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Switching Periodic Event-Triggered H ∞ H ∞ H_(oo)H_{\infty} Control for NCSs and Its Application to UAVs
NCS 的周期性事件触发 H ∞ H ∞ H_(oo)H_{\infty} 控制切换及其在无人机中的应用

Zhiying Wu ® ®  ^("® "){ }^{\text {® }}, Yan Wang ® ®  ^("® "){ }^{\text {® }}, Junlin Xiong ® ®  ^("® "){ }^{\text {® }}, Member, IEEE, and Min Xie ® ®  ^("® "){ }^{\text {® }}, Fellow, IEEE
吴志英 ® ®  ^("® "){ }^{\text {® }} ， 王 ® ®  ^("® "){ }^{\text {® }} 艳 ， 熊俊林 ® ®  ^("® "){ }^{\text {® }} ， IEEE 会员， 谢敏 ® ®  ^("® "){ }^{\text {® }} ， IEEE 会士

Abstract  抽象

Based on the switching periodic event-triggered scheme (ETS), the control problem is investigated for networked control systems (NCSs) and its application to unmanned aerial vehicles (UAVs). Inspired by the switched system framework, a new switching ETS is proposed to reduce data transmission among the components of the NCS. The feature of the switching ETS is that the triggered conditions are scheduled by switching signals. The switching controller is model-dependent such that the studied NCS is modelled as a switching system with two time-delay subsystems. Applying the switching system technique, the criteria under the designed control scheme are established for stability and the H ∞ H ∞ H_(oo)H_{\infty} performance analysis. Finally, an UAV system and a networked invert pendulum are utilized to show the availability of the proposed switching ETS.
基于切换周期性事件触发方案（ETS），研究了网络控制系统（NCS）的控制问题及其在无人机（UAV）中的应用。受交换系统框架的启发，提出了一种新的交换 ETS，以减少 NCS 组件之间的数据传输。开关 ETS 的特点是触发条件由开关信号调度。开关控制器与模型相关，因此所研究的 NCS 被建模为具有两个时延子系统的开关系统。应用开关系统技术，建立了所设计控制方案下的稳定性和 H ∞ H ∞ H_(oo)H_{\infty} 性能分析标准。最后，利用无人机系统和联网倒摆来显示所提出的开关 ETS 的可用性。

Index Terms-Event-triggered control, Lyapunov functional, networked control systems (NCSs), stability, unmanned aerial vehicles (UAVs).
索引术语-事件触发控制、李雅普诺夫功能、网络控制系统 （NCS）、稳定性、无人机 （UAV）。

