Practical Consensus of Leader-following Multi-agent System With Unknown Coupling Weights
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摘要: 一致性理论在许多领域有广泛应用.现有很多研究成果是关于恒同一致的.在实际中，由于任何系统都会不可避免地受到一定的外界扰动，要求误差函数的极限等于0是难以做到的，但当时间充分大时误差函数在可接受区间内是可行的.本文首先给出多智能体系统的实用一致性概念，然后研究含未知耦合权重的一阶非线性领导-跟随多智能体系统的实用一致性问题.通过设计合适的控制协议，运用图论、矩阵理论和强实用稳定性理论，得到该多智能体系统实现实用一致性的充分条件.数值模拟验证了理论结果的正确性.Abstract: Consensus theory has been widely applied to many fields. Most of the research is on identical consensus. In many practical cases, it is impossible for agents to achieve identical consensus because any system has certain disturbance. In this case, it cannot be achieved that the limit of the error function is equal to 0, but it is feasible that the value of the error function is bounded within an interval if the time is sufficiently large. In this paper, the concept of practical consensus is given, then the problem of practical consensus in leader-following multi-agent systems with unknown coupling weights is investigated. By designing an appropriate control protocol and using graph theory, matrix theory and strong practical stability theory, a sufficient condition is given to realize practical consensus of the multi-agent system. Numerical simulations are given to verify the theoretical results.
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Key words:
- Multi-agent system /
- practical consensus /
- leader-following /
- strong practical stability
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近十几年来, 随着人工智能技术的发展, 多智能体的一致性逐渐引起了数学、物理、控制和通信等多个领域中学者的兴趣[1-9].一致性是指通过设计控制协议, 使得智能体的一些状态变量(例如位置或速度等)渐近趋于恒同.目前已经有许多有关多智能体一致性的研究成果.例如, Zeng等[10]研究了通信延时及输入延时并存的情况下, 一般线性动态特性的高阶多智能体在固定且无向的网络拓扑下的一致性; Guan等[11]研究了混合控制下二阶多智能体系统的一致性问题; 杨洪勇等[12]研究了离散时间分数阶多智能体系统的一致性问题; Li等[13]考虑了非线性多智能体系统存在节点故障的问题, 并给出了一致性恢复方法.此外, 领导-跟随多智能体系统一致性问题也获得了一些研究成果[14-17]. Wen等[14]研究了二阶非线性领导-跟随多智能体模型分别在无向和有向网络拓扑下的一致性; Ma等[15]研究了一阶非线性系统的聚类-时延一致性; Peng等[16]研究了具有时变时延的领导-跟随多智能体系统的一致性问题; Wang等[17]研究了二阶非线性系统的时延一致性问题.
以上研究考虑的一致性是所有智能体的状态变量渐近趋于恒同(下文称为恒同一致性, 即).通常人们期望多智能体系统的状态是趋于恒同一致的, 然而在实际应用中要求系统中每个跟随智能体的真实运动状态与领导智能体的运动状态之间的偏差趋于0往往是不可能的.近年来实用一致性的一些研究工作说明了这一点[18-19].实用一致性是指偏差函数最终保持在一个确定的集合中, 而不是趋于0. Dong等运用Lyapunov-Krasovskii方法, 研究了在没有外部扰动时系统达成恒同一致()所需的条件, 并探讨了在有外界扰动时系统达成实用一致()所需的条件, 其中容许误差上界与外部扰动的范数有关[18]. Li等考虑了系统 , 导出每个智能体的运动状态收敛于区间$[ ( k-1/2 )\mu \Delta $, $( k$ $+$ $1/2 )\mu \Delta ]$的充分条件, 其中或$ \lfloor {{{\eta ( 0 )} / A}} \rfloor + 1$ [19].
本文研究一阶非线性领导-跟随多智能体模型的实用一致性问题, 主要内容包括:
1) 给出$\lambda $ -实用一致的概念, $\lambda$为容许误差上界; 运用强实用稳定性理论, 设计新的控制协议, 讨论系统满足何种条件时, 跟随智能体与领导智能体之间的偏差函数在充分大的时间之后能够一直保持在容许误差状态集$Q =\{ {\pmb e}\in {\mathbf{R}^n}|$ $\| {\pmb e} \|$ $\le$ $\lambda \}$之内.这蕴含偏差函数极限不存在的情况(如偏差函数为正弦函数, 虽然有界但极限不存在).
