Adaptive Dynamic Event-Triggered Sliding Mode Tracking Control of Pneumatic Vibration Isolation System

Abstract

In this paper, an adaptive dynamic event-triggered sliding mode control scheme is proposed for a pneumatic vibration isolation platform. First, an experimental platform is designed and constructed, and a corresponding dynamic model is established, which explicitly accounts for the unknown threshold voltage at the input side. Based on this model, an adaptive sliding mode controller is developed. Then, to suppress unnecessary actuator updates, a dynamic event-triggered mechanism is introduced. Lyapunov-based analysis demonstrates the stability of the closed-loop system and guarantees the exclusion of Zeno behavior. Finally, experimental results on the pneumatic platform verify the effectiveness and superiority of the proposed approach.

1. Introduction

Active pneumatic vibration isolation platforms can effectively isolate and attenuate vibration disturbances while supporting heavy payloads with relatively low energy consumption, and have therefore been widely applied in semiconductor manufacturing, precision measurement, ultra-precision machining, and agricultural production [1,2,3,4]. As the core component of such platforms, pneumatic actuators regulate the chamber pressure through control valves, converting compressed air into mechanical motion to ensure fast response capability. In practical applications, pneumatic isolation platforms are not only employed for vibration suppression but also frequently undertake trajectory tracking and precise positioning tasks. However, pneumatic control systems inherently exhibit significant nonlinear characteristics and are further influenced by parameter uncertainties, dead-zone effects, air leakage, and load variations. These complexities make it difficult for conventional control methods to guarantee robustness and stability in high-precision scenarios. Therefore, the development of robust and efficient control strategies is of great importance for improving actuator performance and enhancing the engineering value of pneumatic vibration isolation platforms.

Nonlinear control has been extensively studied, yielding methods such as predictive control [5], operator-based robust right coprime factorization approach [6], and sliding-mode control [7]. Sliding mode control has been extensively acknowledged for its strong robustness against system uncertainties and external disturbances, and has been successfully employed in a wide range of nonlinear systems, and it can be integrated with other methodologies to enhance performance [8,9,10,11,12,13]. In [14], an adaptive integral sliding mode control scheme is proposed for uncertain systems with unknown disturbances and dead zones, where the dead-zone effect is treated as a compounded disturbance addressed by a sliding mode disturbance observer. Furthermore, to cope with unknown parameters in systems, adaptive control has emerged as an effective method. Typically, adaptive control achieves this by parameter estimation and online updating mechanisms, together with adaptive laws designed using Lyapunov stability theory, so as to adjust control gains in real time, thereby compensating for uncertainties caused by unknown parameters [15,16]. Owing to its capability of online learning and parameter adaptation, adaptive control can effectively enhance control accuracy and reduce the conservatism of sliding mode robust designs, making it particularly suitable for complex nonlinear systems. Although these methods improve robustness and adaptivity, they are still implemented under a conventional time-triggered framework, in which the controller is adjusted at every sampling instant. Such frequent updates not only impose unnecessary computational and communication burdens but also accelerate actuator wear, thereby reducing energy efficiency and shortening service life. Therefore, it remains an important problem to investigate how to reduce the control-update frequency while ensuring robustness and stability.

Fortunately, event-triggered control has emerged, in which the control signal is updated or transmitted only according to a predefined triggering mechanism, thereby ensuring system performance from both stability and efficiency perspectives [17,18]. Recently, various types of event-triggered control mechanisms have been proposed, which can be classified according to their triggering properties, such as fixed-threshold event-triggered control [19], relative-threshold event-triggered control [18,20], switching-threshold event-triggered control [21], and static event-triggered control [22]. Recently, the idea of dynamic event-triggered control has been introduced, in which a positive auxiliary variable is incorporated to dynamically adjust the triggering threshold. Compared with static event-triggered schemes, it not only reduces the number of triggering events and obtains larger inter-event time intervals but also achieves a better trade-off between control performance and resource utilization [23,24,25]. In addition, in [26], a backstepping-based dynamic event-triggered mechanism is designed, which ensures zero tracking/stabilization error while effectively excluding Zeno behavior.

