计算机系统应用  2019, Vol. 28 Issue (3): 242-249 PDF

Distributed Event-Triggered H Filtering in Discrete Multi-Agent Systems
LI Jia-Hao, GAO Jin-Feng
Faculty of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, China
Foundation item: National Natural Science Foundation of China (61374083)
Abstract: This study investigates event-triggered H filtering problem for discrete-time multi-agent systems with switching topologies. A novel distributed event-triggered control scheme is constructed to determine whether each agent should transmit the current sampled data to the filter, thus effectively save network resources. Considering the existence of network-induced delay and modeling the switching of network topologies by a Markov process, the distributed filtering error system is modeled as a closed-loop system with multiple time-varying delays by using the event-triggered control scheme and the proposed H filter. By employing Lyapunov-Krasovskii functional and linear matrix inequality method with multi-interval upper and lower bounds information, some sufficient conditions and the design method of H filter parameters are obtained to guarantee the closed-loop system to achieve asymptotic stability with an H performance index. Finally, two numerical examples are given to illustrate the effectiveness of the proposed method.
Key words: event-triggered     multi-agent systems     switching topologies     filtering     linear matrix inequality

1 问题描述 1.1 预备知识

${R^n}$ 表示 $n$ 维的欧式空间, ${R^{n \times m}}$ 表示 $n \times m$ 阶实矩阵集合. ${I_N}$ 表示 $n$ 维单位矩阵, 0表示具有适当维数的零矩阵. 上标 $- 1$ ${\rm T}$ 分别表示矩阵的逆和转置. 符号 $\otimes$ $*$ 分别表示Kronecker积和对称矩阵的对称部分. ${\left( {A \otimes B} \right)^{\rm T}} =$ ${A^{\rm T}} \otimes {B^{\rm T}}$ , $\left( {A \otimes B} \right)\left( {C \otimes D} \right) = AC \otimes BD$ .

$G = \left( {V,E,A} \right)$ 表示一个 $n$ 阶有向加权图, 其中 $V =$ $\left\{ {{v_1},{v_2}, \cdots ,{v_n}} \right\}$ 表示图 $G$ 顶点集合, $E \subseteq V \times V$ 表示图 $G$ 的有向边集合. 一条从节点 ${v_i}$ 到节点 ${v_j}$ 的有向边表示为 $\left( {{v_i},{v_j}} \right)$ . 用 ${N_i} = \left\{ {{v_j} \in V\left| {\left( {{v_j},{v_i}} \right) \in E} \right.} \right\}$ 表示节点 ${v_i}$ 的邻居集. 邻接矩阵 $A = \left[ {{a_{ij}}} \right] \in {R^{N \times N}}$ , 其中矩阵元素 ${a_{ij}}$ 表示为节点 ${v_i}$ 与节点 ${v_j}$ 的连接权重. 如果 ${v_j} \in {N_i}$ , 则 ${a_{ij}} > 0$ , 否则 ${a_{ij}}{\rm{ = }}0$ . 定义度矩阵 $D = diag\left\{ {{d_1},{d_2}, \cdots ,{d_N}} \right\}$ , 其中对角元素 ${d_i} = \displaystyle\sum\nolimits_{j \in {N_i}} {{a_{ij}}}$ 表示节点 ${v_i}$ 的出度. 图 $G$ 的Laplacian矩阵定义为 $L = D - A = \left[ {{l_{ij}}} \right] \in {R^{N \times N}}$ . 其中, ${l_{ii}} = \displaystyle\sum\nolimits_{j \in {N_i}} {{a_{ij}}}$ , 且 ${l_{ij}} = - {a_{ij}},i \ne j$ .

