Dual Event-Triggered Controller Co-Design for Networked Control Systems with Network-Induced Delays

Abstract

To address the presence of network-induced delays in networked control systems (NCSs), a dual event-triggered mechanism (DETM) is used to investigate the problem of reducing network delays and controller co-design. Firstly, the DETM of the sensor–controller (SC) and the controller–actuator (CA) is adopted. By determining whether the sampled data meet the event-triggered threshold conditions for network transmission, we effectively reduce the sampled data transmitted over the network, which can reduce a network delay by reducing occupation of the network resources. Secondly, a dual event-triggered NCS model with a network-induced delay is developed, and a Lyapunov function including a DETM and network-induced delay is chosen. The functional upper limit of the Lyapunov function is estimated by combining the Wirtinger’s-based integral inequality with the reciprocally convex approach. This results in a stability criterion for systems with low conservativeness and a controller co-design method for a DETM. Finally, the availability of this method was verified through a numerical example and case study.

1. Introduction

With the rapid growth of the Internet and network techniques, networked control systems (NCSs) have caused concerns. A network is added to the traditional control system to form an NCSs. NCSs have the merits of cost-effectiveness, simple maintenance, and high security and are broadly used in fields such as aerospace, the national defense industry, remote medical treatment, and others. Since, in practical NCSs, the components of the control system (controller, plant, actuator, et al.) lie in different positions, their signals are connected through the network. The signal is in the transmission process, and, because of restrictions to the network bandwidth, the NCSs inevitably has issues, e.g., network delay, packet loss, and timing disorder [1,2,3,4,5]. Sun, Sun, and Chen [1] studied the stability and controller design of NCSs with network-induced delays and random sampling intervals. Song et al. [5] studied the $H_{\infty }$ controller design problem of NCSs with both a time delay and packet disordering by analyzing and establishing the relationship between the time delay and packet disordering and transforming the NCS with both the time delay and packet disordering into a model with a multi-step time delay. In NCSs, because of the limitation of network bandwidth resources, a network transmission delay will be caused. How to design a reasonable communication scheme has been widely studied [6,7,8]. Time delays are inescapable in NCSs. The existence of a time delay will cause packet loss and timing disorder of the control signals in network transmission, which can also bring poor system stability and even runaway control system stability. Therefore, how to decrease the network delay while guaranteeing the control performance of the control system is an extremely important scientific issue. Yue et al. [6] proposed a new event-triggered mechanism (ETM). By analyzing the influence of a network transmission delay, the delay system model of NCSs is established. Shen et al. [9] analyzed the network transmission problem under this ETM and studied the event-triggered output feedback $H_{\infty }$ control issue of NCSs. Since network resources are limited, it is important to find a reasonable scheme to raise the utilization of communication resources to decrease the network transmission delay. This ETM is a widely used traditional communication scheme. However, it is important to note that the time-triggered scheme uses a constant sampling time during sampling, causing a large amount of redundant information to be transmitted to the network, resulting in limited communication resources. Low utilization also increases network delays. Therefore, a solution that can replace the traditional time-triggered scheme is found to increase the exploitation of network communication resources and achieve the goal of reducing network transmission delays.

Recently, a novel ETM appeared in the literature [10,11,12], which is different from the classic time-triggered scheme. It can greatly decrease the operating ratio of network bandwidth resources. In addition, this ETM reduces the time for uploading the sampled data to the network by pre-set event-triggered threshold conditions, which can ensure the control performance and decrease unnecessary data transmission. In the case of limited communication bandwidth, reducing unnecessary data transmission can increase data transfer efficiency, thereby reducing the problem of data transmission delays in communication networks. Recently, the ETM has triggered much academic research [13,14,15,16,17,18]. Zha et al. [19] studied the $H_{\infty }$ output feedback control issue for event-triggered Markovian jump systems with measurement output quantization. Liu and Yue [20] proposed a new fault model of NCSs based on an ETM. Yue and Hu [21] proposed an innovative time-delay system model to study the control design issue of event-triggered NCSs with state quantization and control input quantization. The event-triggered output feedback $H_{\infty }$ control problem of NCSs is studied in [22,23]. The ETM multi-agent systems with topologies were researched [24,25].

In the above studies, an ETM is introduced into the sensor–controller (SC). These communication schemes determine whether the sensor’s sampling information needs to be uploaded to the communication network based on event-triggering threshold conditions. Only when the current sampled data signal meets or is not satisfied with the conditions set in advance will the signal be transmitted or discarded. In networked control systems, the channel resources from the controller–actuator (CA) are also constrained by bandwidth resources. At present, there are some research studies on introducing an ETM from the CA to raise the level of network resource utilization [26,27,28,29]. Li et al. [30] modeled the closed-loop system as one that exhibits double event-triggered and network-induced delays and studied the dual event-triggered output feedback $H_{\infty }$ control of NCSs with a network-induced delay. Zha, Fang, and Liu [31] analyzed the bandwidth employing issue in SC and CA communication. A model-based event-triggered transmission tactic was proposed to study the design of event-triggered controllers in NCSs. Specifically, when the output signal of the controller is transported to the actuator past the network, the constraints of limited communication resources will inevitably cause network latency, packet loss, and timing disorder. Therefore, it is very important to design an ETM between the control and actuator.

Inspired by the above results, this article concentrates on a dual event-triggered mechanism (DETM) controller co-design for NCSs with network-induced delays. The contributions are as follows. (1) The first contribution is addressing the problem of time delays caused by low utilization of a communication resource-constrained network in networked control systems. Different from [6,19,20], in this article, a dual event-triggered NCS model is established by innovating a DETM in the two channels of the SC and CA. This model can reduce the network delay by improving the utilization of network resources. (2) The second contribution is the construction of a Lyapunov–Krasovskii function (LKF), which includes a time delay and DETM, by taking into account the effects of a delay on the stability of the systematic system. Simultaneously, in comparison to [29,30,31], when estimating the upper limit of the LKF, the upper limit of the single integral term in the LKF is estimated by combining the Wirtinger’s-based integral inequality with the reciprocally convex approach. By dividing the integral interval, as much important information about the delay is retained as possible while avoiding amplification of the delay, degradation of the conservatism of the systematic system, and reduction of the network transmission delay. (3) The third contribution, according to the Lyapunov stability theory analysis, which considers the effects of time-varied delays and the DETM, is that system stability criteria and controller co-design methods are presented by means of a linear matrix inequality (LMI).