I. Introduction  一、引言

NETWORKED control systems (NCSs) exchange information among the components by utilizing a communication network [1], [2], [3]. NCSs are widely applied in smart grids [4], and unmanned aerial vehicle (UAV) systems [5], [6], [7]. The control problem have been investigated in [8], [9] for UAV systems. In control systems, the system measurements are transmitted periodically under time-triggered scheme. This may result in the transmissions of numerous unnecessary signals. Recently, event-triggered schemes (ETSs) can efficiently reduce
网络控制系统（NCS）利用通信网络在组件之间交换信息[1]、[2]、[3]。NCS 广泛应用于智能电网[4]和无人机系统[5]、[6]、[7]。无人机系统的控制问题已在[8]、[9]中进行了研究。在控制系统中，系统测量值在时间触发方案下定期传输。这可能会导致传输大量不必要的信号。最近，事件触发方案 （ETS） 可以有效地减少
data transmissions [10], [11], [12], [13], [14]. In event-triggered control systems, the system measurements are transmitted when the condition of ETS is satisfied [15]. For the ETS, the eventtriggered condition is predefined to ensure the reduction of the amount of data transmission, and the satisfaction of the system performance.
数据传输[10]、[11]、[12]、[13]、[14]。在事件触发控制系统中，当满足 ETS 条件时，系统测量值就会被传输[15]。对于 ETS，预定义了事件触发条件，以确保数据传输量的减少，以及系统性能的满足。
The ETS has received substantial research attentions, and many meaningful results have been presented [11], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. To mention a few, Tabuada [16] proposed a state-dependent ETS based on Lyapunov function. Dynamic ETS was investigated in [18] for linear systems by using a dynamic variable. The static and dynamic ETSs were introduced in [20] with time-varying delays. The control problem was addressed in [21] subject to distributed delay. To relax the triggered conditions, an integral-based ETS is constructed in [22] by using the integrals of system states. In most of these ETSs, continuous supervision is required on the system measurements, which may lead to unnecessary computation.
ETS 受到了广泛的研究关注，并提出了许多有意义的结果[11]，[16]，[17]，[18]，[19]，[20]，[21]，[22]，[23]，[24]，[25]。仅举几例，Tabuada [16]提出了一种基于 Lyapunov 函数的状态依赖性 ETS。[18]中通过使用动态变量研究了线性系统的动态 ETS。静态和动态 ETS 在[20]中引入，具有时变延迟。控制问题在[21]中得到了解决，但存在分布式延迟。为了放宽触发条件，在[22]中利用系统状态的积分构建了一个基于积分的 ETS。在大多数 ETS 中，需要对系统测量进行持续监督，这可能会导致不必要的计算。
To avoid continuous supervision, the periodic ETSs (PETSs) were proposed in [26], [27], [28], [29], [30], [31], [32]. Compared with continuous ETSs, the PETSs only require periodic supervision. Specifically, a periodic ETS was presented in [26] for NCSs. Based on the periodic ETS in [26], an adaptive ETS was proposed in [29] for multi-area power systems. The data-driven control problem under ETS is investigated in [33]. The event-triggered fault detection problem was investigated in [6] for UAVs. The saturated threshold event-triggered control is studied in [34] for UAVs. The cooperative synchronization problem is investigated in [25] for multiple fixed-wing UAVs. The results are extended to adaptive fault-tolerant control in [7]. A switching-like ETS was studied in [35] for NCSs under denial-of-service attacks. However, the triggered condition based on the Lyapunov function still needs further improvement. Inspired by the idea of switched systems [36], [37], [38], [39], a switching ETS is proposed for NCSs and its application to UAVs. The investigated system is modelled as a switched system under the switching ETS. The stability problem is analyzed in [40] for continuous-time switched systems with a random switching signal. Based on a ETS with a waiting time, the event-triggered system is modelled as a switched system in [41]. The dynamic ETC problem is investigated in [42] for UAVs. The approaches used in [40], [41], [42] are not directly applied in this paper. The following challenges are needed to be overcome: Firstly, it is challenging to design a switching ETS to further reduce the data transmissions. Secondly, it is is challenging to ensure the system stability under the proposed switching ETS,
为避免持续监督，[26]、[27]、[28]、[29]、[30]、[31]、[32]提出了定期 ETS（PETS）。与连续 ETS 相比，PETS 只需要定期监督。具体来说，[26]中提出了 NCS 的周期性 ETS。基于[26]中的周期性 ETS，[29]提出了一种针对多区域电力系统的自适应 ETS。ETS 下的数据驱动控制问题在[33]中进行了研究。[6]研究了无人机的事件触发故障检测问题。[34]研究了无人机的饱和阈值事件触发控制。[25]研究了多架固定翼无人机的协同同步问题。结果在[7]中扩展到自适应容错控制。[35]研究了一种类似交换的 ETS，用于拒绝服务攻击下的 NCS。然而，基于李雅普诺夫函数的触发条件仍需进一步改进。受交换系统[36]、[37]、[38]、[39]的启发，提出了一种用于 NCS 的交换 ETS 及其在无人机上的应用。所研究的系统被建模为开关 ETS 下的开关系统。[40]分析了具有随机开关信号的连续时间开关系统的稳定性问题。基于具有等待时间的 ETS，事件触发系统在[41]中被建模为交换系统。[42]研究了无人机的动态 ETC 问题。本文未直接应用[40]、[41]、[42]中使用的方法。需要克服以下挑战：首先，设计开关 ETS 以进一步减少数据传输具有挑战性。其次，在所提出的交换 ETS 下，确保系统稳定性具有挑战性，
where the triggered parameter is regulated by the switching signals. The following primary contributions of this paper are summarized.
其中触发参数由开关信号调节。总结了本文的主要贡献如下。
  1. A switching ETS is proposed for NCSs and its application to UAVs, wherein the triggered conditions are scheduled by switching signal. The proposed ETS outperforms the one in [26] for the reduction of the signal transmission.
    针对 NCS 及其在无人机中的应用，提出了一种开关 ETS，其中触发条件由开关信号调度。所提出的 ETS 在减少信号传输方面优于[26]中的 ETS。
  2. A switching system model with two time-delay subsystems is established for the investigated system under the switching ETS. It provides a unified framework to codesign of switching ETS and controller.
    在开关 ETS 下为所研究系统建立了具有两个时延子系统的开关系统模型。它为交换 ETS 和控制器的协同设计提供了一个统一的框架。
  3. According to the derived stabilization criteria, the trigger parameters of the switching ETS and mode-dependent controller gains are developed to ensure the NCS being stable and achieving a good H ∞ H ∞ H_(oo)H_{\infty} performance.
    根据推导的稳定标准，制定了开关 ETS 的触发参数和与模式相关的控制器增益，以确保 NCS 稳定并实现良好的 H ∞ H ∞ H_(oo)H_{\infty} 性能。
    The rest of the paper is presented. In Section II, the switching ETS is proposed. The stability and H ∞ H ∞ H_(oo)H_{\infty} performance can be guaranteed in Section III for the investigated system under the proposed switching ETS. The examples and the conclusions are shown in Sections IV and V, respectively.
    本文的其余部分将呈现。在第二节中，提出了开关 ETS。在所研究的开关 ETS 下，在第三节中可以保证其稳定性和 H ∞ H ∞ H_(oo)H_{\infty} 性能。示例和结论分别显示在第四节和第五节中。
Notation: R n R n R^(n)\mathbb{R}^{n} represents the Euclidean space with n n nn dimension. ‖ ⋅ ‖ ‖ ⋅ ‖ ||*||\|\cdot\| denotes the Euclidean norm. The superscripts I I II and " T T TT " are the identity matrix, and the matrix transpose, respectively. N N N\mathbb{N} is the set of nonnegative integers. λ max ( ⋅ ) λ max ( ⋅ ) lambda_(max)(*)\lambda_{\max }(\cdot) ( λ min ( ⋅ ) λ min ( ⋅ ) lambda_(min)(*)\lambda_{\min }(\cdot) ) represents the maximum (minimum) eigenvalue value. * represents X T U [ ∗ ] = X T U X X T U [ ∗ ] = X T U X X^(T)U[**]=X^(T)UXX^{T} U[*]=X^{T} U X or [ ∗ ] U X = X T U X [ ∗ ] U X = X T U X [**]UX=X^(T)UX[*] U X=X^{T} U X. Define ‖ x t ‖ h = sup − h ≤ ρ ≤ 0 { ‖ x ˙ ( t + ρ ) ‖ , ‖ x ( t + ρ ) ‖ } x t h = sup − h ≤ ρ ≤ 0   { ‖ x ˙ ( t + ρ ) ‖ , ‖ x ( t + ρ ) ‖ } ||x_(t)||_(h)=s u p_(-h <= rho <= 0){||x^(˙)(t+rho)||,||x(t+rho)||}\left\|x_{t}\right\|_{h}=\sup _{-h \leq \rho \leq 0}\{\|\dot{x}(t+\rho)\|,\|x(t+\rho)\|\}.
表示法： R n R n R^(n)\mathbb{R}^{n} 表示带维度的 n n nn 欧几里得空间。 ‖ ⋅ ‖ ‖ ⋅ ‖ ||*||\|\cdot\| 表示欧几里得范数。上标 I I II 和“ T T TT ”分别是恒等矩阵和矩阵转置。 N N N\mathbb{N} 是非负整数的集合。 λ max ( ⋅ ) λ max ( ⋅ ) lambda_(max)(*)\lambda_{\max }(\cdot) （ λ min ( ⋅ ) λ min ( ⋅ ) lambda_(min)(*)\lambda_{\min }(\cdot) ） 表示最大（最小）特征值。* 表示 X T U [ ∗ ] = X T U X X T U [ ∗ ] = X T U X X^(T)U[**]=X^(T)UXX^{T} U[*]=X^{T} U X 或 [ ∗ ] U X = X T U X [ ∗ ] U X = X T U X [**]UX=X^(T)UX[*] U X=X^{T} U X 。定义 ‖ x t ‖ h = sup − h ≤ ρ ≤ 0 { ‖ x ˙ ( t + ρ ) ‖ , ‖ x ( t + ρ ) ‖ } x t h = sup − h ≤ ρ ≤ 0   { ‖ x ˙ ( t + ρ ) ‖ , ‖ x ( t + ρ ) ‖ } ||x_(t)||_(h)=s u p_(-h <= rho <= 0){||x^(˙)(t+rho)||,||x(t+rho)||}\left\|x_{t}\right\|_{h}=\sup _{-h \leq \rho \leq 0}\{\|\dot{x}(t+\rho)\|,\|x(t+\rho)\|\} 。

II. Problem Statement  二、问题陈述

In Section II-A, the structure under switching ETS is given for NCSs, and the design of switching ETS is described in Section II-B.
在第 II-A 节中，给出了 NCS 的开关 ETS 下的结构，并在第 II-B 节中描述了开关 ETS 的设计。

A. System Model  A. 系统型号

The investigated NCS is given by:
所调查的 NCS 由以下方式给出：
{ x ˙ ( t ) = A x ( t ) + B ω ω ( t ) + B u ( t ) z ( t ) = C x ( t ) + D u ( t ) x ˙ ( t ) = A x ( t ) + B ω ω ( t ) + B u ( t ) z ( t ) = C x ( t ) + D u ( t ) {[x^(˙)(t)=Ax(t)+B_(omega)omega(t)+Bu(t)],[z(t)=Cx(t)+Du(t)]:}\left\{\begin{array}{l} \dot{x}(t)=A x(t)+B_{\omega} \omega(t)+B u(t) \\ z(t)=C x(t)+D u(t) \end{array}\right.
where A , B ω , B , C A , B ω , B , C A,B_(omega),B,CA, B_{\omega}, B, C and D D DD are constant matrices, x ( t ) ∈ R n x ( t ) ∈ R n x(t)inR^(n)x(t) \in \mathbb{R}^{n}, z ( t ) ∈ R p z ( t ) ∈ R p z(t)inR^(p)z(t) \in \mathbb{R}^{p}, and u ( t ) ∈ R m u ( t ) ∈ R m u(t)inR^(m)u(t) \in \mathbb{R}^{m} are system state, controlled output, system input, respectively. In addition, ω ( t ) ∈ L 2 [ 0 , ∞ ) ω ( t ) ∈ L 2 [ 0 , ∞ ) omega(t)inL_(2)[0,oo)\omega(t) \in \mathcal{L}_{2}[0, \infty) is external disturbance.
其中 A , B ω , B , C A , B ω , B , C A,B_(omega),B,CA, B_{\omega}, B, C 和 分别 D D DD 是常量矩阵、 x ( t ) ∈ R n x ( t ) ∈ R n x(t)inR^(n)x(t) \in \mathbb{R}^{n} 、 z ( t ) ∈ R p z ( t ) ∈ R p z(t)inR^(p)z(t) \in \mathbb{R}^{p} 和 u ( t ) ∈ R m u ( t ) ∈ R m u(t)inR^(m)u(t) \in \mathbb{R}^{m} 是系统状态、受控输出、系统输入。此外， ω ( t ) ∈ L 2 [ 0 , ∞ ) ω ( t ) ∈ L 2 [ 0 , ∞ ) omega(t)inL_(2)[0,oo)\omega(t) \in \mathcal{L}_{2}[0, \infty) 还有外界干扰。

