Leader-following consensus of multi-agent systems via event-based impulsive control

Abstract

Leader-following consensus of multi-agent systems with general linear models is investigated via event-based impulsive control approach. Event-based impulsive controller and state-dependent triggering function are designed. Impulsive instants are determined by certain triggering events, that is, impulsive effect performs only at the instants when events occur, and so as controller update. Sufficient conditions on leader-following consensus are proposed by using stability theory of impulsive differential equations, matrix theory, and inequality technique. Zeno-behavior is also excluded for the concerned closed-loop system. Besides, a technique that used the state information of agents only at event instants is proposed to avoid continuous communication among agents. Finally, an example is presented to illustrate the effectiveness of the obtained theoretical results.

Introduction

Over the past decades, control problems for networked multi-agent systems have been extensively studied. Multi-agent systems can be potentially applied in such areas as ecosystems, Internet, social networks, and global economic markets. A particular focus is the consensus of multi-agent systems,^1,2 which requires all agents to achieve the desired common goal. Many researchers have contributed to this topic.^3–7

In recent years, impulsive control problems arise in practical applications, such as orbital transfer of satellites, ⁸ ecosystems management, ⁹ stabilization and synchronization of neural networks, ¹⁰ and so on. Moreover, impulsive control seems more useful than continuous control, since in some practical problems, such as the control or regulation of savings rates of a bank, the continuous control is impossible and only impulsive control can be used. Due to its variety of applications, interest in impulsive control has grown.^11–15 It should be pointed out that most of the existing studies focus on the case that impulsive instants are pre-determined and fixed or only the lower and upper bound of impulsive intervals are given. To ensure the performance of the considered systems, frequency of impulsive effect has to be designed fast enough by taking the worst scenario into consideration, while such high frequency of impulsive effect might not be necessary for achieving the control objective. This partially motivates our study.

On the contrary, event-based control, where event instants are determined by triggering condition and triggering function associated with systems’ states, was proposed for networked control systems due to its advantage in reducing the frequency of controller update and communication between various components/subsystems of control systems. In other words, this control mechanism can improve the usage of limited bandwidth resource. Event-based control for discrete-time multi-agent systems was investigated in Ding et al. ¹⁶ and Zhu et al. ¹⁷ Some results based on event-based control for continuous-time multi-agent systems were presented in Yi et al. ¹⁸ and Li et al. ¹⁹ Event-based consensus of multi-agent systems with linear dynamics was investigated in Zhu and Jiang, ²⁰ Cheng and Ugrinovskii, ²¹ Garcia et al., ²² Yin et al., ²³ and Borgers et al. ²⁴

In order to determine impulsive instants dynamically using state information, event-based impulsive control was studied by various researchers, where impulsive instants are determined by certain events. The issue of exponential stability via event-based impulsive control for continuous-time dynamical systems was investigated in Liu et al. ²⁵ The global uniform exponential stability criteria were derived and the Zeno-behavior was excluded. The exponential input-to-state stability under impulsive events for discrete-time impulsive dynamical networks was investigated by Liu et al. ²⁶ An event-based impulsive control scheme (ETI) for differential evolution (DE) was introduced by Du et al. ²⁷ Pinning consensus in networks with time-varying topology and event-based diffusions via impulsive control was studied by Han et al. ²⁸ Via event-based impulsive control approach, the exponential stabilization of continuous-time dynamic systems was investigated by Zhu et al., ²⁹ and the synchronization problem of memristive neural networks was also investigated by obtained results.

In this paper, consensus for leader-followers multi-agent systems is investigated. The novelties of this paper lie in the following. (1) Event-based impulsive control approach, which can reduce the cost of information exchange and controller update, is used to study leader-following consensus problem. The controller update instants and impulsive instants are determined by the appropriately designed triggering function and triggering condition. (2) Not only leader-following consensus for the considered systems can be achieved, but also the Zeno-behavior can be excluded under the designed event-based impulsive controller. (3) To avoid continuous communication among agents, a novel technique using the state information of agents only at event instants is provided. Besides, an example is presented to illustrate the effectiveness of the obtained theoretical results.

