Observer-Based Disturbance Rejection Control for Switched Nonlinear Networked Systems under Event-Triggered Scheme

Abstract

This paper employs the disturbance rejection technique for a class of switched nonlinear networked control systems (SNNCSs) with an observer-based event-triggered scheme. To estimate the influence of exogenous disturbances on the proposed system, the equivalent input disturbance (EID) technique is employed to construct an EID estimator. To provide adequate disturbance rejection performance, a new control law is built that includes the EID estimation. Furthermore, to preserve communication resources, an event-based mechanism for control signal transmission is devised and implemented. The primary goal of this work is to provide an observer-based event-triggered disturbance rejection controller that ensures the resulting closed-loop form of the examined systems is exponentially stable. Specifically, by employing a Lyapunov–Krasovskii approach, a new set of sufficient conditions in the form of linear matrix inequalities (LMIs) is derived, ensuring the exponential stabilization criteria are met. Eventually, a numerical example is used to demonstrate the efficacy and practicality of the proposed control mechanism.

1. Introduction

The switched system is a very flexible modeling tool consisting of a family of distinct subsystems, operated according to a specific switching rule, that determines which subsystem is active at any given time [1, 2]. Switched systems are employed in a variety of real-world settings, including power systems, networked control systems, communication networks, electrical devices and circuits, underwater vehicle systems, manipulator robots, and many more areas [3, 4]. In recent years, a considerable amount of research has been conducted on the stability and stabilization of switched systems using Lyapunov techniques. For instance, the existence of a common Lyapunov function for individual systems that ensures stability of the switched system in the context of arbitrary switching sequence, see [5, 6]. The authors in [7] discussed the stability and stabilization problem of switched linear systems along with average dwell time approach, where as in [4], the issue of switched nonlinear systems under multiple time delays is discussed.

On the other hand, networked control systems (NCSs) have received considerable attention due to their advantages over traditional point-to-point wired control techniques, including convenience extension, information sharing, affordability, ease of installation and maintenance, and high reliability [8]. As a result, they have recently been widely used in many different fields, such as robotic manipulators, vehicle highway systems, spacecraft systems, teleoperation systems, smart grids, and other areas. It is common for NCSs to use wired and wireless communications networks to link the various components of the system, including sensors, controllers, and actuators, which are situated in separate locations, in order to form a feedback control system [5, 9]. During the past few years, switched NCS’s have received a great deal of attention from researchers [10, 11] due to the fact that these kind of systems can well describe numerous practical systems with abrupt parameter variations. However, the communication network introduces some new difficulties to the controller design of switched NCSs [12].

Furthermore, some interesting issues may arise due to the characteristics of the network, such as congestion on the network, data loss, and disorder, and delay due to the network. In order to reduce the network resources, the time-triggered technique has been established in several versions. Over a specified time period, every data is sampled and transmitted, whether required or not. In the case that if the difference between the latest transmitted signal and the current sampling signal is relatively small, then a significant amount of the same data will be transmitted in the network, which will cause unnecessary traffic on the network and unwanted communication resources. To tackle these issues, quite of few researchers devote themselves in developing extremely effective data transmission techniques, for example, [13, 14]. Motivated by [5], an event-triggered technique has been developed to further reduce the limited bandwidth and enhance the usage of communication resources. The authors in [10] propose an event-triggered $H_{\infty }$ communication control problem for switched networked systems with the assistance of the LKF theory. Further, the authors in [9] discussed an event-triggered $H_{\infty }$ control strategy for NCSs under communication delay. Under an adaptive event-triggered communication scheme framework, an $H_{\infty }$ tracking control problem for nonlinear networked systems is derived with limited network communication [15]. In particular, the distributed $H_{\infty }$ control problem for switched networked linear control system is discussed in [16] subject to quantization and packet dropouts. In [3], an observer-based controller design has been developed for the switched networked nonlinear systems with packet dropouts and average dwell-time mechanisms.

As a result, it is more important from the perspective of the control system to evaluate exogenous disturbances affecting the control input channel since they are a source of not only the collapse of the system’s control performance but also machine malfunctions, vibrations, and other issues. In order to achieve good disturbance rejection performance, the EID method, such as the one described in [17, 18], is an active disturbance rejection technique with two degrees of freedom, which actively processes disturbance information and constructs compensation signals for them. The main advantage of employing the EID technique is that no restrictions are imposed or demand any information concerning disturbances [2]. As a general rule, EID is capable of rejecting and estimating both matched and mismatched external disturbances, and its configuration allows for easy modeling. Several design methodologies have been proposed in order to take into account the energetic performance of EID, as recently examined in the literature, see, for example, [19, 20]. Based on the disturbance rejection method, the switched neutral time-delay system is relaxed in [2] via EID approach. The authors in [18] derived an effective disturbance rejection method for time delay system based on the EID approach. She et al. [21] explored the problem of enhancing the disturbance-rejection performance for servo systems, which involves the estimate of an EID and an improved servo system design using the EID technique.

Moreover, it has been reported that innumerable advanced techniques on an event-triggered control of switched networked systems have already been published, but as far as we are aware, this has not yet been explored for an observer-based event-triggered control scheme for SNNCSs, especially for disturbance rejection technique and ADT approach. As stipulated by the author’s perspective, the main intention of this paper is to investigate the EID method to reject disturbances for SNNCSs in the presence of observer-based event-triggered control mechanism. Roughly stated, the novelty of this paper is contributed as follows:

• An unified disturbance rejection and observer-based control design for SNNCSs with external disturbances and an event-triggered scheme is developed in this study.

• One aspect of the proposed event-triggered DRC scheme is that in contrast to the traditional time-triggered-DRC scheme [22, 23], it can significantly reduce communication frequency while achieving satisfactory closed-loop system performance by utilizing a new event-triggered transmission scheme. However, many existing event-triggered approaches without direct disturbance compensation have a tendency to violate the event triggering conditions when disturbances occur [15], while the ET-ADRC can be effectively used to reduce the occurrence times of triggering conditions by utilizing active disturbance compensation.

• The proposed controller includes an EID estimator to efficiently estimate and reduce external disturbances in the system understudy.

• Furthermore, by resorting to a Lyapunov–Krasovskii functional, adequate constraints are attained in the context of LMIs. It follows that the requisite controller gain matrices are determined by solving the established conditions.

Finally, to show the utility of the proposed control scheme, a realistic example is given.

2. Problem description

In this article, we focus on obtaining an observer-based event-triggered control strategy for SNNCSs with disturbance and achieving disturbance rejection. To save the limited network resources, an event-triggered generator is employed.

2.1 Controlled plant

As illustrated in Figure 1, the controlled plant is modeled by the following SNNCSs with external disturbance:

Figure 1.

Controlled plant.

$$\begin{array}{l}\dot{\mathcal{X}}\left(t\right)={\mathcal{A}}_{\sigma \left(t\right)}\mathcal{X}\left(t\right)+{\mathcal{B}}_{\sigma \left(t\right)}\mathcal{U}\left(t\right)+{\mathcal{B}}_{\mathcal{G}\sigma \left(t\right)}\mathcal{G}\left(\mathcal{X}\left(t\right)\right)+{\mathcal{B}}_{\mathcal{W}\sigma \left(t\right)}\mathcal{W}\left(t\right),\\ \mathcal{Y}\left(t\right)={\mathcal{C}}_{\sigma \left(t\right)}\mathcal{X}\left(t\right),\\ \mathcal{Z}\left(t\right)={\mathcal{E}}_{\sigma \left(t\right)}\mathcal{X}\left(t\right)+{\mathcal{F}}_{\sigma \left(t\right)}\mathcal{W}\left(t\right),\end{array}$$E1

where $\mathcal{X}\left(t\right)\in {\mathbb{R}}^{{\mathcal{N}}_{\mathcal{X}}}$ is the state vector; $\mathcal{Y}\left(t\right)\in {\mathbb{R}}^{{\mathcal{N}}_{\mathcal{Y}}}$ and $\mathcal{Z}\left(t\right)\in {\mathbb{R}}^{{\mathcal{N}}_{\mathcal{Z}}}$ are the measured and the controlled outputs of the system; $\mathcal{W}\left(t\right)\in {\mathbb{R}}^{{\mathcal{N}}_{\mathcal{W}}}$ is the disturbance input signal, which belongs to ${\mathcal{L}}_2\left[0,\infty \right)$; $\mathcal{U}\left(t\right)\in {\mathbb{R}}^{{\mathcal{N}}_{\mathcal{U}}}$ is the ideal control signal generated by the controller; $\sigma \left(t\right)$ represents the switching signal that accepts values from a finite collection $\mathcal{I}=\left\{1,2,\cdots ,\mathcal{R}\right\};$ for example, when $\sigma \left(t\right)=i\in \mathcal{I},$ it implies that the $i^{\mathit{th}}$ subsystem has been enabled, where $\mathcal{R}$ denotes the number of subsystems; The matrices ${\mathcal{A}}_i,{\mathcal{B}}_i,{\mathcal{B}}_{\mathcal{G}i},{\mathcal{B}}_{\mathcal{W}i,}{\mathcal{C}}_i,{\mathcal{E}}_i$ and ${\mathcal{F}}_i$, $i=1,\cdots ,\mathcal{R}$, are known real constant matrices with appropriate dimensions; furthermore, $\mathcal{G}\left(\mathcal{X}\right)={\left[{\mathcal{G}}_1\left(\mathcal{X}\right),{\mathcal{G}}_2\left(\mathcal{X}\right),\dots {\mathcal{G}}_n\left(\mathcal{X}\right)\right]}^T$ is a nonlinear vector function, which fulfills the global Lipschitz requirement $‖\mathcal{G}\left(t,\mathcal{X}\right)-\mathcal{G}(t,\widehat{\mathcal{X}})‖\le {\beta }_{\mathcal{G}}‖\mathcal{X}-\widehat{\mathcal{X}}‖,\forall \mathcal{X},\widehat{\mathcal{X}}\in {\mathbb{R}}^{{\mathcal{N}}_{\mathcal{X}}},$ where ${\beta }_{\mathcal{G}}$ is a positive scalar.

