I. INTRODUCTION

CYBER-PHYSICAL systems (CPSs) are a kind of novel complex systems which tightly integrate the dynamics of the physical processes with communication,computation and control (3C)^[1]. In CPSs,the networks and embedded computers are used to transmit data,process information and control a large number of local physical processes^[2]. CPSs have been used in some practical systems,e.g.,intelligent transportation,smart grids and intelligent robotic systems^{[3, 4]}.

Compared with the traditional control system,CPSs have several new features. Firstly,the local physical processes in CPSs are geographically distributed,so CPSs are a kind of typical systems which are composed of many smaller subsystems (also called "agents"). Each agent gathers information through the communication network and makes control decisions with other agents together. Multi-agent systems need to be controlled by distributed cooperative control strategy to ensure the performance of the whole system. One of the key issues of distributed cooperative control for multi-agent systems is distributed (or decentralized) average consensus control problem^{[5, 6, 7, 8]}. The average consensus problem means to design a network protocol such that,as time goes on,the states of all agents asymptotically reach the average of the initial states^[5].

Secondly,communication is one key component of CPSs,which influences the performance of distributed cooperative control for multi-agent systems. However,communication network is not always reliable,so the communication constraints including communication delay and data dropouts are inevitably introduced and may degrade the system performance or even cause the system instability. Some researches focus on the consensus problems with communication delay for first-order^[5] or second-order systems^[8], having fixed or switching topology^[9]. A necessary and sufficient condition was derived for multi-agent systems with heterogeneous time delays to achieve average consensus^[10]. The consensus problem with dynamically changing topologies, time-varying delays and data packet dropouts was also investigated^[11].

Thirdly,a large number of data are generated by various sensors which represent the different characteristics of physical processes. However,there exist a large number of redundant data within these data which bring greater pressure to the network transmission. It is necessary to propose an event-triggered sampling mechanism for multi-agent systems^{[12, 13]} as an alternative to traditional time-triggered sampling (i.e.,the fixed sampling period). Using the event-triggered mechanism,the data can be transmitted only when the triggered condition is satisfied,which improves the effective transmission of data by reducing communication network burden.

Considering the new characteristics of CPSs mentioned above, researchers studied decentralized event-triggered control for multi-agent systems without communication constraints^[12], where each agent sends its local information to the network only when the norm of measurement error crosses a certain threshold. With decentralized event-triggered control,the triggered instants were reduced without continuous monitoring the states of neighbors^[14]. Moreover,the recent researches of observer-based output feedback event-triggered control^[15] and leader-following exponential consensus problem with distributed event-triggered control^[16] were presented.

Furthermore,there are also some researches about decentralized event-triggered problem with partial communication constraints^{[17, 18, 19]}. Considering communication delays,the novel event-based scheduling strategy was studied^[17],which was considered as the time-dependent function. Considering directed graphs,decentralized event-triggered control of discrete-time heterogeneous multi-agent systems was investigated^[18]. Considering communication delays and data dropouts,the event-triggered data transmission in distributed networked control systems was examined^{[19, 20]}. However,how to get the quantitative relationship between communication constraints and system stability for undirected fixed network is still an open issue. It is very important for system performance of multi-agent systems.

Based on the above observation,the previous research results we focused on were the average consensus problem for multi-agent systems with centralized event-triggered mechanism and only communication delays^{[21, 22]}. Considering the geographically distributed agent and networked-related non-deterministic issues, the further study in this paper is mainly concerned with decentralized event-triggered average consensus problem for multi-agent systems with communication delays and data dropouts in an undirected network.

The generally used decentralized event-triggered functions in the existing work^{[12, 13, 14, 15, 16]},all such functions have the similar form,i.e.,$e_i^2 \le {c_i}z_i^2$. Compared with this form,a different event-triggered mechanism is introduced with the form of $e_i^{\rm{T}}{\Phi _i}{e_i} \le {\sigma _i}z_i^{\rm{T}}{\Phi _i}{z_i}$^{[18, 19]}. It has several parameters to adjust the event-triggered threshold,thus it is more general. Under the event-triggered mechanism^{[18, 19]},we will solve the average consensus problem based on the theoretical framework for multi-agent systems with an undirected,connected network topology as in [9]. Furthermore,using the time-delay system model in [10],this paper will present the design of decentralized event-triggered average consensus for continuous-time multi-agent systems with communication delays and data dropouts.

Two main contributions are as follows: firstly,under the decentralized event-triggered mechanism,a reduced dimension hybrid system model with communication delay and data dropouts is proposed,where the decentralized event-triggered mechanism, communication delays and data dropouts are taken into account within one framework. Secondly,using reciprocally convex approach and Lyapunov-Krasovskii stability theory,sufficient conditions that make all the agents achieve the average consensus asymptotically are presented. Furthermore,the upper bound of communication delay and the maximal allowable number of successive data dropouts (MANSD) can be obtained,which can conveniently provide the relationship between the triggering parameters, communication constraints and the system stability. Specially,the quantitative relationship between the triggering parameters,MANSD and the system stability is derived.

The remainder of the paper is organized as follows. Section II presents the problem statement. System model with decentralized event-triggered mechanism and only communication delays is established in Section III. Section IV presents the proposed approach with decentralized event-triggered mechanism and communication delays. Section V gives the results of decentralized event-triggered mechanism with communication delays and data dropouts. Simulation results are described in Section VI. Section VII concludes the paper.

II. PROBLEM STATEMENT

Supposed that a networked subsystem of CPSs consists of $N$ agents,and each agent is the single-integrator system with dynamics:

\begin{align} {\dot x_i}(t) = {u_i}(t),i \in \left\{ {I = 1,2, \ldots ,N} \right\}, \end{align}

(1)

where ${x_i}(t) \in {{\bf R}^M}$ denotes the state (or value) and ${u_i}(t) \in {{\bf R}^M}$ denotes the control input (or protocol) of the ${i}$- th agent. Supposed that agent $i$ is regarded as a node ${v_i}$,then the network is defined by the graph $G$ with the sets of nodes $V = \left\{ {{v_1},\ldots ,{v_N}} \right\}$ and edges $E = \{ {e_{ij}} = ({v_i},{v_j})\} \subseteq V \times V$. For an undirected graph with nodes $N$,the set of neighbors of node ${v_i}$ is denoted as ${N_i} = \left\{ {\left. {{v_j} \in V} \right|({v_i},{v_j}) \in E} \right\}$. $A = {[{a_{ij}}]_{N \times N}}$ is a communication adjacency matrix with nonnegative adjacency elements ${a_{ij}}$,where ${a_{ii}} = 0$ and ${a_{ij}} > 0$ (${v_j} \in {N_i}$). The degree matrix is $D = {\rm diag}\{{d_{11}},{d_{22}},\ldots ,{d_{NN}}\}$,where ${d_{ii}} = \sum\nolimits_{j = 1}^N {{a_{ij}}}$. The Laplacian graph of $G$ is defined as $L = D - A$. It can be easily obtained that every row sum of $L$ is zero and thus ${1_N} = {[1,\;1,\; \ldots ,\;1]^{\rm{T}}} \in {{\bf{R}}^N}$ is an eigenvector of $L$ associated with the eigenvalue $\lambda = 0$. Therefore,this means that ${\rm Rank}(L) \le N - 1$.

In this paper,we assume that the undirected graph $G$ is connected. The $G$ is called connected if and only if there is a path between any two distinct nodes. Furthermore,it is easy to know that the Laplacian $L$ of an undirected graph $G$ satisfies $L{{\rm{1}}_N}{\rm{ = 0}}$ and $1_N^TL = 0$.

The consensus problem without communication delays has been studied,and the consensus control protocol was given by^[9]

\begin{align} {u_i}(t) = \sum\limits_{{v_j} \in {N_i}} {({x_j} - {x_i})} . \end{align}

(2)

Here with the consensus protocol,each agent consists of a controller,and the dynamics is shown in Fig. 1.