B. Design of Switching ETS
B. 开关 ETS 的设计

In this subsection, we propose a switching ETS for NCSs to reduce the amount of transmissions. Inspired by the switched system framework, the key idea is to construct the relationship between the triggered parameters and Lyapunov function. In this framework, the derivative can be non-negative for Lyapunov function, such that transmission rate can be reduced.
在本小节中，我们提出了一种用于 NCS 的开关 ETS，以减少传输量。受开关系统框架的启发，关键思想是构建触发参数与李雅普诺夫函数之间的关系。在这个框架下，导数对于李雅普诺夫函数可以是非负的，这样传输速率就可以降低。
Inspired by the switched system framework, the trigger parameter is switched between δ 1 δ 1 delta_(1)\delta_{1} and δ 2 δ 2 delta_(2)\delta_{2}, which is scheduled by switching signals. For t ∈ T ↓ [ b n , b n + 1 ) t ∈ T ↓ b n , b n + 1 t inT_(darr)[b_(n),b_(n+1))t \in \mathcal{T}_{\downarrow}\left[b_{n}, b_{n+1}\right), the trigger parameter is chosen as δ 1 δ 1 delta_(1)\delta_{1}; otherwise, the trigger parameter is set as δ 2 δ 2 delta_(2)\delta_{2} for t ∈ T ↑ [ b n , b n + 1 ) t ∈ T ↑ b n , b n + 1 t inT_(uarr)[b_(n),b_(n+1))t \in \mathcal{T}_{\uparrow}\left[b_{n}, b_{n+1}\right), where T ↑ [ b n , b n + 1 ) = [ b n + l n , b n + 1 ) T ↑ b n , b n + 1 = b n + l n , b n + 1 T_(uarr)[b_(n),b_(n+1))=[b_(n)+l_(n),b_(n+1))\mathcal{T}_{\uparrow}\left[b_{n}, b_{n+1}\right)=\left[b_{n}+l_{n}, b_{n+1}\right), and
受开关系统框架的启发，触发参数在 和 之间 δ 1 δ 1 delta_(1)\delta_{1} δ 2 δ 2 delta_(2)\delta_{2} 切换，这是通过开关信号调度的。对于 t ∈ T ↓ [ b n , b n + 1 ) t ∈ T ↓ b n , b n + 1 t inT_(darr)[b_(n),b_(n+1))t \in \mathcal{T}_{\downarrow}\left[b_{n}, b_{n+1}\right) ，触发器参数选择为 δ 1 δ 1 delta_(1)\delta_{1} ;否则，触发器参数设置为 δ 2 δ 2 delta_(2)\delta_{2} t ∈ T ↑ [ b n , b n + 1 ) t ∈ T ↑ b n , b n + 1 t inT_(uarr)[b_(n),b_(n+1))t \in \mathcal{T}_{\uparrow}\left[b_{n}, b_{n+1}\right) 、 、 其中 T ↑ [ b n , b n + 1 ) = [ b n + l n , b n + 1 ) T ↑ b n , b n + 1 = b n + l n , b n + 1 T_(uarr)[b_(n),b_(n+1))=[b_(n)+l_(n),b_(n+1))\mathcal{T}_{\uparrow}\left[b_{n}, b_{n+1}\right)=\left[b_{n}+l_{n}, b_{n+1}\right) 、 和
T ↓ [ b n , b n + 1 ) = [ b n , b n + l n ) T ↓ b n , b n + 1 = b n , b n + l n T_(darr)[b_(n),b_(n+1))=[b_(n),b_(n)+l_(n))\mathcal{T}_{\downarrow}\left[b_{n}, b_{n+1}\right)=\left[b_{n}, b_{n}+l_{n}\right) represent the ( n + 1 n + 1 n+1n+1 )-th interval that V ˙ ( t ) < 0 V ˙ ( t ) < 0 V^(˙)(t) < 0\dot{V}(t)<0 and V ˙ ( t ) ≥ 0 V ˙ ( t ) ≥ 0 V^(˙)(t) >= 0\dot{V}(t) \geq 0, respectively.
分别 T ↓ [ b n , b n + 1 ) = [ b n , b n + l n ) T ↓ b n , b n + 1 = b n , b n + l n T_(darr)[b_(n),b_(n+1))=[b_(n),b_(n)+l_(n))\mathcal{T}_{\downarrow}\left[b_{n}, b_{n+1}\right)=\left[b_{n}, b_{n}+l_{n}\right) 表示 （ n + 1 n + 1 n+1n+1 ）-th 区间 和 V ˙ ( t ) < 0 V ˙ ( t ) < 0 V^(˙)(t) < 0\dot{V}(t)<0 V ˙ ( t ) ≥ 0 V ˙ ( t ) ≥ 0 V^(˙)(t) >= 0\dot{V}(t) \geq 0 。
The following switching ETS is proposed:
提出了以下开关 ETS：
t k + 1 = t k + { inf { l h ∣ e T ( i k h ) Φ 1 [ ∗ ] ≥ δ 1 x T ( t k ) Φ 1 [ ∗ ] } , t ∈ T ↓ [ b n , b n + 1 ) inf { l h ∣ e T ( i k h ) Φ 2 [ ∗ ] ≥ δ 2 x T ( t k ) Φ 2 [ ∗ ] } , t ∈ T ↑ [ b n , b n + 1 ) t k + 1 = t k + inf l h ∣ e T i k h Φ 1 [ ∗ ] ≥ δ 1 x T t k Φ 1 [ ∗ ] , t ∈ T ↓ b n , b n + 1 inf l h ∣ e T i k h Φ 2 [ ∗ ] ≥ δ 2 x T t k Φ 2 [ ∗ ] , t ∈ T ↑ b n , b n + 1 {:[t_(k+1)=t_(k)+],[{[i n f{lh∣e^(T)(i_(k)h)Phi_(1)[**] >= delta_(1)x^(T)(t_(k))Phi_(1)[**]}","t inT_(darr)[b_(n),b_(n+1))],[i n f{lh∣e^(T)(i_(k)h)Phi_(2)[**] >= delta_(2)x^(T)(t_(k))Phi_(2)[**]}","t inT_(uarr)[b_(n),b_(n+1))]:}]:}\begin{aligned} & t_{k+1}=t_{k}+ \\ & \left\{\begin{array}{l} \inf \left\{l h \mid e^{T}\left(i_{k} h\right) \Phi_{1}[*] \geq \delta_{1} x^{T}\left(t_{k}\right) \Phi_{1}[*]\right\}, t \in \mathcal{T}_{\downarrow}\left[b_{n}, b_{n+1}\right) \\ \inf \left\{l h \mid e^{T}\left(i_{k} h\right) \Phi_{2}[*] \geq \delta_{2} x^{T}\left(t_{k}\right) \Phi_{2}[*]\right\}, t \in \mathcal{T}_{\uparrow}\left[b_{n}, b_{n+1}\right) \end{array}\right. \end{aligned}
where t k t k t_(k)t_{k} is the latest triggered instant, i k h = l h + t k , l = 1 , 2 i k h = l h + t k , l = 1 , 2 i_(k)h=lh+t_(k),l=1,2i_{k} h=l h+t_{k}, l=1,2, … … dots\ldots, is the present sampled instant, e ( i k h ) = x ( i k h ) − x ( t k ) e i k h = x i k h − x t k e(i_(k)h)=x(i_(k)h)-x(t_(k))e\left(i_{k} h\right)=x\left(i_{k} h\right)-x\left(t_{k}\right), δ σ ( t ) δ σ ( t ) delta_(sigma(t))\delta_{\sigma(t)} is the parameter of the switching ETS, e T ( i k h ) Φ j [ ∗ ] = e T i k h Φ j [ ∗ ] = e^(T)(i_(k)h)Phi_(j)[**]=e^{T}\left(i_{k} h\right) \Phi_{j}[*]= e T ( i k h ) Φ j e ( i k h ) , x T ( t k ) Φ j [ ∗ ] = x T ( t k ) Φ j x ( t k ) , j = 1 , 2 e T i k h Φ j e i k h , x T t k Φ j [ ∗ ] = x T t k Φ j x t k , j = 1 , 2 e^(T)(i_(k)h)Phi_(j)e(i_(k)h),quadx^(T)(t_(k))Phi_(j)[**]=x^(T)(t_(k))Phi_(j)x(t_(k)),quad