广东工业大学学报  2018, Vol. 35Issue (2): 75-80.  DOI: 10.12052/gdutxb.170125.
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引用本文 

张振华, 彭世国. 二阶多智能体系统拓扑切换下的领导跟随一致性[J]. 广东工业大学学报, 2018, 35(2): 75-80. DOI: 10.12052/gdutxb.170125.
Zhang Zhen-hua, Peng Shi-guo. Leader-Following Consensus of Second-Order Multi-Agent Systems with Switching Topology[J]. JOURNAL OF GUANGDONG UNIVERSITY OF TECHNOLOGY, 2018, 35(2): 75-80. DOI: 10.12052/gdutxb.170125.

基金项目:

国家自然科学基金资助项目(61374081)

作者简介:

张振华(1991–),男,硕士研究生,主要研究方向为多智能体系统及其一致性问题、非线性系统和脉冲控制. E-mail:1435334933@qq.com

文章历史

收稿日期:2017-08-03
二阶多智能体系统拓扑切换下的领导跟随一致性
张振华, 彭世国     
广东工业大学 自动化学院,广东 广州  510006
摘要: 在有积极领导者且系统网络拓扑切换的条件下, 对二阶非线性多智能体系统的领导跟随一致性进行了研究. 协议在非脉冲时刻采取连续控制, 在脉冲时刻采取离散控制,且脉冲时刻创新性地提出各智能体的速度增量既与邻接智能体的位移状态有关又与速度状态有关. 构建新模型的误差系统, 基于Lyapunov稳定性理论, 利用数学归纳法和其他相关知识, 得到系统拓扑图无向且连通时系统实现一致性必须满足的充分条件, 并用Matlab软件实例仿真验证了结果的正确性.
关键词: 二阶非线性多智能体系统    领导跟随一致性    拓扑切换    脉冲控制    
Leader-Following Consensus of Second-Order Multi-Agent Systems with Switching Topology
Zhang Zhen-hua, Peng Shi-guo     
School of Automation, Guangdong University of Technology, Guangzhou 510006, China
Abstract: With the conditions of active leadership and system topology switching, the leader-following consensus in the second-order nonlinear multi-agent systems is studied. While the protocol takes continuous control at non-pulsed moments and discrete control at pulse moments, an innovative view is raised that the speed increment of each agent is not only related to the displacement state of the adjacency agent but also to the velocity state. A New Model of Error System is constructed, based on Lyapunov stability theory, using mathematical induction and other related knowledge, and getting the sufficient conditions to achieve consensus when the system topology is directed and connected, and the result is verified by the Matlab simulation.
Key words: second-order nonlinear multi-agent systems    leader-following consensus    switching topology    impulsive control    

过去20多年间,关于多智能体系统一致性问题的研究引起学界越来越多的关注,对一些经典多智能体系统的建模研究有助于解决自然界和人类社会中一些常见的难题[1]. 譬如:群集和蜂拥,飞行器编队控制,神经网络的稳定性控制,全球经济市场变化等[2-5]. 实际问题中系统状态一般呈非线性变化,因此对非线性多智能体系统的研究尤为必要. 文献[6]研究了一类混沌系统的控制问题,针对此问题还有其他一些有效的方法被提出,包括模糊控制法[7],自适应控制法[8]、间歇性控制法[9]、脉冲控制法[10]、切换控制法[11]等等.

传统的连续控制协议控制多智能体系统会存在一些弊端,例如有时无法连续采集各智能体的状态信息以对系统实施精确控制. 此时可以引入脉冲控制协议对系统进行离散式控制. 文献[12]研究了一类混沌系统在脉冲控制下的稳定性和同步问题. 文献[13]研究了一类二阶非线性系统脉冲控制下的平均一致性问题. 文献[14-17]则研究了在有领导者的情况下二阶非线性多智能体系统的领导跟随一致性.

受文献[13-17]启发,本文设计了不同于文献[14]的全新协议. 对文献[14]的协议作了如下改进:一是智能体自身非线性向量值函数仅与位移状态有关; 二是非脉冲时刻跟随智能体速度状态的更新仅受到邻接智能体影响; 三是在脉冲协议中位移增量仅与位移有关,而速度增量既与位移有关,又与速度有关; 四是脉冲控制协议是从文献[13, 15]的协议中推广得到,和以往协议略有不同; 五是领导者是积极领导者,即领导者和各节点的连接关系是不断变化的(领导节点无需全局可达),且任一固定节点在拓扑切换时不接受领导者信息也能和领导者保持一致; 在该协议下研究了二阶非线性多智能体系统在网络拓扑切换且无向连通时的领导跟随一致性,仿真结果验证了协议的正确性.