We say the nodes of a network have reached a consensus if and only if ${x_i} = {x_j}$ for all $i \ne j,i,j \in I$. Whenever the nodes of a network are all in agreement,the value of all nodes is called the group decision value. Particularly,for all $i,j \in I$,if there exists

\begin{align} \mathop {\lim }\limits_{t \to \infty } {x_i}(t) = \frac{1}{N}\sum\limits_{j = 1}^N {{x_j}(0)} , \end{align}

(3)

then ${u_i}(t)$ asymptotically solves the average consensus problem.

In this paper,we discuss the consensus problem with communication delays and the consensus control protocol was given by^[10]

\begin{align} {u_i}(t) = \sum\limits_{{v_j} \in {N_i}} {({x_j}(t - {\tau _{ij}}) - {x_i}(t - {\tau _{ij}}))} . \end{align}

(4)

In many real multi-agent systems,the control input ${u_i}(t)$ updates its value with fixed period $h$,i.e.,each agent is regularly sampled with period $h$,where the monotonically increasing sampling sequence is described by the set ${M_1} = \{ 0,h,2h,\ldots ,kh\}$,$k \in {\bf N}$.

This time-triggered sampling (i.e.,the fixed sampling period) leads to heavy burden of the network. Therefore,it is necessary to propose an event-triggered sampling mechanism,which determines agent $i$ sends a message or not to its neighbors through network. With event-triggered mechanism,the broadcast release sequence is described by the set ${M_2} = \{ r_0^ih,r_1^ih,r_2^ih,\ldots ,r_k^ih,\ldots \} \subseteq {M_1},$ $k \leq r_k^i \in {\bf N}.$

Remark 1. When the condition ${M_2} \subset {M_1}$ is obtained,the numbers of sampled data are reduced with event-triggered mechanism,which can reduce communication burden. Specially,if ${M_2} = {M_1}$ it means all sampled data are transmitted and the event-triggered mechanism becomes a time-triggered scheme.

Considering the unreliability of communication network,the communication delay and data dropouts are introduced. Assumed that only data packet dropouts happen in communication,the successful broadcast release of agent $i$ is characterized by monotonically increasing sequences ${M_3} = \{ t_0^ih, t_1^ih, t_2^ih,\ldots ,t_k^ih,\ldots \} \subseteq {M_2}$,$r_k^i \leq t_k^i \in {\bf N}$. Furthermore,considering communication delays, the successful transmission finishing instant of sampled value is described by the set ${M_4} = \{ t_0^ih,t_1^ih + \tau _{{t_1}}^i,t_2^ih + \tau _{{t_2}}^i,\ldots ,t_k^ih + \tau _{{t_k}}^i,\ldots \} $,where the $\tau _{{t_k}}^i$ denotes the time delay at successful transmission instant $t_k^ih$ and is bounded with $\tau _{{t_k}}^i \in [0,{\bar \tau ^i}]$.

Remark 2. If all broadcast release data are successfully transmitted without data dropouts,we have $t_k^i = r_k^i$,i.e., ${M_3} = {M_2}$. Then only considering communication delays,we can still use the sequence set ${M_4} $ to describe the successful transmission finishing instant.

III. SYSTEM MODEL WITH DECENTRALIZED EVENT-TRIGGERED MECHANISM AND ONLY COMMUNICATION DELAYS

From Remark 2,considering event-triggered mechanism and only communication delays,we have ${M_3} = {M_2}$. Define the state measurement error of agent $i$ as follows:

\begin{align} {e_i}(n_k^ih) = {x_i}(n_k^ih) - {x_i}(t_k^ih), \end{align}

(5)

$n_k^ih=t_k^ih+lh,l\in {\bf Z}$,and ${e_i}(n_k^ih)$ is the error between the current sampled value ${x_i}(n_k^ih)$ and the latest transmission sampled value ${x_i}(t_k^ih)$.

Then the event-triggered transmission scheme is designed as^{[18, 19]}

\begin{align} \begin{array}{l} {f_i}(n_k^ih,t_k^ih) = e_i^{\rm{T}}(n_k^ih){\Phi _i}{e_i}(n_k^ih)\quad \\ - \gamma x_i^{\rm{T}}(t_k^ih){\Phi _i}{x_i}(t_k^ih) \ge 0, \end{array} \end{align}

(6)

where $0 \le \gamma < 1$ is a given and bounded positive scalar parameter,and ${\Phi _i}$ is a positive definite matrix.

If the condition (6) is satisfied,the current sampled information ${x_i}(n_k^ih)$ is transmitted,or ${f_i}((n_k^ih,t_k^ih) \le 0$, i.e.,${x_i}(n_k^ih)$ will not be transmitted. Obviously,the released states ${x_i}(n_k^ih)$ is the subsequences of the latest transmission sampled value,which is denoted by ${x_i}(t_{k + 1}^ih)$.

Remark 3. The event-triggered transmission scheme (6) is decentralized. Agent $i$ only requires its own state measurement error and state value to update the event-triggered instant $t_0^ih,t_1^ih,\ldots ,t_k^ih,\ldots $. Due to different agents,the event-triggered instants are different. Compared with centralized event-triggered mechanism^[22],it is not necessary to require the local and its neighbor information at the same time in order to implement the event-triggered condition. Moreover,the matrix $\Phi_i$ in (6) is a positive definite parameter matrix,which is applied to adjust the weight of the event-triggered threshold. Furthermore,considering the distributed event-triggered functions in [12, 13, 14, 15, 16],they all have the similar form of $e_i^2 \le {c_i}z_i^2$ and can be obtained by defining ${\Phi _i} = I$ in (6). Therefore,compared with the event-triggered functions in [12, 13, 14, 15, 16],the event-triggered condition in this paper is more general.

Considering decentralized event-triggered mechanism and communication delays,the sampled value is obtained at the sequence of event $t_0^ih,t_1^ih + \tau _{{t_1}}^i,t_2^ih + \tau _{{t_2}}^i,\ldots ,t_k^ih + \tau _{{t_k}}^i,\ldots$ through network. Since the control input at the actuator is generated by a zero-order-holder (ZOH),the value of the input ${u_i}(t)$ with the holding time $t \in [t_k^ih + \tau _{{t_k}}^i, t_{k + 1}^ih + \tau _{{t_{k + 1}}}^i)$ is held constant and equal to the last control update,i.e.,

\begin{align} {u_i}(t) = {u_i}(t_k^ih),t \in [t_k^ih + \tau _{{t_k}}^i,t_{k + 1}^ih + \tau _{{t_{k + 1}}}^i), \end{align}

(7)

and thus the proposed decentralized consensus law of (4) is defined as

\begin{align} \begin{array}{*{20}{l}} {{u_i}(t) = \sum\limits_{{v_j} \in {N_i}} {({x_j}(t_{k'(t)}^jh) - {x_i}(t_k^ih))} ,}\\ {\quad t \in [t_k^ih + \tau _{{t_k}}^i,t_{k + 1}^ih + \tau _{{t_{k + 1}}}^i),} \end{array} \end{align}

(8)

where $k'(t) \triangleq \arg \mathop {\min }\limits_{m \in {{\bf N}}:t \geq t_m^j} \left\{ {t - t_m^j} \right\}$.

Remark 4. With the decentralized consensus law (8),each agent $i$ updates its own control input at event times $t_0^ih, t_1^ih,\ldots ,t_k^ih,\ldots $,as well as at the last event times of its neighboring agents $t_0^j,t_1^jh,\ldots ,{v_j} \in {N_i}.$

The dynamic system (1) with decentralized consensus law (8) is given by

\begin{align} \begin{array}{*{20}{l}} {{{\dot x}_i}(t) = - \sum\limits_{{v_j} \in {N_i}} {({x_i}(t_k^ih) - {x_j}(t_{k'(t)}^jh))} ,}\\ {\quad t \in [t_k^ih + \tau _{{t_k}}^i,t_{k + 1}^ih + \tau _{{t_{k + 1}}}^i).} \end{array} \end{align}

(9)

For system (9),to achieve a detailed timing analysis,the holding time interval is divided as follows^[19]:

\begin{align} \begin{array}{*{20}{l}} {\Omega = [t_k^ih + \tau _{{t_k}}^i,t_{k + 1}^ih + \tau _{{t_{k + 1}}}^i) = \cup {\Omega _l},}\\ {{\Omega _l} = [n_k^ih + \tau _{{n_k}}^i,n_k^ih + h + \tau _{{n_k} + 1}^i),} \end{array} \end{align}

(10)

where $n_k^ih = t_k^ih + lh$,$l = 0,1,\ldots ,t_{k + 1}^i - t_k^i - 1$. If $l = t_{k + 1}^i - t_k^i - 1$,then $\tau _{{n_k} + 1}^i = \tau _{{t_{k + 1}}}^i$,otherwise,$\tau _{{n_k}}^i = \tau _{{t_k}}^i$.

