={\left(\begin{array}{cc}y& \dot{y}\end{array}\right)}^T$ is the state variable of the plant, $\widehat{\mathit{w}}\in {\mathbb{R}}^N$ is the adaptive weight vector of the model, and $\mathit{k}(·):{\mathbb{R}}^2\to {\mathbb{R}}^N$ is the kernel function vector. In this paper, $\mathit{k}$ uses the generalized Gaussian kernel expressed by Equations (15) and (16), where ${\tilde{\mathit{x}}}_i\in {\mathbb{R}}^2$ represents the learning data, $\sigma >0$ is the standard deviation of the kernel, $\gamma >0$ is the order of the norm, and $\mathrm{\Gamma }(·)$ is the gamma function.

The estimation error $\epsilon$ of uncertainty is defined as shown in Equation (17).

$$\begin{array}{cc}\hfill \epsilon \left(t\right)& =∆\left(t\right)-\widehat{∆}\left(t\right)\hfill \\ \hfill & =∆\left(t\right)-\widehat{\mathit{w}}{\left(t\right)}^T\mathit{k}\left(\mathit{x}\right)\left(t\right)\hfill \end{array}$$

(17)

In order to minimize $\epsilon$, an observer based on the evaluation function J shown in Equations (18) and (19) is considered, where $\widehat{·}$ represents the estimated values of each variable, $P=\mathrm{diag}\left(p_y,1\right),p_y>0$ is the weight for the state variable, and $p_{\epsilon }>0$ is the weight for the uncertainty estimation error.

$$\begin{array}{cc}\hfill & J\left(\widehat{\mathit{w}}\right)={\displaystyle \frac{1}{2}}\left({(\mathit{x}-\widehat{\mathit{x}})}^{T}P(\mathit{x}-\widehat{\mathit{x}})+p_{\epsilon }{\epsilon }^2\right)\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & \widehat{\mathit{x}}\left(t\right)={\left(\begin{array}{cc}\widehat{y}\left(t\right)& \widehat{\dot{y}}\left(t\right)\end{array}\right)}^T\hfill \end{array}$$

(19)

The observer is designed as shown in Equation (20), where $v_1,v_2$, and ${\mathit{v}}_3$ are the state correction inputs of the observer.

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(\begin{array}{c}\widehat{y}\\ \widehat{\dot{y}}\\ \widehat{\mathit{w}}\end{array}\right)\left(t\right)=\left(\begin{array}{c}\widehat{\dot{y}}\left(t\right)+v_1\left(t\right)\\ \frac{1}{m}\left\{f_0\left(\widehat{y},\widehat{\dot{y}}\right)\left(t\right)+\widehat{∆}\left(t\right)+f_u\left(t\right)\right\}+v_2\left(t\right)\\ {\mathit{v}}_3\left(t\right)\end{array}\right)\hfill \end{array}$$

(20)

To adaptively minimize the evaluation function shown in Equation (18), it is desirable for it to be a Lyapunov function. Therefore, let us consider the time derivative of it.

$$\begin{array}{cc}\hfill \dot{J}\left(\widehat{\mathit{w}}\right)& ={(\mathit{x}-\widehat{\mathit{x}})}^{T}P(\dot{\mathit{x}}-\dot{\widehat{\mathit{x}}})+p_{\epsilon }\epsilon \dot{\epsilon }\hfill \end{array}$$

(21)

By substituting Equations (1), (17), and (20) into Equation (21), Equation (22) is expressed.

$$\begin{array}{cc}\hfill \dot{J}\left(\widehat{\mathit{w}}\right)& =(y-\widehat{y})p_y(\dot{y}-\widehat{\dot{y}}-v_1)\hfill \\ \hfill & +(\dot{y}-\widehat{\dot{y}})\frac{1}{m}(f_0(y-\widehat{y},\dot{y}-\widehat{\dot{y}})+\epsilon -m{v}_2)\hfill \\ \hfill & +p_{\epsilon }\epsilon \left(\dot{∆}-{\widehat{\mathit{w}}}^T\dot{\mathit{k}}\left(\mathit{x}\right)-{\mathit{v}}_3^T\mathit{k}\left(\mathit{x}\right)\right)\hfill \\ \hfill & =-p_y(y-\widehat{y})v_1\hfill \\ \hfill & +(\dot{y}-\widehat{\dot{y}})\left\{p_y(y-\widehat{y})+\frac{1}{m}\left(f_0(y-\widehat{y},\dot{y}-\widehat{\dot{y}})\right)-v_2\right\}\hfill \\ \hfill & +\epsilon \left\{\frac{1}{m}(\dot{y}-\widehat{\dot{y}})+p_{\epsilon }\left(\dot{∆}-{\widehat{\mathit{w}}}^T\dot{\mathit{k}}\left(\mathit{x}\right)\right)-p_{\epsilon }{\mathit{v}}_3^T\mathit{k}\left(\mathit{x}\right)\right\}\hfill \end{array}$$