1.2 系统描述

 $\left\{ {\begin{array}{*{20}{c}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! {{x_i}\left( {k + 1} \right) = A{x_i}\left( k \right) + B{w_i}\left( k \right)} \\ \!\!{{y_i}\left( k \right) = C\displaystyle\sum\nolimits_{j \in {N_i}} {{a_{ij}}\left( {{x_i}\left( k \right) - {x_j}\left( k \right)} \right) + D{w_i}\left( k \right)} } \\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{{z_i}\left( k \right) = E{x_i}\left( k \right)} \end{array} } \right.$ (1)

${G_{r\left( k \right)}} \in \Phi {\rm{ = }}\left\{ {{G_1},{G_2}, \cdots ,{G_q}} \right\}$ 来表示Markov切换拓扑, 对切换拓扑集 $\Phi$ 中的每一个拓扑 ${G_r}$ , 用 ${A_r}$ , ${D_r}$ 以及 ${L_r}$ 来分别定义相应的邻接矩阵, 度矩阵以及Laplace矩阵. $r\left( k \right)$ 在有限集合 $S = \left\{ {1,2, \cdots ,q} \right\}$ 中定义为Markov过程. 从模态 $r$ 到模态 $s$ 的转移率定义如下:

 $P\left\{ {r\left( {k + 1} \right) = s\left| {r\left( k \right){\rm{ = }}r} \right.} \right\} = {\pi _{rs}}$

 $\Pi {\rm{ = }}\left[ {\begin{array}{*{20}{c}} {{\pi _{11}}}& \cdots &{{\pi _{1q}}} \\ \vdots & \ddots & \vdots \\ {{\pi _{q1}}}& \cdots &{{\pi _{qq}}} \end{array}} \right]$

 $\left\{ {\begin{array}{*{20}{c}} \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! {{x_i}\left( {k + 1} \right) = A{x_i}\left( k \right) + B{w_i}\left( k \right)} \\ \!\!{{y_i}\left( k \right) = C\displaystyle\sum\nolimits_{j \in {N_i}} {a_{_{ij}}^{r(k)}\left( {{x_i}\left( k \right) - {x_j}\left( k \right)} \right) + D{w_i}\left( k \right)} } \\ \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\!\!\!\!{{z_i}\left( k \right) = E{x_i}\left( k \right)} \end{array} } \right.$ (2)

 $\left\{ {\begin{array}{*{20}{c}} \!\! {\mathop {{x_i}}\limits^ \wedge \left( {k + 1} \right) = {F_r}\mathop {{x_i}}\limits^ \wedge \left( k \right) + {G_r}\mathop {{y_i}}\limits^ \wedge \left( k \right)} \\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\mathop {{z_i}}\limits^ \wedge \left( k \right) = E\mathop {{x_i}}\limits^ \wedge \left( k \right)} \end{array} } \right.$ (3)

 $\chi _i^{\rm T}\left( {k_l^i + j} \right){\phi _r}{\chi _i}\left( {k_l^i + j} \right) \leqslant {\sigma _r}y_i^{\rm T}\left( {k_l^i + j} \right){\phi _r}{y_i}\left( {k_l^i + j} \right)$ (4)

 $\begin{gathered} k_{l + 1}^i = k_l^i + \mathop {\min }\limits_{{p_i}} \{ {p_i}|\chi _i^{\rm T}\left( {k_l^i + j} \right){\phi _r}{\chi _i}\left( {k_l^i + j} \right) \\ \geqslant {\sigma _r}y_i^T\left( {k_l^i + j} \right){\phi _r}{y_i}\left( {k_l^i + j} \right)\} \end{gathered}$ (5)

 ${\hat y_i}\left( k \right) = {y_i}\left( {k_l^i} \right)$ (6)

 $\left\{ {\begin{array}{*{20}{c}} \!\!\! {\mathop {{x_i}}\limits^ \wedge \left( {k + 1} \right) = {F_r}\mathop {{x_i}}\limits^ \wedge \left( k \right) + {G_r}{y_i}\left( {k_l^i} \right)} \\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! {\mathop {{z_i}}\limits^ \wedge \left( k \right) = E\mathop {{x_i}}\limits^ \wedge \left( k \right)} \end{array} } \right.$ (7)

 $e_i^{\rm T}\left( k \right){\phi _r}{e_i}\left( k \right) \leqslant {\sigma _r}y_i^{\rm T}\left( {k - \tau \left( k \right)} \right){\phi _r}{y_i}\left( {k - \tau \left( k \right)} \right)$ (8)

