Adaptive RBF Neural Network Cooperative Control for High-order Nonlinear Multi-agent Systems With Uncertainties
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摘要:
针对外界环境的干扰及自身系统参数的不确定性对一类高阶非线性多智能体系统的影响，研究在领导跟随者网络模型下系统一致性的问题.该动力学系统中含有高阶积分器耦合未知非线性动力学和未知外部干扰，采用分布式自适应径向基函数（radial basis function，RBF）神经网络控制算法，确保神经网络对智能体非线性项进行在线逼近，滑模控制消除持续有界扰动等不确定项对稳定性的影响.首先设计出神经网络权值的自适应律，提出一种基于神经网络的自适应滑模控制协议，利用李雅普诺夫稳定性理论，证明该多智能体系统实现领导跟随一致性，并且最终有界跟踪误差的充分条件.在同质和异质多智能体2种条件下，仿真结果验证了提出方法的正确性.
Abstract:The leader-following consensus control problem is considered for a class of high-order nonlinear multi-agent systems with external disturbances and uncertain system parameters. The dynamics of systems with high-order integrator coupling unknown nonlinear dynamics and unknown external disturbance, adopts the distributed adaptive radial basis function(RBF)neural network control algorithm, to ensure that the neural network is employed to approximate the unknown nonlinear system functions on line, and eliminate persistent bounded disturbances such as uncertainties affecting stability. First of all, weights of neural network adaptive tuning law was designed, then a kind of adaptive sliding mode control protocol based on RBF neural network was proposed. Using Lyapunov stability theory, the sufficient condition of high-order nonlinear uncertain multi-agent system had leader-following to achieving consensus, and ultimately bounded residual errors was discussed. The results of umerical simulations of homogeneous and heterogeneous multi-agent systems are given to demonstrate the effectiveness of the proposed control methodology.
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领导者跟随一致性问题是指系统中有一个(或几个)智能体充当领导者，而其余个体都为跟随者智能体，其中领导者具有期望的状态轨线且其动态行为不受其他节点信息的影响.控制目标是为跟随者智能体设计基于邻居信息的分布式协议，使得所有智能体都能够渐近跟踪领导者的动态行为.领导者跟随结构是在许多生物系统中均存在的节能机制，并且其能够加强群体的通信和导向[1].所以，有领导者的多智能体系统也受到了越来越多学者的关注.
文献[1-2]针对一类满足Lipschitz条件的高阶非线性多智能体系统，设计基于观测器的鲁棒控制，解决多智能体系统的状态合围控制问题.文献[3]研究有向拓扑结构下离散时间多智能体系统的协调预见跟踪问题.文献[4]在具有输入时滞条件下，设计了一类滑模观测器，能够在有限时间内为跟随者发送领导者的信息.文献[5]研究了一类含有外部干扰和未建模动态的高阶非线性多智能体系统的分布式模糊自适应控制问题.文献[6]对领导者跟随混合阶多智能体系统的一致性问题进行了更为详尽的研究.文献[7]针对线性系统，提出了领导者跟随网络中异质多智能体系统达到输出一致的解决方案，当状态信息不易直接测量时，设计了一种基于状态观测器的动态调节器来重构状态.文献[8]针对二阶非线性系统，采用滑模控制算法，提出了一个有领导者的多智能体网络在有限时间之内达到状态一致性.文献[9]针对二阶非线性系统，利用神经网络的万能逼近功能对未知函数进行在线估计和逼近，获得一个基于神经网络的自适应协议.
本文研究了高阶非线性系统的协同跟踪控制问题.研究基于以下2个方面：
第一，现有的多智能体系统研究多集中于一阶和二阶系统，然而，在实际工程中，单连杆柔性关节机械手、机器人编队合作和同步发电机协调工作等是以高阶动力学建模的.以协调作战的无人机组为例，迫于环境变化突然改变运动方向，此时不但要求位置和速度一致，而且要求加速度协调统一.因此，研究此类高阶非线性系统不仅具有极为重要的理论价值，而且也有较强的工程实用价值[10].