1 预备知识 1.1 图论

通常,用有向图与无向图表示多智能体系统的网络拓扑. 令 $G = (R,V,{A})$ 表示N阶无向图,其中R= $ \left\{ {{R_1},{R_2}, \cdots \! ,{R_N}} \right\}$ 是拓扑图中N个节点的集合,Vij= $\left\{ {({V_i},{V_j}),i,j \in R} \right\}$ 表示拓扑图中自起始节点i到末节点j的边集. ${A} = [{a_{i,j}}]$ 表示邻接矩阵且元素 ${a_{ij}} \geqslant 0$ 表示节点i和节点j之间边的权值,当两点之间有连接时值大于0. 定义出度矩阵 ${D} = {{diag}}({d_i},i = 1,2, \cdots \! ,N)$ ,其中 ${d_i} = \sum\limits_{j = 1}^n {{a_{ij}}} $ 表示邻接矩阵A的第i行元素之和. 那么图G的Laplacian矩阵用 ${L} = {D} - {A}$ 表示, 矩阵元素可表示为

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

有积极领导者存在时,令连接矩阵 ${B} = $ ${{diag}}({b_1},{b_2}, \cdots \! ,{b_N}) \in {R^{N \times N}}$ ,对角线上元素bi表示领导者与其他节点的连接权值,通常取1或0.

1.2 模型及相关引理

引理1[18](Rayleigh-Ritz)存在Hermite矩阵 ${A} \in {M_n}$ Mnn阶实矩阵的集合,那么A的所有特征值: ${\lambda _{\min }} = {\lambda _1} \leqslant {\lambda _2} \leqslant \cdots \leqslant {\lambda _{n - 1}} \leqslant {\lambda _n} = {\lambda _{\max }}$ ,满足公式 ${\lambda _{{1}}}{{{x}}^{{T}}}{{x}} \leqslant $ $ {{{x}}^{{T}}}{{Ax}} \leqslant {\lambda _n}{{{x}}^{{T}}}{{x}}$ xRnn维列向量.

引理2[19]符号“ $ \otimes $ ”表示Kronecker积,存在常数a和矩阵ABCD满足下列运算性质:

(1) $(a{A}) \otimes {B} = {A} \otimes (a{B})$

(2) $({A} + {B}) \otimes {C} = {A} \otimes {C} + {B} \otimes {C}$

(3) $({A} \otimes {B})({C} \otimes {D}) = ({AC}) \otimes ({BD})$

(4) ${({A} \otimes {B})^{{T}}} = {{A}^{{T}}} \otimes {{B}^{{T}}}$ .

引理3[20] (Gronwall)令 $v(t)$ $g(t)$ 是定义在开区间(a,b)上的非负连续标量函数, $a < {t_0} < b$ , $c \geqslant 0$ . 当 $a < t < $ b时,令 $v(t) \! \leqslant \! c \! + \! \left| {\int_{{t_0}}^t {v(s)g(s)} {{d}}s} \right|$ ,则 $v(t) \! \leqslant \! c\exp \left( {\left| {\int_{{t_0}}^t {g(s){{d}}s} } \right|} \right)$ .

引理4[21] $\delta (t)$ 是Dirac函数,当t≠0时其值为0. 该函数具有如下性质:对所有连续紧函数 $g(t)$ 与常数 $\varepsilon \ne {{0}}$ ,均有 $\int_{a - \varepsilon }^{a + \varepsilon } {g(t)\delta (t - a){{d}}t = g(a)} $ .

本文引入一个二阶领导者,动力学模型如下:

$\left\{ \begin{array}{l}{{\dot{ x}}_0}(t) = {{v}_0}(t);\\[7pt]{{\dot{ v}}_0}(t) = f({{x}_0}(t),t).\end{array} \right.$ (1)

其中,f表示智能体自身动力学的非线性向量值函数;xyn维列向量.