In order to use the EID-based technique to analyze SNNCSs, we must first examine the EID’s definition.

Definition 1 Let the input $\mathcal{U}\left(t\right)$ in the SNNCSs plant Eq. (1) be zero. A signal, ${\mathcal{W}}_e\left(t\right),$ on the control input channel is called an EID of the disturbance $\mathcal{W}\left(t\right),$ if it produces the same impact on the output as the disturbance $\mathcal{W}\left(t\right)$ does for all $t\ge 0.$

2.2 Configuration of the EID-based SNNCSs

For the purpose of enhancing disturbance-rejection performance for the SNNCSs, Liu et al. [18] suggest an EID technique. The EID-based control system setup has two degrees of freedom. As a consequence, it makes it possible for this technique to actively examine the information from the disturbance and develop a compensation for it. A disturbance estimator is crucial in this procedure for compensating the disturbances. It has a significant impact on disturbance rejection.

It should be noted that since ${\mathcal{B}}_{\sigma \left(t\right)}$ and ${\mathcal{B}}_{\mathcal{W}\sigma \left(t\right)}$ may have various dimensions, the disturbance may be imposed on a channel other than the control input channel and that the number of disturbances and related input channels may also be greater than one. However, if we suppose that a disturbance is exclusively placed on the control input channel, as illustrated in Figure 2, then the plant in terms of Definition 1 as

Figure 2.

Controlled plant with EID.

$$\dot{\mathcal{X}}\left(t\right)={\mathcal{A}}_{\sigma \left(t\right)}\mathcal{X}\left(t\right)+{\mathcal{B}}_{\mathcal{G}\sigma \left(t\right)}\mathcal{G}\left(\mathcal{X}\left(t\right)\right)+{\mathcal{B}}_{\sigma \left(t\right)}\left(\mathcal{U}\left(t\right)+{\mathcal{W}}_e\left(t\right)\right).$$E2

In practical implementation, it should be noted that the entire states of the physical plant Eq. (2) are not completely measurable. Taking this fact into account, an observer-based controller can be constructed by means of measurement output to estimate the states. The state observer is:

$$\begin{array}{l}\dot{\widehat{\mathcal{X}}}\left(t\right)={\mathcal{A}}_{\sigma \left(t\right)}\widehat{\mathcal{X}}\left(t\right)+{\mathcal{B}}_{\sigma \left(t\right)}{\mathcal{U}}_{\mathcal{F}}\left(t\right)+{\mathcal{B}}_{\mathcal{G}\sigma \left(t\right)}\mathcal{G}\left(\widehat{\mathcal{X}}\left(t\right)\right)+{\mathcal{L}}_{\sigma \left(t\right)}\left(\mathcal{Y}\left(t\right)-\widehat{\mathcal{Y}}\left(t\right)\right),\\ \widehat{\mathcal{Y}}\left(t\right)={\mathcal{C}}_{\sigma \left(t\right)}\widehat{\mathcal{X}}\left(t\right),\end{array}$$E3

where $\widehat{\mathcal{X}}\left(t\right)\in {\mathbb{R}}^{{\mathcal{N}}_{\mathcal{X}}}$ and $\widehat{\mathcal{Y}}\left(t\right)\in {\mathbb{R}}^{{\mathcal{N}}_{\mathcal{Y}}}$ are the reconstructed state and output vectors of $\mathcal{X}\left(t\right)$ and $\mathcal{Y}\left(t\right),$ respectively; ${\mathcal{U}}_{\mathcal{F}}\left(t\right)$ is the control input vector of the system, introduced below; and ${\mathcal{L}}_i,i=1,\cdots ,\mathcal{R},$ are the observer gain matrix to be determined.

The observer-based feedback control law ${\mathcal{U}}_{\mathcal{F}}\left(t\right),$ shown in Figure 3 is designed as follows:

Figure 3.

Controlled plant with EID-based event-triggered mechanism.

$${\mathcal{U}}_{\mathcal{F}}\left(t\right)={\mathcal{K}}_{\sigma \left(t\right)}\widehat{\mathcal{X}}\left(t\right)$$E4

where ${\mathcal{K}}_i,i=1,\cdots ,\mathcal{R}$ denotes the control gain matrices.

It should be emphasized that SNNCS Eq. (1) may contain disturbances, which could result in the system’s poor performance or instability. In order to estimate and reject both matched and unmatched disturbances, the EID technique does not require a priori knowledge of the disturbances. In this SNNCSs, an EID technique is employed in the control channel, which yields satisfactory disturbance rejection performance. According to discussed in [17, 18], the optimal EID estimated disturbance can be represented by

$$\begin{array}{l}\widehat{\mathcal{W}}\left(t\right)={\mathcal{B}}_{\sigma \left(t\right)}^{+}{\mathcal{L}}_{\sigma \left(t\right)}{\mathcal{C}}_{\sigma \left(t\right)}\left(\mathcal{X}\left(t\right)-\widehat{\mathcal{X}}\left(t\right)\right)+{\mathcal{U}}_{\mathcal{F}}\left(t\right)-\mathcal{U}\left(t\right),\\ \text{where}{\mathcal{B}}_{\sigma \left(t\right)}^{+}={\left({\mathcal{B}}_{\sigma \left(t\right)}^T{\mathcal{B}}_{\sigma \left(t\right)}\right)}^{-1}{\mathcal{B}}_{\sigma \left(t\right)}^T.\end{array}$$E5

In order to filter the measurement noise in the estimated disturbance, let $\widehat{\mathcal{W}}\left(t\right)$ pass through a filter represented as

$$\begin{array}{l}{\dot{\mathcal{X}}}_{\mathcal{F}}\left(t\right)={\mathcal{A}}_{\mathcal{F}\sigma \left(t\right)}{\mathcal{X}}_{\mathcal{F}}\left(t\right)+{\mathcal{B}}_{\mathcal{F}\sigma \left(t\right)}\widehat{\mathcal{W}}\left(t\right),\\ \tilde{\mathcal{W}}\left(t\right)={\mathcal{C}}_{\mathcal{F}\sigma \left(t\right)}{\mathcal{X}}_{\mathcal{F}}\left(t\right)\end{array}$$E6

where ${\mathcal{X}}_{\mathcal{F}}\left(t\right)\in {\mathbb{R}}^{{\mathcal{N}}_{{\mathcal{X}}_{\mathcal{F}}}}$ is the state of the filter $\mathcal{F}\left(s\right),$ whereas $\tilde{\mathcal{W}}\left(t\right)$ represents the output of the filter, and ${\mathcal{A}}_{\mathcal{F}\sigma \left(t\right)},{\mathcal{B}}_{\mathcal{F}\sigma \left(t\right)}$ and ${\mathcal{C}}_{\mathcal{F}\sigma \left(t\right)}$ are known parameters. Moreover, the transfer function of the filter $\mathcal{F}\left(s\right)$ satisfies $\left|\mathcal{F}\left(\mathit{j\omega }\right)\right|\approx 1,\forall \omega \in \left[0,{\omega }_r\right],$ where ${\omega }_r$ is the maximum frequency selected for the disturbance estimation [18]. An appropriate filter has a cutoff frequency that is more than ten times greater than ${\omega }_r$.

2.3 Event-triggered mechanism

To decrease the network usage, an event-triggered sampling method is used in place of the traditional periodic sampling technique [1, 11]. In this technique, sensor measurement is only communicated when the previously transmitted value and the value of a specific function of the current sensor measurement reach a threshold value [5]. The event detector considers the following conditions while deciding whether or not to transmit the current signal to the controller:

$$t_{k+1}h=t_{k}h{+}_s^{\min }\left\{\mathit{sh}|{\widehat{e}}^T\left(t_{k}h+\mathit{sh}\right){\mathrm{\Psi }}_{1\sigma \left(t\right)}\widehat{e}\left(t_{k}h+\mathit{sh}\right)>{\varsigma }_{\sigma \left(t\right)}{\widehat{x}}^T\left(t_{k}h\right){\mathrm{\Psi }}_{2\sigma \left(t\right)}\widehat{x}\left(t_{k}h\right)\right\},$$E7

where $0<{\varsigma }_i<1,i=1,2,\cdots ,\mathcal{R}$ are known parameters, ${\mathrm{\Psi }}_{1\sigma \left(t\right)}$ and ${\mathrm{\Psi }}_{2\sigma \left(t\right)}$ are symmetric positive definite matrices to be constructed, $h>0$ is the sampling period, and $\widehat{e}\left(t_{k}h+\mathit{sh}\right)$ is the difference between the two state estimations at the last transmission instant and the present sampling instant, that is, $\widehat{e}\left(t_{k}h+\mathit{sh}\right)=\widehat{\mathcal{X}}\left(t_{k}h\right)-\widehat{\mathcal{X}}\left(t_{k}h+\mathit{sh}\right).$

When the data provided by the event monitor is transferred to the controller at $t_k,$ a communication delay known as the sensor-to-controller delay ${\tau }_{\mathit{sc}}\left(t_k\right)$ is produced. Similarly, the controller’s transmission of actuation signals to the actuator at $t_k$ causes another communication delay known as the controller-to-actuator delay ${\tau }_{\mathit{ca}}\left(t_k\right).$ The sum of these two delays is represented by ${\tau }_{t_k}={\tau }_{\mathit{sc}}\left(t_k\right)+{\tau }_{\mathit{ca}}\left(t_k\right)$ with $0<{\tau }_{t_k}<{\tau }_M,$ where ${\tau }_M$ represents the maximum delay bounds of ${\tau }_{t_k}$ [9]. Finally, the actuator actuates at the time specified by $t_{k}h+{\tau }_{t_k}$. During the network transmission, the input of the controlled plant at the time interval $\left[t_{k}h+{\tau }_{t_k},t_{k+1}h+{\tau }_{t_{k+1}}\right)$ is denoted by

$${\widehat{\mathcal{U}}}_{\mathcal{F}}\left(t\right)={\mathcal{K}}_{\sigma \left(t\right)}\widehat{\mathcal{X}}\left(t_{k}h\right)$$E8