第二，由于外界环境的干扰及自身系统参数的不确定性，导致研究对象往往具有未知且复杂的非线性动态，从而很难获得控制对象的精确数学模型，这种非线性动态的存在，系统可能由同构向异构转变，为非线性系统的协同控制带来困难[11].
针对上述问题，本文针对Brunovsky型高阶非线性多智能体协同跟踪控制进行了研究.该动力学系统的特点是每个追随者节点都通过一个高阶积分器耦合未知非线性动力学和未知外部干扰，每个节点动态可以完全不同[12].领导者节点是一个高阶非自治非线性系统，所有跟随节点对其动态都是未知的.本文提出分布式自适应径向基函数(radial basis function，RBF)神经网络控制算法，确保神经网络对非线性项进行在线逼近，以及消除持续有界扰动等不确定项对稳定性的影响，并设计出神经网络权值的自适应律，从而最终获得一个基于自适应RBF神经网络的自适应协议，该方法能够解决高阶非线性不确定多智能体系统的有领导者一致性追踪问题，保证最终有界跟踪误差.
本文给出了一种在弱连通的条件下，有领导者的高阶非线性不确定多智能体网络的一致性协议，并给出了相应的理论论证，通过例子和数值仿真都验证了本文提出方法的正确性和有效性.
1. 预备知识和问题描述
1.1 图论基础和记号说明
为了描述由多个智能体组成的通信网络, 基于图论原理, 采用图G=(V, E, A)来表示智能体之间的信息交互关系，其中，顶点V={v1, v2, …, vN}是N个智能体的集合；E∈V×V表示智能体之间信息交互的有向边集；A=[aij]N×N是与信息拓扑相关联的权值矩阵；aij表示边(vj, vi)的权值，对于有向图而言，如果(vj, vi)∈E, 则aij＞0，否则，aij=0；相应的入度矩阵和图的拉普拉斯矩阵分别定义为D=diag{deg1, deg2, …, degN}和L=D-A，式中${\deg _i} = \sum\limits_{j = 1}^N {\;{a_{ij}}} $是顶点i的入度.
本文将采用如下记号：${{\mathbb{R}}^{n}}$表示n维向量实数集合；1n表示元素为1的n维列向量；‖·‖表示向量的欧式范数；In表示为n×n维的单位矩阵；diag{m1, m2, …, mn}表示对角元素为m1，m2，…，mn的对角矩阵；blkdiag{W1，W2，…，Wn}表示对角元素为矩阵W1，W2，…，Wn的分块对角矩阵；σ(P)表示矩阵P的最小奇异值；σ(P)表示矩阵P的最大奇异值；tr{·}表示矩阵的迹.
1.2 问题描述
考虑由N+1个智能体组成的系统，其中第i个智能体具有如下的Brunovsky非线性动态模型：
$$ \left\{ {\begin{array}{*{20}{l}} {{{\dot x}_{i,m}}(t) = {x_{i,m + 1}}(t),m = 1,2, \cdots ,M - 1}\\ {{{\dot x}_{i,m}}(t) = {f_i}(t,{x_i}) + {d_i}(t) + {u_i}(t),m = M} \end{array}} \right. $$ (1) 式中：i=1, 2, …, N；xi, m(t)∈$\mathbb{R}$表示第i个智能体在t时刻的第m阶状态；xi(t)=[xi, 1(t), xi, 2(t), …, xi, M(t)]T∈${{\mathbb{R}}^{M}}$表示第i个智能体状态向量；di(t)∈$\mathbb{R}$表示第i个智能体的不确定项(包括所受到的外界扰动或者未建模动态)；ui(t)∈$\mathbb{R}$表示在t时刻对第i个智能体的控制输入变量；fi(t, xi)是一个连续的函数，表示第i个智能体的固有非线性动力学行为.