The structure of this article is as follows. In Section 2, considering the influence of restraining communication resources and the impact of time-varied delays on the stability of the system, a DETM is introduced, and an NCS model including a time delay and DETM is established. In Section 3, based on the establishment of a networked control system model including a time delay and ETM, the system stability criteria and controller co-design methods are presented by means of an LMI. In Section 4, the availability of this method was verified through a numerical example and case study. In Section 5, the conclusion of this article is summarized, and future work and research directions are emphasized. Finally, the research results of this article are further clarified regarding where and how they should be adopted.

Notation: In this article, ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times m}$ indicate the n-dimensional Euclidean space; the set of $n\times n$ expresses real matrices; the symbol $P>0$ $(P\ge 0)$ represents that P is a real symmetric and positive definite matrix; the superscript ’$-1$’ represents the inverse; and symmetric terms in the matrix are expressed by ${}^{\prime }{\ast }^{\prime }$. $\mathrm{diag}(\mathit{A},\mathit{B})=\left[\begin{array}{cc}\mathit{A}& \mathit{0}\\ \ast & \mathit{B}\end{array}\right]$. The sign E manifests the unit array.

2. Problem Formulation

2.1. System Description

Consider the dual event-triggered NCS structure diagram shown in Figure 1 and establish a system’s mathematical model as follows:

$$\begin{array}{c}\dot{x}(t)=Ax(t)+Bu(t),\hfill \end{array}$$

(1)

where $x(t)\in {\mathbb{R}}^n$ is the state variable, $u(t)\in {\mathbb{R}}^m$ is the controlled input, and A and B are known real constant matrices.

As shown in Figure 1, in NCSs, the sensor samples the control output at a constant sampling time, and the sampled data are transferred to the controller through the network. The output signal of the controller is sampled by the sampler and transferred to the actuator past the network. In fact, the state error between the input signal of the current controller (actuator) and the up-to-date transmission output signal of the sensor (controller) is very small; continuing to transmit the current sampling signal will cause some redundant information to be transmitted in the network. Because the network bandwidth resources are limited, such signal transmission can cause dissipation of the network bandwidth resources, and the transmission of a large number of unnecessary signals will also lead to a delay in useful signal transmission. How to reduce the unnecessary transmission of ’redundant signals’ in the network, spare network bandwidth resources, and decrease network bandwidth resources has attracted extensive attention and research from scholars, and many meaningful research results have been obtained, such as those in references [6,32,33].

2.2. Modeling Based on the Event-Triggered Scheme

In this article, we assume that in NCSs (1), the communication network resources and the energy of each component of the system are constrained. In an effort to resolve the issue of a network delay caused by the constraint of bandwidth resources in the SA communication network, we adopt the dual event-triggered mechanism (i.e., the ETM is introduced to the SC and the CA, respectively, as shown in Figure 1, to minimize the SC side and the CA signal packet transmission. Under the premise of ensuring system stability and the control performance, the delay problem of NCSs is reduced.

In an effort to tackle this issue, we introduce an ETM in Figure 1. On the one hand, the event generator is drawn into the SC, and the sampled data of the sensor is transferred to the controller past the network, which can reduce the network resource occupancy rate of the SC. On the other hand, the event generator is also brought into the CA. The control information output by the controller determines whether to transmit to the actuator via the network according to the event threshold condition so as to decrease the magnitude of information transmitted in the network through the role of two event triggers. In theory, compared with the single ETM used in the present literature, the DETM must further decrease the amount of network communications and save network bandwidth resources, thereby reducing the network’s time problem.

For the purpose of a convenient period analysis, we make the following assumptions:

• The sensor samples the system state using a constant period h. The sampling instant is $lh$, $l\in 1,2,3,\cdots$.
• The function of the logic zero-order holder (ZOH) is to store the latest control input (the actuator input). At present, if there is no latest control signal to reach the controller (actuator), the ZOH will always maintain the control input of the controller (actuator). Until the latest control signal is transferred to the ZOH through the network, this is the control signal that the ZOH will keep before updating. At the same time, since some unordered packets contain outdated information, the ZOH can actively discard these packets.
• The sampling state $x(lh)$ of the system can be used to decide the signal transfer time between the SC $i_{k}h$. The time when the event trigger releases data is expressed as $i_{k}h$, $i_k\in N$. The data release time of the event trigger between the CA is expressed as $t_{k}h$, $t_k\in N$.
• ${\tau }_{t_k}^{sc}$ stands for the network delay of the signal from the SC, and ${\tau }_{t_k}^{ca}$ represents the network transmission delay of the control signal from the CA. The sum of ${\tau }_{t_k}^{sc}$ and ${\tau }_{t_k}^{ca}$, as well as computing the delay and packet waiting delay, is denoted by ${\tau }_{t_k}$, for which ${\tau }_{t_k}\in (0,\overline{\tau }]$ and $\overline{\tau }$ are the upper limits of ${\tau }_{t_k}$.

From the previous analysis, it can be perceived that the transfer of the sensor-sampled data to the actuator via the network depends on whether the sampling data meets the two event-triggered threshold conditions.

The setting of the event-triggered on the SC is as follows:

$$e_k^T(i_{k}h){\Phi }_1{e}_k(i_{k}h)\le {\rho }_1{x}^T(i_{k}h+lh){\Phi }_{1}x(i_{k}h+lh),$$

(2)

where $e_k(i_{k}h)=x(i_{k}h)-x(i_{k}h+lh)$, ${\Phi }_1$ is a symmetric matrix, ${\rho }_1$ is a scalar, ${\rho }_1\in [0,1)$, and $l\in N$. $x(i_{k}h)$ is the up-to-date transmission signal received by the controller, and $x(i_{k}h+lh)$ is the up-to-date sampled data of the current sensor. If the latest sampling signal of the present sensor does not satisfy the event-triggered threshold condition (2), the sampling signal will be transmitted to the controller through the network. If the current sampled data satisfies condition (2), the sampling signal can be actively discarded. Whether the current sampled data of the sensor can be transmitted to the controller is decided by the state error $e_k(i_{k}h)$, and, on the other hand, it is also related to the most recently transmitted control signal $x(i_{k}h)$. The next sampling signal release instant $i_{k+1}$ can be expressed in the following form:

$$\begin{array}{c}{i}_{k+1}=i_{k}h+\underset{l\ge 0}{min}\left\{(l+1)h\right|e_k^T(i_{k}h){\Phi }_1{e}_k(i_{k}h)\le {\rho }_1{x}^T(i_{k}h+lh){\Phi }_{1}x(i_{k}h+lh)\}.\hfill \end{array}$$

(3)

According to the properties of the ZOH, the dwell time of the ZOH at the SC is divided into intervals. We define $d=t_{k+1}-t_k-1$. Then, the ZOH’s holding zone is:

$$[t_{k}h+{\tau }_{t_k},t_{k+1}h+{\tau }_{t_{k+1}}]=\bigcup _{l=0}^n{{\rm Y}}_{l,k},$$

