Stabilization of Nonlinear Vibration of a Fractional-Order Arch MEMS Resonator Using a New Disturbance-Observer-Based Finite-Time Sliding Mode Control

Abstract

This paper deals with chaos control in an arch microelectromechanical system (MEMS) from the fractional calculus perspective. There is a growing need for effective controllers in various technological fields, and it is important to consider disruptions, uncertainties, and control input limitations when designing a practical controller. To address this problem, we propose a novel disturbance-observer-based terminal sliding mode control technique for stabilizing and controlling chaos in a fractional-order arch MEMS resonator. The design of this technique takes into account uncertainty, disturbances, and control input saturation in the fractional-order system. The proposed control technique is practical for real-world applications because it includes control input saturation. The equation for a fractional-order arch MEMS resonator is presented, and its nonlinear vibration and chaotic behavior are studied. The design process for the proposed control technique is then described. The Lyapunov stability theorem is used to prove the finite-time convergence of the proposed controller and disturbance observer. The proposed controller is applied to the arch MEMS resonator, and numerical simulations are used to demonstrate its effectiveness and robustness for uncertain nonlinear systems. The results of these simulations clearly show the effectiveness of the proposed control technique.

1. Introduction

Microelectromechanical systems (MEMSs) are small devices that have gained significant attention in recent years due to their broad potential applications in actuators, sensors, and energy harvesters [1,2,3]. Among the various types of MEMS devices, arch resonators are commonly used as sensors and actuators due to their relatively simple structure, high sensitivity, and good performance in terms of resonance frequency, quality factor, and energy conversion efficiency [4]. However, the nonlinear behavior of arch resonators, including hysteresis, saturation, and friction, can pose significant challenges to the design and control of these devices [5].

The stability and performance of arch resonators can be improved by using advanced control techniques, such as disturbance-observer-based control, sliding mode control, and fractional-order control [6]. Disturbance-observer-based control is a robust control method that can compensate for unknown or uncertain disturbances and uncertainties in the system model [7]. Sliding mode control is a robust control method that can guarantee a finite-time convergence to the desired equilibrium point and reject external perturbations [8].

One key component in the control of complex systems is the disturbance observer, which is able to compensate for unknown or uncertain disturbances [9,10,11,12,13,14,15,16,17,18,19,20,21]. It is typically used in systems that have nonlinear or chaotic dynamics, where traditional control methods may not be effective [22,23,24,25,26,27,28,29,30,31]. The disturbance observer estimates the disturbance based on the system’s output and then generates a control signal that compensates for the disturbance. By compensating for the disturbance, the disturbance observer can improve the system’s performance, stability, and robustness. Additionally, it can also increase the system’s accuracy and reduce the effect of any uncertainties present in the system [10]. Overall, disturbance observers are an important tool in controlling nonlinear and chaotic systems, as they can help to improve the system’s performance and robustness in the presence of disturbances.

Fractional calculus is widely acknowledged as a valuable tool for modeling and analyzing the dynamic behavior of MEMSs [10]. The fractional derivative operator, which is a generalization of the integer-order derivative operator, can capture the memory and non-integer-order derivative effects of the system dynamics, which are often neglected in the classical integer-order modeling approach [32]. As a result, fractional calculus has been applied to various types of MEMS devices, such as microgyroscopes, microaccelerometers, microelectromechanical switches, and microresonators, to improve the accuracy and robustness of the modeling and control of these devices [33].

One of the main advantages of fractional calculus in MEMSs is that it can provide a more realistic and accurate representation of the system dynamics, especially for systems with memory and viscoelastic effects [34]. For example, the fractional derivative operator has been applied to model the dynamic behavior of microgyroscopes, which are commonly used for inertial navigation and motion sensing applications [35]. By using the fractional derivative operator, the memory and viscoelastic effects of the microgyroscope can be represented, which are a result of the internal friction and structural damping of the device. As a result, the fractional-order modeling approach can provide a more accurate representation of the microgyroscope dynamics, especially at low and intermediate frequencies, where the integer-order modeling approach may not be sufficient [36]. The application of fractional calculus in MEMSs has the potential to significantly enhance the accuracy and robustness of the modeling and control of these devices.