系统领导者0的动力学行为可描述为
$$ \left\{ {\begin{array}{*{20}{l}} {{{\dot x}_{0,m}}(t) = {x_{0,m + 1}}(t),m = 1,2, \cdots ,M - 1}\\ {{{\dot x}_{0,m}}(t) = {f_0}(t,{x_0}),m = M} \end{array}} \right. $$ (2) 式中：x0, m(t)∈$\mathbb{R}$表示领导者0在t时刻的第m阶状态；x0(t)=[x0, 1(t), x0, 2(t), …, x0, M(t)]T∈${{\mathbb{R}}^{M}}$表示领导者0状态向量；f0(t, xi)是一个连续的函数，表示领导者0的固有非线性动力学行为.
智能体i第m阶一致性误差为δi, m=xi, m-x0, m.令δm=[δ1, m δ2, m … δN, m]T，进而得到δm=xm-1x0, m，式中：xm=[x1, m x2, m … xN, m]T∈${{\mathbb{R}}^{M}}$，m=1, 2, …, M.本文设计分布式控制器的目标是一致性误差逼近为零，提高跟随者跟踪性能[13].
定义1(协同一致最终有界)[14] 对于任意m(m=1, 2, …, M)，如果存在紧集Ωm⊂${{\mathbb{R}}^{N}}$满足以下3个条件，则称具有领导者的高阶多智能体跟随误差δm是协同一致最终有界(cooperatively uniformly ultimately bounded，CUUB)的.
1) {0}⊂Ωm；
2) ∀δm(t0)⊂Ωm；
3) 存在上确界Δm和时间Tm，在∀t≥t0+Tm时，有‖δm‖≤Δm.
对于智能体i，在t≥t0+Tm时，如果跟随误差是协同一致最终有界的，则跟随者状态xi, m(t)收敛到领导者x0, m(t)的邻域内.
第i个智能体局部邻域误差定义为
$$ \begin{array}{l} {e_{i,m}} = \sum\limits_{j \in {N_i}} {{a_{ij}}} ({x_{j,m}} - {x_{i,m}}) + {b_i}({x_{0,m}} - {x_{i,m}}) = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{j = 1}^N {{L_{ij}}} {x_{j,m}} + {b_i}({x_{0,m}} - {x_{i,m}}) \end{array} $$ (3) 式中：m=1, 2, …, M.如果跟随智能体i与领导智能体0之间有一个有向边(v0, vi)，跟随智能体可以“感知”领导智能体的信息，那么这条边的权值为bi＞0，否则bi=0，定义邻接矩阵B=diag{b1, b2, …, bN}, 第m阶全局邻域误差为em(t)=[e1, m, e2, m, …, eN, m]T∈${{\mathbb{R}}^{N}}$，令f(t, x)=[f1(t, x1), f2(t, x2), …, fN(t, xN)]T∈${{\mathbb{R}}^{N}}$, u(t)=[u1(t), u2(t), …, uN(t)]T∈${{\mathbb{R}}^{N}}$，d(t)=[d1(t), d2(t), …, dN(t)]T∈${{\mathbb{R}}^{N}}$，对式(3)求导，可得
$$ \left\{ {\begin{array}{*{20}{l}} {{{\dot e}_{i,m}}(t) = {e_{i,m + 1}}(t),m = 1,2, \cdots ,M - 1}\\ {{{\dot e}_{i,m}}(t) = - (\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})(\mathit{\boldsymbol{f}} + \mathit{\boldsymbol{d}} + \mathit{\boldsymbol{u}} - {f_0}{\mathit{\boldsymbol{1}}_N}),m = M} \end{array}} \right. $$ 定义领导智能体和跟随智能体组成的扩展有向网络图为G=(V, E, A)，实现网络G与领导智能体0之间的信息交互.领导者跟随一致性问题需要对网络拓扑做出以下假设.