2) 在假设真实系统和理想系统的耦合权重误差为有界的条件下, 用理想耦合权重和真实耦合权重之间误差的上界来描述真实系统的耦合权重需满足的条件.
本文内容安排如下:第1节简述强实用稳定性的定义和相关引理; 第2节先给出模型描述, 再给出实用一致性的定义, 并通过设计合适的控制协议, 运用图论、矩阵理论以及稳定性理论, 得到系统实现实用一致的充分条件; 第3节通过数值模拟验证理论结果的正确性; 第4节给出结论.
1. 强实用稳定性简述
符号说明: $\forall $表示任意, $\exists $表示存在, ${Q^c}$表示$Q$的补集, $\partial Q$表示$Q$的边界.考虑系统
$ \begin{align} \dot {\pmb x} = {\pmb f}\left( {t, {\pmb x}} \right) \end{align} $
(1) 其中, ${\pmb f} \in C[ {I \times D, {\mathbf{R}^n}} ]$, $D$是中包含${\pmb x} = 0$的区域, ${\pmb f}( {t, 0} )$ $\equiv$ $0$.
定义 1.设${Q_0}$和$Q$是包含${\pmb x} = 0$的有界闭集, 其中${Q_0}$是$Q$的子集.若$\forall {{\pmb x}_0} \in {Q_0}$, 系统(1)的解${\pmb x}( {t;{t_0}, {{\pmb x}_0}} )$满足, $t \ge {t_0}$, 则称系统(1)的零解是关于${Q_0}$, $Q$实用稳定的.
定义 2.若系统(1)的零解是关于${Q_0}$, $Q$实用稳定的; 且对$\forall {{\pmb x}_0} \in {\mathbf{R}^n}$, , 使得当$t>T( {{t_0}, {{\pmb x}_0}} )$时, 系统(1)的所有解${\pmb x}( {t;{t_0}, {{\pmb x}_0}} ) \in Q$, 则称系统(1)的零解是关于${Q_0}$, $Q$强实用稳定的.
实用稳定性和Lyapunov意义下的稳定性互不包含.强实用稳定的容许误差状态集$Q$确定了真实状态与理想状态偏差的大小.
类似于文献[20], 可以得到以下定理:
定理1.若存在函数, 满足条件:
H1. $t \ge {t_0}$时, 对, 有;
H2.对$\forall {{\pmb x}_1} \in {Q_0}$, , $\forall {t_2} \ge {t_1} \ge {t_0}$, 有$V( {t_1}$, ${{\pmb x}_1} ) $ $ <$ $V( {{t_2}, {{\pmb x}_2}} )$.
则系统(1)的零解是关于${Q_0}$, $Q$强实用稳定的.
证明.假设$\exists {{\pmb x}_0} \in {Q_0}$, 系统(1)的解${\pmb x}( {t, {t_0}, {{\pmb x}_0}} )$在$t=$ ${t_0}$时在${Q_0}$内, 在某时刻$T > {t_0}$有, 则$\exists {t_1}$, ${t_0}$ $ <$ ${t_1} <T$, 使得
$ \{ {{\pmb x}( {t, {t_0}, {{\pmb x}_0}} )| {{t_1} < t \le T} } \} \subset Q^{c}, ~~~ {\pmb x}( {{t_1}, {t_0}, {{\pmb x}_0}} ) \in {Q_0} $
故根据条件H2有
$ V( {{t_1}, {\pmb x}( {{t_1}, {t_0}, {{\pmb x}_0}} )} ) < V( {T, {\pmb x}( {T, {t_0}, {{\pmb x}_0}} )} ) $
矛盾, 因为由条件H1可知, 是关于$t$的单调减函数, 故系统(1)的零解是关于${Q_0}$, $Q$实用稳定的.
