I. INTRODUCTION

Recently,there has been an increasing interest in event-triggered feedback control systems^{[1, 2, 3, 4, 5, 6, 7, 8, 9]}. In the event-triggered control,data transmission or control actuation is executed after the occurrence of an event generated by an event-triggering mechanism. The event-triggering mechanism often depends on a well-defined event-triggering condition where the measurement error plays an essential role. When the magnitude of the measurement error reaches the prescribed threshold,an event is triggered. It is noted that the event-triggered technique can reduce resource usage and provide a higher degree of robustness. However,in many cases it requires dedicated hardware to monitor the plant permanently,which is not available in many general purpose devices. This motives the development of self-triggered control for digital platforms^{[10, 11, 12, 13]}. In self-triggered control,the next update time of the controller is computed at the previous one,without having to keep track of the measurement error that triggers the execution between two consecutive update instants. Due to its advantages,self-triggered control of multi-agent systems (MASs) has been receiving more and more attention. For example,in ^{[10, 12]},some novel self-triggered control strategies for MASs were proposed,under which MASs achieved average consensus. It can be observed from those existing works that self-triggered controller for MASs available nowadays are based on static state-feedback under a restrictive assumption that all the agents$'$ states can be measured. It can also be observed that individual agent dynamics are assumed to be a single-integrator. Compared with the conventional event-triggered control on MASs^{[14, 15]},continuous measurement and communication are not required in this paper,thus some actuation power and communication resource might be saved. To the best of our knowledge,there are few works either on self-triggered control for MASs with general linear dynamics based on state feedback or on output feedback. This motivates this study.

In this paper,we study the consensus problem of MASs with general linear dynamics via distributed self-triggered control. Some novel distributed self-triggered controllers are proposed for MASs$'$ asymptotically consensus. Both state feedback and output feedback distributed self-triggered consensus problems are investigated respectively,which help save the resource usage while guarantee a satisfactory consensus performance. The rest of the paper is organized as follows. In Section II,the problem under study is formulated. Sections III and IV provide the main results of this paper,that is,distributed self-triggered control laws based on state feedback and output feedback schemes,respectively. An example is presented in Section V to illustrate the effectiveness of the proposed control methods. Finally,the conclusions are given in Section VI.

${\bf Notations 1.}$ For vector $x={\rm col}\left( x_{1},\cdots ,x_{n}\right) \in {\bf R}^{n}$ and matrix $A=[a_{ij}]_{n\times n}\in {\bf R}^{n\times n}$,$\Vert x\Vert$ and $\Vert A\Vert$ denote 2-norms of $x$ and $A$,respectively. A real matrix $P>0$ $(P<0)$ denotes a positive (negative)-definite matrix $P$. $M^{\rm T}$ denotes the transpose of matrix $M$. The identity matrix of order $m$ is denoted as $I_{m}$. Moreover,matrices are assumed to have compatible dimensions if not explicitly stated. $A\otimes B$ denotes the Kronecker product of matrices $A$ and $B$. $\textbf{1}$ denote the column vector with all entries equal to one. ${\bf N}$ denotes the set of positive integers.

II. PROBLEM FORMULATION

The communication topology among agents is represented by an undirected graph $\mathcal {G}=(\mathcal {V},\mathcal {E},\mathcal {A})$,where $\mathcal {V}=$ $\{\upsilon_{1}$, $\cdots,\upsilon_{N}\}$ is the set of nodes with the node indices belonging to a finite index set $\mathcal {I}=\{1,\cdots,N\}$ and $\mathcal {E}\subseteq \mathcal {V}\times \mathcal {V}$ is the set of unordered pairs of nodes,called edges. Two nodes $\upsilon_{i}$,$\upsilon_{j}$ are adjacent,or neighboring,if $(\upsilon_{i},\upsilon_{j})$ is an edge of graph $\mathcal {G}$. A path on $\mathcal {G}$ from node $\upsilon_{i_{1}}$ to node $\upsilon_{i_{l}}$ is a sequence of edges of the form $(\upsilon_{i_{k}},\upsilon_{i_{k+1}})$,$k=1,\cdots,l-1$. A graph is called connected if there exists a path between every pair of distinct nodes. The adjacency matrix $\mathcal {A}=$ $[a_{ij}]$ $\in$ ${\bf R}^{N\times N}$ is the matrix with nonnegative adjacency elements $a_{ij}$ and zero diagonal elements. If edge $({{\upsilon_i}},{{\upsilon_j}})$ $\in$ $\mathcal {E}$,then node $\upsilon_{j}$ is called a neighbor of node $\upsilon_{i}$ and $a_{ij}$ $>$ $0$ $\Leftrightarrow$ $(\upsilon_{i},\upsilon_{j})\in \mathcal {E}$. The neighbor index set of agent $\upsilon_{i}$ is denoted by $\mathcal {N}_{i}=\{j\in\mathcal {I}|(\upsilon_{j},\upsilon_{i})\in\mathcal {E}\}$. The degree matrix of $\mathcal {G}$ is given by $\Delta={\rm diag}\{\Delta_1,\Delta_2,\cdots,\Delta_N\}$,where $\Delta_i=$ $\sum\nolimits_{j\in\mathcal {N}_{i}}a_{ij}$. Matrix $\mathcal {L}=\Delta-\mathcal {A}$ is the Laplacian matrix of graph $\mathcal {G}$. If $\mathcal {G}$ is connected,its Laplacian matrix has a single zero eigenvalue and the corresponding eigenvector is $\textbf{1}$ and the eigenvalues of $\mathcal {L}$ are denoted by $0=$ $\lambda_{1}$ $<$ $\lambda_{2}\leq\cdots\leq\lambda_{N}$.