Preliminaries

Notations

For $y\in R^n$ and $\mathcal{A}\in R^{n\times n}$ , let $\parallel y\parallel$ be any vector norm. ${\lambda }_i(\mathcal{A})$ and $\mathrm{Re}({\lambda }_i(\mathcal{A}))$ represent an eigenvalue and the real part of an eigenvalue of matrix $\mathcal{A}$ , respectively. ${\lambda }_{\max }(\mathcal{A})$ and ${\lambda }_{\min }(\mathcal{A})$ are the maximum and minimum eigenvalue of symmetric real matrix $\mathcal{A}$ , respectively. $\parallel \mathcal{A}\parallel ={\sup }_{y\ne 0}||\mathcal{A}y||/||y||$ and $\mu (\mathcal{A})=\underset{l\to 0^{+}}{lim}(\parallel I+l\mathcal{A}\parallel -1)/l$ are the induced matrix norm and matrix measure, respectively. $I$ is the identity matrix with appropriate dimension. Especially, ${\mu }_2(\mathcal{A})=1/2{\lambda }_{\max }(\mathcal{A}+{\mathcal{A}}^T)$ , which is the induced matrix measure by Euclidean norm of $\mathcal{A}$ . $\overline{N}=\{0,1,2,\dots \}$ .

Graph theory

The communication topology among these agents can be represented by a digraph (directed graph) $G=(\mathcal{V},\mathcal{E},\overline{A})$ . $\mathcal{V}$ is called vertex set and $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ is an edge set. The edge $(i,j)$ in the edge set of a directed graph denotes that agent $j$ can obtain information from agent $i$ . The set of in-neighbors of vertex $i$ is denoted by ${\mathcal{N}}_i=\{j\in \mathcal{V}:(j,i)\in \mathcal{E}\}$ . $\overline{A}=(a_{\mathrm{ij}}{)}_{n\times n}$ is the adjacency matrix of followers. The factor $a_{\mathrm{ij}}$ of $\overline{A}$ denotes the weight of edge $(j,i)$ . $a_{\mathrm{ij}}>0$ if $(j,i)\in \mathcal{E}$ and $a_{\mathrm{ij}}=0$ , otherwise. We define the leader adjacency matrix $\overline{B}=\mathrm{diag}\{b_1,b_2,·,b_N\}$ associated with $G$ to depict whether the agents are connected to the leader called agent $0$ in graph $G$ . $b_i>0$ if agent $0$ is a in-neighbor of agent $i$ and $b_i=0$ , otherwise. Let $D=\mathrm{diag}\{d_1,d_2,\dots ,d_n\}$ be the degree matrix whose diagonal elements are defined by $d_i={\Sigma }_{j\in {\mathcal{N}}_i}{a}_{\mathrm{ij}}$ . The Laplacian is defined as $L=D-\overline{A}$ . All the row sums of $L$ are zero. Let $H=L+\overline{B}$ . According to Ren and Beard, ³⁰ all eigenvalues of $H$ are positive if and only if there is a spanning tree with the leader as the root in digraph $G$ .

Problem formulation

Consider the following linear leader-followers multi-agent systems

$$\{\begin{array}{c}\mathrm{Leader}:{\stackrel{·}{s}}_0(t)=A{s}_0(t)\hfill \\ \multicolumn{1}{c}{\mathrm{Followers}:{\stackrel{·}{s}}_i(t)=A{s}_i(t)+B{u}_i(t)}\end{array},\forall t\ge t_0$$

(1)

where $i=1,2,\dots ,N$ , $A\in R^{n\times n},B\in R^{n\times m}$ , $s_0\in R^n$ is the state of leader, and $s_i\in R^n$ and $u_i\in R^m$ are the state and the controller of agent $i$ , respectively. Since the object of this paper is to design appropriate controller $u_i(t)$ such that the consensus of multi-agent systems (1) can be achieved, we assume that $B\ne 0_{n\times m}$ .