j=1,2e^{T}\left(i_{k} h\right) \Phi_{j} e\left(i_{k} h\right), \quad x^{T}\left(t_{k}\right) \Phi_{j}[*]=x^{T}\left(t_{k}\right) \Phi_{j} x\left(t_{k}\right), \quad j=1,2, where switching signal σ ( t ) ∈ { 1 , 2 } σ ( t ) ∈ { 1 , 2 } sigma(t)in{1,2}\sigma(t) \in\{1,2\} determines the operation mode of current ETS, and Φ 1 Φ 1 Phi_(1)\Phi_{1} and Φ 2 Φ 2 Phi_(2)\Phi_{2} are positive definite matrices to be designed, which is shown in Fig. 1. From the definition of T ↑ [ b n , b n + 1 ) T ↑ b n , b n + 1 T_(uarr)[b_(n),b_(n+1))\mathcal{T}_{\uparrow}\left[b_{n}, b_{n+1}\right) and T ↓ [ b n , b n + 1 ) T ↓ b n , b n + 1 T_(darr)[b_(n),b_(n+1))\mathcal{T}_{\downarrow}\left[b_{n}, b_{n+1}\right), the length of the ( n + 1 n + 1 n+1n+1 )-th interval that V ˙ ( t ) ≥ 0 V ˙ ( t ) ≥ 0 V^(˙)(t) >= 0\dot{V}(t) \geq 0 is denoted by b n + 1 − b n − l n ∈ R ≥ 0 b n + 1 − b n − l n ∈ R ≥ 0 b_(n+1)-b_(n)-l_(n)inR_( >= 0)b_{n+1}-b_{n}-l_{n} \in \mathbb{R}_{\geq 0}. The interval sequences satisfies
式中 t k t k t_(k)t_{k} ，是最新触发瞬时， i k h = l h + t k , l = 1 , 2 i k h = l h + t k , l = 1 , 2 i_(k)h=lh+t_(k),l=1,2i_{k} h=l h+t_{k}, l=1,2 ，， … … dots\ldots 是当前采样瞬时， e ( i k h ) = x ( i k h ) − x ( t k ) e i k h = x i k h − x t k e(i_(k)h)=x(i_(k)h)-x(t_(k))e\left(i_{k} h\right)=x\left(i_{k} h\right)-x\left(t_{k}\right) 是 δ σ ( t ) δ σ ( t ) delta_(sigma(t))\delta_{\sigma(t)} 开关 ETS 的参数， e T ( i k h ) Φ j [ ∗ ] = e T i k h Φ j [ ∗ ] = e^(T)(i_(k)h)Phi_(j)[**]=e^{T}\left(i_{k} h\right) \Phi_{j}[*]= e T ( i k h ) Φ j e ( i k h ) , x T ( t k ) Φ j [ ∗ ] = x T ( t k ) Φ j x ( t k ) , j = 1 , 2 e T i k h Φ j e i k h , x T t k Φ j [ ∗ ] = x T t k Φ j x t k , j = 1 , 2 e^(T)(i_(k)h)Phi_(j)e(i_(k)h),quadx^(T)(t_(k))Phi_(j)[**]=x^(T)(t_(k))Phi_(j)x(t_(k)),quad j=1,2e^{T}\left(i_{k} h\right) \Phi_{j} e\left(i_{k} h\right), \quad x^{T}\left(t_{k}\right) \Phi_{j}[*]=x^{T}\left(t_{k}\right) \Phi_{j} x\left(t_{k}\right), \quad j=1,2 其中开关信号 σ ( t ) ∈ { 1 , 2 } σ ( t ) ∈ { 1 , 2 } sigma(t)in{1,2}\sigma(t) \in\{1,2\} 决定当前 ETS 的工作模式， Φ 1 Φ 1 Phi_(1)\Phi_{1} 和是 Φ 2 Φ 2 Phi_(2)\Phi_{2} 要设计的正定矩阵，如图 1 所示。根据 和 的 T ↑ [ b n , b n + 1 ) T ↑ b n , b n + 1 T_(uarr)[b_(n),b_(n+1))\mathcal{T}_{\uparrow}\left[b_{n}, b_{n+1}\right) 定义，用 b n + 1 − b n − l n ∈ R ≥ 0 b n + 1 − b n − l n ∈ R ≥ 0 b_(n+1)-b_(n)-l_(n)inR_( >= 0)b_{n+1}-b_{n}-l_{n} \in \mathbb{R}_{\geq 0} 表示的 V ˙ ( t ) ≥ 0 V ˙ ( t ) ≥ 0 V^(˙)(t) >= 0\dot{V}(t) \geq 0 第 （ n + 1 n + 1 n+1n+1 ）个区间的 T ↓ [ b n , b n + 1 ) T ↓ b n , b n + 1 T_(darr)[b_(n),b_(n+1))\mathcal{T}_{\downarrow}\left[b_{n}, b_{n+1}\right) 长度。区间序列满足
0 ≤ b 0 ≤ b 0 + l 0 < b 1 < b 1 + l 1 < b 2 < ⋯ 0 ≤ b 0 ≤ b 0 + l 0 < b 1 < b 1 + l 1 < b 2 < ⋯ 0 <= b_(0) <= b_(0)+l_(0) < b_(1) < b_(1)+l_(1) < b_(2) < cdots0 \leq b_{0} \leq b_{0}+l_{0}<b_{1}<b_{1}+l_{1}<b_{2}<\cdots
Remark 1: In the existing adaptive ETS, t k + 1 = t k + t k + 1 = t k + t_(k+1)=t_(k)+t_{k+1}=t_{k}+ inf { l h ∣ e T ( i k h ) Φ 1 e ( i k h ) ≥ δ ( t k ) x T ( t k ) Φ x ( t k ) } inf l h ∣ e T i k h Φ 1 e i k h ≥ δ t k x T t k Φ x t k i n f{lh∣e^(T)(i_(k)h)Phi_(1)e(i_(k)h) >= delta(t_(k))x^(T)(t_(k))Phi x(t_(k))}\inf \left\{l h \mid e^{T}\left(i_{k} h\right) \Phi_{1} e\left(i_{k} h\right) \geq \delta\left(t_{k}\right) x^{T}\left(t_{k}\right) \Phi x\left(t_{k}\right)\right\}, the trigger parameter δ δ delta\delta can be adjusted adaptively, but they need to satisfy V ˙ ( t ) ≤ 0 V ˙ ( t ) ≤ 0 V^(˙)(t) <= 0\dot{V}(t) \leq 0 to ensure the system stability and H ∞ H ∞ H_(oo)H_{\infty} performance. Under the ETSs in [26], [35], the smaller parameter of the proposed ETS is used to maintain the system stability. In this work, inspired by the switched system framework, triggered parameter of the proposed switching ETS is regulated by the switching signals. In some intervals, the derivative can be non-negative for Lyapunov function, i.e., V ˙ ( t ) ≤ 2 α 2 V ( t ) V ˙ ( t ) ≤ 2 α 2 V ( t ) V^(˙)(t) <= 2alpha_(2)V(t)\dot{V}(t) \leq 2 \alpha_{2} V(t) are required only during the time interval t ∈ T ↑ [ b n , b n + 1 ) t ∈ T ↑ b n , b n + 1 t inT_(uarr)[b_(n),b_(n+1))t \in \mathcal{T}_{\uparrow}\left[b_{n}, b_{n+1}\right). The larger parameter of our switching ETS can be expected, which leads to the reduction in the amount of transmissions. The benefit can be verified in the Section IV.
