Filtered Right Coprime Factorization and Its Application to Control a Pneumatic Cylinder

Abstract

The main objective of this research is to expand right coprime factorization based on operator theory in nonlinear systems. A novel method for right coprime factorization is proposed by introducing an operator that can deform the system’s response into an arbitrary shape. This enables the design of control systems that are highly effective against noise. As an application, we use a pneumatic stage. The effectiveness of this method is verified through simulations and real-world experiments.

1. Introduction

Operator theory is a theory in control theory that allows for consideration of control systems in the time domain [1]. This eliminates the need to transform into the frequency domain and makes it a theory applicable to nonlinear systems. Right coprime factorization based on operator theory has been considered to enable reference tracking and stability compensation [2]. By combining the concept of operators with right coprime factorization, it is possible to achieve reference tracking and stability compensation while handling the system in the time domain. However, there is a problem with right coprime factorization, namely that it is weak against uncertainty of the control target model. When uncertainty is present, right coprime factorization does not function well. This gave rise to robust right coprime factorization based on operator theory [3]. This theory ensures robust stability of the system when the conditions are met and achieves target value tracking and stability compensation even in control objects that include uncertainties.

However, there is a problem with right coprime factorization based on operator theory. The main issue is its weakness against noise. When noise is present, it fails to achieve reference tracking. Therefore, this paper proposes right coprime factorization based on operator theory combined with a filter [4]. This makes it possible to suppress the noise in the signal with the filter, thereby enabling reference tracking. Moreover, this method does not require the design of a new disturbance observer, making it easier to design.

As an application of this method, this paper targets a pneumatic-driven stage (hereafter referred to as the pneumatic stage). The pneumatic stage consists of four air cylinders connected in parallel, and the rectangular stage on top of the cylinders moves along with the pistons. Compressed air is injected from both sides of the cylinders, and the pistons are driven by the differential pressure. Additionally, friction occurs when the pistons move. It is also important to note that the density of compressed air constantly changes, causing the pistons not to move at a consistent speed. Therefore, research has been conducted to mathematically express the nonlinear friction of pneumatic cylinders [5]. The LuGre model was proposed as a new dynamic friction model and has been used in various studies. Considering the friction characteristics of hydraulic cylinders, a modified LuGre model based on the LuGre model was proposed in [6]. Additionally, friction models that consider the variation of the lubrication film have been newly proposed [7,8]. Furthermore, the friction characteristics of pneumatic cylinders using the LuGre model have been proposed [9,10].

In previous studies [11,12,13], the pneumatic stage was modeled by separating it into pneumatic and mechanical components, and control was implemented using a ${\mathrm{PDD}}^2$ compensator, achieving high-speed and high-precision positioning. The control model was determined by the response to step signals for both the pneumatic and mechanical components; the pneumatic part was designed with a first-order system, and the mechanical part was designed with a second-order system. Here, friction occurs between the air cylinder and the piston cross-section. The injected air compresses in front of the piston, causing the piston to be pushed out with a slight delay. Therefore, the dynamic friction of the pneumatic cylinder depends on the flow of compressed air and is nonlinear. In a previous study [14], a dynamic friction model considering nonlinear friction was proposed, and a control method combining stability assurance based on operator theory [1] was suggested. However, a PI–D compensator was used. The reason for using the compensator was not clearly explained in either study [11,14].

Therefore, this paper proposes a control system based on operator theory combined with a filter. This control system makes the signals flowing through the real system more stable. We explore the potential application of this new control system in electron beams used in semiconductor manufacturing. Additionally, considering that the control target has nonlinear characteristics, we use sliding mode control to achieve reference tracking [15,16].

The structure of this paper is as follows. In Section 2, we describe the mathematical knowledge necessary for the subsequent discussions. In Section 3, we explain the driving principles and structure of the pneumatic stage targeted in this study and present its mathematical model. In Section 4, we describe the problem setting and the proposed method to solve this problem. In Section 5, we design a control system based on the proposed method. In Section 6, we present the results of simulations and real-world experiments. Finally, in Section 7, we conclude the study.

2. Mathematical Preparation

This section describes the operator theory used to design the control system and defines and introduces the mathematical knowledge required for it.

2.1. Operator Theory

2.1.1. Operator Definition

In this study, the term “operator” refers to a method of representing the temporal “mapping” from any input space to the output space. This allows for representation of the behavior of the nonlinearity of the controlled object as a mapping associated with input–output signals. In this research, the notation $S:U\to Y$, expressed in the subsequent descriptions, refers to the spatial mapping by the operator (S) from the input signal space (U) to the output signal space (Y) in the temporal domain. Hereafter, the input and output signals are denoted as time functions $u\left(t\right),$ and $y\left(t\right)$, respectively, and the output within any time interval is expressed as in Equation (1). Here, $u\left(t\right)ϵU,$ and $y\left(t\right)ϵY$ satisfy the conditions.

$$\begin{array}{c}\hfill y\left(t\right)=S\left(u\right(t\left)\right)\end{array}$$

(1)

As mentioned above, treating the input–output relationship as a mapping allows for a control system design without the need to obtain frequency responses through Laplace transforms from differential equations. This approach enables control system design even for differential equations with nonlinear terms.