假设$\exists {{\pmb x}^c} \in Q_0^c$, $\exists {t_n} \to + \infty $ $( {n \to \infty } )$, 有由条件H1知
$ {{\frac{{\rm d}V}{{\rm d}t}} |_{(1)}} \le - \delta <0 $
故$t \to + \infty $时, 有
$ V( {{t_n}, {\pmb x}( {{t_n}, {t_0}, {{\pmb x}^c}} )} ) \le V( {{t_0}, {{\pmb x}^c}} ) - \delta ( {t - {t_0}} ) \to - \infty $
另一方面, $t \to + \infty $时, 取${{\pmb x}_0} \in {Q_0}$, 由条件H2可知
$ {\rm const} = V( {{t_0}, {{\pmb x}_0}} ) < V( {{t_n}, {\pmb x}( {{t_n}, {t_0}, {{\pmb x}^c}} )} ) $
矛盾, 所以对, $\exists T( {{t_0}, {{\pmb x}^c}} )$使得系统(1)的解, ${t_0}$, ${{\pmb x}^c} )$满足
$ \{ {{\pmb x}( {t, {t_0}, {{\pmb x}^c}} )| {t \ge {t_0} + T} } \} \subseteq Q $
综上, 系统(1)的零解是关于${Q_0}$, $Q$强实用稳定的.
2. 主要理论结果
考虑一阶多智能体的理想系统
$ \begin{align} {\dot {\pmb x}_i}\left( t \right) = {\pmb f}\left( {{{\pmb x}_i}\left( t \right)} \right) + \sum\limits_{j = 1}^N {{a_{ij}}{{\pmb x}_j}\left( t \right) + {{\pmb u}_i}} \end{align} $
(2) 其中, $i = 1, 2, \cdots, N$, 表示第$i$个智能体的状态, 是一个连续映射, 表征第$i$个智能体的自身动力学, 当第$i$个智能体可以接受第$j$个智能体的信息时, ${a_{ij}} $ $>$ $0$, 否则${a_{ij}} = 0$; ${a_{ii}} = - \sum_{j \ne i}^{} {{a_{ij}}} $, ${{\pmb u}_i} \in {{\bf R}^n}$表示第$i$个智能体的控制输入.
由于网络拓扑中的耦合权重可能存在受到扰动的问题, 故真实系统(3)与理想系统(2)可能存在一定差异, 真实系统(3)为
$ \begin{align} {\dot {\pmb y}_i}\left( t \right) = {\pmb f}\left( {{{\pmb y}_i}\left( t \right)} \right) + \sum\limits_{j = 1}^N {{{\tilde a}_{ij}}\left( t \right){{\pmb y}_j}\left( t \right) + {{\pmb u}_i}}, ~~i = 1, 2, \cdots, N \end{align} $
(3) 其中, , 记$L = ( {{a_{ij}}} )$, , $W$ $ =$ $({{L + {L^{\rm T}}}})/{2}$, .
设领导智能体动力学方程
$ \begin{align} {\dot {\pmb x}_0}\left( t \right) = {\pmb f}\left( {{{\pmb x}_0}\left( t \right)} \right) \end{align} $
(4) 下面给出实用一致性的定义.
定义3. 记状态偏差变量, ${\pmb e}( t )$ .若$\exists \lambda > 0$且 $0$, 当时系统(3)和(4)的解满足$\| {{\pmb e}_i}( t, {t_0}$, ${\pmb e}( {{t_0}} ) ) \|$ $\le$ $\lambda $, 则称系统(3)和(4)达到$\lambda $ -实用一致, 为容许误差上界.
接下来研究真实耦合权重与理想耦合权重${a_{ij}}$之间的误差不超过某个上界$\mu $时$( {i \ne j} )$, 多智能体系统达成实用一致性的问题.