备注 1：在现有的自适应 ETS 中， t k + 1 = t k + t k + 1 = t k + t_(k+1)=t_(k)+t_{k+1}=t_{k}+ inf { l h ∣ e T ( i k h ) Φ 1 e ( i k h ) ≥ δ ( t k ) x T ( t k ) Φ x ( t k ) } inf l h ∣ e T i k h Φ 1 e i k h ≥ δ t k x T t k Φ x t k i n f{lh∣e^(T)(i_(k)h)Phi_(1)e(i_(k)h) >= delta(t_(k))x^(T)(t_(k))Phi x(t_(k))}\inf \left\{l h \mid e^{T}\left(i_{k} h\right) \Phi_{1} e\left(i_{k} h\right) \geq \delta\left(t_{k}\right) x^{T}\left(t_{k}\right) \Phi x\left(t_{k}\right)\right\} 触发参数 δ δ delta\delta 可以自适应调整，但需要 V ˙ ( t ) ≤ 0 V ˙ ( t ) ≤ 0 V^(˙)(t) <= 0\dot{V}(t) \leq 0 满足以保证系统的稳定性和 H ∞ H ∞ H_(oo)H_{\infty} 性能。在[26]、[35]中的 ETS 下，所提出的 ETS 的较小参数用于维持系统稳定性。在这项工作中，受开关系统框架的启发，所提出的开关 ETS 的触发参数由开关信号调节。在某些区间内，导数对于 Lyapunov 函数可以是非负的，即仅 V ˙ ( t ) ≤ 2 α 2 V ( t ) V ˙ ( t ) ≤ 2 α 2 V ( t ) V^(˙)(t) <= 2alpha_(2)V(t)\dot{V}(t) \leq 2 \alpha_{2} V(t) 在时间间隔内才需要 t ∈ T ↑ [ b n , b n + 1 ) t ∈ T ↑ b n , b n + 1 t inT_(uarr)[b_(n),b_(n+1))t \in \mathcal{T}_{\uparrow}\left[b_{n}, b_{n+1}\right) 。可以预期我们的开关 ETS 的参数会更大，从而减少传输量。好处可以在第四节中验证。
As described in Fig. 2, the switching ETS is used to decide the transmitted signals. For t ∈ [ b n , b n + l n ) t ∈ b n , b n + l n t in[b_(n),b_(n)+l_(n))t \in\left[b_{n}, b_{n}+l_{n}\right), the signal is transmitted when the condition e T ( i k h ) Φ 1 [ ∗ ] ≥ δ 1 x T ( t k ) Φ 1 [ ∗ ] e T i k h Φ 1 [ ∗ ] ≥ δ 1 x T t k Φ 1 [ ∗ ] e^(T)(i_(k)h)Phi_(1)[**] >= delta_(1)x^(T)(t_(k))Phi_(1)[**]e^{T}\left(i_{k} h\right) \Phi_{1}[*] \geq \delta_{1} x^{T}\left(t_{k}\right) \Phi_{1}[*] is satisfied. For t ∈ [ b n + l n , b n + 1 ) t ∈ b n + l n , b n + 1 t in[b_(n)+l_(n),b_(n+1))t \in\left[b_{n}+l_{n}, b_{n+1}\right), the signal is transmitted when the condition e T ( i k h ) Φ 2 [ ∗ ] ≥ δ 2 x T ( t k ) Φ 2 [ ∗ ] e T i k h Φ 2 [ ∗ ] ≥ δ 2 x T t k Φ 2 [ ∗ ] e^(T)(i_(k)h)Phi_(2)[**] >= delta_(2)x^(T)(t_(k))Phi_(2)[**]e^{T}\left(i_{k} h\right) \Phi_{2}[*] \geq \delta_{2} x^{T}\left(t_{k}\right) \Phi_{2}[*] is satisfied. For the design of the switching ETS (2), the key principle is to construct the relationship between the parameters δ 1 , δ 2 δ 1 , δ 2 delta_(1),delta_(2)\delta_{1}, \delta_{2} and V ˙ ( x ) V ˙ ( x ) V^(˙)(x)\dot{V}(x). Under the proposed switching ETS, the stability can be ensured for the investigated system with desired H ∞ H ∞ H_(oo)H_{\infty} performance, see Theorems 1 and 2.
如图 2 所示，开关 ETS 用于确定发射信号。对于 t ∈ [ b n , b n + l n ) t ∈ b n , b n + l n t in[b_(n),b_(n)+l_(n))t \in\left[b_{n}, b_{n}+l_{n}\right) ，当满足条件 e T ( i k h ) Φ 1 [ ∗ ] ≥ δ 1 x T ( t k ) Φ 1 [ ∗ ] e T i k h Φ 1 [ ∗ ] ≥ δ 1 x T t k Φ 1 [ ∗ ] e^(T)(i_(k)h)Phi_(1)[**] >= delta_(1)x^(T)(t_(k))Phi_(1)[**]e^{T}\left(i_{k} h\right) \Phi_{1}[*] \geq \delta_{1} x^{T}\left(t_{k}\right) \Phi_{1}[*] 时传输信号。对于 t ∈ [ b n + l n , b n + 1 ) t ∈ b n + l n , b n + 1 t in[b_(n)+l_(n),b_(n+1))t \in\left[b_{n}+l_{n}, b_{n+1}\right) ，当满足条件 e T ( i k h ) Φ 2 [ ∗ ] ≥ δ 2 x T ( t k ) Φ 2 [ ∗ ] e T i k h Φ 2 [ ∗ ] ≥ δ 2 x T t k Φ 2 [ ∗ ] e^(T)(i_(k)h)Phi_(2)[**] >= delta_(2)x^(T)(t_(k))Phi_(2)[**]e^{T}\left(i_{k} h\right) \Phi_{2}[*] \geq \delta_{2} x^{T}\left(t_{k}\right) \Phi_{2}[*] 时传输信号。对于开关 ETS（2）的设计，关键原则是构建参数 δ 1 , δ 2 δ 1 , δ 2 delta_(1),delta_(2)\delta_{1}, \delta_{2} 与 V ˙ ( x ) V ˙ ( x ) V^(˙)(x)\dot{V}(x) 之间的关系。在所提出的开关 ETS 下，可以确保所研究系统的稳定性并具有所需的 H ∞ H ∞ H_(oo)H_{\infty} 性能，参见定理 1 和 2。
To establish our main results, the assumptions are stated as follows:
为了确定我们的主要结果，假设如下：
Assumption 1: Let N ( t ) = CARD { q ∈ N ∣ b q + l q ∈ [ 0 , t ) } N ( t ) = CARD q ∈ N ∣ b q + l q ∈ [ 0 , t ) N(t)=CARD{q inN∣b_(q)+l_(q)in[0,t)}N(t)=\operatorname{CARD}\left\{q \in \mathbb{N} \mid b_{q}+l_{q} \in[0, t)\right\}, where N ( t ) N ( t ) N(t)N(t) is the transition times of the proposed switching ETS during the [ 0 , t ) [ 0 , t ) [0,t)[0, t) period, and CARD is the element number of the set. The time interval { T ↑ [ b n , b n + 1 ) } n ∈ N T ↑ b n , b n + 1 n ∈ N {T_(uarr)[b_(n),b_(n+1))}_(n inN)\left\{\mathcal{T}_{\uparrow}\left[b_{n}, b_{n+1}\right)\right\}_{n \in \mathbb{N}} satisfies the switching frequency constraint of event-triggered conditions for a scalar τ D ∈ R > h τ D ∈ R > h tau_(D)inR_( > h)\tau_{D} \in \mathbb{R}_{>h}, and a scalar N 0 ∈ R ≥ 0 N 0 ∈ R ≥ 0 N_(0)inR_( >= 0)N_{0} \in \mathbb{R}_{\geq 0}, the following inequality
假设 1：设 N ( t ) = CARD { q ∈ N ∣ b q + l q ∈ [ 0 , t ) } N ( t ) = CARD q ∈ N ∣ b q + l q ∈ [ 0 , t ) N(t)=CARD{q inN∣b_(q)+l_(q)in[0,t)}N(t)=\operatorname{CARD}\left\{q \in \mathbb{N} \mid b_{q}+l_{q} \in[0, t)\right\} ，其中 N ( t ) N ( t ) N(t)N(t) 是该 [ 0 , t ) [ 0 , t ) [0,t)[0, t) 周期内拟议的切换 ETS 的转换时间，CARD 是集合的元素编号。时间间隔 { T ↑ [ b n , b n + 1 ) } n ∈ N T ↑ b n , b n + 1 n ∈ N {T_(uarr)[b_(n),b_(n+1))}_(n inN)\left\{\mathcal{T}_{\uparrow}\left[b_{n}, b_{n+1}\right)\right\}_{n \in \mathbb{N}} 满足标量 τ D ∈ R > h τ D ∈ R > h tau_(D)inR_( > h)\tau_{D} \in \mathbb{R}_{>h} 和 N 0 ∈ R ≥ 0 N 0 ∈ R ≥ 0 N_(0)inR_( >= 0)N_{0} \in \mathbb{R}_{\geq 0} 标量的事件触发条件的开关频率约束，以下不等式
N ( t ) ≤ N 0 + t τ D N ( t ) ≤ N 0 + t τ D N(t) <= N_(0)+(t)/(tau_(D))N(t) \leq N_{0}+\frac{t}{\tau_{D}}