In this paper,the consensus problem for a group of $N$ identical agents with general linear dynamics is investigated,which can be described by

\begin{align}\label{1} \left\{ \begin{array}{lll} \dot{x}_{i}(t)=Ax_{i}(t)+Bu_{i}(t),\\[1mm] y_{i}(t)=Cx_i(t),~~ i=1,2,\cdots,N, \end{array} \right. \end{align}

(1)

where $x_{i}(t)\in{\bf R}^{n}$ is the state,$u_{i}(t)\in {\bf R}^{p}$ is the control input,$y_{i}(t)\in {\bf R}^{q}$ is the measured output,$A$,$B$ and $C$ are constant matrices with compatible dimensions. Assume that $(A,B)$ is controllable and $(A,C)$ is observable. Protocol $u_i(t)$ is said to solve the consensus problem asymptotically,if the states of agents satisfy

\begin{equation}\label{2} \begin{array}{cll} \lim\limits_{t\rightarrow \infty}\|x_{i}(t)-x_{j}(t)\|=0,~~\forall i,j\in\mathcal {I}, ~i\neq j. \end{array} \end{equation}

(2)

In this paper,both distributed self-triggered state feedback and output feedback control laws are proposed.

III. DISTRIBUTED SELF-TRIGGERED CONTROL BASED ON STATE FEEDBACK

In self-triggered control,each control task triggers its next release based on the value of the last sampled measurement. If all the states of agents are measurable,the distributed self-triggered state feedback control law is designed as

\begin{equation}\label{3} \begin{array}{cll} u_{i}(t)=-\mu F\omega_{i}(t_{k}^{i}),~~t\in[t_{k}^{i},t_{k+1}^{i}), \end{array} \end{equation}

(3)

where $\omega_{i}(t_{k}^{i})=\sum\nolimits_{j\in\mathcal {N}_{i}}a_{ij}(x_{i}(t_{k}^{i})-x_{j}(t_{k}^{i}))$,$\mu$ is a positive scalar and $F\in{\bf R}^{p\times n}$ is the feedback gain matrix.

${\bf Remark 1.}$ It is noted that the controller used in most existing works on event-triggered control of MASs^{[2, 8, 10]} is triggered at the neighbors$'$ event time,i.e.,$u_i(t)$ $=$ $\sum\nolimits_{j\in\mathcal {N}_{i}}a_{ij}(x_i(t_k^{i})-x_j(t_{k'}^{j}))$,where $k'=k'(t)=\arg\max\nolimits_{l\in{\bf N}}\{l|t\geq t_{l}^{j}\}$. Thus for each $t\in[t_{k}^{i},t_{k+1}^{i})$,$t_{k'(t)}^{j}$ is the last event time of agent $j$. However,controller (3) is triggered only at the event time of itself.

The sequence of execution instants for agent $i$ is denoted by $t_0^{i},t_1^{i},\cdots$. The state measurement error is defined as

\begin{align}\label{4} e_{i}(t)=\omega_{i}(t_{k}^{i})-\omega_{i}(t),~~t\in[t_{k}^{i},t_{k+1}^{i}), \end{align}

(4)

where $~\omega_{i}(t)=\sum\nolimits_{j\in\mathcal {N}_{i}}a_{ij}(x_{i}(t)-x_{j}(t))$. The closed-loop system of (1) and (3) is

\begin{align*} \dot{x}_{i}(t)=&\ Ax_{i}(t)-\mu BF\omega_{i}(t_{k}^{i})=\\ &\ Ax_{i}(t)-\mu BF(e_{i}(t)+\omega_{i}(t)), \end{align*}

which can be written in a compact form

\begin{equation}\label{5} \begin{array}{cll} \dot{x}(t)=\bar{A}x(t)-(\mu I_{N}\otimes BF)(e(t)+\omega(t)), \end{array} \end{equation}