 $\left\{ {\begin{array}{*{20}{c}} \!\!\! {\mathop {{x_i}}\limits^ \wedge \left( {k + 1} \right) = {F_r}\mathop {{x_i}}\limits^ \wedge \left( k \right) + {G_r}{y_i}\left( {k - \tau \left( k \right)} \right) - {G_r}{e_i}\left( k \right)} \\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\mathop {{z_i}}\limits^ \wedge \left( k \right) = E\mathop {{x_i}}\limits^ \wedge \left( k \right)} \end{array} } \right.$ (9)

 $\xi _i^{\rm T}\left( k \right) = \left[ {x_i^{\rm T}\left( k \right),{{\hat x}_i}^{\rm T}\left( k \right)} \right]$
 ${\bar z_i}\left( k \right) = {z_i}\left( k \right) - {\hat z_i}\left( k \right)$
 ${\hat w_i}\left( k \right) = {\left[ {w_i^{\rm T}\left( k \right),w_i^{\rm T}\left( {k - \tau \left( k \right)} \right)} \right]^{\rm T}}$

 $\left\{ {\begin{array}{*{20}{c}} \! \! \! {{\xi _i}\left( {k + 1} \right) = \overline A {\xi _i}\left( k \right) + \overline B {H_2}\displaystyle\sum\nolimits_{j \in {N_i}} {a_{ij}^r\left[ {{\xi _i}\left( {k - \tau \left( k \right)} \right)} \right.} } \\ \! \! \! \! \! \!\! \! \! \! \! \!\! \! \! \! \! \!\! \! \! \! \! \! \! \! \! \! \! \!\! \! \! \! \! \!\! \! \! \! \! \!\! \! \! \! \! \!\! \! \! \! \! \!\! \! \! \! {\left. { - {\xi _j}\left( {k - \tau \left( k \right)} \right)} \right] + \overline C {e_i}\left( k \right) + \overline D \mathop {{w_i}}\limits^ \wedge \left( k \right)} \\ \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!\! \! \! \! \! \!\! \! \! \! \! \!\! \! \! \! \! \!\! \! \! \! \! \! \! \! \! \! \! \!\! \! \! \! \! \!\! \! \! \! \! \!\! \! \! \! \! \!\! \! \! \! \! \!\! \! \! \! \! \!\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! {{{\overline z }_i}\left( k \right) = E{H_1}{\xi _i}\left( k \right)} \end{array} } \right.$ (10)

$\overline A = \left[ {\begin{array}{*{20}{c}} A&0 \\ 0&{{F_r}} \end{array}} \right]$ , $\overline B = \left[ {\begin{array}{*{20}{c}} 0 \\ {{G_r}C} \end{array}} \right]$ , $\overline C = \left[ {\begin{array}{*{20}{c}} 0 \\ { - {G_r}} \end{array}} \right]$ , $\overline D = \left[ {\begin{array}{*{20}{c}} B&0 \\ 0&{{G_r}D} \end{array}} \right], \quad$ ${H_1} = \left[ {\begin{array}{*{20}{c}} I&{ - I} \end{array}} \right], \;\;\;\;\;\; \;\;\;$ ${H_2} = \left[ {\begin{array}{*{20}{c}} I&0 \end{array}} \right]$

 $\xi \left( k \right) = {\left[ {{\xi _1}^{\rm T}\left( k \right),\xi _2^{\rm T}\left( k \right), \cdots ,\xi _N^{\rm T}\left( k \right)} \right]^{\rm T}}$
 $e\left( k \right) = {\left[ {{e_1}^{\rm T}\left( k \right),e_2^{\rm T}\left( k \right), \cdots ,e_N^{\rm T}\left( k \right)} \right]^{\rm T}}$
 $\hat w\left( k \right) = {\left[ {\hat w_1^{\rm T}\left( k \right),\hat w_2^{\rm T}\left( k \right), \cdots ,\hat w_N^{\rm T}\left( k \right)} \right]^{\rm T}}$
 $\overline z \left( k \right) = {\left[ {\overline z _1^{\rm T}\left( k \right),\overline z _2^{\rm T}\left( k \right), \cdots ,\overline z _N^{\rm T}\left( k \right)} \right]^{\rm T}}$