Moreover, fractional calculus has enhanced the performance of other complex systems. To name a few, in [37] a fractional-order sliding-mode control, which utilizes a two-hidden-layer recurrent neural network (THLRNN) for a single-phase shunt active power filter has been proposed. It has been demonstrated that the fractional approach effectively compensates for current with acceptable tracking error and provides better compensation properties and robustness compared to traditional neural sliding control. In [38], a coupled fractional-order sliding mode control (CFSMC) and obstacle avoidance scheme is proposed for four-wheeled steerable mobile robots (FSMRs) in dynamic environments to improve collision-free trajectory tracking control. They validated their technique through experiments on a real-time FSMR system, which showed improved performance in terms of significantly mitigated following errors, faster disturbance rejection, and smooth transition as compared to conventional methods. In [39], a permanent magnet synchronous motor (PMSM) system using a fractional-order sliding mode control (SMC) method was proposed. The superiority of that method in terms of faster convergence, better tracking precision, and better disturbance rejection properties has been demonstrated.

In this paper, we propose a new control approach for the stabilization of nonlinear vibration of fractional-order arch MEMS resonators. The proposed approach combines the advantages of disturbance-observer-based control, sliding mode control, and fractional-order control to achieve a high level of robustness and performance in the presence of nonlinearities, uncertainties, and external perturbations.

The main contribution of this paper is the proposal of a new control approach for the stabilization of nonlinear vibration of fractional-order arch MEMS resonators. This approach combines the advantages of three advanced control techniques: disturbance-observer-based control, sliding mode control, and fractional-order control. The proposed control approach is able to handle unknown external disturbances and achieve a high level of robustness and performance in the presence of nonlinearities, uncertainties, and external perturbations.

One of the key innovations of this approach is the tailoring of the control signal and sliding surface specifically for fractional-order systems. This allows for an improved representation of the system dynamics, especially for systems with memory and viscoelastic effects, which are often neglected in traditional integer-order modeling. Additionally, the proposed control approach is supported by a rigorous theoretical analysis, using Lyapunov stability theory and finite-time stability theory to establish the stability conditions and the convergence rate of the controller, even in the presence of unknown input saturation for fractional-order systems. Another important aspect of the proposed control approach is its ability to handle control input saturation. Control input saturation occurs when the control input reaches its maximum or minimum value, which can result in poor performance or even instability of the system. The proposed control approach includes a saturation function that ensures the control input stays within a certain range, allowing for improved performance and stability even in the presence of input saturation. This feature of the proposed approach enhances the robustness and practicality of the proposed control strategy and its ability to handle real-world scenarios and constraints. The effectiveness of the proposed control approach is demonstrated through simulations and experiments on a fractional-order arch MEMS resonator, providing a significant advancement in the field of MEMS control, with potential applications in a wide range of areas such as navigation, motion sensing, and energy harvesting.

The rest of the paper is organized in the following way: In Section 2, we introduce the control approach and derive the stability conditions and the convergence rate. In Section 3, we present the mathematical model of a fractional-order arch MEMS resonator. In Section 4, we present the simulation and numerical results of the stabilization of the proposed system. The last section of the paper, Section 5, includes our concluding remarks and suggestions for future research.

2. Control Design

In this section, at first, some basic definitions and lemma and are expressed, and after that, the proposed control and its finite-time stability are elaborated. Consider the state-space equation of the system in the presence of unknown disturbance as follows:

$$\begin{array}{c}{D}^{{\alpha }_1\left(t\right)}{x}_1=x_2\\ D^{{\alpha }_2\left(t\right)}{x}_2=f\left(x,t\right)+d_0\left(t\right)+g\left(x,t\right)u\left(t\right)\end{array}$$

(1)

where ${\alpha }_1$ and ${\alpha }_2$ denote the fractional derivative. $x_1$ and $x_2$ stand for the states of the system. $f\left(x,t\right)$ and $g\left(x,t\right)$ represent the nonlinear functions of the system, $d_0\left(t\right)$ represents unknown disturbances, and $u\left(t\right)$ denotes control input.

2.1. Control Input Saturation

Input saturation is an important consideration in the design of nonlinear control for MEMS resonators because it ensures that the control input stays within a safe and acceptable range. Without input saturation, the control input could potentially become too large and cause the resonator to become unstable or even damage the system. Input saturation also helps to prevent the control input from reaching its maximum or minimum values, which would lead to a loss of control over the system. Additionally, input saturation helps to prevent the control input from saturating the actuator, which would result in poor control performance. By limiting the range of the control input, input saturation helps to ensure that the system remains stable and performs within acceptable limits, even in the presence of uncertainties and disturbances. This is particularly important in MEMS resonators, which are sensitive to small changes in input and can be easily affected by external disturbances.

