Structural Vibration Suppression Using a Reduced-Order Extended State Observer-Based Nonsingular Terminal Sliding Mode Controller with an Inertial Actuator

Abstract

In this paper, we mainly aimed to design a reduced-order extended state observer-based active vibration controller for a structural vibration control system with total disturbances, i.e., model uncertainties, higher harmonics, and external excitations. A reduced-order extended state observer (RESO)-based nonsingular terminal sliding mode vibration control (RESO–NTSMVC) method is proposed for the vibration suppression of an all-clamped plate structure with an inertial actuator. First, a second-order state space model of the thin plate, with an inertial actuator, was established by solving the dynamic partial differential equation and analyzing the physical model. Second, the total disturbances, i.e., model uncertainties, higher harmonics, and external excitations, were estimated and compensated for by using a RESO via a feedforward part. Third, a NTSMVC based on an estimated value was designed to obtain a fast-tracking rate and effective vibration suppression performance. In addition, the stability of the closed-loop system was proven by using a Lyapunov stability criterion. Finally, a semi-physical experimental instrument was built based on the MATLAB/Simulink real-time environment and the NI-PCIE6343 acquisition card to verify strong anti-disturbance performance and effective vibration control performance of the designed method. The experimental comparison results showed that the vibration amplitudes of the proposed method could be reduced by 11.7 dB, when the traditional extended state observer-based nonsingular terminal sliding mode vibration control (ESO–NTSMVC) method achieved a control effect of only 6.5 dB. The comparative experimental results showed that the proposed method possessed better vibration suppression performance and anti-disturbance performance.

1. Introduction

The all-clamped plate has been broadly recognized in precision instruments, aerospace engineering, shipbuilding, automotive engineering, and other practical industries, due to its light weight, high stiffness, and simple structure [1,2,3]. However, the existence of total disturbances, such as model uncertainty, high order harmonics, system coupling, and external unknown environmental excitations, can inevitably cause structural vibration with large amplitudes in engineering practices. Long-term and large-amplitude vibration of the all-clamped plate may eventually lead to structural fatigue damage, component life reduction, and even hidden danger resulting in serious accidents. Therefore, structural vibration control is a critical problem to be studied in many engineering fields [4,5,6]. In addition, electrodynamic inertial actuators are widely embedded in thin plates as actuators for active vibration control to satisfy strict performance requirements, due to their characteristics of high bandwidth and satisfactory control ability of complex periodic vibration [7,8].

Several control methods have been proposed to suppress structural vibration within inertial actuators [9,10,11,12,13]. Based on direct velocity feedback, structural vibration was effectively minimized in [11]. A robust linear vibration controller based on a direct adaptive control algorithm with an inertial actuator, was introduced to improve the performance for an active vibration control system in the presence of unknown time varying disturbances [12]. In [13], the optimal control method was used to suppress structural vibration to satisfy the performance requirements of the environment. The above-mentioned advanced vibration suppression methods can enhance system vibration suppression performance; however, a mathematical model should be accurately established because of the complex dynamic characteristics [14]. In addition, the vibration suppression performance improvements of the above-mentioned model-based vibration control methods may be limited, since it is difficult to attenuate the total disturbances of these methods without accurate model information. The sliding model-based vibration control strategy is a popular method, due to its strong anti-disturbance ability regarding uncertainties [15,16,17,18]. For instance, a fuzzy sliding mode controller was employed for vibration suppression to improve system robustness when the system was excited [19,20]. In order to obtain ideal active vibration control performance, a sliding mode predictive control strategy for fast vibration suppression was proposed in [21]. Unfortunately, traditional sliding mode control has two disadvantages: the chattering phenomenon and a slow convergence rate [22]. Therefore, nonsingular terminal sliding mode control (NTSMC) was developed with the advantages of fast finite-time convergence and high tracking accuracy [23,24]. In [25], the nonsingular terminal sliding mode control method was applied for vibration reduction. The time required to reach the target value from any deviation position was a finite-time interval. In addition, the vibration system achieved fast convergence and accurate tracking performance under various disturbances. To solve the chattering problem, several disturbance estimation strategies were developed to eliminate disturbances and uncertainties. Considering the uncertainties in a vibration control system, a disturbance estimator (DE)-based fuzzy sliding mode controller (FSMC) has been developed to estimate uncertainties to improve control performances [26,27]. Regarding vibrational problems associated with robots, a generalized proportional integral observer (GPIO)-based sliding mode control was utilized to improve the performance with unknown states and disturbances in [28]. An equivalent input disturbance (EID) approach was developed to estimate, and compensate for, the effect of a seismic shock for vibration suppression in civil structures [29,30].