2.1.2. Feedback Control Based on Operator Theory

Let the input space, the internal state signal space of the plant, and the output space be denoted as U, W, and Y, respectively. Consider a nonlinear plant represented by the operator $P:U\to Y$. Suppose there exist a stable operator ($N:W\to Y$) and a stable and invertible operator ($D:W\to U$) such that the nonlinear plant (P) can be expressed as in Equation (2). In this case, N and D are the right factorization of P and can be represented as shown in Figure 1.

$$\begin{array}{c}\hfill P=N{D}^{-1}\end{array}$$

(2)

Furthermore, suppose there exist a stable operator ($S:Y\to U$) and a stable and invertible operator ($R:U\to U$) such that the unimodular operator (M) is

$$\begin{array}{c}\hfill SN+RD=M\end{array}$$

(3)

When Equation (3) holds, N and D are said to be the right coprime factorization of P. Equation (3) is called the Bezout identity. The unimodular operator (M) refers to an operator that is stable and invertible. If the right coprime factorization (N) and $D^{-1}$ of the nonlinear plant and the operators S and R satisfy the Bezout identity (3), then, as shown in Figure 2, the operators S and R act as the feedback and feedforward compensator, respectively, for the unstable operator ($D^{-1}$). Consequently, the output (w) becomes stable, enabling the design of an overall stable nonlinear control system. Additionally, Figure 2 is transformed into Figure 3. Based on the above design, the plant’s reference response is expressed in Equation (4). Here, when the control input space is denoted as R, the transformed nonlinear plant ($P^{\prime }$) becomes $P^{\prime }:R\to Y$.

$$\begin{array}{c}\hfill P^{\prime }=N{M}^{-1}\end{array}$$

(4)

2.1.3. Robust Stability

Additionally, this control system takes into account uncertainties such as modeling errors. The right coprime factorization of the actual plant ($P+\Delta P$), including uncertainty ($\Delta P$), is represented as follows:

$$\begin{array}{c}\hfill P+\Delta P=(N+\Delta N)D^{-1}\end{array}$$

(5)

It is known that the uncertainty ($\Delta P$) is integrated into the operator ($\Delta N$), which is difficult to determine through direct calculation. However, it is known that the output remains bounded. The Bezout identity for ensuring BIBO stability of the plant with uncertainties ($\Delta P$) is expressed as follows:

$$\begin{array}{c}\hfill S(N+\Delta N)+RD=\tilde{M}\end{array}$$

(6)

The nonlinear feedback system considering uncertainties is illustrated in Figure 4. If M and $\tilde{M}$ are a unimodular operators and it is proven that $N+\Delta N$ and D are coprime, then stability is guaranteed. According to Equations (3) and (6), $\tilde{M}-M$ can be expressed as

$$\begin{array}{cc}\hfill \tilde{M}-M& =S(N+\Delta N)-SN\hfill \\ \hfill \tilde{M}& =\left[I+\left(S(N+\Delta N)-SN\right)M^{-1}\right]M\hfill \end{array}$$

(7)

Here, $M^{-1}$ being a unimodular operator ensures stability, while the term $\left(S(N+\Delta N)-SN\right)M^{-1}$ represents the open-loop operator of the feedback system. If it satisfies Equation (8) and if it satisfies Equation (7), the stability of ${\tilde{M}}^{-1}$ is ensured. Therefore, $\tilde{M}$ being a unimodular operator ensures robust stability of the system.

$$\begin{array}{c}\hfill {∥(S(N+\Delta N)-SN)M^{-1}∥}_{Lip}<1\end{array}$$

(8)

$$\begin{array}{cc}\hfill {\tilde{M}}^{-1}& =M^{-1}{\left[I+\left(S(N+\Delta N)-SN\right)M^{-1}\right]}^{-1}\hfill \end{array}$$

(9)

2.2. Sliding Mode Control

By appropriately designing the switching hyperplane and control input, the sliding mode control system can compensate for uncertainties in the overall system [17].

2.2.1. Basic Concepts of Sliding Mode Control

The theory that changes the structure of the control system is called variable structure control theory. Sliding mode control is a particularly well-structured theory within variable structure control theory and is used for various control objectives, such as system stabilization and servo systems. When using general sliding mode control, the switching surface ($\sigma =0$) is designed to represent the system response of a lower order than the given control target. Furthermore, a variable-structure control input ($r\left(x,t\right)$) is designed to bring the state ($x\left(t\right)$) from any initial condition to the switching surface within a finite time and to constrain it on the switching surface. When the switching surface and the variable-structure control input are appropriately designed, the system can be made to follow the desired dynamics on the switching surface.

2.2.2. Control Input in Sliding Mode Control

In the design of sliding mode control, there are cases where the control input is free and cases where it is predetermined. In either case, the objective is to satisfy the reaching condition. Here, we define a natural number (i). In the case of a free structure, the control input ($r\left(t\right)$) is determined by the switching function ($\sigma$) according to any of the conditions in Equations (10)–(12).

$$\begin{array}{cc}\hfill & {\sigma }_i\dot{{\sigma }_i}<0\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \dot{v}=\frac{d}{dt}\left({\sigma }^T\sigma \right)<0\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & \dot{{\sigma }_i}=-q_i\mathrm{sgn}\left({\sigma }_i\right)-k_i{f}_i\left({\sigma }_i\right)\hfill \end{array}$$

(12)

Equation (10) is used for the direct switching method, Equation (11) is used for the Lyapunov function method, and Equation (12) is used for the reaching law method. On the other hand, when the structure of sliding mode control is predetermined, it is important to determine the controller gain such that the desired reaching law is satisfied. An example of a structure that specifies the control law is relay control. For each element of the control input (r), the sliding mode control input is in relay form. The relay gain either remains constant or varies depending on the state. The control input in relay control is shown in Equation (13).

$$\begin{array}{cc}\hfill & r_i=\left\{\begin{array}{c}\hfill k_i^{+}(x,t)\left({\sigma }_i>0\right)\\ \hfill k_i^{-}(x,t)\left({\sigma }_i\le 0\right)\end{array}\right.\hfill \end{array}$$

(13)

$k_i^{+}$ and $k_i^{-}$ are determined to satisfy the desired reaching conditions. Another example involves the design of control input r using the control law of the equivalent control for the system expanded in this study. Each sliding mode control is represented by Equation (14).

$$\begin{array}{c}\hfill r=r_{eq}+\Delta r\end{array}$$

(14)

where $r_{eq}$ is the control input for equivalent sliding mode control, which constrains the system on the switching surface and induces the sliding mode, and $\Delta r$ is an additional term added to satisfy the reaching condition. The reaching condition refers to the condition where the state ($x\left(t\right)$) starting from any initial state approaches the switching surface within a finite time. $\Delta r$ generally utilizes relay control.