Event-based impulsive controller for agent $i$ is designed as follows

$$\{\begin{array}{c}{u}_i(t)=-K{q}_i(t_k),\mathrm{for}t\in [t_k,t_{k+1}),k\in \overline{N}\hfill \\ \multicolumn{1}{c}{s_i(t)=s_i(t^{-})+C_k{q}_i(t^{-}),\mathrm{for}t=t_k,k\in \overline{N}}\end{array}$$

(2)

where $K\in R^{m\times n}$ and $C_k\in R^{n\times n}$ are control gain matrices to be determined later, and $q_i(t)=\sum _{j=1}^N{a}_{\mathrm{ij}}(s_i(t)-s_j(t))+b_i(s_i(t)-s_0(t))$ is the combined measurement of agent $i$ . $q_i(t_k^{-})$ denotes the left limit of $q_i(t)$ at $t=t_k$ .

Impulsive instants and the controller update instants $t_k$ for agents are defined iteratively by

$$t_{k+1}=inf\{t:t>t_k\mathrm{and}h(t)\ge 0\}$$

(3)

where

$$h(t)=\parallel e(t)\parallel -\eta \parallel q(t_k)\parallel$$

(4)

is said to be the triggering function, where $q(t)=(q_1^T(t),q_2^T(t),\dots ,q_N^T(t){)}^T$ , $e(t)=(e_1^T(t),e_2^T(t),\dots ,e_N^T(t){)}^T$ and $e_i(t)=q_i(t_k)-q_i(t)$ is the measurement error of agent $i$ .

Remark 1

Different with the research results,^13,14,25,29 the event-based impulsive control, where impulsive instants and controller update instants $t_k$ are determined by given state-dependent event condition, is applied to investigate leader-following consensus of multi-agent systems.

Definition 1

The multi-agent systems (1) is said to achieve leader-following consensus under some designed event-based impulsive controller $u_i(t)$ if

$$\underset{t\to \infty }{lim}\parallel s_i(t)-s_0(t)\parallel =0,\forall t\ge t_0$$

holds for any initial conditions.

Definition 2

The closed-loop system does not exhibit Zeno-behavior if $\underset{k\in \overline{N}}{inf}\{t_{k+1}-t_k\}>0$ .

Because $s_0(t)$ is continuous, the state equation of leader can be expressed as follows

$$\{\begin{array}{c}{\stackrel{·}{s}}_0(t)=A{s}_0(t),\mathrm{for}t\in [t_k,t_{k+1})\hfill \\ \multicolumn{1}{c}{s_0(t)=s_0(t^{-}),\mathrm{for}t=t_k,k\in \overline{N}}\end{array}$$

(5)

Let ${\delta }_i(t)=s_i(t)-s_0(t)$ . By equations (1), (2), and (5), we have

$$\{\begin{array}{c}{\stackrel{·}{\delta }}_i(t)=A{\delta }_i(t)-\mathrm{BK}{q}_i(t)-\mathrm{BK}{e}_i(t),\mathrm{for}t\in [t_k,t_{k+1}),k\in \overline{N}\hfill \\ \multicolumn{1}{c}{{\delta }_i(t)={\delta }_i(t^{-})+C_k{q}_i(t^{-}),\mathrm{for}t=t_k,k\in \overline{N}}\end{array}$$

(6)

Denote $\delta (t)=({\delta }_1^T(t),\cdots ,{\delta }_N^T(t){)}^T$ , then equation (6) can be rewritten in the following form

$$\{\begin{array}{c}\stackrel{·}{\delta }(t)=\tilde{A}\delta (t)-\tilde{B}e(t),\mathrm{for}t\in [t_k,t_{k+1}),k\in \overline{N}\hfill \\ \multicolumn{1}{c}{\delta (t)={\tilde{C}}_k\delta (t^{-}),\mathrm{for}t=t_k,k\in \overline{N}}\end{array}$$

(7)

where $\tilde{A}=(I_N\otimes A-H\otimes \mathrm{BK}),\tilde{B}=(I_N\otimes \mathrm{BK}),{\tilde{C}}_k=(I_N\otimes I_n+H\otimes C_k)$ .

Therefore, leader-following consensus of equation (1) is achieved if and only if $\delta (t)\to 0$ as $t\to \infty$ for any initial conditions.

Main results

First, we will proof that there is no Zeno-behavior in the closed-loop system. Then, we will give the consensus conditions.

Theorem 1

Consider the leader-followers multi-agent systems (1) with event-based impulsive controller (2), where impulsive instants sequence $\{t_k\}$ is determined by equation (3). Then, the closed-loop system does not exist Zeno-behavior for $\eta >0$ if there is a spanning tree with the leader as the root in digraph $G$ .