 $\left\{ {\begin{array}{*{20}{c}} \!\! \!\!{\xi \left( {k + 1} \right) = \left( {{I_N} \otimes \overline A } \right)\xi \left( k \right) + \left( {{L_r} \otimes \overline B {H_2}} \right)\xi \left( {k - \tau \left( k \right)} \right)} \\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{ + \left( {{I_N} \otimes \overline C } \right)e\left( k \right) + \left( {{I_N} \otimes \overline D } \right)\hat w\left( k \right)} \\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\overline z \left( k \right) = \left( {{I_N} \otimes E{H_1}} \right)\xi \left( k \right)} \end{array} } \right.$ (11)
2 主要结果

 $\left[ {\begin{array}{*{20}{c}} {{\Xi _{11}}}&{\hat H_2^{\rm T}R}&0&0&0&{{\Xi _{16}}}&{{\Xi _{17}}}\\ * &{{\Xi _{22}}}&R&0&{{\Xi _{25}}}&{{\Xi _{26}}}&{{\Xi _{27}}}\\ * & * &{ - Q - R}&0&0&0&0\\ * & * & * &{ - {{\hat \phi }_r}}&0&{{\Xi _{46}}}&{{\Xi _{47}}}\\ * & * & * & * &{{\Xi _{55}}}&{{\Xi _{56}}}&{{\Xi _{57}}}\\ * & * & * & * & * &{ - {P_r}}&0\\ * & * & * & * & * & * &{ - R} \end{array}} \right] < 0$ (12)
 $\sum\limits_{s = 1}^q {{\pi _{rs}}{P_s} \leqslant {P_r}}$ (13)

 $\begin{array}{c} {\Xi _{11}}{\rm{ = }}{\left( {{I_N} \otimes E{H_1}} \right)^{\rm T}}\left( {{I_N} \otimes E{H_1}} \right) - {P_r} + \hat H_2^{\rm T}Q{{\hat H}_2} - \hat H_2^{\rm T}R{{\hat H}_2} \end{array}$
 ${\Xi _{22}}{\rm{ = }}{\sigma _r}{\left( {{L_r} \otimes C} \right)^{\rm T}}{\hat \phi _r}\left( {{L_r} \otimes C} \right) - 2R$
 ${\Xi _{25}}{\rm{ = }}{\sigma _r}{\left( {{L_r} \otimes C} \right)^{\rm T}}{\hat \phi _r}\left( {{I_N} \otimes D{H_3}} \right)$
 $\begin{gathered} {\Xi _{55}}{\rm{ = }}{\sigma _r}{\left( {{I_N} \otimes D{H_3}} \right)^{\rm T}{\rm T}}{\hat \phi _r}\left( {{I_N} \otimes D{H_3}} \right) - {\gamma ^2}\left( {{I_N} \otimes H_2^{\rm T}{H_2}} \right) \end{gathered}$
 ${\Xi _{16}}{\rm{ = }}{\left( {{I_N} \otimes \overline A } \right)^{\rm T}}{P_r}, {\Xi _{26}}{\rm{ = }}{\left( {{L_r} \otimes \overline B } \right)^{\rm T}}{P_r}$
 ${\Xi _{46}}{\rm{ = }}{\left( {{I_N} \otimes \overline C } \right)^{\rm T}}{P_r}, {\Xi _{56}}{\rm{ = }}{\left( {{I_N} \otimes \overline D } \right)^{\rm T}}{P_r}$
 ${\Xi _{17}}{\rm{ = }}{\tau _M}{\left( {{I_N} \otimes \left( {\bar A - \tilde I} \right)} \right)^{\rm T}}\hat H_2^{\rm T}R$
 ${\Xi _{27}}{\rm{ = }}{\tau _M}{\left( {{L_r} \otimes \bar B} \right)^{\rm T}}\hat H_2^{\rm T}R,{\Xi _{47}}{\rm{ = }}{\tau _M}{\left( {{I_N} \otimes \bar C} \right)^{\rm T}}\hat H_2^{\rm T}R$
 ${\Xi _{57}}{\rm{ = }}{\tau _M}{\left( {{I_N} \otimes \bar D} \right)^{\rm T}}\hat H_2^{\rm T}R,{\hat \phi _r} = {I_N} \otimes {\phi _r},{H_3} = \left[ {\begin{array}{*{20}{c}} 0&I \end{array}} \right]$

