Observer-based event-triggered time-varying formation control of linear multi-agent systems with distributed infinite delays

Abstract

Compared with bounded delays, distributed infinite delays are more general in practical systems. Event-triggered control can effectively reduce energy consumption and communication costs. This paper addresses time-varying formation control of general linear multi-agent systems with distributed infinite delays in both of their inputs and outputs. An observer-based event-triggered formation control protocol considering distributed infinite delays is proposed, which is related to the combined observed information and some formation compensation signals at triggering time instants. By utilizing inequality techniques, the desired time-varying formation can be implemented while Zeno-behavior is excluded. Some numerical simulations are carried out for demonstrating the validity of theoretical results.

Introduction

Formation control, as a crucial embranchment in the field of cooperative problem of multi-agent systems (MASs), drives agents to accomplish the specified missions collaboratively through designing control laws to achieve a prescribed time-invariant or time-varying geometric configuration. However, it is very difficult for fixed formation to adjust relative locations timely according to variable environment, and to expand or contract the movement scales while performing some complex collaborative tasks. Consequently, theoretical and technical studies on time-varying formation (TVF) control have drawn considerable attention,^1–4 and also, the results of TVF are applicable to time-invariant formation and consensus control.

Time-delay phenomena are inevitable in complex networked systems on account of very frequent information exchange among various digital devices, which can bring about deterioration of system performance and instability of system. Until now, much progress has been made in investigating formation problem with time delays.^5–8 Notably, most of the existing studies considered bounded delays. However, many real systems generally possess a spatial extent of time delays and cannot be well modeled by dynamical systems with bounded delays. Distributed infinite delays may better characterize such complex hysteresis phenomena and have some significant applications, such as traffic flow systems, ⁹ biological systems, ¹⁰ mechanics, ¹¹ neural networks, ¹² and so on. Consensus problem with infinite delays of MASs was discussed for single integrator MASs ¹³ and linear MASs, ¹⁴ respectively. The output containment control of heterogeneous MASs with unbounded distributed transmission delays was presented. ¹⁵ The authors ¹⁶ addressed the robust cooperative output regulation problem of heterogeneous uncertain MASs with unbounded distributed transmission delays. The authors ¹⁷ focused on the formation-containment tracking for heterogeneous MASs under distributed infinite transmission delays. Leader-following formation control of MASs with distributed infinite input time delays was studied. ¹⁸ Nevertheless, there are few report on time-varying formation control with distributed infinite delays so far, which motivates us to carry out this work.

The key of implement of the desired formation is to design appropriate distributed control protocol based on states of agents. However, it is more difficult to measure some internal state information of MASs than general output information. Thus, observer-based control mechanism for formation problem has generated growing interest.^19–25 On the other hand, aiming at the limitation of communication bandwidth and data storage capacities of microprocessor in real system, researchers raised event-triggered control (ETC) strategy to reduce interagent communications.^26–30 Furthermore, observer-based event-triggered control technique, combining advantages of event-triggered and observer-based control, was put forward. By adopting complex-valued Laplacian, researchers ³¹ presented a fully distributed event-triggered formation controller depending on the dynamic observed state. A dynamic event-triggered updating rule based on output feedback was put forward for bipartite TVF tracking. ³² The authors ³³ developed two observer-based ETC algorithms to achieve formation in MASs with switching directed topologies. Considering collision avoidance, researchers raised a distributed event-triggered control technique based on state observer to handle formation problem. ³⁴ To form circle formation, an observer-based ETC method was presented. ³⁵ The authors ³⁶ introduced an event-triggered control scheme that utilizes observed states and incorporates certain formation compensation signals.

Motivated by the above discussions, the current work studies observer-based event-triggered TVF for linear MASs with distributed infinite delays in inputs and outputs. Our main contributions are summarized: (i) Compared with state-based controller of linear systems only taking into account distributed infinite input delays,^13,14 an observer-based scheme is explored for linear MASs with distributed infinite delays in the course of both input and output signal transmission, which is more general, rational and also accords with the practical situation. (ii) Event-triggered control schemes^36,37 were related not only to the combined observed information at triggering time instants but also to the continuous monitor of formation compensation signals. However, the ETC protocol utilized in this paper is only in connection with these information at triggering time instants, which can avoid continuous updates of controller. As a result, it conserves more energy. (iii) The TVF problem is transformed into a consensus problem using an orthogonal row matrix. A sufficient condition on TVF along with Zeno-free triggering is derived.

The remaining parts are organized as follows. Graph theories and problem formulation of observer-based event-triggered TVF are provided in Section 2. Detailed analyses of the sufficient condition for TVF, along with discussions on Zeno-free behavior, are presented in Section 3. To illustrate the performance of the controller, numerical simulations are provided in Section 4, followed by the final conclusion.

Notations: Let $R^n$ and $R^{n\times m}$ be the set of $n$ -dimensional real column vectors and $n\times m$ -dimensional real matrices, respectively. For a matrix $A$ , ${\lambda }_i(A)$ represents the $i\mathrm{th}$ eigenvalues of A, $\mathrm{R}e{\lambda }_i(A)$ is the real part of ${\lambda }_i(A)$ , $A^T$ stands for its transpose. The Kronecker product of two matrices $A$ and $B$ is represented by $A\otimes B$ . A diagonal matrix with diagonal elements $a_i$ , $i=1,\cdots ,n$ , is denoted as $\mathrm{diag}\{a_1,\cdots ,a_n\}$ . $\parallel A\parallel$ denotes the induced matrix norm. For $x\in R^n$ , $\parallel x\parallel$ is the Euclidean norm.

Preliminaries

Graph theories

An undirected graph is utilized to depict the communication topology with $N$ agents. Denote $\mathcal{G}=(\overline{O},\overline{E},\overline{M})$ be an undirected graph with a set of nodes $\overline{O}=\{{\overline{o}}_1,{\overline{o}}_2,\cdots ,{\overline{o}}_N\}$ , an edge set $\overline{E}\subseteq \{{\overline{e}}_{\mathrm{ij}}=({\overline{o}}_i,{\overline{o}}_j):{\overline{o}}_i,{\overline{o}}_j\in \overline{O},i\ne j\},$ a weighted adjacency matrix $\overline{M}=[{\overline{m}}_{\mathrm{ij}}]\in R^{N\times N}$ with ${\overline{m}}_{\mathrm{ij}}=1$ if and only if $({\overline{o}}_i,{\overline{o}}_j)\in \overline{E}$ , and ${\overline{m}}_{\mathrm{ij}}=0$ otherwise. The undirected graph means that $({\overline{o}}_i,{\overline{o}}_j)\in \overline{E}$ if and only if $({\overline{o}}_j,{\overline{o}}_i)\in \overline{E}$ . ${\mathcal{N}}_i=\{{\overline{o}}_j\in \overline{O}:({\overline{o}}_i,{\overline{o}}_j)\in \overline{E}\}$ represents the neighbor set of node ${\overline{o}}_i$ . Let $D=\mathrm{diag}\{d_1,\cdots ,d_N\}$ with $d_i=\sum _{j=1}^N{\overline{m}}_{\mathrm{ij}}$ be the degree matrix of $\mathcal{G}$ , then Laplacian matrix $L=D-\overline{W}=\left[l_{\mathrm{ij}}\right]\in R^{N\times N}$ . $\mathcal{G}$ is connected if there exists a path from node ${\overline{o}}_i$ to ${\overline{o}}_j$ .

• Lemma 1. If $\mathcal{G}$ is connected, the eigenvalues of $L$ are $0={\lambda }_1<{\lambda }_2\le \cdots \le {\lambda }_N$ .

Problem formulation

We consider linear multi-agent systems (MASs) consisting of N agents. The dynamics of each agent are of the form

$$\begin{array}{c}{\stackrel{·}{x}}_i(t)=A{x}_i(t)+B{u}_i(t),\hfill \\ y_i(t)=C{\int }_0^{+\infty }{s}_i(\tau )x_i(t-\tau )d\tau ,i=1,\cdots ,N,\hfill \end{array}$$

(1)

where $A\in R^{n\times n}$ , $B\in R^{n\times m}$ , $C\in R^{q\times n}$ are constant matrices, satisfying $\mathrm{rank}(B)=m$ . $x_i(t)\in R^n$ , $y_i(t)\in R^q$ , and $u_i(t)\in R^m$ stand for the state, output and control input, respectively. $s_i(\tau )$ : $[0,+\infty )\to [0,+\infty )$ is the delay kernel satisfying ${\int }_0^{+\infty }{s}_i(\tau )d\tau =1$ and ${\int }_0^{+\infty }{s}_i(\tau )e^{\upsilon \tau }d\tau <+\infty$ for some positive constant $\upsilon$ .

To avoid directly utilizing the internal states of all agents, we reconstruct the system states by designing a state observer. The dynamics of the observer are

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{\hat{x}}}_i(t)}& =A{\hat{x}}_i(t)+B{u}_i(t)+F({\hat{y}}_i(t)-y_i(t)),\hfill \\ \multicolumn{1}{c}{{\hat{y}}_i(t)}& =C{\int }_0^{+\infty }{s}_i(\tau ){\hat{x}}_i(t-\tau )d\tau ,i=1,\cdots ,N,\hfill \end{array}$$

(2)

where ${\hat{x}}_i(t)\in R^n$ is the observer state of $x_i(t)$ , ${\hat{y}}_i\in R^q$ is the measured output of the observer, and $F$ is the observer gain matrix with appropriate dimensions to be determined later.

Let ${\tilde{\vartheta }}_i(t)={\hat{x}}_i(t)-x_i(t)$ be the state estimation error.