Define

\begin{align} {\eta _i}(t) = t - n_k^ih,t \in {\Omega _l}. \end{align}

(11)

From (11),it can be obtained that

\begin{align} \eta _1^i < {\eta _i}(t) \le \eta _3^i,t \in {\Omega _l}, \end{align}

(12)

where $\eta _1^i = \inf \{ \tau _{_{{t_k}}}^i\} > 0$ and $\eta _{_3}^i \buildrel \Delta \over = h + {\bar \tau ^i}$.

According to the aforementioned definitions and analysis,it is noted that this definition of $k'(t)$ implies ${x_j}(t_{k'(t)}^jh) = {x_j}(n_k^jh) - {e_j}(n_k^jh)$. Thus it is seen that (9) can be equivalently written as

\begin{align} \begin{array}{*{20}{l}} {{{\dot x}_i}(t) = - \sum\limits_{{v_j} \in {N_i}} {({x_i}(n_k^ih) - {x_j}(n_k^jh))} }\\ {\qquad + \sum\limits_{{v_j} \in {N_i}} {({e_i}(n_k^ih) - {e_j}(n_k^jh))} }\\ {\qquad = - \sum\limits_{{v_j} \in {N_i}} {({x_i}(t - {\eta _i}(t)) - {x_j}(t - {\eta _j}(t))} }\\ {\qquad + \sum\limits_{{v_j} \in {N_i}} {({e_i}(n_k^ih) - {e_j}(n_k^jh))} ,t \in {\Omega _l}.} \end{array} \end{align}

(13)

Denoting

$$x(t) = {\left[{{x_1}(t),\ldots ,{x_N}(t)} \right]^{\rm T}},$$

$$x(t - \eta (t)) = {\left[{{x_1}(t - {\eta _1}(t)),\ldots ,{x_N}(t - {\eta _N}(t)} \right]^{\rm T}},$$

$$e(n_k^{}h) = {\left[{{e_1}(n_k^1h),\ldots ,{e_N}(n_k^Nh)} \right]^{\rm T}},$$

the system (13) with fixed topology $G$ is described as

\begin{align} \dot x(t) = - \left( {L \otimes {I_M}} \right)\left( {x(t - \eta (t)) - e({n_k}h)} \right). \end{align}

(14)

Moreover,between $t_k^ih$ and $t_k^ih + lh$,no control signal is triggered,i.e.,

\begin{align} \begin{array}{*{20}{l}} {{f_i}(n_k^ih,t_k^ih) = e_i^{\rm{T}}(n_k^ih){\Phi _i}{e_i}(n_k^ih)}\\ {\quad - \gamma x_i^{\rm{T}}(t_k^ih){\Phi _i}{x_i}(t_k^ih) < 0.} \end{array} \end{align}

(15)

Therefore,with decentralized event-triggered transmission scheme, the successful broadcast release instant $t_{k + 1}^ih$ is defined by

\begin{align} t_{k + 1}^ih = t_k^ih + \mathop {\min }\limits_n \{ \left. {nh} \right|e_i^{\rm{T}}(n_k^ih){\Phi _i}{e_i}(n_k^ih) \ge \gamma {\psi _i}\} , \end{align}

(16)

where ${\psi _i} = x_i^{\rm T}(t_k^ih){\Phi _i}{x_i}(t_k^ih), n \in {\bf N}$.

The following task is to find the conditions for system (14) to reach average consensus with the event-triggered communication scheme (16).

IV. THE PROPOSED APPROACH A. Problem Setting

Consider system (15) with undirected,connected network topology. By the definition (3),in order to achieve average consensus,we denote $\alpha (t) = Ave(x) = \frac{1}{N}\sum\nolimits_i {{x_i}(t} )$ by the average of the agents’ states firstly. Due to the undirected graph,the Laplacian $L$ satisfies $1_N^TL = 0$. Therefore,there is $\left( {1_N^{\rm{T}} \otimes 1_M^{\rm{T}}} \right)\left( {L \otimes {I_M}} \right) = 0$. For system (14),it implies $\left( {1_N^{\rm{T}} \otimes 1_M^{\rm{T}}} \right)\dot x = - {\rm{ }}\left( {1_N^{\rm{T}}L \otimes 1_M^{\rm{T}}} \right)\left( {x(t - \eta (t) - e({n_k}h)} \right) = 0,$ i.e.,$\sum\nolimits_{i = 1}^N {{{\dot x}_i}(t)} = 0.$ Then the time derivative of $\alpha (t)$ is given by $\dot \alpha (t) = \frac{1}{N}\sum\nolimits_i {{{\dot x}_i}(t} ) = 0.$ So that $\alpha = Ave(x) = Ave(x(0)) = \frac{1}{N}\sum\nolimits_i {{x_i}(0} ),$ i.e.,$\alpha = Ave(x)$ is an invariant quantity. Then the state vector $x(t)$ can be decomposed as^[9]

\begin{align} x(t) = \alpha \left( {{\rm{ }}{1_N} \otimes {1_M}} \right) + \delta (t), \end{align}

(17)

i.e.,

${x_i}(t) = \alpha {1_M} + {\delta _i}(t),$

where $\sum\limits_i {{\delta _i}(t)} = 0.$

Substituting (17) into (14),we have

\begin{align} \dot \delta (t) = - \left( {L \otimes {I_M}} \right)\left( {\delta (t - \eta (t) - e({n_k}h)} \right), \end{align}

(18)

where $\delta (t) = {\left[{{\delta _1}(t),\ldots ,{\delta _N}(t)} \right]^{\rm T}},$ $\delta (t - \eta (t)) = {\left[ {{\delta _1}(t - {\eta _1}(t)),\ldots ,{\delta _N}(t - {\eta _N}(t)} \right]^{\rm T}},$ $e({n_k}h) ={[{{e_1}(n_k^1h), \ldots ,}}$ ${e_N}(n_k^Nh)]^{\rm T}$,and ${\eta _1} = \min \{ \eta _1^i(t)\} \le \eta (t) \le {\eta _N} = \max \{ \eta _3^i(t)\}$.

