Event-Triggered Control for Flapping-Wing Robot Aircraft System Based on High-Gain Observers

Abstract

In this paper, an event-triggered (ET) control strategy for a flapping-wing robot aircraft system (FWRA) based on high-gain observers is investigated. To solve the vibration problems of bending deformation and torsional deformation that may be encountered in an FWRA during flight, a novel control method is proposed. Firstly, high-gain observers are used to accurately estimate the unmeasured states of the system, and then output feedback ET controllers are designed by combining ET mechanisms. These controllers can effectively suppress the vibrations and ensure the stability of the system, and the occurrence of the Zeno phenomenon is effectively prevented, while the communication burden is reduced. Finally, the simulation results verify the effectiveness of the proposed method.

1. Introduction

A flapping-wing robot aircraft system (FWRA) is an advanced aerial vehicle that integrates many technologies in bionics, aerodynamics, mechanical manufacturing and intelligent control. The system imitates the flight mode of birds and insects in nature [1,2,3] and generates lift and thrust by flapping its wings to realize flight tasks in complex environments. Compared with a traditional fixed-wing structure, the flexible wing structure has significant advantages, such as a light weight, high flexibility, high energy efficiency and low noise [4,5,6], and has great application potential in military reconnaissance, environmental perception, disaster relief and other fields [7,8]. A 3D model of an FWRA is shown in Figure 1. However, FWRAs often experience significant bending and torsional vibrations during flight due to their complex mechanical structure and high-speed flapping action. These vibrations not only affect the stability and control accuracy of the aircraft but may also cause damage to its structure and shorten its service life. Therefore, vibration control becomes one of the key challenges in the design of FWRAs.

Flexible structures include flexible manipulators [9,10,11], flexible beams [12,13,14], flexible risers [15,16,17], etc. Effective vibration suppression in flexible structures is a current research focus. For example, in [18], the problem of vibration control design for an overhead crane system is solved. In [19], a consensus tracking control strategy is proposed for multi-agent systems. The designed controller not only can ensure that the angle and displacement of all agents reach a consensus but can also suppress the elastic deformation and the speed of deformation of each agent. This paper focuses on the vibration control of FWRAs.

In practical engineering, some control signals are not directly measurable due to the influence of noise, and this may result in an inability to accurately control the system [20,21]. In order to solve this problem, observers have come into being. Among these, high-gain observers stand out among the many observers with their unique advantages. Currently, there are many studies on high-gain observers. For example, in [22], a set of high-gain observers is designed to replicate the unmeasurable system states when a system’s states cannot be directly measured. In [23], an enhanced high-gain observer is designed to monitor the posture of a needle tip. In this paper, inspired by the above, high-gain observers are used to estimate the unmeasured states of the system.

In increasingly complex control systems and communication networks, the traditional data transmission and control strategies based on time periodicity are facing many challenges, such as low resource utilization, aggravated network congestion and system response delays. In order to solve these problems, event-triggered (ET) control emerges, as required by the times, and has gradually received extensive attention and research. Compared to traditional periodic sampling, ET control is capable of reducing communication transmissions while maintaining the system’s performance effectively [24,25,26]. At present, many research results have been obtained on ET control. For example, in [27], the communication resources were saved by introducing a relative-threshold-based ET strategy. In [28], an ET output feedback inversion boundary controller is proposed for a 2 × 2 hyperbolic ODE system which determines the trigger time to update the continuous-time control law obtained. In [29], through the design of internal dynamic variables, the communication transmission frequency and control consumption are effectively reduced.

In summary, it is important to develop an output feedback ET controller for an FWRA. The main contributions of this paper are as follows:

(1)

The works [30,31,32] assumed that all system states can be accurately obtained through measurements, and their studies were carried out on this basis. However, in practical engineering, some control signals are difficult to detect directly due to the influence of noise. This may lead to inaccurate system control. To overcome this challenge, this paper uses high-gain observers to estimate the boundary states of the system.

(2)

ET control can effectively lighten the computational burden and communication requirements of the controller and improve the energy efficiency of the system. This control method has received extensive attention and was utilized in [33,34]. However, although ET control has made remarkable achievements in many fields, there is still a lack of relevant research in the specific field of FWRAs. How to optimize the control performance of FWRAs and reduce their resource consumption through an ET mechanism has become a key problem to solve urgently. Therefore, this paper proposes output feedback ET controllers with which the vibrations of an FWRA are effectively suppressed, while communication resources are saved and the communication burden is reduced.

The rest of this paper is organized as follows: Section 2 presents a dynamic model of the FWRA and the reasonable assumptions and control objectives of this paper. Section 3 outlines the design method for the output feedback ET controllers and analyzes the stability of the closed-loop system. In Section 4, simulation results are presented to validate the effectiveness of the proposed approach. The final part presents the main conclusions of the problem studied in this paper.

Notations: To keep this paper concise, the following notations are consistently utilized, $\left(\dot{\ast }\right)=\partial \left(\ast \right)/\partial t$, $\left(\ddot{\ast }\right)={\partial }^2\left(\ast \right)/\partial t^2$, ${\left(\ast \right)}^{\prime }=\partial \left(\ast \right)/\partial l$, ${\left(\ast \right)}^{″}={\partial }^2\left(\ast \right)/\partial l^2$, ${\left(\ast \right)}^{″\prime }={\partial }^3\left(\ast \right)/\partial l^3$, ${\left(\ast \right)}^{⁗}={\partial }^4\left(\ast \right)/\partial l^4$, where $\partial$ represents the partial derivative; $\left(\dot{\ast }\right)$ and $\left(\ddot{\ast }\right)$ represent the first- and second-order partial derivatives of $\left(\ast \right)$ with respect to $t$; and ${\left(\ast \right)}^{\prime }$, ${\left(\ast \right)}^{″}$, ${\left(\ast \right)}^{‴}$, and ${\left(\ast \right)}^{⁗}$ represent the first-, second-, third-, and fourth-order partial derivatives of $\left(\ast \right)$ with respect to $l$. In addition, parts of the variables $t$ and $l$ are omitted, which means that $\left(\ast \right)\left(l,t\right)$, $\left(\ast \right)\left(0,t\right)$, and $\left(\ast \right)\left(S,t\right)$ are simplified into $\left(\ast \right)$, $\left(\ast \right)\left(0\right)$, and $\left(\ast \right)\left(S\right)$, respectively.