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Figure 1: Fig. 1. Structure of switching event-triggered NCSs and its application to UAVs.
图 1：图 1.开关事件触发 NCS 的结构及其在无人机中的应用。

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Figure 2: Fig. 2. Design of switching ETS.
图 2：图 2.开关 ETS 的设计。
is satisfied for all τ D , t ∈ R ≥ 0 τ D , t ∈ R ≥ 0 tau_(D),t inR_( >= 0)\tau_{D}, t \in \mathbb{R}_{\geq 0}.
为所有人 τ D , t ∈ R ≥ 0 τ D , t ∈ R ≥ 0 tau_(D),t inR_( >= 0)\tau_{D}, t \in \mathbb{R}_{\geq 0} 提供满意。
Assumption 2: For the lengths of the first period, there exists a lower bound l min l min  l_("min ")l_{\text {min }}, i.e.,
假设 2：对于第一个周期的长度，存在一个下限 l min l min  l_("min ")l_{\text {min }} ，即
inf n ∈ N { l n } ≥ l min inf n ∈ N   l n ≥ l min i n f_(n inN){l_(n)} >= l_(min)\inf _{n \in \mathbb{N}}\left\{l_{n}\right\} \geq l_{\min }
Similarly, for the lengths of the second period, there exists a upper bound g max g max  g_("max ")g_{\text {max }}, i.e.,
同样，对于第二个周期的长度，存在一个上限 g max g max  g_("max ")g_{\text {max }} ，即
inf n ∈ N { b n + 1 − b n − l n } ≤ g max inf n ∈ N   b n + 1 − b n − l n ≤ g max i n f_(n inN){b_(n+1)-b_(n)-l_(n)} <= g_(max)\inf _{n \in \mathbb{N}}\left\{b_{n+1}-b_{n}-l_{n}\right\} \leq g_{\max }
Remark 2: Assumptions 1 and 2 are reasonable for the design of the switching ETS. In practice, to maintain the system stability, it is required that Lyapunov function V ( t ) V ( t ) V(t)V(t) decreases for some intervals while V ( t ) V ( t ) V(t)V(t) increases for some intervals. Moreover, the duration of the interval T ↑ [ b n , b n + 1 ) T ↑ b n , b n + 1 T_(uarr)[b_(n),b_(n+1))\mathcal{T}_{\uparrow}\left[b_{n}, b_{n+1}\right) cannot be too large. By using the switched system theory in [39], [43], the results can be directly extended to multiple trigger parameters. For simplicity, the proposed switching ETS only contains two trigger parameters δ 1 δ 1 delta_(1)\delta_{1} and δ 2 δ 2 delta_(2)\delta_{2}.
注 2：假设 1 和 2 对于开关 ETS 的设计是合理的。在实践中，为了保持系统稳定性，要求李雅普诺夫函数 V ( t ) V ( t ) V(t)V(t) 在某些区间内降低，在 V ( t ) V ( t ) V(t)V(t) 某些区间内增加。此外，间隔的持续时间 T ↑ [ b n , b n + 1 ) T ↑ b n , b n + 1 T_(uarr)[b_(n),b_(n+1))\mathcal{T}_{\uparrow}\left[b_{n}, b_{n+1}\right) 不能太大。通过使用[39]、[43]中的开关系统理论，可以将结果直接扩展到多个触发参数。为简单起见，所提出的开关 ETS 仅包含两个触发参数 δ 1 δ 1 delta_(1)\delta_{1} 和 δ 2 δ 2 delta_(2)\delta_{2} 。
Denote the set of transmitted signals under the time-triggered scheme S 1 = { 0 , h , … , m h , … } , m ∈ N S 1 = { 0 , h , … , m h , … } , m ∈ N S_(1)={0,h,dots,mh,dots},m inN\mathcal{S}_{1}=\{0, h, \ldots, m h, \ldots\}, m \in \mathbb{N}. Denote the set of transmitted signals determined by the switching ETS (2) S 2 = S 2 = S_(2)=\mathcal{S}_{2}= { t 0 , t 1 , t 2 , … , t k , … } t 0 , t 1 , t 2 , … , t k , … {t_(0),t_(1),t_(2),dots,t_(k),dots}\left\{t_{0}, t_{1}, t_{2}, \ldots, t_{k}, \ldots\right\}. Thus, S 2 ⊆ S 1 S 2 ⊆ S 1 S_(2)subeS_(1)\mathcal{S}_{2} \subseteq \mathcal{S}_{1}. By utilizing Zero Order
表示时间触发方案下的发射信号集 S 1 = { 0 , h , … , m h , … } , m ∈ N S 1 = { 0 , h , … , m h , … } , m ∈ N S_(1)={0,h,dots,mh,dots},m inN\mathcal{S}_{1}=\{0, h, \ldots, m h, \ldots\}, m \in \mathbb{N} 。表示由开关 ETS （2） S 2 = S 2 = S_(2)=\mathcal{S}_{2}= { t 0 , t 1 , t 2 , … , t k , … } t 0 , t 1 , t 2 , … , t k , … {t_(0),t_(1),t_(2),dots,t_(k),dots}\left\{t_{0}, t_{1}, t_{2}, \ldots, t_{k}, \ldots\right\} 确定的发射信号集。因此， S 2 ⊆ S 1 S 2 ⊆ S 1 S_(2)subeS_(1)\mathcal{S}_{2} \subseteq \mathcal{S}_{1} .通过利用零阶
Hold (ZOH), the controller can be updated:
保持 （ZOH），控制器可以更新：
u ( t ) = { K 1 x ( t k ) , t ∈ [ b n , b n + l n ) K 2 x ( t k ) , t ∈ [ b n + l n , b n + 1 ) u ( t ) = K 1 x t k , t ∈ b n , b n + l n K 2 x t k , t ∈ b n + l n , b n + 1 u(t)={[K_(1)x(t_(k))","t in[b_(n),b_(n)+l_(n))],[K_(2)x(t_(k))","t in[b_(n)+l_(n),b_(n+1))]:}u(t)=\left\{\begin{array}{l} K_{1} x\left(t_{k}\right), t \in\left[b_{n}, b_{n}+l_{n}\right) \\ K_{2} x\left(t_{k}\right), t \in\left[b_{n}+l_{n}, b_{n+1}\right) \end{array}\right.
where the controller gains K i , i ∈ { 1 , 2 } K i , i ∈ { 1 , 2 } K_(i),i in{1,2}K_{i}, i \in\{1,2\} are needed to be designed.
需要设计控制器增益 K i , i ∈ { 1 , 2 } K i , i ∈ { 1 , 2 } K_(i),i in{1,2}K_{i}, i \in\{1,2\} 的地方。