Denoting $\delta ({t_k}h) = {\left[{{\delta _1}(t_k^1h),\ldots ,{\delta _N}(t_k^Nh)} \right]^{\rm T}},$ with the transformation

$$\tilde \delta (t) = {(U \otimes {I_M})^{\rm T}}\delta (t),$$ $$\tilde \delta (t - \eta (t)) = {(U \otimes {I_M})^{\rm T}}\delta (t - \eta (t)),$$ $$\tilde e({n_k}h) = {(U \otimes {I_M})^{\rm T}}e({n_k}h),$$ $$\tilde \delta ({t_k}h) = {(U \otimes {I_M})^{\rm T}}\delta ({t_k}h),$$

the system (18) can be described by

\begin{align} \begin{array}{*{20}{c}} \cdot \\ {\tilde \delta } \end{array}(t) = - \left( {\left( {{U^{\rm{T}}}LU} \right) \otimes {I_M}} \right)\left( {\tilde \delta (t - \eta (t)) - \tilde e({n_k}h)} \right). \end{align}

(19)

Lemma 1^[23].Given

$U = {\rm{ }}\left[ {{U_1}\frac{{{1_N}}}{{\sqrt N }}} \right] \in {{\bf{R}}^{N \times N}},$

where $U$ is an orthogonal matrix,i.e.,${U^{\rm T}}U = {I_N}.$ ${U_1} \in {{\bf R} ^{N \times (N - 1)}}$ represents the first $N$-1 columns of $U$,which is the orthogonal complement of ${{\rm{1}}_N}$ satisfying $U_1^{\rm T}{U_1} = {I_{N - 1}}.$ Then for any Laplacian matrix $L \in {{\bf R} ^{N \times N}}$,

$${U^{\rm T}}LU = \left[{\begin{array}{*{20}{c}} {{U_1}^{\rm T}L{U_1}}&{{0_{(N - 1) \times 1}}}\\ {\frac{1}{{\sqrt N }}1_N^{\rm T}L{U_1}}&0 \end{array}} \right].$$

In particular,for any Laplacian matrix $L \in {{\bf R}^{N \times N}}$ of an undirected graph,

$${U^{\rm T}}LU = \left[{\begin{array}{*{20}{c}} {{U^{\rm T}_1}L{U_1}}&{{0_{(N - 1) \times 1}}}\\ {{0_{1 \times (N - 1)}}}&0 \end{array}} \right].$$

Noting that $\sum\nolimits_i {{\delta _i}(t - \eta (t))} = 0,$ $\sum\nolimits_i {{{\dot \delta }_i}(t)} = 0$,it can be obtained that

$$\dot {\tilde {\delta}} (t) = {U^{\rm T}}\dot {\delta} (t){\rm{ = }}{\left[{{{\dot {\bar {\delta}} }^{\rm T}} 0} \right]^{\rm T}},$$ $$\tilde \delta (t - \eta (t)) = {U^{\rm T}}\delta (t - \eta (t)) = {\left[{{{\bar \delta }^{\rm T}} (t - \eta (t)) 0} \right]^{\rm T}},$$ $$\tilde e({n_k}h) = {U^{\rm T}}e({n_k}h) = {\left[{{{\bar e}^{\rm T}}({n_k}h) 0} \right]^{\rm T}},$$ $$\tilde \delta ({t_k}h) = {U^{\rm T}}\delta ({t_k}h) = {\left[{{{\bar \delta }^{\rm T}} ({t_k}h) 0} \right]^{\rm T}}.$$

Then,according to Lemma 1,the system (19) is equivalent to

\begin{align} \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} \cdot \\ {\bar \delta } \end{array}(t) = - \left( {\bar L \otimes {I_M}} \right)\bar \delta (t - \eta (t) + \left( {\bar L \otimes {I_M}} \right)\;\bar \delta ({n_k}h),\;}\\ {\;{\eta _1} \le \eta (t) \le {\eta _N},t \in {\Omega _l},} \end{array} \end{align}

(20)

where $ \bar \delta (t) = {\left[{{{\bar \delta }_1}(t),\ldots ,{{\bar \delta }_{N - 1}}(t)} \right]^{\rm T}},$ $ \bar \delta (t - \eta (t)) = {\left[{{{\bar \delta }_1}(t - {\eta _1}(t)), \ldots ,{{\bar \delta }_{N - 1}}(t - {\eta _{N - 1}}(t)} \right]^{\rm T}},$ $ \bar e({n_k}h) = {\left[{{{\bar e}_1}(n_k^1h),\ldots ,{{\bar e}_{N - 1}}(n_k^{N - 1}h)} \right]^{\rm T}},$ $ \bar L = U_1^{\rm T}L{U_1}$ and ${\rm Rank}(\bar L) = N - 1.$

Because ${\Phi _i}$ is a positive definite matrix,there exists invertible matrix W satisfying ${\Phi _i} = {W^{\rm T}_i}{W_{i}}.$ Substituting ${\Phi _i} = {W^{\rm T}_i}{W_{i}}$ into (15) yields

$$e_i^{\rm T}(n_k^ih){W{\rm T}_i}^{W_{i.}}{e_i}(n_k^ih) - \gamma x_i^{\rm T}(t_k^ih){W^{\rm T}_i}{W_i}{x_i}(t_k^ih) < 0,$$ $$\left\| {{W_{i.}}{e_i}(n_k^ih)} \right\| - \sqrt \gamma \left\| {{W_i}{x_i}(t_k^ih)} \right\| < 0,$$

i.e.,

\begin{align} \left\| {{W_{i.}}{e_i}(n_k^ih)} \right\| < \sqrt \gamma \left\| {{W_i}\alpha } \right\| + \sqrt \gamma \left\| {{W_i}{\delta _i}(t_k^ih)} \right\|. \end{align}

(21)

From (21),if the following condition

\begin{align} \left\| {{W_{i.}}{e_i}(n_k^ih)} \right\|{\rm{ < }}\sqrt \gamma \left\| {{W_i}{\delta _i}(t_k^ih)} \right\|, \end{align}

(22)

is satisfied,the condition (15) is established. Therefore,from (22)

$$e_i^{\rm T}(n_k^ih){\Phi _i}{e_i}(n_k^ih) < \gamma \delta _i^{\rm T}(t_k^ih){\Phi _i}{\delta _i}(t_k^ih),$$ $${e^{\rm T}}({n_k}h)\Phi e({n_k}h) < {\delta^{\rm T}}({t_k}h)\Phi '\delta ({t_k}h),$$

i.e.,

$$\left[ {{{\bar e}^{\rm{T}}}({n_k}h)0} \right]{\left( {U \otimes {I_M}} \right)^{\rm{T}}}\Phi \left( {U \otimes {I_M}} \right){\left[ {{{\bar e}^{\rm{T}}}({n_k}h)0} \right]^{\rm{T}}}$$ $$ - \left[ {{{\bar \delta }^{\rm{T}}}({t_k}h)0} \right]{\left( {U \otimes {I_M}} \right)^{\rm{T}}}\Phi '\left( {U \otimes {I_M}} \right){\left[ {{{\bar \delta }^{\rm{T}}}({t_k}h)0} \right]^{\rm{T}}} < 0,$$

\begin{align} {\bar e^{\rm{T}}}({n_k}h)\tilde \Phi \bar e({n_k}h) - {\bar \delta ^{\rm{T}}}({t_k}h)\tilde \Phi '\bar \delta ({t_k}h) < 0, \end{align}

(23)

and between ${t_k}h$ and ${t_k}h + nh\,(n = 1,2,\ldots ,l)$ no control signal is triggered,where $\Phi = {\rm diag}\{{\Phi _1}, \ldots ,{\Phi _N}\},$ $\Phi ' = \gamma \Phi$,$\bar e({n_k}h) = \bar \delta ({n_k}h) - \bar \delta ({n_k}h),$ $\tilde \Phi = {\left( {{U_1} \otimes {I_M}} \right)^{\rm T}}\Phi \left( {{U_1} \otimes {I_M}} \right),$ and $\tilde \Phi ' = {\left( {{U_1} \otimes {I_M}} \right)^{\rm T}}\Phi '\left( {{U_1} \otimes {I_M}} \right).$

Remark 5. Under decentralized event-triggered mechanism, to solve the average consensus for system (14) with undirected, connected fixed network topology,a reduced dimension hybrid system (20) is proposed with decentralized event-triggered mechanism (23). Then,using reciprocally convex approach^[24], sufficient conditions for average consensus are obtained by a linear matrix inequality (LMI) set. This is proven in the following theorem.