Proof

Computing the upper right-hand Dini derivative of $\parallel e(t)\parallel$ over interval $[t_k,t_{k+1})$ , we derive that

$$\begin{array}{c}\multicolumn{1}{c}{D^{+}\parallel e(t)\parallel \le \parallel \stackrel{·}{e}(t)\parallel =\parallel \stackrel{·}{q}(t)\parallel }\\ \le \parallel H\otimes I_n\parallel \parallel \tilde{A}\parallel \parallel \delta (t)\parallel \hfill \\ +\parallel H\otimes I_n\parallel \parallel \tilde{B}\parallel \parallel e(t)\parallel \hfill \end{array}$$

(8)

Since there is a spanning tree with the leader as the root in digraph $G$ , one has ${\lambda }_{\min }[(H^T\otimes I_n)(H\otimes I_n)]>0$ . Noticing that $q(t)=(H\otimes I_n)\delta (t)$ and $q(t)=q(t_k)-e(t)$ , one can easily obtain that $\parallel \delta (t)\parallel \le 1/\sqrt{{\lambda }_{\min }[(H^T\otimes I_n)(H\otimes I_n)]}(\parallel q(t_k)\parallel +\parallel e(t)\parallel )$ .

Then, invoking equation (8), one can derive that $D^{+}\parallel e(t)\parallel \le b_1\parallel q(t_k)\parallel +(b_1+b_2)\parallel e(t)\parallel$ , where $b_1=\parallel H\otimes I_n\parallel \parallel \tilde{A}\parallel /\sqrt{{\lambda }_{\min }[(H^T\otimes I_n)(H\otimes I_n)]}$ and $b_2=\parallel H\otimes I_n\parallel \parallel \tilde{B}\parallel$ .

It follows from $\parallel e(t_k)\parallel =0$ that

$$\parallel e(t)\parallel \le \frac{b_1\parallel q(t_k)\parallel }{b_1+b_2}[\exp ((b_1+b_2)(t-t_k))-1]$$

(9)

Next event will not be triggered until triggering function (4) crosses zero, that is, $\parallel e(t_{k+1})\parallel =\eta \parallel q(t_k)\parallel$ . Then, by equation (9), one can get that $\eta \parallel q(t_k)\parallel \le b_1\parallel q(t_k)\parallel /(b_1+b_2)[\exp ((b_1+b_2)(t_{k+1}-t_k))-1]$ . Without loss of generality, we assume that $\parallel q(t_k)\parallel \ne 0$ , otherwise, the consensus of the closed-loop system has been achieved. Thus, $t_{k+1}-t_k$ is lower bounded by a strictly positive value $\theta =\ln (1+\eta (b_1+b_2)/b_1)/(b_1+b_2)>0$ , which implies that the Zeno-behavior is excluded.

Lemma 1

Let $P\in R^{n\times n}$ be symmetric and positive definite. ¹¹ Then, $\parallel y{\parallel }_P=\sqrt{y^T\mathrm{Py}}$ is a norm of a vector $y\in R^n$ . The induced norm and measure of matrix $A\in R^{n\times n}$ are, respectively, given by $\parallel A{\parallel }_P=\parallel \mathcal{D}A{\mathcal{D}}^{-1}{\parallel }_2$ and ${\mu }_P(A)={\mu }_2(\mathcal{D}A{\mathcal{D}}^{-1})$ , where ${\mathcal{D}}^T\mathcal{D}=P$ .

Lemma 2

Let $w(t,t_0)$ be the state transition matrix of the linear system ¹¹

$$\{\begin{array}{c}\stackrel{·}{x}(t)=\mathcal{A}x(t)\hfill \\ \multicolumn{1}{c}{x(t_k)={\mathcal{C}}_{k}x(t_k^{-})}\end{array}$$

where $t\ne t_k$ and $k\in \overline{N}=\{0,1,2,\dots ,\}$ . Given a constant $\alpha \ge \parallel {\mathcal{C}}_k\parallel$ for all $k\in \overline{N}$ , we have the following results