假设1 在扩展网络G中存在一棵以领导者0为根节点的有向生成树.
$$ \begin{array}{l} \begin{array}{*{20}{c}} {\mathit{\boldsymbol{q}} = {{\left[ {{q_1},{q_2}, \cdots ,{q_N}} \right]}^{\rm{T}}} = {{(\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})}^{ - 1}}{\mathit{\boldsymbol{1}}_N}} \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{P}} = {\rm{diag}} \{ {p_i}\} = {\rm{diag}} \{ 1/{q_i}\} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{Q}} = \mathit{\boldsymbol{P}}(\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}}) + {(\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})^{\rm{T}}}\mathit{\boldsymbol{P}} \end{array} $$ 可知P和Q为正定矩阵.
根据已知假设1，在扩展网络G中，矩阵L+B是非奇异矩阵.
引理2 ‖δm‖≤‖em‖/σ(L+B), m=1, 2, …, M.
证明：根据式(3)，进一步地，定义多智能体系统全局误差向量
$$ \begin{array}{l} {\left. {\begin{array}{*{20}{c}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{\delta }} = [{\mathit{\boldsymbol{\delta }}_1}}&{{\mathit{\boldsymbol{\delta }}_2}}& \cdots &{{\mathit{\boldsymbol{\delta }}_M}} \end{array}} \right]^{\rm{T}}} = \\ {\left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{x}}_1} - {\mathit{\boldsymbol{1}}_N}{x_{0,1}}}&{{\mathit{\boldsymbol{x}}_2} - {\mathit{\boldsymbol{1}}_N}{x_{0,2}}}& \cdots &{{\mathit{\boldsymbol{x}}_M} - {\mathit{\boldsymbol{1}}_N}{x_{0,M}}} \end{array}} \right]^{\rm{T}}} \end{array} $$ 式中：xm=[x1, m x2, m … xN, m]T∈${{\mathbb{R}}^{N}}$，m=1, 2, …, M.
根据式(3)，扩展有向网络图G的全局误差向量[16]为
$$ \left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{e}}_1} = - (\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})({\mathit{\boldsymbol{x}}_1} - {\mathit{\boldsymbol{1}}_N}{x_{0,1}}) = - (\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}}){\mathit{\boldsymbol{\delta }}_1}}\\ \vdots \\ {{\mathit{\boldsymbol{e}}_m} = - (\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})({\mathit{\boldsymbol{x}}_m} - {\mathit{\boldsymbol{1}}_N}{\mathit{\boldsymbol{x}}_{0,m}}) = - (\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}}){\mathit{\boldsymbol{\delta }}_m}} \end{array}} \right. $$ (4) 根据假设1，矩阵L+B是非奇异的.在式(4)中，em=-(L+B)δm，由此可得
$$ {\mathit{\boldsymbol{\delta }}_m} = - {(\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})^{ - 1}}{\mathit{\boldsymbol{e}}_m} $$ 所以
$$ \left\| {{\mathit{\boldsymbol{\delta }}_m}} \right\| = \left\| {{{(\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})}^{ - 1}}{\mathit{\boldsymbol{e}}_m}} \right\| \le \left\| {{\mathit{\boldsymbol{e}}_m}} \right\|/{\mathrm{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ }\!\!\sigma\!\!\text{ }}} (\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}}) $$ 2. 自适应协同跟踪控制器设计
对存在不确定性的领导者的多智能体系统，采用离线训练的神经网络显然是不合适的.为解决此类问题，本节设计了分布式神经网络控制器，采用在线自适应RBF神经网络控制方法，实现神经网络权值矩阵的自适应调整，以解决领导者跟随一致性问题.
2.1 滑模面函数设计
采用滑模变结构控制，主要是因为该控制算法在保证系统稳定性的同时，还具有快速响应的特性.
选取智能体i(i∈N)的滑模面函数为
$$ {r_i} = {\alpha _1}{e_{i,1}} + {\alpha _2}{e_{i,2}} + \cdots + {\alpha _{M - 1}}{e_{i,M - 1}} + {e_{i,M}} $$ (5) 系数α1, α2, …, αM-1满足以s为自变量的多项式sM-1+αM-1sM-2+…+α1是Hurwitz的.那么如果ri是有界的，则ei也为有界的.进一步，随着ri→0，有ei→0.
