Abstract A new transmission strategy based on distributed event-triggered mechanism and average dwell time approach is proposed for consensus of time-delay free linear multi-agent systems with communication constraints. A subset of agents are selected according to the condition of event-triggered mechanism to communicate with its adjacent agents through network channel at certain sampling instant, which reduces the occupation of network bandwidth. The possible combinations of agents’ communication are considered as modes of multi-agent systems. Therefore, we modelled multi-agent systems as a class of switched systems. The average dwell time method is applied to determine its dwell-time of each mode, which avoids the frequent mode-switching resulted from event-triggered mechanism. An integrated design scheme is presented to achieve the thresholds of event-triggered mechanism, the average dwell time and the consensus controller gain simultaneously by using Lyapunov–Krasovskii functional and linear matrix inequality (LMI) technique. Finally, a simulation example illustrates the effectiveness of theoretical results. 1. Introduction The investigation of cooperative control of multi-agent systems has both theoretical and practical significance and has achieved considerable progress in recent years due to its broad applications in formation control of unmanned air vehicles (see Beard et al., 2002), schedule of automated highway systems (see Bender 1991) and wireless sensor networks (see CORTES 2008) etc. As one of the basis of cooperative control in multi-agent systems, consensus control of distributed multi-agent systems has become an attractive topic in control realm (see Lin & Jia 2011; Chen et al., 2014). Generally speaking, the main purpose of the consensus problem is to design a valid control algorithm such that a group of agents converge to a consistent quantity of interest through information interaction, which is commonly completed through digital network channel at discrete-time instant. Accordingly, to name a few, consensus problem of multi-agent systems under intermittent communication was studied in Gao et al. (2009), Wang & Xie (2012) and Huang et al. (2014). In Gao et al. (2009), consensus control of second-order multi-agent systems with fixed topology and switching topology were studied respectively based on sampled-data control. The average consensus problem of second-order multi-agent sampled control systems with time-varying sampling intervals was investigated in Wang & Xie (2012), in which the continuous-time multi-agent system was transformed into linear discrete-time system by the iterative method, and the process of how to select the time-varying sampling intervals were researched. In Huang et al. (2014), the problem of leader-following consensus of multi-agent systems with second-order nonlinear dynamics was discussed, and the definitions of completely and partly intermittent communication were presented. In addition, the sufficient conditions for consensus tracking under a general fixed topology were obtained by designing distributed consensus policies based on the relative local intermittent information. However, time-triggered transmission strategy instead of event-triggered transmission strategy is used in above results, that means agents communicate with its neighbours regularly by sharing the wireless network regardless of the fluctuation of the sampled data. In practical applications, due to the limitation of bandwidth and power, agents are required to transmit information as less as possible. Apparently, the traditional time-triggered control results in the waste of bandwidth to certain extent. Recently, as an alternative of time-triggered control, event-triggered control is proposed with the idea of updating the controllers’ input only when measurements exceed certain thresholds. The time-dependent and state-dependent event-triggered mechanism were designed in Seyboth et al. (2013), Hu et al. (2015) and Lu et al. (2017), respectively, and only the necessary information was allowed to transmit through the network channel (see Niu & Ho 2014). In Seyboth et al. (2013), the consensus problem of single-integrator agents network and double-integrator agents networks were addressed whether there exists communication delay or not. Average consensus or convergence to a ball centred at the average consensus was guaranteed for each scenario