1. If $0<\alpha <1$ and $\rho =\underset{k\in \overline{N}}{\sup }\{t_{k+1}-t_k\}<\infty$ , then

$$\Vert w(t,t_0)\Vert \le 1/\alpha \exp ((\mu (\mathcal{A})+\ln \alpha /\rho )(t-t_0)),\forall t\ge t_0$$

• 2. If $\alpha \ge 1$ and $\theta =\underset{k\in \overline{N}}{inf}\{t_{k+1}-t_k\}>0$ , then

$$\Vert w(t,t_0)\Vert \le \alpha \exp ((\mu (\mathcal{A})+\ln \alpha /\theta )(t-t_0)),\forall t\ge t_0$$

Lemma 3 (Gronwall’s Inequality)

Let a nonnegative continuous function $v(t)$ satisfies

$$v(t)\le c+\underset{t_0}{\overset{t}{\int }}w(s)v(s)\mathrm{ds},\forall t\ge t_0$$

where $c$ is a positive constant, $w(t)\ge 0$ . Then

$$v(t)\le c\exp (\underset{t_0}{\overset{t}{\int }}w(s)\mathrm{ds})$$

Theorem 2

Consider the leader-followers multi-agent systems (1) with event-based impulsive controller (2), where impulsive instants sequence $\{t_k\}$ is determined by equation (3). Assume that there is a spanning tree with the leader as the root in digraph $G$ . Then, there exist gain matrices $K$ and $C_k$ satisfying $\parallel {\tilde{C}}_k\parallel =\parallel I_N\otimes I_n+H\otimes C_k\parallel \le \alpha$ , $k\in \overline{N}$ such that leader-following consensus of the closed-loop control systems (1) with (2) can be achieved if one of the following conditions is satisfied for $\rho ={\sup }_{k\in \overline{N}}{t}_{k+1}-t_k<\infty$ , $\theta =\underset{k\in \overline{N}}{\inf }{t}_{k+1}-t_k$ and $0<\eta <1$ :

1. For $0<\alpha <1$

$$\mu (\tilde{A})+\frac{\ln \alpha }{\rho }+\frac{\eta \parallel H\parallel \parallel \tilde{B}\parallel }{\alpha (1-\eta )}<0$$

(10)

• 2. For $\alpha \ge 1$

$$\mu (\tilde{A})+\frac{\ln \alpha }{\theta }+\frac{\eta \alpha \parallel H\parallel \parallel \tilde{B}\parallel }{1-\eta }<0$$

(11)

Proof

By Theorem 1, we have $\theta =\underset{k\in \overline{N}}{inf}{t}_{k+1}-t_k>0$ . Next, it will be shown that leader-following consensus of system (1) can be achieved with impulsive controller (2).

By the definition of triggering function $(4)$ and impulsive instants $t_k$ , and $0<\eta <1$ , one can derive that

$$\parallel e(t)\parallel \le \frac{\eta }{1-\eta }\parallel q(t)\parallel \le \frac{\eta \parallel H\parallel }{1-\eta }\parallel \delta (t)\parallel$$

(12)

The solution of (7) can be represented as follows

$$\delta (t)=\mathrm{\Theta }(t,t_0)\delta (t_0)-\underset{t_0}{\overset{t}{\int }}\mathrm{\Theta }(t,s)\tilde{B}e(s)\mathrm{ds}$$

(13)

where $\mathrm{\Theta }(t,t_0)$ is the state transition matrix of the following linear impulsive system

$$\{\begin{array}{c}\stackrel{·}{\delta }(t)=\tilde{A}\delta (t),\mathrm{for}t\in [t_k,t_{k+1})\hfill \\ \multicolumn{1}{c}{\delta (t)={\tilde{C}}_k\delta (t^{-}),\mathrm{for}t=t_k,k\in \overline{N}}\end{array}$$

• Case I: For $0<\alpha <1$ .