From (1) and (2), one has

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\stackrel{~}{\vartheta }}}_i(t)=A{\tilde{\vartheta }}_i(t)+\mathrm{FC}{\int }_0^{+\infty }{s}_i(\tau ){\tilde{\vartheta }}_i(t-\tau )d\tau ,}\end{array}$$

(3)

Denote $\tilde{\vartheta }(t)={({\tilde{\vartheta }}_1^T(t),{\tilde{\vartheta }}_2^T(t),\cdots ,{\tilde{\vartheta }}_N^T(t))}^T$ , $\hat{S}(\tau )=$ diag $\{s_1(\tau ),\cdots ,s_N(\tau )\}$ . According to ${\int }_0^{+\infty }{s}_i(\tau )d\tau =1$ , one gets ${\int }_0^{+\infty }\hat{S}(\tau )d\tau =I_N$ . Then,

$$\begin{array}{l}\stackrel{.}{\stackrel{~}{\vartheta }}(t)=\left(I_N\otimes A\right)\tilde{\vartheta }(t)+{\int }_0^{+\infty }\left(\widehat{S}(\tau )\otimes FC\right)\tilde{\vartheta }\left(t-\tau \right)d\tau \\ =\left(I_N\otimes A+{\int }_0^{+\infty }(\widehat{S}(\tau )\otimes FC)d\tau \right)\tilde{\vartheta }(t)\\ -{\int }_0^{+\infty }\left(\widehat{S}(\tau )\otimes FC\right)\left(\tilde{\vartheta }(t)-\tilde{\vartheta }(t-\tau )\right)d\tau \\ =\left(I_N\otimes A+I_N\otimes FC\right)\tilde{\vartheta }(t)\\ -{\int }_0^{+\infty }\left(\widehat{S}(\tau )\otimes FC\right)(\tilde{\vartheta }(t)-\tilde{\vartheta }(t-\tau ))d\tau .\end{array}$$

(4)

• Theorem 1. If $(A,C)$ is detectable, then one can select a control gain matrix $F$ to make sure that the state estimation error is exponentially stable. Namely, there are positive constants $W$ and $\lambda$ making

$$\parallel \tilde{\vartheta }(t)\parallel \le W\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }{e}^{-\lambda (t-t_0)},t\ge t_0,$$

(5)

where $\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }=\underset{t\in (-\infty ,t_0]}{\sup }\parallel \tilde{\vartheta }(t)\parallel$ .

• Proof. Since $(A,C)$ is detectable, we can choose $F$ to guarantee $A+\mathrm{FC}$ is Hurwitz. Then, we get $I_N\otimes A+I_N\otimes \mathrm{FC}$ is Hurwitz. Hence, for $t\ge t_0$ , we can choose these positive constants $W$ and $\varpi$ satisfying

$$\parallel e^{(I_N\otimes A+I_N\otimes \mathrm{FC})(t-t_0)}\parallel \le W{e}^{-\varpi (t-t_0)}.$$

(6)

Recalling (4), utilizing the variation of parameter formula gives

$$\begin{array}{l}\parallel \tilde{\vartheta }(t)\parallel \le \parallel e^{\left(I_N\otimes A+I_N\otimes FC\right)(t-t_0)}\parallel \parallel \tilde{\vartheta }(t_0)\parallel \\ +{\int }_{t_0}^t{\int }_0^{+\infty }\parallel e^{\left(I_N\otimes A+I_N\otimes FC\right)(t-s)}\parallel \parallel \widehat{S}(\tau )\otimes FC\parallel \\ \left(\parallel \tilde{\vartheta }(s)-\tilde{\vartheta }(s-\tau )\parallel \right)d\tau ds.\end{array}$$

(7)

As a result, due to (6), (7) can be rewritten as

$$\begin{array}{c}\parallel \tilde{\vartheta }(t)\parallel \le W{e}^{-\varpi (t-t_0)}\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }\hfill \\ +W{\int }_{t_0}^t{\int }_0^{+\infty }{e}^{-\varpi (t-s)}\parallel \hat{S}(\tau )\otimes \mathrm{FC}\parallel \hfill \\ (\parallel \tilde{\vartheta }(s)\parallel +\parallel \tilde{\vartheta }(s-\tau )\parallel )d\tau \mathrm{ds}.\hfill \end{array}$$

(8)

Next, for $\rho >1$ , we will prove

$$\parallel \tilde{\vartheta }(t)\parallel <\rho W\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }{e}^{-\lambda (t-t_0)}\stackrel{\mathrm{\Delta }}{=}\chi (t),t\ge t_0.$$

(9)

It is noted that inequality (9) holds for $t\in (-\infty ,t_0]$ .

If (9) is not satisfied, one can find $t^{*}>t_0$ satisfying $\parallel \tilde{\vartheta }(t^{*})\parallel =\chi (t^{*})$ and $\parallel \tilde{\vartheta }(t)\parallel <\chi (t)$ when $t\in (-\infty ,t^{*})$ .

Subsequently, according to (8), one can further derive

$$\begin{array}{c}\multicolumn{1}{c}{\chi (t^{*})=\parallel \tilde{\vartheta }(t^{*})\parallel <\rho W\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }{e}^{-\varpi (t^{*}-t_0)}+\rho W^2\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }\parallel \mathrm{FC}\parallel }\\ \times {\int }_{t_0}^{t^{*}}{\int }_0^{+\infty }{e}^{-\varpi (t^{*}-s)}(e^{-\lambda (s-t_0-\tau )}+e^{-\lambda (s-t_0)})\parallel \hat{S}(\tau )\parallel d\tau \mathrm{ds}\hfill \\ \le \rho W\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }{e}^{-\varpi (t^{*}-t_0)}+\rho W^2\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }\parallel \mathrm{FC}\parallel \hfill \\ \times \left({\int }_{t_0}^{t^{*}}{e}^{-\varpi (t^{*}-s)}{e}^{-\lambda (s-t_0)}\mathrm{ds}\right)\hfill \\ \times ({\int }_0^{+\infty }{(\parallel \hat{S}(\tau )\parallel e^{\upsilon \tau }{)}^{2}d\tau )}^{\frac{1}{2}}{\left({\int }_0^{+\infty }{(e^{(\lambda -\upsilon )\tau }+e^{-\upsilon \tau })}^{2}d\tau \right)}^{\frac{1}{2}}\hfill \\ \le \rho W\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }{e}^{-\varpi (t^{*}-t_0)}\hfill \\ +\rho W^2\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }\parallel \mathrm{FC}\parallel \left({\int }_{t_0}^{t^{*}}{e}^{-\varpi (t^{*}-s)}{e}^{-\lambda (s-t_0)}\mathrm{ds}\right)\hfill \\ \times {\left({\int }_0^{+\infty }{(\parallel \hat{S}(\tau )\parallel e^{\upsilon \tau })}^{2}d\tau \right)}^{\frac{1}{2}}\times {\left(\frac{2}{\upsilon -\lambda }\right)}^{\frac{1}{2}}.\hfill \end{array}$$

(10)

Let $\psi ={\int }_0^{+\infty }{(\parallel \hat{S}(\tau )\parallel e^{\upsilon \tau })}^{2}d\tau$ . Moreover, there is a positive constant $r_1$ satisfying $r_1\lambda \ge 2$ and $r_1\lambda \ge \upsilon -\lambda$ . Thus,

$$\begin{array}{c}\multicolumn{1}{c}{\chi (t^{*})<\rho W\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }{e}^{-\varpi (t^{*}-t_0)}+\rho W^2\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }}\\ \times \frac{r_1\lambda {\psi }^{\frac{1}{2}}}{\upsilon -\lambda }\frac{\parallel \mathrm{FC}\parallel }{\varpi -\lambda }(e^{-\lambda (t^{*}-t_0)}-e^{-\varpi (t^{*}-t_0)})\hfill \\ =\rho W\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }\hfill \\ [e^{-\varpi (t^{*}-t_0)}+\frac{r_1\lambda {\psi }^{\frac{1}{2}}W}{\upsilon -\lambda }\frac{\parallel \mathrm{FC}\parallel }{\varpi -\lambda }(e^{-\lambda (t^{*}-t_0)}-e^{-\varpi (t^{*}-t_0)})]\hfill \end{array}$$

(11)

Let $\varsigma =\frac{\upsilon +\varpi +r_1{\psi }^{\frac{1}{2}}W\parallel \mathrm{FC}\parallel -\sqrt{{(\upsilon +\varpi +r_1{\psi }^{\frac{1}{2}}W\parallel \mathrm{FC}\parallel )}^2-4\upsilon \varpi }}{2}$ and $\max \{\frac{2}{r_1},\frac{\upsilon }{r_1+1}\}<\lambda <\min \{\upsilon ,\varpi ,\varsigma \}$ .

Since $\max \{\frac{2}{r_1},\frac{\upsilon }{r_1+1}\}<\lambda <\varsigma$ , one gets ${\lambda }^2-(\upsilon +\varpi +r_1{\psi }^{\frac{1}{2}}W\parallel \mathrm{FC}\parallel )\lambda +\upsilon \varpi >0$ and then $r_1\lambda {\psi }^{\frac{1}{2}}W\parallel \mathrm{FC}\parallel <(\upsilon -\lambda )(\varpi -\lambda )$ , that is, $\frac{r_1\lambda {\psi }^{\frac{1}{2}}W\parallel \mathrm{FC}\parallel }{(\upsilon -\lambda )(\varpi -\lambda )}<1$ . Thus,

$$\begin{array}{c}\multicolumn{1}{c}{\chi (t^{*})<\rho W\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }[e^{-\varpi (t^{*}-t_0)}+(e^{-\lambda (t^{*}-t_0)}-e^{-\varpi (t^{*}-t_0)})]}\\ =\rho W\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }{e}^{-\lambda (t^{*}-t_0)}=\chi (t^{*}).\hfill \end{array}$$

(12)

This contradiction of (12) implies that (9) is valid for any $\rho >1$ . Letting $\rho \to 1$ , the inequality (5) can be obtained.

The proof is completed.

• Definition 1. Denote

$$h(t)={[h_1^T(t),\cdots ,h_N^T(t)]}^T$$

be the desired TVF, which is a bounded piecewise differentiable vector function. MASs (1) is called to implement the desired TVF $h(t)$ if

$$\underset{t\to \infty }{lim}((x_i(\mathrm{t})-x_j(\mathrm{t}))-(h_i(\mathrm{t})-h_j(\mathrm{t})))=0$$

holds with any given bounded initial conditions.