(5)

where $\bar{A}=I_{N}\otimes A$,$x(t)=[x_{1}^{\rm T}(t),\cdots,x_{N}^{\rm T}(t)]^{\rm T}$,$e(t)=$ $[e_{1}^{\rm T}(t)$,$\cdots$,$e_{N}^{\rm T}(t)]^{\rm T}$,and $ \omega(t)=[\omega_{1}^{\rm T}(t),\cdots,\omega_{N}^{\rm T}(t)]^{\rm T}$. By multiplying both sides of (5) by $\mathcal {L}\otimes I_{m}$,one has

\begin{equation}\label{6} \begin{array}{cll} \dot{\omega}(t)=(\bar{A}-\mu\mathcal {L}\otimes BF)\omega(t)-(\mu\mathcal {L}\otimes BF)e(t), \end{array} \end{equation}

(6)

thus

\begin{equation}\label{7} \begin{array}{cll} \dot{\omega}_{i}(t)=A\omega_{i}(t)-\mu BF\sum\limits_{j\in\mathcal {N}_{i}}a_{ij}(\omega_{i}(t_{k}^{i})-\omega_{j}(t_{k'}^{j})). \end{array} \end{equation}

(7)

Then we have the following result.

${\bf Theorem 1.}$ Assume that the communication graph $\mathcal {G}$ is connected. Given $\delta>0$,consider the controller gain $F$ $=$ $B^{\rm T}P$,where $P>0$ is a solution of the Riccati equation

\begin{equation}\label{8} \begin{array}{cll} A^{\rm T}P+PA-2PBB^{\rm T}P+\delta I=0, \end{array} \end{equation}

(8)

and the triggering instant is chosen such that

\begin{align}\label{9} &t_{k+1}^{i}\leq t_{k}^{i}+\frac{1}{\|A\|}\ln\Bigg(1+\dfrac{d_{i}}{1+d_{i}}\times\notag\\ &\qquad \dfrac{\|A\|\|\omega_{i}(t_{k}^{i})\|}{\|A\omega_{i}(t_{k}^{i})\|+\|\mu BF\sum\limits_{j\in\mathcal {N}_{j}}a_{ij} (\omega_{i}(t_{k}^{i})-\omega_{j}(t_{k'}^{j}))\|}\Bigg), \end{align}

(9)

where $\mu$ is sufficiently large such that $\mu\lambda_{2}\geq 1$,$d_{i}=\Big(\sigma_{i}\frac{\delta-2\mu\theta \Delta_{i}\|PBB^{\rm T}P\|}{\frac{2\mu\Delta_{i}}{\theta}\|PBB^{\rm T}P\|}\Big)^{1/2}$, with $0<\sigma_{i}<1$ and $0<\theta<\frac{\delta}{2\mu\max\{\Delta_{i}\}\|PBB^{\rm T}P\|}$, then $N$ agents in (1) will reach consensus under the control law (3).

${\bf Proof.}$ Consider a Lyapunov function candidate for the closed-loop system as

\begin{equation}\label{10} \begin{array}{cll} V(t)=\omega^{\rm T}(t)\bar{P}\omega(t), \end{array} \end{equation}

(10)

where $\bar{P}=I_{N}\otimes P$ and $P>0$. Calculating the time derivative of $V(t)$ along the solution of (5),one has

\begin{align} \dot{V}(t)=&\ \omega^{\rm T}(t)\big(I_{N}\otimes(PA+A^{\rm T}P)-2\mu\mathcal {L}\otimes PBB^{\rm T}P \big)\times\notag\\ &\ \omega(t)+2\mu\sum\limits_{i=1}^{N}\sum\limits_{j\in\mathcal {N}_{i}}a_{ij}\omega_{i}(t)PBB^{\rm T}P\times\notag\\ &\ (e_j(t)-e_{i}(t)). \end{align}

(11)

Since $\mathcal {G}$ is connected,zero is a simple eigenvalue of $\mathcal {L}$ and all the other eigenvalues are positive. Let $U\in{\bf R}^{N\times N}$ be a unitary matrix such that $U^{\rm T}\mathcal {L}U=\Lambda={\rm diag}\{0,\lambda_{2},\cdots,\lambda_{N}\}$. The right and left eigenvectors of $\mathcal {L}$ corresponding to the zero eigenvalue are $\textbf{1}$ and $\textbf{1}^{\rm T}$,respectively. One can choose $U$ $=$ $[\frac{\textbf{1}}{\sqrt{N}}~~X_1]$ and $U^{\rm T}=\left[ \begin{array}{c} \frac{\textbf{1}^{\rm T}}{\sqrt{N}} \\ X_2 \\ \end{array} \right]$,with $X_{1}\in{\bf R}^{N\times(N-1)}$ and $X_{2}\in{\bf R}^{(N-1)\times N}$. Let $\xi(t)=[\xi_{1}^{\rm T}(t),\cdots,\xi_{N}^{\rm T}(t)]^{\rm T}=(U^{\rm T}\otimes I_n)\omega(t)$. Notice that $\omega_{1}(t)+\cdots +\omega_{N}(t)=0$. Then $\xi_1(t)$ $=$ $0$,and thus

