具有定常输入的二阶多智能体系统的平均一致性滤波
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 智能系统学报  2018, Vol. 13 Issue (3): 399-406  DOI: 10.11992/tis.201612022 0

引用本文

ZHENG Min, LIU Chenglin, LIU Fei. Average-consensus filter of second-order multiagent systems with constant inputs[J]. CAAI Transactions on Intelligent Systems, 2018, 13(3), 399-406. DOI: 10.11992/tis.201612022.

文章历史

Average-consensus filter of second-order multiagent systems with constant inputs
ZHENG Min, LIU Chenglin, LIU Fei
Institute of Automation, Jiangnan University, Wuxi 214122, China
Abstract: Average-consensus filtering of a second-order multiagent system with constant inputs was investigated and a proportional-integral consensus filtering algorithm was proposed. Based on constant input, fixed and symmetrical connection topology, and on the Routh and Nyquist Criteria, the asymptotically consistent convergence conditions of second-order multiagent systems without time delays and with identical communication delays were obtained. In addition, the final consensus state of multiagent systems was the average value of the constant inputs. Finally, a numerical simulation of a multiagent system comprising five agents in the connection topology was used to verify the accuracy of the theoretical results.
Key words: average-consensus filter    proportional-integral algorithm    constant inputs    communication delay    second-order multi-agent systems    without time delay    symmetric and connected topology    frequency-domain analysis

1 问题描述 1.1 连接拓扑图

$n$ 阶有向图 ${{G}} = \left( {{{V}},{{E}},{{A}}} \right)$ 的组成部分包括：节点集 ${{V}} = \{ {v_1},{v_2}, \cdots ,{v_n}\}$ 、边集 ${{E}} \subseteq {{V}} \times {{V}}$ 以及加权邻接矩阵 ${{A}} = [{a_{ij}}] \in {{\bf{R}}^{n \times n}}$ 。便于描述，节点的下标集表示为 ${{\varGamma }} = \{ 1,2, \cdots ,n\}$ 。在图 ${{G}}$ 中，节点 $i$ 指向节点 $j$ 的有向边为 ${e_{ij}} = (i,j) \in {{E}}$ ，对应的连接权值为 ${a_{ij}} > 0$ ，否则， ${a_{ij}} = 0$ 。如果 ${a_{ij}}{\rm{ = }}{a_{ji}} > 0$ ，则称图 ${{G}}$ 是对称的。节点 $i$ 的邻接集合定义为 ${N_i} = \left\{ {j \in {{V}}:\left( {i,j} \right) \in {{E}}} \right\}$ 。根据邻接矩阵写出拉普拉斯矩阵 ${{L}} = [{l_{ij}}]({{L}} \in {{\bf{R}}^{n \times n}})$ ，定义为

 ${l_{ij}}{\rm{ = }}\left\{ \begin{array}{l}\displaystyle\sum\limits_{j = 1}^n {{a_{ij}}} ,\;\;\;\;i = j\\ - {a_{ij}},\;\;\;\;i \ne j\end{array} \right.$

1.2 模型描述

$n$ 个二阶智能体构成的二阶多智能体系统为

 $\left\{ \begin{array}{l}{{\dot x}_i} = {v_i}\\{{\dot v}_i} = {u_i},i \in {{\varGamma }}\end{array} \right.$ (1)

 ${\lim _{t \to \infty }}{x_i}(t) = \frac{1}{n}\sum\limits_{i \in {{\varGamma }}} {{\varphi _i}}$

 $\left\{ \begin{array}{l}{u_i} = - k{v_i} + \gamma ({\varphi _i} - {x_i}) - {k_P}\displaystyle\sum\limits_{j \in {N_i}} {{a_{ij}}({x_i} - {x_j})} - {k_I}{\eta _i}\\{{\dot \eta }_i} = \sum\limits_{j \in {N_i}} {{a_{ij}}({x_i} - {x_j})} \end{array} \right.$ (2)