(4)

where ${{\rm Y}}_{l,k}=[t_{k}h+lh+{\tau }_{t_k},t_{k}h+lh+h+{\tau }_{t_k+l+1}]$, $l=0,1,2,\cdots ,n$. For example, as shown in Figure 2, the holding time $[t_1,t_2)=[t_{1}h+{\tau }_{t_1},t_{2}h+4h+{\tau }_{t_2})$ of the sensor side ZOH can be divided into subintervals: ${{\rm Y}}_{01}=[t_{1}h+{\tau }_{t_1},t_{1}h+h+{\tau }_{t_1+1})$, ${{\rm Y}}_{11}=[t_{1}h+h+{\tau }_{t_1+1},t_{1}h+2h+{\tau }_{t_1+2})$, ${{\rm Y}}_{21}=[t_{1}h+2h+{\tau }_{t_1+2},t_{1}h+3h+{\tau }_{t_1++})$, and ${{\rm Y}}_{31}=[t_{1}h+3h+{\tau }_{t_1+3},t_{1}h+4h+{\tau }_{t_1+4})=[t_{1}h+3h+{\tau }_{t_1+3},t_{2}h+\tau )$.

The setting of the event triggered on the CA is as follows:

$$f^T(t_{k}h){\Phi }_{2}ft(t_{k}h)\le {\rho }_2{u}^T(t_{k}h+(i_{t_k+j}-i_{t_k})h){\Phi }_{2}u(t_{k}h+(i_{t_k+j}-i_{t_k})h),$$

(5)

where $f(t_{k}h)=u(t_{k}h)-u(t_{k}h+(i_{t_k+j}-i_{t_k})h)$, ${\Phi }_2$ is a matrix, ${\rho }_2$ is a scalar, ${\rho }_2\in [0,1)$, and $j\in N$. $u(t_{k}h)$ is the up-to-date control signal transmitted at the actuator, and $u(t_{k}h+(i_{t_k+j}-i_{t_k})h)$ is the most recently sampled control signal at the controller. The up-to-date sampled control signal on the controller will be released and transferred to the actuator through the network only when event-triggered threshold condition (5) is not satisfied; otherwise, the sampled signal can be actively left. The time when the next control signal is released is expressed as:

$$t_{k+1}=t_{k}h+\underset{j\ge 0}{min}\left\{(i_{t_k+j}-i_{t_k})h\right|f^T(t_{k}h){\Phi }_{2}ft(t_{k}h)\le {\rho }_2{u}^T(t_{k}h+(i_{t_k+j}-i_{t_k})h){\Phi }_{2}u(t_{k}h+(i_{t_k+j}-i_{t_k})h)\}.$$

(6)

Remark 1. 

Because of the existence of a network-induced delay in the networked control system, when the sampled signal $x(t_{k}h)$ of the sensor meets event-triggered threshold conditions (2) and (5), it will reach the controller through the network at the $t_{k}h+{\tau }_{t_k}^{sc}$ time and reach the actuator through the network at the $t_{k}h+{\tau }_{t_k}$ time.

Remark 2. 

Through the analysis of the above event-triggered threshold conditions (2) and (5), the relationship between the set of periodically sampled signal-release times at the sensor, the set of the event-triggered signal-release times at the sensor, and the set of event-triggered signal-release times at the controller can be seen: $\{t_{1}h,t_{2}h,t_{3}h,\cdots \}\subseteq \{i_{1}h,i_{2}h,i_{3}h,\cdots \}\subseteq \{1h,2h,3h,\cdots \}$. The values of $\{i_{1}h,i_{2}h,i_{3}h,\cdots \}$ and $\{t_{1}h,t_{2}h,t_{3}h,\cdots \}$ are not only related to the change of the sensor sampling system state and controller output but also to the two event-triggered parameters ${\rho }_1$ and ${\rho }_2$.

According to the properties of the ZOH, the dwell time of the ZOH on the CA is divided into intervals. Then, the ZOH’s holding interval is:

$$[t_{k}h+{\tau }_{t_k},t_{k+1}h+{\tau }_{t_{k+1}}]=\bigcup _{l=0}^n{\Omega }_{l^j,j},$$

(7)

with ${\Omega }_{l^j,j}=[t_{k}h+(i_{t_k+j}-i_{t_k})h+l^{j}h+{\tau }_{t_k+i_{t_k+j}-i_{t_k}+l^j},t_{k}h+(i_{t_k+j+1}-i_{t_k})h+l^{j}h$

$+{\tau }_{t_k+i_{t_k+1+j}-i_{t_k}+l^j}]$, $j=0,1,\cdots ,n$, $l^j=0,1,\cdots ,l_M^j$, $i_{t_k+n}-i_{t_k}+l_M^j=t_{k+1}-t_k-1$. For $l_M^j$, there are different maximum values of $[i_{t_k+j},i_{t_k+j+1}]$ for different $l_M^j=i_{t_k+j+1}-i_{t_k+j}-1$. In order to facilitate an understanding of the ZOH’s interval division in Equation (7), one can view the schematic diagram of the ZOH’s holding time division at the SA in Figure 3.

Define $\eta (t)=t-t_{k}h-(i_{t_k+j}-i_{t_k})h+l^{j}h$. It is clear that

$$0\le {\tau }_{t_k+i_{t_k+j}-i_{t_k}+l^j}\le \eta (t)\le h+\overline{\tau }\triangleq \overline{\eta }.$$

(8)

In this article, we design a controller $u(t)=Kx(t)$ that meets the performance demands of the system, where K is the gain matrix to be confirmed.

According to the previous analysis, it can be perceived that under the dual event-triggered mechanism, the actual input of the actuator is

$$u(t)=Kx(t_{k}h),t\in [t_{k}h+{\tau }_{t_k},t_{k+1}h+{\tau }_{t_{k+1}}).$$

(9)

When $t\in [t_{k}h+{\tau }_{t_k},t_{k+1}h+{\tau }_{t_{k+1}})$, we make the following provisions: $g_k(t)=x(t_{k}h)-x(t_{k}h+(i_{t_k+j}-i_{t_k})h)$, $e_k(t)=x(t_{k}h+(i_{t_k+j}-i_{t_k})h)-x(t_{k}h+(i_{t_k+j}-i_{t_k})h+l^{j}h)$.