under event-triggered condition with time-dependent and exponentially decreasing threshold. In Hu et al. (2015), the centralized and decentralized state-dependent event-triggered protocols were proposed for mean square consensus of first-order multi-agent systems with noises. Nevertheless, all of the agents are triggered simultaneously to transmit information to their neighbours and update their control actuation, which is a drawback of centralized event-triggered mechanism since it may increase the occupation of communication resource and computation burden. In Lu et al. (2017), an event-driven robust output feedback model predictive control was proposed for constrained linear systems subject to bounded disturbances, which can ensure state convergence of the closed-loop system. In Niu & Ho (2014), a new control strategy with on-line updating the quantizer’s parameter is proposed to ensure the controlled system to attain the satisfying dynamic performance and H-infinity disturbance attenuation level. Furthermore, the model of multi-agent systems is extended to more general high-order linear dynamics in Zhang et al. (2014), Garcia et al. (2014) and Guo et al. (2014), and in which each agent can determine the information to be transmitted or not only according to its own event-triggered condition. The asynchronous event-triggered algorithms were provided based on the triggering time sequences of all agents in Zhang et al. (2014). Additionally, the triggering conditions were presented by adopting variable substitution approach, which guaranteed the consensus of system with fixed and switched topology. In Garcia et al. (2014), consensus protocol was proposed by using decoupled dynamics of neighbour agents instead of zero-order holder (ZOH), and the design of event-trigger thresholds which only relies on local information was presented. A distributed event-triggered sampled-data transmission algorithm was proposed in Guo et al. (2014), which avoided the continuous measurement by computing the event-triggered condition at each sampling instant. However, the frequent mode-switching problem induced by event-triggered mechanism is not investigated in the results mentioned above. On the other hand, multi-agent systems under event-triggered mechanism can be modelled as switched systems due to a subset of agents are selected by the event-triggered mechanism to transmit their states at sampling instant. In addition, event-triggered mechanism might generate frequent mode-switching which may influence the dynamic performance of multi-agent systems. Motivated by this, the dwell time method is introduced in this article to reduce the unnecessary switching induced by the event-triggered condition. Relevant papers have discussed the switched system with dwell time approach. For example, the mean square stability of a sort of Markov jump linear systems was studied in Bolzern et al. (2010) based on dwell time approach. For a class of switched positive linear continuous and discrete systems with average dwell time switching, the average dwell time condition that ensures the system globally uniformly exponentially stable was provided by using multiple linear co-positive Lyapunov function approach in Zhao et al. (2012). To the best of the authors’ knowledge, the transmission strategy in multi-agent systems under event-triggered mechanism and dwell time method has not been investigated in the existing results. Although event-triggered transmission strategy reduces the amount of information being transmitted, it may result in high-frequent mode switching which has a harmful effect on system dynamic performance. Additionally, the dwell time approach can decrease the switching frequency to certain degree for switched system. Therefore, a hybrid transmission strategy is proposed for linear multi-agent systems with communication constraints. The main contributions of this article are as follows: (1) A new transmission strategy based on distributed event-triggered mechanism and average dwell time approach is proposed in this article. On one hand, each agent can determine its data to be transmitted or not according to the distributed event-triggered mechanism at certain sampling instant in order to reduce communication traffic and save bandwidth. On the other hand, average dwell time approach is used for reducing the frequency of mode-switching caused by event-triggered