B. Main Results

Theorem 1. Consider a system of $N$ agents with undirected,connected network topology and uncertain communication delays. For some given positive constants ${\eta _1}$,${\eta _N}$ and $0 < \gamma < 1 ,$ under the decentralized event-triggered mechanism (23),the system (20) is asymptotically stable,if there exist matrix ${S_2} \in {{\bf R}^{M \times M}}$ and positive definite matrices ${\Phi _i} \in {{\bf R}^{M \times M}} (i = 1, \ldots,N),$ $P,{Q_i},{R_i}\,(i = 1,2) \in {{\bf R}^{M \times M}},$ such that

\begin{align} \left[ {\begin{array}{*{20}{l}} {\begin{array}{*{20}{c}} {{R_2}}&{S_2^{\rm{T}}} \end{array}}\\ {\begin{array}{*{20}{c}} {{S_2}}&{{R_2}} \end{array}} \end{array}} \right] < 0, \end{align}

(24)

\begin{align} \left[{\begin{array}{*{20}{c}} {{\Pi _{11}}}& * & * & * & * & * & * \\ {{\Pi _{21}}}&{{\Pi _{22}}}& * & * & * & * & * \\ {{\Pi _{31}}}&{{\Pi _{32}}}&{{\Pi _{33}}}& * & * & * & * \\ 0&{{\Pi _{42}}}&{{\Pi _{43}}}&{{\Pi _{44}}}& * & * & * \\ {{\Pi _{51}}}&{{\Pi _{52}}}&0&0&{{\Pi _{55}}}& * & * \\ 0&{{\Pi _{62}}}&0&0&{{\Pi _{65}}}&{{\Pi _{66}}}& * \\ 0&{{\Pi _{72}}}&0&0&{{\Pi _{75}}}&0&{{\Pi _{77}}} \end{array}} \right]\ \label{eq26} \end{align}

(25)

where ${\Pi _{11}} = {I_{N - 1}} \otimes \left( {{Q_1} + {Q_2} - {R_1}} \right)$,${\Pi _{21}} = - \left( {{{\bar L}^{\rm T}} \otimes P} \right)$,${\Pi _{22}} = {I_{N - 1}} \otimes \left( { - 2{R_2} + {S_2} + S_2^{\rm T}} \right) + \tilde \Phi ',$ ${\Pi _{31}} = {I_{N - 1}} \otimes {R_1},$ ${\Pi _{32}} = {I_{N - 1}} \otimes \left( {{R_2} - S_2^{\rm T}} \right),$ ${\Pi _{33}} = - {I_{N - 1}} \otimes \left( {{Q_1} + {R_1} + {R_2}} \right),$ ${\Pi _{42}} = {I_{N - 1}} \otimes \left( {{R_2} - {S_2}} \right),$ ${\Pi _{43}} = {I_{N - 1}} \otimes {S_2},$ ${\Pi _{44}} = - {I_{N - 1}} \otimes \left( {{Q_2} + {R_2}} \right),$ ${\Pi _{51}} = {\bar L^{\rm T}} \otimes P$,${\Pi _{52}} = - \tilde \Phi ',$ ${\Pi _{55}} = \tilde \Phi ' - \tilde \Phi ,$ ${\Pi _{62}} = - {\eta _1}\left( {\bar L \otimes {R_1}} \right),$ ${\Pi _{65}} = {\eta _1}\left( {\bar L \otimes {R_1}} \right),$ ${\Pi _{66}} = - {I_{N - 1}} \otimes {R_1},$ ${\Pi _{72}} = - \left( {{\eta _N} - {\eta _1}} \right)\left( {\bar L \otimes {R_2}} \right),$ ${\Pi _{75}} = \left( {{\eta _N} - {\eta _1}} \right)\left( {\bar L \otimes {R_2}} \right),$ ${\Pi _{77}} = - {I_{N - 1}} \otimes {R_2},$ $\bar L = {U_1}^{\rm T}L{U_1},$ $\Phi = {\rm diag}\left\{{\Phi _1},\ldots ,{\Phi _N}\right\},$ $\Phi ' = \gamma \Phi ,$ $\tilde \Phi = {\left( {{U_1} \otimes {I_M}} \right)^{\rm T}}\Phi \left( {{U_1} \otimes {I_M}} \right),$ and $\tilde \Phi ' = {\left( {{U_1} \otimes {I_M}} \right)^{\rm T}}\Phi '\left( {{U_1} \otimes {I_M}} \right).$ ${U_1} \in {{\bf R}^{N \times (N - 1)}}$ represents the first $N-1$ columns of $U$, which is the orthogonal complement of ${{\rm{1}}_N}$ satisfying $U_1^{\rm T}{U_1} = {I_{N - 1}}.$ $U$ is an orthogonal matrix.

The proof of Theorem 1 is presented in Appendix A.

Corollary 1. With the conditions (24) and (25),the system (14) can asymptotically achieve average consensus under the event-triggered mechanism (16),i.e.,

$$\mathop {\lim }\limits_{t \to \infty } {x_i}(t) = \alpha = \frac{1}{N}\sum\limits_i {{x_i}(0)}.$$

Remark 6. Using the proposed method in Theorem 1,for the given positive constants $0 < \gamma < 1$,an upper bound ${\eta _N} = \max \{ \eta _3^i(t)\}$ of communication delays can be obtained for all agents. Note that the communication delay of each agent is uncertain and cannot exceed the ${\eta _N}$. We also obtain that the maximum sampling time $h$ does not exceed ${\eta _N},$ where $\eta _{_3}^i \triangleq h + {\bar \tau ^i}.$ When the delay ${\bar \tau ^i}$ of each agent is small enough,the upper bound ${\eta _N}$ gives one possible maximum sampling period. Similarly,when the sampling period $h$ is small enough, the conservative upper bound ${\eta _N}$ is the maximum delay.

V. DECENTRALIZED EVENT-TRIGGERED MECHANISM AND MODELING OF MULTI-AGENT SYSTEMS WITH COMMUNICATION DELAYS AND DATA DROPOUTS

If communication constraints include communication delays and data dropouts,it means that not all broadcast release data are successfully transmitted,then we have ${M_3} \subset {M_2}.$ For data dropouts,they are sometimes modeled as a homogeneous Markov chain^{[25, 26]},which cannot give the MANSD. Therefore, considering the MANSD model^{[19, 20]},the decentralized event-triggered transmission scheme (6) is changed as

\begin{align} \begin{array}{*{20}{l}} {{f_i}(n_k^ih,r_k^ih) = \hat e_i^{\rm{T}}(n_k^ih){\Phi _i}{{\hat e}_i}(n_k^ih)}\\ {\qquad - \sigma x_i^{\rm{T}}(r_k^ih){\Phi _i}{x_i}(r_k^ih) \ge 0,} \end{array} \end{align}

(26)

where $0 \le \sigma < 1$ and ${\hat e_i}(n_k^ih) = {x_i}(n_k^ih) - {x_i}(r_k^ih)$.

Compared with scheme (6),the state measurement error of agent in (26) is different and $0 < \sigma \leq \delta < 1$ is satisfied in order to consider the extra communication delays by data dropouts.

Similarly,with decentralized event-triggered transmission scheme (26),the broadcast release instant $r_{k + 1}^ih$ is defined by

\begin{align} r_{k + 1}^ih = r_k^ih + \mathop {\min }\limits_m \{ \left. {mh} \right|\hat e_i^{\rm{T}}(m_k^ih){\Phi _i}{\hat e_i}(m_k^ih) \ge \sigma {\hat \psi _i}\} , \end{align}

(27)

where ${\hat e_i}(m_k^ih) = {x_i}(m_k^ih) - {x_i}(r_k^ih),$ ${\hat \psi _i} = x_i^{\rm T}(r_k^ih)$ ${\Phi _i}{x_i}(r_k^ih), m \in {\bf N},$ and $m_k^ih = r_k^ih + \upsilon h.$

The following Theorem 2 is developed for achieving decentralized event-triggered average consensus with communication delays and data dropouts.

Theorem 2. Consider a system of $N$ agents with undirected, connected network topology,uncertain communication delays and data dropouts. For some given positive constants $h$,${\eta _1}$,${\eta _N}$ and $\gamma $,$\sigma $,under the decentralized event-triggered mechanism (27),the system (20) is asymptotically stable,if there exist matrix ${S_2} \in {{\bf R}^{M \times M}}$ and positive definite matrices ${\Phi _i} \in {{\bf R}^{M \times M}}\,(i = 1,\ldots,N),$ $P,{Q_i},{R_i}\,(i = 1,2) \in {{\bf R} ^{M \times M}},$ such that (24) and (25) hold,and the MANSD defined by ${d_{MANSD}}$ satisfies

\begin{align} {d_k} \le {d_{MANSD}} \buildrel \Delta \over = \left\lfloor {{{\log }_{(1 + \theta )(1 + \sqrt \sigma )}}\frac{{1 + \sqrt \gamma }}{{1 + \sqrt \sigma }}} \right\rfloor , \end{align}

(28)

where $\theta = \frac{h}{{1 - \sqrt \gamma }}$ and $\left\lfloor \Delta \right\rfloor $ gives the largest integer smaller than or equal to $\Delta $.