2.2.3. Chattering Prevention Measures

In sliding mode control, one of the fundamental assumptions is that the control input can switch infinitely fast from one value to another. However, switching control inputs at infinite speed is impossible, leading to chattering between sliding mode and steady-state mode. Chattering in the steady state manifests as high-frequency oscillations near equilibrium points. Ideal relay control, as depicted in Figure 5a, is described by Equation (15).

$$\begin{array}{cc}\hfill & r\left(\sigma \right)=\mathrm{sgn}\left(\sigma \right)=\left\{\begin{array}{c}\hfill +1\left(\sigma >0\right)\\ \hfill 0\left(\sigma =0\right)\\ \hfill -1\left(\sigma <0\right)\end{array}\right.\hfill \end{array}$$

(15)

In ideal relay control, instantaneous switching occurs at $\sigma =0$, and the ideal sliding mode resides at $\sigma =0$. Additionally, the control input can switch infinitely close to $\sigma =0$, thereby eliminating chattering entirely. There is no steady-state error, and furthermore, the control system constrains the system at $\sigma =0$, ensuring invariance. However, achieving ideal relay characteristics is impossible in practice. Therefore, a method involving the replacement of discontinuous functions with continuous approximations like the saturation function shown in Figure 5b can be considered. An example of saturation control is given in Equation (16).

$$\begin{array}{cc}\hfill & r\left(\sigma \right)=\mathrm{sat}\left(\sigma \right)=\left\{\begin{array}{cc}\hfill +1& \left(\sigma >T\right)\hfill \\ \hfill \frac{\sigma }{T}& \left(\left|\sigma \right|\le T\right)\hfill \\ \hfill -1& \left(\sigma <T\right)\hfill \end{array}\right.\hfill \end{array}$$

(16)

In state-space representations, boundary layers are employed on switching surfaces. In this context, the robustness of the system is characterized by the width of the boundary layer. Let $\mu >0$ define the amount of hysteresis related to $\sigma$. When employing hysteresis characteristics, the control input does not switch at $\sigma =0$ but at $\sigma =\pm \mu$. When using actual saturation functions, the sliding mode control system is widely asymptotically stable. While the property of invariance does not exist, it becomes possible to eliminate high-frequency chattering. In all cases depicted in Figure 5, the sliding mode control system is stable, and each state trajectory is constrained within a band in the state space. When the width of the band is sufficiently narrow, the dynamics within the band approach quasi-sliding mode.

3. Model Design

In this section, we model the friction of a pneumatic cylinder and a pneumatic stage. Dynamic friction modeling of a pneumatic cylinder is presented in Section 3.1.

Table 1. Parameters of the pneumatic stage.

Symbol	Description	Unit
z	Average deflection of sliding surface	m
${\sigma }_0$	Bristle stiffness	N/m
${\sigma }_1$	Microviscous friction coefficient	N $·$ s/m
${\sigma }_2$	Coefficient of viscous friction	N $·$ s/m
v	Relative velocity of piston and cylinder	m/s
$v_s$	Stribeck speed	m/s
$v_b$	Maximum speed of lubrication film thickness change	m/s
h	Non-dimensional non-stationary lubricant film thickness	m
$h_{ss}$	Non-dimensional steady-state lubricant film thickness	m
T	Time constants for fluid friction dynamics	s
$F_r$	Frictional force	N
$F_c$	Coulomb’s friction force	N
$F_s$	Maximum static friction	N
${\tau }_h$	Time constant of lubricant film dynamics	s
$K_f$	Proportionality constant of lubricant film thickness	${(\mathrm{m}/\mathrm{s})}^{-2/3}$
m	Mass of stage	kg
$S_d$	Piston cross-sectional area	${\mathrm{m}}^2$
k	Spring constant	N/m
$T_{air}$	Servo valve time constant	s
$K_{air}$	Servo valve gain	-

shows the description and numerical values of each parameter used in the model.

3.1. Friction Model of Pneumatic Cylinder

This section describes the mathematical model of dynamic friction in the pneumatic cylinder used in this study. It begins by explaining the conceptual framework of the model. Subsequently, the equations of the model are presented, followed by explanations for each equation. Finally, it incorporates the model with an equation representing frictional force against velocity.

3.1.1. LuGre Model

When designing a model of a pneumatic stage considering physical phenomena, it is necessary to consider the friction between the cross-sections of the piston and cylinder. As the friction model of the cross section of the piston and cylinder, the LuGre model is used, in which the pre-slip phenomenon in friction is taken into account [5].

The LuGre model, as depicted in Figure 6, imagines two rigid bodies in contact through compliant bristles. In this model, the friction interface between the two surfaces is considered the contact between these rigid bristles. Surfaces in the micro-world are highly irregular, and contact occurs between the surfaces at several asperities. When a force is applied tangentially, the bristles deform like springs, and the repulsive force of these springs is considered frictional force. The average deformation of these bristles is denoted as z. Note that in Figure 6, for explanatory purposes, the lower bristles are depicted as rigid.

3.1.2. Friction Model Equation

When the force is sufficiently large, some of the rigid bristles deform significantly and slip. This phenomenon is highly random due to the irregular shape of the surfaces. Based on references [5,6], the modified LuGre model incorporates the dynamics of fluid friction, as expressed in Equations (17)–(23) below. These equations account for the dynamic friction behavior of pneumatic cylinders.