\begin{align}\label{12} \omega^{\rm T}(t)&\big(I_{N}\otimes(PA+A^{\rm T}P)-2\mu\mathcal {L}\otimes PBB^{\rm T}P \big)\omega(t)=\notag\\ &\ \xi^{\rm T}(t)\big(I_{N}\otimes(PA+A^{\rm T}P)-2\mu\Lambda\otimes PBB^{\rm T}P \big)\xi(t)=\notag\\ &\sum\limits_{i=2}^{N}\xi_{i}^{\rm T}(t)\big(PA+A^{\rm T}P-2\mu\lambda_{i} PBB^{\rm T}P \big)\xi_{i}(t). \end{align}

(12)

By choosing sufficiently large $\mu$ such that $\mu\lambda_{2}\geq 1$,one has

\begin{align} &PA+A^{\rm T}P-2\mu\lambda_{i} PBB^{\rm T}P\leq\notag\\ &\qquad PA+A^{\rm T}P-2 PBB^{\rm T}P =-\delta I, \end{align}

(13)

where the last equation is derived by using (8). It follows from (12) that

\begin{align} &\omega^{\rm T}(t)\big(I_{N}\otimes(PA+A^{\rm T}P)-2\mu\mathcal {L}\otimes PBB^{\rm T}P \big)\omega(t)\leq\notag\\ &\qquad\ -\delta \|\xi(t)\|^{2}=-\delta \|\omega(t)\|^{2}. \end{align}

(14)

From (12) to (14) and noticing $a_{ij}=a_{ji}$,one has

\begin{align} \dot{V}(t)\leq& -\delta\sum\limits_{i=1}^{N}\|\omega_{i}(t)\|^{2}\;+\notag\\ &\ 2\mu\sum\limits_{i=1}^{N}\sum\limits_{j\in\mathcal {N}_{i}}a_{ij}\omega_{i}(t)PBB^{\rm T}P(e_j(t)-e_{i}(t))\leq\notag\\ & -\delta\sum\limits_{i=1}^{N}\|\omega_{i}(t)\|^{2}+\mu\|PBB^{\rm T}P\|\sum\limits_{i=1}^{N}\Delta_{i}(\theta\|\omega_{i}(t)\|^{2}\;+\notag\\ &\ \dfrac{1}{\theta}\|e_{i}(t)\|^{2})+\mu\|PBB^{\rm T}P\|\sum\limits_{i=1}^{N}\Delta_{i}\theta\|\omega_{i}(t)\|^{2}\;+\notag\\ &\ \mu\|PBB^{\rm T}P\|\sum\limits_{i=1}^{N}\sum\limits_{j\in\mathcal {N}_{i}}a_{ij}\dfrac{1}{\theta}\|e_{j}(t)\|^{2}=\notag\\ &\ \sum\limits_{i=1}^{N}\Big\{\big(-\delta +2\mu\|PBB^{\rm T}P\|\Delta_{i}\theta \big)\|\omega_{i}(t)\|^{2}\;+\notag\\ &\ 2\mu\|PBB^{\rm T}P\|\dfrac{\Delta_{i}}{\theta}\|e_{i}(t)\|^{2}\Big\}, \end{align}

(15)

where $\theta$ is a positive scalar defined in Theorem 1. For each $i$,define the triggering condition as

\begin{equation}\label{16} \begin{array}{cll} \|e_{i}(t)\|\leq\dfrac{d_{i}}{1+d_{i}}\|\omega_{i}(t_{k}^{i})\|, \end{array} \end{equation}

(16)

where $d_i$ is defined in Theorem 1. It follows from (4) and (16) that $\|e_{i}(t)\|\leq d_{i}\|\omega_{i}(t)\|$. Thus

\begin{eqnarray*} \begin{array}{cll} \dot{V}(t)\leq\sum\limits_{i=1}^{N}(\sigma_{i}-1)(\delta-2\mu\theta\Delta_{i}\|PBB^{\rm T}P\|)\|\omega_{i}(t)\|^{2}, \end{array} \end{eqnarray*}

then $\dot{V}(t)<0$,for any $0<\sigma_{i}<1$ and $0<\theta<\frac{\delta}{2\mu\max\nolimits_{i}\{\Delta_{i}\}\|PBB^{\rm T}P\|}$. It follows from (4) and (7) that

\begin{align}\label{17} \|\dot{e}_{i}(t)\|=&\ \|\dot{\omega}_{i}(t)\|\leq\notag\\ &\ \|A\|\|e_{i}(t)\|+\|A\omega_{i}(t_{k}^{i})\|+\notag\\ &\ \|\mu BF\sum\limits_{j\in\mathcal {N}_{i}}a_{ij}(\omega_{i}(t_{k}^{i})-\omega_{j}(t_{k'}^{j}))\|. \end{align}