In this paper, an adaptive dynamic event-triggered sliding mode control scheme is proposed for the pneumatic vibration isolation system. The controller design accounts for unknown threshold voltages induced by air leakage, payload mass, and dead zones. Moreover, the scheme can significantly reduce the controller update frequency and achieve effective trajectory tracking. The main contributions are as follows:

• A pneumatic vibration isolation platform is designed and constructed, and the corresponding dynamic model is established, where the model considers the unknown threshold voltage at the input side.
• An adaptive dynamic event-triggered sliding mode tracking control strategy is proposed, wherein a novel dynamic event-triggered mechanism is constructed accompanied by a rigorous Lyapunov-based stability proof, which also guarantees the exclusion of Zeno behavior. Unlike existing methods, the designed controller provides adaptive compensation for the unknown threshold voltage. Moreover, the introduced dynamic event-triggered mechanism effectively suppresses unnecessary actuator switching, thereby improving energy efficiency and extending actuator lifespan.
• Experimental results verify the effectiveness and superiority of the proposed approach, showing a significant reduction in control updates compared with conventional time-triggered methods.

The remainder of this paper is organized as follows. Section 2 gives the experimental setup and system modeling. Section 3 develops the proposed controllers and provides the theoretical analysis. Section 4 presents the experimental validation and comparisons. Section 5 gives the conclusions.

2. Experimental Setup and Dynamic Modeling

2.1. Experimental Setup

The experimental setup is constructed to evaluate the proposed control strategy, as shown in Figure 1. The system consists of an air compressor, an electro-pneumatic regulator, a pneumatic actuator mechanically connected to a single-link arm structure, a displacement sensor, regulated DC power supplies, a National Instruments USB-6000 series data acquisition (DAQ) device, and a computer. The compressed air is regulated before driving the pneumatic actuator, whose output is measured by the displacement sensor. The regulated DC power supplies provide stable voltage sources for powering the displacement sensor. The DAQ device serves as the interface between the computer and the regulator for both control input and signal acquisition. The experimental computer is a Lenovo Xiaoxin laptop running Microsoft Windows, and all control algorithms are implemented in MATLAB. The specifications of the main hardware components are summarized in

Table 1. Specifications of the experimental setup components.

Component	Specification
Air compressor	ANEST IWATA Corporation (Yokohama, Japan); COMGPAC CFP37-8.5D
Computer	Lenovo Group Ltd. (Beijing, China); Xiaoxin-15IIL laptop
Software	MATLAB R2024b (MathWorks, Natick, MA, USA)
DAQ device	NI (National Instruments Corporation), Austin, TX, USA; USB-6002 multifunction DAQ
Electro-pneumatic regulator	KOGANEI Corporation (Tokyo, Japan); KTR200-2, input 0–10 V, output 0.02–0.84 MPa
Sensor	OMRON Corporation (Kyoto, Japan); ZX-LD40L laser displacement sensor, red laser (650 nm), reference distance 40 mm, measuring range 30–50 mm
Power supply	Kikusui Electronics Corporation (Yokohama, Japan); PMM18-2.5DU (±18 V, 2.5 A)
Pneumatic actuator	Driving a single-link arm structure as the load

.

2.2. Pneumatic System Modeling

Figure 2 illustrates the overall schematic of the pneumatic vibration isolation system, including the interactions among the air spring, compressor, electro-pneumatic regulator, sensor, and controller, together with the force transmission pathways acting on the payload.

According to the force analysis in Figure 2, the dynamic model of the pneumatic vibration isolation system is given by

$$m\ddot{x}=-c\left(\dot{x}-{\dot{x}}_b\right)-k\left(x-x_b\right)+f_{\mathrm{air}}-mg,$$

(1)

where x is the displacement, m is the equivalent load mass, c is the damping coefficient, and k is the equivalent spring stiffness. The base is assumed to remain stationary during the experiments, implying $x_b={\dot{x}}_b={\ddot{x}}_b=0$. Consequently, the dynamic model (1) can be simplified to the linear form.

The pneumatic actuation force $f_{\mathrm{air}}$ is related to the effective command through an identified gain:

$$f_{\mathrm{air}}=K_{\mathrm{gain}}{u}_{\mathrm{eff}}.$$

(2)

where $K_{\mathrm{gain}}$ is the actuator gain identified from experimental data through curve fitting and approximated as constant.