The proof of Theorem 2 is presented in Appendix B.

Remark 7. Theorem 2 can conveniently provide the relationship between triggering parameters,communication constraints and system stability,Specially,the quantitative relationship between triggering parameters,MANSD and system stability is derived. Therefore,considering the given time delays ${\eta _1}$,${\eta _N}$ and ${d_{MANSD}}$ together,the parameters $\gamma,$ $\sigma,$ ${\Phi _i}$ in the event-triggered mechanism can be obtained using the proposed approach.

Algorithm 1 to get the parameters is given here:

Algorithm 1.

Step 1. For the given time delay ${\eta _N}$,set $\gamma = \gamma + \mu $ and the step size $\mu = {\mu _0},$ where ${\mu _0}$ is a specified constant and small enough and the initial value of $\gamma $ is set to 0.

Step 2. Based on LMI approach,the feasible solutions that satisfy the matrix inequalities (24) and (25) are obtained. If the feasible solutions can be found,then go to Step 3. Otherwise,go back to Step 1.

Step 3. Using LMI approach,the corresponding ${\Phi _i}$ is found. For the given sampling period $h$,${d_{MANSD}}$ and the current $\gamma $,the maximum $\sigma (\gamma ,{d_{MANSD}})$ is obtained under the condition (28). If $\sigma (\gamma , {d_{MANSD}}) \leq 0,$ return to Step 1.

Step 4. Go to Step 1 for another value of $\gamma$ and $\sigma$,until $\gamma \geq 1$ the search is stopped and the maximum $\gamma $ is found.

VI. NUMERICAL EXAMPLE

To verify the effectiveness of proposed method,we consider an undirected and fixed network of five agents^[17],where each agent can represent a mobile robot in a simple CPS consisting of five mobile robots and is geographically distributed. The communication graph $G$ of the undirected and fixed network is shown in Fig. 2. It has connected digraph with 0-1 weights and ${x_i}(t) \in {{\bf R}^2}$ is the state vector of agent $i$. The initial conditions of each agent are set as ${x_1}(0) = {[-0.5,-0.6]^{\rm T}},$ ${x_2}(0) = {[-1,-1.2]^{\rm T}},$ ${x_3}(0) = {[3,3]^{\rm T}},$ ${x_4}(0) = {[-3,-2.8]^{\rm T}},$ ${x_5}(0) = {[1.5,1.6]^{\rm T}}.$

In the following,in order to discuss the average consensus problem with decentralized event-triggered mechanism and constraints in CPS,the two cases are analyzed.

1) With decentralized event-triggered mechanism and only communication delays,set $\gamma = 0.40,$ ${\eta _1} = 0$ and $h=0.01$. Using Theorem 1,maximum allowable delay ${\eta _N} = 0.054$ is gained. With ${\eta _N} = 0.05$,the state trajectories of five agents are shown in Fig. 3. It is seen each component of state vector asymptotically achieves average consensus for $t \in [0,6)$.

The effectiveness of event-triggered mechanism becomes clear in Fig. 4,which shows that the sampling frequency of each agent is remarkably reduced. The specific total event-triggered instants of each agent are shown in Table I.

Remark 8. Note that the numerical example has been used in [17],where the time-dependent trigger conditions with exponentially decreasing thresholds on the measurement errors can guarantee asymptotic convergence to average consensus. However,it is seen from Fig. 3 that the system has achieved a faster asymptotic convergence speed to average consensus by using the proposed method. Moreover,compared with our previous work of the centralized event-triggered mechanism^[22],the decentralized event-triggered mechanism in this paper only requires its own local measurement error and the current sampled-data to update the event-triggered instant and no longer requires the global information. Therefore,the total number of decentralized event-triggered instants of each agent are different from the centralized event-triggered instants. Moreover,due to uncertainty of the communication delays,the results of each simulation in Fig. 4 and Table I are not identical.

2) Considering decentralized event-triggered mechanism with communication delays and data dropouts,set ${\eta _1} = 0$ and $h=0.01$. If different $\eta _N$ and $d_{MANSD}$ are given,based on Algorithm 1,the different maximum $\gamma $,$\sigma$ are obtained which are listed in Table II.

From Table II,the simulation results have shown the quantitative relationship between triggering parameters and MANSD. With the allowed MANSD,the values of triggering parameters $\gamma $ and $\sigma$ cannot exceed the maximum values in Table II. We also found that the maximum allowable delay decreases with the increasing of triggering parameter $\gamma $.

For ${\eta _1} = 0,$ ${\eta _N} = 0.10$,${\gamma} = 0.10$, ${\sigma} = 0.02$,$h$=0.01 and ${d_{MANSD}}{\rm{ = }}1$,the event-triggered instants,successful broadcast release instants and state trajectories of Agent 1 are shown in Fig. 5. It shows that, with data dropouts,the successful broadcast release instants are less than the event-triggered instants and Agent 1 asymptotically achieves average consensus. Due to the uncertain communication delays,the successful broadcast release instants are not entirely included in event-triggered instants.

VII. CONCLUSION

The decentralized event-triggered average consensus problem for multi-agent systems in CPSs with communication delays and data dropouts has been investigated in this paper. Considering the communication and control characteristics of CPSs,the multi-agent systems are modeled as a reduced dimension hybrid system with decentralized event-triggered mechanism,communication delays and data dropouts. Some sufficient conditions for average consensus are presented, and an upper bound of time delay and MANSD are derived. Furthermore,due to CPSs are the complex systems of 3C,how to solve decentralized event-triggered mechanism with computation issue is another important direction in the future.

APPENDIX A PROOF OF THEOREM 1

Based on the system (20),we have constructed a Lyapunov-Krasovskii functional candidate as

\begin{align} \begin{array}{l} V(t) = {{\bar \delta }^{\rm T}}(t)\left( {{I_{N - 1}} \otimes P} \right)\bar \delta (t)\\ \quad +\int_{t - {\eta _1}}^t {{{\bar \delta }^{\rm T}} (\alpha )\left( {{I_{N - 1}} \otimes {Q_1}} \right)\bar \delta (\alpha ){\rm d}\alpha } \\ \quad +\int_{t - {\eta _N}}^t {{{\bar \delta }^{\rm T}} (\alpha )\left( {{I_{N - 1}} \otimes {Q_2}} \right)\bar \delta (\alpha ){\rm d}\alpha } \\ \quad + {\eta _1}\int_{ - {\eta _1}}^0 {\int_{t + \alpha }^t {{{\dot {\bar {\delta}} }^{\rm T}}(\beta )\left( {{I_{N - 1}} \otimes {R_1}} \right)\dot {\bar {\delta}} (\beta ){\rm d}\beta {\rm d}\alpha } } \\ \quad + {\eta _{N1}}\int_{ - {\eta _N}}^{ - {\eta _1}} {\int_{t + \alpha }^t {{{\dot {\bar {\delta}} }^{\rm T}}(\beta )\left( {{I_{N - 1}} \otimes {R_2}} \right)\dot {\bar {\delta}} (\beta ){\rm d}\beta {\rm d}\alpha ,} } \end{array} \end{align}

(A.1)

where $P,{Q_1},{Q_2},{R_1},{R_2} \in {{\bf R}^{M \times M}}$ are positive definite matrices and ${\eta _{N1}} = {\eta _N} - {\eta _1}$.