We decompose the holding interval of zero-order-hold (ZOH) $\left[t_{k}h+{\tau }_{t_k},t_{k+1}h+{\tau }_{t_{k+1}}\right)$ into the following subintervals as in [24]: $\left[t_{k}h+{\tau }_{t_k},t_{k+1}h+{\tau }_{t_{k+1}}\right)={\cup }_{s=0}^{t_{k+1}-t_k-1}{T}_s,$ where $T_s=\left[i_{k}h+{\tau }_{i_k},i_{k}h+h+{\tau }_{i_k+1}\right)$ and $i_{k}h=t_{k}h+\mathit{sh},s=0,1,\cdots ,t_{k+1}-t_k-1$ signify the sampling instants that occur between the present sampling instant $t_{k}h$ and the future sampling instant $t_{k+1}h.$ If $s$ takes the value of $t_{k+1}-t_k-1$, then ${\tau }_{i_k+1}={\tau }_{t_k+1}$, otherwise ${\tau }_{i_k}={\tau }_{t_k}$. The network permissible equivalent delay $\tau \left(t\right)$ is now defined as $\tau \left(t\right)=t-i_{k}h,t\in T_s$_, then we have $0\le \tau \left(t\right)\le {\tau }_M.$

2.4 Analysis and design of the closed-loop SNNCSs

Eventually, the filter estimated disturbance $\tilde{\mathcal{W}}\left(t\right)$ together with observer-based control law ${\mathcal{U}}_{\mathcal{F}}\left(t\right)$ yields a new controller, which is given by

$$\mathcal{U}\left(t\right)={\widehat{\mathcal{U}}}_{\mathcal{F}}\left(t\right)-\tilde{\mathcal{W}}\left(t\right)={\mathcal{K}}_{\sigma \left(t\right)}\widehat{\mathcal{X}}\left(t_{k}h\right)-\tilde{\mathcal{W}}\left(t\right)$$E9

The system internal stability condition does not depend on exogenous signals, hence the exogenous signals $\mathcal{W}\left(t\right)$ assumed to be zero when analyzing the internal stability. Furthermore, we define the error between the state Eq. (1) and the observer Eq. (3) with the configuration $\mathrm{\Delta }\mathcal{X}\left(t\right)=\mathcal{X}\left(t\right)-\widehat{\mathcal{X}}\left(t\right).$ We use the states of the control system $\left(\widehat{\mathcal{X}}\left(t\right),\mathrm{\Delta }\mathcal{X}\left(t\right),{\mathcal{X}}_{\mathcal{F}}\left(t\right)\right)$ to define $\phi \left(t\right)={\left[\widehat{\mathcal{X}}\left(t\right)\mathrm{\Delta }\mathcal{X}\left(t\right){\mathcal{X}}_{\mathcal{F}}\left(t\right)\right]}^T$ and use it to describe the closed-loop system. Since

$$\begin{array}{l}\dot{\widehat{\mathcal{X}}}\left(t\right)={\mathcal{A}}_{\sigma \left(t\right)}\widehat{\mathcal{X}}\left(t\right)+{\mathcal{B}}_{\sigma \left(t\right)}{\mathcal{K}}_{\sigma \left(t\right)}\widehat{\mathcal{X}}\left(t-\tau \left(t\right)\right)+{\mathcal{B}}_{\sigma \left(t\right)}{\mathcal{K}}_{\sigma \left(t\right)}\widehat{e}\left(i_{k}h\right)\\ +{\mathcal{B}}_{\mathcal{G}\sigma \left(t\right)}\mathcal{G}\left(\widehat{\mathcal{X}}\left(t\right)\right)+{\mathcal{L}}_{\sigma \left(t\right)}{\mathcal{C}}_{\sigma \left(t\right)}\mathrm{\Delta }\mathcal{X}\left(t\right),\\ \mathrm{\Delta }\dot{\mathcal{X}}\left(t\right)=\left({\mathcal{A}}_{\sigma \left(t\right)}-{\mathcal{L}}_{\sigma \left(t\right)}{\mathcal{C}}_{\sigma \left(t\right)}\right)\mathrm{\Delta }\mathcal{X}\left(t\right)+{\mathcal{B}}_{\mathcal{G}\sigma \left(t\right)}\mathfrak{G}\left(t\right)-{\mathcal{B}}_{\sigma \left(t\right)}{\mathcal{C}}_{\mathcal{F}\sigma \left(t\right)}{\mathcal{X}}_{\mathcal{F}}\left(t\right),\\ {\dot{\mathcal{X}}}_{\mathcal{F}}\left(t\right)=\left({\mathcal{A}}_{\mathcal{F}\sigma \left(t\right)}+{\mathcal{B}}_{\mathcal{F}\sigma \left(t\right)}{\mathcal{C}}_{\mathcal{F}\sigma \left(t\right)}\right){\mathcal{X}}_{\mathcal{F}}\left(t\right)+{\mathcal{B}}_{\mathcal{F}\sigma \left(t\right)}{\mathcal{B}}_{\sigma \left(t\right)}^{+}{\mathcal{L}}_{\sigma \left(t\right)}{\mathcal{C}}_{\sigma \left(t\right)}\mathrm{\Delta }\mathcal{X}\left(t\right).\end{array}$$E10

where $\mathfrak{G}\left(t\right)=\mathcal{G}\left(\mathcal{X}\left(t\right)\right)-\mathcal{G}\left(\widehat{\mathcal{X}}\left(t\right)\right)$ and the state-space representation of the closed-loop system is

$$\dot{\phi }\left(t\right)={\mathcal{A}}_{\sigma \left(t\right)}\phi \left(t\right)+{\mathcal{B}}_{1\sigma \left(t\right)}\phi \left(t-\tau \left(t\right)\right)+{\mathcal{B}}_{2\sigma \left(t\right)}\widehat{e}\left(i_{k}h\right)+{\mathcal{B}}_{3\sigma \left(t\right)}\phi \left(\mathcal{G}\left(\mathcal{X}\left(t\right)\right)\right)$$E11

and the system matrices are given as

$$\begin{array}{l}{\mathcal{A}}_{\sigma \left(t\right)}=\left[\begin{array}{ccc}{\mathcal{A}}_{\sigma \left(t\right)}& {\mathcal{L}}_{\sigma \left(t\right)}{\mathcal{C}}_{\sigma \left(t\right)}& 0\\ 0& {\mathcal{A}}_{\sigma \left(t\right)}-{\mathcal{L}}_{\sigma \left(t\right)}{\mathcal{C}}_{\sigma \left(t\right)}& -{\mathcal{B}}_{\sigma \left(t\right)}{\mathcal{C}}_{\mathcal{F}\sigma \left(t\right)}\\ 0& {\mathcal{B}}_{\mathcal{F}\sigma \left(t\right)}{\mathcal{B}}_{\sigma \left(t\right)}^{+}{\mathcal{L}}_{\sigma \left(t\right)}{\mathcal{C}}_{\sigma \left(t\right)}& {\mathcal{A}}_{\mathcal{F}\sigma \left(t\right)}+{\mathcal{B}}_{\mathcal{F}\sigma \left(t\right)}{\mathcal{C}}_{\mathcal{F}\sigma \left(t\right)}\end{array}\right],\end{array}$$

$$\begin{array}{c}{\mathcal{B}}_{1\sigma \left(t\right)}=\left[\begin{array}{ccc}{\mathcal{B}}_{\sigma \left(t\right)}{\mathcal{K}}_{\sigma \left(t\right)}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],{\mathcal{B}}_{2\sigma \left(t\right)}=\left[\begin{array}{c}{\mathcal{B}}_{\sigma \left(t\right)}{\mathcal{K}}_{\sigma \left(t\right)}\\ 0\\ 0\end{array}\right],\\ {\mathcal{B}}_{3\sigma \left(t\right)}=\left[\begin{array}{ccc}{\mathcal{B}}_{\mathcal{G}\sigma \left(t\right)}& 0& 0\\ 0& {\mathcal{B}}_{\mathcal{G}\sigma \left(t\right)}& 0\\ 0& 0& 0\end{array}\right],\phi \left(\mathcal{G}\left(\mathcal{X}\left(t\right)\right)\right)=\left[\begin{array}{c}\mathcal{G}\left(\mathcal{X}\left(t\right)\right)\\ \mathfrak{G}\left(t\right)\\ 0\end{array}\right].\end{array}$$

Moreover, the upcoming definitions and lemma are more significant for substantiating the required results in the forthcoming section.

Definition 2 [7] For any ${\mathcal{T}}_2>{\mathcal{T}}_1>0,$ let ${\mathcal{N}}_{\sigma }\left({\mathcal{T}}_1,{\mathcal{T}}_2\right)$ denote the switching number of $\sigma \left(t\right)$ on an interval $\left({\mathcal{T}}_1,{\mathcal{T}}_2\right).$ If ${\mathcal{N}}_{\sigma }\left({\mathcal{T}}_1,{\mathcal{T}}_2\right)\le N_0+\frac{{\mathcal{T}}_2-{\mathcal{T}}_1}{{\tau }_a},$ holds for given ${\tau }_a>0$ and $N_0\ge 0,$ then ${\tau }_a$ is the ADT, and ${\mathcal{N}}_0$ be the chatter bound.

Definition 3 [7] The augmented error system Eq. (11) is said to be exponentially stable under the switching signal $\sigma \left(t\right),$ if there exist two constants $\mathcal{M}>0$ and $\aleph >0$ such that the following inequality holds:

$$‖\phi \left(t\right)‖\le \mathcal{M}{e}^{-\aleph \left(t-t_0\right)}{‖\varphi ‖}_L,t\ge t_0,$$E12

${‖\varphi ‖}_L={\mathit{sup}}_{0\le \theta \le t_0}‖\phi \left(\theta \right)‖$ and $\aleph$ is the decay rate.