The effective input $u_{\mathrm{eff}}$ is subject to a unknown threshold voltage, which can be modeled as

$$u_{\mathrm{eff}}={[u_s-u_b]}_{+}=max\{u_s-u_b,0\},$$

(3)

where $u_s$ is the command voltage and $u_b>0$ is the unknown threshold voltage. Figure 3 illustrates the characteristic of Equation (3).

To facilitate controller design, the dynamic model (1) can be rewritten as the following linear state-space form. Specifically, let $x_1=x$ and $x_2=\dot{x}$, and the dynamics can be expressed as

$$\begin{array}{cc}\hfill {\dot{x}}_1& =x_2,\hfill \\ \hfill {\dot{x}}_2& =-\frac{k}{m}{x}_1-\frac{c}{m}{x}_2+\frac{K_{\mathrm{gain}}}{m}{u}_{\mathrm{eff}}-g,\hfill \end{array}$$

(4)

2.3. Tracking-Error System

Based on the state-space representation in (4), we now derive the tracking-error dynamics with respect to a given reference trajectory.

We consider a desired reference trajectory $x_d\left(t\right)$, and it is assumed to be twice-differentiable. Then, the tracking errors can be defined as

$$\begin{array}{c}\hfill e_1=x_1-x_d.\end{array}$$

(5)

$$\begin{array}{c}\hfill e_2=x_2-{\dot{x}}_d.\end{array}$$

(6)

Then, the error dynamics can be expressed as

$$\begin{array}{cc}\hfill {\dot{e}}_1& =e_2,\hfill \\ \hfill {\dot{e}}_2& =-\frac{k}{m}{e}_1-\frac{c}{m}{e}_2+\frac{K_{\mathrm{gain}}}{m}{u}_{\mathrm{eff}}+\mathsf{\Psi }\left(t\right),\hfill \end{array}$$

(7)

where

$$\mathsf{\Psi }\left(t\right)=-\frac{k}{m}{x}_d\left(t\right)-\frac{c}{m}{\dot{x}}_d\left(t\right)-{\ddot{x}}_d\left(t\right)-g.$$

(8)

Based on the developed experimental pneumatic platform and the derived error system, the control objectives of this work are (1) to compensate for the unknown input bias and external disturbances through an adaptive method; (2) to achieve high-precision trajectory tracking of the reference signal; (3) to reduce the control-update frequency by employing a dynamic event-triggered strategy, thereby improving energy efficiency and extending actuator durability.

3. Design of Controllers

In order to solve the trajectory tracking control problem of the developed experimental pneumatic platform, this section designs two adaptive sliding mode controllers. The first employs a conventional time-triggered mechanism, while the second adopts a dynamic event-triggered mechanism.

3.1. Time-Triggered Adaptive Sliding Mode Controller

To design the controller, we construct the following sliding mode surface by integrating proportional–integral–derivative error terms

$$s=K_d{e}_2+K_p{e}_1+K_i{\int }_0^t{e}_1\left(\tau \right)d\tau ,$$

(9)

where $K_d$, $K_p$, and $K_i$ are positive design parameters.

Differentiating (9) and substituting the error dynamics (7) and (8), we obtain

$$\begin{array}{cc}\hfill \dot{s}& =K_d{\dot{e}}_2+K_p{\dot{e}}_1+K_i{e}_1\hfill \\ \hfill & ={\displaystyle \frac{K_d{K}_{\mathrm{gain}}}{m}}{u}_{\mathrm{eff}}+\left(-{\displaystyle \frac{K_{d}c}{m}}+K_p\right)e_2+\left(-{\displaystyle \frac{K_{d}k}{m}}+K_i\right)e_1+K_d\mathsf{\Psi }\left(t\right).\hfill \end{array}$$

(10)

Based on (10), the controller is designed as

$$u_s={\widehat{u}}_b+u_{eq}+u_h,$$

(11)

where ${\widehat{u}}_b$ denotes the estimate of the unknown threshold voltage, $u_{eq}$ is the equivalent control component, and $u_h$ ensures robustness against uncertainties.