(22)

In order to make the first and second terms of Equation (22) negative and the third term sufficiently small, $v_1,v_2$, and ${\mathit{v}}_3$ are designed as shown in Equations (23)–(25), where $G_1>0$ and $G_2>0$ are design parameters.

$$\begin{array}{cc}\hfill & v_1=G_1(y-\widehat{y})\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & v_2=p_y\left(y-\widehat{y}\right)+\frac{1}{m}{f}_0(y-\widehat{y},\dot{y}-\widehat{\dot{y}})+G_2\left(\dot{y}-\widehat{\dot{y}}\right)\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & {\mathit{v}}_3={\displaystyle \frac{\mathit{k}\left(\mathit{x}\right)}{m{p}_{\epsilon }{∥\mathit{k}\left(\mathit{x}\right)∥}^2}}\left(\dot{y}-\widehat{\dot{y}}\right)\hfill \end{array}$$

(25)

By substituting Equations (23)–(25) into Equation (22), Equation (26) is expressed.

$$\begin{array}{cc}\hfill \dot{J}\left(\widehat{\mathit{w}}\right)& =-G_1{(y-\widehat{y})}^2-G_2{(\dot{y}-\widehat{\dot{y}})}^2+p_{\epsilon }\epsilon \left(\dot{∆}-{\widehat{\mathit{w}}}^T\dot{\mathit{k}}\left(\mathit{x}\right)\right)\hfill \end{array}$$

(26)

Since $p_{\epsilon }$ is an arbitrary positive constant, if $p_{\epsilon }$ is sufficiently small, the third term of Equation (26) can be ignored, and Equation (26) becomes locally negative. From Assumption 2, the uncertainty $∆$ is assumed to be small, so $\epsilon$ is sufficiently small and the third term in Equation (26) can be ignored. Therefore, Equation (18) becomes a Lyapunov function if $\epsilon$ is sufficiently small, and $\epsilon$ can be minimized.

5.2. SMC with Uncertainty Compensation

The tracking error $e^{*}$ with respect to the command value r of the ideal model is defined as shown in Equation (27).

$$\begin{array}{c}\hfill e^{*}\left(t\right)=y^{*}\left(t\right)-r\left(t\right)\end{array}$$

(27)

The dynamics of $e^{*}$ are expressed as shown in Equations (3) and (28).

$$\begin{array}{cc}\hfill {\ddot{e}}^{*}\left(t\right)& ={\ddot{y}}^{*}\left(t\right)-\ddot{r}\left(t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{m}}\left(f_0(y^{*},{\dot{y}}^{*})\left(t\right)+f_u^{*}\left(t\right)\right)-\ddot{r}\left(t\right)\hfill \end{array}$$

(28)

For these two variables, e and $e^{*}$, the sliding surface s is defined by using the constants $p_1,p_2,p_3,p_4,p_5\in {\mathbb{R}}_{+}$, as shown in Equation (29).

$$\begin{array}{cc}\hfill s\left(t\right)& =p_{1}e\left(t\right)+p_2\dot{e}\left(t\right)+p_3\ddot{e}\left(t\right)+p_4{e}^{*}\left(t\right)+p_5{\dot{e}}^{*}\left(t\right)+{\ddot{e}}^{*}\left(t\right)\hfill \end{array}$$

(29)

For the switching surface s shown in Equation (29), the Lyapunov function V is defined as shown in Equation (30).

$$\begin{array}{c}\hfill V\left(s\right)={\displaystyle \frac{K_1}{{\gamma }_1+1}}{\left|s\right|}^{{\gamma }_1+1}+{\displaystyle \frac{1}{2}}{\dot{s}}^2>0\end{array}$$

(30)

where $0<{\gamma }_1\le 1$ and $K_1>0$ are constants. The time derivative of V is represented as follows:

$$\begin{array}{cc}\hfill \dot{V}\left(s\right)& =\dot{s}\left(K_1{\left|s\right|}^{{\gamma }_1}\mathrm{sign}\left(s\right)+\ddot{s}\right)\hfill \end{array}$$

(31)

Let, $f_u^{*}$ and $f_u$ be determined as shown in Equations (32)–(34).