Motivated by the TVF control law,^36–38 we design the following ETC strategy:

$$\begin{array}{c}\multicolumn{1}{c}{u_i(t)=-H{\int }_0^{+\infty }{k}_i(\tau )q_i(t_k^i-\tau )d\tau +{\mathrm{\Upsilon }}_i(t_k^i),t\in [t_k^i,t_{k+1}^i),}\end{array}$$

(13)

in which $H$ is the gain matrix to be determined later, $k_i(\tau )$ : $[0,+\infty )\to [0,+\infty )$ is the delay kernel satisfying ${\int }_0^{+\infty }{k}_i(\tau )d\tau =1$ and ${\int }_0^{+\infty }{k}_i(\tau )e^{\epsilon \tau }d\tau <+\infty$ for some positive constant $\epsilon$ , $q_i(t)=\sum _{j\in {\mathcal{N}}_i}({\mathrm{\Gamma }}_i(t)-{\mathrm{\Gamma }}_j(t))$ with ${\mathrm{\Gamma }}_i(t)={\hat{x}}_i(t)-h_i(t)$ ,

${\mathrm{\Upsilon }}_i(t)$ represents the formation compensation signal to expand the feasible formation set. ³⁸ $t_k^i$ stands for the triggering instant defined by the following equation

$$\begin{array}{c}\multicolumn{1}{c}{t_{k+1}^i=inf\{t:t>t_k^i\mathrm{and}{g}_i(t)>0\},}\end{array}$$

(14)

in which

$$\begin{array}{cc}\multicolumn{1}{c}{g_i(t)}& =\Vert {\delta }_i(t)\Vert +\Vert e_i(t)\Vert -{\beta }_1\Vert q_i(t_k^i)\Vert -{\beta }_2{e}^{-\varrho (t-t_0)}\hfill \end{array}$$

(15)

is the triggering function with ${\beta }_1>0,{\beta }_2>0$ and $\varrho >0$ , the formation compensation signal error ${\delta }_i(t)={\mathrm{\Upsilon }}_i(t_k^i)-{\mathrm{\Upsilon }}_i(t)$ , the combined measurement error $e_i(t)=q_i(t_k^i)-q_i(t)$ . $e_i(t)$ and ${\delta }_i(t)$ are reset to zero when $g_i(t)\ge 0$ .

Let $M$ be an $(N-1)\times N$ matrix with orthogonal rows which are orthogonal to $1_N$ , that is

$$\begin{array}{c}M{1}_N=0,M{M}^T=I_{N-1}.\hfill \end{array}$$

(16)

Because $I_N-M^{T}M$ is an orthogonal projection matrix onto the span of $1_N$ according to the property of $M$ , and that $1_N$ is in the null space of $L$ , one gets $L(I-M^{T}M)=0$ , then $L=L{M}^{T}M$ .

• Lemma 2. The matrix $\mathrm{ML}{M}^T$ and $L$ have the same nonzero eigenvalues. ³⁹

• Lemma 3. If $(A,B)$ is stabilizable, for any $\mu >0$ , there is a symmetric positive definite matrix $Q$ making the following algebraic Riccati equality (ARE) satisfy

$$\begin{array}{c}\mathrm{QA}+A^{T}Q-\mathrm{QB}{B}^{T}Q+\mu I=0.\hfill \end{array}$$

(17)

Consequently, according to Lemma 2, we can derive that all eigenvalues of $\mathrm{ML}{M}^T$ are positive. It is evident that there exists $\sigma >0$ such that $\sigma {\lambda }_i\left(\mathrm{ML}{M}^T\right)\ge 1,i=1,2,\cdots ,n-1$ . Referring to (17), it can be deduced that all eigenvalues of $A-B{B}^{T}Q$ are negative. Letting $H=\sigma B^{T}Q$ , all eigenvalues of $A-{\lambda }_i\left(\mathrm{ML}{M}^T\right)\mathrm{BH},i=1,2,\cdots ,n-1$ are negative. To proceed, all eigenvalues of $I_{N-1}\otimes A-\mathrm{ML}{M}^T\otimes \mathrm{BH}$ are negative. Thus, for $t\ge t_0$ , there are positive constants $\alpha$ and $\gamma$ such that

$$\begin{array}{c}\multicolumn{1}{c}{\Vert e^{(I_{N-1}\otimes A-\mathrm{ML}{M}^T\otimes \mathrm{BH})(t-t_0)}\Vert \le \alpha e^{-\gamma (t-t_0)}.}\end{array}$$

(18)

• Definition 2. The event-triggering time sequence $\{t_k^i\}$ is free of Zeno-behavior if $\underset{k}{inf}\{t_{k+1}^i-t_k^i\}>0$ for all $i$ .

Main results

Based on the above preliminaries, a sufficient condition on TVF is put forward under the designed ETC scheme (13) while Zeno-behavior in $\{t_k^i\}$ is excluded in the following.

In order to read conveniently, denote

$$\begin{array}{c}\xi =\frac{\gamma }{\alpha (\varphi \parallel \mathrm{BH}\parallel \parallel M\parallel +\parallel M\otimes B\parallel )\sqrt{N}\parallel L{M}^T\otimes I\parallel +\gamma },\hfill \\ \kappa =\gamma +\epsilon +\alpha {\phi }^{\frac{1}{2}}{r}_2\parallel \mathrm{BH}\parallel \parallel M\parallel \parallel M^T\parallel \hfill \\ -\alpha (a_1\varphi \parallel \mathrm{BH}\parallel \parallel M\parallel +a_2\parallel M\otimes B\parallel ),\hfill \\ \iota =\frac{\kappa -\sqrt{{\kappa }^2-4(\gamma \epsilon -\alpha (a_1\varphi \parallel \mathrm{BH}\parallel \parallel M\parallel +a_2\parallel M\otimes B\parallel )\epsilon )}}{2},\hfill \end{array}$$

and $z=max\{\frac{2}{r_2},\frac{\epsilon }{r_2+1}\}$ , where $r_2$ is a positive constant.

• Theorem 2. Consider linear MASs (1) with observer (2) and ETC protocol (13) for $H=\sigma B^{T}Q$ . The triggering time instants are determined by (14). Assuming that there exists differentiable vector function ${\mathrm{\Upsilon }}_i(t)$ and ${\stackrel{·}{\mathrm{\Upsilon }}}_i(t)$ is bounded, $h_i(t)$ satisfies

$$\begin{array}{c}A{h}_i(t)+B{\mathrm{\Upsilon }}_i(t)-{\stackrel{·}{h}}_i(t)=0.\hfill \end{array}$$

(19)

Then, for ${\beta }_1\in (0,\xi )$ , ${\beta }_2>0$ and $\varrho \in (z,\min \{\lambda ,\epsilon ,\gamma ,\iota \})$ , the desired TVF $h(t)$ is feasible if $(A,B)$ is stabilizable, $(A,C)$ is detectable and $\mathcal{G}$ is connected.

• Proof. In term of (1) and (2), it holds

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\mathrm{\Gamma }}}_i(t)={\stackrel{·}{\hat{x}}}_i(t)-{\stackrel{·}{h}}_i(t)}\\ =A{\hat{x}}_i(t)+B{u}_i(t)+F({\hat{y}}_i(t)-y_i(t))-{\stackrel{·}{h}}_i(t)\hfill \\ =A{\mathrm{\Gamma }}_i(t)-\mathrm{BH}{\int }_0^{+\infty }{k}_i(\tau )q_i(t_k^i-\tau )d\tau +B{\delta }_i(t)\hfill \\ +\mathrm{FC}{\int }_0^{+\infty }{s}_i(\tau ){\tilde{\vartheta }}_i(t-\tau )d\tau \hfill \\ =A{\mathrm{\Gamma }}_i(t)-\mathrm{BH}{\int }_0^{+\infty }{k}_i(\tau )q_i(t-\tau )d\tau +B{\delta }_i(t)\hfill \\ -\mathrm{BH}{\int }_0^{+\infty }{k}_i(\tau )e_i(t-\tau )d\tau +\mathrm{FC}{\int }_0^{+\infty }{s}_i(\tau ){\tilde{\vartheta }}_i(t-\tau )d\tau .\hfill \end{array}$$

(20)

Let $\mathrm{\Gamma }(t)={({\mathrm{\Gamma }}_1^T(t),\cdots ,{\mathrm{\Gamma }}_N^T(t))}^T$ , $\delta (t)={({\delta }_1^T(t),\cdots ,{\delta }_N^T(t))}^T$ , $e(t)={(e_1^T(t),\cdots ,e_N^T(t))}^T$ . (20) becomes

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\mathrm{\Gamma }}(t)=(I_N\otimes A)\mathrm{\Gamma }(t)-{\int }_0^{+\infty }(\hat{L}(\tau )\otimes \mathrm{BH})\mathrm{\Gamma }(t-\tau )d\tau }\\ +(I_N\otimes B)\delta (t)-{\int }_0^{+\infty }(\hat{K}(\tau )\otimes \mathrm{BH})e(t-\tau )d\tau \hfill \\ +{\int }_0^{+\infty }(\hat{S}(\tau )\otimes \mathrm{FC})\tilde{\vartheta }(t-\tau )d\tau ,\hfill \end{array}$$

(21)

in which $\hat{K}(\tau )=\mathrm{diag}$ $\{k_1(\tau ),\cdots ,k_N(\tau )\}$ and $\hat{L}(\tau )=[{\hat{l}}_{\mathrm{ij}}(\tau ){]}_{N\times N}$ with

$${\hat{l}}_{\mathrm{ij}}(\tau )=\{\begin{array}{cc}\sum _{j=1}^N{\overline{m}}_{\mathrm{ij}}{k}_i(\tau ),\hfill & i=j,\hfill \\ \multicolumn{1}{c}{-{\overline{m}}_{\mathrm{ij}}{k}_i(\tau ),}& i\ne j.\hfill \end{array}$$