mechanism. (2) Integrated design approach for consensus control of multi-agent systems based on hybrid transmission strategy is presented. We convert the consensus problem of multi-agent systems into the stability problem of a class of switched systems by defining state synchronization error. Furthermore, the event-triggered mechanism, average dwell time and consensus control protocol are designed simultaneously by using Lyapunov–Krasovskii functional and LMI technique. Notation. Suppose undirected graph |$G = (v,\varepsilon, A)$| of order |$N$| denotes the communication topology of multi-agent systems with |$N$| agents. Where |$\nu = {\rm{\{ 1}},{\rm{2}}, \ldots \ldots, N{\rm{\} }}$| denotes the set of |$N$| nodes, |$\varepsilon \subseteq \nu \times \nu$| denotes the edge of graph, and |$A = [{a_{ij}}] \in {R^{N \times N}}$| is the weighted adjacent matrix. The element |${a_{ij}}$| in |$A$| is determined by the edge between |$i$| and |$j$|, that is |${a_{ij}} > 0$| if |$(i,j) \in v$| ; otherwise |${a_{ij}} = 0$|. In undirected graph |$G$|, |${a_{ij}} = {a_{ji}}$| holds. The set of the adjacent neighbours of node |$i$| is denoted by |${N_i} = \{ j:(i,j) \in \varepsilon \}$|. The Laplacian matrix of graph |$G$| is defined as |$L = [{l_{ij}}] \in {R^{N \times N}}$| with |${l_{ii}} = \sum\limits_{j = 1}^N {{a_{ij}}}$| and |${l_{ij}} = - {a_{ij}}$| if |$i \ne j$|. |${R^n}$| is the |$n$|-dimensional Euclidean space, and |${R^{m \times n}}$| denotes the set of all |$m \times n$| real matrices. |${I_n}$| is the |$n$|-dimensional identity matrix. For a symmetric matrix |$P$|, |$P > 0$| mains |$P$| is positive definite. |$\overline \lambda (A)$|(or |$\underline \lambda (A)$|) represents the largest (or smallest ) eigenvalue of matrix |$A$|. |$diag({a_1},{a_2}, \cdots, {a_n})$| stands for a diagonal matrix with diagonal elements |${a_1},{a_2}, \cdots, {a_n}$|. |${\rm{||}}{\rm{.||}}$| indicates the Euclidean norm. Notation |$\otimes$| denotes the Kronecker product. |${A^T}$| denotes the transpose of matrix |$A$|. The sign |$*$| represents the symmetric term in a symmetric matrix. |$C_m^n = {{m!} \over {n!(m - n)!}}$|, |$m! = m \times (m - 1) \times \cdots \times 1$|. 2. Problem formulation Consider a multi-agent system composed of |$N$| agents, the model of the |$i$| th agent is described by: x˙i(t)=Axi(t)+Bui(t), i=1,2,......,N, (2.1) where |${x_i}(t) = {({x_{i1}}(t),{x_{i2}}(t), \ldots \ldots, {x_{i{\rm{n}}}}(t))^T} \in {R^n}$|, |${u_i}(t) = {({u_{i1}}(t),{u_{i2}}(t), \ldots \ldots, {u_{i{\rm{m}}}}(t))^T} \in {R^{\rm{m}}}$| denote the state and control input of agent |$i$|, respectively. |$A,B$| are constant matrices of appropriate dimensions. The structure of agent is shown in Fig. 1. Fig. 1. View largeDownload slide The structure of agent |$i$|. Fig. 1. View largeDownload slide The structure of agent |$i$|. Assumption 2.1 The communication topology |$G$| of multi-agent system is undirected and connected graph. Assumption 2.2 The controllers and actuators are event-driven. The sensors are clock driven with sampling period |$h$|, |$h > 0$|. As shown in Fig. 1, a distributed event-triggered mechanism is designed for the multi-agent system due to the limitation of network bandwidth and agents’ power. Whether or not the sampled data being transmitted to its controller and adjacent neighbours through network channel is determined by the following distributed event-triggered conditions. At |$k$| th sampling instant, the input of controller |$i$| is denoted as |${\hat x_i}(kh)$|, which will be defined in (2.3). The set of the adjacent neighbours of node |$i$| is denoted as |${N_i} = \{ j:(i,j) \in \varepsilon \}$|. For |$\forall i \in \nu$|, the sampled data is transmitted when: ||ei(kh)||≥ωi||μi(kh)||, (2.2) where |${\omega _i} > 0$| is the threshold being designed in the following sections. |${e_i}(kh) = {x_i}(kh) - {\hat x_i}((k - 1)h)$| is the error between the current state and the input of controller|$i$| at the last sampling time instant. |${\mu _i}(kh) = \sum\limits_{j \in {N_i}} {{a_{ij}}({{\hat x}_i}((k - 1)h) - {{\hat x}_j}(k - 1)h)}$| depends on the controller input of agent|$i$| and its neighbour agents at the last sampling instant. For |$i$|th agent, the current state |${x_i}(kh)$| is transmitted when the error |${e_i}(kh)$| satisfies event-triggered condition (2.2). Otherwise the data is discarded, and the corresponding controller adopts ZOH to keep the last value. The