 $\left\{ \begin{array}{l}{{\dot x}_i} = {v_i}\\{{\dot v}_i} = - k{v_i} + \gamma ({\varphi _i} - {x_i}) - {k_P}\displaystyle\sum\limits_{j \in {N_i}} {{a_{ij}}({x_i} - {x_j})} - {k_I}{\eta _i}\\{{\dot \eta }_i} = \sum\limits_{j \in {N_i}} {{a_{ij}}({x_i} - {x_j})} \end{array} \right.$ (3)
2 一致性收敛分析 2.1 无时延情况

 $\left\{ \begin{array}{l}{\dot{ x}} = {{v}}\\{\dot{ v}} = - k{{v}} + \gamma ({{\varphi }} - {{x}}) - {k_P}{{Lx}} - {k_I}{{\eta }}\\{\dot{ \eta }} = {{Lx}}\end{array} \right.$ (4)

 $\begin{array}{c}{{x}}(s) = {{P}}(s){{\varphi }}(s) + \displaystyle\frac{{s + k}}{\gamma }{{P}}(s){{x}}(0) + \\[8pt]\displaystyle\frac{1}{\gamma }{{P}}(s){{v}}(0) - \displaystyle\frac{{{k_I}}}{\gamma }\frac{{{{P}}(s)}}{s}{{\eta }}(0)\end{array}$ (5)

 ${{P}}(s) = \frac{{\gamma s{{I}}}}{{({s^3} + k{s^2} + \gamma s){{I}} + {k_p}{{L}}s + {k_I}{{L}}}}$ (6)

${{x}}(s) \!=\!\! {[{x_1}(s) \,{x_2}(s) \cdots {x_n}(s)]^{\rm{T}} }$ ${{v}}(s) \!=\!\! {[{v_1}(s)\, {v_2}(s) \cdots {v_n}(s)]^{\rm{T}} }$ ${{\eta }}(s) =$ ${[{\eta _1}(s)\, {\eta _2}(s) \, \cdots \, {\eta _n}(s)]^{\rm{T}} }$ ${{\varphi }}(s) = {[\displaystyle\frac{{{\varphi _1}}}{s} \, \displaystyle\frac{{{\varphi _2}}}{s} \, \cdots \, \displaystyle\frac{{{\varphi _n}}}{s}]^{\rm{T}} }$ ${{x}}(0)$ ${{v}}(0)$ ${{\eta }}(0)$ 分别为智能体的位置、速度和内部状态初始值。

1) $k > 0,\gamma > 0$

2) ${k_I} > 0,k(\gamma + {k_p}{\lambda _i}) > {k_I}{\lambda _i}$

1) 令传递函数矩阵 ${{P}}(s)$ 的特征方程为

 $\det (({s^3} + k{s^2} + \gamma s){{I}} + {k_p}{{L}}s + {k_I}{{L}}) = 0$ (7)

 $\prod\limits_{i = 1}^n {{s^3} + k{s^2} + (\gamma + {k_p}{\lambda _i})s + {k_I}{\lambda _i} = 0}$ (8)

 ${s^3} + k{s^2} + (\gamma + {k_p}{\lambda _i})s + {k_I}{\lambda _i} = 0,\;i = 1, \cdots ,n$ (9)

${\lambda _1} = 0$ 时，式(9)表示为

 $s({s^2} + ks + \gamma ) = 0$ (10)

${\lambda _i},\;i = 2,3, \cdots ,n$ 时，方程(9)的Routh阵列表为

 $\begin{array}{*{20}{c}} {{s^3}}&1&{\gamma + {k_p}{\lambda _i}} \\ {{s^2}}&k&{{k_I}{\lambda _i}} \\ s&{\displaystyle\frac{{k(\gamma + {k_p}{\lambda _i}) - {k_I}{\lambda _i}}}{k}}&{} \\ {{s^0}}&{{k_I}{\lambda _i}}&{} \end{array}$

2)为了证明多智能体系统式(3)渐近达到平均一致滤波，分别证明式(5)满足：

${{P}}(s){{\varphi }}(s) \to \displaystyle\frac{{{{\bf{1}}_n}{\bf{1}}_n^{\rm{T}} }}{n}{{\varphi }}$