The equivalent controller $u_{eq}$ is designed as

$$\begin{array}{cc}\hfill u_{eq}& =-{\displaystyle \frac{1}{L}}\left[\left(-{\displaystyle \frac{K_{d}c}{m}}+K_p\right)e_2+\left(-{\displaystyle \frac{K_{d}k}{m}}+K_i\right)e_1+K_d\mathsf{\Psi }\left(t\right)\right],\hfill \end{array}$$

(12)

with $L={\displaystyle \frac{K_d{K}_{\mathrm{gain}}}{m}}.$

The robust term $u_h$ is designed to guarantee stability in the presence of residual uncertainties:

$$u_h=-{\displaystyle \frac{1}{L}}\left(k_1\mathrm{sign}\left(s\right)+k_{2}s\right),$$

(13)

where $k_1$ and $k_2$ are positive design parameters.

Furthermore, the adaptive law for ${\widehat{u}}_b$ is chosen as

$${\dot{\widehat{u}}}_b=-{\gamma }_{b}Ls,$$

(14)

with ${\gamma }_b>0$ denoting the adaptation gain.

Theorem 1.

Consider the closed-loop system composed of the error dynamics (7), the sliding variable (9), and the control law (11)–(14). Then the sliding variable $s\left(t\right)$ converges asymptotically to zero.

Proof. 

Consider the Lyapunov function candidate

$$V_1=\frac{1}{2}{s}^2+\frac{1}{2{\gamma }_b}{\tilde{u}}_b^2.$$

(15)

where ${\tilde{u}}_b=u_b-{\widehat{u}}_b$ is the estimation error.

Its time derivative along the closed-loop trajectories is

$${\dot{V}}_1=s\dot{s}+{\displaystyle \frac{1}{{\gamma }_b}}{\tilde{u}}_b{\dot{\tilde{u}}}_b.$$

(16)

Since ${\dot{\tilde{u}}}_b=-{\dot{\widehat{u}}}_b$, using the adaptive law (14) yields

$${\dot{\tilde{u}}}_b={\gamma }_{b}Ls.$$

(17)

To evaluate $\dot{s}$, substitute the control law (11) together with (12) and (13) into (10), so one has

$$\dot{s}=-k_{2}s-k_1\mathrm{sign}\left(s\right)-L{\tilde{u}}_b.$$

(18)

Substituting (17) and (18) into (16), we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_1& =s\left(-k_{2}s-k_1\mathrm{sign}\left(s\right)-L{\tilde{u}}_b\right)+{\displaystyle \frac{1}{{\gamma }_b}}{\tilde{u}}_b\left({\gamma }_{b}Ls\right)\hfill \\ \hfill & =-k_2{s}^2-k_1\left|s\right|\le 0.\hfill \end{array}$$

(19)

Thus, s can converge to the origin, and according to its definition, the tracking error $e_1$ and $e_2$ can also converge to zero as $t\to \infty$ [27]. The proof is complete. □

In the time-triggered scheme, the actuator must be updated at every sampling instant, and such high-frequency operations accelerate its wear. To address this issue, a dynamic event-triggered mechanism is introduced in the next section.

3.2. Dynamic Event-Triggered Sliding Mode Control

In this framework, the control input is updated only when a triggering condition is violated, rather than at every sampling instant. This strategy substantially reduces communication and actuation efforts while preserving stability and excluding Zeno behavior.

The event-triggered controller is defined as

$$\left\{\begin{array}{cc}\hfill u_{si}\left(t\right)& =u_s\left(t_i\right),\forall t\in [t_i,t_{i+1}],\hfill \\ \hfill t_{i+1}& =inf\left\{t\in \mathbb{R}|t>t_i,\begin{array}{cc}\hfill & \chi \left(t\right)+{\kappa }_1\left((1-{\eta }_d^2)\left((k_2-\frac{L}{2}){\left|s\right|}^2+k_1\left|s\right|\right)-\frac{L}{2}{\left|e_u\right|}^2\right)\le 0\hfill \end{array}\right\},\hfill \end{array}\right.$$