From the definition of $e_k(t)$ and $\eta (t)$, combined with the event-triggered threshold condition (2), we can obtain:

$$e_k^T(i_{k}h){\Phi }_1{e}_k(i_{k}h)\le {\rho }_1{x}^T(t-\eta (t)){\Phi }_{1}x(t-\eta (t)),t\in [t_{k}h+{\tau }_{t_k},t_{k+1}h+{\tau }_{t_{k+1}}).$$

(10)

Similarly, from the definitions of $g_k(t)$ and $\eta (t)$, combined with event-triggered threshold condition (5), we gain:

$$g_k^T(t)K^T{\Phi }_{2}K{g}_k(t)\le {\rho }_2{[x(t-\eta (t)+e_k(t))]}^T{\Phi }_2[x(t-\eta (t)+e_k(t))],\in [t_{k}h+{\tau }_{t_k},t_{k+1}h+{\tau }_{t_{k+1}}).$$

(11)

Remark 3. 

Through the above analysis, it can be seen that whether it is the sampling signal of the plant or the sampling signal of the controller, only the event-triggered threshold condition that is unmet should be transmitted through the network, and the event-triggered sampling signal will be actively discarded. When the sampling signal of a sensor does not meet event-triggered threshold conditions (10) and (11), it will be transferred to the actuator by the network. If event-triggered threshold condition (10) or (11) is satisfied, it will be actively discarded. Therefore, the DETM can further degrade the transmission of signals in the network and reduce the occupation of network resources, thereby decreasing the network transmission delay in the network.

On the basis of the above analysis, the real control input of the actuator can be indicated as follows:

$$x(t_{k}h)=g_k(t)+e_k(t)+x(t-\eta (t)).$$

(12)

By combining Equations (1), (8), and (12), Equation (1) is rewritten as

$$\dot{x}(t)=Ax(t)+BK{g}_k(t)+BK{e}_k(t)+BKx(t-\eta (t)).$$

(13)

The system’s initial states $x(t)$ on $[t_0-\overline{\eta },t_0]$ are $x(t+\vartheta )=\phi (\vartheta )$, $\vartheta \in [-\overline{\eta },0]$, $\phi (0)=x_0$, where $\phi (0)$ is a continuous function of $[t_0-\overline{\eta },t_0]$.

Remark 4. 

Through analysis, it can be noted that compared to the availability of research results, the DETM can better save network resources and decrease the network delay in the network because event-triggered condition (3) can reduce the network transmission data on the SC, and event-triggered condition (5) can further select the control signal transferred by the network to the CA.

2.3. Preliminaries

The lemmas below will be used to prove the following main results:

Please refer to Appendix A.

3. Main Results

Theorem 1. 

Given the scalars ${\eta }_M,{\rho }_1,and{\rho }_2$ and the control matrix K, suppose that $P>0$, $Q>0$, R > 0, ${\Phi }_1$, and ${\Phi }_2$. Then, the following LMI meets the condition that systematic system (13) is asymptotically stable.

$${\Xi }^l=\left[\begin{array}{ccc}{\Xi }_{11}& \ast & \ast \\ {\Xi }_{21}& -P{R}^{-1}P& \ast \\ {\Xi }_{31}& 0& -{\rho }_2{K}^T{\Phi }_{2}K\end{array}\right]<0.$$

(14)

where

$${\Xi }_{11}=\left[\begin{array}{ccccccc}{\Pi }_{11}& \ast & \ast & \ast & \ast & \ast & \ast \\ {\Pi }_{21}& {\Pi }_{22}& \ast & \ast & \ast & \ast & \ast \\ 0& -\frac{2}{{\eta }_M}R& {\Pi }_{33}& \ast & \ast & \ast & \ast \\ K^T{B}^{T}P& 0& 0& -{\Phi }_1& \ast & \ast & \ast \\ K^T{B}^{T}P& 0& 0& 0& -K^T{\Phi }_{2}K& \ast & \ast \\ \frac{6}{{\eta }_M}R& \frac{6}{{\eta }_M}R& 0& 0& 0& -\frac{12}{{\eta }_M}R& \ast \\ 0& \frac{6}{{\eta }_M}R& \frac{6}{{\eta }_M}R& 0& 0& 0& -\frac{12}{{\eta }_M}R\end{array}\right],$$

$${\Pi }_{11}=A^{T}P+PA+Q-\frac{4}{{\eta }_M}R,$$

$${\Pi }_{21}=K^T{B}^{T}P-\frac{2}{{\eta }_M}R,$$

$${\Pi }_{22}={\rho }_1{\Phi }_1-\frac{8}{{\eta }_M}R,$$

$${\Pi }_{33}=-Q-\frac{4}{{\eta }_M}R,$$

$${\Xi }_{21}=\sqrt{{\eta }_M}[PA,PBK,0,PBK,PBK,0,0],$$

$${\Xi }_{31}=[0,E,0,E,0,0,0].$$

Proof. 

We construct a Lyapunov–Krasovskii function

$$V(t)=x^T(t)Px(t)+{\int }_{t-{\eta }_M}^t{x}^T(s)Qx(s)\mathrm{d}s+{\int }_{t-{\eta }_M}^t{\int }_s^t{\dot{x}}^T(v)R\dot{x}(v)\mathrm{d}v\mathrm{d}s,$$

(15)

where P, Q, and R are positive definite matrices, ${\xi }^T(t)=[x^T(t),x^T(t-\eta (t)),x^T(t-{\eta }_M),e_k^T,g_k^T,\frac{1}{\eta (t)}{\int }_{t-\eta (t)}^t{x}^T(s)\mathrm{d}s,\mathrm{and}\frac{1}{{\eta }_M-\eta (t)}{\int }_{t-{\eta }_M}^{t-\eta (t)}{x}^T(s)\mathrm{d}s]$. Using the coefficient of $V(t)$ in Equation (15) with the trajectory of system (13):

$$\begin{array}{cc}\dot{V}(t)\hfill & =2{\dot{x}}^T(t)Px(t)+x^T(t)Qx(t)-x^T(t-{\eta }_M)Qx(t-{\eta }_M)\hfill \\ \hfill & +{\eta }_M{\dot{x}}^T(t)R\dot{x}(t)-{\int }_{t-{\eta }_M}^t{\dot{x}}^T(s)R\dot{x}(s)\mathrm{d}s.\hfill \end{array}$$

(16)

Add the event-tiggered condition to Equation (16), we can gain

$$\begin{array}{cc}\dot{V}(t)\hfill & =2{\dot{x}}^T(t)Px(t)+x^T(t)Qx(t)-x^T(t-{\eta }_M)Qx(t-{\eta }_M)\hfill \\ \hfill & +{\eta }_M{\dot{x}}^T(t)R\dot{x}(t)-{\int }_{t-{\eta }_M}^t{\dot{x}}^T(s)R\dot{x}(s)\mathrm{d}s+e_k^T(t){\Phi }_{1}e(t)\hfill \\ \hfill & -e_k^T(t){\Phi }_{1}e(t)+g_k^T(t)K^T{\Phi }_{2}K{g}_k(t)-g_k^T(t)K^T{\Phi }_{2}K{g}_k(t).\hfill \end{array}$$