Lemma 1 [25] Assume $\tau \left(t\right)\in \left[0,{\tau }_M\right],$ for any matrices ${\mathcal{R}}_i\in {\mathbb{R}}^{n\times n}$ and ${\mathcal{M}}_i\in {\mathbb{R}}^{n\times n}$ that satisfy $\left[\begin{array}{ll}{\mathcal{R}}_i& {\mathcal{M}}_i\\ {\mathcal{M}}_i^T& {\mathcal{R}}_i\end{array}\right]\ge 0,$ the following inequality holds:

$$-{\tau }_M{\int }_{t-{\tau }_M}^t{\dot{x}}^T\left(s\right){\mathcal{R}}_i\dot{x}\left(s\right)\mathit{ds}\le {\zeta }_1^T\left(t\right){\mathrm{\Gamma }}_i\zeta \left(t\right)$$E13

where

$${\zeta }_1\left(t\right)=\left[\begin{array}{c}x\left(t\right)\\ x\left(t-\tau \left(t\right)\right)\\ x\left(t-{\tau }_M\right)\end{array}\right],{\mathrm{\Gamma }}_i=\left[\begin{array}{ccc}-{\mathcal{R}}_i& \ast & \ast \\ {\mathcal{R}}_i^T-{\mathcal{M}}_i^T& -2{\mathcal{R}}_i+{\mathcal{M}}_i+{\mathcal{M}}_i^T& \ast \\ {\mathcal{M}}_i^T& {\mathcal{R}}_i^T-{\mathcal{M}}_i^T& -{\mathcal{R}}_i\end{array}\right].$$

3. Main results

The prime intention of this section is to design the observer-based event-triggered scheme for SNNCSs by dint of EID technique. More precisely, based on the appropriate Lyapunov stability theory, we establish a new set of sufficient criterion for the existence of observer based disturbance rejection event-triggered control designs that can be expressed in terms of LMIs, which makes the augmented error system Eq. (11) is exponential stable. For analytical convenience, we define the positive definite matrices as follows: ${\mathcal{P}}_i=diag\left\{{\mathcal{P}}_{1i},{\mathcal{P}}_{2i},{\mathcal{P}}_{3i}\right\},{\mathcal{Q}}_i=diag\left\{{\mathcal{Q}}_{1i},{\mathcal{Q}}_{2i},{\mathcal{Q}}_{3i}\right\},{\mathcal{R}}_i=diag\left\{{\mathcal{R}}_{1i},{\mathcal{R}}_{2i},{\mathcal{R}}_{3i}\right\}.$

3.1 Average dwell-time analysis

In this subsection, we derive sufficient conditions for the exponential stability of the closed-loop SNNCS Eq. (11) along with observer-based disturbance rejection event-triggered scheme by using the average dwell time technique.

Theorem 3.1 For given trigger parameter ${\varsigma }_i,$ nonnegative real scalar ${\tau }_M,$ and parameters $\alpha ,\mu ,$ the controller gain matrices ${\mathcal{K}}_i,{\mathcal{L}}_i,$ the closed-loop SNNCS Eq. (11) with event-triggered sampling scheme is exponentially stable if there exist matrices ${\mathcal{P}}_i,{\mathcal{Q}}_i,{\mathcal{R}}_i,{\mathrm{\Psi }}_{1i},{\mathrm{\Psi }}_{2i}$ and scalars ${\beta }_{\mathcal{G}},{\beta }_{\mathcal{H}}$ such that the following LMIs hold for all $i\in \mathcal{I}:$

$${\mathrm{\Theta }}_i=\left[\begin{array}{cccc}{\mathrm{\Theta }}_{i\mathrm{,1,1}}& {\mathrm{\Theta }}_{i\mathrm{,1,2}}& \cdots & {\mathrm{\Theta }}_{i\mathrm{,1,17}}\\ {\mathrm{\Theta }}_{i\mathrm{,2,1}}& {\mathrm{\Theta }}_{i\mathrm{,2,2}}& \cdots & {\mathrm{\Theta }}_{i\mathrm{,2,17}}\\ ⋮& ⋮& & ⋮\\ {\mathrm{\Theta }}_{i\mathrm{,17,1}}& {\mathrm{\Theta }}_{i\mathrm{,17,2}}& \cdots & {\mathrm{\Theta }}_{i\mathrm{,17,17}}\end{array}\right]<0,$$E14

where,

$$\begin{array}{l}{\mathrm{\Theta }}_{i\mathrm{,1,1}}=\alpha {\mathcal{P}}_{1i}+{\mathcal{Q}}_{1i}-e^{-{\mathit{\alpha \tau }}_M}{\mathcal{R}}_{1i},{\mathrm{\Theta }}_{i\mathrm{,2,2}}=\alpha {\mathcal{P}}_{2i}+{\mathcal{Q}}_{2i}-e^{-{\mathit{\alpha \tau }}_M}{\mathcal{R}}_{2i},\\ {\mathrm{\Theta }}_{i\mathrm{,3,3}}=\alpha {\mathcal{P}}_{3i}+{\mathcal{Q}}_{3i}-e^{-{\mathit{\alpha \tau }}_M}{\mathcal{R}}_{3i},{\mathrm{\Theta }}_{i\mathrm{,4,1}}=e^{-{\mathit{\alpha \tau }}_M}\left({\mathcal{R}}_{1i}-{\mathcal{M}}_{1i}\right),\\ {\mathrm{\Theta }}_{i\mathrm{,4,4}}={\varsigma }_i{\mathrm{\Psi }}_{2i}+e^{-{\mathit{\alpha \tau }}_M}\left(-2{\mathcal{R}}_{1i}+{\mathcal{M}}_{1i}++{\mathcal{M}}_{1i}^T\right),{\mathrm{\Theta }}_{i\mathrm{,5,2}}=e^{-{\mathit{\alpha \tau }}_M}\left({\mathcal{R}}_{2i}-{\mathcal{M}}_{2i}\right),\\ {\mathrm{\Theta }}_{i\mathrm{,5,5}}=e^{-{\mathit{\alpha \tau }}_M}\left(-2{\mathcal{R}}_{2i}+{\mathcal{M}}_{2i}+{\mathcal{M}}_{2i}^T\right),{\mathrm{\Theta }}_{i\mathrm{,6,3}}=e^{-{\mathit{\alpha \tau }}_M}\left({\mathcal{R}}_{3i}-{\mathcal{M}}_{3i}\right),\\ {\mathrm{\Theta }}_{i\mathrm{,6,6}}=e^{-{\mathit{\alpha \tau }}_M}\left(-2{\mathcal{R}}_{3i}+{\mathcal{M}}_{3i}+{\mathcal{M}}_{3i}^T\right),{\mathrm{\Theta }}_{i\mathrm{,7,1}}=e^{-{\mathit{\alpha \tau }}_M}{\mathcal{M}}_{1i},\\ {\mathrm{\Theta }}_{i\mathrm{,7,4}}=e^{-{\mathit{\alpha \tau }}_M}\left({\mathcal{R}}_{1i}-{\mathcal{M}}_{1i}\right),{\mathrm{\Theta }}_{i\mathrm{,7,7}}=-e^{-{\mathit{\alpha \tau }}_M}{\mathcal{Q}}_{1i}-e^{-{\mathit{\alpha \tau }}_M}{\mathcal{R}}_{1i},\\ {\mathrm{\Theta }}_{i\mathrm{,8,2}}=e^{-{\mathit{\alpha \tau }}_M}{\mathcal{M}}_{2i},{\mathrm{\Theta }}_{i\mathrm{,8,5}}=e^{-{\mathit{\alpha \tau }}_M}\left({\mathcal{R}}_{2i}-{\mathcal{M}}_{2i}\right),{\mathrm{\Theta }}_{i\mathrm{,8,8}}=-e^{-{\mathit{\alpha \tau }}_M}{\mathcal{Q}}_{2i}-e^{-{\mathit{\alpha \tau }}_M}{\mathcal{R}}_{2i},\\ {\mathrm{\Theta }}_{i\mathrm{,9,3}}=e^{-{\mathit{\alpha \tau }}_M}{\mathcal{M}}_{3i},{\mathrm{\Theta }}_{i\mathrm{,9,6}}=e^{-{\mathit{\alpha \tau }}_M}\left({\mathcal{R}}_{3i}-{\mathcal{M}}_{3i}\right),{\mathrm{\Theta }}_{i\mathrm{,9,9}}=-e^{-{\mathit{\alpha \tau }}_M}{\mathcal{Q}}_{3i}-e^{-{\mathit{\alpha \tau }}_M}{\mathcal{R}}_{3i},\\ {\mathrm{\Theta }}_{i\mathrm{,10,1}}=2{\mathcal{P}}_{1i}^T+2{\mathcal{N}}_{1i}{\mathcal{A}}_i,{\mathrm{\Theta }}_{i\mathrm{,10,2}}=2{\mathcal{N}}_{1i}{\mathcal{L}}_i{\mathcal{C}}_i,{\mathrm{\Theta }}_{i\mathrm{,10,4}}=2{\mathcal{N}}_{1i}{\mathcal{B}}_i{\mathcal{K}}_i,\\ {\mathrm{\Theta }}_{i\mathrm{,10,10}}={\tau }_M^2{\mathcal{R}}_{1i}-2{\mathcal{N}}_{1i},{\mathrm{\Theta }}_{i\mathrm{,11,2}}=2{\mathcal{P}}_{2i}^T+2{\mathcal{N}}_{2i}{\mathcal{A}}_i-2{\mathcal{N}}_{2i}{\mathcal{L}}_i{\mathcal{C}}_i,\\ {\mathrm{\Theta }}_{i\mathrm{,11,3}}=-2{\mathcal{N}}_{2i}{\mathcal{B}}_i{\mathcal{C}}_{\mathcal{F}i},{\mathrm{\Theta }}_{i\mathrm{,11,11}}={\tau }_M^2{\mathcal{R}}_{2i}-2{\mathcal{N}}_{2i},{\mathrm{\Theta }}_{i\mathrm{,12,2}}=2{\mathcal{N}}_{3i}{\mathcal{B}}_{\mathcal{F}i}{\mathcal{B}}_i^{+}{\mathcal{L}}_i{\mathcal{C}}_i,\\ {\mathrm{\Theta }}_{i\mathrm{,12,3}}=2{\mathcal{P}}_{3i}^T+2{\mathcal{N}}_{3i}{\mathcal{A}}_{\mathcal{F}i}+2{\mathcal{N}}_{3i}{\mathcal{B}}_{\mathcal{F}i}{\mathcal{C}}_{\mathcal{F}i},{\mathrm{\Theta }}_{i\mathrm{,12,12}}={\tau }_M^2{\mathcal{R}}_{3i}-2{\mathcal{N}}_{3i},\\ {\mathrm{\Theta }}_{i\mathrm{,13,10}}=2{\mathcal{B}}_{\mathcal{F}i}^T{\mathcal{N}}_{1i}^T,{\mathrm{\Theta }}_{i\mathrm{,13,13}}=-{\beta }_{\mathcal{G}},{\mathrm{\Theta }}_{i\mathrm{,14,11}}=2{\mathcal{B}}_{\mathcal{F}i}^T{\mathcal{N}}_{2i}^T,{\mathrm{\Theta }}_{i\mathrm{,14,14}}=-{\beta }_{\mathcal{H}},\\ {\mathrm{\Theta }}_{i\mathrm{,15,10}}=2{\mathcal{K}}_i^T{\mathcal{B}}_i^T{\mathcal{N}}_{1i}^T,{\mathrm{\Theta }}_{i\mathrm{,15,15}}=-{\mathrm{\Psi }}_{1i},{\mathrm{\Theta }}_{i\mathrm{,16,1}}={\mathcal{G}}^T,{\mathrm{\Theta }}_{i\mathrm{,16,16}}=-{\beta }_{\mathcal{G}},\\ {\mathrm{\Theta }}_{i\mathrm{,17,2}}={\mathcal{G}}^T,{\mathrm{\Theta }}_{i\mathrm{,17,17}}=-{\beta }_{\mathcal{H}},\end{array}$$