Equation (17) represents the friction model for the cylinder cross-section. The first and second terms on the right-hand side of Equation (17) depict the frictional force generated by the deformation of elastic bristles, the third term represents viscous friction, and the fourth term represents fluid friction of the lubricating film due to viscous friction. Equation (18) describes the dynamics of the average deformation of the bristles.

$$\begin{array}{cc}\hfill & F_r={\sigma }_{0}z+{\sigma }_1\frac{dz}{dt}+{\sigma }_2\left(v+T\frac{dv}{dt}\right)\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & \frac{dz}{dt}=v-\frac{{\sigma }_{0}z}{g\left(v,h\right)}v\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & g\left(v,h\right)=F_c+[\left(1-h\right)F_s-F_c]e^{-{\left(v/v_s\right)}^n}\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & \frac{dh}{dt}=\frac{1}{{\tau }_h}\left(h_{ss}-h\right)\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & {\tau }_h=\left\{\begin{array}{cc}\hfill {\tau }_{hp}& (v\ne 0,h\le h_{ss})\hfill \\ \hfill {\tau }_{hn}& (v\ne 0,h>h_{ss})\hfill \\ \hfill {\tau }_{h0}& (v=0)\hfill \end{array}\right.\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & h_{ss}=\left\{\begin{array}{cc}\hfill & K_f{\left|v\right|}^{2/3}\left(\right|v|\le |v_b\left|\right)\hfill \\ \hfill & K_f|v_b{|}^{2/3}\left(\right|v|>|v_b\left|\right)\hfill \end{array}\right.\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & K_f=\left(1-\frac{F_c}{F_s}\right){\left|v_b\right|}^{-2/3}\hfill \end{array}$$

(23)

3.1.3. Friction and Stribeck Functions

Here, the denominator ($g(v,h)$) in Equation (18) represents the Stribeck function. The velocity defining the boundary value for a specific low-speed region is the Stribeck velocity ($v_s$), and n is an appropriate exponent. The Stribeck function exhibits Coulomb friction and the Stribeck effect. The Stribeck function is defined as shown in Equation (19). This equation mathematically expresses the characteristic whereby friction decreases as velocity increases in the low-speed region. The following Figure 7a outlines the general shape of the Stribeck curve. Equation (19) incorporates the dynamic behavior of the lubricating oil film into the Stribeck function.

3.1.4. Dynamic Behavior of Lubricant Film

Equations (20)–(23) depict the dynamics of the lubricating oil film (h). Equation (21) indicates the time constants (${\tau }_h$) for the dynamics of the lubricating oil film in each scenario, where ${\tau }_{hp}$ is the time constant during acceleration, ${\tau }_{hn}$ is the time constant during deceleration, and ${\tau }_{h0}$ is the time constant during dwell time (when $v=0$). In Equation (22), the lubricating film increases with cylinder velocity (v) only when the cylinder velocity is lower than the velocity at which the friction force is minimized in the negative friction region ($v_b$), i.e., during the steady state. Otherwise, it is maintained at its maximum value. Figure 7b outlines the profile of the lubricating oil film (h).

3.2. Pneumatic Stage Model

This section divides the model of the pneumatic stage into pneumatic and mechanical components. The model of the pneumatic components specifies that the pressure ($p\left(t\right)$) is output in response to the applied voltage ($u\left(t\right)$), as given by Equation (24).

The model of the mechanical component outputs the position ($y\left(t\right)$) in response to the input pressure ($p\left(t\right)$). Following the dynamics of mechanics described by the equation of motion ($m\ddot{y}=F$), incorporating the restoring force due to the spring and the frictional force ($F_r$) against velocity, the model is given by Equation (25). The mathematical model for $F_r$ is described by Equations (17)–(23), as mentioned earlier. It is important to note that the product of the supply pressure ($p\left(t\right)$) and the piston’s cross-sectional area ($S_d$) gives the force (F). Considering the pneumatic stage where four pneumatic cylinders are connected in parallel, the force due to pressure is multiplied by four.

$$\begin{array}{cc}\hfill & \frac{dp\left(t\right)}{dt}=-\frac{1}{T_{air}}p\left(t\right)+\frac{K_{air}}{T_{air}}u\left(t\right)\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & \frac{d^{2}y\left(t\right)}{d{t}^2}=-\frac{F_r\left(\dot{y}\right)}{m}-\frac{k}{m}y\left(t\right)+\frac{4S_d}{m}p\left(t\right)\hfill \end{array}$$

(25)

4. Problem Setting and Proposed Method

This section describes the problem formulation and proposes a method to solve it.

4.1. Problem Setting

Existing operator theory-based right coprime factorizations are vulnerable to noise. When noise is present, existing methods fail to operate effectively. Therefore, this study proposes an extension of the operator theory-based right coprime factorization, incorporating desired operators. In this study, the desired operators are those capable of mitigating noise, aiming to integrate filters. This approach aims to propose a control system that functions effectively even in the presence of noise.

4.2. Proposed Method

Based on Section 2.1, we consider a new operator theory incorporating the operator (Q) and defining input space (U), internal state signal space within the plant ($\tilde{W}$), and output space (Y). Assume a nonlinear plant represented by the operator $P:U\to Y$, with stable operators $\tilde{N}:\tilde{W}\to Y$ and stable and invertible operator $\tilde{D}:\tilde{W}\to U$ existing. Here, $\tilde{N}$ and $\tilde{D}$ form the right factorization of P. Next, consider a stable operator, $Q:W\to \tilde{W}$, where W represents the estimated state signal space. Through operator Q, we define operators N and D as $N=\tilde{N}Q:W\to Y$ and $D=\tilde{D}Q:W\to U$, respectively. Considering the right factorization after passing through operator Q, we obtain the following Equation (26).

$$\begin{array}{cc}\hfill P& =N{D}^{-1}\hfill \\ \hfill & =\left(\tilde{N}Q\right)\left(Q^{-1}{\tilde{D}}^{-1}\right)\hfill \\ \hfill & =\tilde{N}{\tilde{D}}^{-1}\hfill \end{array}$$

(26)

From Equation (26), it is evident that the results of the right factorization remain unchanged, even when using operators filtered through a filter. The block diagram illustrating this right factorization is shown in Figure 8. Additionally, the operator $R^{-1}$ when using the new right factorization is depicted in Figure 9, indicating that the signal entering the plant is passed through the operator (Q).

Thus, by designing an appropriate operator (Q), it becomes feasible to apply operators characterized by operator theory. If the operator (Q) is appropriately designed, it can possess various characteristics. In this study, however, it is designed with filter characteristics to address noise issues.

5. Control System Design

In this section, we combine the control theory explained in Section 2 with the control methods proposed in Section 4 to perform control system design using the model derived in Section 3. The overall structure of the control system is shown in Figure 10. C represents the sliding mode controller designed based on the proposed method, while $S,R^{-1},N,$ and $D^{-1}$ denote operators. In Figure 10, $r,$ and $r^{*}$ denote reference values, e represents the error, u is the input, w is the quasi-state, y is the output, and s indicates feedback.