 $V\left( k \right) = {V_1}\left( k \right) + {V_2}\left( k \right) + {V_3}\left( k \right)$

 ${V_1}\left( k \right) = {\xi ^{\rm T}}\left( k \right){P_r}\xi \left( k \right)$
 ${V_2}\left( k \right) = \displaystyle\sum\nolimits_{s = k - {\tau _M}}^{k - 1} {{\xi ^{\rm T}}\left( s \right)\hat H_2^{\rm T}Q{{\hat H}_2}\xi \left( s \right)}$
 ${V_3}\left( k \right) = {\tau _M}\displaystyle\sum\nolimits_{s = - {\tau _M} + 1}^0 {\displaystyle\sum\nolimits_{l = k + s - 1}^{k - 1} {{\delta ^{\rm {\rm T}}}\left( l \right)\hat H_2^{\rm T}R{{\hat H}_2}\delta \left( l \right)} }$
 $\delta \left( l \right) = \xi \left( {k + 1} \right) - \xi \left( k \right), {\hat H_2} = {I_N} \otimes {H_2}$

 $\begin{gathered} {\rm E}\left\{ {\Delta {V_1}\left( k \right)} \right\} = {\rm E}\left\{ {{V_1}\left( {k + 1} \right) - {V_1}\left( k \right)} \right\} \\ {\rm{ = }}{\rm E}\left\{ {{\xi ^{\rm T}}\left( {k + 1} \right)} \right.\sum\nolimits_{j = 1}^N {{\pi _{rs}}{P_j}\xi \left( {k + 1} \right)} \left. { - {\xi ^{\rm T}}\left( k \right){P_r}\xi \left( k \right)} \right\} \end{gathered}$
 $\begin{array}{l} {\rm{E}}\left\{ {\Delta {V_2}\left( k \right)} \right\} = {\rm{E}}\left\{ {{V_2}\left( {k + 1} \right) - {V_2}\left( k \right)} \right\}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{ = E}}\{ {\xi ^{\rm T}}\left( k \right)\hat H_2^{\rm T}Q{{\hat H}_2}\xi \left( k \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - {\xi ^{\rm T}}\left( {k - {\tau _M}} \right)\hat H_2^{\rm T}Q{{\hat H}_2}\xi \left( {k - {\tau _M}} \right)\} \end{array}$
 $\begin{array}{l} {\rm{E}}\left\{ {\Delta {V_3}\left( k \right)} \right\}{\rm{ = E}}\left\{ {{V_3}\left( {k + 1} \right) - {V_3}\left( k \right)} \right\}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{ = E}}\{ \tau _M^2{\delta ^{\rm T}}\left( k \right)\hat H_2^{\rm T}R{{\hat H}_2}\delta \left( k \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - {\tau _M}\displaystyle\sum\nolimits_{l = k - {\tau _M}}^{k - 1} {{\delta ^{\rm T}}\left( l \right)\hat H_2^{\rm T}R{{\hat H}_2}\delta \left( l \right)} \} \end{array}$

 $\begin{gathered} {\tau _M}\displaystyle\sum\nolimits_{l = k - {\tau _M}}^{k - 1} {{\delta ^{\rm T}}\left( l \right)\hat H_2^{\rm T}R{{\hat H}_2}\delta \left( l \right)} \\ {\rm{ = }}{\tau _M}\displaystyle\sum\nolimits_{l = k - {\tau _M}}^{k - 1 - \tau \left( k \right)} {{\delta ^{\rm T}}\left( l \right)\hat H_2^{\rm T}R{{\hat H}_2}\delta \left( l \right)} \\ \;\;\;{\rm{ + }}{\tau _M}\displaystyle\sum\nolimits_{l = k - \tau \left( k \right)}^{k - 1} {{\delta ^{\rm T}}\left( l \right)\hat H_2^{\rm T}R{{\hat H}_2}\delta \left( l \right)} \\ \end{gathered}$