In addition to ensuring stability and performance, control input saturation is also important in the design of nonlinear control for MEMS resonators because it takes into account the physical limitations of the system. MEMS resonators are often designed with specific actuators that have limited range of motion and force capabilities. If the control input is not limited by saturation, the controller may generate inputs that exceed the physical capabilities of the actuator, which would result in unrealistic control performance. This would lead to an unrealistic controller, which cannot be implemented in real-world scenarios. Therefore, control input saturation is essential in the design of nonlinear control for MEMS resonators because it ensures that the control input stays within the physical limitations of the actuator, making the controller realistic and implementable in real-world systems.

The limited control input is obtained by enforcing constraints on the input saturation as follows:

$$u=\{\begin{array}{cc}{u}_{max}& if u_r>u_{max}\\ u_r& \\ u_{min}& if u_r<u_{min}\end{array}$$

(2)

The limited control input, represented by “$u$”, is applied to the system, and “$u_r$” represents the desired control input command. The lower and upper bounds of control input saturation are represented by $u_{min}$ and $u_{max},$ respectively. Defining $\tilde{u}$ as $u-u_r$ and substituting it in Equation (2) into Equation (1) gives

$$\begin{array}{c}{D}^{{\alpha }_1\left(t\right)}{x}_1=x_2\\ D^{{\alpha }_2\left(t\right)}{x}_2=f\left(x,t\right)+d_0\left(t\right)+g\left(x,t\right)\left(u_r+\tilde{u}\right)=f\left(x,t\right)+d\left(t\right)+g\left(x,t\right)u_r\end{array}$$

(3)

where $d\left(t\right)=g\left(x,t\right)\tilde{u}+d_0 \left(t\right)$ denotes the compound disturbance imposed on the system by applying input saturation.

Lemma 1. 

[40]). Let V(t) be a continuous positive definite function that satisfies the following inequality.

$$\dot{V}\left(t\right)+\vartheta V\left(t\right)+\xi V^{\chi }\le 0, \forall t>t_0$$

(4)

Therefore, $V\left(t\right)$ reaches its equilibrium point in a finite amount of time,$t_s$,

$$t_s\le t_0+\frac{1}{\vartheta \left(1+\chi \right)}\ln \frac{\vartheta V^{1-\chi }\left(t_0\right)+\xi }{\xi }$$

(5)

where$0<\chi <1$ and $\vartheta >$ $\xi$ > 0.

2.2. Fast Disturbance Observer

The use of a disturbance observer allows for improved accuracy in complex systems [9,10,41]. This is because the disturbance observer takes into account factors that may affect the accuracy of the measurement, such as the use of only the first mode of vibration of the MEMS resonator or imprecise dimensions of the micro/nanostructures [42,43]. By considering these factors, the disturbance observer is able to provide a more accurate measurement than would be possible using the first mode of vibration alone or without accounting for imprecise dimensions. This can be particularly important when working with micro/nanostructures, as the small size of these structures can make them difficult to measure accurately. In the process of designing this type of finite-time disturbance observer for the nonlinear system, the following auxiliary variables should be established [44]:

$$\sigma =D^{\alpha -1}z-D^{\alpha -1}x$$

(6)

and $z$ is given by

$$D^{\alpha }z=-k\sigma -\beta sign\left(\sigma \right)-\epsilon {\sigma }^{p_0/q_0}-\left|f\left(x\right)\right|sign\left(\sigma \right)+g\left(x\right)u$$

(7)

where $p_0$ and $q_0$ are odd positive integers, such that $p_0<q_0$. Furthermore, $k \mathrm{and} \epsilon$ are positive constants and $\beta >\left|d\right|$. The disturbance observer $\widehat{d}$ is represented by

$$\widehat{d}=-k\sigma -\beta sign\left(\sigma \right)-\epsilon {\sigma }^{\frac{p_0}{q_0}}-\left|f\left(x\right)\right|sign\left(\sigma \right)-f\left(x\right)$$

(8)