Let $\tilde{\mathrm{\Psi }}(t)=(M\otimes I)\mathrm{\Gamma }(t)$ . By ${\int }_0^{+\infty }{k}_i(\tau )d\tau =1$ , we have ${\int }_0^{+\infty }\hat{L}(\tau )d\tau =L$ . Then system (21) becomes

$$\begin{array}{c}\stackrel{·}{\stackrel{~}{\mathrm{\Psi }}}(t)=(I_{N-1}\otimes A)\tilde{\mathrm{\Psi }}(t)+(M\otimes B)\delta (t)\hfill \\ -{\int }_0^{+\infty }(M\hat{L}(\tau )M^T\otimes \mathrm{BH})\tilde{\mathrm{\Psi }}(t-\tau )d\tau \hfill \\ -{\int }_0^{+\infty }(M\hat{K}(\tau )\otimes \mathrm{BH})e(t-\tau )d\tau \hfill \\ +{\int }_0^{+\infty }(M\hat{S}(\tau )\otimes \mathrm{FC})\tilde{\vartheta }(t-\tau )d\tau \hfill \\ =(I_{N-1}\otimes A-{\int }_0^{+\infty }(M\hat{L}(\tau )M^T\otimes \mathrm{BH})d\tau )\tilde{\mathrm{\Psi }}(t)\hfill \\ +{\int }_0^{+\infty }(M\hat{L}(\tau )M^T\otimes \mathrm{BH})(\tilde{\mathrm{\Psi }}(t)-\tilde{\mathrm{\Psi }}(t-\tau ))d\tau \hfill \\ +(M\otimes B)\delta (t)\hfill \\ -{\int }_0^{+\infty }(M\hat{K}(\tau )\otimes \mathrm{BH})e(t-\tau )d\tau \hfill \\ +{\int }_0^{+\infty }(M\hat{S}(\tau )\otimes \mathrm{FC})\tilde{\vartheta }(t-\tau )d\tau \hfill \end{array}$$

$$\begin{array}{l}=\mathcal{W}\tilde{\mathrm{\Psi }}(t)+\left(M\otimes B\right)\delta (t)\\ +{\int }_0^{+\infty }\left(M\widehat{L}(\tau )M^T\otimes BH\right)\left(\tilde{\mathrm{\Psi }}(t)-\tilde{\mathrm{\Psi }}(t-\tau )\right)d\tau \\ -{\int }_0^{+\infty }\left(M\widehat{K}(\tau )\otimes BH\right)e\left(t-\tau \right)d\tau \\ +{\int }_0^{+\infty }\left(M\widehat{S}(\tau )\otimes FC\right)\tilde{\vartheta }\left(t-\tau \right)d\tau ,\end{array}$$

(22)

where $\mathcal{W}=I_{N-1}\otimes A-\mathrm{ML}{M}^T\otimes \mathrm{BH}$ .

Utilizing variation of parameter formula yields

$$\begin{array}{c}\tilde{\mathrm{\Psi }}(t)=e^{\mathcal{W}(t-t_0)}\tilde{\mathrm{\Psi }}(t_0)\hfill \\ +{\int }_{t_0}^t{\int }_0^{+\infty }{e}^{\mathcal{W}(t-s)}(M\hat{L}(\tau )M^T\otimes \mathrm{BH})\hfill \\ (\tilde{\mathrm{\Psi }}(s)-\tilde{\mathrm{\Psi }}(s-\tau ))d\tau \mathrm{ds}\hfill \\ -{\int }_{t_0}^t{\int }_0^{+\infty }{e}^{\mathcal{W}(t-s)}(M\hat{K}(\tau )\otimes \mathrm{BH})e(s-\tau )d\tau \mathrm{ds}\hfill \\ +{\int }_{t_0}^t{\int }_0^{+\infty }{e}^{\mathcal{W}(t-s)}(M\hat{S}(\tau )\otimes \mathrm{FC})\tilde{\vartheta }(s-\tau )d\tau \mathrm{ds}\hfill \\ +{\int }_{t_0}^t(M\otimes B)e^{\mathcal{W}(t-s)}\delta (s)\mathrm{ds}.\hfill \end{array}$$

(23)

Combining (18) and (23) yields

$$\begin{array}{c}\parallel \tilde{\mathrm{\Psi }}(t)\parallel \le \alpha e^{-\gamma (t-t_0)}\parallel \tilde{\mathrm{\Psi }}(t_0)\parallel \hfill \\ +\alpha \parallel \mathrm{BH}\parallel {\int }_{t_0}^t{\int }_0^{+\infty }{e}^{-\gamma (t-s)}\parallel M\hat{L}(\tau )M^T\parallel \parallel \tilde{\mathrm{\Psi }}(s)-\tilde{\mathrm{\Psi }}(s-\tau )\parallel d\tau \mathrm{ds}\hfill \\ +\alpha \parallel \mathrm{BH}\parallel {\int }_{t_0}^t{\int }_0^{+\infty }{e}^{-\gamma (t-s)}\parallel M\hat{K}(\tau )\parallel \parallel e(s-\tau )\parallel d\tau \mathrm{ds}\hfill \\ +\alpha \parallel \mathrm{FC}\parallel {\int }_{t_0}^t{\int }_0^{+\infty }{e}^{-\gamma (t-s)}\parallel M\hat{S}(\tau )\parallel \parallel \tilde{\vartheta }(s-\tau )\parallel d\tau \mathrm{ds}\hfill \\ +\alpha \parallel M\otimes B\parallel {\int }_{t_0}^t{e}^{-\gamma (t-s)}\delta (s)\mathrm{ds}.\hfill \end{array}$$

(24)

According to triggering condition (15), one has

$$\begin{array}{c}\multicolumn{1}{c}{\Vert {\delta }_i(t)\Vert +\Vert e_i(t)\Vert \le {\beta }_1\Vert q_i(t_k^i)\Vert +{\beta }_2{e}^{-\varrho (t-t_0)}}\\ \multicolumn{1}{c}{\le {\beta }_1\Vert e_i(t)\Vert +{\beta }_1\Vert q_i(t)\Vert +{\beta }_2{e}^{-\varrho (t-t_0)}.}\end{array}$$

(25)

As a consequence,

$$\begin{array}{c}\begin{array}{c}\Vert e_i(t)\Vert \le \frac{{\beta }_1}{1-{\beta }_1}\Vert L\otimes I\Vert \Vert \mathrm{\Gamma }(t)\Vert +\frac{{\beta }_2}{1-{\beta }_1}{e}^{-\varrho (t-t_0)},\hfill \end{array}\hfill \\ \begin{array}{c}\Vert {\delta }_i(t)\Vert \le {\beta }_1\Vert L\otimes I\Vert \Vert \mathrm{\Gamma }(t)\Vert +{\beta }_2{e}^{-\varrho (t-t_0)},\hfill \end{array}\hfill \end{array}$$

then,

$$\begin{array}{c}\Vert e(t)\Vert \le \frac{\sqrt{N}{\beta }_1\Vert L\otimes I\Vert }{1-{\beta }_1}\Vert \mathrm{\Gamma }(t)\Vert +\frac{\sqrt{N}{\beta }_2}{1-{\beta }_1}{e}^{-\varrho (t-t_0)}\hfill \\ =a_1\Vert \tilde{\mathrm{\Psi }}(t)\Vert +b_1{e}^{-\varrho (t-t_0)},\hfill \\ \Vert \delta (t)\Vert \le a_2\Vert \tilde{\mathrm{\Psi }}(t)\Vert +b_2{e}^{-\varrho (t-t_0)},\hfill \end{array}$$

(26)

in which $a_1=\frac{\sqrt{N}{\beta }_1\Vert L{M}^T\otimes I\Vert }{1-{\beta }_1},b_1=\frac{\sqrt{N}{\beta }_2}{1-{\beta }_1},a_2=\sqrt{N}{\beta }_1\Vert L{M}^T\otimes I\Vert ,b_2=\sqrt{N}{\beta }_2$ .

By Theorem 1 and (26), from (24), we have

$$\begin{array}{c}\parallel \tilde{\mathrm{\Psi }}(t)\parallel \le \alpha e^{-\gamma (t-t_0)}\parallel \tilde{\mathrm{\Psi }}(t_0)\parallel \hfill \\ +\alpha \parallel \mathrm{BH}\parallel \parallel M\parallel \parallel M^T\parallel {\int }_{t_0}^t{\int }_0^{+\infty }{e}^{-\gamma (t-s)}\hfill \\ \parallel \hat{L}(\tau )\parallel \parallel \tilde{\mathrm{\Psi }}(s)-\tilde{\mathrm{\Psi }}(s-\tau )\parallel d\tau \mathrm{ds}\hfill \\ +\alpha a_1\parallel \mathrm{BH}\parallel \parallel M\parallel {\int }_{t_0}^t{\int }_0^{+\infty }{e}^{-\gamma (t-s)}\hfill \\ \times \parallel \hat{K}(\tau )\parallel \parallel \tilde{\mathrm{\Psi }}(s-\tau )\parallel d\tau \mathrm{ds}\hfill \\ +\alpha b_1\parallel \mathrm{BH}\parallel \parallel M\parallel {\int }_{t_0}^t{\int }_0^{+\infty }{e}^{-\gamma (t-s)}\hfill \\ \parallel \hat{K}(\tau )\parallel e^{-\varrho (s-t_0-\tau )}d\tau \mathrm{ds}\hfill \\ +\alpha W\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }\parallel \mathrm{FC}\parallel \parallel M\parallel {\int }_{t_0}^t{\int }_0^{+\infty }{e}^{-\gamma (t-s)}\hfill \\ \parallel \hat{S}(\tau )\parallel e^{-\lambda (s-t_0-\tau )}d\tau \mathrm{ds}\hfill \\ +\alpha a_2\parallel M\otimes B\parallel {\int }_{t_0}^t{e}^{-\gamma (t-s)}\parallel \tilde{\mathrm{\Psi }}(s)\parallel \mathrm{ds}\hfill \\ +\alpha b_2\parallel M\otimes B\parallel {\int }_{t_0}^t{e}^{-\gamma (t-s)}{e}^{-\varrho (s-t_0)}\mathrm{ds}.\hfill \end{array}$$