The extended state observer (ESO)-based controller is another practical disturbance estimation-based control method, originally proposed by Han [31]. It defines the total disturbances, i.e., internal and external disturbances, as new system state variables without an accurate mathematical model. The ESO-based control framework can estimate and compensate for total disturbances via a feedforward channel. Over the past decade, the ESO-based control method has been widely used in complex structural vibration systems, due to its few adjustable parameters and strong anti-disturbance ability [32]. In this paper, the force of an inertial actuator acting on a plate was regarded as disturbance, which reduced the number of sensors. A larger bandwidth of the ESO observer could obtain a better disturbance rejection performance, but the anti-noise ability would deteriorate with an increase in bandwidth [33]. Similar to [34], a RESO, which was easier to implement without an extra sensor, was designed to obtain excellent vibration suppression performance by solving the poor anti-noise ability caused by higher bandwidth.

Table 1. Methods comparison list.

Methods	References	Performance Characteristics.
Sliding mode	[15,16]	1. Strong anti-disturbance ability. 2. Chattering problem.
Fuzzy sliding mode	[19,20]	1. Improved system robustness. 2. Chattering problem.
Nonsingular terminal sliding mode	[23,24,25]	1. Fast finite-time convergence and high tracking accuracy. 2. Chattering problem.
Disturbance estimator (DE)	[26,27]	1. Estimate uncertainties.
Generalized proportional integral observer (GPIO)	[28]	1. Estimate unknown states and disturbances.
Equivalent input disturbance (EID)	[29,30]	1. Estimate and compensate the disturbances.
Extended state observer (ESO)	[31,32,33]	1. Estimate and compensate the disturbances without model information.
Reduced-order extended state observer (RESO)	[34]	1. Estimate and compensate the disturbances without model. 2. Easier than ESO. 3. Solving the poor anti-noise ability caused by higher bandwidth.

is a method comparison list.

A reduced-order extended state observer-based nonsingular terminal sliding mode vibration control (RESO–NTSMVC) method was proposed to enhance the performance of the plate vibration control system. The main characteristics of this strategy were as follows: (1) The RESO was proposed to estimate total disturbances and solve the problem of poor anti-noise ability caused by high bandwidth gain; (2) The secondary excitation of the inertial actuator to the plate was regarded as disturbance, which simplified the controller and reduced the number of displacement sensors; (3) The designed controller could obtain fast convergence rate and stronger robustness.

The remaining sections of this paper are summarized as follows: The dynamics of an all-clamped plate with an inertial actuator vibration system are stated in Section 2. A novel RESO–NTSMVC controller is designed in Section 3. Moreover, the stability of the designed controller is proven in Section 4. To assess the proposed RESO–NTSMVC against the ESO–NTSMVC, the experimental comparative results are shown in Section 5. Finally, the work is summarized in Section 6.

2. System Description

2.1. Dynamic Analysis of an All-Clamped Plate

According to dynamics theory, the damped forced vibration of a thin plate can be determined by solving the following basic equation [35,36]:

$${\nabla }^{4}w+\frac{\rho h}{D}\frac{{\partial }^{2}w}{\partial t^2}+\frac{\delta }{D}\frac{\partial w}{\partial t}=\frac{F\left(x,y,t\right)}{D},$$

(1)

where the operator ${\nabla }^4=\frac{{\partial }^4}{\partial x^4}+2\frac{{\partial }^2}{\partial x^2}\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^4}{\partial y^4}$. The symbols $w$, $\rho$, and $h$ are the deflection, density, and thickness of the thin plate, respectively. In addition, the symbols $\delta$, $F\left(x,y,t\right)$, and $D$ are the viscous damping coefficient, external force, and flexural stiffness, respectively.

According to the superposition principle of vibration theory, the dynamic system can be described by the linear combination of modes:

$$w={\displaystyle \sum _{m=1}^{\infty }{w}_m}={\displaystyle \sum _{m=1}^{\infty }{\eta }_m(t)W_m(x,y)},$$

(2)

where $W_m(x,y)$ and ${\eta }_m(t)$ are the mode function and the displacement associated with the m-th mode, respectively.

In addition, the force $F(x,y,t)$ can be described as:

$$F(x,y,t)={\displaystyle \sum _{m=1}^{\infty }{F}_m(t)W_m}(x,y),$$

(3)

where $W_m(x,y)$ can be described as follows:

$${\nabla }^4{W}_m(x,y)=\frac{{\omega }_m^2\rho h}{D}{W}_m(x,y).$$

(4)

By substituting Equations (2)–(4) into Equation (1), the equation of damped forced vibration can be rewritten as follows:

$${\displaystyle \sum _{m=1}^{\infty }{\eta }_m\frac{{\omega }_m{}^2\rho h}{D}{W}_m+\frac{\rho h}{D}}{\displaystyle \sum _{m=1}^{\infty }\frac{{\partial }^2{\eta }_m}{\partial t^2}}{W}_m+\frac{\delta }{D}{\displaystyle \sum _{m=1}^{\infty }\frac{\partial {\eta }_m}{\partial t}}{W}_m=\frac{1}{D}{\displaystyle \sum _{m=1}^{\infty }{F}_m{W}_m}.$$