(17)

The following combinatorial Equations (10), (11), and (17) can be obtained:

$$\begin{array}{cc}\dot{V}(t)\hfill & \le 2{\dot{x}}^T(t)Px(t)+x^T(t)Qx(t)-x^T(t-{\eta }_M)Qx(t-{\eta }_M)+{\eta }_M{\dot{x}}^T(t)R\dot{x}(t)\hfill \\ \hfill & -{\int }_{t-{\eta }_M}^t{\dot{x}}^T(s)R\dot{x}(s)\mathrm{d}s+{\rho }_1{x}^T(t-\eta (t)){\Phi }_{1}x(t-\eta (t))-e_k^T(t){\Phi }_{1}e(t)\hfill \\ \hfill & -g_k^T(t)K^T{\Phi }_{2}K{g}_k(t)+{\rho }_2{[x(t-\eta (t)+e_k(t))]}^T{\Phi }_2[x(t-\eta (t)+e_k(t))].\hfill \end{array}$$

(18)

Considering the influence of time-varied delays on the stability of the system, for the estimation of the upper limit of ${\int }_{t-{\eta }_M}^t{\dot{x}}^T(s)Rx(s)\mathrm{d}s$ with the upper limit of the delay in Equation (18), the Wirtinger’s-based inequality in Lemma 1 is used to estimate the upper limit of the integral term ${\int }_{t-{\eta }_M}^t{\dot{x}}^T(s)Rx(s)\mathrm{d}s$. In the process of dealing with Equation (18), the delay intermediate variable ${\eta }_M$ and the derivative of the difference between the upper limit of the delay and the delay ${\eta }_M-\eta (t)$ appear, respectively. The two terms are scaled by using the reciprocally convex approach in Lemma 2. The process is as follows:

$$\begin{array}{cc}-\hfill & {\int }_{t-{\eta }_M}^t{\dot{x}}^T(s)Rx(s)\mathrm{d}s\hfill \\ =\hfill & {\int }_{t-\eta (t)}^t{\dot{x}}^T(s)Rx(s)\mathrm{d}s-{\int }_{t-{\eta }_M}^{t-\eta (t)}{\dot{x}}^T(s)Rx(s)\mathrm{d}s\hfill \\ \le \hfill & -{\xi }^T(t)\left(\frac{1}{\eta (t)}{\left[\begin{array}{c}{G}_1\\ G_2\end{array}\right]}^T\overline{R}\left[\begin{array}{c}{G}_1\\ G_2\end{array}\right]\right.\left.+\frac{1}{{\eta }_M-\eta (t)}{\left[\begin{array}{c}{G}_3\\ G_4\end{array}\right]}^T\overline{R}\left[\begin{array}{c}{G}_3\\ G_4\end{array}\right]\right)\hfill \\ \le \hfill & -{\xi }^T(t)\left(\frac{1}{{\eta }_M}{\left[\begin{array}{c}{G}_{12}\\ G_{34}\end{array}\right]}^T\tilde{R}\left[\begin{array}{c}{G}_{12}\\ G_{34}\end{array}\right]\right)\xi (t).\hfill \end{array}$$

(19)

where

$$\overline{R}=\left[\begin{array}{cc}R& 0\\ \ast & 3R\end{array}\right],$$

$$G_1=[E,-E,0,0,0,0,0],$$

$$G_2=[0,E,E,0,0,-2E,0],$$

$$G_3=[0,E,-E,0,0,0,0],$$

$$G_4=[0,E,E,0,0,0,-2E],$$

$$G_{12}=\left[\begin{array}{c}{G}_1\\ G_2\end{array}\right],G_{34}=\left[\begin{array}{c}{G}_3\\ G_4\end{array}\right],\tilde{R}=\left[\begin{array}{cccc}R& \ast & \ast & \ast \\ 0& 3R& \ast & \ast \\ 0& 0& R& \ast \\ 0& 0& 0& 3R\end{array}\right].$$

Remark 5. 

For the upper limit of ${\int }_{t-{\eta }_M}^t{\dot{x}}^T(s)Rx(s)\mathrm{d}s$ in Equation (18), the LKF is estimated by combining the Wirtinger’s-based integral inequality with the reciprocally convex approach. When the integral interval is divided, as much important information about the delay can be retained as possible. At the same time, delay amplification can be avoided, the conservatism of the system can be reduced, and the network transfer delay must be reduced.

The term ${\eta }_M{\dot{x}}^T(t)R\dot{x}(t)$ in Equation (18) can be processed as follow:

$$\begin{array}{cc}\hfill & {\eta }_M{\dot{x}}^T(t)R\dot{x}(t)\hfill \\ =\hfill & {[Ax(t)+BK{e}_k(t)+BK{g}_k(t)+BKx(t-\eta (t))]}^T{\eta }_{M}R[Ax(t)+BK{e}_k(t)+BK{g}_k(t)+BKx(t-\eta (t))]\hfill \\ =\hfill & {\xi }^T(t)\sqrt{{\eta }_M}{[PA,PBK,0,PBK,0,0]}^T{P}^{-1}R{P}^{-1}\sqrt{{\eta }_M}[PA,PBK,0,PBK,0,0]\xi (t)\hfill \\ =\hfill & {\xi }^T(t){\Xi }_{21}^T{P}^{-1}R{P}^{-1}{\Xi }_{21}\xi (t).\hfill \end{array}$$

(20)

For the purpose of convenience for the use of LMI to solve, ${\rho }_2{[x(t-\eta (t)+e_k(t))]}^T$${\Phi }_2[x(t-\eta (t)+e_k(t))]$, existing in the inequality (18), combined with Equation (13), can be employed under the formal denotation:

$$\begin{array}{cc}\hfill & {\rho }_2{[x(t-\eta (t)+e_k(t))]}^T{K}^T{\Phi }_{2}K[x(t-\eta (t)+e_k(t))]\hfill \\ \hfill & ={\rho }_2{\xi }^T(t){[0,E,0,E,0,0]}^T{K}^T{\Phi }_{2}K[0,E,0,E,0,0]\xi (t)\hfill \\ \hfill & ={\rho }_2{\xi }^T(t){\Xi }_{31}^T{K}^T{\Phi }_{2}K{\Xi }_{31}\xi (t).\hfill \end{array}$$

(21)