information updating of agent |$i$| under event-triggered mechanism is: x^i(kh)={xi(kh)x^i((k−1)h) if (2.2) is satisfiedotherwise. (2.3) The transmission vector of multi-agent system at |$t = kh$| is denoted as |${S_m} = {[{T_{{\sigma _{_1}}}},{T_{{\sigma _{_2}}}}, \cdots, {T_{{\sigma _{_N}}}}]^T}$|. Where |$m = 1,2, \cdots, M$|, |$M = C_N^0 + C_N^1 + \cdots + C_N^N$|, and |${T_{{\sigma _i}}} = 1$|, |$i = 1,2, \ldots \ldots, N$| if the current state of agent |$i$| is transmitted, otherwise |${T_{{\sigma _i}}} = 0$|. |$\sigma (t):[0, + \infty ) \to S = {\rm{\{ }}{S_1},{S_2}, \cdots, {S_M}\}$| denotes the switching signal under the event-triggered mechanism and |$S$| is the system mode matrix. Let |$M_{{\sigma _i}}^k,$||$i = 1,2, \ldots \ldots, N$| denotes the state transmission matrix of agent |$i$| at |$t = kh$| and let |$M_\sigma ^k = diag\{ M_{{\sigma _i}}^k\}$|, |$i = 1,2, \ldots \ldots, N$|. Mσik={In0 if (2.2) is satisfiedotherwise. (2.4) Then the input of controller |$i$| at |$t = kh$| is: x^i(kh)=Mσikxi(kh)+(In−Mσik)x^i((k−1)h). (2.5) The consensus control protocol is designed as: ui(t)=−K∑j∈Niaij(x^i(kh)−x^j(kh)),t∈[kh,(k+1)h), (2.6) where |$K$| is the consensus controller gain matrix being determined later. Remark 2.1 Whether or not the current states of agents are transmitted is determined by the event-triggered mechanism at each sampling instant based on each agents’ current state and the last state being transmitted. The inter-event intervals of each agent are at least one sampling period. Additionally, multi-agent system under the distributed event-triggered mechanism selects different group of agents to transmit their current states, thus forms a switched system with |$M$| modes. Definition 2.1 (Hespanha & Morse 1999) For any switching signal |$\sigma$| and |$t > {t_0} \ge 0$|, let |${N_\sigma }({t_0},t)$| denote the switching number of |$\sigma$| over |$[{t_0},t)$|, if there exist |${N_0} \ge 0$| and |${\tau _a} > 0$|, such that Nσ(t,t0)≤N0+t−t0τa (2.7) holds, then |${\tau _a}$| is called the average dwell time. Definition 2.2 (Olfati-Saber et al. 2007) The consensus of multi-agent systems (2.1) is said to be achieved, if and only if |$\mathop {{\rm{lim}}}\limits_{t \to \infty } ||{x_i}(t) - {x_j}(t)|| = 0$|, |$\forall i,j = 1,2, \cdots, N$| is satisfied. Definition 2.3 (Liberzon 2012) The equilibrium |$x = 0$| of the switched system |$\dot x(t) = {A_{\sigma (t)}}x(t)$| is globally uniformly exponentially stable under a proper switching signal |$\sigma (t)$| if there exist constants |$\alpha > 0$|, |$\beta > 0$| such that the state response satisfies |$||x(t)|| \le \alpha {e^{ - \beta (t - {t_0})}}||x({t_0})||$|, |$\forall t \ge {t_0}$| with arbitrary initial conditions |$x({t_0})$|. In order to reduce transmission traffic and save the limited bandwidth and power, we design the distributed event-triggered mechanism in this article. However, the event-triggered mechanism may increase the frequency of mode-switching among system modes. For mitigating the limitation and decreasing the impact of mode-switching on dynamic performance, we integrate the event-triggered mechanism and the average dwell time approach to construct a new hybrid transmission strategy, which contains three steps: (i) Choosing system mode and updating state information based on the event-triggered mechanism. (ii) Determining the dwell-time of the mode in (i) by the condition of average dwell time. (iii) Repeating above steps when the dwell-time is over. The hybrid transmission strategy is denoted as: {⋯,(S(tl),τa(tl)),(S(tl+1),τa(tl+1)),⋯}, (2.8) where |${t_l}$| is the |$l$|th switching instant of multi-agent system, |$S({t_l})$| is the corresponding system mode, |$S({t_l}) \in S$|. |${\tau _a}({t_l})$| is the dwell-time of mode |$S({t_l})$|. The objective of this article is to design the distributed event-triggered mechanism (2.2), consensus control protocol (2.6), and the average dwell time|${\tau _a}$| in (2.8) such that multi-agent system (2.1) can asymptotically achieve consensus. 