(20)

where $t_i$ denotes the i-th triggering instant, $u_s\left(t\right)$ is the sliding mode control law, $u_{si}\left(t\right)$ is the piecewise constant control signal actually applied to the plant, $k_1>0$ and $k_2>0$ are design constants, and ${\kappa }_1>0$ is the event-triggered parameter. The scalar ${\eta }_d\in (0,1)$ is a design parameter that adjusts the event-triggered condition. $e_u\left(t\right)$ is the sampling error defined as

$$e_u\left(t\right)=u_{si}\left(t\right)-u_s\left(t\right),$$

(21)

which represents the difference between the implemented control and the continuous control law. The auxiliary variable $\chi \left(t\right)$ in (20) provides additional flexibility in the triggering condition and is generated by the following differential equation:

$$\dot{\chi }\left(t\right)=-{\kappa }_2\chi \left(t\right)+(1-{\eta }_d^2)\left((k_2-\frac{L}{2}){\left|s\right|}^2+k_1\left|s\right|\right)-\frac{L}{2}{\left|e_u\right|}^2,$$

(22)

where ${\kappa }_2>0$ is a design parameter and the initial condition satisfies $\chi \left(0\right)>0$.

Lemma 1.

For the dynamic variable $\chi \left(t\right)$ defined in (22), under the event-triggered mechanism (20), it holds that

$$\chi \left(t\right)>0,\forall t\ge 0.$$

(23)

Proof. 

From the triggering condition (20), we have

$$(1-{\eta }_d^2)\left((k_2-\frac{L}{2}){\left|s\right|}^2+k_1\left|s\right|\right)-\frac{L}{2}{\left|e_u\right|}^2\ge -\frac{1}{{\kappa }_1}\chi \left(t\right).$$

(24)

Substituting (24) into the dynamics (22) gives

$$\dot{\chi }\left(t\right)\ge -\left({\kappa }_2+\frac{1}{{\kappa }_1}\right)\chi \left(t\right).$$

(25)

By the comparison lemma, it follows that

$$\chi \left(t\right)\ge \chi \left(0\right)exp(-({\kappa }_2+\frac{1}{{\kappa }_1})t)>0$$

(26)

Hence, $\chi \left(t\right)>0$ holds for all $t\ge 0$, which completes the proof.

□

Then, we give the following Theorem.

Theorem 2.

Consider the closed-loop system composed of the error dynamics (7), the sliding variable (9), and the adaptive sliding mode control law (11)–(14) implemented under the dynamic event-triggered mechanism (20)–(22). Then all closed-loop trajectories are bounded, and the sliding variable $s\left(t\right)$ converges to zero.

Proof. 

Consider the composite Lyapunov candidate

$$V_2=\frac{1}{2}{s}^2+\frac{1}{2{\gamma }_b}{\tilde{u}}_b^2+\chi \left(t\right),$$

(27)

Differentiating (27) along the trajectories of the closed-loop system yields

$${\dot{V}}_2=s\dot{s}+\frac{1}{{\gamma }_b}{\tilde{u}}_b{\dot{\tilde{u}}}_b+\dot{\chi }.$$

(28)

With $u_{si}\left(t\right)=u_s\left(t_i\right)=u_s\left(t\right)+e_u\left(t\right)$ applied to the plant, the dynamics of s can be expressed as

$$\dot{s}=-k_{2}s-k_1\mathrm{sign}\left(s\right)-L{\tilde{u}}_b+L{e}_u,$$

(29)

From the adaptive update law (14), we obtain

$${\dot{\tilde{u}}}_b=-{\dot{\widehat{u}}}_b={\gamma }_{b}Ls.$$

(30)

Substituting (29) and (30) into (28) gives

$${\dot{V}}_2=-k_2{s}^2-k_1\left|s\right|+L{e}_{u}s+\dot{\chi }.$$

(31)

By applying the inequality $L{e}_{u}s\le \frac{L}{2}|e_u{|}^2+\frac{L}{2}{\left|s\right|}^2$, we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_2\le & -(k_2-\frac{L}{2}){\left|s\right|}^2-k_1\left|s\right|+\frac{L}{2}{\left|e_u\right|}^2-{\kappa }_2\chi \left(t\right)\hfill \\ & +(1-{\eta }_d^2)\left((k_2-\frac{L}{2}){\left|s\right|}^2+k_1\left|s\right|\right)-\frac{L}{2}{\left|e_u\right|}^2.\hfill \end{array}$$

(32)