In Section 5.1, we conducted control system design considering stability assurance based on the operator theory with the proposed filter integration. In Section 5.2, we proceeded with the design of the sliding mode controller.

5.1. Control System Design Based on Proposed Method

The pneumatic stage control plant ($P\left[u\right]\left(t\right)$) is determined by the following Equation (27).

$$\begin{array}{c}\hfill P:\left\{\begin{array}{cc}& \ddot{y}\left(t\right)=-\frac{F_r\left(\dot{y}\left(t\right)\right)}{m}-\frac{k}{m}y\left(t\right)+\frac{4S_d}{m}p\left(t\right)\hfill \\ & \dot{p}\left(t\right)=-\frac{1}{T_{air}}p\left(t\right)+\frac{K_{air}}{T_{air}}u\left(t\right)\hfill \\ & F_r={\sigma }_{0}z+{\sigma }_1\dot{z}+{\sigma }_2\left(\dot{y}\left(t\right)+T\ddot{y}\left(t\right)\right)\hfill \\ & \dot{z}=\dot{y}\left(t\right)-\frac{{\sigma }_{0}z}{g\left(v,h\right)}\dot{y}\left(t\right)\hfill \\ & g\left(\dot{y}\left(t\right),h\right)=F_c+[\left(1-h\right)F_s-F_c]e^{-{\left(\dot{y}/v_s\right)}^n}\hfill \end{array}\begin{array}{cc}& \dot{h}=\frac{1}{{\tau }_h}\left(h_{ss}-h\right)\hfill \\ & {\tau }_h=\left\{\begin{array}{cc}\hfill {\tau }_{hp}& (v\ne 0,h\le h_{ss})\hfill \\ \hfill {\tau }_{hn}& (v\ne 0,h>h_{ss})\hfill \\ \hfill {\tau }_{h0}& (v=0)\hfill \end{array}\right.\hfill \\ & h_{ss}=\left\{\begin{array}{cc}\hfill K_f{\left|v\right|}^{2/3}& \left(\right|v|\le |v_b\left|\right)\hfill \\ \hfill K_f{\left|v_b\right|}^{2/3}& \left(\right|v|>|v_b\left|\right)\hfill \end{array}\right.\hfill \\ & K_f=\left(1-\frac{F_c}{F_s}\right){\left|v_b\right|}^{-2/3}\hfill \end{array}\right.\end{array}$$

(27)

From Equation (27), we decomposed the plant P into ${\tilde{D}}^{-1}$ and $\tilde{N}$ on the right. We designed $\tilde{N}\left[w\right]\left(t\right)$ and $\tilde{D}\left[w\right]\left(t\right)$ as shown in Equations (28) and (29), respectively. The unstable element is denoted as $\tilde{w}$ and treated as a quasi-state signal. In this context, according to operator theory, $\tilde{N}$ remains to achieve the desired operation akin to a mass–spring–damper system, while $\tilde{D}$ disappears to incorporate complex elements. Here, f is the coefficient of the damper system modeled as a second-order lag system in previous research [13]. Furthermore, subsequent equations involving $F_r$ remain unchanged, and detailed equations are omitted for brevity.

$$\begin{array}{cc}\hfill \tilde{N}:& \left\{\begin{array}{cc}\hfill & {\ddot{x}}_n\left(t\right)=-\frac{f}{m}{\dot{x}}_n\left(t\right)-\frac{k}{m}{x}_n\left(t\right)+\tilde{w}\left(t\right)\hfill \\ \hfill & y\left(t\right)=x_n\left(t\right)\hfill \end{array}\right.\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill \tilde{D}:& \left\{\begin{array}{c}\begin{array}{cc}\hfill & {\ddot{x}}_{d2}\left(t\right)=-\frac{f}{m}{\dot{x}}_{d2}\left(t\right)-\frac{k}{m}{x}_{d2}\left(t\right)+\tilde{w}\left(t\right)\hfill \\ \\ \hfill & x_{d1}\left(t\right)=-\frac{m}{4S_d}{\ddot{x}}_{d2}\left(t\right)-\frac{F_r\left({\dot{x}}_{d2}\left(t\right)\right)}{4S_d}+\frac{k}{4S_d}{x}_{d2}\left(t\right)\hfill \\ \\ \hfill & u\left(t\right)=-\frac{T_{air}}{K_{air}}{\dot{x}}_{d1}\left(t\right)+\frac{1}{K_{air}}{x}_{d1}\left(t\right)\hfill \end{array}\end{array}\right.\hfill \end{array}$$

(29)

We apply the proposed method here. As a noise mitigation measure, we design the operator (Q) as a first-order low-pass filter as shown in Equation (30). Here, $T_{LPF}$ represents the time constant of the first-order low-pass filter. Furthermore, we define the operators N and D as the operators passed through the first-order low-pass filter (Q), as shown in Equations (31) and (32), respectively.

$$\begin{array}{cc}\hfill & Q:\left\{\begin{array}{cc}\hfill \dot{\tilde{w}}\left(t\right)& =\frac{1}{T_{LPF}}(-\tilde{w}\left(t\right)+w\left(t\right))\hfill \end{array}\right.\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & N:\left\{\begin{array}{cc}\hfill & {\dot{\tilde{w}}}_n\left(t\right)=\frac{1}{T_{LPF}}(-{\tilde{w}}_n\left(t\right)+w\left(t\right))\hfill \\ \hfill & {\ddot{x}}_n\left(t\right)=-\frac{f}{m}{\dot{x}}_n\left(t\right)-\frac{k}{m}{x}_n\left(t\right)+{\tilde{w}}_n\left(t\right)\hfill \\ \hfill & y\left(t\right)=x_n\left(t\right)\hfill \end{array}\right.\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & D:\left\{\begin{array}{c}\begin{array}{cc}\hfill & {\dot{\tilde{w}}}_d\left(t\right)=\frac{1}{T_{LPF}}(-{\tilde{w}}_d\left(t\right)+w\left(t\right))\hfill \\ \hfill & {\ddot{x}}_{d2}\left(t\right)=-\frac{f}{m}{\dot{x}}_{d2}\left(t\right)-\frac{k}{m}{x}_{d2}\left(t\right)+{\tilde{w}}_d\left(t\right)\hfill \\ \\ \hfill & x_{d1}\left(t\right)=-\frac{m}{4S_d}{\ddot{x}}_{d2}\left(t\right)-\frac{F_r\left({\dot{x}}_{d2}\left(t\right)\right)}{4S_d}+\frac{k}{4S_d}{x}_{d2}\left(t\right)\hfill \\ \\ \hfill & u\left(t\right)=-\frac{T_{air}}{K_{air}}{\dot{x}}_{d1}\left(t\right)+\frac{1}{K_{air}}{x}_{d1}\left(t\right)\hfill \end{array}\end{array}\right.\hfill \end{array}$$