(5)

Equation (5) can be further simplified as follows:

$$\frac{{\partial }^2{\eta }_m}{\partial t^2}+\frac{\delta }{\rho h}\frac{\partial {\eta }_m}{\partial t}+{\eta }_m{\omega }_m{}^2=\frac{1}{\rho h}{F}_m.$$

(6)

When the external vibration is excited at the modal frequency, consider that $F_m$ represents the external excitation and the control force of the inertial actuator. In addition, the modal damping ratio satisfies the expression ${\zeta }_m=\frac{\delta }{2\rho h{\omega }_m}$, and, Equation (6) can be rewritten as a second-order differential equation:

$${\ddot{\eta }}_m+2{\zeta }_m{\omega }_m{\dot{\eta }}_m+{\omega }_m^2{\eta }_m=b_m{f}_{cm}+f_{dm},$$

(7)

where ${\eta }_m$, ${\zeta }_m$, and ${\omega }_m$ are the displacement, damping ratio, and natural frequency associated with the m-th mode, respectively. The symbols of $f_{cm}$, $b_m$, and $f_{dm}$ are the control force of the actuator, the control gain, and the modal force of the excitations, respectively.

The out-of-bandwidth modes can be appropriately neglected. Hence, similar to [37], the transfer function between ${\eta }_m$ and $b_m{f}_{cm}+f_{dm}$ within the bandwidth of interest can be expressed as follows:

$$G_p(s)={\displaystyle \sum _m^n\frac{b_m}{s^2+2{\zeta }_m{\omega }_{m}s+{\omega }_m^2}}.$$

(8)

2.2. Inertial Actuator Dynamics

The typical schematic diagram of the structure with the condition of the all-clamped plate is shown in Figure 1. The inertial actuator generates force to the plate structure where the acceleration sensor is attached to measure the vibration signal. An inertial actuator mass $m_a$ is attached to the structure via a spring $k_a$, a damper $c_a$, and magnetic induction coil voltage $u_e$, in parallel. A current $i$ is generated when an input voltage $u$ is applied to the electromagnetic coil. According to Lorentz’s law, the control force $f_a$ has a coefficient relation with the current $i$. Therefore, the electrical equation of the actuator coil can be obtained as follows:

$$\{\begin{array}{l}{f}_a=Bli\hfill \\ u=L_e\frac{di}{dt}+R_{e}i+Bl({\dot{x}}_a-{\dot{x}}_s)\hfill \end{array},$$

(9)

where $B$ and $l$ are the electromagnetic induction coefficient and coil length, respectively. The variables $x_a$ and $x_s$ represent the displacement of the inertial actuator and the structure, respectively. The parameters of the inertial actuator applied can be obtained as $L_e$ = 0.02 mH and $R_e$ = 7.5 $\mathsf{\Omega }$. Similar to [32], the transfer function of the inertial actuator is $G(s)=\frac{R_e}{s{L}_e+R_e}$. Therefore, the Bode diagram of G(s) is shown in Figure 2. The frequency response of the electrical circuit can be regarded as equal to 1 within the frequency bandwidth of interest in Figure 2, i.e., less than 1000 rad/s. In addition, the control force $f_a$ can be regarded as $g_a$ times the input voltage, and therefore, the model of the inertial actuator can be expressed as follows [38]:

$$\{\begin{array}{l}{f}_a=g_{a}u\hfill \\ f_c=f_a+c_a({\dot{x}}_s-{\dot{x}}_a)+k_a(x_s-x_a)\hfill \\ f_c=m_a{\ddot{x}}_a\hfill \end{array}.$$

(10)

Similar to [39], the inertial actuator displacement in Laplace domain can be expressed by Equation (11) as follows:

$$X_a(s)=\frac{1}{m_a{s}^2+c_{a}s+k_a}{F}_a(s)+\frac{c_{a}s+k_a}{m_a{s}^2+c_{a}s+k_a}{X}_s(s),$$

(11)

where $X_a(s)$, $F_a(s)$, and $X_s(s)$ are the Laplace transforms of $x_a$, $f_a$, and $x_s$, respectively.

According to Equations (10) and (11), the transfer function of the transmitted force $f_c$ can be derived as:

$$\begin{array}{ll}{F}_c(s)& =(\frac{m_a{s}^2}{m_a{s}^2+c_{a}s+k_a}{g}_a)U(s)+(\frac{m_a{s}^2(c_{a}s+k_a)}{m_a{s}^2+c_{a}s+k_a})X_s(s)\\ & =G_a(s)U(s)+G_s(s)X_s(s),\end{array}$$

(12)

where $F_c(s)$, $U(s)$, and $X_s(s)$ are the Laplace transform of $f_c$, $u$, and $x_s$, respectively. The symbol $G_a(s)$ is defined as the transfer function of the inertial actuator when the structure displacement is equal to 0. In addition, the transfer function $G_s(s)$ refers to the sensitivity of coupling force to structure displacement.