定义滑模全局误差向量为r=[r1, r2, …, rN]T，那么，r=α1e1+…+αM-1eM-1+eM.
定义E1=[e1, …, eM-1]∈${{\mathbb{R}}^{N\times \left( M-1 \right)}}$，E2=${\mathit{\boldsymbol{\dot E}}_1}$=[e2, …, eM]，l=[0 … 0 1]T∈${{\mathbb{R}}^{M-1}}$，以及
(6) 则有
$$ {\mathit{\boldsymbol{E}}_2} = {\mathit{\boldsymbol{E}}_1}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}^{\rm{T}}} + \mathit{\boldsymbol{r}}{\mathit{\boldsymbol{l}}^{\rm{T}}} $$ (7) 由于Λ是Hurwitz矩阵, 给出任何正数β, 存在一个对称矩阵P1＞0，使得李雅普诺夫方程成立，即
$$ {\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}^{\rm{T}}}{\mathit{\boldsymbol{P}}_1} + {\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{ \boldsymbol{\varLambda} }} = - \beta {\mathit{\boldsymbol{I}}_N} $$ (8) 动态滑模误差r的导数为
$$ \mathit{\boldsymbol{\dot r}} = \mathit{\boldsymbol{\rho }} - (\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})(\mathit{\boldsymbol{f}} + \mathit{\boldsymbol{d}} + \mathit{\boldsymbol{u}} - {f_0}{\mathit{\boldsymbol{1}}_N}) $$ 式中：ρ=α1e2+α2e3+…+αM-1eM=E2 α；α=[α1, α2, …, αM-1]T.
下面引理3表明了ri(i∈N)的最终有界性，则ei也是有界性的.
引理3 对于智能体i(i=1, 2, …, N)，假设
$$ \begin{array}{*{20}{l}} {|{r_i}(t)| \le {\psi _i},\forall t \ge {t_0}}\\ {|{r_i}(t)| \le {\xi _i},\forall t \ge {T_{{\xi _i}}}} \end{array} $$ 时间Tξi＞t0，上确界ψi＞0，ξi＞0.存在时间TΘi＞t0，上确界Ψi＞0，Θi＞0，使得
$$ \begin{array}{*{20}{c}} {\left\| {{\mathit{\boldsymbol{e}}_i}(t)} \right\| \le {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_i},\forall t \ge {t_0}}\\ {\left\| {{\mathit{\boldsymbol{e}}_i}(t)} \right\| \le {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_i},\forall t \ge {T_{{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_i}}}} \end{array} $$ 证明：令ei(t)=[ei, 1, ei, 2, …, ei, M-1]T∈${{\mathbb{R}}^{M-1}}$，根据式(5)可得
$$ {\mathit{\boldsymbol{\dot e}}_i} = \mathit{\boldsymbol{ \boldsymbol{\varLambda} }}{\mathit{\boldsymbol{e}}_i} + \mathit{\boldsymbol{l}}{r_i} $$ (9) 式(9)的通解为
$$ {\mathit{\boldsymbol{e}}_i}(t) = {{\rm{e}}^{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}(t - {t_0})}}{\mathit{\boldsymbol{e}}_i}({t_0}) + \int_{{t_0}}^t {{{\rm{e}}^{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}(t - \tau )}}} \mathit{\boldsymbol{l}}r{ _i}(\tau ){\rm{d}}\tau $$ 式中t0为初始时刻.