(27)

Then,

$$\begin{array}{c}\parallel \tilde{\mathrm{\Psi }}(t)\parallel \le \alpha e^{-\gamma (t-t_0)}\parallel \tilde{\mathrm{\Psi }}(t_0)\parallel \hfill \\ +\alpha \parallel \mathrm{BH}\parallel \parallel M\parallel \parallel M^T\parallel {\int }_{t_0}^t{\int }_0^{+\infty }{e}^{-\gamma (t-s)}\hfill \\ \parallel \hat{L}(\tau )\parallel \parallel \tilde{\mathrm{\Psi }}(s)-\tilde{\mathrm{\Psi }}(s-\tau )\parallel d\tau \mathrm{ds}\hfill \\ +\alpha a_1\parallel \mathrm{BH}\parallel \parallel M\parallel {\int }_{t_0}^t{\int }_0^{+\infty }{e}^{-\gamma (t-s)}\hfill \\ \parallel \hat{K}(\tau )\parallel \parallel \tilde{\mathrm{\Psi }}(s-\tau )\parallel d\tau \mathrm{ds}+\alpha b_1\parallel \mathrm{BH}\parallel \parallel M\parallel \hfill \\ \times \left({\int }_{t_0}^t{e}^{-\gamma (t-s)}{e}^{-\varrho (s-t_0)}\mathrm{ds}\right)\hfill \\ (({\int }_0^{+\infty }\parallel \hat{K}(\tau )\parallel e^{\epsilon \tau }d\tau )\times \underset{\tau \ge 0}{\sup }\left(e^{(\varrho -\epsilon )\tau }\right))\hfill \\ +\alpha W\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }\parallel \mathrm{FC}\parallel \parallel M\parallel \left({\int }_{t_0}^t{e}^{-\gamma (t-s)}{e}^{-\lambda (s-t_0)}\mathrm{ds}\right)\hfill \\ \times (({\int }_0^{+\infty }\parallel \hat{S}(\tau )\parallel e^{\upsilon \tau }d\tau )\times \underset{\tau \ge 0}{\sup }\left(e^{(\lambda -\upsilon )\tau }\right))\hfill \\ +\alpha a_2\parallel M\otimes B\parallel {\int }_{t_0}^t{e}^{-\gamma (t-s)}\parallel \tilde{\mathrm{\Psi }}(s)\parallel \mathrm{ds}\hfill \\ +\alpha b_2\parallel M\otimes B\parallel {\int }_{t_0}^t{e}^{-\gamma (t-s)}{e}^{-\varrho (s-t_0)}\mathrm{ds}.\hfill \end{array}$$

(28)

Let $\varphi ={\int }_0^{+\infty }\parallel \hat{K}(\tau )\parallel e^{\epsilon \tau }d\tau$ , $\overline{\psi }={\int }_0^{+\infty }\parallel \hat{S}(\tau )\parallel e^{\upsilon \tau }d\tau$ .

Since $\underset{\tau \ge 0}{\sup }\left(e^{(\varrho -\epsilon )\tau }\right)=1$ and $\underset{\tau \ge 0}{\sup }\left(e^{(\lambda -\upsilon )\tau }\right)=1$ , we get

$$\begin{array}{c}\parallel \tilde{\mathrm{\Psi }}(t)\parallel \le \alpha e^{-\gamma (t-t_0)}\parallel \tilde{\mathrm{\Psi }}(t_0)\parallel +\alpha \parallel \mathrm{BH}\parallel \parallel M\parallel \parallel M^T\parallel \hfill \\ \times {\int }_{t_0}^t{\int }_0^{+\infty }{e}^{-\gamma (t-s)}\parallel \hat{L}(\tau )\parallel (\parallel \tilde{\mathrm{\Psi }}(s)\parallel +\parallel \tilde{\mathrm{\Psi }}(s-\tau )\parallel )d\tau \mathrm{ds}\hfill \\ +\alpha a_1\parallel \mathrm{BH}\parallel \parallel M\parallel {\int }_{t_0}^t{\int }_0^{+\infty }{e}^{-\gamma (t-s)}\parallel \hat{K}(\tau )\parallel \parallel \tilde{\mathrm{\Psi }}(s-\tau )\parallel d\tau \mathrm{ds}\hfill \\ +\alpha a_2\parallel M\otimes B\parallel {\int }_{t_0}^t{e}^{-\gamma (t-s)}\parallel \tilde{\mathrm{\Psi }}(s)\parallel \mathrm{ds}\hfill \\ +\frac{\alpha b_1\parallel \mathrm{BH}\parallel \parallel M\parallel \varphi }{\gamma -\varrho }(e^{-\varrho (t-t_0)}-e^{-\gamma (t-t_0)})\hfill \\ +\frac{\alpha W\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }\parallel \mathrm{FC}\parallel \parallel M\parallel \overline{\psi }}{\gamma -\lambda }(e^{-\lambda (t-t_0)}-e^{-\gamma (t-t_0)})\hfill \\ +\frac{\alpha b_2\parallel M\otimes B\parallel }{\gamma -\varrho }(e^{-\varrho (t-t_0)}-e^{-\gamma (t-t_0)})\hfill \\ \le \alpha e^{-\gamma (t-t_0)}\parallel \tilde{\mathrm{\Psi }}(t_0)\parallel +\alpha \parallel \mathrm{BH}\parallel \parallel M\parallel \parallel M^T\parallel \hfill \\ \times {\int }_{t_0}^t{\int }_0^{+\infty }{e}^{-\gamma (t-s)}\parallel \hat{L}(\tau )\parallel (\parallel \tilde{\mathrm{\Psi }}(s)\parallel +\parallel \tilde{\mathrm{\Psi }}(s-\tau )\parallel )d\tau \mathrm{ds}\hfill \\ +\alpha a_1\parallel \mathrm{BH}\parallel \parallel M\parallel {\int }_{t_0}^t{\int }_0^{+\infty }{e}^{-\gamma (t-s)}\parallel \hat{K}(\tau )\parallel \parallel \tilde{\mathrm{\Psi }}(s-\tau )\parallel d\tau \mathrm{ds}\hfill \\ +\alpha a_2\parallel M\otimes B\parallel {\int }_{t_0}^t{e}^{-\gamma (t-s)}\parallel \tilde{\mathrm{\Psi }}(s)\parallel \mathrm{ds}\hfill \\ +\alpha c(e^{-\varrho (t-t_0)}-e^{-\gamma (t-t_0)}),\hfill \end{array}$$

(29)

where $c=\frac{b_1\parallel \mathrm{BH}\parallel \parallel M\parallel \varphi +b_2\parallel M\otimes B\parallel }{\gamma -\varrho }+\frac{W\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }\parallel \mathrm{FC}\parallel \parallel M\parallel \overline{\psi }}{\gamma -\lambda }$ .

Since ${\beta }_1\in (0,\xi )$ , one has $\frac{\sqrt{N}{\beta }_1\parallel L{M}^T\otimes I\parallel }{1-{\beta }_1}\alpha (\varphi \parallel \mathrm{BH}\parallel \parallel M\parallel +\parallel M\otimes B\parallel )<\gamma$ , and then, $\alpha (a_1\varphi \parallel \mathrm{BH}\parallel \parallel M\parallel +a_2\parallel M\otimes B\parallel )<\gamma$ . Furthermore, $\alpha (a_1\varphi \parallel \mathrm{BH}\parallel \parallel M\parallel +a_2\parallel M\otimes B\parallel )\epsilon <\gamma \epsilon$ and $\alpha (a_1\varphi \parallel \mathrm{BH}\parallel \parallel M\parallel +a_2\parallel M\otimes B\parallel )<\gamma +\epsilon +\alpha {\phi }^{\frac{1}{2}}{r}_2\parallel \mathrm{BH}\parallel \parallel M\parallel \parallel M^T\parallel$ . Thus, a positive constant $\varrho$ satisfying $z<\varrho <\min \{\epsilon ,\gamma ,\iota \}$ can be found to make sure $ϵ=\{\alpha c(ϵ-\varrho )(\gamma -\varrho )\}/$ . $\{{\varrho }^2-[\gamma +ϵ+\alpha {\phi }^{\frac{1}{2}}{r}_2\parallel \mathrm{BH}\parallel \parallel M\parallel \parallel M^T\parallel -\alpha (a_1\varphi \parallel \mathrm{BH}\parallel \parallel M\parallel +a_2\parallel M\otimes B\parallel )]\varrho +\gamma ϵ-\alpha (a_1\varphi \parallel \mathrm{BH}\parallel \parallel M\parallel +a_2\parallel M\otimes B\parallel )ϵ\}>0$

Let $\parallel \tilde{\mathrm{\Psi }}(t_0){\parallel }_{\tau }=\underset{t\in (-\infty ,t_0]}{\sup }\parallel \tilde{\mathrm{\Psi }}(t)\parallel$ .

Let $Z=max\{ϵ,\alpha \parallel \tilde{\mathrm{\Psi }}(t_0){\parallel }_{\tau }\}$ .

Next, we’ll prove

$$\parallel \tilde{\mathrm{\Psi }}(t)\parallel <\eta Z{e}^{-\varrho (t-t_0)}\stackrel{\mathrm{\Delta }}{=}\zeta (t),t\ge t_0,$$

(30)

holds for $\eta >1$ . It is noted that inequality (30) holds for $t\in (-\infty ,t_0]$ .

Otherwise, we can find $t^{*}>t_0$ satisfying $\parallel \tilde{\mathrm{\Psi }}(t^{*})\parallel =\zeta (t^{*})$ and $\parallel \tilde{\mathrm{\Psi }}(t)\parallel <\zeta (t)$ when $t\in (-\infty ,t^{*})$ .