3. Consensus analysis and design of multi-agent systems 3.1. Model transformation The state synchronization error of multi-agent systems is defined as: δi(t)=xi(t)−1N∑j=1Nxj(t). (3.1) Let |$\delta (t) = {[\delta _{_1}^T(t),\delta _2^T(t), \cdots \delta _N^T(t)]^T}$|. According to (2.1), (2.6) and (3.1), we have δ˙i(t) =Aδi(t)−BK∑j∈Niaij(x^i(kh)−x^j(kh)) +1N∑i=1N∑j∈NiBKaij(x^i(kh)−x^j(kh)),t∈[kh,(k+1)h). (3.2) Due to the symmetry of undirected topology graph, namely |${a_{ij}} = {a_{ji}}$|, we obtain 1N∑i=1N∑j∈NiBKaij(x^i(kh)−x^j(kh))=0. (3.3) From (2.5) and |${e_i}(kh) = {x_i}(kh) - {\hat x_i}((k - 1)h)$|, we have ∑j∈Niaij(x^i(kh)−x^j(kh)) =∑j∈Niaij(xi(kh)−xj(kh))−∑j∈Niaij(ei(kh)−ej(kh)) +∑j∈Niaij(Mσikei(kh)−Mσikej(kh)). (3.4) Combining (3.3), (3.4) and (3.2), then δ˙i(t) =Aδi(t)−BK∑j∈Niaij(δi(kh)−δj(kh))+BK∑j∈Niaij(ei(kh)−ej(kh)) −BK∑j∈Niaij(Mσikei(kh)−Mσikej(kh)),t∈[kh,(k+1)h). (3.5) Let |$e(t) = {[e_{_1}^T(t),e_2^T(t), \cdots, e_N^T(t)]^T}$|, |$x(t) = {[x_{_1}^T(t),x_2^T(t), \cdots, x_N^T(t)]^T}$|, we obtain the following closed-loop system: δ˙(t) =(IN⊗A)δ(t)−(L⊗BK)δ(kh)+(L⊗BK)e(kh) −(L⊗BK)Mσke(kh),t∈[kh,(k+1)h). (3.6) For the convenience of stability analysis, let |$\tau (t) = t - kh$|, |$kh \le t \le (k + 1)h$|, |$k \in N$|. Obviously, |$\dot \tau (t) = 1$| when |$t \ne kh$|. |$\tau (t)$| is piece-wise linear and is discontinuous at |$t = kh$|. Clearly, |$0 \le \tau (t) < h$|. Consequently, system (3.6) can be rewritten as a switched system: δ˙(t) =(IN⊗A)δ(t)−(L⊗BK)δ(t−τ(t))+(L⊗BK)e(t−τ(t)) −(L⊗BK)Mσke(t−τ(t)),t∈[kh,(k+1)h). (3.7) According to the state synchronization error (3.1) and Definition 2.2, the consensus of multi-agent systems (2.1) is asymptotically achieved if and only if the closed-loop switched system (3.7) is asymptotically stable. 3.2. Design of hybrid transmission strategy and consensus control protocol Lemma 3.1 (Schur complement lemma) (Ji et al., 2004) For matrix $${S}=\left[\begin{array}{@{}cc@{}} S_{11} &S_{12}\\ S_{21} & S_{22} \end{array} \right]$$ , where |${S_{11}}$| is |$r \times r$| matrix, three inequalities as follows are equivalent: (1) S<0 (2) S11<0,S22−S12TS11−1S12<0 (3) S22<0,S11−S12S22−1S12T<0. Lemma 3.2 (Zhang & Han 2013) For any constant matrix |$R \in {R^{n \times n}}$|, |$R = {R^T} > 0$|, |$H \in {R^{n \times k}}$|, time-varying function |$\tau (t)$| satisfying |$0 < \tau (t) \le h$|, and vector function |$\dot \delta :[ - h,0] \to {R^n}$| such that the following integral is well defined, let |$\int_{t - \tau (t)}^t {\dot \delta (s)} = F\phi (t)$|, where |$F \in {R^{n \times k}}$| and |$\phi (t) \in {R^k}$|. Then the following inequality holds −∫t−τ(t)tδ˙T(t)Rδ˙(t)ds≤ϕT(t)(τ(t)HTR−1H−FTH−HTF)ϕ(t). Theorem 3.1 Given constant |$h > 0$|, |$\alpha > 0$|, and |$\forall \sigma ({t_l}) = p \in S$|, |$\forall \sigma ({t_{l{\rm{ - }}1}}) = q \in S$|, where |$\sigma$| is determined by event-triggered mechanism (2.2). Then the closed-loop switched system (3.7) is globally uniformly exponentially stable and has decay rate |$\rho$| if there exist a series of positive definite matrices |${P_p},{Q_p},{R_p},{X_p},{Y_p}$|, |${P_q},{Q_q},{R_q},{X_q},{Y_q}$|, |${H_d}(d = 1,2)$| with appropriate dimensions and a set of scalars |${\omega _i}(i = 1,2,...,N)$| such that following inequalities hold {Γp+Π3+h(J+Π1+Π4)<0Γp+Π3+hΠ2<0 (3.8) Pp≤βPq,Qp≤βQq,Rp≤βRq,Xp≤βXq,Yp≤βYq (3.9) and the average dwell time of|$p$| th sub-system satisfies: τa>τa∗=lnβα, (3.10) where |$\gamma = {e^{ - \alpha h}}$|, |$\beta \ge 1$|, |$E = ({I_N} \otimes A){\varepsilon _1} + (L \otimes BK){\varepsilon _{42}} - (L \otimes BK)M_\sigma ^k{\varepsilon _4}$|, |$J = \varepsilon _{12}^T{X_p}E + E{X_p}{\varepsilon _{12}}$|, |${{\it {\Gamma}} _p} = \varepsilon _1^T{P_p}E + {E^T}{P_p}{\varepsilon _1} + \varepsilon _1^T{Q_p}{\varepsilon _1} + \alpha \varepsilon _1^T{P_p}{\varepsilon _1} - \varepsilon _{12}^T{X_p}{\varepsilon _{12}} - \gamma \varepsilon _3^T{Q_p}{\varepsilon _3} - \varepsilon _{23}^T{H_1} - H_{_1}^T{\varepsilon _{23}} - \varepsilon _{12}^T{H_2} - H_2^T{\varepsilon _{12}} + \varepsilon _{24}^T({L^T}{\it {\Lambda}} L \otimes {I_n}){\varepsilon _{24}} - \varepsilon _4^T({I_N} \otimes {I_n}){\varepsilon _4}$|, |${{\it {\Pi}} _1} = H_1^TR_p^{ - 1}{H_1}$|, |${{\it {\Pi}} _2} = H_2^TR_p^{ - 1}{H_2}$|, |${{\it {\Pi}} _3} = h{E^T}({R_p} + {Y_p})E$|, |${{\it {\Pi}} _4} = \alpha \varepsilon _{12}^T{X_p}{\varepsilon _{12}}$|, |${\it {\Lambda}} = diag({\omega _1},{\omega _2}, \cdots, {\omega _N})$|. Where |${\varepsilon _i}(i = 1,2,3,4)$| is block identity matrix, and |${\varepsilon _{ij}} = {\varepsilon _i} - {\varepsilon _j}$|. Such as $${\varepsilon _1}= \left[\begin{array}{@{}cccc@{}} I_{nN} & 0 & 0 & 0\\ \end{array}\right]^T, \varepsilon _{14} = \left[ \begin{array}{@{}cccc@{}} I_{nN} & 0 & 0 & - I_{nN}\\ \end{array} \right]^T$$ . Proof. Set |$\phi (t) = {[{\delta ^T}(t),{\delta ^T}(t - \tau (t)),{\delta ^T}(t - h),{e^T}(t - \tau (t))]^T}$|. (1) Suppose multi-agent system is located at |$p$|th mode when |$t \in (kh,(k + 1)h)$|. Choose a Lyapunov–Krasovskii functional as: Vp(t)=Vp1(t)+Vp2(t), (3.11) where Vp1(t) =Vp11(t)+Vp12(t)+Vp13(t) =δT(t)Ppδ(t)+∫t−hteα(s−t)δT(s)Qpδ(s)ds+∫−h0∫t+θteα(s−t)δ˙T(s)Rpδ˙(s)dsdθVp2(t) =Vp21(t)+Vp22(t)=(h−τ(t))×{[δ(t)−δ(t−τ(t))]TXp[δ(t)−δ(t−τ(t))]} +∫t−τ(t)t(s−t+h)eα(s−t)δ˙T(s)Ypδ˙(s)ds. According to (3.7) and |$\dot \tau (t) = 1$| when |$t \ne kh$|, we have V˙p1(t) =2δT(t)Ppδ˙(t)+δT(t)Qpδ(t) −e−αhδT(t−h)Qpδ(t−h)+hδ˙T(t)Rpδ˙(t)−αVp12(t)−αVp13(t)+ηV˙p2(t) =−[δ(t)−δ(t−τ(t))]TXp[δ(t)−δ(t−τ(t))]+2(h−τ(t)) (3.12) {[δ(t)−δ(t−τ(t))]TXp[δ˙(t)−δ˙(t−τ(t))]}+hδ˙T(t)Ypδ˙(t)−αVp22(t), (3.13) where |$\eta = - \int_{t - h}^t {{e^{\alpha (s - t)}}{{\dot \delta }^T}(s){R_p}\dot \delta (s)ds}$|. From Lemma 3.2 and |$\int_{t - h}^{t - \tau (t)} {\dot \delta (s)ds} = {\varepsilon _{23}}\phi (t)$|, |$\int_{t - \tau (t)}^t {\dot \delta (s)ds} = {\varepsilon _{12}}\phi (t)$|, we obtain η =−∫t−hteα(s−t)δ˙T(s)Rpδ˙(s)ds ≤e−αh[−∫t−ht−τ(t)δ˙T(s)Rpδ˙(s)ds−∫t−τ(t)tδ˙T(s)Rpδ˙(s)ds] ≤e−αh[(h−τ(t))H1TRp−1H1−ε23TH1−H1Tε23+τ(t)H2TRp−1H2−ε12TH2−H2Tε12]. (3.14) Furthermore, from (3.12)–(3.14), we have V˙p(t)+αVp(t) ≤ϕT(t){2ε1TPpE+ε1TQpε1−γε3TQpε3+hETRpE+αε1TPpε1 +(h−τ(t))H1TRp−1H1−ε23TH1−H1Tε23+τ(t)H2TRp−1H2 −ε12TH2−H2Tε12−ε12TXpε12+2(h−τ(t))ε12TXpE+hETYpE +α(h−τ(t))ε12TXpε12−ε4T(IN⊗In)ε4}ϕ(t)+eT(kh)(IN⊗In)e(kh). (3.15) When the event-triggered condition is not satisfied, from (2.2), we have eT(kh)(IN⊗In)e(kh) ≤(x(kh)−e(kh))T(LT⊗In)(Λ⊗In)(L⊗In)(x(kh)−e(kh)) ≤(δ(kh)−e(kh))T(LTΛL⊗In)(δ(kh)−e(kh)) =ϕT(t)ε24T(LTΛL⊗In)ε24ϕ(t). (3.16) From (3.15) and (3.16), we have V˙p(t)+αVp(t)≤ϕT(t)Σϕ(t), (3.17) where |${\it {\Sigma}} = {{\it {\Gamma}} _p} + {{\it {\Pi}} _3} + (h - \tau (t))(J + {{\it {\Pi}} _1} + {{\it {\Pi}} _4}) + \tau (t){{\it {\Pi}} _2}$|. Notice that |${\it {\Sigma}}$| is a convex combination of |${{\it {\Pi}} _1}$| and |${{\it {\Pi}} _2}$| on |$\tau (t) \in [0,h]$|. Hence, if (3.8) holds, then |${\it {\Sigma}} < 0$|. That is |${\dot V_p}(t) \le - \alpha {V_p}(t)$|. By the integral of |${\dot V_p}(t) \le - \alpha {V_p}(t)$|, we obtain Vp(t)≤e−α(t−t0)Vp(t0). (3.18) Therefore, |${V_p}$| is exponentially decayed on the |$p$| th mode. (2) For arbitrarily switching instant |${t_l}$|, based on (3.9), we have Vσ(tl)(tl)≤βVσ(tl−)(tl−). (3.19) Thus, when |$t \in [{t_l},{t_{l + 1}})$|, from (3.18) and (3.19), we obtain Vσ(t)(t) ≤e−α(t−tl)Vσ(tl)(tl)≤e−α(t−tl)βVσ(tl−)(tl−) ≤e−α(t−tl)βe−α(tl−tl−1)Vσ(tl−1)(tl−1)≤e−α(t−tl)β2e−α(tl−tl−1)Vσ(tl−1−)(tl−1−) ≤⋯≤e−α(t−t0)βlVσ(t0)(t0). By Definition 2.1, |$l = {N_\sigma }(t,{t_0}) \le {N_0} + {{t - {t_0}} \over {{\tau _a}}}$|, we have Vσ(t)(t) ≤βN0+t−t0tae−α(t−t0)Vσ(t0)(t0)≤βN0e−α(t−t0)e(t−t0)lnβτaVσ(t0)(t0) =βN0e−(α−lnβτa)(t−t0)Vσ(t0)(t0). (3.20) According to (3.11), we have |${\kappa _1}{\rm{||}}\delta (t){\rm{|}}{{\rm{|}}^{\rm{2}}} \le {V_{\sigma (t)}}(t) \le {\beta ^{{N_0}}}{e^{ - (\alpha - {{\ln \beta } \over {{\tau _a}}})(t - {t_0})}}{\kappa _2}{\rm{||}}\delta ({t_0}){\rm{|}}{{\rm{|}}^{\rm{2}}}$|. Further,we can obtain |δ(t)||≤κ2κ1βN02e−12(α−lnβτa)(t−t0)||δ(t0)||=ϖ||δ(t0)||e−ρ(t−t0), (3.21) where |$\varpi = \sqrt {{{{\kappa _2}} \over {{\kappa _1}}}} {\beta ^{{{{N_0}} \over 2}}}$|, |$\rho = {1 \over 2}(\alpha - {{\ln \beta } \over {{\tau _a}}})$|, |${\kappa _1} = \mathop {\min }\limits_{p \in S} (\underline \lambda ({P_p}) + \underline \lambda ({X_p}))$|, |${\kappa _2} = \mathop {\max }\limits_{p \in S} (\overline \lambda ({P_p}) + \overline \lambda ({X_p}) + h\overline \lambda ({Q_p}) + h\overline \lambda ({Y_p}) + {{{h^2}} \over 2}\overline \lambda ({R_p}))$|. From inequality (3.10) and |$\alpha > 0$|, |$\beta \ge 1$|, we have |$\varpi > 0$| and |$\rho > 0$|. According to Definition 2.3, we can conclude that the closed-loop switched system is globally uniformly exponentially stable and has the decay rate |$\rho$|. The proof is completed. □ Corollary 3.1 Given constant |$\lambda > 0$|, |$\mu > 0$|, |$h > 0$|, |$\alpha > 0$|, and |$\forall \sigma ({t_l}) = p \in S$|,|$\forall \sigma ({t_{l{\rm{ - }}1}}) = q \in S$|, where |$\sigma$| is determined by event-triggered mechanism (2.2). Consensus of multi-agent systems (2.1) with control protocol (2.6) under distributed event-triggered mechanism (2.2) can be asymptotically achieved if there exist a set of positive definite