2.3. Problem Statement

Intractable nonlinearity factors, regarded as the external disturbance, inevitably exist in the system. In order to reduce the number of sensors, the motion of the inertial actuator, considered as the secondary excitation of the plate structure, does not need to be measured directly by the sensor, since, in this paper, it is defined as the disturbance generated. In addition, due to parameter uncertainty, the experimental model must deviate from the actual physical system model. According to Equation (7), the system differential equation can be reconstructed as Equation (13) by taking the displacement of the all-clamp plate $x_s$ as a system state variable as follows:

$${\ddot{x}}_s=-2{\zeta }_0{\omega }_0{\dot{x}}_s-{\omega }_0^2{x}_s+b{f}_c+f_d.$$

(13)

By substituting Equation (12) into Equation (13), the system can be rewritten in the following cascade form:

$${\ddot{x}}_s=\underset{\mathrm{total}\mathrm{disturbance}f}{\underbrace{\underset{\mathrm{model}\mathrm{error}}{\underbrace{-2{\zeta }_0{\omega }_0{\dot{x}}_s-{\omega }_0^2{x}_s}}+\underset{\mathrm{actuator}\mathrm{effect}}{\underbrace{f_c^{\prime }}}+\underset{\mathrm{external}\mathrm{disturbance}}{\underbrace{f_d}}}}+\underset{\mathrm{control}\mathrm{gain}b}{\underbrace{b{g}_a}}u.$$

(14)

Remark 1.

The polynomial$-2{\zeta }_0{\omega }_0{\dot{x}}_s-{\omega }_0^2{x}_s$is related to the velocity and displacement of the plate, which is considered to be model data. The polynomial$f_c^{\prime }$is the excitation generated by the motion of the inertial actuator. The external disturbances$f_d$are produced by the external excitation. The term$b{g}_a$is the control gain. In addition, the total disturbance$f$is introduced by defining$f=-2{\zeta }_0{\omega }_0{\dot{x}}_s-{\omega }_0^2{x}_s+f_c^{\prime }+f_d$. In a real physical vibration system, the displacement of the vibration, the external excitation, and the input of the inertial actuator are all bounded, so$f$is bounded. Moreover, the polynomial$f$is a time-varying continuous and bounded function, and, therefore,$\dot{f}$is also bounded. The parameter$b_0$is the estimation of control gain. The state space error model can be deduced as follows in Equation (15) by defining the state variable as$x_1=x_s-r$and$x_2={\dot{x}}_s-\dot{r}$:

$$\{\begin{array}{l}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2=f+b_{0}u\hfill \end{array}.$$

(15)

3. Composite Controller Design

3.1. Reduced-Order Extended State Observer Design

Considering the total disturbance $f$ as a new system state variable $x_3$, the state space Equation (15) can be further extended as follows:

$$\{\begin{array}{l}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2=b_{0}u+x_3\hfill \\ {\dot{x}}_3=\dot{f}\hfill \end{array}.$$

(16)

According to Equation (16), a third-order ESO is designed as follows [32]:

$$\{\begin{array}{l}\begin{array}{l}{\dot{z}}_1=z_2-{\iota }_1(z_1-x_1)\hfill \\ {\dot{z}}_2=z_3-{\iota }_2(z_1-x_1)+b_{0}u\hfill \\ {\dot{z}}_3=-{\iota }_3(z_1-x_1)\hfill \end{array}\\ {\widehat{x}}_1=z_1,{\widehat{x}}_2=z_2,{\widehat{x}}_3=z_3\end{array},$$

(17)

where ${\widehat{x}}_i$ ($i=1,2,3$) are the estimation values of system state variables $x_i$, and ${\iota }_i$ values are the gains of the ESO.

Based on the bandwidth method [40], the gains can be set as ${\iota }_1=3\omega$, ${\iota }_2=3{\omega }^2$, ${\iota }_3={\omega }^3$. From [41], a contradictory conclusion can be drawn that the whole control performance can be improved with the large bandwidth gain of ESO, while the system anti-noise ability decreases due to the high-pass filter characteristics. Considering that the vibrational output value can be directly measured by a sensor, a RESO-based vibration controller was designed following Equation (18):

$$\{\begin{array}{l}{\dot{z}}_2=z_3+{\beta }_2{x}_1-{\beta }_1(z_2+{\beta }_1{x}_1)+b_{0}u\hfill \\ {\dot{z}}_3=-{\beta }_2(z_2+{\beta }_1{x}_1)\hfill \\ {\widehat{x}}_2=z_2+{\beta }_1{x}_1,{\widehat{x}}_3=z_3+{\beta }_2{x}_1\hfill \end{array},$$

(18)

where ${\beta }_1$, ${\beta }_2$ are the gains of the RESO. The gains can be set as ${\beta }_1=2{\omega }_0$ and ${\beta }_2={\omega }_0{}^2$ based on the bandwidth method.