由系统(3)和(4)得
$ \begin{align} {\dot {\pmb e}_i}\left( t \right) = {\pmb f}\left( {{{\pmb y}_i}\left( t \right)} \right) - {\pmb f}\left( {{{\pmb x}_0}\left( t \right)} \right) + \sum\limits_{j = 1}^N {{{\tilde a}_{ij}}\left( t \right){{\pmb e}_j}\left( t \right) + {{\pmb u}_i}} \end{align} $
(5) 若${\dot {\pmb e}_i}\left( t \right) = 0$时${{\pmb u}_i} = 0$, 则${{\pmb e}_i}\left( t \right) \equiv 0$是系统(5)的平衡点, 那么系统(3)和(4)的 -实用一致问题转化为系统(5)的强实用稳定问题.
定义4. QUAD条件[21-22].若, 对$\forall {\pmb x} \in {{\bf R}^n}$, ${\pmb y}$ $\in$ , 使得函数${\pmb f}$满足
$ \begin{align} {\left( {{\pmb x} - {\pmb y}} \right)^{\rm T}}\left( {{\pmb f}\left( {{\pmb x}, t} \right) - {\pmb f}\left( {{\pmb y}, t} \right)} \right) \le \varepsilon {\left( {{\pmb x} - {\pmb y}} \right)^{\rm T}}\left( {{\pmb x} - {\pmb y}} \right) \end{align} $
(6) 则称${\pmb f}$满足QUAD条件.
注1.相较于Lipschitz条件, QUAD条件是弱化的条件.
定理2.如果矩阵是负定矩阵, 那么对$\forall t$, 也是负定矩阵, 其中$B=$ ${{\rm diag}}\{ {b_1}$, ${b_2}, {b_3} , \cdots, {b_N} \}$.
证明.记, $( {{{\tilde c}_{ij}}( t )} ) $ $=$ $\varepsilon {I_N} + \tilde W( t ) - B$.那么, ${\tilde c_{ii}}( t )$ , , ${\tilde c_{ij}}( t ) =({{\tilde a}_{ij}}( t )$ $+$ ${{\tilde a}_{ji}}( t ))/{2} $, $i \ne j$.
根据Gershgorin圆盘定理[23], 第$i$个行Gershgorin圆圆心为$( {{c_{ii}}{\rm{, }}0} )$, 半径为$\sum_{j = 1, i \ne j}^N {| {{c_{ij}}} |}$, 由于是负定矩阵, 所有Gershgorin圆包含的区域都在负半平面, 那么且${c_{ii}} + \sum_{j = 1, i \ne j}^N {| {{c_{ij}}} |} < 0$.那么, $i \ne j$时
$ \begin{align*} {\tilde c_{ii}}\left( t \right) - {c_{ii}} = {\tilde a_{ii}}\left(t \right) - {a_{ii}} - 2\left( {N - 1} \right)\mu= \end{align*} $
$ \begin{align*} \qquad \sum\limits_{j = 1, i \ne j}^N {\left( {{a_{ij}} - {{\tilde a}_{ij}}\left( t \right)} \right)} - 2\left( {N - 1} \right)\mu \le - \left( {N - 1} \right)\mu \\[2mm] \left| {{{\tilde c}_{ij}}\left( t \right)} \right| - \left| {{c_{ij}}} \right| \le \left| {\left| {{{\tilde c}_{ij}}\left( t \right)} \right| - \left| {{c_{ij}}} \right|} \right|\le \\ \qquad \left| {\frac{{{{\tilde a}_{ij}}\left( t \right) + {{\tilde a}_{ji}}\left( t \right)}}{2} - \frac{{{a_{ij}} + {a_{ji}}}}{2}} \right|\le \\ \qquad \left| {\frac{{{{\tilde a}_{ij}}\left( t \right) - {a_{ij}}}}{2}} \right| + \left| {\frac{{{{\tilde a}_{ji}}\left( t \right) - {a_{ji}}}}{2}} \right| \le \mu \end{align*} $
则
$ {\tilde c_{ii}}\left( t \right) + \sum\limits_{j = 1, i \ne j}^N {\left| {{{\tilde c}_{ij}}\left( t \right)} \right|} \le {c_{ii}} + \sum\limits_{j = 1, i \ne j}^N {\left| {{c_{ij}}} \right|} < 0 $
显然${\tilde c_{ii}}( t ) <0$. 的Gershgorin圆圆心为$( {{\tilde c}_{ii}}( t )$, $0 )$, 半径为$\sum_{j = 1, i \ne j}^N {| {{{\tilde c}_{ij}}( t )} |} $.