Considering Equations (6)–(8), one can obtain

$$\dot{\sigma }=D^{\alpha }z-D^{\alpha }x=-k\sigma -\beta sign\left(\sigma \right)-\epsilon {\sigma }^{p_0/q_0}-\left|f\left(x\right)\right|sign\left(\sigma \right)-f\left(x\right)-d$$

(9)

According to Equations (7) and (8), the following equation can be achieved:

$$\begin{array}{cc}\tilde{d}=\widehat{d}-d& =-k\sigma -\beta sign\left(\sigma \right)-\epsilon {\sigma }^{\frac{p_0}{q_0}}-\left|f\left(x\right)\right|sign\left(\sigma \right)-f\left(x\right)-d\\ & =-k\sigma -\beta sign\left(\sigma \right)-\epsilon {\sigma }^{\frac{p_0}{q_0}}-\left|f\left(x\right)\right|sign\left(\sigma \right)-f\left(x\right)-D^{\alpha }x+f\left(x\right)\\ & +g\left(x\right)u=-k\sigma -\beta sign\left(\sigma \right)-\epsilon {\sigma }^{\frac{p_0}{q_0}}-\left|f\left(x\right)\right|sign\left(\sigma \right)+g\left(x\right)u-D^{\alpha }x\\ & =D^{\alpha }z-D^{\alpha }x=\dot{\sigma }\end{array}$$

(10)

Theorem 1. 

When the disturbance observer (6)–(8) is applied to the uncertain system (1), the disturbance estimation error (9) will converge to zero in a finite time.

Proof. 

The Lyapunov function candidate is assumed to be

$$V_0={\sigma }^2$$

(11)

The time derivative of $V_0$ is given as

$$\begin{array}{c}{\dot{V}}_0=\sigma \dot{\sigma }=\sigma (-k\sigma -\beta sign\left(\sigma \right)-\epsilon {\sigma }^{p_0/q_0}-\left|f\left(x\right)\right|sign\left(\sigma \right)-f\left(x\right)-d)\\ \le -k{\sigma }^2-\beta \sigma sign\left(\sigma \right)-\epsilon {\sigma }^{\frac{p_0+q_0}{q_0}}-\left|f\left(x\right)\right|\sigma sign\left(\sigma \right)-\sigma f\left(x\right)-\sigma d\\ \le -k{\sigma }^2-\beta \left|\sigma \right|-\epsilon {\sigma }^{\frac{p_0+q_0}{q_0}}-\left|f\left(x\right)\right|\left|\sigma \right|-\sigma f\left(x\right)-\left|\sigma \right|\left|d\right|\\ \le -k{\sigma }^2-\epsilon {\sigma }^{\frac{p_0+q_0}{q_0}}=-k{V}_0-\epsilon V_0^{(p_0+q_0)/2q_0}\end{array}$$

(12)

Thus, based on Lemma 1 and 2, as well as Equations (12), we can conclude that the auxiliary vector $\sigma$ will converge to zero in a finite time, resulting in the convergence of the disturbance approximation error $\tilde{d}$ to zero in a finite time, which is also given as follows:

$$t_s<t_0+\frac{q_0}{k\left(p_0+3q_0\right)}\ln \left(\frac{k{s}_0^{(q_0-p_0)/ q_0}{t}_0}{\epsilon }+1\right)$$

(13)

where $t_{0\Delta }$ denotes the initial time.

Consider the desired value as y. By defining $e=x-y$ as the tracking error, we present a new fractional nonsingular terminal sliding surface, which is defined as follows:

$$\begin{array}{c}s\left(t\right)=D^{\alpha -1}e+D^{\alpha -2}\left({\delta }_{1}e+{\delta }_2 e^{p_1/q_1}\right)\\ \dot{s}\left(t\right)=D^{\alpha }e+D^{\alpha -1}\left({\delta }_1 e+{\delta }_2 e^{p_1/q_1}\right)\end{array}$$

(14)

As long as the system is on the sliding surface, it will adhere to the following equations:

$$s\left(t\right)=0 , \dot{s}\left(t\right)=0$$

(15)

Hence, the dynamics of the proposed fractional nonsingular terminal sliding mode can be achieved by using (14) as follows:

$$\begin{array}{c}\dot{s}\left(t\right)=D^{\alpha }e+D^{\alpha -1}\left({\delta }_1 e+{\delta }_2 e^{\frac{p_1}{q_1}}\right)=0\\ D^{\alpha }e=-D^{\alpha -1}\left({\delta }_1 e+{\delta }_2 e^{\frac{p_1}{q_1}}\right)\end{array}$$

(16)

The fractional-order control input is defined as

$$u=\left(g\left(x\right)-f\left(x\right)-\widehat{d}-D^{\alpha -1}\left({\delta }_1 e+{\delta }_2 e^{\frac{p_1}{q_1}}\right)-{\theta }_{1}s-{\theta }_2{s}^{\frac{p_1}{q_1}}\right)$$

(17)

where $K$, $\delta$, and $\alpha$ are all positive design parameters, with δ being greater than α. □

Theorem 2. 