(32)

Based on Equations (31) and (32), the following Bezout identity can be expressed.

$$\begin{array}{cc}\hfill & SN+RD=M\hfill \end{array}$$

(33)

The operator (R) is designed to satisfy Equation (34). Here, the operator (S) is assumed to be the identity operator (I) to simplify the design. Equation (33) becomes Equation (34).

$$\begin{array}{c}\hfill R=(M-N)D^{-1}\end{array}$$

(34)

where M is the proportional operator, and $g_d$ is the design parameter. Then, Equation (35) is expressed as follows.

$$\begin{array}{cc}\hfill & M:\left\{\begin{array}{cc}\hfill & x_m\left(t\right)=\frac{1}{g_d}w\left(t\right)\hfill \\ \hfill & r\left(t\right)=x_m\left(t\right)\hfill \end{array}\right.\hfill \end{array}$$

(35)

With the above design ($N,D^{-1}$, and M), the operator (R) becomes Equation (36).

$$\begin{array}{cc}\hfill & R:\left\{\begin{array}{c}\begin{array}{cc}\hfill & {\dot{x}}_{rd1}\left(t\right)=-\frac{1}{T_{air}}{x}_{rd1}\left(t\right)+\frac{K_{air}}{T_{air}}u\left(t\right)\hfill \\ \hfill & {\ddot{x}}_{rd2}\left(t\right)=-\frac{F_r\left({\dot{x}}_{rd2}\left(t\right)\right)}{m}-\frac{k}{m}{x}_{rd2}\left(t\right)+\frac{4S_d}{m}{x}_{rd1}\left(t\right)\hfill \\ \hfill & {\tilde{w}}_{rd}\left(t\right)={\ddot{x}}_{rd2}\left(t\right)-\frac{f}{m}{\dot{x}}_{rd2}\left(t\right)+\frac{k}{m}{x}_{rd2}\left(t\right)\hfill \\ \hfill & w_{rd}\left(t\right)=T_{LPF}{\dot{\tilde{w}}}_{rd}\left(t\right)+{\tilde{w}}_{rd}\left(t\right)\hfill \\ \hfill & {\ddot{x}}_{rn}\left(t\right)=-\frac{f}{m}{x}_{rn}\left(t\right)-\frac{k}{m}{x}_{rn}\left(t\right)+{\tilde{w}}_{rd}\left(t\right)\hfill \\ \hfill & x_m\left(t\right)=\frac{1}{g_d}{w}_{rd}\left(t\right)\hfill \\ \hfill & e^{*}\left(t\right)=x_m\left(t\right)-x_{rn}\left(t\right)\hfill \end{array}\end{array}\right.\hfill \end{array}$$

(36)

With the above design, we were able to convert the control target from $P=N{D}^{-1}\left[u\right]\left(t\right)$ to $P^{\prime }=N{M}^{-1}\left[r\right]\left(t\right)$. From now on, the target of control is expressed as Equation (37).

$$\begin{array}{cc}\hfill & P^{\prime }:\left\{\begin{array}{cc}\hfill \dot{\tilde{w}}\left(t\right)& =\frac{1}{T_{LPF}}(-\tilde{w}\left(t\right)+g_{d}r\left(t\right))\hfill \\ \hfill \ddot{y}\left(t\right)& =-\frac{f}{m}\dot{y}\left(t\right)-\frac{k}{m}y\left(t\right)+\tilde{w}\left(t\right)\hfill \end{array}\right.\hfill \end{array}$$

(37)

5.2. Proof of Stability of Control System

In this study, the control system is designed using operator theory. The right factorization is used to decompose the plant (P) into operators $N,$ and D. Stability is guaranteed by satisfying the Bezoo equation. First, we show $SN,$ and $RD$ using Equations (31), (32) and (36).

$$\begin{array}{cc}\hfill SN:& \left\{\begin{array}{cc}\hfill & {\dot{\tilde{w}}}_n\left(t\right)=\frac{1}{T_{LPF}}(-{\tilde{w}}_n\left(t\right)+w\left(t\right))\hfill \\ \hfill & {\ddot{x}}_n\left(t\right)=-\frac{f}{m}{\dot{x}}_n\left(t\right)-\frac{k}{m}{x}_n\left(t\right)+{\tilde{w}}_n\left(t\right)\hfill \\ \hfill & b\left(t\right)=x_n\left(t\right)\hfill \end{array}\right.\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill RD:& \left\{\begin{array}{c}\begin{array}{cc}\hfill & {\dot{\tilde{w}}}_d\left(t\right)=\frac{1}{T_{LPF}}(-{\tilde{w}}_d\left(t\right)+w\left(t\right))\hfill \\ \hfill & {\ddot{x}}_{d2}\left(t\right)=-\frac{f}{m}{\dot{x}}_{d2}\left(t\right)-\frac{k}{m}{x}_{d2}\left(t\right)+{\tilde{w}}_d\left(t\right)\hfill \\ \hfill & {\tilde{w}}_{rd}\left(t\right)={\ddot{x}}_{rd2}\left(t\right)-\frac{f}{m}{\dot{x}}_{rd2}\left(t\right)+\frac{k}{m}{x}_{rd2}\left(t\right)\hfill \\ \hfill & w_{rd}\left(t\right)=T_{LPF}{\dot{\tilde{w}}}_{rd}\left(t\right)+{\tilde{w}}_{rd}\left(t\right)\hfill \\ \hfill & {\ddot{x}}_{rn}\left(t\right)=-\frac{f}{m}\dot{x_{rn}}\left(t\right)-\frac{k}{m}{x}_{rn}\left(t\right)+{\tilde{w}}_{rd}\left(t\right)\hfill \\ \hfill & x_m\left(t\right)=\frac{1}{g_d}{w}_{rd}\left(t\right)\hfill \\ \hfill & e^{*}\left(t\right)=x_m\left(t\right)-x_{rn}\left(t\right)\hfill \end{array}\end{array}\right.\hfill \end{array}$$