By Lemma 2 (equations (12) and (13)), we have

$$\begin{array}{c}\parallel \delta (t)\parallel \le \parallel \mathrm{\Theta }(t,t_0)\parallel \parallel \delta (t_0)\parallel \hfill \\ +\underset{t_0}{\overset{t}{\int }}\parallel \mathrm{\Theta }(t,s)\parallel \parallel \tilde{B}\parallel \parallel e(s)\parallel \mathrm{ds}\hfill \\ \le \frac{1}{\alpha }\exp ((\mu (\tilde{A})+\frac{\ln \alpha }{\rho })(t-t_0))\parallel \delta (t_0)\parallel \hfill \\ +\frac{\eta \parallel \tilde{B}\parallel \parallel H\parallel }{\alpha (1-\eta )}\underset{t_0}{\overset{t}{\int }}\exp ((\mu (\tilde{A})+\frac{\ln \alpha }{\rho })(t-s))\parallel \delta (s)\parallel \mathrm{ds}\hfill \end{array}$$

(14)

Let $\mu =-(\mu (\tilde{A})+\ln \alpha /\rho )$ , $\tilde{a}=\eta \parallel H\parallel \parallel \tilde{B}\parallel /(\alpha (1-\eta ))$ . Then, equation (14) can be written as follows

$$\begin{array}{c}\multicolumn{1}{c}{\parallel \delta (t)\parallel \le \frac{1}{\alpha }\exp (-\mu (t-t_0))\parallel \delta (t_0)\parallel }\\ +\tilde{a}\underset{t_0}{\overset{t}{\int }}{e}^{-\mu (t-s)}\parallel \delta (s)\parallel \mathrm{ds}\hfill \end{array}$$

(15)

that is

$$\parallel \delta (t)\parallel e^{\mu (t-t_0)}\le \tilde{c}+\tilde{a}\underset{t_0}{\overset{t}{\int }}\exp (\mu (s-t_0))\parallel \delta (s)\parallel \mathrm{ds}$$

where $\tilde{c}=1/\alpha$ . Then, in terms of Lemma 3, one can easily get that

$$\parallel \delta (t)\parallel \le \tilde{c}\exp (-(\mu -\tilde{a})(t-t_0))$$

(16)

Thus, invoking equation (10), there is a positive constant $\lambda =-(\mu (\tilde{A})+\ln \alpha /\rho +\eta \parallel H\parallel \parallel \tilde{B}\parallel /(\alpha (1-\eta )))>0$ such that

$$\parallel \delta (t)\parallel \le \frac{1}{\alpha }\exp (-\lambda (t-t_0)),\forall t\ge t_0$$

(17)

• Case II: For $\alpha \ge 1$ .

By Lemma 1, Lemma 2 (equations (11), (13), (14)) and following a similar procedure as for case I, we can show that there exits a positive constant $\lambda =-(\mu (\tilde{A})+\ln \alpha /\theta +\eta \alpha \parallel H\parallel \parallel \tilde{B}\parallel /(1-\eta ))>0$ such that

$$\parallel \delta (t)\parallel \le \alpha \exp (-\lambda (t-t_0)),\forall t\ge t_0$$

(18)

Therefore, it follows from equations (17) and (18) that the consensus of the closed-loop system (1) with controller (2) can be achieved.

Remark 2

Theorem 2 shows that whether $\mu (\tilde{A})$ is positive or negative, there always exists an event-based impulsive controller such that the consensus of the closed-loop control system can be achieved. Equation (10) implies that if $\mu (\tilde{A})>0$ , one can design proper impulsive gain matrices $C_k$ , $k\in \overline{N}$ , and $0<\eta <1$ such that $\parallel {\tilde{C}}_k\parallel =\parallel I_N\otimes I_n+H\otimes C_k\parallel \le \alpha <1$ . Then, leader-following consensus can be achieved. Equation (11) implies that if there is a matrix $K$ such that $\mu (\tilde{A})<0$ , then matrices $C_k$ , $k\in \overline{N}$ can be designed such that $\parallel {\tilde{C}}_k\parallel =\parallel I_N\otimes I_n+H\otimes C_k\parallel \le \alpha$ ( $\alpha \ge 1$ ). Under this case, $\alpha$ is the upper bound of the impulsive disturbances.