The interval Θ = [ t k , t k + 1 ) Θ = t k , t k + 1 Theta=[t_(k),t_(k+1))\Theta=\left[t_{k}, t_{k+1}\right) is split to Θ m = [ i k h , h + i k h ) Θ m = i k h , h + i k h Theta_(m)=[i_(k)h,h+i_(k)h)\Theta_{m}=\left[i_{k} h, h+i_{k} h\right), i.e. Θ = ⋃ Θ m Θ = ⋃ Θ m Theta=uuuTheta_(m)\Theta=\bigcup \Theta_{m}, where i k h = l h + t k h , l = 0 , 1 , … , l max i k h = l h + t k h , l = 0 , 1 , … , l max i_(k)h=lh+t_(k)h,l=0,1,dots,l_(max)i_{k} h=l h+t_{k} h, l=0,1, \ldots, l_{\max } with l max = t k + 1 − t k − 1 l max  = t k + 1 − t k − 1 l_("max ")=t_(k+1)-t_(k)-1l_{\text {max }}=t_{k+1}-t_{k}-1. Denote τ ( t ) ≜ t − i k h , t ∈ Θ m τ ( t ) ≜ t − i k h , t ∈ Θ m tau(t)≜t-i_(k)h,t inTheta_(m)\tau(t) \triangleq t-i_{k} h, t \in \Theta_{m}, thus we obtain h ≥ τ ( t ) ≥ 0 h ≥ τ ( t ) ≥ 0 h >= tau(t) >= 0h \geq \tau(t) \geq 0, i.e., h h hh is the upper bound of τ ( t ) τ ( t ) tau(t)\tau(t).
区间 Θ = [ t k , t k + 1 ) Θ = t k , t k + 1 Theta=[t_(k),t_(k+1))\Theta=\left[t_{k}, t_{k+1}\right) 被拆分为 Θ m = [ i k h , h + i k h ) Θ m = i k h , h + i k h Theta_(m)=[i_(k)h,h+i_(k)h)\Theta_{m}=\left[i_{k} h, h+i_{k} h\right) ，即 Θ = ⋃ Θ m Θ = ⋃ Θ m Theta=uuuTheta_(m)\Theta=\bigcup \Theta_{m} ，其中 i k h = l h + t k h , l = 0 , 1 , … , l max i k h = l h + t k h , l = 0 , 1 , … , l max i_(k)h=lh+t_(k)h,l=0,1,dots,l_(max)i_{k} h=l h+t_{k} h, l=0,1, \ldots, l_{\max } 与 l max = t k + 1 − t k − 1 l max  = t k + 1 − t k − 1 l_("max ")=t_(k+1)-t_(k)-1l_{\text {max }}=t_{k+1}-t_{k}-1 。表示 τ ( t ) ≜ t − i k h , t ∈ Θ m τ ( t ) ≜ t − i k h , t ∈ Θ m tau(t)≜t-i_(k)h,t inTheta_(m)\tau(t) \triangleq t-i_{k} h, t \in \Theta_{m} ，因此我们得到 h ≥ τ ( t ) ≥ 0 h ≥ τ ( t ) ≥ 0 h >= tau(t) >= 0h \geq \tau(t) \geq 0 ，即 h h hh 是 的 τ ( t ) τ ( t ) tau(t)\tau(t) 上限。
Thus, the investigated system is described by
因此，所研究的系统由
{ x ˙ ( t ) = A x ( t ) + B ω ω ( t ) + B K 1 [ e ( i k h ) + x ( t − τ ( t ) ) ] t ∈ [ b n , b n + l n ) x ˙ ( t ) = A x ( t ) + B ω ω ( t ) + B K 2 [ e ( i k h ) + x ( t − τ ( t ) ) ] t ∈ [ b n + l n , b n + 1 ) x ˙ ( t ) = A x ( t ) + B ω ω ( t ) + B K 1 e i k h + x ( t − τ ( t ) ) t ∈ b n , b n + l n x ˙ ( t ) = A x ( t ) + B ω ω ( t ) + B K 2 e i k h + x ( t − τ ( t ) ) t ∈ b n + l n , b n + 1 {[x^(˙)(t)=Ax(t)+B_(omega)omega(t)+BK_(1)[e(i_(k)h)+x(t-tau(t))]],[t in[b_(n),b_(n)+l_(n))],[x^(˙)(t)=Ax(t)+B_(omega)omega(t)+BK_(2)[e(i_(k)h)+x(t-tau(t))]],[t in[b_(n)+l_(n),b_(n+1))]:}\left\{\begin{array}{l} \dot{x}(t)=A x(t)+B_{\omega} \omega(t)+B K_{1}\left[e\left(i_{k} h\right)+x(t-\tau(t))\right] \\ t \in\left[b_{n}, b_{n}+l_{n}\right) \\ \dot{x}(t)=A x(t)+B_{\omega} \omega(t)+B K_{2}\left[e\left(i_{k} h\right)+x(t-\tau(t))\right] \\ t \in\left[b_{n}+l_{n}, b_{n+1}\right) \end{array}\right.
The two conditions in the switching ETS (2) are regulated by the switching signal, then the investigated system is modelled as a switched system. The paper aims to obtain the parameters δ 1 δ 1 delta_(1)\delta_{1} and δ 2 δ 2 delta_(2)\delta_{2} of switching ETS (2) and mode-dependent controller gains K 1 K 1 K_(1)K_{1} and K 2 K 2 K_(2)K_{2}. The objectives need to be satisfied are listed as follows:
开关 ETS（2）中的两个条件由开关信号调节，然后将所研究的系统建模为开关系统。该文旨在获得开关 ETS（2）和模式相关控制器增益的参数 δ 1 δ 1 delta_(1)\delta_{1} 和 δ 2 δ 2 delta_(2)\delta_{2} K 2 K 2 K_(2)K_{2} 。 K 1 K 1 K_(1)K_{1} 需要满足的目标如下：
  1. The asymptotic stability is maintained when ω ( t ) = 0 ω ( t ) = 0 omega(t)=0\omega(t)=0;
    当 ω ( t ) = 0 ω ( t ) = 0 omega(t)=0\omega(t)=0 时保持渐近稳定性;
  2. γ 2 ∫ 0 ∞ ω T ( t ) ω ( t ) d t ≥ ∫ 0 ∞ z T ( t ) z ( t ) d t γ 2 ∫ 0 ∞   ω T ( t ) ω ( t ) d t ≥ ∫ 0 ∞   z T ( t ) z ( t ) d t gamma^(2)int_(0)^(oo)omega^(T)(t)omega(t)dt >= int_(0)^(oo)z^(T)(t)z(t)dt\gamma^{2} \int_{0}^{\infty} \omega^{T}(t) \omega(t) d t \geq \int_{0}^{\infty} z^{T}(t) z(t) d t holds for any x ( 0 ) = 0 x ( 0 ) = 0 x(0)=0x(0)=0 and ω ( t ) ≠ 0 ω ( t ) ≠ 0 omega(t)!=0\omega(t) \neq 0.
    对任何和 x ( 0 ) = 0 x ( 0 ) = 0 x(0)=0x(0)=0 ω ( t ) ≠ 0 ω ( t ) ≠ 0 omega(t)!=0\omega(t) \neq 0 都 γ 2 ∫ 0 ∞ ω T ( t ) ω ( t ) d t ≥ ∫ 0 ∞ z T ( t ) z ( t ) d t γ 2 ∫ 0 ∞   ω T ( t ) ω ( t ) d t ≥ ∫ 0 ∞   z T ( t ) z ( t ) d t gamma^(2)int_(0)^(oo)omega^(T)(t)omega(t)dt >= int_(0)^(oo)z^(T)(t)z(t)dt\gamma^{2} \int_{0}^{\infty} \omega^{T}(t) \omega(t) d t \geq \int_{0}^{\infty} z^{T}(t) z(t) d t 成立。
    Remark 3: In [41], [42], due to the ETS with a waiting time, the investigated system is modelled as a switching between the system under periodic sampling and the system under continuous ETS. In [30], [44], the studied system is modelled as a switching between between a delay-free system and a time-delay system. In this paper, two switching subsystems with time-delay and error is established under the switching ETS.
    注 3：在[41]、[42]中，由于 ETS 有等待时间，因此将所研究的系统建模为定期采样下的系统和连续 ETS 下的系统之间的切换。在[30]、[44]中，所研究的系统被建模为无延迟系统和延时系统之间的切换。本文在开关 ETS 下建立了两个具有时延和误差的开关子系统。