由于Λ是Hurwitz矩阵, 存在φ＞0和λ＞0，使得
$$ \left\| {{{\rm{e}}^{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}(t - {t_0})}}} \right\| \le \varphi {{\rm{e}}^{ - \lambda (t - {t_0})}} $$ 成立.进一步，结合‖l‖=1可得
$$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} {\left\| {{\mathit{\boldsymbol{e}}_i}(t)} \right\| \le \varphi {{\rm{e}}^{ - \lambda (t - {t_0})}}\left\| {{\mathit{\boldsymbol{e}}_i}({t_0})} \right\| + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \int_{{t_0}}^t \varphi {{\rm{e}}^{ - \lambda (t - \tau )}}\left\| \mathit{\boldsymbol{l}} \right\|\left| {{r_i}(\tau )} \right|{\rm{d}}\tau \le } \end{array}\\ \begin{array}{*{20}{c}} {\varphi {{\rm{e}}^{ - \lambda (t - {t_0})}}\left\| {{\mathit{\boldsymbol{e}}_i}({t_0})} \right\| + \frac{{\varphi \left\| \mathit{\boldsymbol{B}} \right\|}}{\lambda }\mathop {{\rm{sup}}}\limits_{{t_1} \le \tau \le t} |{r_i}(\tau )| = }\\ {\varphi {{\rm{e}}^{ - \lambda (t - {t_0})}}\left\| {{\mathit{\boldsymbol{e}}_i}({t_0})} \right\| + \frac{\varphi }{\lambda }\mathop {{\rm{sup}}}\limits_{{t_1} \le \tau \le t} |{r_i}(\tau )|} \end{array} \end{array} $$ (10) 从式(10)可以看出，如果ri(t)有界，则‖ei(t)‖ < ∞，从而得到ei, m(t)对于所有的m=1, 2, …, M都是有界的.因而由方程(5)可得
$$ {e_{i,M}}(t) = {r_i} - {\alpha _1}{e_{i,1}}(t) - {\alpha _2}{e_{i,2}}(t) - \cdots - {\alpha _{M - 1}}{e_{i,M - 1}}(t) $$ 也为有界的.因此，如果ri(t) < ∞，那么‖ei(t)‖ < ∞，即‖ei(t)‖有界.
接着将证明若ri(t)→0，则有‖ei(t)‖→0成立.由于ri(τ)→0，故对任意给定的充分小常数εr>0，存在时刻t1使得当τ≥t1时，(φ/λ)|ri(τ)|≤εr成立，由此得到$\left( {\varphi /\lambda } \right)\mathop {\sup }\limits_{{t_1} \le \tau \le t} \left| {{r_i}\left( \tau \right)} \right| \le {\varepsilon _r}$.类似地，由e-λ(t-t1)的指数稳定性可以得到，对任意给定的充分小常数εe>0，存在时刻t2使得当t-t1≥t2时，有φe-λ(t-t1)‖ei(t1)‖≤εe成立.用t1替换不等式(10)中的变量t0可得：当t≥t1+t2时，有
$$ \begin{array}{*{20}{c}} {\left\| {{\mathit{\boldsymbol{e}}_i}(t)} \right\| \le \varphi {{\rm{e}}^{ - \lambda (t - {t_1})}}\left\| {{\mathit{\boldsymbol{e}}_i}({t_1})} \right\| + \frac{\varphi }{\lambda }\mathop {{\rm{sup}}}\limits_{{t_1} \le \tau \le t} |{r_i}(\tau )| \le }\\ {\varphi {{\rm{e}}^{ - \lambda (t - {t_1})}}\left\| {{\mathit{\boldsymbol{e}}_i}({t_1})} \right\| + \frac{\varphi }{\lambda }\mathop {{\rm{sup}}}\limits_{\tau \ge {t_1}} |{r_i}(\tau )| \le {\varepsilon _e} + {\varepsilon _r}} \end{array} $$ 成立.由εe和εr的任意性可知，当t→∞时，有‖ei(t)‖→0成立，由此得到ei, m(t)→0对于所有m=1, 2，…，M-1成立.因而由方程(5)可得ei, M(t)=ri-α1ei, 1(t)-α2ei, 2(t)-…-αM-1·ei, M-1(t)→0.综上，当ri(t)→0时有ei(t)→0成立.引理得证.