matrices |${\hat P_p},{\hat Q_p},{\hat R_p},{\hat Y_p}$| and some matrices |${\hat H_d}(d = 1,2)$| with appropriate dimensions and |${\it {\Lambda}} = diag({\omega _1},{\omega _2}, \cdots, {\omega _N})$|, such that the inequality (3.9) and the following linear matrix inequalities hold [Γ^p+h(J^+Π^4)hE^ThH^1T∗−hΘ0∗∗−hR^p]<0 (3.22) [Γ^phE^ThH^2T∗−hΘ0∗∗−hR^p]<0 (3.23) and the average dwell time satisfies (3.10). Moreover, the consensus controller gain is |$K = \hat K\hat P_p^{ - 1}$|. where |$\gamma = {e^{ - \alpha h}}$|, |$\hat E = ({I_N} \otimes A{\hat P_p}){\varepsilon _1} + (L \otimes B\hat K){\varepsilon _{42}} - (L \otimes B\hat K)M_\sigma ^k{\varepsilon _4}$|, |${{\hat {\it {\Gamma}} }_p} = \varepsilon _1^T\hat E + {{\hat E}^T}{\varepsilon _1} + \varepsilon _1^T{{\hat Q}_p}{\varepsilon _1} + \alpha \varepsilon _1^T({I_N} \otimes {{\hat P}_p}){\varepsilon _1} - \mu \varepsilon _{12}^T({I_N} \otimes {{\hat P}_p}){\varepsilon _{12}} - \gamma \varepsilon _3^T{{\hat Q}_p}{\varepsilon _3} - \varepsilon _{23}^T{{\hat H}_1} - \hat H_{_1}^T{\varepsilon _{23}} - \varepsilon _{12}^T{{\hat H}_2} - \hat H_2^T{\varepsilon _{12}} + \varepsilon _{24}^T({L^T}{\it {\Lambda}} L \otimes {{\hat \Phi }_p}){\varepsilon _{24}} - \varepsilon _4^T({I_N} \otimes {{\hat \Phi }_p}){\varepsilon _4}$|, |$\hat J = \mu (\varepsilon _{12}^T\hat E + {\hat E^T}{\varepsilon _{12}})$|, |${\hat {\it {\Pi}} _4} = \alpha \mu \varepsilon _{12}^T({I_N} \otimes {\hat P_p}){\varepsilon _{12}}$|, |$\Theta = 2\lambda {I_N} \otimes {\hat P_p} - {\lambda ^2}({\hat R_p} + {\hat Y_p})$|. Proof. Based on Schur complement lemma, (3.8) can be transformed into [Γp+h(J+Π4)hET(Rp+Yp)hH1T∗−h(Rp+Yp)0∗∗−hRp]<0 (3.24) [ΓphET(Rp+Yp)hH2T∗−h(Rp+Yp)0∗∗−hRp]<0. (3.25) Let |${P_p} = {I_N} \otimes U$|, and by the following variable substitutions |${\hat P_p} = {U^{ - 1}}$|, |${\hat Q_p} = P_p^{ - 1}{Q_p}P_p^{ - 1}$|, |${\hat R_p} = P_p^{ - 1}{R_p}P_p^{ - 1}$|, |${X_p} = \mu {P_p}$|, |${\hat Y_p} = P_p^{ - 1}{Y_p}P_p^{ - 1}$|, |$V = diag{\rm{\{ }}P_p^{ - 1},P_p^{ - 1},P_p^{ - 1},P_p^{ - 1}{\rm{\} }}$|, |${\hat H_d} = P_p^{ - 1}{H_d}V$|, |$(d = 1,2)$|, |${\hat \Phi _p} = {U^{ - 1}}{U^{ - 1}}$|, |$\hat K = K{U^{ - 1}}$|. Pre- and post-multiplying both the sides of (3.24) and (3.25) by |$Z = diag{\rm{\{ }}V,{({R_p} + {Y_p})^{ - 1}},P_p^{ - 1}{\rm{\} }}$|, respectively, where |$- P{R^{ - 1}}P \le {\lambda ^2}R - 2\lambda P$| is applied to the term |$- {({R_p} + {Y_p})^{ - 1}} = - P_p^{ - 1}{({\hat R_p} + {\hat Y_p})^{ - 1}}P_p^{ - 1}$|, yields |$- {({R_p} + {Y_p})^{ - 1}} \le - \Theta$|. we obtain (3.22) and (3.23). The proof is completed. □ 4. Numerical example Consider the multi-agent system composed of four agents described by: x˙i(t)=[−414−2]xi(t)+[13−21]ui(t),i=1,2,3,4, (4.1) where the sampling period of multi-agent system is |$h = 0.1s$|. The communication topology is shown in Fig. 2, whose Laplacian matrix is Fig. 2. View largeDownload slide The communication topology of multi-agent systems. Fig. 2. View largeDownload slide The communication topology of multi-agent systems. L=[3−1−1−1−12−10−1−120−1001]. Taking the proposed hybrid transmission strategy into account, which is based on the distributed event-triggered mechanism and the average dwell time approach, there are |$M$| combinations of agents communication, where |$M = C_4^0 + C_4^1 + \cdots + C_4^4 = 16$|. The system mode matrix determined by the event-triggered mechanism is denoted as: S=(S1,S2,⋯,S16)=[0100011100011101001001001101101100010010101101110000100101101111]. Set |$\alpha = 0.2$|, |$\lambda = 1$|, |$\mu = 1$|. We obtain $$K = \left[ \begin{array}{cc} 0.2098 & - 0.0574\\ - 0.0917 & 0.0684\\ \end{array} \right]$$ , |${\it {\Lambda}} = diag(0.180, 0.090, 0.180, 0.240)$| by solving linear matrix inequalities (3.22) and (3.23). Moreover, when |$\beta = 1.12$|, inequalities in (3.9) are satisfied. From Corollary 3.1, the average dwell time is required to satisfy |${\tau _a} > \tau _a^ * = {{\ln \beta } \over \alpha } = 0.5666$|. The initial states of four agents are assumed as $${x_1}(0) = \left[\begin{array}{cc} {0.5} & {1.5}\\ \end{array}\right]^T$$ , $${x_2}(0) = \left[ \begin{array}{cc} { - 1} & 0\\ \end{array} \right]^T$$ , $${x_3}(0) = \left[ \begin{array}{cc} 0 & 1\\ \end{array}\right]^T$$ , $${x_4}(0) = \left[ \begin{array}{cc} 0.5 & - 0.5\\ \end{array}\right]^T$$ , respectively. Based on the hybrid transmission strategy and the consensus