and the remaining terms are zero. Then the average dwell-time scheme

$${\tau }_a>{\tau }_a^{\ast }=\frac{ln\mu }{\alpha },$$E15

where $\mu \ge 1$ satisfies (for all $i,j\in \mathcal{I}$)

$${\mathcal{P}}_i\le \mu {\mathcal{P}}_j,{\mathcal{Q}}_i\le \mu {\mathcal{Q}}_j,{\mathcal{R}}_i\le \mu {\mathcal{R}}_j.$$E16

Moreover, the estimation of state decay is given by

$$‖\phi \left(t\right)‖\le \sqrt{\frac{{\beta }_2}{{\beta }_1}}{e}^{-\aleph \left(t-t_0\right)}{‖\phi \left(t_0\right)‖}_L,\sqrt{\frac{{\beta }_2}{{\beta }_1}}\ge 1$$E17

where

$$\begin{array}{l}\aleph =\frac{1}{2}\left(\alpha -\frac{ln\mu }{{\tau }_a}\right),{\beta }_1{=}_{i\in \mathcal{I}}^{\min }{\lambda }_{\max }\left({\mathcal{P}}_i\right),\\ {\beta }_2{=}_{i\in \mathcal{I}}^{\max }{\lambda }_{\max }\left({\mathcal{P}}_i\right)+{\tau }_M{e^{-{\mathit{\alpha \tau }}_M}}_{i\in \mathcal{I}}^{\max }{\lambda }_{\max }\left({\mathcal{Q}}_i\right)+\left({\tau }_M^2/2\right){e^{-{\mathit{\alpha \tau }}_M}}_{i\in \mathcal{I}}^{\max }{\lambda }_{\max }\left({\mathcal{R}}_i\right).\end{array}$$

Proof: To achieve the desired result, the LKF for the closed-loop SNNCS Eq. (11) is constructed in the following form:

$$\begin{array}{l}{\mathcal{V}}_i\left(\phi \left(t\right)\right)={\phi }^T\left(t\right){\mathcal{P}}_i\phi \left(t\right)+{\int }_{t-{\tau }_M}^t{e}^{\alpha \left(s-t\right)}{\phi }^T\left(s\right){\mathcal{Q}}_i\phi \left(s\right)\mathit{ds}\\ +{\tau }_M{\int }_{t-{\tau }_M}^t{\int }_{\theta }^t{e}^{\alpha \left(s-t\right)}{\dot{\phi }}^T\left(s\right){\mathcal{R}}_i\dot{\phi }\left(s\right)\mathit{\text{dsdθ}}.\end{array}$$E18

Then, computing the time derivatives of $V_i\left(\phi \left(t\right)\right)$ along the trajectories of SNNCS Eq. (11), we can get

$$\begin{array}{l}{\dot{\mathcal{V}}}_i\left(\phi \left(t\right)\right)+\alpha {\mathcal{V}}_i\left(\phi \left(t\right)\right)=2{\phi }^T\left(t\right){\mathcal{P}}_i\dot{\phi }\left(t\right)+{\phi }^T\left(t\right)\left(\alpha {\mathcal{P}}_i+{\mathcal{Q}}_i\right)\phi \left(t\right)\\ -e^{-{\mathit{\alpha \tau }}_M}{\phi }^T\left(t-{\tau }_M\right){\mathcal{Q}}_i\phi \left(t-{\tau }_M\right)+{\tau }_M^2{\dot{\phi }}^T\left(t\right){\mathcal{R}}_i\dot{\phi }\left(t\right)\\ -{\tau }_M{e}^{-{\mathit{\alpha \tau }}_M}{\int }_{t-{\tau }_M}^t{\dot{\phi }}^T\left(s\right){\mathcal{R}}_i\dot{\phi }\left(s\right)\mathit{ds}.\end{array}$$E19

According to Lemma 1 for matrices ${\mathcal{M}}_{\mathit{ia}}\left(a=\mathrm{1,2,3}\right),$ the integral terms in Eq. (19) can be expressed as

$$-{\tau }_M{e}^{-{\mathit{\alpha \tau }}_M}{\int }_{t-{\tau }_M}^t{\dot{\phi }}^T\left(s\right){\mathcal{R}}_i\dot{\phi }\left(s\right)\mathit{ds}\le e^{-{\mathit{\alpha \tau }}_M}{\zeta }_a^T\left(t\right){\mathrm{\Gamma }}_a{\zeta }_a\left(t\right),\left(a=\mathrm{1,2,3}\right)$$E20

where

$$\begin{array}{c}{\zeta }_1\left(t\right)=\left[\begin{array}{c}x\left(t\right)\\ x\left(t-\tau \left(t\right)\right)\\ x\left(t-{\tau }_M\right)\end{array}\right],{\zeta }_2\left(t\right)=\left[\begin{array}{c}\mathrm{\Delta }\left(x\left(t\right)\right)\\ \mathrm{\Delta }\left(x\left(t-\tau \left(t\right)\right)\right)\\ \mathrm{\Delta }\left(x\left(t-{\tau }_M\right)\right)\end{array}\right],{\zeta }_3\left(t\right)=\left[\begin{array}{c}{\mathcal{X}}_{\mathcal{F}}\left(t\right)\\ {\mathcal{X}}_{\mathcal{F}}\left(t-\tau \left(t\right)\right)\\ {\mathcal{X}}_{\mathcal{F}}\left(t-{\tau }_M\right)\end{array}\right],\\ {\mathrm{\Gamma }}_a=\left[\begin{array}{ccc}-{\mathcal{R}}_{\mathit{ia}}& \ast & \ast \\ {\mathcal{R}}_{\mathit{ia}}-{\mathcal{M}}_{\mathit{ia}}& -2{\mathcal{R}}_{\mathit{ia}}+{\mathcal{M}}_{\mathit{ia}}+{\mathcal{M}}_{\mathit{ia}}^T& \ast \\ {\mathcal{M}}_{\mathit{ia}}& {\mathcal{R}}_{\mathit{ia}}-{\mathcal{M}}_{\mathit{ia}}& -{\mathcal{R}}_{\mathit{ia}}\end{array}\right].\end{array}$$

Furthermore, it is to be specified that

$${\beta }_{\mathcal{G}}^{-1}{\mathcal{X}}^T\left(t\right){\mathcal{G}}^T\mathcal{GX}\left(t\right)-{\beta }_{\mathcal{G}}{\mathcal{G}}^T\left(\mathcal{X}\left(t\right)\right)\mathcal{G}\left(\mathcal{X}\left(t\right)\right)>0,$$E21

$${\beta }_{\mathcal{H}}^{-1}{\mathrm{\Delta }}^T\left(\mathcal{X}\left(t\right)\right){\mathcal{G}}^T\mathcal{G}\mathrm{\Delta }\left(\mathcal{X}\left(t\right)\right)-{\beta }_{\mathcal{H}}{\mathfrak{G}}^T\left(t\right)\mathfrak{G}\left(t\right)>0.$$E22

On the other hand, for any matrices ${\mathcal{N}}_i$ the following equality holds

$${\dot{\phi }}^T\left(t\right){\mathcal{N}}_i\left[{\mathcal{A}}_i\phi \left(t\right)+{\mathcal{B}}_{1i}\phi \left(t-\tau \left(t\right)\right)+{\mathcal{B}}_{2i}\widehat{e}\left(i_{k}h\right)+{\mathcal{B}}_{3i}\phi \left(\mathcal{G}\left(\mathcal{X}\left(t\right)\right)\right)-\dot{\phi }\left(t\right)\right]=0$$E23

where ${\mathcal{N}}_i={\left[{\mathcal{N}}_{1i}^T{\mathcal{N}}_{2i}^T{\mathcal{N}}_{3i}^T\right]}^T$.