2. System Modeling and Preliminaries

The FWRA studied in this paper is characterized by the following PDE dynamic model system for $\forall \left(l,t\right)\in \left(0,S\right)\times \left[0,\infty \right)$ [2]:

$$m\ddot{D}-m{x}_{e}c\ddot{\theta }+\eta E{I}_b{\dot{D}}^{⁗}+E{I}_b{D}^{⁗}=F_b$$

(1)

$$I_p\ddot{\theta }-m{x}_{e}c\ddot{D}-\eta GJ{\dot{\theta }}^{″}-GJ{\theta }^{″}=-x_{a}c{F}_b$$

(2)

$$D\left(0\right)=D^{\prime }\left(0\right)=D^{″}\left(S\right)=\theta \left(0\right)=0$$

(3)

$$E{I}_b{D}^{‴}\left(S\right)+\eta E{I}_b{\dot{D}}^{‴}\left(S\right)=-H_1\left(t\right)$$

(4)

$$GJ{\theta }^{\prime }\left(S\right)+\eta GJ{\dot{\theta }}^{\prime }\left(S\right)=H_2\left(t\right)$$

(5)

where $S$ is the length of the flapping wing; $m$ represents the mass per unit of span; the polar moment of inertia, denoted by $I_p$, characterizes the resistance of the wing’s cross-section to torsional or rotational deformation; $E{I}_b$ represents the bending stiffness; $GJ$ represents the torsional stiffness; $x_{e}c$ represents the distance between the wing’s center of mass and its shear center; $x_{a}c$ can be referred to as the distance between the aerodynamic center and the shear center; the term $\eta$ denotes the Kelvin–Voigt damping coefficient; the wing tip is denoted by $l=S$; and $H_1\left(t\right)$ and $H_2\left(t\right)$ correspond to the boundary control inputs that influence the bending and torsional deformations, respectively. Along the flexible wing, a distributed disturbance exists, which is denoted by $F_b$.

Assumption 1

([30]). For an unknown time-varying distributed disturbance $F_b$, we suppose the constant $F_{b\max }\in {ℝ}^{+}$ is such that it satisfies $\left|F_b\right|\le F_{b\max }$, $\forall \left(l,t\right)\in \left[0,S\right]\times \left[0,+\infty \right)$.

Control objectives: In this paper, combined with ET mechanisms, output feedback ET controllers for an FWRA are designed to achieve the following objectives: (1) Output feedback ET controllers are proposed to effectively suppress the vibration of bending and torsional deformation of the FWRA. (2) We make sure that all signals are bounded. (3) The Zeno phenomenon is avoided.

3. Controller Design and Stability Analysis

To attain the control objectives, output feedback ET controllers are designed, and the stability of the closed-loop system is analyzed. Figure 2 shows a diagram of the control framework, which clearly shows the control idea in this paper.

3.1. Design of the Output Feedback ET Controllers

The boundary signals $D\left(S\right)$ and $\theta \left(S\right)$ can be measured using sensors, and the backward difference method can be used to obtain the signals $\dot{D}\left(S\right)$ and $\dot{\theta }\left(S\right)$. The presence of noise in the actual system leads to the amplification of the measurement noise generated by the sensors during the calculation of the differentiation, which in turn affects the controller implementation. To mitigate the influence of noise on the system’s signal acquisition, high-gain observers are employed to estimate the unmeasurable states $\dot{D}\left(S\right)$ and $\dot{\theta }\left(S\right)$ of the system. Before embarking on the design process, it is essential to consider the following critical lemma.

Lemma 1

([20]). Suppose that we have a system with an output $\wp \left(t\right)$, and its first $p$ derivatives are bounded by $\left|{\wp }^c\right|<Y_c$, where $Y_c,c=1,\dots ,p$ are positive constants; then, the following linear system is considered:

$$\left\{\begin{array}{l}\omega {\hslash }_{\mathrm{ƛ}t}={\hslash }_{\mathrm{ƛ}+1},\mathrm{ƛ}=1,\dots ,p-1,\\ \omega {\hslash }_{pt}=-{\overline{\aleph }}_1{\hslash }_p-{\overline{\aleph }}_2{\hslash }_{p-1}-\cdots -{\overline{\aleph }}_{p-1}{\hslash }_2-{\hslash }_1+o_1\left(t\right)\end{array}\right.$$

(6)

where $\omega$ is any small positive constant, and the parameters ${\overline{\aleph }}_i$, for $\mathrm{ƛ}=1,\dots ,p-1$, are chosen such that the polynomial $s^p+{\overline{\aleph }}_1{s}^{p-1}+\cdots +{\overline{\aleph }}_{p-1}s+1$ is a Hurwitz polynomial. We can observe the following property:

$${\alpha }_c={\displaystyle \frac{{\hslash }_c}{{\omega }^{c-1}}}-o_1^{\left(c-1\right)}=-\omega {\psi }^{\left(c\right)},c=1,\dots ,p-1$$

(7)

where $\psi ={\hslash }_p+{\overline{\aleph }}_1{\hslash }_{p-1}+\cdots +{\overline{\aleph }}_{p-1}{\hslash }_1$, with ${\psi }^{\left(c\right)}$ denoting the $cth$ derivative of $\psi$. In addition, the constants $\mathcal{T}>0$ and $f_c>0$ are such that $\forall t>\mathcal{T}$, and one has $‖{\alpha }_c‖\le \omega n_c,c=1,\dots ,p$, where $‖·‖$ denotes the standard Euclidean norm.

By applying Lemma 1, it is clear that ${\displaystyle \frac{{\hslash }_{c+1}}{{\omega }^c}}$ converges to $o_1^{\left(c\right)}$, which is the $c\mathrm{th}$ derivative of $o_1$; this implies that ${\alpha }_c$ converges to zero due to the high gain ${\displaystyle \frac{1}{\omega }}$. Consequently, it is appropriate to select ${\displaystyle \frac{{\hslash }_{c+1}}{{\omega }^c}}$ as an observer to estimate the output signals up to the $p\mathrm{th}$-order derivative.

Define $J_1\left(t\right)=D\left(S\right)$ and $J_2\left(t\right)=\dot{D}\left(S\right)$; the estimate of the state $J_2\left(t\right)$ is designed as

$${\widehat{J}}_2\left(t\right)={\displaystyle \frac{{\hslash }_2\left(t\right)}{\omega }}$$

(8)

where ${\hslash }_2\left(t\right)$ and the estimate error ${\tilde{J}}_2\left(t\right)$ are designed as

$$\left\{\begin{array}{l}\omega {\hslash }_{1t}={\hslash }_2\hfill \\ \omega {\hslash }_{2t}=-{\overline{\aleph }}_1{\hslash }_2-{\hslash }_1+J_1\left(t\right)\hfill \\ {\tilde{J}}_2={\widehat{J}}_2-J_2\hfill \end{array}\right.$$

(9)

In the same manner, we define $E_1\left(t\right)=\theta \left(S\right)$ and $E_2\left(t\right)=\dot{\theta }\left(S\right)$, and the estimate of the state $E_2\left(t\right)$ is designed as

$${\widehat{E}}_2\left(t\right)={\displaystyle \frac{{\varphi }_{02}\left(t\right)}{{\omega }_0}}$$

(10)

where ${\varphi }_{02}\left(t\right)$ and the estimate error ${\tilde{E}}_2\left(t\right)$ are designed as

$$\left\{\begin{array}{l}{\omega }_0{\varphi }_{01t}={\varphi }_{02}\hfill \\ {\omega }_0{\varphi }_{02t}=-{\overline{\aleph }}_{01}{\varphi }_{02}-{\varphi }_{01}+E_1\left(t\right)\hfill \\ {\tilde{E}}_2={\widehat{E}}_2-E_2\hfill \end{array}\right.$$