As a matter of fact, if (A, B) is stabilizable and the communication topology of agents has a directed spanning tree with the leader as the root, then there is a matrix $K=\tilde{h}{B}^T\tilde{P}$ such that all the real parts of the eigenvalues of $\tilde{A}=(I_N\otimes A-H\otimes \mathrm{BK})$ are in the left half-open plane (refer to Zhu and Jiang ²⁰ ), where $\tilde{h}\mathrm{Re}{\lambda }_i(H)>1,\forall i$ and $\tilde{P}$ is the solution of algebraic Riccati equation (ARE): $A^T\tilde{P}+\tilde{P}A-\tilde{P}B{B}^T\tilde{P}+\tilde{Q}=0$ with any given positive definite matrix $\tilde{Q}={\tilde{Q}}^T$ . Thus, by Lyapunov stability theory, there is a symmetric and positive matrix $P$ such that ${\tilde{A}}^{T}P+P\tilde{A}=-Q$ for any given symmetric and positive matrix $Q$ . Therefore, by Lemma 1, we have ¹¹

$$\begin{array}{c}\multicolumn{1}{c}{{\mu }_P(\tilde{A})=\underset{l\to 0^{+}}{lim}\frac{\parallel I+l\tilde{A}{\parallel }_P-1}{l}}\\ =\underset{y\ne 0}{\sup }\frac{y^T({\tilde{A}}^{T}P+P\tilde{A})y}{2y^T\mathrm{Py}}=\underset{y\ne 0}{\sup }\frac{-y^T\mathrm{Qy}}{2y^T\mathrm{Py}}\hfill \\ \le \underset{y\ne 0}{\sup }\frac{-{\lambda }_{\min }(Q)y^{T}y}{2{\lambda }_{\min }(P)y^{T}y}=\frac{-{\lambda }_{\min }(Q)}{2{\lambda }_{\min }(P)}<0\hfill \end{array}$$

which implies that if $(A,B)$ is stabilizable and the communication topology of the system has a directed spanning tree with the leader as the root, then there exists a matrix $K=\tilde{h}{B}^T\tilde{P}$ such that ${\mu }_P(\tilde{A})<0$ . In other words, we can use ${\mu }_P(\tilde{A})$ to replace $\mu (\tilde{A})$ in equation (11).

Remark 3

It should be pointed out that in order to monitor triggering condition (3) to computing $\parallel e(t)\parallel$ , continuous communication among agents is still required. In the following, we will provide a technique to avoid this situation.

Theorem 3

Assume that the conditions in Theorem 2 are satisfied and impulsive instants $t_k$ are defined by the following equation

$$t_{k+1}=t_k+\frac{\ln (1+\frac{(b_1+b_2)\eta \parallel q(t_k)\parallel }{b_1\parallel H\parallel \parallel \delta (t_k)\parallel })}{b_1+b_2}$$

(19)

Then, leader-following consensus of the closed-loop control systems (1) with (2) can be achieved. Furthermore, there is no Zeno-behavior for the concerned closed-loop control system.

Proof

In fact, by the proof of Theorem 1, we have

$$\parallel e(t)\parallel \le \frac{b_1\parallel H\parallel \parallel \delta (t_k)\parallel }{b_1+b_2}[\exp ((b_1+b_2)(t-t_k))-1]$$

(20)

By equation (20) and the definition of $t_k$ using equation (19), we can get that

$$\begin{array}{c}\parallel e(t)\parallel \le \frac{b_1\parallel H\parallel \parallel \delta (t_k)\parallel }{b_1+b_2}\hfill \\ [\exp ((b_1+b_2)(t_{k+1}-t_k))-1]\le \eta \parallel q(t_k)\parallel \hfill \end{array}$$

Then, by Theorem 2, we can get that leader-following consensus of the closed-loop control systems (1) with (2) is achieved. And it is obvious that there is no Zeno-behavior in the closed-loop system. The proof is completed.

Simulation example

In this section, we will give one example to illustrate the feasibility and effectiveness of theoretical result obtained in previous section.

Consider leader-followers multi-agent systems (1) with five followers, and $s_i(t)=[s_i^1(t),s_i^2(t),s_i^3(t){]}^T\in R^3,i=0,1,2,3,4,5$ , $A\in R^{3\times 3},B\in R^{3\times 1}$ , $u_i(t)\in R^{1\times 1}$ . The connectivity weight are given as follows: $a_{15}=a_{21}=a_{23}=a_{31}=a_{34}=a_{42}=a_{45}=a_{52}=a_{54}=1,b_1=b_2=1$ and other weights are all equal to zero. Obviously, there is a directed spanning tree with the leader as the root.