$$\begin{array}{cc}\hfill f_u^{*}\left(t\right)& =p_3^{-1}m\left(-p_1{e}^{*}\left(t\right)-p_2{\dot{e}}^{*}\left(t\right)\right)-f_0(y^{*},{\dot{y}}^{*})\left(t\right)+m\ddot{r}\left(t\right)\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill f_u\left(t\right)& =f_u^{*}\left(t\right)+m\left(-p_{4}e\left(t\right)-p_5\dot{e}\left(t\right)\right)-f_0(e,\dot{e})\left(t\right)-K^{*}\widehat{∆}\left(t\right)-m{f}_q\left(t\right)\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\ddot{f}}_q\left(t\right)& =K_1{\left|s\left(t\right)\right|}^{{\gamma }_1}\mathrm{sign}\left(s\left(t\right)\right)+\left(K_2+\overline{\ddot{∆}}\right)\mathrm{sign}\left(\dot{s}\left(t\right)\right)\hfill \end{array}$$

(34)

where $0<K^{*}<1$ and $K_2>0$ are design parameters, $\overline{\ddot{∆}}>0$ is the upper limit of the absolute value of uncertainty $\ddot{∆}$, and $\widehat{∆}$ is the estimated value of uncertainty.

By substituting Equations (32)–(34) into Equation (31), Equation (35) is expressed. Here, since $K^{*}<1$, $\left|\ddot{∆}\left(t\right)-K^{*}\ddot{\widehat{∆}}\left(t\right)\right|\le \left|\ddot{∆}\left(t\right)\right|$ holds, and the absolute value is greater than the original value, so Equation (35) is expressed as Equation (36). Since Equation (36) holds, V is a Lyapunov function, and the asymptotic stability of the control system is guaranteed. Therefore, $e\to 0$ and $e^{*}\to 0$ are guaranteed, and the response of the ideal model and the actual system follows the command value r.

$$\begin{array}{cc}\hfill \dot{V}\left(s\right)& =-K_2\left|\dot{s}\right|+\left(\left(\ddot{∆}\left(t\right)-K^{*}\ddot{\widehat{∆}}\left(t\right)\right)\dot{s}-\overline{\ddot{∆}}\left|\dot{s}\right|\right)\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & \le -K_2\left|\dot{s}\right|+\left(\left|\ddot{∆}\left(t\right)\right|-\overline{\ddot{∆}}\right)\left|\dot{s}\right|<0\hfill \end{array}$$

(36)

In actual implementation, Equation (34) is transformed into Equation (37) to implement it for chattering removal. Through this implementation, chattering is suppressed by making the control input continuous.

$$\begin{array}{cc}\hfill {\ddot{f}}_q\left(t\right)& =K_1{\left|s\left(t\right)\right|}^{{\gamma }_1}\mathrm{sign}\left(s\left(t\right)\right)+\left(K_2+\overline{\ddot{∆}}\right){\left|\dot{s}\left(t\right)\right|}^{{\gamma }_2}\mathrm{sign}\left(\dot{s}\left(t\right)\right)\hfill \end{array}$$

(37)

5.3. Shape Parameter Identification of Generalized Gaussian Kernel

In this section, the method of identifying the shape parameters from the observation information is explained. In the generalized Gaussian kernel, $\sigma$ and $\gamma$ exist as shape parameters. The $n-\mathrm{th}$ central moment $M^n(·),n\in \mathbb{N}$ of the generalized Gaussian distribution is used for the identification of the shape parameters. $M^n(·)$ is expressed as shown in Equation (38).

$$\begin{array}{c}\hfill M^n\left(\mathit{x}\right)=\sum _{i=1}^N{x}_i^n\end{array}$$

(38)

In the case of the generalized Gaussian distribution, the even-order central moment $M^{2m}(·),m\in \mathbb{N}$ is expressed as shown in Equation (39).

$$\begin{array}{c}\hfill M^{2m}\left(\mathit{x}\right)={\left\{{\sigma }^2\frac{\mathrm{\Gamma }\left(1/\gamma \right)}{\mathrm{\Gamma }\left(3/\gamma \right)}\right\}}^m\frac{\mathrm{\Gamma }\left((2m+1)/\gamma \right)}{\mathrm{\Gamma }\left(1/\gamma \right)}\end{array}$$

(39)

From this equation, kurtosis $\varphi$ is expressed as shown in Equation (40).

$$\begin{array}{c}\hfill \varphi ={\displaystyle \frac{M^4\left(\mathit{x}\right)}{{\left(M^2\left(\mathit{x}\right)\right)}^2}}={\displaystyle \frac{\mathrm{\Gamma }(5/\gamma )\mathrm{\Gamma }(1/\gamma )}{{\left(\mathrm{\Gamma }(3/\gamma )\right)}^2}}\end{array}$$