(11)

Based on the utilization of the high-gain observers, the following ET mechanisms are proposed for the controller design:

$$H_1\left(t\right)={\upsilon }_1\left(t_k\right),\forall t\in \left[t_k,t_{k+1}\right)$$

(12)

$$t_{k+1}=\inf \left\{t\in ℝ‖e_1\left(t\right)\ge {\zeta }_1\right\}$$

(13)

$$H_2\left(t\right)={\upsilon }_2\left(t_s\right),\forall t\in \left[t_s,t_{s+1}\right)$$

(14)

$$t_{s+1}=\inf \left\{t\in ℝ‖e_2\left(t\right)\ge {\zeta }_2\right\}$$

(15)

where the intermediate control signals are denoted as ${\upsilon }_1\left(t_k\right)$ and ${\upsilon }_2\left(t_s\right)$, and the timing for updating the control input $H_1\left(t\right)$ and $H_2\left(t\right)$ is represented by $t_k$ and $t_s$, respectively. When ET mechanisms (13) and (15) are triggered, the current times are immediately recorded as $t_{k+1}$ and $t_{s+1}$, and the control inputs $H_1\left(t\right)$ and $H_2\left(t\right)$ at this time are sent to the actuator. $0<{\zeta }_1<{\overline{\zeta }}_1$, $0<{\zeta }_2<{\overline{\zeta }}_2$, and ${\zeta }_i,i=1,2$ are unknown positive constants. $e_1\left(t\right)={\upsilon }_1\left(t\right)-H_1\left(t\right)$ and $e_2\left(t\right)={\upsilon }_2\left(t\right)-H_2\left(t\right)$ are the measurement errors, ${\upsilon }_1\left(t\right)$ and ${\upsilon }_2\left(t\right)$ are the control laws to be designed.

There are continuous time-varying coefficients ${\mu }_1\left(t\right)$ and ${\mu }_2\left(t\right)$ which satisfy [32]

$$\left\{\begin{array}{l}{\upsilon }_1\left(t\right)=H_1\left(t\right)+{\mu }_1\left(t\right){\zeta }_1\\ {\upsilon }_2\left(t\right)=H_2\left(t\right)+{\mu }_2\left(t\right){\zeta }_2\end{array}\right.$$

(16)

where

$${\mu }_1\left(t\right)\triangleq \left\{\begin{array}{l}{\mu }_1\left(t_k\left(i\right)\right)=0\\ {\mu }_1\left(t_k\left(i+1\right)\right)=\pm 1\end{array}\right.,\left|{\mu }_1\left(t\right)\right|\le 1$$

(17)

$${\mu }_2\left(t\right)\triangleq \left\{\begin{array}{l}{\mu }_2\left(t_s\left(j\right)\right)=0\\ {\mu }_2\left(t_s\left(j+1\right)\right)=\pm 1\end{array}\right.,\left|{\mu }_2\left(t\right)\right|\le 1$$

(18)

In conjunction with the ET strategies provided, the output feedback ET controllers are formulated as follows:

$${\upsilon }_1\left(t\right)=-k_1\left(a{\widehat{J}}_2+b{J}_1\right)$$

(19)

$${\upsilon }_2\left(t\right)=-k_2\left(a{\widehat{E}}_2+b{E}_1\right)$$

(20)

where $k_i\left(i=1,2\right)$ and $a,b>0$ are constants.

3.2. The Stability Analysis

To achieve the control objectives, the Lyapunov function is chosen as

$$V\left(t\right)=V_1\left(t\right)+V_2\left(t\right)$$

(21)

where

$$V_1\left(t\right)={\displaystyle \frac{a}{2}}m{\displaystyle {\int }_0^S{\dot{D}}^{2}dl}+{\displaystyle \frac{a}{2}}E{I}_b{\displaystyle {\int }_0^S{D^{″}}^{2}dl}+{\displaystyle \frac{a}{2}}{I}_p{\displaystyle {\int }_0^S{\dot{\theta }}^{2}dl}+{\displaystyle \frac{a}{2}}GJ{\displaystyle {\int }_0^S{{\theta }^{\prime }}^{2}dl}$$

(22)

$$\begin{array}{ll}{V}_2\left(t\right)=& bm{\displaystyle {\int }_0^S\dot{D}Ddl}+b{I}_p{\displaystyle {\int }_0^S\dot{\theta }\theta dl}\\ & -bm{x}_{e}c{\displaystyle {\int }_0^S\left[D\dot{\theta }+\theta \dot{D}\right]dl}-am{x}_{e}c{\displaystyle {\int }_0^S\dot{D}\dot{\theta }dl}\end{array}$$

(23)

Let us create a new function

$$\mathsf{\Omega }\left(t\right)={\displaystyle {\int }_0^S\left\{{\dot{\theta }}^2+{\dot{D}}^2+{\left[{\theta }^{\prime }\right]}^2+{\left[D^{″}\right]}^2\right\}dl}$$

(24)

From (22) and (24), one acquires

$${\eta }_2\mathsf{\Omega }\left(t\right)\le V_1\left(t\right)\le {\eta }_1\mathsf{\Omega }\left(t\right)$$

(25)

where ${\eta }_1={\displaystyle \frac{a}{2}}\max \left(m,I_p,E{I}_b,GJ\right)>0$, ${\eta }_2={\displaystyle \frac{a}{2}}\min \left(m,I_p,E{I}_b,GJ\right)>0$.

Thanks to Lemma 2 from [35] and using Young’s inequality, (23) is written as follows:

$$\begin{array}{l}\left|V_2\left(t\right)\right|\le \left(bm+bm{x}_{e}c+am{x}_{e}c\right){\displaystyle {\int }_0^S{\dot{D}}^{2}dl}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+\left(b{I}_p+bm{x}_{e}c+am{x}_{e}c\right){\displaystyle {\int }_0^S{\dot{\theta }}^{2}dl}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+\left(bm+bm{x}_{e}c\right)S^4{\displaystyle {\int }_0^S{\left[D^{″}\right]}^{2}dl}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+\left(b{I}_p+bm{x}_{e}c\right)S^2{\displaystyle {\int }_0^S{\left[{\theta }^{\prime }\right]}^{2}dl}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\le {\eta }_3\mathsf{\Omega }\left(t\right)\end{array}$$

(26)

where ${\eta }_3=\max \left(bm+m{x}_{e}c\left(a+b\right),b{I}_p+m{x}_{e}c\left(a+b\right),\left(bm+bm{x}_{e}c\right)S^4,\left(b{I}_p+bm{x}_{e}c\right)S^2\right)>0$.

From (25) and (26), it is demonstrated that $V\left(t\right)$ possesses both an upper bound and a lower bound, i.e.,

$$0\le {\beta }_2\mathsf{\Omega }\left(t\right)\le V\left(t\right)\le {\beta }_1\mathsf{\Omega }\left(t\right)$$

(27)

We select suitable parameters that satisfy the following two inequalities: ${\beta }_1={\eta }_1+{\eta }_3>0,{\beta }_2={\eta }_2-{\eta }_3>0$.