Finally, substituting the dynamics of $\chi \left(t\right)$ from (22) into (32), we obtain

$${\dot{V}}_2\le -{\eta }_d^2\left((k_2-\frac{L}{2}){\left|s\right|}^2+k_1\left|s\right|\right)-{\kappa }_2\chi \left(t\right).$$

(33)

Then, it follows that

$${\dot{V}}_2\le 0\mathrm{if}{k}_2>\frac{L}{2}.$$

Hence, we can conclude that $s\to 0$ as $t\to \infty$, and according to the definition of s in (9), the tracking errors $e_1$ and $e_2$ converge to zero, and that $x\to x_d$ and $\dot{x}\to {\dot{x}}_d$ as $t\to \infty$. The proof is complete. □

Based on the above theoretical results, the overall control architecture can be illustrated in Figure 4.

Remark 1.

In practical implementation, the discontinuous function $\mathrm{sign}(·)$ in (13) is usually replaced by its smooth approximation $tanh(s/{\varphi }_s)$ with ${\varphi }_s>0$ in order to alleviate chattering.

3.3. Exclusion of Zeno Behavior

Theorem 3.

Under the proposed event-triggered mechanism, the inter-event times ${\tau }_i=t_{i+1}-t_i$ satisfy the lower bound

$${\tau }_i\ge {\displaystyle \frac{{\delta }_i}{U}}>0,$$

which guarantees the exclusion of Zeno behavior. Here, ${\delta }_i$ and U are positive constants, with their specific definitions provided in subsequent derivations.

Taking the derivative of $e_u\left(t\right)$, one has

$${\dot{e}}_u\left(t\right)={\dot{u}}_s\left(t_i\right)-{\dot{u}}_s\left(t\right)=-{\dot{u}}_s\left(t\right),t\in [t_i,t_{i+1}).$$

(34)

According to (11)–(14) and Remark 1, the derivative of $u_s\left(t\right)$ is uniformly bounded, and there exists a constant $U>0$ such that

$$|{\dot{u}}_s\left(t\right)|\le U.$$

(35)

Therefore, under (35), it has

$$|e_u\left(t\right)|=\left|{\int }_{t_i}^t{\dot{e}}_u\left(\tau \right)d\tau \right|\le {\int }_{t_i}^t\left|{\dot{u}}_s\left(\tau \right)\right|d\tau \le U(t-t_i).$$

(36)

Accordingly, the next triggering instant occurs when the right-hand side of (36) exceeds the following triggering threshold:

$$e_m\left(t_{j+1}\right)=\sqrt{\frac{2}{L}\left((1-{\eta }_d^2)\left((k_2-\frac{L}{2}){\left|s\right|}^2+k_1\left|s\right|\right)+\frac{\chi \left(t_{j+1}\right)}{{\kappa }_1}\right)}.$$

(37)

Due to $\chi \left(t\right)\ge 0$ guaranteed by Lemma 1, there exists a positive lower bound ${\delta }_i>0$ such that

$$e_m\left(t_{j+1}\right)\ge {\delta }_i>0,\forall t\in [t_i,t_{i+1}).$$

(38)

Combining (36) and (38), every inter-event time ${\tau }_i=t_{i+1}-t_i$ admits the lower bound

$${\tau }_i\ge {\displaystyle \frac{{\delta }_i}{U}}>0.$$

(39)

Therefore, Zeno behavior is rigorously excluded.

4. Experiments

In this section, to validate the effectiveness and superiority of the proposed control scheme, we conduct three comparative experiments: a time-triggered control scheme, a time-triggered control scheme without adaptive parameters, and a dynamic event-triggered control scheme. The system and design parameters are given in

Table 2. Values of plant, controller, and event-triggered parameters.

Parameters	Values	Parameters	Values
m	5	$K_d$	$0.025$
c	200	$K_p$	3
k	$30,000$	$K_i$	25
$k_1$	50	$k_2$	2
$\Delta t$	$0.01\mathrm{s}$	${\eta }_d$	$0.88$
${\varphi }_s$	$0.5$	${\gamma }_b$	60
${\kappa }_1$	4	${\kappa }_2$	1
$K_{\mathrm{gain}}$	$300\mathrm{N}/\mathrm{V}$	$\chi \left(0\right)$	5

, in which the design parameters are selected via trial and error method, consistent with the theoretically derived design ranges.