(17)

So the evolution of $\|e_{i}(t)\|$ for $t\in[t_{k}^{i},t_{k+1}^{i})$ is bounded by the solution of

\begin{align}\label{18} \|\dot{p}_{i}(t)\|=&\ \|A\|\|p_{i}(t)\|+\|A\omega_{i}(t_{k}^{i})\|+\notag\\ &\ \|\mu BF\sum\limits_{j\in\mathcal {N}_{i}}a_{ij}(\omega_{i}(t_{k}^{i})-\omega_{j}(t_{k'}^{j}))\| \end{align}

(18)

with $p_{i}(t_{k}^{i})=0$. Thus the corresponding solution of (18) is given by

\begin{align}\label{19} &\|p_{i}(t)\|=\notag\\ &\qquad\dfrac{\|A\omega_{i}(t_{k}^{i})\|+\|\mu BF\sum\limits_{j\in\mathcal {N}_{i}}a_{ij}(\omega_{i}(t_{k}^{i})-\omega_{j}(t_{k'}^{j}))\|}{\|A\|}\times\notag\\ &\qquad({\rm e}^{\|A\|(t-t_{k}^{i})}-1). \end{align}

(19)

From (16) and (19),one has that an upper bound of the time for $\|e_{i}(t)\|$ to evolve from $0$ to $\frac{d_{i}}{1+d_{i}}\|\omega_{i}(t_{k}^{i})\|$ satisfies

\begin{align}\label{20} &\dfrac{\|A\omega_{i}(t_{k}^{i})\|+\|\mu BF\sum\limits_{j\in\mathcal {N}_{i}}a_{ij}(\omega_{i}(t_{k}^{i})-\omega_{j}(t_{k'}^{j}))\|}{\|A\|}\times\notag\\ &\qquad (e^{\|A\|(t-t_{k}^{i})}-1)=\dfrac{d_{i}}{1+d_{i}}\|\omega_{i}(t_{k}^{i})\|. \end{align}

(20)

Thus the triggering time can be chosen as

\begin{align*} &t_{k+1}^{i}\leq t_{k}^{i}+\dfrac{1}{\|A\|}\ln\Big(1+\\ &\qquad\dfrac{d_{i}}{1+d_{i}}\dfrac{\|A\|\|\omega_{i}(t_{k}^{i})\|}{\|A\omega_{i}(t_{k}^{i})\|+\|\mu BF\sum\limits_{j\in\mathcal {N}_{j}}a_{ij}(\omega_{i}(t_{k}^{i})-\omega_{j}(t_{k'}^{j}))\|}\Big). \end{align*}

IV. DISTRIBUTED SELF-TRIGGERED OUTPUT FEEDBACK CONTROL

If some states of the system cannot be measured,control strategies based on state feedback cannot be used. In this case, control schemes based on output feedback should be used. A state observer can be designed as

\begin{equation}\label{21} \left\{ \begin{array}{lll} \dot{\hat{x}}_{i}(t)=A\hat{x}_{i}(t)+Bu_{i}(t)+L(y_{i}(t)-\hat{y}_{i}(t)),\\ \hat{y}_{i}(t)=C\hat{x}_i(t),~~ i=1,2,\cdots,N, \end{array} \right. \end{equation}

(21)

where $\hat{x}_{i}(t)\in{\bf R}^{n}$ is the observer state, $\hat{y}_{i}(t)\in {\bf R}^{q}$ is the observer measured output, $L\in {\bf R}^{n\times q}$ is a constant matrix to be designed. Define

\begin{align}\label{22} &\tilde{x}_{i}(t)=x_{i}(t)-\hat{x}_{i}(t),\notag\\%~\hat{\omega}_{i}(t)=\sum\limits_{j\in\mathcal{N}_{i}}a_{ij}(\hat{x}_{i}(t)-\hat{x}_{j}(t)),\notag\\ &\tilde{\omega}_{i}(t)= \sum\limits_{j\in\mathcal {N}_{i}}a_{ij}(\tilde{x}_{i}(t)-\tilde{x}_{j}(t)). \end{align}

(22)