(39)

When the initial values of all the internal states are 0, ${\tilde{w}}_n={\tilde{w}}_d={\tilde{w}}_{rd}$, ${\dot{\tilde{w}}}_n={\dot{\tilde{w}}}_d={\dot{\tilde{w}}}_{rd}$, $x_n=x_{d2}=x_{rd2}=x_{rn}$, ${\dot{x}}_n={\dot{x}}_{d2}={\dot{x}}_{rd2}={\dot{x}}_{rn}$, and $w=w_{rd}$. Therefore, $RD$ become

$$\begin{array}{cc}\hfill RD:& \left\{\begin{array}{c}\begin{array}{cc}\hfill & {\ddot{x}}_n\left(t\right)=-\frac{f}{m}{\dot{x}}_n\left(t\right)-\frac{k}{m}{x}_n\left(t\right)+{\tilde{w}}_n\left(t\right)\hfill \\ \hfill & w\left(t\right)=T_{LPF}{\dot{\tilde{w}}}_n\left(t\right)+{\tilde{w}}_n\left(t\right)\hfill \\ \hfill & x_m\left(t\right)=\frac{1}{g_d}{w}_{rd}\left(t\right)\hfill \\ \hfill & e^{*}\left(t\right)=x_m\left(t\right)-x_n\left(t\right)\hfill \end{array}\end{array}\right.\hfill \end{array}$$

(40)

Therefore, according to Equations (38) and (40), $SN+RD$ becomes

$$\begin{array}{cc}\hfill SN+RD:& \left\{\begin{array}{cc}\hfill & {\dot{\tilde{w}}}_n\left(t\right)=\frac{1}{T_{LPF}}(-{\tilde{w}}_n\left(t\right)+w\left(t\right))\hfill \\ \hfill & {\ddot{x}}_n\left(t\right)=-\frac{f}{m}{\dot{x}}_n\left(t\right)-\frac{k}{m}{x}_n\left(t\right)+{\tilde{w}}_n\left(t\right)\hfill \\ \hfill & b\left(t\right)=x_n\left(t\right)\hfill \\ \hfill & w\left(t\right)=T_{LPF}{\dot{\tilde{w}}}_n\left(t\right)+{\tilde{w}}_n\left(t\right)\hfill \\ \hfill & x_m\left(t\right)=\frac{1}{g_d}{w}_{(}t)\hfill \\ \hfill & e^{*}\left(t\right)=\frac{1}{g_d}w\left(t\right)-x_n\left(t\right)\hfill \\ \hfill & r^{*}\left(t\right)=e^{*}\left(t\right)+b\left(t\right)=\frac{1}{g_d}w\left(t\right)\hfill \end{array}\right.\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill \to SN+RD& \left\{\begin{array}{cc}\hfill & x_m\left(t\right)=\frac{1}{g_d}w\left(t\right)\hfill \\ \hfill & r\left(t\right)=x_m\left(t\right)\hfill \end{array}\right.\hfill \end{array}$$

(42)

5.3. Sliding Mode Controller Design

First, from Equation (37).

$$\begin{array}{cc}\hfill \ddot{y}\left(t\right)& =-\frac{f}{m}\dot{y}\left(t\right)-\frac{k}{m}y\left(t\right)+g_d(1-e^{-\frac{t}{T_{LPF}}})r\left(t\right)\hfill \\ \hfill & =-a_1\dot{y}\left(t\right)-a_{2}y\left(t\right)+f\left(t\right)r\left(t\right)\hfill \end{array}$$

(43)

However, let $a_1=\frac{f}{m},a_2=\frac{k}{m},f\left(t\right)=g_d\left(1-e^{-\frac{t}{T_{LPF}}}\right)$. Next, define the error function ($e\left(t\right)$) for tracking of the reference value as $e\left(t\right)=y_d-y\left(t\right)$, where $y_d$ is the reference value. For sliding mode control, set the switching surface (s) as $s=\dot{e}+Ke$, where K is a design parameter. Consider the Lyapunov function in this case.

$$\begin{array}{cc}\hfill \dot{V}& =s\dot{s}\hfill \\ \hfill & =s(\ddot{e}+Ke)\hfill \\ \hfill & =s\{({\ddot{y}}_d-\ddot{y})+K\dot{e}\}\hfill \\ \hfill & =s({\ddot{y}}_d-f\left(t\right)r+a_1\dot{y}+a_{2}y+K\dot{e})\hfill \\ \hfill & =s{\ddot{y}}_d+s{a}_1\dot{y}+s{a}_{2}y+sK\dot{e}-sf\left(t\right)r\hfill \end{array}$$

where we set $r=r_1+r_2+r_3+r_4$.

$$\begin{array}{cc}\hfill \dot{V}=& (s{\ddot{y}}_d-sf\left(t\right)r_1)+(s{a}_1\dot{y}-sf\left(t\right)r_2)\hfill \\ \hfill & +(s{a}_{2}y-sf\left(t\right)r_3)+(sK\dot{e}-sf\left(t\right)r_4)\hfill \end{array}$$

(44)

We design each r to satisfy the Lyapunov stability condition, the sum of which is shown in Equation (45). Here, the function $g\left(t\right)$ satisfies $g\left(t\right)<f\left(t\right)$ for $t\ge 0$, and ${\tilde{\ddot{y}}}_d,b_0$ are design parameters satisfying ${\ddot{y}}_d<{\tilde{\ddot{y}}}_d$ and $0<b_0<1$, respectively.

$$\begin{array}{c}\hfill r=\frac{\mathrm{sat}\left(s\right)}{b_{0}g\left(t\right)}({\tilde{\ddot{y}}}_d+a_1\left|\dot{y}\right|+a_2\left|y\right|+K\left|\dot{e}\right|)\end{array}$$

(45)

6. Simulation and Experiment

In this section, based on the control system designed in Section 5, simulations and experiments of pneumatic stage position control are conducted to verify the effectiveness of the proposed method.