对含有N个智能体的二阶非线性多智能体系统,本文设计了如下动力学模型(2)与控制协议(3)、(4):

$\left\{ \begin{array}{l}{{\dot{ x}}_i}(t) = {{v}_i}(t) + {u_{i1}}(t),\\[7pt]{{\dot{ v}}_i}(t) = f({{x}_i}(t),t) + {u_{i2}}(t);\end{array} \right.$ (2)
$\begin{split}{u_{i1}}(t) = & \sum\limits_{{{k}} = 1}^{ + \infty } {\delta (t - {t_k})} ({{B}_k}a\sum\limits_{j \in {N_i}s(t)} {{a_{ij}}(s(t))({{x}_j}(t) - } \\[7pt]&{{x}_i}(t)) +{b_i}({{B}_k} - {{I}_n})({{x}_i}(t) - {{x}_0}(t)));\end{split}$ (3)
$\begin{split}{u_{i2}}&(t) = a\left[ {\sum\limits_{j \in {N_I}s(t)} {{a_{ij}}(s(t))({{v}_j}(t) - {{v}_i}(t))} } \right] + \\ &\sum\limits_{k = 1}^{ + \infty } {\delta (t - {t_k}){{B}_k}a\sum\limits_{j \in {N_i}s(t)} {{a_{ij}}(s(t))} ({{x}_j}(t) - {{x}_i}(t) +} \\ & {{v}_j}(t) - {{v}_i}(t)) + \sum\limits_{k = 1}^{ + \infty } {\delta (t - {t_k}){b_i}({{B}_k} - {{I}_n})} ({{v}_i}(t) - {{v}_0}(t)) .\end{split}$ (4)

备注:

(1) 当多智能体系统的网络拓扑切换时,定义分段定常切换信号 $s(t):[{t_0}, + \infty ) \to P$ ,该信号决定系统的拓扑结构. $s(t)$ 为常数时表示固定拓扑,反之,令 $P = {{\{ 1}},{{2}}, \cdots \! ,{{m\} }}$ ,那么切换信号下所有可能切换的拓扑图用 $\left\{ {{{\bar G}_1},{{\bar G}_2}, \cdots \! ,{{\bar G}_m}} \right\}$ 表示[14]. 同理,当积极领导者和各智能体的连接关系顺序切换时,定义分段定常切换信号 $\omega (t):[{t_0}, + \infty ) \to {P^{{1}}}$ ,该信号决定系统的拓扑结构. $\omega (t)$ 为常数时表示固定拓扑,反之,令 ${P^{{1}}} =$ $ {{\{ 1}},{{2}}, \cdots \! {{,}}m{{\} }}$ ,那么切换信号下所有可能切换的拓扑图也用 $\left\{ {{{\bar G}_1},{{\bar G}_2}, \cdots \! ,{{\bar G}_m}} \right\}$ 表示.

(2) ${{v}_i}(t) \!=\! {({{v}_{i1}}(t),{{v}_{i2}}(t), \cdots \! ,{{v}_{in}}(t))^{{T}}} \! \in \! {{\bf{{R}}}^{{n}}}$ ${{x}_i}(t) \!=\! {({{x}_{i1}}(t),}$ $ {{{x}_{i2}}(t), \cdots \! ,{{x}_{in}}(t))^{{T}}} \in {{\bf{{R}}}^{{n}}}$ $\delta (t)$ 是Dirac函数, ${{B}_k} \in {{\bf{{R}}}^{{{n}} \times {{n}}}}$ 是脉冲增益矩阵, $a( > 0)$ 是耦合强度, ${u_i}(t)$ 是控制输入.

结合引理4将式(2)、(3)、(4)转化为如下动力学模型 $(\forall i,mj = 1,2, \cdots \! ,N)$

$\left\{ \begin{aligned} &{{\dot{ x}}_i}(t) = {{v}_i}(t),t \ne {t_k};\\ & \begin{array}{l} \!\!\!\! {{{\dot v}}_i}(t) = f({{{\dot x}}_i}(t),t) + a\displaystyle\sum\limits_{j \in {N_i}s(t)} {{a_{ij}}(s(t))({{{v}}_j}(t)}- \\ \quad \!\!\!\!{{{v}}_i}(t)),t \ne {t_k}; \end{array}\\& \Delta {{x}_i}({t_k}) = {{x}_i}(t_k^ + ) - {{x}_{{i}}}(t_k^ - ) = \\ & \quad {{B}_k}a\sum\limits_{j \in {N_i}s(t)} {{a_{ij}}(s(t))({{x}_j}(t_{{k}}^ - ) - {{x}_i}(t_k^ - )) + } \\ & \quad{b_i}(\omega (t))({{B}_k} - {{I}_n})({{x}_i}(t_k^ - ) - {{x}_0}(t_k^ - )),t = {t_k};\\ & \Delta {{v}_i}({t_k}) = {{v}_i}(t_k^ + ) - {{v}_i}(t_{k}^ - ) = \\ & \quad{{B}_k}a\sum\limits_{j \in {N_i}s(t)} {{a_{ij}}(s(t))({{x}_j}(t_k^ - ) - {{x}_i}(t_k^ - ) + {{v}_j}(t_k^ - ) -} \\ & \quad{{v}_i}(t_k^ - )) + {b_i}(\omega (t))({{B}_k} - {{I}_n})({{v}_i}(t_k^ - ) - {{v}_0}(t_k^ - )),t = {t_k}. \end{aligned} \right.$ (5)