② 保证式(5)中 $\displaystyle\frac{{s + k}}{\gamma }{{P}}(s){{x}}(0)$ $\displaystyle\frac{1}{\gamma }{{P}}(s){{v}}(0)$ $\displaystyle\frac{{{k_I}}}{\gamma }\displaystyle\frac{{{{P}}(s)}}{s}{{\eta }}(0)$ 渐近趋于零状态。

①的证明　在无向连通拓扑结构下，Laplacian矩阵 ${{L}}$ 是实对称的。 ${\lambda _1} = 0$ ${{L}}$ 的一个单一特征值，对应的右特征向量为 ${{r}} = \displaystyle\frac{{{{\bf{1}}_n}}}{{\sqrt n }}$ ，其他 $n - 1$ 个特征值对应的特征向量为 ${{S}} = [{S_2}\,\,{S_3}\,\, \cdots \,\, {S_n}]$ 。假定 ${{Q}} = [\displaystyle\frac{{{{\bf{1}}_n}}}{{\sqrt n }}\,\,\,\,{{S}}]$ ，则 ${{Q}}{{{Q}}^{\rm{T}} } = {{I}}$ ${{QL}}{{{Q}}^{\rm{T}} } =$ $\rm{diag} \left\{ {{\lambda _1}} \right., \cdots ,\left. {{\lambda _n}} \right\}$

 $\begin{array}{c}{{P}}(s) = {{{Q}}^{\mathop{\rm T}\nolimits} }{{QP}}(s){{{Q}}^{\mathop{\rm T}\nolimits} }{{Q}} = \\{{{Q}}^{\mathop{\rm T}\nolimits} }\displaystyle\frac{{\gamma s{{I}}}}{{({s^3} + k{s^2} + \gamma s){{I}} + {k_p}{{QL}}{{{Q}}^{\mathop{\rm T}\nolimits} }s + {k_I}{{QL}}{{{Q}}^{\mathop{\rm T}\nolimits} }}}{{Q}} = \\[8pt]{{{Q}}^{\mathop{\rm T}\nolimits} }\left( {\begin{array}{*{20}{c}}{{h_1}(s)} & {{\bf{0}}_{n - 1}^{\rm{T}}}\\{{{\bf{0}}_{n - 1}}} & {{\bar{ P}}(s)}\end{array}} \right){{Q}}\end{array}$ (11)

 ${h_1}(s) = \frac{\gamma }{{{s^2} + ks + \gamma }}$ (12)
 $\begin{array}{c}{\bar{ P}}(s) = {\mathop{\rm diag}\nolimits} \left\{ {\displaystyle\frac{{\gamma s}}{{{s^3} + k{s^2} + (\gamma + {k_p}{\lambda _2})s + {k_I}{\lambda _2}}}} \right., \cdots ,\\[8pt]\left. {\displaystyle\frac{{\gamma s}}{{{s^3} + k{s^2} + (\gamma + {k_p}{\lambda _n})s + {k_I}{\lambda _n}}}} \right\}\end{array}$ (13)

 $\begin{array}{c}{\lim _{s \to 0}}s{{P}}(s)\displaystyle\frac{{{\varphi }}}{s} = {{P}}(0){{\varphi }} = \left( {\begin{array}{*{20}{c}}{\displaystyle\frac{{{{\bf{1}}_n}}}{{\sqrt n }}} & {{S}}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{{h_1}(0)} & {{\bf{0}}_{n - 1}^{\rm{T}}}\\{{{\bf{0}}_{n - 1}}} & {{\bar{ P}}(0)}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{\displaystyle\frac{{{\bf{1}}_n^{\rm{T}}}}{{\sqrt n }}}\\{{{{S}}^{\rm{T}}}}\end{array}} \right){{\varphi }} = \\[8pt]\displaystyle\frac{{{{\bf{1}}_n}{\bf{1}}_n^{\rm{T}}}}{n}{h_1}(0){{\varphi }} + {{S\bar P}}(0){{{S}}^{\mathop{\rm T}\nolimits} }{{\varphi }}\end{array}$ (14)