由${\tilde c_{ii}}( t ) + $ $ <$ $0$可知, 所有Gershgorin圆包含的区域都在负半轴, 即负定.
定理3.若系统(2), (3)和(4)中的智能体的自身动力学满足QUAD条件, 是负定矩阵, ${\tilde a_{ij}}( t )$是关于$t$连续变化的, 令控制器
$ \begin{align*} {{\pmb u}_i} = \begin{cases} - {b_i}\left( {{{\pmb y}_i}\left( t \right) - {{\pmb x}_0}\left( t \right)} \right), &\left\| {\pmb e} \right\| \ge \lambda \\ - {b_i}\left( {{{\pmb y}_i}\left( t \right) - {{\pmb x}_0}\left( t \right)} \right)\exp \left( {k\left\| {\pmb e} \right\| - k\lambda } \right), &\left\| {\pmb e} \right\| < \lambda \end{cases} \end{align*} $
称正常数$k$为控制衰减系数, 则系统(3)和(4)达到$\lambda$ -实用一致.
证明.显然${{\pmb u}_i}$连续.令.取${Q_0}=Q=$ $\{{\pmb e}$ , 对于 $ \ge$ $\lambda \}$, 有
$ \begin{array}{l} {\left. {\frac{{{\rm d}V\left( {t, {\pmb e}} \right)}}{{\rm d}t}} \right|_{(5)}} = \frac{1}{2}\sum\limits_{i = 1}^N {\dot {\pmb e}_i^{\rm T}\left( t \right){{\pmb e}_i}\left( t \right)} + \frac{1}{2}\sum\limits_{i = 1}^N {{\pmb e}_i^{\rm T}\left( t \right){{\dot {\pmb e}}_i}\left( t \right)}= \\ \quad \quad \sum\limits_{i = 1}^N {\pmb e}_i^{\rm T}\left( t \right)\Bigg[ {\pmb f}\left( {{\pmb y}_i}\left( t \right)\right) - {\pmb f}\left( {{\pmb x}_0\left( t \right)} \right)+ \\ \frac{1}{2}\sum\limits_{j=1}^{N}{{}}\left( {{{\tilde{a}}}_{ij}}\left( t \right)+{{{\tilde{a}}}_{ji}}\left( t \right) \right){{\pmb{e}}_{j}}\left( t \right)-{{b}_{i}}{{\pmb{e}}_{i}}\left( t \right)]\le \\ \quad \quad \sum\limits_{i = 1}^N {\varepsilon {\pmb e}_i^{\rm T}\left( t \right){{\pmb e}_i}\left( t \right)} + \sum\limits_{i = 1}^N {{\pmb e}_i^{\rm T}\left( t \right)} \Bigg[ \frac{1}{2}\sum\limits_{j = 1}^N ( {\tilde a}_{ij}\left( t \right)+\\ \quad \quad {\tilde a}_{ji}\left( t \right) ){{\pmb e}_j}\left( t \right) - {b_i}{{\pmb e}_i}\left( t \right) \Bigg]=\\ \quad \quad \sum\limits_{i = 1}^N {\varepsilon {\pmb e}_i^{\rm T}\left( t \right){{\pmb e}_i}\left( t \right)} + \sum\limits_{i = 1}^N {{\pmb e}_i^{\rm T}\left( t \right)} \frac{1}{2}\sum\limits_{j = 1}^N ( {{\tilde a}_{ij}}\left( t \right)+ \\ \quad \quad {{\tilde a}_{ji}}\left( t \right) ){{\pmb e}_j}\left( t \right) - \sum\limits_{i = 1}^N {{\pmb e}_i^{\rm T}\left( t \right)} {b_i}{{\pmb e}_i}\left( t \right)=\\ \quad \quad {{\pmb e}^{\rm T}}\left( t \right)\left( {\varepsilon {I_{Nn}} + \tilde W \otimes {I_n} - B \otimes {I_n}} \right){\pmb e}\left( t \right) \end{array} $
由定理2可知, 当负定时, $\varepsilon {I_N} + \tilde W( t ) - B$负定.而矩阵的特征值等于矩阵的特征值乘以矩阵${I_n}$的特征值.因此 $\otimes$ ${I_n}-B \otimes {I_n}$是负定的, 从而在时负定.另一方面, 记$m$为的最大特征值, 显然仍为负定矩阵.那么对于, 有
$ \begin{align*} &{{\pmb e}^{\rm T}}\left( t \right)\left( {\varepsilon {I_{Nn}} + \tilde W \otimes {I_n} - B \otimes {I_n}} \right){\pmb e}\left( t \right) <\\ &\qquad \frac{m}{2}{{\pmb e}^{\rm T}}\left( t \right){\pmb e}\left( t \right)=\frac{m}{2}{\left\| {\pmb e} \right\|^2} \end{align*} $
则存在正数$\delta = - \frac{m}{2}{\lambda ^2}$, 使得, 满足条件H1.