In this case, the base is fixed, and thus we directly plot the platform displacement. First, the experimental results under the time-triggered control scheme are summarized in Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9. To evaluate the effectiveness and robustness of the proposed method, a load disturbance is applied to the air spring at approximately $t=6\mathrm{s}$, and a open-loop constant feedforward method is included for comparison. Figure 5 depicts the tracking trajectory, while Figure 6 presents the tracking error. Although small tracking errors remain, they are mainly attributable to sensor measurement noise and lie within an acceptable range. Furthermore, it can be seen that after the disturbance is applied and dissipates, the system state can quickly return to the desired value. In contrast, under the open-loop constant-input baseline without sliding mode control, the trajectory fails to converge or maintain the target position, thereby further demonstrating the effectiveness of the proposed scheme. Figure 7 depicts the control input. Figure 8 shows the sliding surface, which converges to a small neighborhood around zero. Finally, Figure 9 presents the adaptive estimate parameter ${\widehat{u}}_b$, whose estimated value is consistent with the actual condition.

For comparison, the results of a time-triggered controller without adaptive parameters are presented in Figure 10, Figure 11 and Figure 12. Specifically, Figure 10 illustrates the tracking trajectory. Figure 11 depicts the corresponding tracking error. Finally, Figure 12 reports the control input. From the comprehensive comparison of Figure 5, Figure 6 and Figure 7 and Figure 10, Figure 11 and Figure 12, it can be observed that the incorporation of adaptive control effectively improves the convergence speed and robustness, while ensuring satisfactory tracking accuracy.

Finally, the results obtained with the dynamic event-triggered control scheme are shown in Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19. Figure 13 shows the system trajectory, and Figure 14 shows the tracking error, which is slightly inferior to the time-triggered method in terms of accuracy. However, as observed from the control input result in Figure 15, dynamic event-triggered method significantly reduces the actuator update frequency, which helps prolong the actuator’s service life. Figure 16 confirms that the sliding surface still converges near zero. Figure 17 displays the adaptive estimation of uncertain parameters.Figure 18 presents the event-triggered condition with a dynamically adjustable threshold. Figure 19 shows the auxiliary dynamic variable, which is always positive. Overall, dynamic event-triggered control method achieves a favorable trade-off by significantly reducing the control-update burden at the cost of a small loss in tracking accuracy.

To further demonstrate the effectiveness and superiority of the dynamic event-triggered mechanism, we compare it with the conventional time-triggered scheme and the static event-triggered scheme. When the auxiliary variable is fixed as $\chi \equiv 0$, the dynamic event-triggered condition reduces to the traditional static form. Figure 20 shows the static event-triggered condition, and the comparison results are summarized in

Table 3. Triggering statistics of different control schemes.

Control Scheme	Number of Triggers	Average Trigger Interval	Trigger Rate
Time-Triggered	2000	0.01 s	100%
Static Event-Triggered	403	0.0496 s	20.15%
Dynamic Event-Triggered	321	0.0623 s	16.05%

. Under the time-triggered control, the number of triggers reaches 2000, with an average trigger interval of 0.01 s and a trigger rate of 100%, since the control input is updated at every sampling instant. In contrast, the static event-triggered control reduces the number of triggers to 403, increases the average trigger interval to 0.0496 s, and decreases the trigger rate to 20.15%, indicating that the triggering condition can effectively suppress unnecessary updates. Furthermore, the dynamic event-triggered control exhibits the most significant improvement, with only 321 triggers, an average trigger interval extended to 0.0623 s, and the trigger rate further reduced to 16.05%. These results clearly verify that the dynamic event-triggered mechanism can substantially reduce the actuator update frequency while maintaining system performance, thereby lowering energy consumption and prolonging actuator lifespan.

5. Conclusions

In this paper, an adaptive dynamic event-triggered sliding mode control scheme was proposed for a pneumatic vibration isolation platform. The stability of the closed-loop system was demonstrated through Lyapunov theory, and the effectiveness of the control scheme was experimentally validated on the platform. The results confirmed that the proposed approach achieves accurate trajectory tracking while significantly reducing actuator update frequency compared with conventional time-triggered control. Future work will focus on exploring other control schemes for the pneumatic vibration isolation system to further enhance performance.