Subsequently, inequality (29) leads to

$$\begin{array}{c}\zeta (t^{*})=\parallel \tilde{\mathrm{\Psi }}(t^{*})\parallel <\alpha \eta \parallel \tilde{\mathrm{\Psi }}(t_0){\parallel }_{\tau }{e}^{-\gamma (t^{*}-t_0)}\hfill \\ +\alpha \eta Z\parallel \mathrm{BH}\parallel \parallel M\parallel \parallel M^T\parallel \hfill \\ \times {\int }_{t_0}^{t^{*}}{\int }_0^{+\infty }{e}^{-\gamma (t^{*}-s)}(e^{-\varrho (s-t_0-\tau )}+e^{-\varrho (s-t_0)})\parallel \hat{L}(\tau )\parallel d\tau \mathrm{ds}\hfill \\ +\alpha \eta Z{a}_1\parallel \mathrm{BH}\parallel \parallel M\parallel \hfill \\ {\int }_{t_0}^{t^{*}}{\int }_0^{+\infty }{e}^{-\gamma (t^{*}-s)}{e}^{-\varrho (s-t_0-\tau )}\parallel \hat{K}(\tau )\parallel d\tau \mathrm{ds}\hfill \\ +\alpha \eta Z{a}_2\parallel M\otimes B\parallel {\int }_{t_0}^{t^{*}}{e}^{-\gamma (t^{*}-s)}{e}^{-\varrho (s-t_0)}\mathrm{ds}\hfill \\ +\alpha \eta c(e^{-\varrho (t^{*}-t_0)}-e^{-\gamma (t^{*}-t_0)})\hfill \\ \le \alpha \eta \parallel \tilde{\mathrm{\Psi }}(t_0){\parallel }_{\tau }{e}^{-\gamma (t^{*}-t_0)}\hfill \\ +\alpha \eta Z\parallel \mathrm{BH}\parallel \parallel M\parallel \parallel M^T\parallel \left({\int }_{t_0}^{t^{*}}{e}^{-\gamma (t^{*}-s)}{e}^{-\varrho (s-t_0)}\mathrm{ds}\right)\hfill \\ {\times ({\int }_0^{+\infty }\mathrm{lpar}(\parallel \hat{L}(\tau )\parallel e^{\epsilon \tau }d\tau {)}^2)}^{\frac{1}{2}}\times {\left({\int }_0^{+\infty }{(e^{(\varrho -\epsilon )\tau }+e^{-\epsilon \tau })}^{2}d\tau \right)}^{\frac{1}{2}}\hfill \\ +\alpha \eta Z{a}_1\parallel \mathrm{BH}\parallel \parallel M\parallel \left({\int }_{t_0}^{t^{*}}{e}^{-\gamma (t^{*}-s)}{e}^{-\varrho (s-t_0)}\mathrm{ds}\right)\hfill \\ \times (({\int }_0^{+\infty }\parallel \hat{K}(\tau )\parallel e^{\epsilon \tau }d\tau )\times \underset{\tau \ge 0}{\sup }\left(e^{(\varrho -\epsilon )\tau }\right))\hfill \\ +\frac{\alpha \eta Z{a}_2\parallel M\otimes B\parallel }{\gamma -\varrho }(e^{-\varrho (t^{*}-t_0)}-e^{-\gamma (t^{*}-t_0)})\hfill \\ +\alpha \eta c(e^{-\varrho (t^{*}-t_0)}-e^{-\gamma (t^{*}-t_0)}).\hfill \end{array}$$

(31)

Let $\phi ={\int }_0^{+\infty }{(\parallel \hat{L}(\tau )\parallel e^{\epsilon \tau })}^{2}d\tau$ . Moreover, there is a positive constant $r_2$ satisfying $r_2\varrho \ge 2$ and $r_2\rho \ge \epsilon -\varrho$ . Then,

$$\begin{array}{c}\zeta (t^{*})<\alpha \eta \parallel \tilde{\mathrm{\Psi }}(t_0){\parallel }_{\tau }{e}^{-\gamma (t^{*}-t_0)}\hfill \\ +\frac{\alpha \eta Z{\phi }^{\frac{1}{2}}{r}_2\varrho \parallel \mathrm{BH}\parallel \parallel M\parallel \parallel M^T\parallel }{(\epsilon -\varrho )(\gamma -\varrho )}(e^{-\varrho (t^{*}-t_0)}-e^{-\gamma (t^{*}-t_0)})\hfill \\ +\frac{\alpha \eta Z{a}_1\varphi \parallel \mathrm{BH}\parallel \parallel M\parallel }{\gamma -\varrho }(e^{-\varrho (t^{*}-t_0)}-e^{-\gamma (t^{*}-t_0)})\hfill \\ +\frac{\alpha \eta Z{a}_2\parallel M\otimes B\parallel }{\gamma -\varrho }(e^{-\varrho (t^{*}-t_0)}-e^{-\gamma (t^{*}-t_0)})\hfill \\ +\alpha \eta c(e^{-\varrho (t^{*}-t_0)}-e^{-\gamma (t^{*}-t_0)})\hfill \\ =\eta \{\alpha \parallel \tilde{\mathrm{\Psi }}(t_0){\parallel }_{\tau }-(\alpha Z\mathrm{\Xi }+\alpha c)\}{e}^{-\gamma (t^{*}-t_0)}\hfill \\ +\eta (\alpha Z\mathrm{\Xi }+\alpha c)e^{-\varrho (t^{*}-t_0)},\hfill \end{array}$$

(32)

where $\mathrm{\Xi }=\frac{{\phi }^{\frac{1}{2}}{r}_2\varrho \parallel \mathrm{BH}\parallel \parallel M\parallel \parallel M^T\parallel +(\epsilon -\varrho )(a_1\varphi \parallel \mathrm{BH}\parallel \parallel M\parallel +a_2\parallel M\otimes B\parallel )}{(\epsilon -\varrho )(\gamma -\varrho )}$ .

Case $I$ : $Z=ϵ$ , which indicates that

$$\begin{array}{c}\alpha \parallel \tilde{\mathrm{\Psi }}(t_0){\parallel }_{\tau }\le \alpha Z\mathrm{\Xi }+\alpha c.\hfill \end{array}$$

Then,

$$\begin{array}{c}\zeta (t^{*})<\eta (\alpha Z\mathrm{\Xi }+\alpha c)e^{-\rho (t^{*}-t_0)}=\zeta (t^{*})\hfill \end{array}$$

(33)

Case $\mathrm{II}$ : $Z=\alpha \parallel \tilde{\mathrm{\Psi }}(t_0){\parallel }_{\tau }$ , which indicates that

$$\begin{array}{c}\alpha \parallel \tilde{\mathrm{\Psi }}(t_0){\parallel }_{\tau }\ge \alpha Z\mathrm{\Xi }+\alpha c.\hfill \end{array}$$

Then, one has

$$\begin{array}{c}\zeta (t^{*})<\eta \{\alpha \parallel \tilde{\mathrm{\Psi }}(t_0){\parallel }_{\tau }-(\alpha Z\mathrm{\Xi }+\alpha c)\}{e}^{-\varrho (t^{*}-t_0)}\hfill \\ +\eta (\alpha Z\mathrm{\Xi }+\alpha c)e^{-\varrho (t^{*}-t_0)}\hfill \\ =\eta \alpha \parallel \tilde{\mathrm{\Psi }}(t_0){\parallel }_{\tau }{e}^{-\varrho (t^{*}-t_0)}=\zeta (t^{*}).\hfill \end{array}$$

(34)

Under the contradictions in (33) and (34), inequality (30) is valid. Consequently, letting $\eta \to 1$ leads to

$$\parallel \tilde{\mathrm{\Psi }}(t)\parallel \le Z{e}^{-\varrho (t-t_0)},t\ge t_0.$$

(35)

Inequality (35) implies $\tilde{\mathrm{\Psi }}(t)$ approaches to zero. Based on the above results, $\mathrm{\Gamma }(t)$ converge to $c{1}_N$ . Thus, $\underset{t\to \infty }{lim}[({\hat{x}}_i(t)-h_i(t))-({\hat{x}}_j(t)-h_j(t))]=0,\forall i,j=1,2,\cdots ,N$ . Noticing $\underset{t\to \infty }{lim}({\hat{x}}_i(t)-x_i(t))=0$ for $\forall i=1,2,\cdots ,N$ from (5), we have

$$\underset{t\to \infty }{\lim }\left[\left(x_i(t)-h_i(t)\right)-\left(x_j(t)-h_j(t)\right)\right]=0,$$

which indicates that linear MASs (1) ultimately implement the desired TVF $h(t)$ .

The proof is completed.

• Theorem 3. Under the assumption that the conditions of Theorem 2 hold, $\{t_k^i\}$ is free of Zeno-behavior.