对于$\forall {{\pmb e}_1} \in {Q_0}$, , 显然$\| {{{\pmb e}_2}} \| > \| {{{\pmb e}_1}} \|$, 又因, ${\pmb e} )$ $=$ $\frac{1}{2}{\| {\pmb e} \|^2}$, 那么对, 有, 满足条件H2.
综上, 由定理1可知, 系统(5)的零解是关于${Q_0}$, $Q$强实用稳定的.故$\exists T( {{t_0}, {{\pmb e(t_0)}}} ) > 0$, 当时, , 显然, 则根据定义3可知系统(3)和(4)达到$\lambda$ -实用一致.
注2. 由是负定矩阵可以得到$\varepsilon {I_N} + W - B$也是负定矩阵, 那么在定理3条件满足的情况下, 用类似的方法可以得到理想系统和领导智能体系统也可以达到 -实用一致.
注3.系统采用控制器${{\pmb u}_i}$意味着越小时系统越不需要控制.控制衰减系数$k$越大表明减小时控制强度的减弱速度越快.
3. 数值模拟
本节通过数值模拟表明第2节中理论结果的正确性.
考虑多智能体理想系统(2)及真实系统(3)中$N = 4$, $n$ $=$ $3$的情况, 令多智能体自身的动力学满足Chua电路方程.
Chua电路[24]可以表示为
$ \begin{align*} &\begin{cases} {{\dot x}_i} = \alpha \left( {{y_i} - { x_i} - \gamma \left( {{ x_i}} \right)} \right)\\ {{\dot y}_i} = {x_i} - {y_i} + {z_i}\\ {{\dot z}_i} = - \beta {y_i} \end{cases}, &\qquad i = 0, 1, \cdots , 4\end{align*} $
其中, , 取$\alpha$ $ =$ $10$, $\beta = 15$, ${m_0} = - 1.25$, ${m_1} = - 0.75$.由于Chua电路全局Lipschitz是连续的, 经计算可知, $\varepsilon = 10$时, ${\pmb f}$满足QUAD条件.
那么式(2)和式(3)分别满足
$ \begin{align*} &\begin{cases} {{\dot x}_i} = \alpha \left( {{y_i} - {x_i} - \gamma \left( {{x_i}} \right)} \right) + \sum\limits_{j = 1}^4 {{a_{ij}}{x_j} + {u_{ix}}} \\ {{\dot y}_i} = {x_i} - {y_i} + {z_i} + \sum\limits_{j = 1}^4 {{a_{ij}}{y_j} + {u_{iy}}} \\ {{\dot z}_i} = - \beta {y_i} + \sum\limits_{j = 1}^4 {{a_{ij}}{z_j} + {u_{iz}}} \end{cases}, \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad {i = 1, \cdots , 4} \end{align*} $
和
$ \begin{align*} &\begin{cases} {{\dot x}_i} = \alpha \left( {{y_i} - {x_i} - \gamma \left( {{x_i}} \right)} \right) + \sum\limits_{j = 1}^4 {{{\tilde a}_{ij}}{x_j} + {u_{ix}}} \\ {{\dot y}_i} = {x_i} - {y_i} + {z_i} + \sum\limits_{j = 1}^4 {{{\tilde a}_{ij}}{y_j} + {u_{iy}}} \\ {{\dot z}_i} = - \beta {y_i} + \sum\limits_{j = 1}^4 {{{\tilde a}_{ij}}{z_j} + {u_{iz}}} \end{cases}, \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad {i = 1, \cdots , 4} \end{align*} $
其中, ${u_{ix}}$, ${u_{iy}}$, ${u_{iz}}$是${{\pmb u}_i}$的分量.