6.1. Simulation Parameters

The simulation results for the target response are presented in this section. The parameters for the LuGre model, the parameters for the pneumatic stage, and the parameters used for the sliding mode controller are listed in

Table 2. Simulation parameters.

Symbol	Value	Unit	Symbol	Value	Unit
${\sigma }_0$	$1.5\times {10}^4$	$\mathrm{N}/\mathrm{m}$	m	15	$\mathrm{kg}$
${\sigma }_1$	$0.01$	$\mathrm{N}·\mathrm{s}/\mathrm{m}$	$S_d$	198	${\mathrm{mm}}^2$
${\sigma }_2$	174	$\mathrm{N}·\mathrm{s}/\mathrm{m}$	k	$0.01$	$\mathrm{N}/\mathrm{m}$
$v_s$	$0.01$	$\mathrm{m}/\mathrm{s}$	$T_{air}$	$0.08$	$\mathrm{s}$
$v_b$	$0.005$	$\mathrm{m}/\mathrm{s}$	$K_{air}$	$4.2\times {10}^4$	$\mathrm{Pa}/\mathrm{V}$
T	0	$\mathrm{s}$	$g_d$	$1.0\times {10}^2$	−
n	$0.5$	[-]	K	34	−
$F_c$	$0.4$	$\mathrm{N}$	f	218	$\mathrm{N}·\mathrm{s}/\mathrm{m}$
$F_s$	$0.41$	$\mathrm{N}$	$g\left(t\right)$	99	−
${\tau }_{hp}$	$0.01$	$\mathrm{s}$	$b_0$	$0.6$	−
${\tau }_{hn}$	$0.2$	$\mathrm{s}$	${\tilde{\ddot{y}}}_d$	5	−
${\tau }_{h0}$	20	$\mathrm{s}$	$h_d$	100	−
$T_{LPF}$	$0.01$	$\mathrm{s}$

. Each parameter was determined based on the experimental setup used in this study.

6.2. Simulation Results

In this section, the effectiveness of the controllers (S and R) designed based on operator theory is verified by simulating the position displacement of a pneumatic stage using MATLAB and Simulink.The sampling time of this simulation is 1 ms. The pneumatic stage is operated at $t>0$. To verify the position change when the pneumatic stage is moved from $0\mathrm{cm}$ to $1.0\mathrm{cm}$, the control input is applied after 1 s. Figure 11a shows the simulation results when the pneumatic stage position was changed from $0\mathrm{cm}$ to $1.0\mathrm{cm}$. The voltage u applied to the servo valve in this case is shown in Figure 11b.

As shown in Figure 11, the target position of 1.0 cm is reached in approximately 0.3 s. Moreover, no overshoot or steady-state error was observed. This confirms the effectiveness of the control system based on the proposed method.

Furthermore, simulations were conducted under the same conditions, even when Gaussian noise was added. The noise was generated using Simulink’s Random Number block. The noise magnitudes were $\pm 4[\%],\pm 40[\%]$, and $\pm 80[\%]$ of the target value, with variances of $1.0\times {10}^{-7},1.0\times {10}^{-5},$ and $4.0\times {10}^{-5}$, respectively. The results with and without the filter were compared under these conditions. As shown in Figure 12, the system with the filter successfully tracked the target, even in the presence of disturbances, whereas the system without the filter failed to do so. The steady-state deviations, according to the data sheet, were $\pm 10\left[\mathsf{\mu }\mathrm{m}\right],\pm 100\left[\mathsf{\mu }\mathrm{m}\right],$ and $\pm 250\left[\mathsf{\mu }\mathrm{m}\right]$. This is likely due to the continuous presence of noise and its varying intensity. However, since the deviations are within $\pm 5\%$ of the target value, it can be said that target tracking was achieved. These results confirm the effectiveness of the proposed method against noise.

6.3. Experimental Results

In this section, we present the results of experiments using the actual setup. The parameters used in the experiments are identical to those listed in Table 2 and used for simulations. We verify the position change of the pneumatic stage from $0\mathrm{m}$ to $1.0\mathrm{cm}$, similar to simulations. The control input starts at $1s$. The experimental results of the pneumatic stage’s position change from $0\mathrm{m}$ to $1.0\mathrm{cm}$ and the corresponding errors are shown in Figure 13. From Figure 13, it can be observed that the target position of 1.0 cm is achieved around 0.9 s.

Comparing the experimental and simulation results, while simulations show no overshoot, the experimental setup exhibits overshoot. This discrepancy is likely due to model inaccuracies or errors. However, it is evident that the output does not diverge, indicating that the system behaves as expected theoretically. This confirms the tracking capability of the pneumatic stage’s position control. Additionally, it is noted that the maximum deviation is $4\mathsf{\mu }\mathrm{m}$, demonstrating the effectiveness of the proposed method in practical experiments.

7. Conclusions

In this study, we extended right coprime factorization based on operator theory. Specifically, we performed a right coprime factorization combined with a filter for noise reduction. To verify the effectiveness of the proposed method, we applied it to a nonlinear pneumatic stage and designed a control system based on the proposed method. We confirmed its effectiveness through simulations and experiments. The proposed method proved to be effective against noise, which could not be mitigated by the previous right coprime factorization. In the future, we would like to apply this proposal not only to right coprime factorization but also to left coprime factorization. If this can be achieved, it will be possible to design control systems that are effective not only against noise but also against disturbances.