• Proof. Denote

$${\stackrel{⌣}{u}}_i(t)=-H{\int }_0^{+\infty }{k}_i(\tau )q_i(t_k^i-\tau )d\tau ,t\in [t_k^i,t_{k+1}^i).$$

Then,

$${\stackrel{⌣}{u}}_i(t)=-H{\int }_0^{+\infty }{k}_i(\tau )q_i(t-\tau )d\tau -H{\int }_0^{+\infty }{k}_i(\tau )e_i(t-\tau )d\tau .$$

Thus,

$$\begin{array}{l}\stackrel{⌣}{u}(t)=-{\int }_0^{+\infty }(\widehat{L}(\tau )\otimes H)\mathrm{\Gamma }(t-\tau )d\tau -{\int }_0^{+\infty }(\widehat{K}(\tau )\otimes H)e(t-\tau )d\tau \\ =-{\int }_0^{+\infty }(\widehat{L}(\tau )M^T\otimes H)\tilde{\mathrm{\Psi }}(t-\tau )d\tau -{\int }_0^{+\infty }(\widehat{K}(\tau )\otimes H)e(t-\tau )d\tau .\end{array}$$

(36)

Taking the upper right-hand Dini derivative of $\Vert e_i(t)\Vert$ over $[t_k^i,t_{k+1}^i)$ gives

$$\begin{array}{c}{D}^{+}\Vert e_i(t)\Vert \le \Vert {\stackrel{·}{e}}_i(t)\Vert =\Vert {\stackrel{·}{q}}_i(t)\Vert =\Vert \sum _{j\in {\mathcal{N}}_i}({\stackrel{·}{\mathrm{\Gamma }}}_i(t)-{\stackrel{·}{\mathrm{\Gamma }}}_j(t))\Vert \hfill \\ =\Vert \sum _{j\in {\mathcal{N}}_i}[({\stackrel{·}{\hat{x}}}_i(t)-{\stackrel{·}{h}}_i(t))-({\stackrel{·}{\hat{x}}}_j(t)-{\stackrel{·}{h}}_j(t))]\Vert \hfill \\ =\Vert \sum _{j\in {\mathcal{N}}_i}(A({\hat{x}}_i(t)-{\hat{x}}_j(t))+B(u_i(t)-u_j(t))\hfill \\ +\mathrm{FC}({\tilde{\vartheta }}_i(t)-{\tilde{\vartheta }}_j(t))-({\stackrel{·}{h}}_i(t)-{\stackrel{·}{h}}_j(t)))\Vert \hfill \\ =\Vert \sum _{j\in {\mathcal{N}}_i}(A({\hat{x}}_i(t)-{\hat{x}}_j(t))+B(u_i(t)-u_j(t))\hfill \\ -(A{h}_i(t)+B{\mathrm{\Upsilon }}_i(t)-A{h}_j(t)-B{\mathrm{\Upsilon }}_j(t))+\mathrm{FC}({\tilde{\vartheta }}_i(t)-{\tilde{\vartheta }}_j(t))\Vert \hfill \\ =\Vert \sum _{j\in {\mathcal{N}}_i}(A({\mathrm{\Gamma }}_i(t)-{\mathrm{\Gamma }}_j(t))+B({\stackrel{⌣}{u}}_i(t)-{\stackrel{⌣}{u}}_j(t))\hfill \\ +B({\delta }_i(t)-{\delta }_j(t))+\mathrm{FC}\hfill \\ ({\int }_0^{+\infty }{s}_i(\tau ){\tilde{\vartheta }}_i(t-\tau )d\tau -{\int }_0^{+\infty }{s}_j(\tau ){\tilde{\vartheta }}_j(t-\tau )d\tau ))\Vert \hfill \\ \le \parallel L\otimes A\parallel \parallel M^T\otimes I\parallel \parallel \tilde{\mathrm{\Psi }}(t)\parallel +\parallel L\otimes B\parallel \parallel \stackrel{⌣}{u}(t)\parallel \hfill \\ +\parallel L\otimes B\parallel \parallel \delta (t)\parallel \hfill \\ +2(N-1){\int }_0^{+\infty }\parallel \hat{S}(\tau )\otimes \mathrm{FC}\parallel \parallel \tilde{\vartheta }(t-\tau )\parallel d\tau .\hfill \end{array}$$

(37)

Substituting (5), (26), (35), and (36) into (37) yields

$$\begin{array}{c}{D}^{+}\parallel e_i(t)\parallel \le \parallel L\otimes A\parallel \parallel M^T\otimes I\parallel \parallel \tilde{\mathrm{\Psi }}(t)\parallel \hfill \\ +\parallel L\otimes B\parallel \parallel M^T\parallel \parallel H\parallel {\int }_0^{+\infty }\parallel \hat{L}(\tau )\parallel \parallel \tilde{\mathrm{\Psi }}(t-\tau )\parallel d\tau \hfill \\ +a_1\parallel L\otimes B\parallel \parallel H\parallel {\int }_0^{+\infty }\parallel \hat{K}(\tau )\parallel \parallel \tilde{\mathrm{\Psi }}(t-\tau )\parallel d\tau \hfill \\ +b_1\parallel L\otimes B\parallel \parallel H\parallel {\int }_0^{+\infty }\parallel \hat{K}(\tau )\parallel \parallel e^{-\rho (t-t_0-\tau )}\parallel d\tau \hfill \end{array}$$

$$\begin{array}{c}+a_2\parallel L\otimes B\parallel \tilde{\mathrm{\Psi }}(t)\parallel +b_2\parallel L\otimes B\parallel e^{-\varrho (t-t_0)}\hfill \\ +2(N-1)\parallel \mathrm{FC}\parallel W\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }{\int }_0^{+\infty }\parallel \hat{S}(\tau )\parallel e^{-\lambda (t-t_0-\tau )}d\tau .\hfill \end{array}$$

(38)

Let $\overline{\phi }={\int }_0^{+\infty }\parallel \hat{L}(\tau )\parallel e^{\epsilon \tau }d\tau$ . Then,

$$\begin{array}{c}{D}^{+}\parallel e_i(t)\parallel \le \parallel L\otimes A\parallel \parallel M^T\otimes I\parallel Z{e}^{-\varrho (t-t_0)}\hfill \\ +\parallel L\otimes B\parallel \parallel M^T\parallel \parallel H\parallel Z{e}^{-\varrho (t-t_0)}\hfill \\ (({\int }_0^{+\infty }\parallel \hat{L}(\tau )\parallel e^{\epsilon \tau }d\tau )\times \underset{\tau \ge 0}{\sup }\left(e^{(\varrho -\epsilon )\tau }\right))\hfill \\ +a_1\parallel L\otimes B\parallel \parallel H\parallel Z{e}^{-\varrho (t-t_0)}\hfill \\ (({\int }_0^{+\infty }\parallel \hat{K}(\tau )\parallel e^{\epsilon \tau }d\tau )\times \underset{\tau \ge 0}{\sup }\left(e^{(\varrho -\epsilon )\tau }\right))\hfill \\ +b_1\parallel L\otimes B\parallel \parallel H\parallel e^{-\varrho (t-t_0)}\hfill \\ (({\int }_0^{+\infty }\parallel \hat{K}(\tau )\parallel e^{\epsilon \tau }d\tau )\times \underset{\tau \ge 0}{\sup }\left(e^{(\varrho -\epsilon )\tau }\right))\hfill \\ +a_2\parallel L\otimes B\parallel Z{e}^{-\varrho (t-t_0)}+b_2\parallel L\otimes B\parallel e^{-\varrho (t-t_0)}\hfill \\ +2(N-1)\parallel \mathrm{FC}\parallel W\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }{e}^{-\varrho (t-t_0)}\hfill \\ (({\int }_0^{+\infty }\parallel \hat{S}(\tau )\parallel e^{\upsilon \tau }d\tau )\times \underset{\tau \ge 0}{\sup }\left(e^{(\lambda -\upsilon )\tau }\right))\hfill \\ =\mathrm{\Theta }{e}^{-\varrho (t-t_0)},\hfill \end{array}$$

(39)

where $\mathrm{\Theta }=\parallel L\otimes A\parallel \parallel M^T\otimes I\parallel Z+\parallel L\otimes B\parallel \parallel M^T\parallel \parallel H\parallel Z\overline{\phi }+\parallel L\otimes B\parallel \parallel H$ . $(a_{1}Z+b_1)\varphi +a_2\parallel L\otimes B\parallel Z+b_2\parallel L\otimes B\parallel +2(N-1)\parallel \mathrm{FC}\parallel W\parallel \tilde{\vartheta }(t_0){\parallel }_{\tau }\overline{\psi }$

Computing the upper right-hand Dini derivative of $\Vert {\delta }_i(t)\Vert$ over interval $[t_k^i,t_{k+1}^i)$ and noticing that ${\stackrel{·}{\mathrm{\Upsilon }}}_i(t)$ is bounded, one gets

$$D^{+}\Vert {\delta }_i(t)\Vert \le \parallel {\stackrel{·}{\delta }}_i(t)\parallel =\Vert {\stackrel{·}{\mathrm{\Upsilon }}}_i(t)\Vert \le \theta ,$$

(40)

in which $\theta$ is the bound of ${\stackrel{·}{\mathrm{\Upsilon }}}_i(t)$ .

Combining (39), (40), ${\delta }_i(t_k^i)=0$ and $e_i(t_k^i)=0$ , one has

$$\parallel {\delta }_i(t)\parallel +\parallel e_i(t)\parallel \le \frac{\mathrm{\Theta }}{\varrho }(e^{-\varrho (t_k^i-t_0)}-e^{-\varrho (t-t_0)})+\theta (t-t_k^i).$$

(41)

The next event is triggered so long as $g(t)$ crosses zero, that is

$$\begin{array}{c}{\beta }_1\parallel q_i(t_k^i)\parallel +{\beta }_2{e}^{-\varrho (t_{k+1}^i-t_0)}=\parallel e_i(t_{k+1}^i)\parallel +\parallel {\delta }_i(t_{k+1}^i)\parallel \hfill \\ \le \frac{\mathrm{\Theta }}{\rho }(e^{-\varrho (t_k^i-t_0)}-e^{-\varrho (t_{k+1}^i-t_0)})+\theta (t_{k+1}^i-t_k^i).\hfill \end{array}$$

Hence,

$${\beta }_2{e}^{-\varrho (t_{k+1}^i-t_0)}\le \frac{\mathrm{\Theta }}{\varrho }(e^{-\varrho (t_k^i-t_0)}-e^{-\varrho (t_{k+1}^i-t_0)})+\theta (t_{k+1}^i-t_k^i).$$

Letting $T_k^i=t_{k+1}^i-t_k^i$ , we can obtain that

$${\beta }_2{e}^{-\rho T_k^i}\le \frac{\mathrm{\Theta }}{\varrho }(1-e^{-\rho T_k^i})+\frac{\theta T_k^i}{e^{-\varrho (t_k^i-t_0)}},$$

that is,

$$({\beta }_2+\frac{\mathrm{\Theta }}{\varrho })e^{-\rho T_k^i}\le \frac{\mathrm{\Theta }}{\varrho }+\frac{\theta T_k^i}{e^{-\rho (t_k^i-t_0)}},$$

(42)

which implies $\underset{k}{\mathrm{i}\mathrm{nf}}\{T_k^i\}>0$ , $\forall i$ . Consequently, there is no Zeno-behavior for any $i$ .