 $\begin{gathered} {\tau _M}\displaystyle\sum\nolimits_{l = k - {\tau _M}}^{k - 1 - \tau \left( k \right)} {{\delta ^{\rm T}}\left( l \right)\hat H_2^{\rm T}R{{\hat H}_2}\delta \left( l \right)} \leqslant \\ - {\left[ {\displaystyle\sum\nolimits_{l = k - {\tau _M}}^{k - 1 - \tau \left( k \right)} {\delta \left( l \right)} } \right]^{\rm T}}\hat H_2^{\rm T}R{\hat H_2}\left[ {\displaystyle\sum\nolimits_{l = k - {\tau _M}}^{k - 1 - \tau \left( k \right)} {\delta \left( l \right)} } \right] \\ \end{gathered}$
 $\begin{gathered} {\tau _M}\displaystyle\sum\nolimits_{l = k - \tau \left( k \right)}^{k - 1} {{\delta ^{\rm T}}\left( l \right)\hat H_2^{\rm T}R{{\hat H}_2}\delta \left( l \right)} \leqslant \\ - {\left[ {\displaystyle\sum\nolimits_{l = k - \tau \left( k \right)}^{k - 1} {\delta \left( l \right)} } \right]^{\rm T}}\hat H_2^{\rm T}R{\hat H_2}\left[ {\displaystyle\sum\nolimits_{l = k - \tau \left( k \right)}^{k - 1} {\delta \left( l \right)} } \right] \\ \end{gathered}$

 $\begin{gathered} {\rm E}\left\{ {\Delta V\left( k \right)} \right\} \leqslant {\rm E}\left\{ {{\xi ^{\rm T}}\left( {k + 1} \right){P_r}\xi \left( {k + 1} \right) - } \right.{\xi ^{\rm T}}\left( k \right){P_r}\xi \left( k \right) \\ \quad \quad \quad \quad \quad \quad \quad + {\xi ^{\rm T}}\left( k \right)\hat H_2^{\rm T}Q{\hat H_2}\xi \left( k \right) + \tau _M^2{\delta ^{\rm T}}\left( k \right)\hat H_2^{\rm T}R{\hat H_2}\delta \left( k \right) \\ \quad \quad - {\xi ^{\rm T}}\left( {k - {\tau _M}} \right)\hat H_2^{\rm T}Q{\hat H_2}\xi \left( {k - {\tau _M}} \right) \\ \quad \quad - {\tau _M}\displaystyle\sum\nolimits_{l = k - {\tau _M}}^{k - 1} {{\delta ^{\rm T}}\left( l \right)\hat H_2^{\rm T}R{{\hat H}_2}\delta \left( l \right)} \\ \quad \quad+ {\sigma _r}{y^{\rm T}}\left( {k - \tau \left( k \right)} \right){\hat \phi _r}y\left( {k - \tau \left( k \right)} \right) \\ - {e^{\rm T}}\left( k \right){\hat \phi _r}e\left( k \right) + {\overline z ^{\rm T}}\left( k \right)\overline z \left( k \right) \\ \quad \quad - {\gamma ^2}{\hat w^{\rm T}}\left( k \right)\hat w\left( k \right)\left. {} \right\}{\rm{ = }}{\eta ^{\rm T}}\left( k \right)\Omega \eta \left( k \right) \\ \end{gathered}$