Then taking the derivative of (21), one has

$$\dot{V}\left(t\right)={\dot{V}}_1\left(t\right)+{\dot{V}}_2\left(t\right)$$

(28)

First of all, taking the derivative of $V_1\left(t\right)$, one has

$$\begin{array}{l}{\dot{V}}_1\left(t\right)=am{\displaystyle {\int }_0^S\dot{D}\ddot{D}dl}+a{I}_p{\displaystyle {\int }_0^S\dot{\theta }\ddot{\theta }dl}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+aGJ{\displaystyle {\int }_0^S{\dot{\theta }}^{\prime }{\theta }^{\prime }dl}+aE{I}_b{\displaystyle {\int }_0^S{\dot{D}}^{″}{D}^{″}dl}\end{array}$$

(29)

To facilitate the calculation, we define the following symbols $A_1=am{\displaystyle {\int }_0^S\dot{D}\ddot{D}dl}$, $A_2=a{I}_p{\displaystyle {\int }_0^S\dot{\theta }\ddot{\theta }dl}$, and the analysis of $A_1$ and $A_2$ is as follows.

After substituting (1) into $A_1$, one obtains

$$\begin{array}{ll}{A}_1& =am{\displaystyle {\int }_0^S\dot{D}\ddot{D}dl}\\ & =a{\displaystyle {\int }_0^S\dot{D}\left(F_b+m{x}_{e}c\ddot{\theta }-\eta E{I}_b{\dot{D}}^{⁗}-E{I}_b{D}^{⁗}\right)dl}\\ & =a{\displaystyle {\int }_0^S\dot{D}{F}_{b}dl}-aE{I}_b{\displaystyle {\int }_0^S\dot{D}{D}^{⁗}dl}\\ & \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}-a\eta E{I}_b{\displaystyle {\int }_0^S\dot{D}{\dot{D}}^{⁗}dl}+am{x}_{e}c{\displaystyle {\int }_0^S\dot{D}\ddot{\theta }dl}\end{array}$$

(30)

Integrating (30) by parts and applying Young’s inequality, Lemma 2 from [35], and boundary condition (4), (30) can be represented in the following manner:

$$\begin{array}{ll}{A}_1\le & -\left({\displaystyle \frac{a\eta E{I}_b}{2S^4}}-a{\sigma }_1\right){\displaystyle {\int }_0^S{\dot{D}}^{2}dl}-{\displaystyle \frac{a\eta E{I}_b}{2}}{\displaystyle {\int }_0^S{{\dot{D}}^{″}}^{2}dl}\\ & +{\displaystyle \frac{a}{{\sigma }_1}}S{F}_{b\max }^2+a{J}_2{H}_1\left(t\right)-aE{I}_b{\displaystyle {\int }_0^S{\dot{D}}^{″}{D}^{″}dl}+am{x}_{e}c{\displaystyle {\int }_0^S\dot{D}\ddot{\theta }dl}\end{array}$$

(31)

After substituting (2) into $A_2$, one obtains

$$\begin{array}{ll}{A}_2& =a{I}_p{\displaystyle {\int }_0^S\dot{\theta }\ddot{\theta }dl}\\ & =a{\displaystyle {\int }_0^S\dot{\theta }\left(-x_{a}c{F}_b+m{x}_{e}c\ddot{D}+\eta GJ{\dot{\theta }}^{″}+GJ{\theta }^{″}\right)dl}\\ & =-a{x}_{a}c{\displaystyle {\int }_0^S\dot{\theta }{F}_{b}dl}+am{x}_{e}c{\displaystyle {\int }_0^S\dot{\theta }\ddot{D}dl}\\ & \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+a\eta GJ{\displaystyle {\int }_0^S\dot{\theta }{\dot{\theta }}^{″}dl}+aGJ{\displaystyle {\int }_0^S\dot{\theta }{\theta }^{″}dl}\end{array}$$

(32)

The same algorithm as in $A_1$, (32), can be represented in the following manner:

$$\begin{array}{ll}{A}_2\le & \hspace{0.33em}-\left({\displaystyle \frac{a\eta GJ}{2S^2}}-a{x}_{a}c{\sigma }_2\right){\displaystyle {\int }_0^S{\dot{\theta }}^{2}dl}+{\displaystyle \frac{a{x}_{a}c}{{\sigma }_2}}S{F}_{b\max }^2\\ & \hspace{0.33em}+a{E}_2{H}_2\left(t\right)-aGJ{\displaystyle {\int }_0^S{\dot{\theta }}^{\prime }{\theta }^{\prime }dl}\\ & \hspace{0.33em}-{\displaystyle \frac{a\eta GJ}{2}}{\displaystyle {\int }_0^S{{\dot{\theta }}^{\prime }}^{2}dl}+am{x}_{e}c{\displaystyle {\int }_0^S\dot{\theta }\ddot{D}dl}\end{array}$$

(33)

After integrating systems (31) and (33) into (29), the following can be obtained:

$$\begin{array}{ll}{\dot{V}}_1\left(t\right)\le & -\left({\displaystyle \frac{a\eta E{I}_b}{2S^4}}-a{\sigma }_1\right){\displaystyle {\int }_0^S{\dot{D}}^{2}dl}-{\displaystyle \frac{a\eta E{I}_b}{2}}{\displaystyle {\int }_0^S{{\dot{D}}^{″}}^{2}dl}\\ & \hspace{0.33em}-{\displaystyle \frac{a\eta GJ}{2}}{\displaystyle {\int }_0^S{{\dot{\theta }}^{\prime }}^{2}dl}-\left({\displaystyle \frac{a\eta GJ}{2S^2}}-a{x}_{a}c{\sigma }_2\right){\displaystyle {\int }_0^S{\dot{\theta }}^{2}dl}\\ & \hspace{0.33em}+am{x}_{e}c{\displaystyle {\int }_0^S\left[\dot{D}\ddot{\theta }+\dot{\theta }\ddot{D}\right]dl}+a{J}_2{H}_1\left(t\right)\\ & \hspace{0.33em}+\left({\displaystyle \frac{a}{{\sigma }_1}}+{\displaystyle \frac{a{x}_{a}c}{{\sigma }_2}}\right)S{F}_{b\max }^2+a{E}_2{H}_2\left(t\right)\end{array}$$

(34)

where ${\sigma }_1,{\sigma }_2>0$ are constants.