式(5)中,脉冲序列 $\left\{ {{t_k}} \right\}$ 满足 ${{0}} \leqslant {t_0} < {t_1} < \cdots < $ $ {t_k} < {t_{k + 1}}$ $\mathop {\lim }\limits_{k \to + \infty } {t_k} = + \infty $ ,脉冲间隔为 $\Delta {t_k} = {t_{k + 1}} - {t_k}$ ;在脉冲时刻智能体位移和速度状态左连续,即 ${{x}_i}({t_k}) = {{x}_i}(t_k^ - )$ ${{v}_i}({t_k}) = {{v}_i}(t_k^ - )$ ;当 $t \in ({t_{k - 1}},{t_k}]$ $s(t) = s(k) \in$ $\left\{ {1,2, \cdots \! ,m} \right\}$ 时,表明拓扑网络随时间切换;

同时, $\left\| {{{B}_k}} \right\| < 1$ 时,脉冲增益矩阵对系统实现领导跟随一致性有积极作用; $\left\| {{{B}_k}} \right\| > 1$ 时,脉冲增益矩阵的存在会使系统状态发散从而无法趋于一致; $\left\| {{{B}_k}} \right\| = 1$ 时,脉冲增益矩阵对系统一致性没有任何影响[13].

定义跟随智能体和领导者之间的状态差值为

${\tilde{ x}_i}(t) \!=\! {{x}_i}(t) \!-\! {{x}_0}(t)$ ${\tilde{ v}_i}{{(t)}} \!=\! {{v}_i}(t) \!-\! {{v}_0}(t)$ . 则 $\tilde{ x} \!=\! (\tilde{ x}_1^{{T}},\tilde{ x}_2^{{T}}, \cdots \! ,$ $\tilde{ x}_N^{{T}})$ $\tilde{ v} = (\tilde{ v}_1^{{T}},\tilde{ v}_2^{{T}}, \cdots \! ,\tilde{ v}_N^{{T}})$ . 令 $F({x},t) = ({f^{{T}}}({{{x}}_1},t),{f^{{T}}}({{x}_2},t), \cdots \! , $

${f^{{T}}}({{{x}}_N},t){)^{{T}}} \in {{\bf{R}}^{N \times n}},$ $F({{x}_0},t) = {({f^{{T}}}({{x}_0},t), \cdots ,{f^{{T}}}({{x}_0},t))^{{T}}} \in $ RN×n.

因此,由式(2)得到误差动力系统如下:

(6)

其中,对于 $\forall t \in ({t_{k - 1}},{t_k}]$ $s(t) = s(k) \in \left\{ {1,2, \cdots \! ,m} \right\}$ $\omega (t) = $ $ \omega (k) \in \left\{ {1,2, \cdots \! ,m} \right\}$ ${{L}_{s(k)}}$ 表示第 $s(k)$ 个拓扑网络的Laplacian矩阵, ${{B}_{w(k)}}$ 表示第 $\omega (k)$ 个拓扑下积极领导者和各节点的连接矩阵;若误差系统(6)在原点处渐近稳定,则可得 $t \to \infty $ 时, $\tilde{ x}(t) = {{0}}$ $\tilde{ v}(t) = 0$ .

2 主要结果

定义:若在定理和假设条件下误差系统(6)的解使 $\mathop {\lim }\limits_{t \to + \infty } \left\| {{{\tilde{ x}}_i}(t)} \right\| = 0$ $\mathop {\lim }\limits_{t \to + \infty } \left\| {{{\tilde{ v}}_i}(t)} \right\| = 0$ ,则称二阶非线性多智能体系统(5)在脉冲控制协议(3, 4)控制下实现领导跟随一致性.

假设1:二阶非线性多智能体系统的网络拓扑图是无向图,领导者在其中可以不是全局可达节点且与智能体的连接伴随拓扑切换而同步切换.