control protocol, the simulation results are shown in Figs 3–8. Fig. 3. View largeDownload slide State trajectories of four agents. Fig. 3. View largeDownload slide State trajectories of four agents. The state trajectories of four agents are shown in Figs 3 and 4, which implies the convergence of each state. The triggering time sequences of each agent under event-triggered mechanism without dwell time and under the proposed hybrid transmission strategy are illustrated in Figs 5 and 7, respectively. The information transmission intervals in both of them are enlarged obviously compared with the intervals under time-triggered mechanism (commonly one sampling period), which implies the occupation of network bandwidth is reduced under event-triggered mechanism. By comparing Fig. 7 with Fig. 5, the unnecessary updating and computation are avoided with the hybrid transmission strategy. Moreover, Figure 8 presents the mode switching of the multi-agent system under event-triggered mechanism without dwell time, which shows quite high mode-switching frequency. Furthermore, in comparison with the mode-switching frequency under the proposed hybrid transmission strategy which is shown in Fig. 6, the frequent mode-switching resulted from event-triggered mechanism is prohibited by introducing average dwell time method, which improves the dynamic performance of the multi-agent system. Fig. 4. View largeDownload slide State trajectories of four agents. Fig. 4. View largeDownload slide State trajectories of four agents. Fig. 5. View largeDownload slide The trigger time sequence under the event-triggered mechanism without dwell time. Fig. 5. View largeDownload slide The trigger time sequence under the event-triggered mechanism without dwell time. Fig. 6. View largeDownload slide The mode-switching of MAS under the event-triggered mechanism without dwell time. Fig. 6. View largeDownload slide The mode-switching of MAS under the event-triggered mechanism without dwell time. Fig. 7. View largeDownload slide The trigger time sequence under the proposed hybrid transmission strategy. Fig. 7. View largeDownload slide The trigger time sequence under the proposed hybrid transmission strategy. Fig. 8. View largeDownload slide The mode-switching of MAS under the proposed hybrid transmission strategy. Fig. 8. View largeDownload slide The mode-switching of MAS under the proposed hybrid transmission strategy. 5. Conclusions In this article, we propose a new integrated design of hybrid transmission strategy and consensus control protocol for networked multi-agent systems with communication resources constraints. The multi-agent system is modelled as a switched system based on the hybrid transmission strategy. Additionally, by adopting the Lyapunov–Krasovskii functional and LMI (linear matrix inequality) technique, the distributed event-triggered mechanism, average dwell time and state feedback controller are provided to guarantee the consensus of multi-agent systems. Numerical example is given to illustrate the effectiveness of proposed approach. Since our work only concerns the linear multi-agent systems with fixed topology, In future work, we will investigate the multi-agent systems with nonlinear dynamics and time-varying topology. Funding National Natural Science Foundation of China (61673219); Tianjin Major Projects of Science and Technology (15ZXZNGX00250); Jiangsu Six Talents Peaks Project of Province (XNYQC-CXTD-001). References Beard, R. W. , McLain, T. W. & Goodrich, M. A. ( 2003 ) Coordinated target assignment and intercept for unmanned air vehicles. IEEE Trans. Robot. Autom., 18, 911 – 922 . Google Scholar Crossref Search ADS Bender, J. G. ( 1991 ) An overview of systems studies of automated highway systems, IEEE Trans. Veh. Technol., 40, 82 – 99 . Google Scholar Crossref Search ADS Bolzern, P. , Colaneri, P. & Nicolao, G. D. ( 2010 ) Markov jump linear systems with switching transition rates: mean square stability with dwell-time, Automatica, 46, 1081 – 1088 . Google Scholar Crossref Search ADS Chen, K. , Wang, J. & Zhang, Y. ( 2014 ) Second-order consensus of nonlinear multi-agent systems with restricted switching topology and time delay, Nonlinear Dynam., 78, 881 – 887 . 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IMA Journal of Mathematical Control and Information – Oxford University Press
Published: Dec 18, 2018
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