The derivative of $V_2\left(t\right)$ with respect to the time $t$ can be obtained as

$$\begin{array}{ll}{\dot{V}}_2\left(t\right)=& bm{\displaystyle {\int }_0^{S}D\ddot{D}dl}+bm{\displaystyle {\int }_0^S{\dot{D}}^{2}dl}+b{I}_p{\displaystyle {\int }_0^S\theta \ddot{\theta }dl}\\ & -bm{x}_{e}c{\displaystyle {\int }_0^S\left[D\ddot{\theta }+2\dot{D}\dot{\theta }+\ddot{D}\theta \right]dl}\\ & -am{x}_{e}c{\displaystyle {\int }_0^S\left[\dot{D}\ddot{\theta }+\ddot{D}\dot{\theta }\right]dl}+b{I}_p{\displaystyle {\int }_0^S{\dot{\theta }}^{2}dl}\end{array}$$

(35)

For ease of description, the symbols $B_1=bm{\displaystyle {\int }_0^{S}D\ddot{D}dl}$, $B_2=b{I}_p{\displaystyle {\int }_0^S\theta \ddot{\theta }dl}$ are defined, and the analysis of $B_1$ and $B_2$ is as follows:

After substituting (1) into $B_1$, one obtains

$$\begin{array}{ll}{B}_1& =bm{\displaystyle {\int }_0^{S}D\ddot{D}dl}\\ & =b{\displaystyle {\int }_0^{S}D\left(F_b+m{x}_{e}c\ddot{\theta }-E{I}_b{D}^{⁗}-\eta E{I}_b{\dot{D}}^{⁗}\right)dl}\\ & =b{\displaystyle {\int }_0^{S}D{F}_{b}dl}-bE{I}_b{\displaystyle {\int }_0^{S}D{D}^{⁗}dl}\\ & \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}-b\eta E{I}_b{\displaystyle {\int }_0^{S}D{\dot{D}}^{⁗}dl}+bm{x}_{e}c{\displaystyle {\int }_0^{S}D\ddot{\theta }dl}\end{array}$$

(36)

By integrating (36) by parts and utilizing Young’s inequality, Lemma 2 from [35], and boundary condition (4), it is obtained that

$$\begin{array}{ll}{B}_1\le & {\displaystyle \frac{b}{{\sigma }_3}}S{F}_{b\max }^2-\left(bE{I}_b-b{\sigma }_3{S}^4-b\eta E{I}_b{\sigma }_4\right){\displaystyle {\int }_0^S{D^{″}}^{2}dl}\\ & +b{J}_1{H}_1\left(t\right)+{\displaystyle \frac{b\eta E{I}_b}{{\sigma }_4}}{\displaystyle {\int }_0^S{{\dot{D}}^{″}}^{2}dl}+bm{x}_{e}c{\displaystyle {\int }_0^{S}D\ddot{\theta }dl}\end{array}$$

(37)

After substituting (2) into $B_2$, one obtains

$$\begin{array}{ll}{B}_2& =b{I}_p{\displaystyle {\int }_0^S\theta \ddot{\theta }dl}\\ & =b{\displaystyle {\int }_0^S\theta \left(-x_{a}c{F}_b+GJ{\theta }^{″}+\eta GJ{\dot{\theta }}^{″}+m{x}_{e}c\ddot{D}\right)dl}\\ & =-b{x}_{a}c{\displaystyle {\int }_0^S\theta F_{b}dl}+bGJ{\displaystyle {\int }_0^S\theta {\theta }^{″}dl}\\ & \hspace{0.33em}\hspace{0.33em}+b\eta GJ{\displaystyle {\int }_0^S\theta {\dot{\theta }}^{″}dl}+bm{x}_{e}c{\displaystyle {\int }_0^S\theta \ddot{D}dl}\end{array}$$

(38)

The same algorithm as in $B_1$, (38), can be expressed in the following manner:

$$\begin{array}{ll}{B}_2\le & {\displaystyle \frac{b{x}_{a}c}{{\sigma }_5}}S{F}_{b\max }^2-\left(bGJ-b{x}_{a}c{\sigma }_5{S}^2-b\eta GJ{\sigma }_6\right){\displaystyle {\int }_0^S{{\theta }^{\prime }}^{2}dl}\\ & +b{E}_1{H}_2\left(t\right)+{\displaystyle \frac{b\eta GJ}{{\sigma }_6}}{\displaystyle {\int }_0^S{{\dot{\theta }}^{\prime }}^{2}dl}+bm{x}_{e}c{\displaystyle {\int }_0^S\theta \ddot{D}dl}\end{array}$$

(39)

After putting systems (37) and (39) into (35), the following can be obtained:

$$\begin{array}{ll}{\dot{V}}_2\left(t\right)\le & -\left(bE{I}_b-b{\sigma }_3{S}^4-b\eta E{I}_b{\sigma }_4\right){\displaystyle {\int }_0^S{D^{″}}^{2}dl}+b{J}_1{H}_1\left(t\right)\\ & -\left(bGJ-b{x}_{a}c{\sigma }_5{S}^2-b\eta GJ{\sigma }_6\right){\displaystyle {\int }_0^S{{\theta }^{\prime }}^{2}dl}+b{E}_1{H}_2\left(t\right)\\ & +{\displaystyle \frac{b\eta GJ}{{\sigma }_6}}{\displaystyle {\int }_0^S{{\dot{\theta }}^{\prime }}^{2}dl}+\left(bm+{\displaystyle \frac{2bm{x}_{e}c}{{\sigma }_7}}\right){\displaystyle {\int }_0^S{\dot{D}}^{2}dl}\\ & +\left(b{I}_p+2bm{x}_{e}c{\sigma }_7\right){\displaystyle {\int }_0^S{\dot{\theta }}^{2}dl}+{\displaystyle \frac{b\eta E{I}_b}{{\sigma }_4}}{\displaystyle {\int }_0^S{{\dot{D}}^{″}}^{2}dl}\\ & -am{x}_{e}c{\displaystyle {\int }_0^S\left[\dot{D}\ddot{\theta }+\dot{\theta }\ddot{D}\right]dl}+\left({\displaystyle \frac{b}{{\sigma }_3}}+{\displaystyle \frac{b{x}_{a}c}{{\sigma }_5}}\right)S{F}_{b\max }^2\end{array}$$

(40)

where ${\sigma }_i\left(i=3,\cdots ,7\right)$ are positive constants.