②的证明　由①的证明可得：

 ${\lim _{s \to 0}}s\frac{{s + k}}{\gamma }{{P}}(s){{x}}(0) = {\lim _{s \to 0}}s\frac{{s + k}}{\gamma }s{{P}}(s)\frac{{{{x}}(0)}}{s} = {{\bf{0}}_n}$
 ${\lim _{s \to 0}}s\frac{1}{\gamma }{{P}}(s){{v}}(0) = {\lim _{s \to 0}}s\frac{1}{\gamma }s{{P}}(s)\frac{{{{v}}(0)}}{s} = {{\bf{0}}_n}$

 $\begin{array}{c}{\lim _{s \to 0}}s\displaystyle\frac{{{k_I}}}{\gamma }\frac{{{{P}}(s)}}{s}{{\eta }}(0) = {\lim _{s \to 0}}\frac{{{k_I}}}{\gamma }s{{P}}(s)\frac{{{{\eta }}(0)}}{s} = \\\displaystyle\frac{{{k_I}}}{\gamma }\frac{{{{\bf{1}}_n}{\bf{1}}_n^{\mathop{\rm T}\nolimits} }}{n}{{\eta }}(0) = \frac{{{k_I}{{\bf{1}}_n}}}{{n\gamma }}\sum\limits_{i \in {{\Gamma }}} {{\eta _i}(0) = {{\bf{0}}_n}} \end{array}$

2.2 有时延情况

 $\begin{array}{c}{u_i}(t) = - k{v_i}(t) + \gamma ({\varphi _i} - {x_i}(t)) - \\{k_p}\displaystyle\sum\limits_{j \in {N_i}} {{a_{ij}}} ({x_i}(t - \tau ) - {x_j}(t - \tau )) - {k_I}{\eta _i}(t)\\[8pt]{{\dot \eta }_i}(t) = \displaystyle\sum\limits_{j \in {N_i}} {{a_{ij}}} ({x_i}(t - \tau ) - {x_j}(t - \tau ))\end{array}$ (15)

 $\left\{ \begin{array}{l}{{\dot x}_i}(t) = {v_i}(t)\\{{\dot v}_i}(t) = - k{v_i}(t) + \gamma ({\varphi _i} - {x_i}(t)) - \\\quad\quad{k_p}\displaystyle\sum\limits_{j \in {N_i}} {{a_{ij}}} ({x_i}(t - \tau ) - {x_j}(t - \tau )) - {k_I}{\eta _i}(t)\\[6pt]{{\dot \eta }_i}(t) = \displaystyle\sum\limits_{j \in {N_i}} {{a_{ij}}} ({x_i}(t - \tau ) - {x_j}(t - \tau ))\end{array} \right.$ (16)

 $\left\{ \begin{array}{l}\!\!\! s{x_i}(s) - {x_i}(0) = {v_i}(s)\\\!\!\! s{v_i}(s) - {v_i}(0) = - k{v_i}(s) + \gamma ({\varphi _i}(s) - {x_i}(s)) - \\\quad{k_p}\displaystyle\sum\limits_{j \in {N_i}} {{a_{ij}}} ({x_i}(s) - {x_j}(s)){{\rm{e}}^{ - s\tau }} - {k_p}{w_i}(s) - {k_I}{\eta _i}(s)\\\!\!\! s{\eta _i}(s) - {\eta _i}(0) = \displaystyle\sum\limits_{j \in {N_i}} {{a_{ij}}} ({x_i}(s) - {x_j}(s)){{\rm{e}}^{ - s\tau }} - {w_i}(s)\end{array} \right.$ (17)

 ${w_i}(s) = \sum\limits_{j \in {N_i}} {{a_{ij}}} \int_{ - \tau }^0 {({x_i}(\theta ) - {x_j}(\theta ))} {{\rm{e}}^{ - s\theta }}{\rm{d}}\theta$

 $\begin{array}{c}{{x}}(s) = {{{G}}_\tau }(s){{\varphi }}(s) + \\\displaystyle\frac{{s + k}}{\gamma }{{{G}}_\tau }(s){{x}}(0) + \frac{1}{\gamma }{{{G}}_\tau }(s){{v}}(0) - \\[6pt]\displaystyle\frac{{{k_I}}}{\gamma }\frac{{{{{G}}_\tau }(s)}}{s}{{\eta }}(0) + \frac{{{k_I} - {k_p}s}}{{\gamma s}}{{{G}}_\tau }(s){{w}}(s)\end{array}$ (18)