 $\begin{gathered} {\eta ^{\rm T}}\left( k \right) = \left[ {\begin{array}{*{20}{c}} {{\xi ^{\rm T}}\left( k \right)}&{{\xi ^{\rm T}}\left( {k - \tau \left( k \right)} \right)\hat H_2^{\rm T}}&{{\xi ^{\rm T}}\left( {k - {\tau _M}} \right)\hat H_2^{\rm T}} \end{array}} \right. \\ \left. {\begin{array}{*{20}{c}} {{e^{\rm T}}\left( k \right)}&{\mathop {{w^{\rm T}}}\limits^ \wedge \left( k \right)} \end{array}} \right] \\ \end{gathered}$
 $\Omega {\rm{ = }}\Upsilon + {\Gamma ^{\rm T}}{P_r}\Gamma + \tau _M^2\Gamma _1^{\rm T}R{\Gamma _1}$
 $\Gamma {\rm{ = }}\left[ {\begin{array}{*{20}{c}} {{I_N} \otimes \overline A }&{L \otimes \overline B }&0&{{I_N} \otimes \overline C }&{{I_N} \otimes \overline D } \end{array}} \right]$
 ${\Gamma _1} = \left[ {\begin{array}{*{20}{c}} {{I_N} \otimes \left( {\overline A - I} \right)}&{L \otimes \overline B }&0&{{I_N} \otimes \overline C } \end{array}} \right.\left. {\;\;{I_N} \otimes \overline D } \right]$
 $\Upsilon {\rm{ = }}\left[ {\begin{array}{*{20}{c}} {{\Xi _{11}}}&{\hat H_2^{\rm T}R}&0&0&0 \\ * &{{\Xi _{22}}}&R&0&{{\Xi _{25}}} \\ * & * &{ - Q - R}&0&0 \\ * & * & * &{ - {{\hat \phi }_r}}&0 \\ * & * & * & * &{{\Xi _{55}}} \end{array}} \right]$

 $\left[ {\begin{array}{*{20}{c}} {{{\hat \Xi }_{11}}}&{\hat H_2^{\rm T}R}&0&0&0&{{{\hat \Xi }_{16}}}&{{\Xi _{17}}} \\ * &{{\Xi _{22}}}&R&0&{{\Xi _{25}}}&{{{\hat \Xi }_{26}}}&{{\Xi _{27}}} \\ * & * &{ - Q - R}&0&0&0&0 \\ * & * & * &{ - {{\hat \phi }_r}}&0&{{{\hat \Xi }_{46}}}&{{\Xi _{47}}} \\ * & * & * & * &{{\Xi _{55}}}&{{{\hat \Xi }_{56}}}&{{\Xi _{57}}} \\ * & * & * & * & * &{{{\hat \Xi }_{66}}}&0 \\ * & * & * & * & * & * &{ - R} \end{array}} \right] < 0$ (14)
 $\displaystyle\sum\limits_{s = 1}^q {{\pi _{rs}}({U_{1s}} - {W_s}) \leqslant {U_{1r}} - {W_r}}$ (15)

 ${\hat \Xi _{11}}{\rm{ = }}{I_N} \otimes \left[ {\begin{array}{*{20}{c}} {{E^{\rm T}}E - {U_{1r}} + Q - R}&{ - {E^{\rm T}}E - {W_r}} \\ { - {E^{\rm T}}E - {W_r}}&{{E^{\rm T}}E - {W_r}} \end{array}} \right]$
 $\begin{array}{l} {\hat \Xi _{16}} = {I_N} \otimes \left[ {\begin{array}{*{20}{c}} {{A^{\rm T}}{U_{1r}}}&{{A^{\rm T}}{W_{1r}}} \\ {\hat F_{_r}^{\rm T}}&{\hat F_{_r}^{\rm T}} \end{array}} \right] \\ {\hat \Xi _{26}}{\rm{ = }}{L^{\rm T}} \otimes \left[ {\begin{array}{*{20}{c}} {{C^{\rm T}}\hat G_r^{\rm T}}&{{C^{\rm T}}\hat G_r^{\rm T}} \end{array}} \right] \end{array}$
 $\begin{array}{l} {\hat \Xi _{46}}{\rm{ = }}{I_N} \otimes \left[ {\begin{array}{*{20}{c}} {\hat G_r^{\rm T}}&{\hat G_r^{\rm T}} \end{array}} \right] \\ {\hat \Xi _{56}}{\rm{ = }}{I_N} \otimes \left[ {\begin{array}{*{20}{c}} {{B^{\rm T}}{U_{1r}}}&{{B^{\rm T}}{W_r}} \\ {{D^{\rm T}}\hat G_r^{\rm T}}&{{D^{\rm T}}\hat G_r^{\rm T}} \end{array}} \right] \end{array}$
 ${\hat \Xi _{66}}{\rm{ = }}{I_N} \otimes \left[ {\begin{array}{*{20}{c}} { - {U_{1r}}}&{ - {W_r}} \\ * &{ - {W_r}} \end{array}} \right]$