Substituting (34) and (40) into (28) yields

$$\begin{array}{ll}\dot{V}\left(t\right)& \le -\left({\displaystyle \frac{a\eta E{I}_b}{2S^4}}-a{\sigma }_1-bm-{\displaystyle \frac{2bm{x}_{e}c}{{\sigma }_7}}\right){\displaystyle {\int }_0^S{\dot{D}}^{2}dl}-\left({\displaystyle \frac{a\eta E{I}_b}{2}}-{\displaystyle \frac{b\eta E{I}_b}{{\sigma }_4}}\right){\displaystyle {\int }_0^S{{\dot{D}}^{″}}^{2}dl}\\ & \hspace{0.33em}\hspace{0.33em}-\left({\displaystyle \frac{a\eta GJ}{2}}-{\displaystyle \frac{b\eta GJ}{{\sigma }_6}}\right){\displaystyle {\int }_0^S{{\dot{\theta }}^{\prime }}^{2}dl}+\left({\displaystyle \frac{a}{{\sigma }_1}}+{\displaystyle \frac{a{x}_{a}c}{{\sigma }_2}}+{\displaystyle \frac{b}{{\sigma }_3}}+{\displaystyle \frac{b{x}_{a}c}{{\sigma }_5}}\right)S{F}_{b\max }^2\\ & \hspace{0.33em}\hspace{0.33em}-\left({\displaystyle \frac{a\eta GJ}{2S^2}}-a{x}_{a}c{\sigma }_2-b{I}_p-2bm{x}_{e}c{\sigma }_7\right){\displaystyle {\int }_0^S{\dot{\theta }}^{2}dl}\\ & \hspace{0.33em}\hspace{0.33em}-\left(bE{I}_b-b{\sigma }_3{S}^4-b\eta E{I}_b{\sigma }_4\right){\displaystyle {\int }_0^S{D^{″}}^{2}dl}+\left(a{J}_2+b{J}_1\right)H_1\left(t\right)\\ & \hspace{0.33em}\hspace{0.33em}-\left(bGJ-b{x}_{a}c{\sigma }_5{S}^2-b\eta GJ{\sigma }_6\right){\displaystyle {\int }_0^S{{\theta }^{\prime }}^{2}dl}+\left(a{E}_2+b{E}_1\right)H_2\left(t\right)\\ & \le -{\mathsf{\Theta }}_1{\displaystyle {\int }_0^S{\dot{D}}^{2}dl}-{\mathsf{\Theta }}_2{\displaystyle {\int }_0^S{D^{″}}^{2}dl}-{\mathsf{\Theta }}_3{\displaystyle {\int }_0^S{\dot{\theta }}^{2}dl}-{\mathsf{\Theta }}_4{\displaystyle {\int }_0^S{{\theta }^{\prime }}^{2}dl}\\ & \hspace{0.33em}\hspace{0.33em}+\left({\displaystyle \frac{a}{{\sigma }_1}}+{\displaystyle \frac{a{x}_{a}c}{{\sigma }_2}}+{\displaystyle \frac{b}{{\sigma }_3}}+{\displaystyle \frac{b{x}_{a}c}{{\sigma }_5}}\right)S{F}_{b\max }^2\\ & \hspace{0.33em}\hspace{0.33em}+\left(a{J}_2+b{J}_1\right)H_1\left(t\right)\hspace{0.33em}+\left(a{E}_2+b{E}_1\right)H_2\left(t\right)\end{array}$$

(41)

We select suitable parameters to meet the following conditions:

$$\left\{\begin{array}{l}{\mathsf{\Theta }}_1={\displaystyle \frac{a\eta E{I}_b}{2S^4}}-a{\sigma }_1-bm-{\displaystyle \frac{2bm{x}_{e}c}{{\sigma }_7}}>0\\ {\mathsf{\Theta }}_2=bE{I}_b-b{\sigma }_3{S}^4-b\eta E{I}_b{\sigma }_4>0\\ {\mathsf{\Theta }}_3={\displaystyle \frac{a\eta GJ}{2S^2}}-a{x}_{a}c{\sigma }_2-b{I}_p-2bm{x}_{e}c{\sigma }_7>0\\ {\mathsf{\Theta }}_4=bGJ-b{x}_{a}c{\sigma }_5{S}^2-b\eta GJ{\sigma }_6>0\\ {\displaystyle \frac{a\eta E{I}_b}{2}}-{\displaystyle \frac{b\eta E{I}_b}{{\sigma }_4}}>0\\ {\displaystyle \frac{a\eta GJ}{2}}-{\displaystyle \frac{b\eta GJ}{{\sigma }_6}}>0\end{array}\right.$$

(42)

To facilitate the analysis, $C_1$ and $C_2$ represent $\left(a{J}_2+b{J}_1\right)H_1\left(t\right)$ and $\left(a{E}_2+b{E}_1\right)H_2\left(t\right)$, respectively. The analysis of $C_1$ and $C_2$ is as follows.

Substituting (16) and (19) into $C_1$, $C_1$ is rewritten as follows:

$$\begin{array}{ll}{C}_1& =\left(a{J}_2+b{J}_1\right)H_1\left(t\right)\\ & =\left(a{J}_2+b{J}_1\right)\left({\upsilon }_1\left(t\right)-{\mu }_1\left(t\right){\zeta }_1\right)\hspace{0.33em}\\ & =\left(a{J}_2+b{J}_1\right)\left(-k_1\left(a{\widehat{J}}_2+b{J}_1\right)-{\mu }_1\left(t\right){\zeta }_1\right)\hspace{0.33em}\\ & =\left(a{J}_2+b{J}_1\right)\left(-k_1\left(a\left(J_2+{\tilde{J}}_2\right)+b{J}_1\right)-{\mu }_1\left(t\right){\zeta }_1\right)\hspace{0.33em}\\ & =-\left(a{J}_2+b{J}_1\right){\mu }_1\left(t\right){\zeta }_1-k_1{\left(a{J}_2+b{J}_1\right)}^2-k_{1}a\left(a{J}_2+b{J}_1\right){\tilde{J}}_2\end{array}$$

(43)

Using Young’s inequality, (43) can be written as follows:

$$C_1\le -\left(k_1-k_{1}a{\epsilon }_1-{\displaystyle \frac{1}{2}}\right){\left(a{J}_2+b{J}_1\right)}^2+{\displaystyle \frac{{\overline{\zeta }}_1^2}{2}}+{\displaystyle \frac{k_{1}a}{{\epsilon }_1}}{\tilde{J}}_2^2$$

(44)

Similarly, Substituting (16) and (20) into $C_2$, $C_2$ is rewritten as follows:

$$\begin{array}{ll}{C}_2& =\left(a{E}_2+b{E}_1\right)H_2\left(t\right)\\ & =\left(a{E}_2+b{E}_1\right)\left({\upsilon }_2\left(t\right)-{\mu }_2\left(t\right){\zeta }_2\right)\hspace{0.33em}\\ & =\left(a{E}_2+b{E}_1\right)\left(-k_2\left(a{\widehat{E}}_2+b{E}_1\right)-{\mu }_2\left(t\right){\zeta }_2\right)\hspace{0.33em}\\ & =\left(a{E}_2+b{E}_1\right)\left(-k_2\left(a\left(E_2+{\tilde{E}}_2\right)+b{E}_1\right)-{\mu }_2\left(t\right){\zeta }_2\right)\hspace{0.33em}\\ & =-\left(a{E}_2+b{E}_1\right){\mu }_2\left(t\right){\zeta }_2-k_2{\left(a{E}_2+b{E}_1\right)}^2-k_{2}a\left(a{E}_2+b{E}_1\right){\tilde{E}}_2\end{array}$$

(45)

Using Young’s inequality, (45) can be written as follows:

$$C_2\le {\displaystyle \frac{{\overline{\zeta }}_2^2}{2}}+{\displaystyle \frac{k_{2}a}{{\epsilon }_2}}{\tilde{E}}_2^2-\left(k_2-{\displaystyle \frac{1}{2}}-k_{2}a{\epsilon }_2\right){\left(a{E}_2+b{E}_1\right)}^2$$

(46)