(40)

From these equations, the evaluation function $J\left(\gamma \right)$ with respect to $\gamma$ is defined as the difference between the observed value of kurtosis and the value calculated from $\gamma$, as shown in Equation (41).

$$\begin{array}{c}\hfill J\left(\gamma \right)={\displaystyle \frac{1}{2}}{\left(\varphi -\frac{\mathrm{\Gamma }(1/\gamma )\mathrm{\Gamma }(5/\gamma )}{\mathrm{\Gamma }{(3/\gamma )}^2}\right)}^2\end{array}$$

(41)

$\gamma$ is updated by Equation (42) so that the evaluation function $J\left(\gamma \right)$ is monotonically decreased, where ${\eta }_{\gamma }>0$ is the update gain, and ${\mathrm{\Gamma }}^{\prime }(·)$ is the first-order derivative of the gamma function.

$$\begin{array}{c}\hfill \dot{\left(\frac{1}{\gamma }\right)}={\eta }_{\gamma }{\displaystyle \frac{\mathrm{\Gamma }{(3/\gamma )}^3\left(\varphi -\frac{\mathrm{\Gamma }(1/\gamma )\mathrm{\Gamma }(5/\gamma )}{\mathrm{\Gamma }{(3/\gamma )}^2}\right)}{{\mathrm{\Gamma }}^{\prime }(1/\gamma )\mathrm{\Gamma }(3/\gamma )\mathrm{\Gamma }(5/\gamma )+5\mathrm{\Gamma }(1/\gamma )\mathrm{\Gamma }(3/\gamma ){\mathrm{\Gamma }}^{\prime }(5/\gamma )-6\mathrm{\Gamma }(1/\gamma ){\mathrm{\Gamma }}^{\prime }(3/\gamma )\mathrm{\Gamma }(5/\gamma )}}\end{array}$$

(42)

By updating $\gamma$ according to Equation (42), the updated equation of the evaluation function $J\left(\gamma \right)$ becomes Equation (43), and $J\left(\gamma \right)$ monotonically decreases, so the convergence of $\gamma$ is guaranteed.

$$\begin{array}{c}\hfill \dot{J}\left(\gamma \right)=-{\eta }_{\gamma }{\left(\varphi -\frac{\mathrm{\Gamma }(1/\gamma )\mathrm{\Gamma }(5/\gamma )}{\mathrm{\Gamma }{(3/\gamma )}^2}\right)}^2=-2{\eta }_{\gamma }J\left(\gamma \right)<0\end{array}$$

(43)

Summarizing the above, the following steps are performed for each control cycle for the identification of the shape parameters.

Step 1: Set the initial values of $\sigma$ and $\gamma$.

Step 2: Observe the state variable $\mathit{x}$ using the sensor.

Step 3: Calculate the 2nd and 4th central moments using Equation (39) when $m=1,2$.

Step 4: Calculate the kurtosis $\varphi$ using Equation (40).

Step 5: Update $\sigma$ as the square root of the 2nd moment.

Step 6: Update $\gamma$ using Equation (42).

5.4. Summary of Proposed Method

Summarizing the above, a block diagram of the control system is shown in Figure 4, where C represents Equation (32), P represents Equation (1), $P_n$ represents Equation (3), S represents Equations (33) and (37), and Adaption Law represents Equation (20) and Section 5.3.

In actual program, the following steps are executed for each control cycle.

Step 1: 

Observe the stage position y using the sensor.

Step 2: 

Observe the stage velocity $\dot{y}$ by differentiating y.

Step 3: 

Update the shape parameters, $\sigma$ and $\gamma$, according to the procedure in Section 5.3.

Step 4: 

Estimate the uncertainty $\widehat{∆}$ using Equation (20). Here, the initial value of the state variable is 0, and $v_1,v_2$, and $v_3$ are the inputs to the observer, as shown in Equations (23)–(25).

Step 5: 

Calculate the ideal operation amount $f_u^{*}$ using Equation (32).

Step 6: 

Calculate the actual operation amount $f_u$ using Equations (33) and (37), and input it to the stage.

6. Verification

In this section, the proposed method is verified. The verification is divided into two parts: a simulation and an experiment.

The model of $f_0$ used in the verification is given by

$$\begin{array}{c}\hfill f_0(y,\dot{y})=-c\dot{y}-ky\end{array}$$

(44)

where $c>0$ and $k>0$ are parameters set so that $f_0\approx f$.