The control law proposed in Equation (17) ensures that all signals of the fractional-order uncertain system (1) will converge to the desired value in a finite time.

Proof. 

Let the Lyapunov function candidate be

$$V=\frac{1}{2}{s}^2$$

(18)

The Lyapunov function candidate’s derivative with respect to time is

$$\begin{array}{c}\dot{V}=s\dot{s}=s\left(D^{\alpha }e+D^{\alpha -1}\left({\delta }_{1}e+{\delta }_2 e^{\frac{p_1}{q_1}}\right)+\dot{\sigma }\right)\\ =s\left(g\left(x\right)-f\left(x\right)-d-u+D^{\alpha -1}\left({\delta }_1 \mathrm{e}+{\delta }_2 e^{\frac{p_1}{q_1}}\right)+\dot{\sigma }\right)\end{array}$$

(19)

According to the proposed control law, we have

$$\begin{array}{c}\dot{V}=s(g\left(x\right)-f\left(x\right)-d-\left(g\left(x\right)-f\left(x\right)-\widehat{d}-D^{\alpha -1}\left({\delta }_1 e+{\delta }_2 e^{\frac{p_1}{q_1}}\right)-{\theta }_{1}s-{\theta }_2{s}^{\frac{p_1}{q_1}}\right)\\ +D^{\alpha -1}\left({\delta }_1 e+{\delta }_2 e^{\frac{p_1}{q_1}}\right)+\dot{\sigma })\\ \dot{V}\le -{\theta }_1{s}^2-{\theta }_2{s}^{\left(p_1+q_1\right)/q_1}\\ \dot{V}\le -2{\theta }_1{V}_0-{\theta }_2{2}^{\frac{p_1+q_1}{2q_1}}{V_0}^{\frac{p_1+q_1}{2q_1}}\end{array}$$

(20)

Based on Lemma 1 and Equation (20), we can conclude that the uncertain fractional-order system (1) will converge to desired value in a finite time. This ends the proof. The convergence time of the system ($t_{s\Delta }$) is also given as follows:

$$t_s<t_0+\frac{q_1}{k\left(p_1+3q_1\right)}\ln \left(\frac{k{s}^{(q_1-p_1)/ q_1}\left(t_0\right)}{\epsilon }+1\right)$$

(21)

where $t_0$ denotes the initial time. □

3. Arch MEMS Resonator

In this paper, we apply the proposed control technique to an initially curved, doubly clamped nano-beam length $L$, width $b,$ and thickness $d_b$, as depicted in Figure 1. The initial deflection of the arch is assumed to be given by $w_0\left(x\right)$ as shown in Figure 1. The transverse deflection of the arch from the initial resting condition in the positive z direction is represented by $w\left(x, t\right)$, as illustrated in Figure 1. An electrostatic transverse load distributed along the length of the arch actuates the arch. As shown in the figure, this load is generated by applying a voltage V to the beam. The schematic diagram of the arch MEMS resonator is depicted in Figure 1.

The nondimensional governing equation for the arch, based on the Euler–Bernoulli beam theory, is given by [45,46,47].

$$\ddot{x}+\mu \dot{x}+\left(1+2h^2{\alpha }_m\right)x+{\alpha }_m{x}^3-3{\alpha }_{m}hx=\frac{\beta \left(1+2Rcos\left({\omega }_{0}t\right)\right)}{2b_{11}\sqrt{{\left(1+h-x\right)}^3}}+u$$