The distributed self-triggered observer-based output control law is designed as

\begin{align}\label{23} u_{i}(t)=-\mu F\hat{\omega}_{i}(t_{k}^{i}), \end{align}

(23)

where $\hat{\omega}_{i}(t_{k}^{i})=\sum\nolimits_{j\in\mathcal {N}_{i}}a_{ij}(\hat{x}_{i}(t_{k}^{i})-\hat{x}_{j}(t_{k}^{i}))$. The state measurement error is defined as

\begin{align}\label{24} e_{i}(t)=\hat{\omega}_{i}(t_{k}^{i})-\hat{\omega}_{i}(t). \end{align}

(24)

By multiplying both sides of the closed-loop system of (1) and (23) by $\mathcal {L}\otimes I_{m}$, one has

\begin{equation}\label{25} \left\{ \begin{array}{lll} \dot{\omega}(t)=\bar{A}\omega(t)-(\mu \mathcal {L}\otimes BF)\hat{\omega}(t)-(\mu \mathcal {L}\otimes BF)e(t),\\ \dot{\hat{\omega}}(t)=(\bar{A}-\mu \mathcal {L}\otimes BF)\hat{\omega}(t)-(\mu \mathcal {L}\otimes BF)e(t)+\\ \qquad\quad (I_{N}\otimes LC)\tilde{\omega}(t),\\ \dot{\tilde{\omega}}(t)=(I_{N}\otimes(A- LC))\tilde{\omega}(t), \end{array} \right. \end{equation}

(25)

where $\tilde{\omega}(t)=[\tilde{\omega}_{1}^{\rm T}(t),\cdots,\tilde{\omega}_{N}^{\rm T}(t)]^{\rm T}$ and $\hat{\omega}(t)=$ $[\hat{\omega}_{1}^{\rm T}(t)$,$\cdots$, $\hat{\omega}_{N}^{\rm T}(t)]^{\rm T}$. It can be observed from the third equation in (25) that $\tilde{\omega}(t)$ will approach zero asymptotically if $L$ is designed to make $A-LC$ Hurwitz. Thus the stability of the second equation in (25) is equivalent to the stability of the following system

\begin{equation}\label{26} \begin{array}{cll} \dot{\hat{\omega}}(t)=(\bar{A}-\mu\mathcal {L}\otimes BF)\hat{\omega}(t)-(\mu\mathcal {L}\otimes BF)e(t). \end{array} \end{equation}

(26)

Then we get the following result.

${\bf Theorem 2.}$ Assume that the communication graph $\mathcal {G}$ is connected. Let $L$ be any gain matrix such that $A-LC$ is Hurwitz. If the triggering time is chosen such that

\begin{equation}\label{27} \begin{array}{cll} t_{k+1}^{i}\leq t_{k}^{i}+\tau, \end{array} \end{equation}

(27)

where $\tau$ satisfies the equation

\begin{equation}\label{28} \begin{array}{cll} b\tau+\dfrac{c}{a}=\delta {\rm e}^{-a\tau}, \end{array} \end{equation}

(28)

with $a=\|A\|$,$ b=\frac{\|LC\|}{\|A\|}\|\mu BF\sum\nolimits_{j\in\mathcal {N}_{i}}(\hat{\omega}_{i}(t_{k}^{i})-\hat{\omega}_{j}(t_{k'}^{j}))\|$, $c=(\|A\|+\|LC\|(1+\frac{d_{i}}{1+d_{i}}))\|\hat{\omega}_{i}(t_{k}^{i})\|$ $+$ $(1$ - $\frac{\|LC\|}{\|A\|})\|\mu BF\sum\nolimits_{j\in\mathcal {N}_{i}}(\hat{\omega}_{i}(t_{k}^{i})-\hat{\omega}_{j}(t_{k'}^{j}))\|$, $\delta=\frac{c}{a}+\frac{d_{i}}{1+d_{i}}\|\hat{\omega}_{i}(t_{k}^{i})\|$, $\mu$ is sufficiently large such that $\mu\lambda_{2}\geq 1$, $F$ $=$ $B^{\rm T}P$,with $P>0$ being a solution of the Riccati equation (8),and $d_i$ is defined in Theorem 1,then $N$ agents in (1) will reach consensus under the control law (23).

${\bf Proof.}$ Consider a Lyapunov function candidate for the closed-loop system as

\begin{eqnarray*} \begin{array}{cll} V(t)=\hat{\omega}^{\rm T}(t)\bar{P}\hat{\omega}(t), \end{array} \end{eqnarray*}

where $\bar{P}$ is defined in (10). Calculating the time derivative of $V(t)$ along the solution of (26),one has

\begin{align}\label{29} &\dot{V}(t)=\notag\\ &\qquad \hat{\omega}^{\rm T}(t)\big(I_{N}\otimes(PA+A^{\rm T}P)-2\mu\mathcal {L}\otimes PBB^{\rm T}P \big)\hat{\omega}(t)+\notag\\ &\qquad 2\mu\sum\limits_{i=1}^{N}\sum\limits_{j\in\mathcal {N}_{i}}a_{ij}\hat{\omega}_{i}(t)PBB^{\rm T}P(e_j(t)-e_{i}(t)). \end{align}