By unifying Eq. (18)-Eq. (23) along with Eq. (7) and applying Schur complement, it can be easy to obtain that

$$\begin{array}{l}{\dot{\mathcal{V}}}_i\left(\phi \left(t\right)\right)+\alpha {\mathcal{V}}_i\left(\phi \left(t\right)\right)-{\widehat{e}}^T\left(i_{k}h\right){\mathrm{\Psi }}_{1i}\widehat{e}\left(i_{k}h\right)+{\varsigma }_i{\widehat{x}}^T\left(t-\tau \left(t\right)\right){\mathrm{\Psi }}_{2i}\widehat{x}\left(t-\tau \left(t\right)\right)\\ ={\mathrm{\Xi }}^T\left(t\right)\mathrm{\Theta \Xi }\left(t\right)\end{array}$$E24

where ${\mathrm{\Xi }}^T\left(t\right)={\left[{\phi }^T\left(t\right){\phi }^T\left(t-\tau \left(t\right)\right){\phi }^T\left(t-{\tau }_M\right){\dot{\phi }}^T\left(t\right){\mathcal{G}}^T\left(\mathcal{X}\left(t\right)\right){\mathfrak{G}}^T\left(t\right){\widehat{e}}^T\left(i_{k}h\right)\right]}^T$ and the element of $\mathrm{\Theta }$ is detailed in Eq. (14).

Similar to the work in [26] the inequality Eq. (24) can equivalently be rephrased in the following form:

$${\dot{\mathcal{V}}}_i\left(\phi \left(t\right)\right)+\alpha {\mathcal{V}}_i\left(\phi \left(t\right)\right)\le 0.$$E25

Suppose, $t\in \left[t_k,t_{k+1}\right]$ and from Eq. (18), we can get

$${\mathcal{V}}_i\left(\phi \left(t\right)\right)\le e^{-\alpha \left(t-t_k\right)}{\mathcal{V}}_i\left(\phi \left(t_k\right)\right).$$E26

From Eq. (16), at switching instant $t_k,$ we have ${\mathcal{V}}_i\left(\phi \left(t_i\right)\right)\le \mu {\mathcal{V}}_{i\left(t_k^{-}\right)}\left(\phi \left(t_k^{-}\right)\right).$ Then, it follows from Eq. (10) and the relation $\rho ={\mathcal{N}}_{\sigma }\left(t_0,t\right)\le \frac{t-t_0}{{\tau }_a},t_0=0$ that

$$\begin{array}{c}{\mathcal{V}}_i\left(\phi \left(t_i\right)\right)\le e^{-\alpha \left(t-t_k\right)}\mu {\mathcal{V}}_{i\left(t_k^{-}\right)}\left(\phi \left(t_k^{-}\right)\right)\\ \le e^{-\alpha \left(t-t_k\right)}{\mu }^2{\mathcal{V}}_{i\left(t_k^{-}\right)}\left(\phi \left(t_k^{-}\right)\right)\\ \cdots \\ \le e^{-\alpha \left(t-t_k\right)}{\mu }^{\rho }{\mathcal{V}}_{i\left(t_0\right)}\left(\phi \left(t_0\right)\right)\\ \le e^{-\left(\alpha -\frac{\mathit{ln\mu }}{{\tau }_a}\right)\left(t-t_0\right)}{\mathcal{V}}_{i\left(t_0\right)}\left(\phi \left(t_0\right)\right).\end{array}$$E27

Furthermore, from Eq. (18) and Eq. (27), we can get

$$\begin{array}{l}{\beta }_1{‖\phi \left(t\right)‖}^2\le {\mathcal{V}}_i\left(\phi \left(t\right)\right)\le e^{-\left(\alpha -\frac{\mathit{ln\mu }}{{\tau }_a}\right)\left(t-t_0\right)}{\mathcal{V}}_i\left(\phi \left(t_0\right),i\left(t_0\right)\right)\le {\beta }_2{‖\phi \left(t_0\right)‖}_L^2,\\ {‖\phi \left(t\right)‖}^2\le \frac{{\beta }_2}{{\beta }_1}{e}^{-\left(\alpha -\frac{\mathit{ln\mu }}{{\tau }_a}\right)\left(t-t_0\right)}{‖\phi \left(t_0\right)‖}_L^2,\\ ‖\phi \left(t\right)‖\le \sqrt{\frac{{\beta }_2}{{\beta }_1}}{e}^{-\aleph \left(t-t_0\right)}{‖\phi \left(t_0\right)‖}_L.\end{array}$$E28

Hence, by using ${\tau }_a$ one can easily obtain $\aleph <1.$ Then, from Definition 3, it can be concluded that the SNNCS Eq. (11) is exponentially stable. The proof is now completed.

Theorem 3.2 For given trigger parameter ${\varsigma }_i,$ nonnegative real scalar ${\tau }_M,$ parameters $\alpha ,\mu ,$ and a very small $\rho >0,$ the closed-loop SNNCS Eq. (11) with event-triggered sampling scheme is exponentially stabilizable if there exist matrices ${\mathcal{X}}_i,{\tilde{\mathcal{Q}}}_i,{\tilde{\mathcal{R}}}_i,{\tilde{\mathrm{\Psi }}}_{1i},{\tilde{\mathrm{\Psi }}}_{2i}$ appropriate dimensioned matrices ${\mathcal{Y}}_i,{\mathcal{Z}}_i,$ and scalars ${\beta }_{\mathcal{G}},{\beta }_{\mathcal{H}}$ such that the following LMIs hold for all $i\in \mathcal{I}:$

$${\tilde{\mathrm{\Theta }}}_i=\left[\begin{array}{cccc}{\tilde{\mathrm{\Theta }}}_{i\mathrm{,1,1}}& {\tilde{\mathrm{\Theta }}}_{i\mathrm{,1,2}}& \cdots & {\tilde{\mathrm{\Theta }}}_{i\mathrm{,1,17}}\\ {\tilde{\mathrm{\Theta }}}_{i\mathrm{,2,1}}& {\tilde{\mathrm{\Theta }}}_{i\mathrm{,2,2}}& \cdots & {\tilde{\mathrm{\Theta }}}_{i\mathrm{,2,17}}\\ ⋮& ⋮& & ⋮\\ {\tilde{\mathrm{\Theta }}}_{i\mathrm{,17,1}}& {\tilde{\mathrm{\Theta }}}_{i\mathrm{,17,2}}& \cdots & {\tilde{\mathrm{\Theta }}}_{i\mathrm{,17,17}}\end{array}\right]<0,$$E29

$$\left[\begin{array}{cc}-\mathit{\rho I}& {\mathcal{C}}_i^T{\overline{\mathcal{X}}}_{2i}-{\mathcal{X}}_{2i}{\mathcal{C}}_i^T\\ 0& -I\end{array}\right]<0$$E30