假设2:非线性向量值函数 $f({{x}_i}(t),t)$ 对任意向量 ${x},\tilde{ x} \in {{\bf{R}}^n}$ 存在非负常量 $\phi $ 满足公式: $\left\| {f({x},t) - f(\tilde{ x},t)} \right\| \leqslant $ $ \varphi \left\| {{x} - \tilde{ x}} \right\|$ ,即该函数满足Lipschitz条件.

定理:在假设1、假设2的条件下,若存在常数 $\vartheta > 1$ 使不等式 ${{{e}}^{\varepsilon ({t_k} - {t_{k - 1}})}} \leqslant \displaystyle\frac{1}{{\vartheta {\lambda _k}}}$ 恒成立,则称脉冲动力系统(5)在控制协议(3, 4)作用下实现领导跟随一致性,各智能体状态最终与领导者状态(1)保持一致.

证明 构造Lyapunov函数

$V(t) = \frac{1}{2}{\tilde{ x}^{{T}}}(t)\tilde{ x}(t) + \frac{1}{2}{\tilde{ v}^{{T}}}(t)\tilde{ v}(t),$ (7)

$\forall t \in ({t_{k - 1}},{t_k}]$ $V(t)$ 沿着(5)的解轨迹的导数为

$\begin{aligned}\dot V(t) = & {{\tilde{ x}}^{{T}}}(t)\tilde{ v}(t) + {{\tilde{ v}}^{{T}}}(t)(({{I}_N} \otimes {{I}_n})(F({x},t) - F({{x}_0},t)) - \\ &(a{{L}_{s(k)}} \otimes {{I}_n})\tilde{ v}(t)).\end{aligned}$

由假设2得

$\begin{aligned}\dot V(t) = & - {{\tilde{ v}}^{{T}}}(t)(a{{L}_{s(k)}} \otimes {{I}_n})\tilde{ v}(t) + {{\tilde{ x}}^{{T}}}(t)\tilde{ v}(t) + \\ &{{\tilde{ v}}^{{T}}}(t)({{I}_{Nn}}\varphi )\tilde{ x}(t).\end{aligned}$

$ - {\tilde{ v}^{{T}}}(t)(a{{L}_{s(k)}} \otimes {{I}_n})\tilde{ v}(t) \leqslant 0$ ,得

$\dot V(t) \leqslant {\tilde{ x}^{{T}}}(t)\tilde{ v}(t) + {\tilde{ v}^{{T}}}(t)({{I}_{Nn}}\varphi )\tilde{ x}(t).$

因为

$\begin{aligned}{{\tilde{ x}}^{{T}}}(t)\tilde{ v}(t) = & {{\tilde{ v}}^{{T}}}(t)\tilde{ x}(t) = {{\tilde{ v}}_{11}}(t){{\tilde{ x}}_{11}}(t) + \\ &{{\tilde{ v}}_{12}}(t){{\tilde{ x}}_{12}}(t) + \cdots + {{\tilde{ v}}_{1n}}(t){{\tilde{ x}}_{1n}}(t) + \\ &{{\tilde{ v}}_{21}}(t){{\tilde{ x}}_{21}}(t) + \cdots + {{\tilde{ v}}_{Nn}}(t){{\tilde{ x}}_{Nn}}(t) \leqslant \\ &\frac{{{1}}}{{{2}}}(\tilde{ v}_{11}^2(t) + \tilde{ x}_{11}^2(t) + \cdots + \tilde{ v}_{Nn}^2(t) + \\ &\tilde{ x}_{Nn}^2(t)) = V(t),\end{aligned}$

同理可得 ${\tilde{ v}^{{T}}}(t)({{I}_{Nn}}\varphi )\tilde{ x}(t) \leqslant \varphi V(t).$

综上,

$\dot V(t) \leqslant \varepsilon V(t)\text{且}\varepsilon = 1 + \phi .$ (8)

由引理3在时间区间 $t \in ({t_{k - 1}},{t_k}],k \in {N_ + }$ 上对式(8)积分得

$V(t) \leqslant V(t_{k - 1}^ + ){e^{\varepsilon (t - {t_{k - 1}})}}.$ (9)

在脉冲时刻tk处,可得

$V(t_k^ + ) = \frac{1}{2}{\tilde{ x}^{{T}}}(t_k^ + )\tilde{ x}(t_k^ + ) + \frac{1}{2}{\tilde{ v}^{{T}}}(t_k^ + )\tilde{ v}(t_k^ + ).$