III. Main Results  三、主要成果

Theorems 1 and 2 analyze the stability and H ∞ H ∞ H_(oo)H_{\infty} performance for NCSs under the proposed switching ETS (2), respectively.
定理 1 和定理 2 分别分析了 NCS 在所提出的开关 ETS 下的稳定性和 H ∞ H ∞ H_(oo)H_{\infty} 性能（2）。
  1. Manuscript received 6 July 2023; revised 14 October 2023; accepted 15 November 2023. Date of publication 28 November 2023; date of current version 22 April 2024. The work was supported in part by the National Natural Science Foundation of China under Grant 62303131, in part by the Science Center Program of National Natural Science Foundation of China under Grant 62188101, in part by InnoHK program, National Natural Science Foundation of China under Grants 61773357, 71971181, and 72032005, in part by the Research Grant Council of Hong Kong under Grants 11203519 and 11200621, in part by Sichuan Science and Technology Program under Grant 2023YFSY0003, and in part by Hong Kong Innovation and Technology Commission (InnoHK Project CIMDA). The review of this article was coordinated by Prof. Shima Nazari. (Corresponding author: Yan Wang.)
    2023 年 7 月 6 日收到稿件;2023 年 10 月 14 日修订;2023 年 11 月 15 日接受。发布日期：2023 年 11 月 28 日;当前版本的日期为 2024 年 4 月 22 日。这项工作部分由国家自然科学基金委员会 62303131 资助，部分由国家自然科学基金科学中心计划资助 62188101 资助，部分由 InnoHK 创新香港研发平台计划、国家自然科学基金委员会资助 61773357、71971181 和 72032005 资助，部分由香港研究资助局资助 11203519 和 11200621 资助。 部分由四川省科技计划（2023YFSY0003）资助，部分由香港创新科技署（InnoHK 创新香港研发平台项目）资助。本文的审阅由 Shima Nazari 教授协调。（通讯作者：王艳。
    Zhiying Wu is with Centre for Artificial Intelligence and Robotics, Hong Kong Institute of Science and Innovation, Chinese Academy of Sciences, Hong Kong (e-mail: wzy100@mail.ustc.edu.cn).
    吴志英现任中国科学院香港科学创新研究院人工智能与机器人研究中心工作（电子邮件：wzy100@mail.ustc.edu.cn）。
    Yan Wang is with the School of Mechanical Engineering and Automation, Harbin Institute of Technology Shenzhen, Shenzhen 518055, China (e-mail: wangyany@mail.ustc.edu.cn).
    王艳现就职于中国 518055 深圳哈尔滨工业大学机械工程与自动化学院（电子邮件：wangyany@mail.ustc.edu.cn）。
    Junlin Xiong is with the Department of Automation, University of Science and Technology of China, Hefei 230026, China (e-mail: junlin.xiong@gmail.com).
    熊俊林就读于中国科学技术大学自动化系，合肥 230026（邮箱：junlin.xiong@gmail.com）。
    Min Xie is with the Department of Advanced Design and Systems Engineering, City University of Hong Kong, Kowloon, Hong Kong, and also with the Center for Intelligent Multidimensional Data Analysis, Hong Kong Science Park, Shatin, Hong Kong (e-mail: minxie@cityu.edu.hk).
    谢敏现任教于香港九龙市立大学先进设计与系统工程系，以及香港沙田香港科学园智能多维数据分析中心（电子邮件：minxie@cityu.edu.hk）。
    Digital Object Identifier 10.1109/TVT.2023.3336573
    数字对象标识符 10.1109/TVT.2023.3336573