After substituting (44) and (46) into (41) and rearranging, (41) can be written as

$$\begin{array}{ll}\dot{V}\left(t\right)& \le -{\mathsf{\Theta }}_1{\displaystyle {\int }_0^S{\dot{D}}^{2}dl}-{\mathsf{\Theta }}_2{\displaystyle {\int }_0^S{D^{″}}^{2}dl}-{\mathsf{\Theta }}_3{\displaystyle {\int }_0^S{\dot{\theta }}^{2}dl}-{\mathsf{\Theta }}_4{\displaystyle {\int }_0^S{{\theta }^{\prime }}^{2}dl}\\ & \hspace{0.33em}\hspace{0.33em}+\left({\displaystyle \frac{a}{{\sigma }_1}}+{\displaystyle \frac{a{x}_{a}c}{{\sigma }_2}}+{\displaystyle \frac{b}{{\sigma }_3}}+{\displaystyle \frac{b{x}_{a}c}{{\sigma }_5}}\right)S{F}_{b\max }^2+{\displaystyle \frac{{\overline{\zeta }}_1^2}{2}}+{\displaystyle \frac{k_{1}a}{{\epsilon }_1}}{\omega }^2{n}^2\\ & \hspace{0.33em}\hspace{0.33em}-\left(k_1-{\displaystyle \frac{1}{2}}-k_{1}a{\epsilon }_1\right){\left(a{J}_2+b{J}_1\right)}^2+{\displaystyle \frac{{\overline{\zeta }}_2^2}{2}}+{\displaystyle \frac{k_{2}a}{{\epsilon }_2}}{\omega }_0^2{n}_0^2\\ & \hspace{0.33em}\hspace{0.33em}-\left(k_2-{\displaystyle \frac{1}{2}}-k_{2}a{\epsilon }_2\right){\left(a{E}_2+b{E}_1\right)}^2\\ & \le -{\beta }_3\mathsf{\Omega }\left(t\right)+r\end{array}$$

(47)

where $r=\left({\displaystyle \frac{a}{{\sigma }_1}}+{\displaystyle \frac{a{x}_{e}c}{{\sigma }_2}}+{\displaystyle \frac{b}{{\sigma }_3}}+{\displaystyle \frac{b{x}_{a}c}{{\sigma }_5}}\right)S{F}_{b\max }^2+{\displaystyle \frac{k_{1}a}{{\epsilon }_1}}{\omega }^2{n}^2+{\displaystyle \frac{{\overline{\zeta }}_2^2}{2}}+{\displaystyle \frac{k_{2}a}{{\epsilon }_2}}{\omega }_0^2{n}_0^2+{\displaystyle \frac{{\overline{\zeta }}_1^2}{2}}$, ${\beta }_3=\min \left\{{\mathsf{\Theta }}_1,{\mathsf{\Theta }}_2,\right.$ $\left.{\mathsf{\Theta }}_3,{\mathsf{\Theta }}_4\right\}$, and ${\epsilon }_1$ and ${\epsilon }_2$ are positive constants.

The remaining pertinent parameters adhere to the following inequalities:

$$\left\{\begin{array}{l}{k}_1-{\displaystyle \frac{1}{2}}-k_{1}a{\epsilon }_1>0\\ k_2-{\displaystyle \frac{1}{2}}-k_{2}a{\epsilon }_2>0\end{array}\right.$$

(48)

Therefore, an upper bound for $\dot{V}\left(t\right)$ exists, i.e.,

$$\dot{V}\left(t\right)\le -\beta V\left(t\right)+r$$

(49)

where $\beta ={\displaystyle \frac{{\beta }_3}{{\beta }_1}}>0$.

Theorem 1.

For the FWRA (1)–(5) and given that the initial conditions are bounded, thanks to Lemma 2 in [35], the ET mechanisms (12)–(15), and the output feedback ET controllers (19)–(20), we achieve the following:

(1)

All of the signals are bounded;

(2)

The vibrations of bending deformation and torsional deformation in the FWRA are effectively suppressed;

(3)

The Zeno phenomenon is avoided.

Proof of Theorem 1.

Multiplying both sides of (49) by $e^{\beta t}$ and then integrating on $\left[0,t\right]$, one has

$$V\left(t\right)\le \left(V\left(0\right)-{\displaystyle \frac{r}{\beta }}\right)e^{-\beta t}+{\displaystyle \frac{r}{\beta }}\le V\left(0\right)e^{-\beta t}+{\displaystyle \frac{r}{\beta }}$$

(50)

According to Lemma 2 in [35], it follows that

$${\displaystyle \frac{1}{S^3}}{D}^2\le {\displaystyle \frac{1}{S^2}}{\displaystyle {\int }_0^S{\left[D^{\prime }\right]}^{2}dl}\le {\displaystyle {\int }_0^S{\left[D^{″}\right]}^{2}dl}\le \mathsf{\Omega }\left(t\right)\le {\displaystyle \frac{V\left(t\right)}{{\beta }_2}}$$

(51)

$${\displaystyle \frac{1}{S}}{\theta }^2\le {\displaystyle {\int }_0^S{\left[{\theta }^{\prime }\right]}^{2}dl}\le \mathsf{\Omega }\left(t\right)\le {\displaystyle \frac{V\left(t\right)}{{\beta }_2}}$$

(52)

Rewriting (51) and (52), one has

$$\left|D\right|\le \sqrt{{\displaystyle \frac{S^3}{{\beta }_2}}\left(V\left(0\right)e^{-\beta t}+{\displaystyle \frac{r}{\beta }}\right)}$$

(53)

$$\left|\theta \right|\le \sqrt{{\displaystyle \frac{S}{{\beta }_2}}\left(V\left(0\right)e^{-\beta t}+{\displaystyle \frac{r}{\beta }}\right)}$$

(54)

Therefore, as $t$ tends to infinity, one has

$$\underset{t\to \infty }{\lim }\left|D\right|\le \sqrt{{\displaystyle \frac{S^{3}r}{{\beta }_2\beta }}}$$

(55)

$$\underset{t\to \infty }{\lim }\left|\theta \right|\le \sqrt{{\displaystyle \frac{Sr}{{\beta }_2\beta }}}$$

(56)

From (53) and (54), it can be seen that due to the bounded initial values, as $t$ approaches infinity, $D\left(S\right)$ and $\theta \left(S\right)$ remain within a small neighborhood. This implies that with $\exists T$, when $t>T$, the system states $D$ and $\theta$ will be close to a small neighborhood around zero.

Since $V_1\left(t\right)$ and $V_2\left(t\right)$ are bounded $\forall t\in \left[0,\infty \right)$, $V_1\left(t\right)+V_2\left(t\right)$ is also bounded. From (22) and (23), it can be seen that $\dot{D}$, $D^{″}$, $\dot{\theta }$, and ${\theta }^{\prime }$ are bounded: $\forall \left(l,t\right)\in \left[0,S\right]\times$ $\left[0,\infty \right)$. Furthermore, according to the boundedness of the kinetic and potential energy in [2], it can be concluded that the states ${\dot{D}}^{″}$ and ${\dot{\theta }}^{\prime }$ are also bounded. Therefore, from (1) and (2), along with the above statements, it can be inferred that $\ddot{D}$ and $\ddot{\theta }$ are bounded $\forall \left(l,t\right)\in \left[0,S\right]\times \left[0,\infty \right)$. Thus, all of the signals are bounded: $\forall \left(l,t\right)\in \left[0,S\right]\times \left[0,\infty \right)$.