The derivative of (A.1) with respect to $t$ is

\begin{align} \begin{array}{l} \dot V(t) = 2{{\bar \delta }^{\rm T}}(t)\left( {{I_{N - 1}} \otimes P} \right)\dot {\bar {\delta}} (t)\\ \qquad \quad + \bar \delta {(t)^{\rm T}}\left( {{I_{N - 1}} \otimes {Q_1}} \right)\bar \delta (t)\\ \qquad \quad - {{\bar \delta }^{\rm T}}(t - {\eta _1})\left( {{I_{N - 1}} \otimes {Q_1}} \right)\bar \delta (t - {\eta _1})\\ \qquad \quad + \bar \delta {(t)^{\rm T}}\left( {{I_{N - 1}} \otimes {Q_2}} \right)\bar \delta (t)\\ \qquad \quad - {{\bar \delta }^{\rm T}}(t - {\eta _N})\left( {{I_{N - 1}} \otimes {Q_2}} \right)\bar \delta (t - {\eta _N})\\ \qquad \quad + \eta _1^2{{\dot {\bar {\delta}} }^{\rm T}}(t)\left( {{I_{N - 1}} \otimes {R_1}} \right)\dot {\bar {\delta}} (t)\\ \qquad \quad - {\eta _1}\int_{t - {\eta _1}}^t {{{\dot {\bar {\delta}} }^{\rm T}}(\alpha )} \left( {{I_{N - 1}} \otimes {R_1}} \right)\dot {\bar {\delta}} (\alpha ){\rm d}\alpha \\ \qquad \quad + \eta _{N1}^2{{\dot {\bar {\delta}} }^{\rm T}}(t)\left( {{I_{N - 1}} \otimes {R_2}} \right)\dot {\bar {\delta}} (t) \\ \qquad \quad - {\eta _{N1}}\int_{t - {\eta _N}}^{t - {\eta _1}} {{{\dot {\bar {\delta}} }^{\rm T}}(\alpha )} \left( {{I_{N - 1}} \otimes {R_2}} \right)\dot {\bar {\delta}} (\alpha ){\rm d}\alpha . \end{array} \end{align}

(A.2)

Definition 1.

$$\chi (t) = {[{\bar \delta ^{\rm T}}(t) {\bar \delta ^{\rm T}}(t - \eta (t)) {\bar \delta ^{\rm T}} (t - {\eta _1}) {\bar \delta ^{\rm T}}(t - {\eta _N}) {\bar e^{\rm T}}({n_k}h)]^{\rm T}},$$ $${e_1} = {\left[{\begin{array}{*{20}{c}} {{I_r}}&0&0&{\begin{array}{*{20}{c}} 0&0 \end{array}} \end{array}} \right]^{\rm T}},$$ $${e_2} = {\left[{\begin{array}{*{20}{c}} 0&{{I_r}}&0&{\begin{array}{*{20}{c}} 0&0 \end{array}} \end{array}} \right]^{\rm T}},$$ $${e_3} = {\left[{\begin{array}{*{20}{c}} 0&0&{{I_r}}&{\begin{array}{*{20}{c}} 0&0 \end{array}} \end{array}} \right]^{\rm T}},$$ $${e_4} = {\left[{\begin{array}{*{20}{c}} 0&0&0&{\begin{array}{*{20}{c}} {{I_r}}&0 \end{array}} \end{array}} \right]^{\rm T}},$$ $${e_5} = {\left[{\begin{array}{*{20}{c}} 0&0&0&{\begin{array}{*{20}{c}} 0&{{I_r}} \end{array}} \end{array}} \right]^{\rm T}},$$ $${e_6} = - {\left[{\left( {\bar L \otimes {I_M}} \right)({e^{\rm T}_2} - {e^{\rm T}_5})} \right]^{\rm T}},$$

where ${I_r} \in {{\bf R} ^{(N - 1)M \times (N - 1)M}}$ is an unit matrix.

Using Jensen's inequality^[27] and reciprocally convex approach^[24] to handle the integral items in (A.2),we get

\begin{align} \begin{array}{l} \dot V(t) \le {\chi ^{\rm T}}[{e_6}\left( {{I_{N - 1}} \otimes P} \right)e_1^{\rm T} + {e_1}\left( {{I_{N - 1}} \otimes P} \right)e_6^{\rm T}\\ \qquad \quad +{e_1}\left( {{I_{N - 1}} \otimes {Q_1}} \right)e_1^{\rm T} - {e_3}\left( {{I_{N - 1}} \otimes {Q_1}} \right)e_3^{\rm T}\\ \qquad \quad +{e_1}\left( {{I_{N - 1}} \otimes {Q_2}} \right)e_1^{\rm T} - {e_4}\left( {{I_{N - 1}} \otimes {Q_2}} \right)e_4^{\rm T}\\ \qquad \quad +\eta _1^2{e_6}\left( {{I_{N - 1}} \otimes {R_1}} \right)e_6^{\rm T}\\ \qquad \quad +\eta _{N1}^2{e_6}\left( {{I_{N - 1}} \otimes {R_2}} \right)e_6^{\rm T}\\ \qquad \quad -({e_1} - {e_3})\left( {{I_{N - 1}} \otimes {R_1}} \right){({e_1} - {e_3})^{\rm T}} - {\Lambda _1}]\chi (t) , \end{array} \end{align}

(A.3)

where

$${\Lambda _1} = {\left[\begin{array}{l} {({e_3} - {e_2})^{\rm T}}\\ {({e_2} - {e_4})^{\rm T}} \end{array} \right]^{\rm T}}\left( {{I_{N - 1}} \otimes {\Lambda _2}} \right)\left[\begin{array}{l} {({e_3} - {e_2})^{\rm T}}\\ {({e_2} - {e_4})^{\rm T}} \end{array} \right]$$

and ${\Lambda _2} = \left[{\begin{array}{*{20}{c}} {{R_2}}&{S_2^{\rm T}}\\ {{S_2}}&{{R_2}} \end{array}} \right] > 0.$

Since between ${t_k}h$ and ${t_k}h + ih ,$ no control signal is triggered,the condition (A.3) is satisfied,then

\begin{align} \begin{array}{l} \dot V(t) \le {\chi ^{\rm T}}[{e_6}\left( {{I_{N - 1}} \otimes P} \right)e_1^{\rm T}+ {e_1}\left( {{I_{N - 1}} \otimes P} \right)e_6^{\rm T}\\ \quad +{e_1}\left( {{I_{N - 1}} \otimes {Q_1}} \right)e_1^{\rm T} - {e_3}\left( {{I_{N - 1}} \otimes {Q_1}} \right)e_3^{\rm T}\\ \quad + {e_1}\left( {{I_{N - 1}} \otimes {Q_2}} \right)e_1^{\rm T} - {e_4}\left( {{I_{N - 1}} \otimes {Q_2}} \right)e_4^{\rm T}\\ \quad + \eta _1^2{e_6}\left( {{I_{N - 1}} \otimes {R_1}} \right)e_6^{\rm T}\\ \quad + \eta _{N1}^2{e_6}\left( {{I_{N - 1}} \otimes {R_2}} \right)e_6^{\rm T}\\ \quad - ({e_1} - {e_3})\left( {{I_{N - 1}} \otimes {R_1}} \right){({e_1} - {e_3})^{\rm T}} - {\Lambda _1}]\chi (t) \\ \quad + {{\bar \delta }^{\rm T}}({t_k}h)\tilde \Phi ' \bar \delta ({t_k}h) - {{\bar e}^{\rm T}}({n_k}h) \tilde \Phi \bar e({n_k}h). \end{array} \end{align}

(A.4)