Assume that

$$A=\left[\begin{array}{ccc}0.0200\hfill & 0.0044\hfill & 0\hfill \\ \multicolumn{1}{c}{0.0300}& 0.0200\hfill & 0.0200\hfill \\ \multicolumn{1}{c}{0}& 0.0020\hfill & 0.0200\hfill \end{array}\right],B=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{0}\end{array}\right]$$

It is clear that $(A,B)$ is unstabilizable. Assume $\rho =0.06$ and

$$K=\left[\begin{array}{ccc}-0.1310\hfill & 0.1068\hfill & 0.1000\hfill \end{array}\right]$$

In fact, $K$ can be chosen arbitrarily. By computing, we can get that $\mu (\tilde{A})=0.1983>0$ . Then, one can design

$$C_k=C=\left[\begin{array}{ccc}-0.1367\hfill & -0.0671\hfill & -0.0999\hfill \\ \multicolumn{1}{c}{-0.1121}& -0.2510\hfill & -0.1633\hfill \\ \multicolumn{1}{c}{-0.00761}& -0.0259\hfill & -0.1994\hfill \end{array}\right]$$

and $\eta =0.0022$ such that $\mu (\tilde{A})+\ln \alpha /\rho +(\eta \parallel H\parallel \parallel \tilde{B}\parallel )/(\alpha (1-\eta ))=-0.2742<0$ . Therefore, by Theorem 2, leader-following consensus can be achieved.

The simulation results with initial conditions $x_0(0)=[0.1,0.2,0.3{]}^T$ , $x_i(0)=0.1\times i\times [1,2,3{]}^T,i=1,\dots ,5$ are depicted in Figures 1–10. In fact, the initial conditions can be chosen arbitrarily.

Figure 1.

States of agents (leader and followers) without impulsive.

Figure 2.

Consensus error between leader and followers without impulsive.

Figure 3.

States of agents (leader and followers) with impulsive (Theorem 2).

Figure 4.

Consensus error between leader and followers with impulsive (Theorem 2).

Figure 5.

Controller update of followers with impulsive (Theorem 2).

Figure 6.

States of agents (leader and followers) with impulsive (Theorem 3).

Figure 7.

Consensus error between leader and followers with impulsive (Theorem 3).

Figure 8.

Controller update of followers with impulsive (Theorem 3).

Figure 9.

Event instants of followers (Theorem 2).

Figure 10.

Event instants of followers (Theorem 3).

Remark 4

In simulation results, dotted lines and solid lines represent the states of leader and that of followers, respectively, and different colors represent different components. Figures 1 and 2 are the states of all agents and the consensus error between leader and followers without impulsive, respectively. Figures 3–5 are simulation results with event-triggered impulsive controller (Theorem 2). While Figures 6–8 are simulation results of Theorem 3, in which impulsive instants are determined by equation (19). Besides, Figures 9 and 10 show that there is no Zeno-behavior in the closed-loop system whether impulsive instants are determined by equation (3) or by equation (19).

Remark 5

Figures 1 and 2 show that the leader-following consensus cannot be achieved without impulsive, while Figures 3–5 (Figures 6–8) show that impulsive effect can make leader-following consensus achieved. So, the impulsive control plays an important role for the consensus of system, where $(A,B)$ is unstabilizable. Comparing Figure 9 with Figure 10, it can be observed that Theorem 3 results in more events.

Conclusion

In this paper, leader-following consensus of multi-agent systems is studied via event-based impulsive control. Impulsive instants are determined by certain state-dependent triggering condition, so as the controller update instants. The design proposed in this paper is simpler and can reduce communication among agents and frequency of controller update. In order to avoid continuous communication among agents in systems, a novel manner, in which the state information of agents only at event instants is used, is proposed to determine the even instants. The leader-following consensus can be achieved under designed event-based impulsive controller. Moreover, the Zeno-behavior can also be excluded. In fact, impulsive instants for each agent in this paper are synchronous. How to deal the case that the consensus can be achieved in asynchronous impulsive instants without using global information, for example, $H$ or $L$ , is more important and will be discussed in future work.