 ${{{G}}_\tau }(s) = \frac{{\gamma s{{I}}}}{{({s^3} + k{s^2} + \gamma s){{I}} + {k_p}{{L}}{{\rm{e}}^{ - s\tau }}s + {k_I}{{L}}{{\rm{e}}^{ - s\tau }}}}$ (19)

1) ${k^2} - 2\gamma \geqslant 0$

2) $\tau < \mathop {\min }\limits_{i = 2,3, \cdots ,n} (\displaystyle\frac{{\arctan (\sqrt {{y_i}} \displaystyle\frac{{{k_p}}}{{{k_I}}}) - {\rm{arccot}} (\displaystyle\frac{{\gamma - {y_i}}}{{ k \sqrt {{y_i}} }}) + \displaystyle\frac{{\text{π}}}{2}}}{{\sqrt {{y_i}} }})$

 ${y^3} + ({k^2} - 2\gamma ){y^2} + ({\gamma ^2} - k_p^2\lambda _i^2)y - k_I^2\lambda _i^2 = 0,i = 2,3, \cdots ,n$ (20)

 $f(x) = \frac{{{c_1}x + {c_2}}}{{{x^3} + {c_3}{x^2} + {c_4}x}}$

 $f'(x) = \frac{{ - 2{c_1}{x^3} - ({c_1}{c_3} + 3{c_2}){x^2} - 2{c_2}{c_3}x - {c_2}{c_4}}}{{{{({x^3} + {c_3}{x^2} + {c_4}x)}^2}}}$

 ${f_1}(x) = {x^3} + {d_1}{x^2} + {d_2}x + {d_3}$

1)根据式(19) ${{{G}}_\tau }(s)$ 特征方程为

 $\det (({s^3} + k{s^2} + \gamma s){{I}} + {k_p}{{L}}{{\rm{e}}^{ - s\tau }}s + {k_I}{{L}}{{\rm{e}}^{ - s\tau }}) = 0$

 $\prod\limits_{i = 1}^n {{s^3} + k{s^2} + \gamma s + {k_p}{\lambda _i}{{\rm{e}}^{ - s\tau }}s + {k_I}{\lambda _i}{{\rm{e}}^{ - s\tau }} = 0}$ (21)

 ${s^3} + k{s^2} + \gamma s + {k_p}{\lambda _i}{{\rm{e}}^{ - s\tau }}s + {k_I}{\lambda _i}{{\rm{e}}^{ - s\tau }}{\rm{ = }}0,i = 1,2, \cdots ,n$ (22)

$s = 0$ 时，特征方程式(22)变为

 ${0^3} + k{0^2} + \gamma 0 + {k_p}{\lambda _i}{{\rm{e}}^{ - 0\tau }}0 + {k_I}{\lambda _i}{{\rm{e}}^{ - 0\tau }}{\rm{ = }}0$

$s \ne 0$ 时，式(22)整理为闭环特征函数的形式：

 $1 + \frac{{{k_p}{\lambda _i}(s + \displaystyle\frac{{{k_I}}}{{{k_p}}})}}{{{s^3} + k{s^2} + \gamma s}}{{\rm{e}}^{ - s\tau }} = 0$ (23)

 $k(s) = \frac{{{k_p}{\lambda _i}(s + \displaystyle\frac{{{k_I}}}{{{k_p}}})}}{{{s^3} + k{s^2} + \gamma s}}{{\rm{e}}^{ - s\tau }}$ (24)

$s = {\rm{j}} w$ ，开环特征函数式(24)的频率特性为

 $k({\rm{j}} w) = \frac{{{k_p}{\lambda _i}({\rm{j}} w + \displaystyle\frac{{{k_I}}}{{{k_p}}})}}{{{{({\rm{j}} w)}^3} + k{{({\rm{j}} w)}^2} + {\rm{j}} \gamma w}}{{\rm{e}} ^{ - {\rm{j}} w\tau }}$ (25)