(29)

Similar to the analysis from (12) to (15),one has

\begin{align}\label{30} \dot{V}(t)\leq&\ \sum\limits_{i=1}^{N}\Big\{\big(-\delta +2\mu\|PBB^{\rm T}P\|\Delta_{i}\theta \big)\|\hat{\omega}_{i}(t)\|^{2}+\notag\\ &\ 2\mu\|PBB^{\rm T}P\|\frac{\Delta_{i}}{\theta}\|e_{i}(t)\|^{2}\Big\}. \end{align}

(30)

Thus for agent $i$ the triggering condition can be designed as

\begin{equation}\label{31} \begin{array}{cll} \|e_{i}(t)\|\leq\dfrac{d_{i}}{1+d_{i}}\|\hat{\omega}_{i}(t_{k}^{i})\|, \end{array} \end{equation}

(31)

where $d_i$ is defined in Theorem 1. It follows from (24) and (30) that

\begin{equation}\label{32} \begin{array}{cll} \|e_{i}(t)\|\leq d_{i}\|\hat{\omega}_{i}(t)\|. \end{array} \end{equation}

(32)

Thus $$ \dot{V}(t)\leq\sum\limits_{i=1}^{N}(\sigma_{i}-1)(\delta-2\mu\theta\Delta_{i}\|PBB^{\rm T}P\|)\|\hat{\omega}_{i}(t)\|^{2}, $$ then $\dot{V}(t)<0$,for any $0<\sigma_{i}<1$ and $0<\theta<\frac{\delta}{2\mu\max\nolimits_{i}\{\Delta_{i}\}\|PBB^{\rm T}P\|}$.

It follows from (24) and (30) that $$ \|\hat{\omega_{i}}(t_{k}^{i})-\hat{\omega_{i}}(t)\|\leq\dfrac{d_{i}}{1+d_{i}}\|\hat{\omega_{i}}(t_{k}^{i})\|, $$ which implies $$ \|\hat{\omega_{i}}(t)\|<\left(1+\dfrac{d_{i}}{1+d_{i}}\right)\|\hat{\omega_{i}}(t_{k}^{i})\|. $$ One further has

\begin{equation}\label{33} \begin{array}{cll} \|\tilde{\omega_{i}}(t)\|-\|\omega_{i}(t)\|\leq\left(1+\dfrac{d_{i}}{1+d_{i}}\right)\|\hat{\omega_{i}}(t_{k}^{i})\|, \end{array} \end{equation}

(33)

thus

\begin{equation}\label{34} \begin{array}{cll} \|\tilde{\omega_{i}}(t)\|\leq \|\omega_{i}(t)\|+\left(1+\dfrac{d_{i}}{1+d_{i}}\right)\|\hat{\omega_{i}}(t_{k}^{i})\|. \end{array} \end{equation}

(34)

From (24) and the first equation in (25),one has

\begin{equation}\label{35} \begin{array}{cll} \dot{\omega}_{i}(t)=A\omega_{i}(t)-\mu BF\sum\limits_{j\in\mathcal {N}_{i}}(\hat{\omega}_{i}(t_{k}^{i})-\hat{\omega}_{j}(t_{k'}^{j})), \end{array} \end{equation}

(35)

then

\begin{align}\label{36} \|\omega_{i}(t)\|\leq&\ \dfrac{1}{\|A\|}\|\mu BF\sum\limits_{j\in\mathcal {N}_{i}}a_{ij}(\hat{\omega}_{i}(t_{k}^{i})-\hat{\omega}_{j}(t_{k}^{j}))\|\times\notag\\ &\ \left({\rm e}^{\|A\|(t-t_{k}^{i})}-1\right). \end{align}

(36)

From (24) and the second equation in (25),one has

\begin{align}\label{37} \|\dot{e}_{i}(t)\|=&\ \|\dot{\hat{\omega_{i}}}(t)\|=\notag\\ &\ \|A\|\|e_{i}(t)\|+\|A\|\|\hat{\omega}_{i}(t_{k}^{i})\|+\notag\\ &\ \|\mu BF\sum\limits_{j\in\mathcal {N}_{i}}a_{ij}(\hat{\omega}_{i}(t_{k}^{i})-\hat{\omega}_{j}(t_{k'}^{j}))\|\;+\notag\\ &\ \|LC\|\|\tilde{\omega_{i}}(t)\|. \end{align}

(37)

Substituting (34) and (36) into (37),one has

\begin{equation}\label{38} \begin{array}{cll} \|\dot{e}_{i}(t)\|\leq a\|e_{i}(t)\|+b{\rm e}^{a(t-t_{k}^{i})}+c,~~t\in[t_{k}^{i},t_{k+1}^{i}), \end{array} \end{equation}

(38)

where $a,b$ and $c$ are defined in Theorem 2.