6.1. Simulation

A simulation of the proposed method was conducted. The simulation parameters are presented in

Table 1. Simulation parameters.

Symbol	Value	Unit	Symbol	Value	Unit
m	$0.0755$	$\mathrm{kg}$	$p_1$	625,000	-
c	$0.105$	$\mathrm{kg}/\mathrm{s}$	$p_2$	5000	-
k	$2.5$	$\mathrm{kg}/{\mathrm{s}}^2$	$p_3$	10	-
R	$17.5$	$\mathrm{\Omega }$	$p_4$	5,250,000	-
L	$27.5\times {10}^{-3}$	$\mathrm{H}$	$p_5$	5000	-
$\widehat{a}$	$31.7$	$\mathrm{N}/\mathrm{A}$	$K_1$	500	-
$\widehat{b}$	$12.0$	$\mathrm{N}/\mathrm{A}$	$K_2$	200	-
$\widehat{k}$	100	$/\mathrm{m}$	${\gamma }_1$	$0.8$	-
$L_x$	$0.01$	$\mathrm{m}$	${\gamma }_2$	$0.7$	-
$L_z$	$0.05$	$\mathrm{m}$	$K^{*}$	$0.9$	-
${\mu }_f$	$6.3\times {10}^{-3}$	$\mathrm{H}/\mathrm{m}$	$\overline{\ddot{∆}}$	100	$\mathrm{N}/{\mathrm{s}}^2$
${\mu }_0$	$4\pi \times {10}^{-5}$	$\mathrm{H}/\mathrm{m}$	${\eta }_{\gamma }$	$0.1$	-
$l_y$	$0.05$	$\mathrm{m}$	$\sigma$	$0.1$	-
$G_1$	10	-	$G_2$	10	-
$p_y$	100	-	$p_{\epsilon }$	7	-
N	100	-

, and the results are shown in Figure 5. Figure 5a–c show a comparison of the stage position, input voltage, and tracking error with and without uncertainty compensation. Figure 5d shows the magnitude of uncertainty with respect to the command force. The parameters were determined through trial and error. The reference value $r\left(t\right)$ is given by Equation (45). This is a sine wave with a frequency of $2\mathrm{Hz}$ and an amplitude of $1\mathrm{mm}$.

$$\begin{array}{c}\hfill r\left(t\right)=\left(1-cos\left(4\pi t\right)\right)\left[\mathrm{mm}\right]\end{array}$$

(45)

In this simulation, Gaussian noise with a standard deviation of 1 was used for ${\tilde{\mathit{x}}}_i$.

From the simulation results, it can be observed that perfect tracking is achieved with respect to the reference value, regardless of whether uncertainty compensation is applied or not. The effectiveness of perfect tracking control using sliding mode control is confirmed. Furthermore, the proposed method of uncertainty compensation reduces the tracking error compared to the case where no compensation is applied, demonstrating its effectiveness in achieving high-precision stage control. Additionally, it can be seen from Figure 5d that the uncertainty satisfies the assumptions in Section 3.

6.2. Experiment

An experiment was conducted on the designed control system. The same parameters as those used in the simulation were employed in the experiment. A sine wave with an amplitude of $1\mathrm{mm}$ and a frequency of $2\mathrm{Hz}$ was used as the command value r. The experimental results are presented in Figure 6. It can be seen from Figure 6d that the uncertainty is sufficiently small compared to the command force, and it satisfies the assumptions in Section 3. From Figure 6, it can be observed that without uncertainty compensation, sufficient tracking cannot be confirmed due to the uncertainties, and an error of up to $0.5\mathrm{mm}$ can be observed. However, by applying uncertainty compensation, it can be improved to within $0.2\mathrm{mm}$ at maximum. The effectiveness of the proposed method for uncertainty compensation is verified. However, it can be seen that higher-order vibrations occur in the proposed method. This is believed to be due to the fact that the uncertainty identifier picks up the sensor noise. Therefore, appropriately filtering the sensor value is a future task that can be performed to suppress higher-order vibrations.

7. Conclusions

In this paper, we proposed a method to compensate for uncertainty using sliding mode control with an estimated uncertainty value, which represents the motion model of a linear slider as a linear operator and uncertainty. We designed a new observer using Kernel LMS for estimating uncertainty and succeeded in identifying uncertainty without directly observing it. By conducting experiments, we confirmed that it is possible to track the position of the stage even when the uncertainty is large in many cases. Specifically, under conditions that typically result in an error of about $0.5\mathrm{mm}$ using general sliding mode control, it was possible to reduce the error to about $0.2\mathrm{mm}$ using the proposed method.