考虑多智能体的拓扑结构, 如图 1所示.
Laplace矩阵$L = \left( {{a_{ij}}} \right)$为
$ \left[ {\begin{array}{*{20}{c}} { - 4}&4&0&0\\ 2&{ - 2}&0&0\\ 1&0&{ - 4}&3\\ 0&0&{17}&{ - 17} \end{array}} \right] $
令误差上界为$\mu = 1$, 取为
$ \left[ {\begin{array}{*{20}{c}} { - 3.1}&{3.1}&0&0\\ {1.2}&{ - 1.2}&0&0\\ {0.2}&0&{ - 2.4}&{2.2}\\ 0&0&{17.5}&{ - 17.5} \end{array}} \right] $
取, 也就是说第4个智能体不可以接收到领导的信息, 满足定理3的条件, 从而使得理想系统和真实系统分别与领导智能体达到 -实用一致.
取初始条件${{\pmb \xi} _i} = {( {{x_i}, {y_i}, {z_i}} )^{\rm T}}$, 其中${{\pmb \xi} _0}$ $=$ , ${{\pmb \xi} _1} = {( {1.1, 2, 3} )^{\rm T}}$, , ${{\pmb \xi} _3}=( 25$, $30$, , ${{\pmb \xi} _4} = {( {50, 100, 20} )^{\rm T}}$,
$ {{\pmb u}_1} = \begin{cases} - 16{{\pmb e}_1}, &\left\| {\pmb e} \right\| \ge \lambda \\ - 16{{\pmb e}_1}\exp \left( {4\left\| {\pmb e} \right\| - 4\lambda } \right), & \left\| {\pmb e} \right\| < \lambda \end{cases} \\ {{\pmb u}_2} = \begin{cases} - 19{{\pmb e}_1}, & \left\| {\pmb e} \right\| \ge \lambda \\ - 19{{\pmb e}_1}\exp \left( {4\left\| {\pmb e} \right\| - 4\lambda } \right), &\left\| {\pmb e} \right\| < \lambda \end{cases} \\ {{\pmb u}_3} = \begin{cases} - 47{{\pmb e}_1}, &\left\| {\pmb e} \right\| \ge \lambda \\ - 47{{\pmb e}_1}\exp \left( {4\left\| {\pmb e} \right\| - 4\lambda } \right), &\left\| {\pmb e} \right\| < \lambda \end{cases} $
${{\pmb u}_4} = 0$, 容许误差上界$\lambda = 2$.通过数值计算, 得到理想系统随时间变化的偏差轨迹(图 2)和真实系统随时间变化的偏差轨迹(图 3).图 2和图 3中的实线是容许误差上界, 表明理想系统和真实系统与领导智能体的偏差在时间充分大时都可以保证小于容许误差上界.
4. 结论
本文首先给出多智能体系统的实用一致性概念, 针对当耦合权重受到扰动时的一阶非线性领导-跟随多智能体系统, 通过设计合适的控制器, 建立相应的偏差系统, 将多智能体系统的实用一致性转化为偏差系统的强实用稳定性, 并运用矩阵理论和稳定性理论, 得到该多智能体真实系统实现实用一致性的充分条件.运用数值模拟, 对理想系统和真实系统的偏差函数进行了对比.然而现实中的一些现象是难以通过一阶多智能体系统模型来解释的, 所以我们下一步计划讨论二阶多智能体系统的实用一致性问题.
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