3.2. Nonsingular Terminal SMC Design

A nonsingular sliding model surface of system model (15) was designed as Equation (19) with the proposed RESO [42]:

$$s=x_1+\frac{1}{\beta }{\widehat{x}}_2{}^{\frac{p}{q}},$$

(19)

where $\beta$ > 0 is a designed constant, and the parameters $p$, $q$ are positive odd integers that satisfy the condition 1 < $p/q$ < 2.

According to Equation (15), we knew ${\dot{x}}_1$ = $x_2$. Based on the RESO proposed in this paper, the estimation of $x_2$ was ${\widehat{x}}_2$. Therefore, the time derivative of Equation (19) can be deduced as follows:

$$\dot{s}={\widehat{x}}_2+\frac{1}{\beta }\frac{p}{q}{\widehat{x}}_2^{\frac{p}{q}-1}\cdot {\dot{\widehat{x}}}_2$$

(20)

According to the third equation in Equation (18), we knew ${\widehat{x}}_2=z_2+{\beta }_1{x}_1$ and ${\widehat{x}}_3=z_3+{\beta }_2{x}_1$, therefore, ${\dot{\widehat{x}}}_2={\dot{z}}_2+{\beta }_1{\dot{x}}_1$. Therefore, based on the first and second equations in Equation (18), the observer state variable ${\dot{\widehat{x}}}_2$ can be derived as follows:

$$\begin{array}{ll}{\dot{\widehat{x}}}_2& ={\dot{z}}_2+{\beta }_1{\dot{x}}_1\\ & =\underset{{\dot{z}}_2}{\underbrace{\stackrel{{\widehat{x}}_3}{\overbrace{z_3+{\beta }_2{x}_1}}-{\beta }_1\stackrel{{\widehat{x}}_2}{\overbrace{(z_2+{\beta }_1{x}_1)}}+b_{0}u}}+{\beta }_1\underset{{\dot{x}}_1}{\underbrace{{\widehat{x}}_2}}\\ & ={\widehat{x}}_3+b_{0}u\end{array}$$

(21)

By substituting Equation (21) into Equation (20), Equation (20) can be further derived as follows:

$$\dot{s}={\widehat{x}}_2+\frac{1}{\beta }\frac{p}{q}{\widehat{x}}_2^{\frac{p}{q}-1}({\widehat{x}}_3+b_{0}u).$$

(22)

The RESO-NTSMVC control law can be designed as follows:

$$u=\frac{1}{b_0}\left[-\beta \frac{q}{p}{\widehat{x}}_2^{2-\frac{p}{q}}-\eta \mathrm{sgn}(s)-{\widehat{x}}_3\right],$$

(23)

where the parameter $\eta$ is a positive switching gain of the sign function.

By substituting Equation (23) into Equation (20), Equation (24) can be obtained as follows:

$$\dot{s}=-\frac{1}{\beta }\frac{p}{q}{\widehat{x}}_2^{\frac{p}{q}-1}\eta \mathrm{sgn}(s).$$

(24)

The RESO–NTSMVC-based vibration control system for the all-clamped plate with inertial actuator is shown in Figure 3. The total disturbance $f$ was estimated by RESO and eliminated by the feedforward part. It can be seen from Equation (23) that the control law $u$ contained a sign function $\eta \mathrm{sgn}(s)$, which would cause the system states to cross back and forth on both sides of the sliding mode surface, thus, generating the chattering phenomenon. Therefore, the chattering problem caused by NTSMVC would be alleviated. In addition, RESO could solve the high frequency noise problem caused by large bandwidth gain.

4. Stability Analysis

4.1. Stability Analysis of RESO

Assumption 1.

According to Remark 1, the total disturbances$f$in the all-clamped plate system (15) are bounded. There exist Lipschitz constants$L$> 0 and$t_j$that satisfy$\left|e_1\right|<L$for all$t>t_j$.