Substituting Equations (13) and (19)–(21) into (18), we obtain the following inequality:

$$\begin{array}{cc}\dot{V}(t)\hfill & \le 2{\dot{x}}^T(t)Px(t)+x^T(t)Qx(t)-x^T(t-{\eta }_M)Qx(t-{\eta }_M)\hfill \\ \hfill & -{\int }_{t-{\eta }_M}^t{\dot{x}}^T(s)R\dot{x}(s)\mathrm{d}s+{\rho }_1{x}^T(t-\eta (t)){\Phi }_{1}x(t-\eta (t))\hfill \\ \hfill & +{\eta }_M{\dot{x}}^T(t)R\dot{x}(t)-e_k^T(t){\Phi }_{1}e(t)-g_k^T(t)K^T{\Phi }_{2}K{g}_k(t)\hfill \\ \hfill & +{\rho }_2{[x(t-\eta (t)+e_k(t))]}^T{K}^T{\Phi }_{2}K[x(t-\eta (t)+e_k(t))]\hfill \\ \hfill & \le {\xi }^T(t)[{\Xi }_{11}+{\Xi }_{21}^{T}R{\Xi }_{21}+{\rho }_2{\Xi }_{31}^T{K}^T{\Phi }_{2}K{\Xi }_{31}]\xi (t).\hfill \end{array}$$

(22)

The parameters ${\Xi }_{11}$, ${\Xi }_{21}$, and ${\Xi }_{31}$ are shown in Theorem 1.

By applying Lemma 3, inequality (22) can be transformed into inequality (14). By using the Lyapunov–Krasovskii stability theory, it can be perceived from Equation (14) that systematic system (13) is asymptotically stable under zero initial conditions, and the stability proof is completed. □

Controller Co-Design for Dual Event-Triggered Networked Control Systems

On the basis of system stability analysis, we derive a controller co-design design method for the DETM.

Theorem 2. 

For the scalars ${\eta }_M$, ${\rho }_1$, and ${\rho }_2$ and the control matrix K, suppose that $P>0$, $Q>0$, $R>0$, ${\Phi }_1$, and ${\Phi }_2$ and that the following LMI holds. Systematic system (13) is asymptotically stable. The gains matrix is $K=X{Y}^{-1}$.

$${\tilde{\Xi }}^l=\left[\begin{array}{ccc}{\tilde{\Xi }}_{11}& \ast & \ast \\ {\tilde{\Xi }}_{21i}& -2\epsilon X+{\epsilon }^{2}R& \ast \\ {\tilde{\Xi }}_{31i}& 0& -{\rho }_2{\tilde{\Phi }}_2\end{array}\right]<0,$$

(23)

where

$${\tilde{\Xi }}_{11}=\left[\begin{array}{ccccccc}{\tilde{\Pi }}_{11}& \ast & \ast & \ast & \ast & \ast & \ast \\ {\tilde{\Pi }}_{21}& {\tilde{\Pi }}_{22}& \ast & \ast & \ast & \ast & \ast \\ 0& -\frac{2}{{\eta }_M}\tilde{R}& {\tilde{\Pi }}_{33}& \ast & \ast & \ast & \ast \\ Y^T{B}^T& 0& 0& -{\tilde{\Phi }}_1& \ast & \ast & \ast \\ Y^T{B}^T& 0& 0& 0& -{\tilde{\Phi }}_2& \ast & \ast \\ \frac{6}{{\eta }_M}\tilde{R}& \frac{6}{{\eta }_M}\tilde{R}& 0& 0& 0& -\frac{12}{{\eta }_M}\tilde{R}& \ast \\ 0& \frac{6}{{\eta }_M}\tilde{R}& \frac{6}{{\eta }_M}\tilde{R}& 0& 0& 0& -\frac{12}{{\eta }_M}\tilde{R}\end{array}\right],$$

$${\tilde{\Pi }}_{11}=A^{T}X+XA+\tilde{Q}-\frac{4}{{\eta }_M}\tilde{R},$$

$${\tilde{\Pi }}_{21}=Y^T{B}^T-\frac{2}{{\eta }_M}\tilde{R},$$

$${\tilde{\Pi }}_{22}={\rho }_1{\tilde{\Phi }}_1-\frac{8}{{\eta }_M}\tilde{R},$$

$${\tilde{\Pi }}_{33}=-Q-\frac{4}{{\eta }_M}\tilde{R},$$

$${\tilde{\Xi }}_{21}=\sqrt{{\eta }_M}[AX,BX,0,BX,XB,0,0],$$

$${\tilde{\Xi }}_{31}=[0,E,0,E,0,0,0],$$

Proof. 

Based on

$$(R-{\epsilon }_1^{-1}P)R^{-1}(R-{\epsilon }_1^{-1}P)\ge 0,$$

we can obtain:

$$-2{\epsilon }_{1}P+{\epsilon }_1^{2}R\ge -P{R}^{-1}P.$$

The $-P{R}^{-1}P$ in inequality (14) is replaced by $-2{\epsilon }_{1}P+{\epsilon }_1^{2}R$, and the following inequality can be obtained:

$$\begin{array}{c}{\Xi }^l=\left[\begin{array}{ccc}{\Xi }_{11}& \ast & \ast \\ {\Xi }_{21}& -P{R}^{-1}P& \ast \\ {\Xi }_{31}& 0& -{\rho }_2{K}^T{\Phi }_{2}K\end{array}\right]=\left[\begin{array}{ccc}{\Xi }_{11}& \ast & \ast \\ {\Xi }_{21}& -2\epsilon P+{\epsilon }^{2}R& \ast \\ {\Xi }_{31}& 0& -{\rho }_2{K}^T{\Phi }_{2}K\end{array}\right]<0.\hfill \end{array}$$

(24)

Define $\Psi =\mathrm{diag}\{X,X,X,X,X,X,X,X,X\}$, by means of $\Psi$ and ${\Psi }^T$. By multiplying the left and right by inequality (24), respectively, we obtain:

$$\begin{array}{cc}{\widehat{\Xi }}^l\hfill & =\left[\begin{array}{ccc}{\widehat{\Xi }}_{11}& \ast & \ast \\ {\widehat{\Xi }}_{21}& -2\epsilon XPX+{\epsilon }^{2}XRX& \ast \\ {\widehat{\Xi }}_{31}& 0& -{\rho }_{2}X{K}_2^{\Phi }KX\end{array}\right]<0,\hfill \end{array}$$