 ${A_i}(w) = \frac{{{k_p}{\lambda _i}\sqrt {{w^2} + \displaystyle\frac{{k_I^2}}{{k_P^2}}} }}{{w\sqrt {{k^2}{w^2} + {{(\gamma - {w^2})}^2}} }}$ (26)

 $\beta (w) = \arctan (\frac{{{k_p}}}{{{k_I}}}w) - {\rm{arccot}} (\frac{{\gamma - {w^2}}}{{kw}}) - w\tau - \frac{\text{π}}{2}$ (27)

 $f(y) = \frac{{k_p^2y + k_I^2}}{{{y^3} + ({k^2} - 2\gamma ){y^2} + {\gamma ^2}y}}$ (28)

 ${y^3} + ({k^2} - 2\gamma ){y^2} + ({\gamma ^2} - k_p^2\lambda _i^2)y - k_I^2\lambda _i^2 = 0$ (29)

 $\beta ({w_i}) = \beta (\sqrt {{y_i}} ) > - {\rm{\pi }}$ (30)

 ${\tau _i} < \frac{{\arctan (\sqrt {{y_i}} \displaystyle\frac{{{k_p}}}{{{k_I}}}) - {\rm{arccot}} (\displaystyle\frac{{\gamma - {y_i}}}{{k\sqrt {{y_i}} }}) + \displaystyle\frac{{\rm{\pi }}}{2}}}{{\sqrt {{y_i}} }}$ (31)

2)根据定理1中①的证明，可得

 $\begin{array}{l}{{{G}}_\tau }(s) = {{{Q}}^{\rm{T}}}{{Q}}{{{G}}_\tau }(s){{{Q}}^{\rm{T}}}{{Q}} = \\[5pt]{{{Q}}^{\rm{T}}}\left[ {\begin{array}{*{20}{c}}{{h_1}(s)} & {{\bf{0}}_{n - 1}^{\rm{T}}}\\[5pt]{{{\bf{0}}_{n - 1}}} & {{{{\bar{ G}}}_\tau }(s)}\end{array}} \right]{{Q}}\end{array}$ (32)

 $\begin{array}{c}{{{\bar{ G}}}_\tau }(s) = {\mathop{\rm diag}\nolimits} \left\{ {\displaystyle\frac{{\gamma s}}{{{s^3} + k{s^2} + (\gamma + {k_p}{\lambda _2}{{\rm{e}}^{ - s\tau }})s + {k_I}{\lambda _2}{{\rm{e}}^{ - s\tau }}}}} \right. , \cdots , \\\left. {\displaystyle\frac{{\gamma s}}{{{s^3} + k{s^2} + (\gamma + {k_p}{\lambda _n}{{\rm{e}}^{ - s\tau }})s + {k_I}{\lambda _n}{{\rm{e}}^{ - s\tau }}}}} \right\}\end{array}$ (33)
 ${{\bar{ G}}_\tau }(0) = {{\bf{0}}_{n \times n}}$ (34)

 ${{{G}}_\tau }(0) = {{{Q}}^{\rm{T}} }{{Q}}{{{G}}_\tau }(0){{{Q}}^{\rm{T}} }{{Q}} = \frac{{{{\bf{1}}_n}{\bf{1}}_n^{\rm{T}} }}{n}$ (35)

 ${\lim _{s \to 0}}s{{{G}}_\tau }(s)\frac{{{\varphi }}}{s} = {{{G}}_\tau }(0){{\varphi }} = \frac{{{{\bf{1}}_n}{\bf{1}}_n^{\rm{T}}}}{n}{{\varphi }}$ (36)

 ${\lim _{s \to 0}}s\frac{{s + k}}{\gamma }{{{G}}_\tau }(s){{x}}(0) = {{\bf{0}}_n}$ (37)
 ${\lim _{s \to 0}}s\frac{1}{\gamma }{{{G}}_\tau }(s){{v}}(0) = {{\bf{0}}_n}$ (38)