We can see from (55) and (56) that $D$ and $\theta$ are bounded; therefore, the vibrations of bending deformation and torsional deformation in the FWRA are effectively suppressed.

Based on the above analysis, it is clear that by appropriately selecting the control parameters, the state variables of the FWRA can converge to the desired values, thereby ensuring the stability of the closed-loop system.

Next, this paper demonstrates in detail that the proposed control scheme can effectively avoid the Zeno phenomenon, which is proven as follows:

Considering $e_1\left(t\right)={\upsilon }_1\left(t\right)-H_1\left(t\right)$, $\forall t\in \left[t_k,t_{k+1}\right)$, $e_2\left(t\right)={\upsilon }_2\left(t\right)-H_2\left(t\right)$, and $\forall t\in \left[t_s,t_{s+1}\right)$, from (19) and (20), it can be further deduced that ${\upsilon }_1\left(t\right)$ and ${\upsilon }_2\left(t\right)$ exhibit continuity as a function of $\left[D\left(S\right),\dot{D}\left(S\right)\right]$ and $\left[\theta \left(S\right),\dot{\theta }\left(S\right)\right]$, respectively. Therefore, positive constants $a$ and $b$ must exist which satisfy $\left|{\dot{e}}_1\left(t\right)\right|=\left|{\dot{\upsilon }}_1\left(t\right)\right|\le a$, $\forall t\in \left[t_k,t_{k+1}\right)$ and $\left|{\dot{e}}_2\left(t\right)\right|=\left|{\dot{\upsilon }}_2\left(t\right)\right|\le b$, and $\forall t\in \left[t_s,t_{s+1}\right)$. Again, here, positive constants ${\mathsf{\Pi }}_1$ and ${\mathsf{\Pi }}_2$ mist exist that permit $t_{k+1}-t_k\ge {\mathsf{\Pi }}_1$ and $t_{s+1}-t_s\ge {\mathsf{\Pi }}_2$, and one can find ${\mathsf{\Pi }}_1\ge {\displaystyle \frac{{\zeta }_1}{a}}$ and ${\mathsf{\Pi }}_2\ge {\displaystyle \frac{{\zeta }_2}{b}}$.

According to the above analysis, the Zeno phenomenon is avoided effectively.

Hence, the proof for Theorem 1 is completed. □

4. The Simulation Example

In this section, the finite difference method is used to simulate the FWRA, and the effectiveness of the designed output feedback ET controllers (19) and (20) is verified using simulation examples.

The parameters for the FWRA (1)–(5) are provided as follows: $S=2 \mathrm{m}$, $m=55 \mathrm{kg}/\mathrm{m}$, $E{I}_b=11.53 \mathrm{N}{\mathrm{m}}^2$, $GJ=$ $15.5 \mathrm{N}{\mathrm{m}}^2$, $x_{e}c=0.13 \mathrm{m}$, $x_{a}c=0.55 \mathrm{m}$, $I_p=5 \mathrm{kg}/\mathrm{m}$, and $\eta =0.09$. In the simulation, the parameters related to the output feedback ET controllers (19) and (20) are designed: $a=3000$, $b=1.1$, $k_1=0.003$, $k_2=5\times {10}^{-4}$, ${\zeta }_1=0.7$, and ${\zeta }_2=0.035$. The observer’s parameters are $\omega =1$, ${\omega }_0=1$, ${\overline{\iota }}_1=1$, and ${\overline{\iota }}_{01}=0.95$. The distributed disturbance $F_b=0.01l\left[1+\sin \left(\pi \tau \right)+3\cos \left(3\pi \tau \right)\right]$.

Considering the given initial conditions $D\left(l,0\right)=0.015\left(l/S\right)$, $\theta \left(l,0\right)=0.01\left(\pi l/3S\right)$. Here, the simulation results are shown in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, which provide a visual representation of our findings.

As depicted in Figure 3 and Figure 4, the bending deformation $D$ and the torsional deformation $\theta$ of the FWRA under output feedback ET control ultimately converge to zero. Therefore, the output feedback ET control scheme proposed in this paper is successful in suppressing vibrations. That is, (1) and (2) in the control objectives are verified in the simulation results.

Figure 5, Figure 6, Figure 7 and Figure 8 provide valuable insights into the control inputs $H_1\left(t\right)$ and $H_2\left(t\right)$ of the FWRA, as well as the ET frequency. Figure 5 and Figure 6 show a comparison of the trajectories of the output feedback ET controllers and the trajectories of the controllers before the event is triggered. It can be seen from the figures that the trajectories of the ET controllers are step-like, while the trajectories of the controllers before the ET mechanisms are smooth curves. Therefore, the frequency of transmission of the control values is significantly reduced when utilizing the ET controllers proposed in this paper. The presence of ET trigger nodes is evident in Figure 7 and Figure 8. Therefore, the ET control scheme presented in this paper effectively lowers the redundancy, reduces the reliance on frequent communication channels, and greatly alleviates the communication burden. That is, (3) in the control objectives is verified in the simulation results.

Figure 9 and Figure 10 display the trajectories followed by the actual and estimated values of the system states $J_2\left(t\right)$ and $E_2\left(t\right)$, respectively. It can be seen from the figures that the estimated values for $J_2\left(t\right)$ and $E_2\left(t\right)$ track their actual values well.

In addition, to more intuitively demonstrate the improvement in the efficiency with ET control in terms of the resource utilization compared with traditional time-triggered control, a comparison of the number of ET and time-triggered triggers is made, as shown in

Table 1. Number of ET and time-triggered triggers.

Controller	ET Controllers		Time-Triggered Controllers
	${\mathit{H}}_1\left(\mathit{t}\right)$	${\mathit{H}}_2\left(\mathit{t}\right)$	${\mathit{\upsilon }}_1\left(\mathit{t}\right)$	${\mathit{\upsilon }}_2\left(\mathit{t}\right)$
Number of triggers	140	194	1998	1998

.

In summary, for an FWRA, the output feedback ET controllers designed can suppress bending deformation and torsional deformation vibrations, the communication burden is effectively reduced, and the working ability of the system is improved in cases of unpredictable system states.

5. Conclusions

In this paper, the vibration suppression problem for an FWRA is studied. High-gain observers are used to estimate certain unmeasured system states. Secondly, the ET mechanism is given, and the output feedback ET controllers are designed based on this ET mechanism to effectively suppress bending and twisting deformation vibrations in FWRAs. In addition, energy consumption for communication is saved, and the Zeno phenomenon is avoided. Finally, the simulation results verify the effectiveness of the proposed control method. However, the ET mechanism proposed in this paper is based on a fixed threshold, which is more conservative. In future work, a dynamic event-triggered technique will be considered to reduce the control consumption further.