The observer estimated errors of RESO are defined as $e_2={\widehat{x}}_2-x_2$, $e_3={\widehat{x}}_3-x_3$, therefore, the error differential equation can be obtained as follows:

$$\{\begin{array}{c}{\dot{e}}_2=-{\beta }_1{e}_2+e_3\\ {\dot{e}}_3=-{\beta }_2{e}_2-\dot{f}\end{array}.$$

(25)

For the convenience of analysis, Equation (25) can be further written as the following matrix form:

$$\left[\begin{array}{c}{\dot{e}}_2\\ {\dot{e}}_3\end{array}\right]=\underset{A}{\underbrace{\left[\begin{array}{cc}-{\beta }_1& 1\\ -{\beta }_2& 0\end{array}\right]}}\left[\begin{array}{c}{e}_2\\ e_3\end{array}\right]+\left[\begin{array}{c}0\\ -\dot{f}\end{array}\right].$$

(26)

The characteristic polynomial of Equation (26) can be described as follows:

$$\det (\lambda I-A)={\lambda }^2+{\beta }_1\lambda +{\beta }_2=(\lambda +{\omega }_0{)}^2,$$

(27)

where $\lambda =-{\omega }_0<0$.

According to Remark 1, $\dot{f}$ is bounded. It is observed that the system poles are distributed in the left half plane, thus, the errors $e_2$ and $e_3$ eventually converge to the equilibrium point ${\left[\begin{array}{cc}-\frac{\dot{f}}{{\omega }_0{}^2}& -\frac{2\dot{f}}{{\omega }_0}\end{array}\right]}^T$. Assuming the all-clamped plate system satisfies Assumption 1, the error states of system (16) converge to the equilibrium point in finite time $t_f$.

4.2. Closed-Loop System Stability Analysis

Similar to [43], a finite-time bounded (FTB) function can be defined as $V_1=\frac{1}{2}(s^2+x_1^2+{\widehat{x}}_2^2)$. Considering $1<p/q<2$, the inequality holds $0<2-p/q<1$. It can be noted that ${\left|{\widehat{x}}_2\right|}^{2-\frac{p}{q}}<1+\left|{\widehat{x}}_2\right|$ and ${\left|{\widehat{x}}_2\right|}^{\frac{p}{q}-1}<1+\left|{\widehat{x}}_2\right|$. The derivative of $V_1$ can be expressed as:

$$\begin{array}{ll}{\dot{V}}_1& =s\dot{s}+x_1{\dot{x}}_1+{\widehat{x}}_2{\dot{\widehat{x}}}_2\\ & =-\frac{1}{\beta }\frac{p}{q}{\widehat{x}}_2^{\frac{p}{q}-1}\eta \left|s\right|+x_1{\widehat{x}}_2+{\widehat{x}}_2(-\beta \frac{q}{p}{\widehat{x}}_2^{2-\frac{p}{q}}-\eta \mathrm{sgn}s)\\ & \le \left|-\frac{1}{\beta }\frac{p}{q}{\widehat{x}}_2^{\frac{p}{q}-1}\eta \left|s\right|+x_1{\widehat{x}}_2+{\widehat{x}}_2(-\beta \frac{q}{p}{\widehat{x}}_2^{2-\frac{p}{q}}-\eta \mathrm{sgn}s)\right|\\ & \le \frac{1}{\beta }\frac{p}{q}\left|{\widehat{x}}_2^{\frac{p}{q}-1}\right|\eta \left|s\right|+\left|x_1\right|\left|{\widehat{x}}_2\right|+\left|{\widehat{x}}_2\right|(\beta \frac{q}{p}\left|{\widehat{x}}_2^{2-\frac{p}{q}}\right|+\eta )\\ & \le \frac{1}{\beta }\frac{p}{q}(1+\left|{\widehat{x}}_2\right|)\eta \left|s\right|+\left|x_1\right|\left|{\widehat{x}}_2\right|+\left|{\widehat{x}}_2\right|[\beta \frac{q}{p}(1+\left|{\widehat{x}}_2\right|)+\eta ]\\ & \le \frac{1}{\beta }\frac{p}{q}\eta \left|s\right|+\frac{1}{\beta }\frac{p}{q}\left|{\widehat{x}}_2\right|\eta \left|s\right|+\left|x_1\right|\left|{\widehat{x}}_2\right|+\beta \frac{q}{p}\left|{\widehat{x}}_2\right|+\beta \frac{q}{p}{\left|{\widehat{x}}_2\right|}^2+\eta \left|{\widehat{x}}_2\right|\\ & \le [\frac{1}{\beta }\frac{p}{q}(1+\eta )]\frac{s^2}{2}+\frac{x_1^2}{2}+(\frac{1}{\beta }\frac{p}{q}\eta +\frac{q}{p}+2\beta \frac{q}{p}+2)\frac{{\widehat{x}}_2^2}{2}+(\frac{1}{\beta }\frac{p}{q}\frac{{\eta }^2}{2}+\frac{q}{p}\frac{{\beta }^2}{2}+\frac{{\eta }^2}{2})\\ & =K_{V1}{V}_1+L_{V1},\end{array}$$

(28)

where $K_{V1}=\max \left\{\frac{1}{\beta }\frac{p}{q}(1+\eta ),1,\frac{1}{\beta }\frac{p}{q}\eta +\frac{q}{p}+2\beta \frac{q}{p}+2\right\},$ and $L_{V1}=\frac{1}{\beta }\frac{p}{q}\frac{{\eta }^2}{2}+\frac{q}{p}\frac{{\beta }^2}{2}+\frac{{\eta }^2}{2}$ are bounded constants.