(22)

where

$$\begin{array}{c}{\alpha }_m=\frac{b{d}_b{g}_0^2}{2I_y{{\displaystyle \int }}_0^1{\left(\frac{d^2\varphi }{d{\widehat{x}}^2}\right)}^{2}d\widehat{x}}{\left({{\displaystyle \int }}_0^1{\left(\frac{d\varphi }{d\widehat{x}}\right)}^{2}d\widehat{x}\right)}^2,\beta =\frac{{\epsilon }_{a0}b{L}^4{V}_{DC}^2}{2g_0^3\tilde{E}{I}_y},R=\frac{V_{AC}}{V_{DC}},h=\frac{h_0}{g_0},\\ {\omega }_o=\frac{{\Omega }_{0}t}{\widehat{\tau }}\sqrt{\frac{{{\displaystyle \int }}_0^1{\left(\frac{d\varphi }{d\widehat{x}}\right)}^{2}d\widehat{x}}{{{\displaystyle \int }}_0^1{\varphi }^{2}d\widehat{x}}},b_{11}={{\displaystyle \int }}_0^1{\left(\frac{d\varphi }{d\widehat{x}}\right)}^{2}d\widehat{x}, \mu =\frac{C_v{L}^2}{\sqrt{\tilde{E}{I}_{y}b{d}_b\rho }}\sqrt{\frac{{{\displaystyle \int }}_0^1{\left(\frac{d\varphi }{d\widehat{x}}\right)}^{2}d\widehat{x}}{{{\displaystyle \int }}_0^1{\left(\frac{d^2\varphi }{d{\widehat{x}}^2}\right)}^{2}d\widehat{x}}}\end{array}$$

(23)

Table 1. The parameters of the fractional-order MEMS resonator [5].

Parameter	Denotation	Parameter	Denotation
L	Length of microbeam	A	Cross-sectional area
$b$	Width of microbeam	${\epsilon }_{a0}$	Vacuum permittivity
$d_b$	Thickness of microbeam	$C_v$	Viscous damping coefficient
Iy	Moment of inertia	$\tilde{E}$	Young’s modulus
${\Omega }_0$	Harmonic load frequency	ρ	Mass density
$V_{DC}$	Direct voltage	$V_{AC}$	Alternating voltage
$\varphi$	Spatial part of the transverse deflection

lists the denotations for the parameters of the fractional-order MEMS resonator [5].

Let $x_1=x, x_2=\dot{x}$; this means the state-space equation of the arch MEMS resonator is as follows:

$$\begin{array}{c}{\dot{x}}_1=x_2\\ {\dot{x}}_2=\frac{\beta \left(1+2Rcos\left({\omega }_{0}t\right)\right) }{2b_{11}\sqrt{{\left(1+h-x_1\right)}^3}}-\left(1+2h^2{\alpha }_m\right)x_1-\mu x_2-{\alpha }_m{x}_1^3+3{\alpha }_{m}h{x}_1^2+u\end{array}$$

(24)

Some studies suggest modeling the MEMS using fractional-order equations because they exhibit complex dynamics that cannot be accurately represented using only integer-order models. The dynamics of MEMSs are characterized by a combination of memory effects, viscoelastic behavior, and nonlocal interactions. These phenomena are not captured by traditional integer-order models but can be accurately represented using fractional-order models. Motivated by this, the current study investigates a fractional version of the arch MEMS resonator, which is described by

$$\begin{array}{c}{D}^{{\alpha }_1\left(t\right)}{x}_1=x_2\\ D^{{\alpha }_2\left(t\right)}{x}_2=\frac{\beta \left(1+2Rcos\left({\omega }_{0}t\right)\right) }{2b_{11}\sqrt{{\left(1+h-x_1\right)}^3}}-\left(1+2h^2{\alpha }_m\right)x_1-\mu x_2-{\alpha }_m{x}_1^3+3{\alpha }_{m}h{x}_1^2\end{array}$$

(25)

where ${\alpha }_1 \mathrm{and} {\alpha }_2 \in \left(0, 1\right)$ denote the fractional-order derivatives. By setting ${\alpha }_m=7.993,\beta =119.9883,h=0.3, \mu =0.1,u=0,{ b}_{11}$= 198.462, and ${\omega }_0=0.4706$ [45,46,47], and ${\alpha }_1={\alpha }_2=0.98,$ we investigated the nonlinear dynamics of the fractional-order arch MEMS resonator in this study.