${A} \!=\! {{I}_{Nn}} \!+\! {{B}_{w(k)}} \! \otimes \! ({{B}_k} \!-\! {{I}_n}) \!-\! (a{{L}_{s(k)}} \! \otimes \! {{B}_k})$ , ${E} \!=\! (a{{L}_{s(k)}} \! \otimes \! {B}_k^{{T}})$ $({{I}_{Nn}} \!+\! {{B}_{w(k)}} \! \otimes \! ({{B}_k} \!-\! {{I}_n}) \!-\! (a{{L}_{s(k)}} \! \otimes \! {B}_k^{{T}}))$ , ${H} \!=\! (a{{L}_{s(k)}} \! \otimes \! {{B}_k})$ ,得

$\begin{aligned}V(t_k^ + ) = & \frac{1}{2}{{\tilde{ x}}^{{T}}}(t_k^ - )({{A}^{{T}}}{A})\tilde{ x}(t_k^ - ) + \frac{1}{2}{{\tilde{ v}}^{{T}}}(t_k^ - )({{A}^{{T}}}{A})\tilde {{v}}(t_k^ - ) - \\ &\;\;\;\;\;\;\;\;\frac{1}{2}{{\tilde{ v}}^{{T}}}(t_k^ - )({E} + {{E}^{{T}}})\tilde{ x}(t_k^ - ) + \frac{1}{2}{{\tilde{ x}}^{{T}}}(t_k^ - )({{H}^{{T}}}{H})\tilde{ x}(t_k^ - ).\end{aligned}$

${\lambda _{k1}} \!=\! {\lambda _{\max }}({{A}^{{T}}}{A})$ , ${\lambda _{k2}} \!=\! {\lambda _{\max }}({E} + {{E}^{{T}}})$ , ${\lambda _{k3}} \!=\! {\lambda _{\max }}({{H}^{{T}}}{H})$ ,得

$\begin{aligned}V(t_k^ + ) \leqslant & {\lambda _{k1}}(\frac{1}{2}{{\tilde{ x}}^{{T}}}(t_k^ - )\tilde{ x}(t_k^ - ) + \frac{1}{2}{{\tilde{ v}}^{{T}}}(t_k^ - )\tilde{ v}(t_k^ - )) - \\ & \frac{1}{2}{\lambda _{k2}}{{\tilde{ v}}^{{T}}}(t_k^ - )\tilde{ x}(t_k^ - ) + \frac{1}{2}{\lambda _{k3}}{{\tilde{ x}}^{{T}}}(t_k^ - )\tilde{ x}(t_k^ - ) \leqslant \\ & {\lambda _{k1}}V(t_k^ - ) - \frac{1}{2}{\lambda _{k2}}{{\tilde{ v}}^{{T}}}(t_k^ - )\tilde{ x}(t_k^ - ) + \frac{1}{2}{\lambda _{k3}}{{\tilde{ x}}^{{T}}}(t_k^ - )\tilde{ x}(t_k^ - ).\end{aligned}$

因为

$\begin{aligned}{{\tilde{ x}}^{{T}}}(t_k^ - )\tilde{ v}(t_k^ - ) = & {{\tilde{ v}}^{{T}}}(t_k^ - )\tilde{ x}(t_k^ - ) \leqslant \frac{{{1}}}{{{2}}}(\tilde{ v}_{11}^2(t_k^ - ) + \tilde{ x}_{11}^2(t_k^ - ) + \cdots + \\ &\tilde{ v}_{Nn}^2(t_k^ - ) + \tilde{ x}_{Nn}^2(t_k^ - )) = V(t_k^ - ),\end{aligned}$

综上得

$V(t_k^ + ) \leqslant {\lambda _k}V(t_k^ - ),\text{且}{\lambda _k} = ({\lambda _{k1}} - \frac{1}{2}{\lambda _{k2}} + {\lambda _{k3}}) \in (0,1).$ (10)

下面计算 $t = ({t_k},{t_{k + 1}}]$ $V(t)$ 的表达式. 当 $t \in ({t_0},{t_1}]$ 时,由式(9)得 $V(t) \leqslant V(t_0^ + ){{{e}}^{\varepsilon (t - {t_0})}}$ ,则 $V({t_{{1}}}) \leqslant V(t_0^ + ){{{e}}^{\varepsilon ({t_{{1}}} - {t_0})}}$ . 由于 $V(t)$ 左连续,即 $V(t_1^ - ) = V({t_1})$ .