It can be shown from Equation (19) that $s$, $x_1$, ${\widehat{x}}_2$ would not escape in finite time. Hence, the overall control system with control law (23) is finite-time stable, and the observer errors $e_2$ and $e_3$ converge to zero in finite time.

Consider the following Lyapunov function:

$$V_2=\frac{1}{2}{s}^2.$$

(29)

Combined with Equation (19), and by taking the derivative of the Lyapunov function $V_2$, the following can be obtained:

$${\dot{V}}_2=s\dot{s}=s(-\frac{1}{\beta }\frac{p}{q}{\widehat{x}}_2^{\frac{p}{q}-1}\eta \mathrm{sgn}(s))=-\frac{1}{\beta }\frac{p}{q}{\widehat{x}}_2^{\frac{p}{q}-1}\eta \left|s\right|,$$

(30)

where $\rho ({\widehat{x}}_2)=\frac{1}{\beta }\frac{p}{q}{\widehat{x}}_2^{\frac{p}{q}-1}\eta \left|s\right|$.

Considering that $p$ and $q$ are positive odd integers and $1<p/q<2$, it has $\rho ({\widehat{x}}_2)>0$ for the case ${\widehat{x}}_2\ne 0$. Next, it can be shown that the sliding variable is finite-time stable and converges to zero in a finite time interval [44]. For the case ${\widehat{x}}_2=0$, it can be obtained that ${\dot{\widehat{x}}}_2=-\beta \frac{q}{p}{\widehat{x}}_2^{2-\frac{p}{q}}-\eta \mathrm{sgn}(s)$. Similar to the proof in [40], it is easy to ascertain that ${\widehat{x}}_2=0$ is not an attractor. Therefore, the states in system (19) would reach zero in finite time. According to the above analysis, once the system states reach the sliding plane, $s=0$ from any original conditions in finite time. Let $s=0$, and satisfy ${\widehat{x}}_2=x_2$, ${\widehat{x}}_3=x_3$, then, after a finite time, the sliding motion can be described as follows:

$$s=x_1+\frac{1}{\beta }{x}_2^{p/q}=x_1+\frac{1}{\beta }{\dot{x}}_1^{p/q}=0.$$

(31)

The time $t_s$ that the system state variable converges to the equilibrium point can be described as follows:

$${\dot{x}}_1=-{\beta }^{q/p}{x}_1^{q/p}.$$

(32)

The finite time $t_s$ is given by integrating Equation (32):

$$t_s=-{\beta }^{p/q}{\displaystyle {\int }_{x_{1,t_r}}^0{x}^{-q/p}}d{x}_1=\frac{p}{{\beta }^{q/p}(p-q)}{\left|x_{1,t_r})\right|}^{\frac{p-q}{p}},$$

(33)

where $t_r$ is defined as the time when the system state variables are in steady-state. With the chosen control parameters $\beta >0$, $1<p/q<2$, the state of system (15) can be driven to the equilibrium point along the sliding plane in finite time. The proof is completed.

5. Experimental Verifications

5.1. Experimental Set-Up

In order to verify the effectiveness of the proposed control method RESO-NTSMVC, a thin plate, sized 500 mm × 500 mm × 1 mm, was taken as the research object, as shown in Figure 4. The property data of the inertial actuator (EX45S, VISTON, German) applied in this paper are shown in

Table 2. Inertial actuator data.

Parameter	Value
Mass, $m_a$	0.06 (kg)
Stiffness, $r_a$	710.61 (N/m)
Damping ratio, $k_a$	0.02
Force constant, $BL$	3.4 (N/A)
Coil resistance, $R_e$	7.5 ($\mathsf{\Omega }$)
Coil inductance, $L_e$	0.002 (H)
Natural angular frequency, ${\omega }_c$	16.3 (Hz)

. The Hev-20 exciter was arranged on the thin plate as an external disturbance. The signal generator (Tektronix AFG1062) gave a sinusoidal excitation signal with an amplitude of 1 V and first resonant frequency of 48.5 Hz. The actuator and acceleration sensor were placed closely together. The vibration signal passes through an acceleration sensor, a constant current source voltage regulator (YE3821), a signal conditioning circuit, and an NI PCI-6343 module, sequentially, to the analog input module of the Simulink simulation. The controller designed in Simulink was based on RESO-NTSMC. The control quantity generated by the designed controller, which was amplified by the power amplifier (HVP-300), drove the actuator to suppress vibration.