where

$$\begin{array}{l}{\tilde{\mathrm{\Theta }}}_{i\mathrm{,1,1}}=\alpha {\mathcal{X}}_{1i}+{\tilde{\mathcal{Q}}}_{1i}-e^{-{\mathit{\alpha \tau }}_M}{\tilde{\mathcal{R}}}_{1i},{\tilde{\mathrm{\Theta }}}_{i\mathrm{,2,2}}=\alpha {\mathcal{P}}_{2i}+{\tilde{\mathcal{Q}}}_{2i}-e^{-{\mathit{\alpha \tau }}_M}{\tilde{\mathcal{R}}}_{2i},\\ {\tilde{\mathrm{\Theta }}}_{i\mathrm{,3,3}}=\alpha {\mathcal{P}}_{3i}+{\tilde{\mathcal{Q}}}_{3i}-e^{-{\mathit{\alpha \tau }}_M}{\tilde{\mathcal{R}}}_{3i},{\tilde{\mathrm{\Theta }}}_{i\mathrm{,4,1}}=e^{-{\mathit{\alpha \tau }}_M}\left({\tilde{\mathcal{R}}}_{1i}-{\tilde{\mathcal{M}}}_{1i}\right),\\ {\tilde{\mathrm{\Theta }}}_{i\mathrm{,4,4}}={\varsigma }_i{\tilde{\mathrm{\Psi }}}_{2i}+e^{-{\mathit{\alpha \tau }}_M}\left(-2{\tilde{\mathcal{R}}}_{1i}+{\tilde{\mathcal{M}}}_{1i}+{\tilde{\mathcal{M}}}_{1i}^T\right),{\tilde{\mathrm{\Theta }}}_{i\mathrm{,5,2}}=e^{-{\mathit{\alpha \tau }}_M}\left({\tilde{\mathcal{R}}}_{2i}-{\tilde{\mathcal{M}}}_{2i}\right),\\ {\tilde{\mathrm{\Theta }}}_{i\mathrm{,5,5}}=e^{-{\mathit{\alpha \tau }}_M}\left(-2{\tilde{\mathcal{R}}}_{2i}+{\tilde{\mathcal{M}}}_{2i}+{\tilde{\mathcal{M}}}_{2i}^T\right),{\tilde{\mathrm{\Theta }}}_{i\mathrm{,6,3}}=e^{-{\mathit{\alpha \tau }}_M}\left({\tilde{\mathcal{R}}}_{3i}-{\tilde{\mathcal{M}}}_{3i}\right),\\ {\tilde{\mathrm{\Theta }}}_{i\mathrm{,6,6}}=e^{-{\mathit{\alpha \tau }}_M}\left(-2{\tilde{\mathcal{R}}}_{3i}+{\tilde{\mathcal{M}}}_{3i}+{\tilde{\mathcal{M}}}_{3i}^T\right),{\tilde{\mathrm{\Theta }}}_{i\mathrm{,7,1}}=e^{-{\mathit{\alpha \tau }}_M}{\tilde{\mathcal{M}}}_{1i},\\ {\tilde{\mathrm{\Theta }}}_{i\mathrm{,7,4}}=e^{-{\mathit{\alpha \tau }}_M}\left({\tilde{\mathcal{R}}}_{1i}-{\tilde{\mathcal{M}}}_{1i}\right),{\tilde{\mathrm{\Theta }}}_{i\mathrm{,7,7}}=-e^{-{\mathit{\alpha \tau }}_M}{\tilde{\mathcal{Q}}}_{1i}-e^{-{\mathit{\alpha \tau }}_M}{\tilde{\mathcal{R}}}_{1i},\\ {\tilde{\mathrm{\Theta }}}_{i\mathrm{,8,2}}=e^{-{\mathit{\alpha \tau }}_M}{\tilde{\mathcal{M}}}_{2i},{\tilde{\mathrm{\Theta }}}_{i\mathrm{,8,5}}=e^{-{\mathit{\alpha \tau }}_M}\left({\tilde{\mathcal{R}}}_{2i}-{\tilde{\mathcal{M}}}_{2i}\right),{\tilde{\mathrm{\Theta }}}_{i\mathrm{,8,8}}=-e^{-{\mathit{\alpha \tau }}_M}{\tilde{\mathcal{Q}}}_{2i}-e^{-{\mathit{\alpha \tau }}_M}{\tilde{\mathcal{R}}}_{2i},\\ {\tilde{\mathrm{\Theta }}}_{i\mathrm{,9,3}}=e^{-{\mathit{\alpha \tau }}_M}{\tilde{\mathcal{M}}}_{3i},{\tilde{\mathrm{\Theta }}}_{i\mathrm{,9,6}}=e^{-{\mathit{\alpha \tau }}_M}\left({\tilde{\mathcal{R}}}_{3i}-{\tilde{\mathcal{M}}}_{3i}\right),{\tilde{\mathrm{\Theta }}}_{i\mathrm{,9,9}}=-e^{-{\mathit{\alpha \tau }}_M}{\tilde{\mathcal{Q}}}_{3i}-e^{-{\mathit{\alpha \tau }}_M}{\tilde{\mathcal{R}}}_{3i},\\ {\tilde{\mathrm{\Theta }}}_{i\mathrm{,10,1}}=2{\mathcal{X}}_{1i}^T+2{\beta }_{1i}{\mathcal{A}}_i{\mathcal{X}}_i,{\tilde{\mathrm{\Theta }}}_{i\mathrm{,10,2}}=2{\beta }_{1i}{\mathcal{C}}_i{\mathcal{Z}}_i,{\tilde{\mathrm{\Theta }}}_{i\mathrm{,10,4}}=2{\beta }_{1i}{\mathcal{B}}_i{\mathcal{Y}}_i,\\ {\tilde{\mathrm{\Theta }}}_{i\mathrm{,10,10}}={\tau }_M^2{\tilde{\mathcal{R}}}_{1i}-2{\beta }_{1i}{\mathcal{X}}_{1i},{\tilde{\mathrm{\Theta }}}_{i\mathrm{,11,2}}=2{\mathcal{X}}_{2i}^T+2{\beta }_{2i}{\mathcal{A}}_i{\mathcal{X}}_{2i}-2{\beta }_{2i}{\mathcal{C}}_i{\mathcal{Z}}_i,\\ {\tilde{\mathrm{\Theta }}}_{i\mathrm{,11,3}}=-2{\beta }_{2i}{\mathcal{B}}_i{\mathcal{C}}_{\mathcal{F}i}{\mathcal{X}}_{3i},{\tilde{\mathrm{\Theta }}}_{i\mathrm{,11,11}}={\tau }_M^2{\tilde{\mathcal{R}}}_{2i}-2{\beta }_{2i}{\mathcal{X}}_{2i},\\ {\tilde{\mathrm{\Theta }}}_{i\mathrm{,12,2}}=2{\beta }_{3i}{\mathcal{B}}_{\mathcal{F}i}{\mathcal{B}}_i^{+}{\mathcal{C}}_i{\mathcal{Z}}_i,{\tilde{\mathrm{\Theta }}}_{i\mathrm{,12,3}}=2{\mathcal{X}}_{3i}^T+2{\beta }_{3i}{\mathcal{A}}_{\mathcal{F}i}{\mathcal{X}}_{3i}+2{\beta }_{3i}{\mathcal{B}}_{\mathcal{F}i}{\mathcal{C}}_{\mathcal{F}i}{\mathcal{X}}_{3i},\\ {\tilde{\mathrm{\Theta }}}_{i\mathrm{,12,12}}={\tau }_M^2{\tilde{\mathcal{R}}}_{3i}-2{\beta }_{3i}{\mathcal{X}}_{3i},{\tilde{\mathrm{\Theta }}}_{i\mathrm{,13,10}}=2{\beta }_{1i}{\mathcal{B}}_{\mathcal{F}i}^T,{\tilde{\mathrm{\Theta }}}_{i\mathrm{,13,13}}=-{\beta }_{\mathcal{G}},{\tilde{\mathrm{\Theta }}}_{i\mathrm{,14,11}}=2{\beta }_{2i}{\mathcal{B}}_{\mathcal{F}i}^T,\\ {\tilde{\mathrm{\Theta }}}_{i\mathrm{,14,14}}=-{\beta }_{\mathcal{H}},{\tilde{\mathrm{\Theta }}}_{i\mathrm{,15,10}}=2{\beta }_{1i}{\mathcal{Y}}_i^T{\mathcal{B}}_i^T,{\tilde{\mathrm{\Theta }}}_{i\mathrm{,15,15}}=-{\tilde{\mathrm{\Psi }}}_{1i},{\tilde{\mathrm{\Theta }}}_{i\mathrm{,16,1}}={\mathcal{G}}^T{\mathcal{X}}_{1i},\\ {\tilde{\mathrm{\Theta }}}_{i\mathrm{,16,16}}=-{\beta }_{\mathcal{G}},{\tilde{\mathrm{\Theta }}}_{i\mathrm{,17,2}}={\mathcal{G}}^T{\mathcal{X}}_{2i},{\tilde{\mathrm{\Theta }}}_{i\mathrm{,17,17}}=-{\beta }_{\mathcal{H}},\end{array}$$

and the remaining terms are zero. If the above LMIs are feasible, the state and observer controller gain matrices are computed by ${\mathcal{K}}_i={\mathcal{Y}}_i{\mathcal{X}}_{1i}^{-1},{\mathcal{L}}_i={\mathcal{Z}}_i{\overline{\mathcal{X}}}_{2i}^{-1}.$ Then the average dwell-time scheme

$${\tau }_a>{\tau }_a^{\ast }=\frac{ln\mu }{\alpha },$$E31

where $\mu \ge 1$ satisfies (for all $i,j\in \mathcal{I}$)

$${\mathcal{X}}_i\le \mu {\mathcal{X}}_j,{\tilde{\mathcal{Q}}}_i\le \mu {\tilde{\mathcal{Q}}}_j,{\tilde{\mathcal{R}}}_i\le \mu {\tilde{\mathcal{R}}}_j.$$E32

Moreover, the estimation of state decay is given by

$$‖\phi \left(t\right)‖\le \sqrt{\frac{{\beta }_2}{{\beta }_1}}{e}^{-\aleph \left(t-t_0\right)}{‖\phi \left(t_0\right)‖}_L,\sqrt{\frac{{\beta }_2}{{\beta }_1}}\ge 1$$E33

where

$$\begin{array}{l}\aleph =\frac{1}{2}\left(\alpha -\frac{ln\mu }{{\tau }_a}\right),{\beta }_1{=}_{i\in \mathcal{I}}^{\min }{\lambda }_{\max }\left({\tilde{\mathcal{X}}}_i\right),\\ {\beta }_2{=}_{i\in \mathcal{I}}^{\max }{\lambda }_{\max }\left({\tilde{\mathcal{X}}}_i\right)+{\tau }_M{e^{-{\mathit{\alpha \tau }}_M}}_{i\in \mathcal{I}}^{\max }{\lambda }_{\max }\left({\tilde{\mathcal{Q}}}_i\right)+\left({\tau }_M^2/2\right){e^{-{\mathit{\alpha \tau }}_M}}_{i\in \mathcal{I}}^{\max }{\lambda }_{\max }\left({\tilde{\mathcal{R}}}_i\right).\end{array}$$

Proof: The proof of this theorem is obtained by following the similar technique together with the same Lyapunov–Krasovskii functional Eq. (18) as in Theorem 3.1. For obtaining the controller gain matrices, let us define ${\mathcal{N}}_i={\beta }_{\mathit{ai}}{\mathcal{P}}_{\mathit{ai}},\left(a=\mathrm{1,2,3}\right),$ here ${\beta }_{\mathit{ai}}$ are the designing parameter. Then pre- and post- multiplying the matrix ${\mathrm{\Theta }}_i$ in Eq. (14) by $\left\{{\mathcal{X}}_i,{\mathcal{X}}_i,{\mathcal{X}}_i,{\mathcal{X}}_i,I,I,{\mathcal{X}}_{1i},I,I\right\},$ and its transpose, respectively, where ${\mathcal{X}}_i=diag\left\{{\mathcal{X}}_{1i},{\mathcal{X}}_{2i},{\mathcal{X}}_{3i}\right\}.$ Note that if (30) holds for a very small $\rho >0$, then ${\mathcal{C}}_i{\mathcal{X}}_{2i}$ almost equals to ${\overline{\mathcal{X}}}_{2i}{\mathcal{C}}_i.$ Further, by setting ${\mathcal{C}}_i{\mathcal{X}}_{2i}={\overline{\mathcal{X}}}_{2i}{\mathcal{C}}_i,{\mathcal{X}}_i={\mathcal{P}}_i^{-1},{\mathcal{X}}_i{\mathcal{Q}}_i{\mathcal{X}}_i={\tilde{\mathcal{Q}}}_i,{\mathcal{X}}_i{\mathcal{R}}_i{\mathcal{X}}_i={\tilde{\mathcal{R}}}_i,{\mathcal{X}}_i{\mathcal{M}}_{\mathit{ai}}{\mathcal{X}}_i={\tilde{\mathcal{M}}}_{\mathit{ai}},\left(a=\mathrm{1,2,3}\right),{\mathcal{X}}_{1i}{\mathrm{\Psi }}_{1i}{\mathcal{X}}_{1i}={\tilde{\mathrm{\Psi }}}_{1i},{\mathcal{X}}_{1i}{\mathrm{\Psi }}_{2i}{\mathcal{X}}_{1i}={\tilde{\mathrm{\Psi }}}_{2i},{\mathcal{Y}}_i={\mathcal{K}}_i{\mathcal{X}}_{1i},{\mathcal{Z}}_i={\mathcal{L}}_i{\overline{\mathcal{X}}}_{2i}.$ The LMI in Eq. (29) can thus be easily obtained. As a consequence, if the LMI-based condition Eq. (29) holds, the closed-loop augmented system Eq. (11) is exponentially stable. This concludes the proof.