由式(10)得 $V(t_1^ + ) \leqslant {\lambda _1}V(t_{{1}}^ - ) = {\lambda _1}V({t_1})$ ,即 $V(t_1^ + ) \leqslant$ $ {\lambda _1}V(t_{{0}}^ + ){{{e}}^{\varepsilon ({t_1} - {t_0})}}$ . 当 $t \in ({t_{{1}}},{t_{{2}}}]$ 时,同理有     $V(t) \leqslant $ $ V(t_1^ + ){{{e}}^{\varepsilon (t - {t_1})}} \leqslant{\lambda _1}V(t_0^ + ){{{e}}^{t - {t_0}}}$ .

k=2得

$\begin{aligned}V(t_2^ + ) \leqslant {\lambda _2}V(t_2^ - ) = & {\lambda _2}V({t_2}) \leqslant {\lambda _2}V(t_1^ + ){{{e}}^{\varepsilon ({t_2} - {t_1})}} \leqslant \\ &{\lambda _{{1}}}{\lambda _{{2}}}V(t_0^ + ){{{e}}^{\varepsilon ({t_2} - {t_0})}}.\end{aligned}$

$t \in ({t_{{k}}},{t_{{{k}} + {{1}}}}]$ 时,由数学归纳法得

$\begin{aligned}V(t) \leqslant & {\lambda _{{1}}}{\lambda _{{2}}} \cdots {\lambda _k}V(t_0^ + ){{{e}}^{\varepsilon (t - {t_0})}} = \frac{{{1}}}{{{\vartheta ^k}}}V(t_0^ + ){{{e}}^{\varepsilon (t - {t_k})}} \times \\ & \vartheta {\lambda _k}{{{e}}^{\varepsilon ({t_k} - {t_{k - 1}})}}\vartheta {\lambda _{(k - 1)}}{{{e}}^{\varepsilon ({t_{k - 1}} - {t_{k - 2}})}} \cdots \vartheta {\lambda _1}{{{e}}^{\varepsilon ({t_1} - {t_0})}}.\end{aligned}$

如果满足条件 $\vartheta > 1$ , ${{{e}}^{\varepsilon ({t_k} - {t_{k - 1}})}} \leqslant \frac{1}{{\vartheta {\lambda _k}}}$ , ${\lambda _k} \in (0,1)$ ,那么可得到

$V(t) \leqslant \frac{{{1}}}{{{\vartheta ^{{k}}}}}V(t_0^ + ){{{e}}^{\varepsilon (t - {t_k})}}.$ (11)

因此,当 $t \to \infty $ 时, $V(t) \to 0$ . 此时 $\mathop {\lim }\limits_{t \to + \infty } \left\| {\tilde{ x}(t)} \right\| = 0$ $\mathop {\lim }\limits_{t \to + \infty } \left\| {\tilde{ v}(t)} \right\| = 0$ ,可得误差动力系统式(6)在原点处渐近稳定, 进一步得出式(5)能够实现领导跟随一致性. 证毕.

备注: $\vartheta = {{1}}$ ,可得 $V(t) \leqslant V(t_0^ + ){{{e}}^{\varepsilon (t - {t_k})}}$ ,此时误差动力系统式(6)在原点处稳定,但不是渐近稳定.

推论:在假设1和假设2的条件下,若脉冲增益矩阵 ${{B}_k} = \hat{ B}$ ,脉冲间隔 $\Delta {t_k} = {t_{k + 1}} - {t_k} = \Delta t$ ,即二者在任意时间内固定不变时,存在 $\vartheta > 1$ 使条件 ${{{e}}^{\varepsilon \Delta t}} \leqslant \displaystyle\frac{1}{{\vartheta {\lambda _k}}}$ 成立,则系统式(5)能够实现领导跟随一致性.

3 数值仿真

针对有5个多智能体的二阶非线性多智能体系统,取各智能体状态初值分别为(–15, 5)、(5, –3)、(18, –9)、(10, 7)、(3, 13),取领导者状态初值为(5, 4);令非线性函数 $f({{x}_i}(t),t) = {{x}_i}(t)\sin ({t^2})$ ,则 $\left| {f({{x}_i},t) - f({{\tilde{ x}}_i},t)} \right| \leqslant $ $ \left| {{{x}_i}(t) - {{\tilde{ x}}_i}(t)} \right|$ ;显然,可取 $\phi = {{1}}$ 使函数满足Lipschitz条件.