注1 从式(10)可以看出：较大的λ值会导致状态xi以较快的速率收敛到零.注意到λ的值取决于式(6)中的系统矩阵Λ，而Λ又与ri系数相关.因此，可以通过选取适当的系数α1, α2, …, αM-1, 得到令人满意的系统收敛速率.
2.2 基于RBF神经网络逼近未知函数f(·)
采用RBF神经网络逼近未知函数f(·)，隐含层第j个节点的输出为
$$ {{\mathit{\boldsymbol{\phi }}} _j} = {\rm{exp}}\left( {\frac{{{{\left\| {{\mathit{\boldsymbol{x}}_i} - {\mathit{\boldsymbol{c}}_j}} \right\|}^2}}}{{2{{\hat b}_j}^2}}} \right) $$ 式中：j=1, 2, …, vi，vi为神经网络隐含层节点数；${\hat b_j}$为方差.
网络的输出由加权函数
$$ {f_i}({x_i}) = \mathit{\boldsymbol{W}}_i^{\rm{T}}{{\mathit{\boldsymbol{\phi }}} _i}({x_i}) + {\varepsilon _i} $$ 实现.式中：Wi=[Wi, 1, Wi, 2, …, Wi, vi]，ϕi(xi)=[ϕi, 1(xi), ϕi, 2(xi), …, ϕi, vi(xi)]T，逼近误差εi∈$\mathbb{R}$.根据Stone-Weierstrass逼近定理, 给定紧集Ω，对任何正数εh, 存在一个足够大的正整数vi*，这样对于任何vi＞vi*，总能找到一个理想权值向量Wi和一个合适的径向基函数向量ϕi，使得|εi|≤εh.
对未知函数fi(t, xi)进行逼近，采用系统状态xi, 1(t)，xi, 2(t)，…，xi, M(t)作为网络的输入，网络输出为
$$ {\hat f_i}({x_i}) = \mathit{\boldsymbol{\hat W}}_i^{\rm{T}}(t){{\mathit{\boldsymbol{\phi }}} _i}({x_i}) $$ 式中第i个智能体的神经网络估计权值为${\mathit{\boldsymbol{\hat W}}_i}\left( t \right) \in $${{\mathbb{R}}^{{{v}_{i}}}}$.
定义最优权值矩W=blkdiag{W1, W2, …, WN}, 估计权值$\mathit{\boldsymbol{\hat W}} = {\rm{blkdiag}}\left\{ {{{\mathit{\boldsymbol{\hat W}}}_1}, {{\mathit{\boldsymbol{\hat W}}}_2}, \cdots , {{\mathit{\boldsymbol{\hat W}}}_N}} \right\}$, 逼近误差ε=[ε1, ε2, …, εN]T, 径向基函数向量ϕ(x)=[ϕ1T(x1), ϕ2T(x2), …, ϕNT(xN)]T，可得全局非线性函数f(·)具有如下的形式:
$$ \mathit{\boldsymbol{f}}(x) = {\mathit{\boldsymbol{W}}^{\rm{T}}}{\mathit{\boldsymbol{\phi }}} (x) + \mathit{\boldsymbol{\varepsilon }} $$ (11) f(x)的估计值为
$$ \mathit{\boldsymbol{\hat f}}(x) = {\mathit{\boldsymbol{\hat W}}^{\rm{T}}}{\mathit{\boldsymbol{\phi }}} (x) $$ (12) 最优权值矩阵W是未知的，需要事先估计.可得理想权值与估计权值之间的误差$\mathit{\boldsymbol{\tilde W}} = \mathit{\boldsymbol{W}} - \mathit{\boldsymbol{\hat W}}$.从而基于估计误差设计权值矩阵W的在线更新，使得神经网络对函数f(·)的逼近能够达到精度要求，最终实现神经网络的自适应学习功能.