 ${\lim _{s \to 0}}s\frac{{{k_I}}}{\gamma }\frac{{{{{G}}_\tau }(s)}}{s}{{\eta }}(0) = \frac{{{k_I}{{\bf{1}}_n}}}{{\gamma n}}{\bf{1}}_n^{\rm{T}} {{\eta }}(0) = {{\bf{0}}_n}$ (39)
 $\begin{array}{c}{\lim _{s \to 0}}s\displaystyle\frac{{{k_I} - {k_p}s}}{{\gamma s}}{{{G}}_\tau }(s){{w}}(s) = \displaystyle\frac{{{k_I}}}{\gamma }{{{G}}_\tau }(0){{w}}(0) = \frac{{{k_I}{{\bf{1}}_n}}}{{\gamma n}}{\bf{1}}_n^{\rm{T}}{{w}}(0) = \\[10pt]\displaystyle\frac{{{k_I}{{\bf{1}}_n}}}{{\gamma n}}\sum\limits_{i \in {{\varGamma }}} {\left(\sum\limits_{j \in {N_i}} {{a_{ij}}} \int_{ - T}^0 {({x_i}(\tau ) - {x_j}(\tau ))} {\rm{d}}\tau \right)} \end{array}$ (40)

 ${a_{ij}}\int_{ - T}^0 {({x_i}(\tau ) - {x_j}(\tau ))} {\rm{d}}\tau + {a_{ji}}\int_{ - T}^0 {({x_j}(\tau ) - {x_i}(\tau ))} {\rm{d}}\tau = 0$

 $\sum\limits_{i \in {{\varGamma }}} {\left(\sum\limits_{j \in {N_i}} {{a_{ij}}} \int_{ - T}^0 {({x_i}(\tau ) - {x_j}(\tau ))} {\rm{d}}\tau \right)} = 0$ (41)

 ${\lim _{s \to 0}}s\frac{{{k_I} - {k_p}s}}{{\gamma s}}{{{G}}_\tau }(s){{w}}(s) = {{\bf{0}}_n}$ (42)

3 数值仿真 3.1 无时延的二阶多智能体系统

 Download: 图 1 包含4个多智能连接拓扑G Fig. 1 Graphical topology of four agents

 Download: 图 2 智能体的位置和速度( ${k_I} =$ 0.03) Fig. 2 Positions and velocities of agents with ${k_I} =$ 0.03
 Download: 图 3 智能体的位置和速度( ${k_I} =$ 0.483) Fig. 3 Positions and velocities of agents with ${k_I} =$ 0.483
 Download: 图 4 智能体的位置和速度( ${k_I} =$ 0.5) Fig. 4 Positions and velocities of agents with ${k_I} =$ 0.5
3.2 时延二阶多智能体系统

 $\begin{array}{c}\displaystyle\frac{{{\rm{d}}\varphi (w)}}{{{\rm{d}}w}} = - T + \displaystyle\frac{1}{{\left[ {1 + {{\left( {\displaystyle\frac{{{k_p}}}{{{k_I}}}} \right)}^2}{w^2}} \right][{k^2}{w^2} + {{(\gamma - {w^2})}^2}]}} \times \\\left\{ \! {\displaystyle\frac{{{k_p}}}{{{k_I}}}(1 - k\frac{{{k_p}}}{{{k_I}}}){w^4} \!+\! [\frac{{{k_p}}}{{{k_I}}}({k^2} \!-\! 2\gamma ) - k - k{{(\displaystyle\frac{{{k_p}}}{{{k_I}}})}^2}\gamma ]{w^2} \!+\! \displaystyle\frac{{{k_p}}}{{{k_I}}}{\gamma ^2} \!-\! k\gamma } \! \right\} \!\!= \\ - T + \displaystyle\frac{{ - 164{w^4} - 49{w^2} + 0.9}}{{(1 + 177.8{w^2})({w^4} + 0.4{w^2} + 0.09)}}\end{array}$

 Download: 图 5 智能体的位置和速度( $T =$ 0.32 s) Fig. 5 Positions and velocities of agents with communication delay $T =$ 0.32 s
 Download: 图 6 智能体的位置和速度( $T =$ 0.75 s) Fig. 6 Positions and velocities of agents with communication delay $T =$ 0.75 s
4 结束语

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