取耦合强度a=0.1,脉冲间隔 ${t_{k + 1}} - {t_k} = {{0}}{{.05}}$ ,脉冲增益矩阵 ${{B}_k} = {{diag}}(0.7,\, 0.7,\, 0.7,\, 0.7,\, 0.7)$ ,连接矩阵分别为 ${B}_{{1}} = {{diag}}(1,\, 1,\, 0,\, 0,\, 1)$ ${B}_{{2}} = {{diag}}(1,\, {{0}},\, {{1}},\, 0,\, 1)$ ${B}_{{3}} = {{diag}}({{0}},\, 1,\, 0,\, 0,\, {{0}})$ ;当 $t \in [{t_{k - 1}}, {t_k})$ ,定义切换信号 $s(t) = \omega (t) = ((k - 1)od 3) + 1$ ,则拓扑切换顺序可用集合 $\left\{ {{{\mathop G\limits^ - }_1},\,{{\mathop G\limits^ - }_2},\,{{\mathop G\limits^ - }_3},\,{{\mathop G\limits^ - }_1},\,{{\mathop G\limits^ - }_2} \cdots } \right\}$ 表示. 见图1.

图 1 拓扑图切换顺序 $({{\mathop G\limits^ - }_1} \to {{\mathop G\limits^ - }_2} \to {{\mathop G\limits^ - }_3})$ Figure 1 The sequence of topology switching $({{\mathop G\limits^ - }_1} \to {{\mathop G\limits^ - }_2} \to {{\mathop G\limits^ - }_3})$

经计算:当拓扑图为 \normalsize${{\mathop G\limits^ - }_1}$ 时, ${\lambda _{k1}} ={{0}}{{.864\;3}} $ ${\lambda _{k2}} = $ $ {{0}}{{.383}}\;{{2}}$ ${\lambda _{k3}} = {{0}}{{.122}}\;{{5}}$ $\varepsilon = {{2}}$ ${t_{k + 1}} - {t_k} = {{0}}.{{05}}$ $\vartheta \leqslant {{1}}{{.137}}\;{{9}}$ ,满足定理条件;当拓扑图为 ${{\mathop G\limits^ - }_{{2}}}$ 时, ${\lambda _{k1}} = {{0}}{{.865}}\;{{4}}$ ${\lambda _{k2}} = $ $ {{0}}{{.383}}\;{{4}}$ ${\lambda _{k3}} = 0.104\;5$ $\varepsilon = {{2}}$ ${t_{k + 1}} - {t_k} = {{0}}{{.05}}$ $\vartheta \leqslant {{1}}{{.162}}\;{{7}}$ ,满足定理条件;当拓扑图为 ${{\mathop G\limits^ - }_{{3}}}$ 时, ${\lambda _{k1}} = {{0}}{{.959}}\;{{8}}$ ${\lambda _{k2}} = $ $ 0.420\;4$ ${\lambda _{k3}} = 0.104\;5$ $\varepsilon = {{2}}$ ${t_{k + 1}} - {t_k} = {{0}}{{.05}}$ $\vartheta \leqslant {{1}}{{.294}}\;{{0}}$ ,满足定理条件. 最后用Matlab软件仿真,得到图2图3.

图2图3不难看出,随着时间不断变化在非脉冲时刻连续控制协议与脉冲时刻脉冲控制协议的双重作用下系统可以实现领导跟随一致性. 同时,在初始阶段的脉冲时刻各智能体状态变化较为剧烈,而约0.5 s以后脉冲时刻的状态增量较小并逐渐趋于0.

图4图5表明:随着时间的增加,各智能体与领导者的状态误差逐渐趋于并稳定于0,实例证明了系统在脉冲控制协议作用下可以实现领导跟随一致性,即本文协议的正确性得到验证.

图 2 协议(3, 4)控制下跟随智能体与领导者的位移状态曲线 Figure 2 The displacement status curves of followers with leader under the control protocol (3, 4)
图 3 协议(3, 4)控制下跟随智能体与领导者的速度状态曲线 Figure 3 The velocity status curves of followers with leader under the control protocol (3, 4)
图 4 协议(3,4)控制下跟随智能体与领导者的位移状态误差 Figure 4 The displacement status errors of followers with leader under the control protocol (3, 4)
4 结论

本文采用一种新的控制协议对一类含有积极领导者的二阶非线性多智能体系统在网络拓扑切换情形下的领导跟随一致性进行了研究. 利用相关理论,得到系统拓扑图无向且连通时系统实现一致性必须满足的充分条件,并用Matlab软件实例仿真验证了结果的正确性.

图 5 协议(3, 4)控制下跟随智能体与领导者的速度状态误差 Figure 5 The velocity status errors of followers with leader under the control protocol (3, 4)
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