So the evolution of $\|e_{i}(t)\|$ for $t\in[t_{k}^{i},t_{k+1}^{i})$ is bounded by the solution of

\begin{equation}\label{39} \begin{array}{cll} \|\dot{p}_{i}(t)\|=a\|p_{i}(t)\|+b{\rm e}^{a(t-t_{k}^{i})}+c. \end{array} \end{equation}

(39)

With $p(t_{k}^{i})=0$,the corresponding solution of (39) is given by

\begin{equation}\label{40} \begin{array}{cll} \|p_{i}(t)\|={\rm e}^{a(t-t_{k}^{i})}\left(\dfrac{c}{a}+b(t-t_{k}^{i})\right)-\dfrac{c}{a}. \end{array} \end{equation}

(40)

From (31) and (40),one has that the upper bound of the time for $\|e_{i}(t)\|$ to evolve from $0$ to $\frac{d_{i}}{1+d_{i}}\|\hat{\omega}_{i}(t_{k}^{i})\|$ satisfies

\begin{equation}\label{41} \begin{array}{cll} {\rm e}^{a(t-t_{k}^{i})}\left(\dfrac{c}{a}+b(t-t_{k}^{i})\right)-\dfrac{c}{a}=\dfrac{d_{i}}{1+d_{i}}\|\hat{\omega_{i}}(t_{k}^{i})\|, \end{array} \end{equation}

(41)

which can be rewritten as

\begin{equation}\label{42} \begin{array}{cll} \dfrac{c}{a}+b\tau=\delta {\rm e}^{-a\tau}, \end{array} \end{equation}

(42)

where $\tau=t-t_{k}^{i}$. Since $\delta>\frac{c}{a}$,$\delta {\rm e}^{-a\tau}$ approaches zero and $b\tau$ approaches positive infinity as $\tau$ goes to infinity. Then there exists a positive scalar $\tau$ that solves the equation. Thus the triggering time can be chosen as $t_{k+1}^{i}\leq t_{k}^{i}+\tau$. The proof is thus completed.

V. A SIMULATION EXAMPLE

In this section,an example is provided to validate the effectiveness of the proposed control approaches. Consider a network described as follows \begin{align*} \left\{ \begin{array}{cll} \dot{x}_{i}(t)=&\!\!\!\!\!\left[ \begin{array}{cccc} 0 & 1 & 0 & 0 \\ -48.6 & -1.25 & 48.6 & 0 \\ 0 & 0 & 0 & 10 \\ 1.95 & 0 & -1.95 & 0 \\ \end{array} \right] x_{i}(t)+\\[7mm] &\!\!\!\!\!\left[ \begin{array}{c} 0 \\ 21.6 \\ 0 \\ 0 \\ \end{array} \right] u_{i}(t),\\[7mm] y_{i}(t)=&\!\!\!\!\!\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{array} \right]x_{i}(t),~~ i=1,2,3,4. \end{array} \right. \end{align*} The Laplacian matrix of the network is given by $$\mathcal {L}=\left[ \begin{array}{rrrr} 1 & -1 & 0 & 0 \\ -1 & 3 & -1 & -1 \\ 0 & -1 & 2 & -1 \\ 0 & -1 & -1 & 2 \\ \end{array} \right]. $$ Choose $\sigma_{1}=0.1$,$\sigma_{2}=0.2$,$\sigma_{3}=0.3$, $\sigma_{4}=0.4$,$\delta=0.01$,$\theta$ $=0.002$. By solving Riccati (8),one has $F=[0.2663$ ~$0.2214$~ $0.0499$ ~$1.2022]$. Using controller (23) and triggering condition (27),the states and states measurement errors of multi-agent systems are shown in Figs.1 and 2,respectively. Choose $L=\left[ \begin{array}{cccc} 10 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{array} \right]^{\rm T}$,$\sigma_{1}=0.1$,$\sigma_{2}$ $=$ $0.2$,$\sigma_{3}$ $=$ $0.3$,$\sigma_{4}=0.4$,$\alpha=0.01$,$\theta=0.002$, for output feedback control. Using controller (23) and triggering condition (27),the states and states measurement errors of multi-agent systems are shown in Figs.3 and 4,respectively.

VI. CONCLUSION

This paper provides some solutions for the consensus problem of identical agents with an event-based control and communication. In order to reduce the control update times and the communication effort,distributed self-triggered cooperative control strategies based on state feedback and output feedback are proposed, respectively,and it is shown that consensus can be reached in both cases for all connected communication graphs. Future work includes extending the proposed approach to MASs with heterogeneous dynamics and agent with uncertainty.