(25)

where

$${\widehat{\Xi }}_{11}=\left[\begin{array}{ccccccc}{\widehat{\Pi }}_{11}& \ast & \ast & \ast & \ast & \ast & \ast \\ {\widehat{\Pi }}_{21}& {\widehat{\Pi }}_{22}& \ast & \ast & \ast & \ast & \ast \\ 0& -\frac{2}{{\eta }_M}XRX& {\widehat{\Pi }}_{33}& \ast & \ast & \ast & \ast \\ X{K}^T{B}^{T}PX& 0& 0& {\widehat{\Pi }}_{44}& \ast & \ast & \ast \\ X{K}^T{B}^{T}PX& 0& 0& 0& {\widehat{\Pi }}_{55}& \ast & \ast \\ \frac{6}{{\eta }_M}\tilde{R}& \frac{6}{{\eta }_M}\tilde{R}& 0& 0& 0& -\frac{12}{{\eta }_M}\tilde{R}& \ast \\ 0& \frac{6}{{\eta }_M}\tilde{R}& \frac{6}{{\eta }_M}\tilde{R}& 0& 0& 0& -\frac{12}{{\eta }_M}\tilde{R}\end{array}\right],$$

$${\widehat{\Pi }}_{11}=X{A}^{T}PX+XXAX+XQX-\frac{4}{{\eta }_M}XRX,$$

$${\widehat{\Pi }}_{21}=X{K}^T{B}^{T}PX-\frac{2}{{\eta }_M}XRX,$$

$${\widehat{\Pi }}_{22}=X{\rho }_1{\Phi }_{1}X-\frac{8}{{\eta }_M}XRX,$$

$${\widehat{\Pi }}_{33}=-XQX-\frac{4}{{\eta }_M}XRX,$$

$${\widehat{\Pi }}_{44}=-X{\Phi }_{1}X,$$

$${\widehat{\Pi }}_{55}=-X{K}^T{\Phi }_{2}KX,$$

$${\widehat{\Xi }}_{21}=\sqrt{{\eta }_M}[XPAX,XPBKX,0,XPBKX,XPBKX,,0],$$

$${\widehat{\Xi }}_{31}=[0,XEX,0,XEX,0,0,0].$$

There are variable coupling terms $X{K}^T{B}^{T}PX$ and $-X{K}^T{\Phi }_{2}KX$ in inequality (25). In order to express these in terms of LMIs, We make the following agreements: $Y=KX$, $X=P^{-1}$, $\tilde{Q}=XQX$, $\tilde{R}=XRX$, ${\tilde{\Phi }}_1=X{\Phi }_{1}X$, and ${\tilde{\Phi }}_2=X{\Phi }_{2}X$, where X is a matrix of proper dimensions. Applying Lemma 2, inequality (25) can be tantamount to the form below, namely inequality (23). □

Remark 6. 

In the derivation process of the controller co-design, there are variable coupling terms $X{K}^T{B}^{T}PX$ and $-X{K}^T{\Phi }_{2}KX$ in inequality (25) that must not be directly solved by the LMI toolbox. To deal with the variable coupling term in inequality (25), we were inspired by the literature [34,35]. By setting the intermediate variable method, the variable coupling term’s transformation is decoupled and then solved by the LMI toolbox in MATLAB.

If only the triggered event is inserted between the SC, then ${\rho }_1\ne 0$ in inequality (2) and ${\rho }_2=0$ in inequality (5). At this time, system (13) degenerates to single triggered event.

$$\dot{x}(t)=Ax(t)+BK{e}_k(t)+BKx(t-\eta (t)).$$

(26)

If only the triggered event is introduced between the SC, then ${\rho }_1=0$ in inequality (2) and ${\rho }_2\ne 0$ in inequality (5). At this time, system (13) degenerates to single triggered event.

$$\dot{x}(t)=Ax(t)+BK{g}_k(t)+BKx(t-\eta (t)).$$

(27)

When the system (13) is only subject to the event-triggered threshold condition (2), according to Corollary 1, we can obtain the controller design approach for the asymptotically stability of systematic system (26).

Corollary 1. 

Given scalars ${\eta }_M$, ${\rho }_1,$, and ε, and supposing the matrices $X>0$, $\tilde{Q}>0$, $\tilde{R}>0$, ${\tilde{\Phi }}_1$, and Y, the following LMI is met. On the condition of event-triggered threshold condition (2), systematic system (13) is asymptotically stable. The controller gains matrix is $K=X{Y}^{-1}$.

$${\tilde{\Xi }}^l=\left[\begin{array}{ccc}{\tilde{\Xi }}_{11}& \ast & \ast \\ {\tilde{\Xi }}_{21i}& -2\epsilon X+{\epsilon }^{2}R& \ast \end{array}\right]<0,$$

(28)

where

$${\tilde{\Xi }}_{11}=\left[\begin{array}{cccccc}{\tilde{\Pi }}_{11}& \ast & \ast & \ast & \ast & \ast \\ {\tilde{\Pi }}_{21}& {\tilde{\Pi }}_{22}& \ast & \ast & \ast & \ast \\ 0& -\frac{2}{{\eta }_M}\tilde{R}& {\tilde{\Pi }}_{33}& \ast & \ast & \ast \\ Y^T{B}^T& 0& 0& -{\tilde{\Phi }}_1& \ast & \ast \\ \frac{6}{{\eta }_M}\tilde{R}& \frac{6}{{\eta }_M}\tilde{R}& 0& 0& 0& -\frac{12}{{\eta }_M}\tilde{}R\\ 0& \frac{6}{{\eta }_M}\tilde{R}& \frac{6}{{\eta }_M}\tilde{R}& 0& 0& -\frac{12}{{\eta }_M}\tilde{}R\end{array}\right],$$

$${\tilde{\Pi }}_{11}=A^{T}X+XA+\tilde{Q}-\frac{4}{{\eta }_M}\tilde{R},$$

$${\tilde{\Pi }}_{21}=Y^T{B}^T-\frac{2}{{\eta }_M}\tilde{R},$$

$${\tilde{\Pi }}_{22}={\rho }_1{\tilde{\Phi }}_1-\frac{8}{{\eta }_M}\tilde{R},$$

$${\tilde{\Pi }}_{33}=-Q-\frac{4}{{\eta }_M}\tilde{R},$$

$${\tilde{\Xi }}_{21}=\sqrt{{\eta }_M}[AX,BX,0,BX,XB,0,0].$$

When system (29) is only subject to the event-triggered threshold condition (5), according to Corollary 2, we can gain the controller design method for the asymptotic stability of systematic system (26).

Corollary 2. 