5.2. Experimental Results

Comparison experiments between RESO–NTSMVC and ESO–NTSMVC are carried out on the vibration control equipment. As a fair comparison, signals of uniform frequency and amplitude were used to excite the thin plate. As can be seen from Figure 5, when the structure was not controlled by a controller, the vibration amplitude was about 1 V in the time domain. First, when the proposed RESO–NTSMVC method was applied for the experiment, the vibration amplitude significantly reduced to about 0.275 V. Second, when the ESO–NTSMVC method was applied for the experiment, the vibration amplitude reduced to 0.42 V with oscillation. It was obvious that the vibration control performance based on RESO–NTSMVC was superior to ESO–NTSMVC. The parameters of the two control methods are selected as follows: the parameters for RESO–NTSMVC were configured as $w=170$, $p=5$, $q=3$, $\beta =4000$, $\eta =25$, $b_0=\mathrm{50,000}$, when the parameters for ESO–NTSMVC were configured as $w=300$, $p=5$, $q=3$, $\beta =80$, $\eta =100$, $b_0=\mathrm{60,000}$.

For clear comparison, similar to [6], the normalized signal spectrum was defined as follows:

$$\mathrm{The}\mathrm{decibel}\mathrm{value}=20{\log }_{10}(FFT(y/y_R)),$$

(34)

where the function $FFT(\cdot )$ was Fourier transformation.

The vibration output reference signal was described by $y_R$. Set $y_R$ to the standard value 1 V, i.e., 0 dB was equal to 1 V and −20 dB was equal to 0 V. When the all-clamped plate was excited by the first natural frequency signal, the first-order mode and other higher-order harmonic frequency mode frequency domain responses could be obtained, as shown in Figure 6.

As can be seen from Figure 5 and Figure 6, the vibration control performance based on RESO–NTSMVC was superior to the controller based on ESO–NTSMVC, especially at the first natural frequency. Moreover, the control curve of the RESO–NTSMVC method was very smooth without any oscillation. In addition, as shown in Figure 6a, the vibration amplitude of the RESO–NTSMVC controller decreased in almost each frequency range, due to the simple calculations, fewer parameters, and high efficiency of the RESO controller. As shown in the frequency domain diagram in Figure 6b, the vibration amplitudes of the RESO–NTSMVC method in the first mode could be reduced by 11.8 dB, while the ESO–NTSMVC method achieved a control effect of only 6.8 dB. Meanwhile, the amplitudes of RESO–NTSMVC at two to six times the first natural modal frequency were reduced by 6.2 dB (at 97 Hz), 10 dB (at 194 Hz), 5dB (at 242.5 Hz), 10.5 dB (at 399.5 Hz), which were superior to those of the ESO–NTSMVC controller, as seen in Figure 6b–f.

In addition, as shown in Figure 7, the control quantities of the two methods were also compared. Compared with the ESO–NTSMVC method, the RESO–NTSMVC method proposed in this paper had a smaller control voltage without oscillation. Therefore, the experimental results showed that the designed controller achieved better vibration suppression performance with lower control voltage. The vibration control effects of the two controllers are shown in

Table 3. Vibration control effect of different controllers.

Controller/Frequency (Hz)	48.5	97	145.5	194	242.5	291	399.5
Without control (dB)	−1.2	−20	−58	−38.5	−56	−67	−62
ESO-NTSMVC (dB)	−8	−27.6	−48.5	−43.5	−57	−75	−68
RESO-NTSMVC (dB)	−13	−26.2	−49.5	−48.5	−61	−69	−72.5

. It can be seen from Figure 6 and Table 3 that the RESO–NTSMVC controller exhibited superiority in dealing with higher harmonic problems.

6. Conclusions

Aimed at resolving the internal uncertain dynamics, higher harmonics, and strong nonlinearity in an all-clamped plate with an inertial actuator, a novel RESO–NTSMVC controller was proposed and verified by a real-time structural vibration suppression experimental set-up. The experimental comparison results showed that the proposed RSEO–NTSMCV controller had strong anti-disturbance performance and effective vibration control performance. The following conclusions are pertinent:

(1)

The nonlinear dynamics of the inertial actuator were considered to be disturbances to the plate; therefore, the vibration controller could be simplified with only one sensor.

(2)

The designed RESO could solve the problem of poor anti-noise ability caused by the high bandwidth gain of the traditional ESO.

(3)

The RESO-NTSMVC controller could obtain a fast convergence rate and strong anti-disturbance ability.

Although the proposed RESO can obtain satisfactory deterministic estimation performance and vibration suppression performance, it is difficult to ensure the stability time of the system, when the external vibration environment is complex and changeable, especially regarding the uncertainties of the initial state of each sliding surface. In the future, we plan to introduce a fixed-time convergence method into the sliding mode control to constrain the convergence time, and therefore, the structural vibration suppression performance can be enhanced with the overall convergence rate.