This completes the proof.

Simulations

MASs under consideration consist of six agents in 2D space. Suppose

$$L=\left[\begin{array}{cccccc}4\hfill & -1\hfill & -1\hfill & 0\hfill & -1\hfill & -1\hfill \\ \multicolumn{1}{c}{-1}& 2\hfill & 0\hfill & -1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-1}& 0\hfill & 3\hfill & -1\hfill & -1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -1\hfill & -1\hfill & 3\hfill & 0\hfill & -1\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -1\hfill & 0\hfill & 2\hfill & -1\hfill \\ \multicolumn{1}{c}{-1}& -1\hfill & 0\hfill & -1\hfill & -1\hfill & 4\hfill \end{array}\right].$$

The delay kernels are given as: $k_1(\tau )=2e^{-2\tau },k_2(\tau )=2e^{-2\tau },k_3(\tau )=3e^{-3\tau },k_4(\tau )=e^{-\tau },k_5(\tau )=e^{-\tau }$ , $k_6(\tau )=3e^{-3\tau },s_1(\tau )=4e^{-4\tau },s_2(\tau )=e^{-\tau },s_3(\tau )=2e^{-2\tau },s_4(\tau )=e^{-\tau },s_5(\tau )=2e^{-2\tau },s_6(\tau )=e^{-\tau }$ which satisfy ${\int }_0^{+\infty }{k}_i(\tau )d\tau =1$ , ${\int }_0^{+\infty }{s}_i(\tau )d\tau =1$ , ${\int }_0^{+\infty }{k}_i(\tau )e^{\epsilon \tau }d\tau <+\infty$ and ${\int }_0^{+\infty }{s}_i(\tau )e^{\upsilon \tau }d\tau <+\infty$ by taking $\epsilon <1$ and $\upsilon <1$ , $i=1,\cdots ,6$ . Furthermore, one can compute that $\psi =1.5119$ , $\varphi =3.3333$ and $\phi =18.0924$ by choosing $\epsilon =0.1$ and $\upsilon =0.5$ .

Assume that

$$A=\left[\begin{array}{cc}0\hfill & 2\hfill \\ \multicolumn{1}{c}{-1}& -2\hfill \end{array}\right],B=\left[\begin{array}{cc}0\hfill & 1\hfill \\ \multicolumn{1}{c}{1}& 1\hfill \end{array}\right],C=\left[\begin{array}{cc}1\hfill & 0\hfill \end{array}\right].$$

The desired TVF shape is

$$\begin{array}{c}\multicolumn{1}{c}{h(t)=\left[\begin{array}{cc}\cos t\hfill & \sin t\hfill \\ \multicolumn{1}{c}{\cos (t+\frac{2\pi }{6})}& \sin (t+\frac{2\pi }{6})\hfill \\ \multicolumn{1}{c}{\cos (t+\frac{4\pi }{6})}& \sin (t+\frac{4\pi }{6})\hfill \\ \multicolumn{1}{c}{\cos (t+\frac{6\pi }{6})}& \sin (t+\frac{6\pi }{6})\hfill \\ \multicolumn{1}{c}{\cos (t+\frac{8\pi }{6})}& \sin (t+\frac{8\pi }{6})\hfill \\ \multicolumn{1}{c}{\cos (t+\frac{10\pi }{6})}& \sin (t+\frac{10\pi }{6})\hfill \end{array}\right].}\end{array}$$

Putting condition (19) into consideration, $\mathrm{\Upsilon }(t)$ can be chosen as

$$\begin{array}{c}\multicolumn{1}{c}{\left[\begin{array}{cc}5\sin t+2\cos t\hfill & -3\sin t\hfill \\ \multicolumn{1}{c}{5\sin (t+\frac{2\pi }{6})+2\cos (t+\frac{2\pi }{6})}& -3\sin (t+\frac{2\pi }{6})\hfill \\ \multicolumn{1}{c}{5\sin (t+\frac{4\pi }{6})+2\cos (t+\frac{4\pi }{6})}& -3\sin (t+\frac{4\pi }{6})\hfill \\ \multicolumn{1}{c}{5\sin (t+\frac{6\pi }{6})+2\cos (t+\frac{6\pi }{6})}& -3\sin (t+\frac{6\pi }{6})\hfill \\ \multicolumn{1}{c}{5\sin (t+\frac{8\pi }{6})+2\cos (t+\frac{8\pi }{6})}& -3\sin (t+\frac{8\pi }{6})\hfill \\ \multicolumn{1}{c}{5\sin (t+\frac{10\pi }{6})+2\cos (t+\frac{10\pi }{6})}& -3\sin (t+\frac{10\pi }{6})\hfill \end{array}\right].}\end{array}$$

To guarantee A + FC is Hurwitz, we select $F={\left[\begin{array}{cc}0.0197\hfill & -0.0324\hfill \end{array}\right]}^T$ .

Letting $\mu =0.01$ , by ARE, we have

$$Q=\left[\begin{array}{cc}0.0087\hfill & 0.0049\hfill \\ \multicolumn{1}{c}{0.0049}& 0.0073\hfill \end{array}\right].$$

Then,

$$H=\sigma B^{T}Q=\left[\begin{array}{cc}0.0044\hfill & 0.0066\hfill \\ \multicolumn{1}{c}{0.0122}& 0.0100\hfill \end{array}\right]$$

with $\sigma =0.9$ . Given

$$M={\left[\begin{array}{ccccc}0.088\hfill & 0.584\hfill & 0.518\hfill & -0.242\hfill & -0.396\hfill \\ \multicolumn{1}{c}{0.175}& -0.044\hfill & 0.062\hfill & -0.466\hfill & 0.761\hfill \\ \multicolumn{1}{c}{0.263}& -0.672\hfill & 0.428\hfill & 0.328\hfill & -0.152\hfill \\ \multicolumn{1}{c}{0.000}& 0.404\hfill & -0.223\hfill & 0.738\hfill & 0.274\hfill \\ \multicolumn{1}{c}{0.351}& -0.807\hfill & -0.698\hfill & -0.242\hfill & -0.396\hfill \\ \multicolumn{1}{c}{-0.877}& -0.187\hfill & -0.087\hfill & -0.116\hfill & -0.091\hfill \end{array}\right]}^T,$$

one has $\alpha =4.4060$ , $\gamma =1.0247$ according to (18). By (6), we have $W=3.3621$ and $\varpi =0.9901$ . Hence, we get $\xi =0.0093$ and $\varsigma =0.0541$ through calculation. Then $\iota =0.0121$ by choosing ${\beta }_1=0.008<\xi$ . Thus, by Theorem 2, the desired formation can be reached for with $0.05<\lambda <\min \{\upsilon ,\varpi ,\varsigma \}$ and ${\beta }_2>0$ .

We randomly choose initial states $x(0)=[-2.09,2.55,-2.24,1.80,1.55,-3.37,-3.81,-0.02,4.59,-1.60,0.85,-2.76{]}^T$ and $\hat{x}(0)=[2.51,-2.45,0.06,1.99,3.91,4.59,0.47,-3.61,-3.51,-2.42,3.41,-2.46{]}^T$ . In view of the given conditions in Theorem 2, let ${\beta }_2=0.02$ and $\varrho =0.01$ . Figures 1–5 illustrate that the implement of desired TVF and Zeno-free behavior. Figures 1 and 2 show the motion trajectories of all agents and the state evolution of the observers. Figure 3 exhibits triggering instants. Figure 4 depicts the control input. The trajectories of six agents coupled with their shape at $t=1s$ , $t=11s$ and $t=16s$ are shown in Figure 5.

Figure 1.

Dynamics of each agent and the observer (the first component).

Figure 2.

Dynamics of each agent and the observer (the second component).

Figure 3.

Events for each agent.

Figure 4.

Control input of each agent.

Figure 5.

State snapshots: (a) the trajectory for each agent, (b) t = 1 s, (c) t = 11 s, and (d) t = 16 s.

Consider linear MASs (1) with observer (2) and the following continuous control protocol:

$$\begin{array}{c}\multicolumn{1}{c}{u_i(t)=-H{\int }_0^{+\infty }{k}_i(\tau )q_i(t-\tau )d\tau +{\mathrm{\Upsilon }}_i(t).}\end{array}$$

(43)

The proof of the implement of the desired TVF of MASs (1) with controller (43) is similar to that of Theorems 1 and 2, thus omitted here.

To compare our event-triggered control (13) with the continuous control (43), the variation trends of formation error based on the above same parameters and initial conditions for event-triggered control and continuous control are depicted in Figure 6(a) and (b), respectively. It is obvious that ${\mathrm{\Delta }}_i$ converges to the bounded region with the increase of time in both cases. Although it can be observed by comparing Figure 6(a) and (b) that the desired TVF can be achieved within the same time, the number of control update is reduced by event-triggered controller. In fact, the number of control update is 1200 in $[0,20]s$ with sampling step of $0.1s$ under continuous controller, while it is 600 under the event-triggered controller. It is, thus, demonstrated that the event-triggered mechanism proposed in this paper can reduce communication frequency effectively while maintaining control performance.

Figure 6.

Formation error ${\mathrm{\Gamma }}_i$ : (a) event-triggered control and (b) continuous control.

Conclusion

Time-varying formation control off linear MASs with distributed infinite input and output delays has been discussed in this study. To circumvent the direct utilization of individual agents’ internal states, a state observer has been developed to reconstruct system states. Aiming at reducing energy consumption and communication costs, an observer-based ETC scheme in consideration of the distributed infinite delays is then proposed, which can ensure the implement of TVF and the exclusion of Zeno-behavior in triggering time sequence. In future research, for a more comprehensive and realistic representation of MASs, we aim to explore TVF in heterogeneous MASs under uncertain disturbances.