Given scalars ${\eta }_M$, ${\rho }_2$, and ε, and supposing the matrices $X>0$, $\tilde{Q}>0$, $\tilde{R}>0$, ${\tilde{\Phi }}_2$, and Y, the following LMI is met. On the condition that event-triggered threshold condition (5), systematic system (13) is asymptotically stable. The gain matrix is $K=X{Y}^{-1}$.

$${\tilde{\Xi }}^l=\left[\begin{array}{ccc}{\tilde{\Xi }}_{11}& \ast & \ast \\ {\tilde{\Xi }}_{21i}& -2\epsilon X+{\epsilon }^{2}R& \ast \\ {\tilde{\Xi }}_{31i}& 0& -{\rho }_2{\tilde{\Phi }}_2\end{array}\right]<0,$$

(29)

where

$${\tilde{\Xi }}_{11}=\left[\begin{array}{cccccc}{\tilde{\Pi }}_{11}& \ast & \ast & \ast & \ast & \ast \\ {\tilde{\Pi }}_{21}& {\tilde{\Pi }}_{22}& \ast & \ast & \ast & \ast \\ 0& -\frac{2}{{\eta }_M}\tilde{R}& {\tilde{\Pi }}_{33}& \ast & \ast & \ast \\ Y^T{B}^T& 0& 0& -{\tilde{\Phi }}_2& \ast & \ast \\ \frac{6}{{\eta }_M}\tilde{R}& \frac{6}{{\eta }_M}\tilde{R}& 0& 0& -\frac{12}{{\eta }_M}\tilde{R}& \ast \\ 0& \frac{6}{{\eta }_M}\tilde{R}& \frac{6}{{\eta }_M}\tilde{R}& 0& 0& -\frac{12}{{\eta }_M}\tilde{R}\end{array}\right],$$

$${\tilde{\Pi }}_{11}=A^{T}X+XA+\tilde{Q}-\frac{4}{{\eta }_M}\tilde{R},$$

$${\tilde{\Pi }}_{21}=Y^T{B}^T-\frac{2}{{\eta }_M}\tilde{R},$$

$${\tilde{\Pi }}_{22}=-\frac{8}{{\eta }_M}\tilde{R},$$

$${\tilde{\Pi }}_{33}=-Q-\frac{4}{{\eta }_M}\tilde{R},$$

$${\tilde{\Xi }}_{21}=\sqrt{{\eta }_M}[AX,BX,0,XB,0,0],$$

$${\tilde{\Xi }}_{31}=[0,E,0,0,0,0].$$

4. Numerical Example and Case Study

Considering system (13), the system matrix parameters are as follows:

$$A=\left[\begin{array}{cc}-2& -0.1\\ -0.1& 0.01\end{array}\right],B=\left[\begin{array}{c}0.05\\ 0.02\end{array}\right].$$

It can be discovered that the system without a controller is unstable. The initial state value of $x_0$ is ${[-0.3,0.3]}^T$.

Next, we offer a comparison of three cases to prove the validation of the presented DETM.

Case1: Considering system (13) under conditions (2) and (5), the parameters are set to ${\eta }_M=0.12$, ${\rho }_1=0.02$, ${\rho }_2=0.01$, and $\epsilon =1$. Applying Theorem 2, by making use of the LMI toolbox, the gain matrix is obtained as $K=[0.0717,-1.4401]$, the allowable maximum network delay is ${\eta }_M=1.995s$, and Figure 4 is the system state response curve.

Case2: Considering system (13) under condition (2), the parameters are set to ${\eta }_M=0.12$, ${\rho }_1=0.02$, ${\rho }_2=0$, and $\epsilon =1$. Applying Corollary 1, by making use of the LMI toolbox, the gain matrix is obtained as $K=[0.0719,-1.4447]$, and Figure 5 is the system state response curve.

Case3: Considering system (13) under condition (5), the parameters are set to ${\eta }_M=0.12$, ${\rho }_1=0$, ${\rho }_2=0.01$, and $\epsilon =1$. Applying Corollary 2, by making use of the LMI toolbox, the gain matrix is obtained as $K=[0.0464,-0.9304]$, and Figure 5 is the system state response curve.

From

Table 1. When ${\rho }_1$, ${\rho }_2$ take different values, the maximum allowable network delay.

${\mathit{\rho }}_1$, ${\mathit{\rho }}_2$	${\mathit{\eta }}_{\mathit{M}}$	Controller Gain Matrices ${\mathit{K}}_{\mathit{i}}$
${\rho }_1=0.02,{\rho }_2=0.01$	1.995 s	$K_1=0.0717,K_2=-1.4401$
${\rho }_1=0.02,{\rho }_2=0$	1.890 s	$K_1=0.6558,K_2=-2.7301$
${\rho }_1=0,{\rho }_2=0.01$	1.903 s	$K_1=0.6368,K_2=-2.7627$

, we can see that when ${\rho }_1=0.02,{\rho }_2=0.01$, system (13) is under the action of the DETM, that is, Case1, and the maximum network transmission delay allowed at this time is 1995s. When ${\rho }_1=0.02,{\rho }_2=0$, system (13) adopts the ETM in the single ETM, the SC adopts the ETM, that is, Case2; the maximum network transmission delay allowed is 1.890 s. When ${\rho }_1=0,{\rho }_2=0.01$, system (13) adopts the ETM in the single ETM, namely, the CA adopts the ETM, that is Case2, and the maximum allowable network transmission delay is 1.903 s. By analyzing the maximum network transmission delay allowed in the three cases of Case1, Case2, and Case3, it can be perceived that we adopt the DETM over the single ETM. The DETM can reduce the network transmission delay in the case of restrained network bandwidth.

From a comparison of Figure 4, Figure 5 and Figure 6, it can be seen that under the action of the DETM and the single ETM, when the DETM is adopted, system (13) can reach the steady-state faster under the condition of maintaining the control performance index, and the response time of the systematic system to reach the steady-state is shorter.

5. Conclusions

To address the presence of network-induced delays in NCSs, a DETM is used to approach the problem of reducing network delays and the controller co-design. Firstly, the DETM of the SC and CA is adopted, which can decrease the transfer ratio of the sampled data in the SC and CA. This mechanism can be implemented by software and does not require the use of load hardware devices to continuously detect the state of the systematic system. Then, the NCS model with the DETM and network-induced delay information is established. According to the Lyapunov stability theory, the method of combining the Wirtinger’s-based integral inequality and the reciprocally convex approach is used to degrade the conservatism of the systematic system. The conditions sufficient for the asymptotic stability of the dual event-triggered NCS and controller co-design are gained in the form of the LMI. Finally, the availability of this method was verified through a numerical example and case study. In this article, the delay issue of NCSs under the DETM has been researched. At the same time, this article yields an approach to studying the device of the DETM’s co-design controller, in theory. In practical applications, when the controller and the plant in the network are distributed in different spatial locations, the network delay requirements are relatively small, and various control information is transmitted through the network. At this time, we must consider the impact of network delays on the system stability. On the precondition of security for the stability of the control systematic system, the DETM can be used as a method to reduce the network latency in practical application. In future work, the related NCSs will be studied in terms of event-triggered predictive control.