Then,a sufficient condition for $\dot V(t) < 0$ is

\begin{align} \begin{array}{l} {e_6}\left( {{I_{N - 1}} \otimes P} \right)e_1^{\rm T} + {e_1} \left( {{I_{N - 1}} \otimes P} \right)e_6^{\rm T} \\ \quad + {e_1}\left( {{I_{N - 1}} \otimes {Q_1}} \right)e_1^{\rm T} - {e_3} \left( {{I_{N - 1}} \otimes {Q_1}} \right)e_3^{\rm T} \\ \quad + {e_1}\left( {{I_{N - 1}} \otimes {Q_2}} \right)e_1^{\rm T}- {e_4}\left( {{I_{N - 1}} \otimes {Q_2}} \right)e_4^{\rm T}\\ \quad - ({e_1} - {e_3})\left( {{I_{N - 1}} \otimes {R_1}} \right){({e_1}- {e_3})^{\rm T}}\\ \quad -{\Lambda _1} + ({e_2} - {e_5})\tilde \Phi '{({e_2} - {e_5})^{\rm T}} - {e_5}\tilde \Phi {e_5}^{\rm T}\\ \quad +\eta _1^2{e_6}\left( {{I_{N - 1}} \otimes {R_1}} \right)e_6^{\rm T}\\ \quad +\eta _{N1}^2{e_6}\left( {{I_{N - 1}} \otimes {R_2}} \right)e_6^{\rm T}\\ \quad =\left[{\begin{array}{*{20}{c}} {{\Pi _{11}}}& * & * & * & * \\ {{\Pi _{21}}}&{{\Pi _{22}}}& * & * & * \\ {{\Pi _{31}}}&{{\Pi _{32}}}&{{\Pi _{33}}}& * & * \\ 0&{{\Pi _{42}}}&{{\Pi _{43}}}&{{\Pi _{44}}}& * \\ {{\Pi _{51}}}&{{\Pi _{52}}}&0&0&{{\Pi _{55}}} \end{array}} \right]\\ \quad +\eta _1^2{e_6}\left( {{I_{N - 1}} \otimes {R_1}} \right)e_6^{\rm T}\\ \quad + \eta _{N1}^2{e_6}\left( {{I_{N - 1}} \otimes {R_2}} \right)e_6^{\rm T} < 0. \end{array} \end{align}

(A.5)

Then,by Schur complement formula,the matrix inequality (A.5) is equivalent to (25).

APPENDIX B PROOF OF THEOREM 2

Consider a successful broadcast release interval $[t_k^ih,t_{k + 1}^ih)$,and suppose that in this interval there are ${d_k}$ unsuccessfully transmitted broadcast packets: $t_k^i = r_0^i < r_1^i < r_2^i < \cdots < r_{{d_k}}^i < r_{{d_k} + 1}^i = t_{k + 1}^i.$ For $l = 0,1,\ldots ,{d_k}$,using the communication scheme (27) yields

\begin{align} \left| {{x_i}(r_{l + 1}^ih - h) - {x_i}(r_l^ih)} \right| \leq\sqrt \sigma \left| {{x_i}(r_l^ih)} \right|. \end{align}

(B.1)

For (B.1),the state error $t \in [r_{{d_k}}^ih,r_{{d_k} + 1}^ih)$ between $t$ and ${t_k}$ is

\begin{align} \begin{array}{l} \left| {{x_i}(t) - {x_i}(t_k^ih)} \right|{\rm{}}\\ \quad = \left| {{x_i}(t) - {x_i}(r_{{d_k}}^ih) + \sum\limits_{l = 0}^{{d_k} - 1} {({x_i}(r_{l + 1}^ih) - {x_i}(r_l^ih))} } \right| \\ \quad \le\sum\limits_{l = 0}^{{d_k}} {\sqrt {{\sigma _i}} {x_i}(r_l^ih))} \\ \quad + \sum\limits_{l = 0}^{{d_k} - 1} {\left| {({x_i}(r_{l + 1}^ih) - {x_i}(r_{l + 1}^ih - h))} \right|} . \end{array} \end{align}

(B.2)

Since

$$\begin{array}{l} \left| {{x_i}(t_k^ih)} \right| = \left| {{x_i}(r_{l + 1}^ih - h) - {x_i}(r_{l + 1}^ih - h) + {x_i}(t_k^ih)} \right| \\ \quad \le\left| {{x_i}(r_{l + 1}^ih - h)} \right| + \sqrt \gamma \left| {{x_i}(t_k^ih)} \right|, \end{array}$$

and

$$\begin{array}{l} \left| {{x_i}(r_{l + 1}^ih - h)} \right| \\ \quad \le\left| {{x_i}(r_{l + 1}^ih - h) - {x_i}(r_l^ih)} \right| + \left| {{x_i}(r_l^ih)} \right|\\ \quad \le (1 + \sqrt \sigma )\left| {{x_i}(r_l^ih)} \right|, \end{array}$$

it can be got that

\begin{align} \left| {{x_i}(t_k^ih)} \right| \le \frac{1}{{1 - \sqrt \gamma }}\left| {{x_i}(r_{l + 1}^ih - h)} \right|\quad \le \frac{{1 + \sqrt \sigma }}{{1 - \sqrt \gamma }}\left| {{x_i}(r_l^ih)} \right|. \end{align}

(B.3)

For system (9),we have:

\begin{align} \left| {{x_i}(r_{l + 1}^ih) - {x_i}(r_{l + 1}^ih - h)} \right| = h\left| {{x_i}(t_k^ih)} \right|. \end{align}

(B.4)

Considering (B.3) and (B.2),it follows that

\begin{align} \begin{array}{l} \left| {({x_i}(r_{l + 1}^ih) - {x_i}(r_{l + 1}^ih - h))} \right| \le \frac{{h(1 + \sqrt \sigma )}} {{1 - \sqrt \gamma }}\left| {{x_i}(r_l^ih)} \right| \\ \quad =\theta (1 + \sqrt \sigma )\left| {{x_i}(r_l^ih)} \right|, \end{array} \end{align}

(B.5)

where $\theta = \frac{h}{{1 - \sqrt \gamma }}$.

Substituting (B.5) into (B.2) yields

\begin{align} \begin{array}{l} \left| {{x_i}(t) - {x_i}(t_k^ih)} \right| \\ \qquad \le \sum\limits_{l = 0}^{{d_k}} {\sqrt \sigma {x_i}(r_l^ih))}+ \sum\limits_{l = 0}^{{d_k} - 1} {\theta (1 + \sqrt \sigma )\left| {{x_i}(r_l^ih)} \right|} . \end{array} \end{align}

(B.6)

From (B.1) and (B.4),we get

$$\left| {{x_i}(r_{l + 1}^ih)} \right|$$ $$quad \le\left| {{x_i}(r_{l + 1}^ih) - {x_i}(r_{l + 1}^ih - h)} \right|$$ $$quad +\left| {{x_i}(r_{l + 1}^ih - h) - {x_i}(r_l^ih)} \right| + \left| {{x_i}(r_l^ih)} \right|$$ $$quad \le {[(1 + \theta )(1 + \sqrt \sigma )]^{l + 1}}\left| {{x_i}(t_k^ih)} \right|,$$

i.e.,

\begin{align} \left| {{x_i}(r_l^ih)} \right| \le {[(1 + \theta )(1 + \sqrt \sigma )]^l}\left| {{x_i}(t_k^ih)} \right|. \end{align}

(B.7)

Substituting (B.7) into (B.6) yields

\begin{align} \begin{array}{l} \left| {{x_i}(t) - {x_i}(t_k^ih)} \right|\le \sum\limits_{l = 0}^{{d_k}} {\sqrt \sigma {x_i}(r_l^ih))} \\ \quad + \sum\limits_{l = 0}^{{d_k} - 1} {\theta (1 + \sqrt \sigma )\left| {{x_i}(r_l^ih)} \right|} \\ \quad \le({(1 + \sqrt \sigma )^{{d_k} + 1}}{(1 + \theta )^{{d_k}}} - 1)\left| {{x_i}(t_k^ih)} \right|. \end{array} \end{align}

(B.8)

Therefore,considering (B.8) and (28) together leads to

\begin{align} \left| {{x_i}(t) - {x_i}(t_k^ih)} \right| \le \sqrt \gamma \left| {{x_i}(t_k^ih)} \right|. \end{align}

(B.9)

Inequality (B.9) ensures the condition (14) in the Theorem 1,this reveals that Theorem 2 can be derived from Theorem 1 if the the communication scheme (27) is applied.