In this paper, we used the Caputo fractional derivative to model the dynamics of the MEMS resonator. The Caputo fractional derivative is a widely used approach for modeling systems with memory effects and nonlocal interactions, which are commonly observed in micro/nanomechanical systems. Unlike the classical integer-order derivatives, the Caputo fractional derivative allows for the modeling of these complex dynamics by introducing a nonlocal term into the governing equations. This nonlocal term describes the history of the system and accounts for the memory effects. The Caputo fractional derivative is defined in terms of the standard integer-order derivative and is appropriate for modeling systems with initial conditions, unlike the Riemann–Liouville fractional derivative. By using this approach, we have been able to accurately capture the dynamics of the MEMS resonator and simulate its behavior under different conditions. Figure 2 and Figure 3 illustrate the periodic and chaotic attractors of a fractional-order arch MEMS resonator, respectively.

Next, we investigated the effects of the fractional order on the behavior of the system.

4. Numerical Simulation of Applied Control Technique

In the previous sections, we presented the mathematical model of the system and developed a controller to suppress the vibrations of a fractional-order arch MEMS resonator. To evaluate the performance and effectiveness of the proposed control scheme, we simulated the time-response of the fractional-order system. At first, we did not apply the control input saturation.

Figure 4 illustrates the stabilized states of the system, and Figure 5 displays the control input determined by the employed control scheme. The states and control input approach zero within a finite amount of time using the proposed disturbance-observer-based TSMC (time-scale modification control) method. For this case, in less than 2 time units the system entirely converged to the desired value; compared with the traditional method, this is a luminous achievement.

As shown in Figure 5, at the starting point, the system needs a large control input; in real-world systems, sometimes applying such a control signal is impossible. In order to provide a realistic simulation, all limitations due to the physical limitations should be taken to account. Hence, here, we consider input saturation. Based on our analyses in Section 2, the proposed controller is able to stabilize the fractional-order system even when there is control input saturation. Here, we add the control saturation, which acts as a bound and prevents the amplitude of control signals from being larger than 100. Figure 6 and Figure 7 show the states and control input of the system when the proposed fractional-order controller is applied, while the control input is subjected to saturation. Although in this case, the control signal is saturated, the error of systems is zero in less than 3 time units.

Next, to test the other capabilities of the proposed technique, we applied a nonlinear, unknown disturbance to the system. Here, based on the stability analyses in Section 2, we expect the controller be able to accurately estimate the unknown function and control the system. Here, we set the bound of the control signal to [−100,100]. Figure 8 and Figure 9, respectively, show the time history of the states of the system and control signal. The time history of the disturbance and its estimated value have been provided in Figure 10. As demonstrated in this figure, although the applied disturbance is complicated and nonlinear, the proposed control scheme accurately estimated it in less than 1 time unit. By accurately estimating the disturbance, the control scheme was able to effectively manipulate or influence the behavior of the system in order to bring it to a desired state or maintain it in a desired state.

The extension of the proposed control approach has potential applications in the design and control of various types of fractional-order MEMS devices, as well as in other fields, such as aerospace, automotive, biomedical, and renewable energy systems.

5. Conclusions

This paper has presented a new technique for the control of a fractional chaotic arch microelectromechanical system (MEMS) resonator. The control method is based on the use of a disturbance observer, which is employed to overcome the lack of accuracy in the system due to the presence of disturbances, which are known to be present in micro/nanostructures. The disturbance observer is used in conjunction with a sliding model control signal, which is used to control the vibration of the system. It is worth noting that the proposed technique for the sliding model control (TSMC) and its sliding surface are specifically tailored for fractional-order systems. This is significant because there are relatively few studies in the literature that propose finite-time sliding surfaces for fractional-order systems. Additionally, the proposed control scheme is able to handle nonlinear control input saturations, which is an important feature for this type of system. Numerical simulation results were presented in the paper to demonstrate the effective performance of the proposed control scheme. The simulation results were presented for a variety of different situations, including both with and without control input saturation. Additionally, in order to demonstrate the practical performance of the proposed control technique in real-world applications, the control schemes were also tested on a chaotic arch MEMS resonator in the presence of an unknown, time-varying external disturbance. The results of this study demonstrate that the proposed control technique is a valuable choice for addressing the inherent uncertainties and potential for faults in micro/nanomechanical systems. In real-world applications, it is often the case that the state of the system is not available for measurement. As such, one potential area of future work is to develop an online finite-time observer-based Kalman filter algorithm, which would allow for the estimation of the state of the system in real time. Finally, the results of this work could be extended to discrete-time systems, which would enable the application of this control technique to a wider range of systems.