 ${F_r} = W_r^{ - 1}{\hat F_r},{G_r} = W_r^{ - 1}{\hat G_r}$ (16)

 ${U_r} = \left[ {\begin{array}{*{20}{c}} {{U_{1r}}}&{{U_{2r}}} \\ * &{{U_{3r}}} \end{array}} \right]$

 ${J_2} = diag\left\{ {\begin{array}{*{20}{c}} {{I_N} \otimes {J_1}}&{\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} I&I&I&I \end{array}}&{{I_N} \otimes {J_1}}&I \end{array}} \end{array}} \right\}$

 ${\hat F_r} = {U_{2r}}{F_r}U_{3r}^{ - 1}U_{2r}^{\rm T}, {\hat G_r} = {U_{2r}}{G_r}$

3 数值仿真

 $A = \left[ {\begin{array}{*{20}{c}} {0.1}&{0.4} \\ 0&{ - 0.5} \end{array}} \right], B = \left[ {\begin{array}{*{20}{c}} {0.2}, \\ {0.5} \end{array}} \right], C = \left[ {\begin{array}{*{20}{c}} {0.1}&0 \end{array}} \right]$
 $D = 0.2, E = \left[ {\begin{array}{*{20}{c}} {0.1}&{0.1} \end{array}} \right]$

 ${L_1} = \left[ {\begin{array}{*{20}{c}} 1&{ - 1}&0&0 \\ 0&1&0&{ - 1} \\ 0&{ - 1}&1&0 \\ { - 1}&0&0&1 \end{array}} \right]$
 图 1 网络通信拓扑图

 ${F_1} = \left[ {\begin{array}{*{20}{c}} {{\rm{ - 0}}{\rm{.0456}}}&{{\rm{0}}{\rm{.2285}}} \\ {{\rm{ - 0}}{\rm{.0539}}}&{{\rm{ - 0}}{\rm{.4570}}} \end{array}} \right], {G_1} = \left[ {\begin{array}{*{20}{c}} {{\rm{ - 0}}{\rm{.660}}} \\ {{\rm{ - 0}}{\rm{.2361}}} \end{array}} \right]$

 ${L_1} = \left[ {\begin{array}{*{20}{c}} 1&{ - 1}&0&0 \\ 0&1&0&{ - 1} \\ 0&{ - 1}&1&0 \\ { - 1}&0&0&1 \end{array}} \right], {L_2} = \left[ {\begin{array}{*{20}{c}} 1&0&0&{ - 1} \\ { - 1}&1&0&0 \\ 0&{ - 1}&1&0 \\ 0&{ - 1}&0&1 \end{array}} \right]$
 图 2 每个智能体的滤波误差

 图 3 每个智能体的触发时刻和触发时间间隔

 图 4 网络通信拓扑图

 $\Pi {\rm{ = }}\left[ {\begin{array}{*{20}{c}} {0.3}&{0.7} \\ {0.4}&{0.6} \end{array}} \right]$

 ${F_1} = \left[ {\begin{array}{*{20}{c}} {{\rm{ - 0}}{\rm{.6257}}}&{{\rm{0}}{\rm{.2758}}} \\ {{\rm{ - 0}}{\rm{.1215}}}&{{\rm{ - 0}}{\rm{.0650}}} \end{array}} \right], {G_1} = \left[ {\begin{array}{*{20}{c}} {{\rm{ - 0}}{\rm{.1253}}} \\ {{\rm{ - 0}}{\rm{.0328}}} \end{array}} \right]$
 ${F_2} = \left[ {\begin{array}{*{20}{c}} {{\rm{0}}{\rm{.7681}}}&{{\rm{0}}{\rm{.5518}}} \\ {{\rm{0}}{\rm{.1453}}}&{{\rm{ - 0}}{\rm{.0218}}} \end{array}} \right], {G_2} = \left[ {\begin{array}{*{20}{c}} {{\rm{0}}{\rm{.1217}}} \\ {{\rm{0}}{\rm{.0328}}} \end{array}} \right]$

 图 5 切换信号

 图 6 每个智能体的滤波误差

4 结论

 图 7 每个智能体的触发时刻和触发时间间隔

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