Remark 1 It should be noted that the condition ${\mathcal{C}}_i{\mathcal{X}}_{2i}={\overline{\mathcal{X}}}_{2i}{\mathcal{C}}_i$ is not a strict LMI and is difficult to deal with using an existing LMI package, such as Matlab. In order to tackle such complication, let us consider the optimization strategy procedure for ${\mathcal{C}}_i{\mathcal{X}}_{2i}={\overline{\mathcal{X}}}_{2i}{\mathcal{C}}_i,$ it may be appropriately rewritten as ${\left({\mathcal{C}}_i{\mathcal{X}}_{2i}-{\overline{\mathcal{X}}}_{2i}{\mathcal{C}}_i\right)}^T\left({\mathcal{C}}_i{\mathcal{X}}_{2i}-{\overline{\mathcal{X}}}_{2i}{\mathcal{C}}_i\right)-{\rho }^{2}I<0,$ where $\rho >0$ is very small [27]. Hence, the aforementioned optimization issue can be rewritten as Eq. (30), using the Schur Complement.

4. Simulation results

Example 1 The mass-spring-damping system is one of the most common simplified models that are used in mechanical engineering, such as the human exoskeleton back frame model [28] or the bridge dynamics model [29], etc. This is one of the most practical systems in regard to our lives and technology. Therefore, it is extremely beneficial to study the system of mass-spring-damping in more detail.

In this study, we consider the following mass-spring-damping system Eq. (34) as shown in Figure 4a.

Figure 4.

(a) Mass-spring-damping system, (b) Restoring force ${\mathcal{R}}_1$ [30].

$$\mathcal{M}\ddot{x}\left(t\right)+{\mathcal{R}}_1+{\mathcal{R}}_2=\mathcal{U}\left(t\right)$$E34

where $\mathcal{M}$ indicates the mass; ${\mathcal{R}}_1=c\dot{x}$ with $c>0$ and ${\mathcal{R}}_2$ denotes the friction force and the restoring force, respectively; $\mathcal{U}\left(t\right)$ represents for the external input. According to Figure 4b, the restoring force ${\mathcal{R}}_2$ contains a linear component and a hardening spring force. In other words, ${\mathcal{R}}_2=\mathit{kx}+{\mathit{ka}}^2{x}^3$ with constants $k$ and $a,$ where $x$ signifies the displacement from a reference point.

Further, let $x\left(t\right)={\left[x_1\left(t\right)x_2\left(t\right)\right]}^T,$ in the meantime, consider $a\ne 0$ and $a=0$ the nonlinear system Eq. (34) could be described by the following two subsystems as given in [30]:

$${\mathcal{A}}_1=\left[\begin{array}{cc}0& 1\\ \frac{-k-4{\mathit{ka}}^2}{m}& -\frac{c}{m}\end{array}\right],{\mathcal{B}}_1=\left[\begin{array}{c}0\\ \frac{1}{m}\end{array}\right],{\mathcal{A}}_2=\left[\begin{array}{cc}0& 1\\ -\frac{k}{m}& -\frac{c}{m}\end{array}\right],{\mathcal{B}}_2=\left[\begin{array}{c}0\\ \frac{1}{m}\end{array}\right].$$

The following parameter values are employed in this scenario: $m=1\mathrm{kg},c=2\mathrm{N}.\mathrm{m}/\mathrm{s},k=8\mathrm{N}/\mathrm{s}$ and $a=0.3m^{-1}$ [30]. Furthermore, the following system parameters have been listed:

$$\begin{array}{l}{\mathcal{B}}_{\mathcal{G}1}=\left[\begin{array}{cc}0.2974& 0\\ 0& 0.2648\end{array}\right],{\mathcal{B}}_{\mathcal{W}1}=\left[\begin{array}{c}0.0737\\ 0.0637\end{array}\right],{\mathcal{C}}_1={\left[\begin{array}{c}0.1536\\ 0.5438\end{array}\right]}^T,\\ {\mathcal{B}}_{\mathcal{G}2}=\left[\begin{array}{c}0.08750\\ 00.0691\end{array}\right],{\mathcal{B}}_{\mathcal{W}2}=\left[\begin{array}{c}0.1391\\ 0.0139\end{array}\right],{\mathcal{C}}_2={\left[\begin{array}{c}0.4167\\ 0.2345\end{array}\right]}^T.\end{array}$$

Let us consider the remaining parameters in Theorem 3.2 to be $\alpha =0.1,\mu =1.1,{\varsigma }_1=0.5,{\varsigma }_2=0.5,\rho =0.5,{\beta }_{1i}=0.2114,{\beta }_{2i}=0.3835,{\beta }_{3i}=0.1885,{\tau }_M=0.1,$ the nonlinear term defined as $\mathcal{G}\left(\mathcal{X}\left(t\right)\right)=0.1\mathit{tan}\left(\mathcal{X}\left(t\right)\right).$ Moreover, the low-pass filter parameters are determined as follows: $w_c=100,{\mathcal{A}}_{\mathcal{F}i}=-101,{\mathcal{B}}_{\mathcal{F}i}=100,{\mathcal{C}}_{\mathcal{F}i}=1.$ Furthermore, the network-induced exogenous disturbance is addressed by $\mathcal{W}\left(t\right)=0.4sin\left(\mathit{\pi t}/2\right)+0.35tanh\left(t-37\right)-0.35tanh\left(t-27\right).$ The LMI constraints derived in Eq. (29) are then solved using the MATLAB LMI toolbox, and the feasibility with the aforementioned parameter values can be determined. The feedback and observer gain matrices are provided below based on the parameters discussed in this article:

$$\begin{array}{l}{\mathcal{K}}_1=\left[3.6688-0.8039\right],{\mathcal{L}}_1={\left[\begin{array}{cc}-5.7747& 0.0034\end{array}\right]}^T,\\ {\mathcal{K}}_2=\left[\begin{array}{cc}1.4785& -0.4722\end{array}\right],{\mathcal{L}}_2={\left[\begin{array}{cc}-4.2569& -0.0143\end{array}\right]}^T.\end{array}$$

Furthermore, during the simulation process, the initial values for the state and observer systems are determined as follows: $\mathcal{X}\left(0\right)={\left[\begin{array}{cc}0.7& -0.8\end{array}\right]}^T$ and $\widehat{\mathcal{X}}\left(0\right)={\left[\begin{array}{cc}-0.7& 0.8\end{array}\right]}^T.$ The simulation results for the SNNCS Eq. (11) are computed from the aforesaid gain values, and the simulation results are displayed in Figures 5–10.

Figure 5.

Simulation results of state and its observer signal. (a) State X₁(t). (b) State X₂(t).

Figure 6.

Simulation results of switching and disturbance signals. (a) Switching signals, (b) disturbance signals.

Figure 7.

Simulation results of error signals.

Figure 8.

Simulation results of state responses with and without EID. (a) State X₁(t), (b) State X₂(t).

Figure 9.

Responses of release instants and release interval, error, and theresholds for SNNCS Eq. (11).

Figure 10.

Phase portrait.

So based on the aforementioned controller gain matrices, the real concentration of the actual state and its observer are shown in Figure 5a and Figure 5b, respectively, concluding that the estimated state closely resembles the actual state. The corresponding switching signal $\sigma \left(t\right)$ and the disturbance signals are then displayed Figure 6a and Figure 6b, respectively. Additionally, the estimated error between real state and their observer is displayed in Figure 7. Moreover, Figure 8a and Figure 8b depict the real state and the observer state in the presence and absence of the EID estimator. The trajectories for an event-triggered release’s instants, intervals, error, and threshold values are shown in Figure 9a and Figure 9b, respectively. The phase portrait of the actual and observer state are then shown in Figure 10a and Figure 10b, respectively.

Additionally, the ADT is determined to be ${\tau }_a>{\tau }_a^{\ast }=0.9531$ for the same parameter values used above. Also, by setting ${\tau }_a=1,$ the decay rate for the system is determined as $\aleph =0.0023$ and the state decay is estimated as $‖\phi \left(t\right)‖\le 1.6003e^{-0.0023\left(t-t_0\right)}{‖\phi \left(t_0\right)‖}_L,\forall t\ge t_0$ which means that the suggested observer-based event-triggered controller ensures the exponential stability of the resulting closed-loop augmented SNNCS Eq. (11).

Eventually, we have clearly concluded from the above simulation results that the proposed controller with observer-based event-driven controller has better disturbance rejection performance for the considered SNNCSs.

5. Conclusion

This paper examined an EID-based disturbance rejection approach for SNNCSs with an event-triggered mechanism under observer-based control. A particular focus was placed on the problem of rejecting disturbances in SNNCSs and proposed an EID-based framework to address this issue. By using our method, we are able to reject both matched and unmatched disturbances with no prior knowledge about their characteristics. A new delay-dependent condition is developed utilizing the Lyapunov-Krasovskii functional in conjunction with the ADT technique to ensure the exponential stability of even-triggered closed-loop SNNCSs. It is possible to achieve the appropriate controller gain matrices through the use of LMIs. Ultimately, a numerical example is presented in order to demonstrate the significance of the suggested control strategy.

Acknowledgments

This paper was supported by the National Science and Technology Council of Taiwan under Grant:110-2811-E-019-505-MY2, 110-2221-E-019-073-MY2.

Conflict of interest

The authors declare no conflict of interest.