为了方便下面自适应RBF神经网络控制系统稳定性证明，记高斯函数输出的最大值ϕi=maxxi∈Ω‖ϕi(xi)‖，理想权值的最大值Wi=max‖Wi‖.根据文献[17]对W、ϕ和ε的定义，可知存在正数ϕ、W和ε，满足‖ϕ‖≤ϕ、‖W‖≤ W和‖ε‖≤ε.
2.3 分布式控制器设计
设计每个智能体i分布式控制律ui和RBF神经网络的权值自适应律做如下假设.
假设2 存在正数X＞0，领导者状态满足‖x0(t)‖≤X.
假设3 存在连续函数g(·)：${{\mathbb{R}}^{m}}$→$\mathbb{R}$，满足|f0(t, x0(t))|≤|g(x0(t))|.
假设4 每个多智能体外部扰动di(t)未知并且有界，即sup{‖d1(t)‖, ‖d2(t)‖, …, ‖dN(t)‖, ‖d0(t)‖}≤ d，d可以为未知正常数.
注2 假设3中表明领导者的非线性部分f0(t, x0(t))存在上确界F，∀t≥t0.假设存在参数X、F和d，对于设计者而言，不必知道，即这些上确界并不直接用于控制器设计，而是用于Lyapunov方法分析系统稳定性.
2.3.1 RBF神经网络的权值自适应律
设计RBF神经网络的权值自适应律为
$$ {\mathit{\boldsymbol{\dot {\hat W}}}_i} = - {\mathit{\boldsymbol{F}}_i}{{\mathit{\boldsymbol{\phi }}} _i}{\mathit{\boldsymbol{r}}_i}{p_i}({\rm{ deg}}{ _i} + {b_i}) - \kappa {\mathit{\boldsymbol{F}}_i}{\mathit{\boldsymbol{\hat W}}_i} $$ (13) 可写成如下的简洁向量形式：
$$ \mathit{\boldsymbol{\dot {\hat W}}} = - \mathit{\boldsymbol{F}}{\mathit{\boldsymbol{\phi }}} \mathit{\boldsymbol{rP}}(\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}}) - \kappa \mathit{\boldsymbol{F\hat W}} $$ (14) 式中：Fi=FiT∈${{\mathbb{R}}^{{{v}_{i}}\times {{v}_{i}}}}$为任意正定矩阵；正数κ为可调整标量；矩阵P在引理1已经定义.
2.3.2 多智能体系统分布式控制协议
较大的外界扰动需要较大的切换增益，引起系统的抖振，控制协议采用神经网络对滑模控制进行补偿，为解决这一问题提供了有效的途径.
根据2.2节介绍，每个智能体i使用隐含层节点数为vi，设计每个智能体i分布式控制协议
$$ \begin{array}{*{20}{c}} {{u_i} = \frac{1}{{ {\rm{deg}}{ _i} + {b_i}}}({\alpha _1}{e_{i,2}} + {\alpha _2}{e_{i,3}} + \cdots + {\alpha _{M - 1}}{e_{i,M}}) - }\\ {{{\hat f}_i}({x_i}) + k{r_i}} \end{array} $$ (15) 分布式控制协议可等价地写成简洁向量形式
$$ \mathit{\boldsymbol{u}} = - {(\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}})^{ - 1}}\mathit{\boldsymbol{\rho }} - \mathit{\boldsymbol{\hat f}}(x) + k\mathit{\boldsymbol{r}} $$ (16) 控制增益满足
$$ k > \frac{2}{{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\sigma }}} (\mathit{\boldsymbol{Q}})}}\left( {\frac{{{\gamma ^2}}}{\kappa } + \frac{2}{\beta }{g^2} + h} \right) $$ (17) 式中：$\gamma = - \frac{1}{2}\overline \phi \overline \sigma \left( \mathit{\boldsymbol{P}} \right)\overline \sigma \left( \mathit{\boldsymbol{A}} \right), h = \frac{{\overline \sigma \left( \mathit{\boldsymbol{P}} \right)\overline \sigma \left( \mathit{\boldsymbol{A}} \right)}}{{\underline \sigma \left( {\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}}} \right)}}\left\| {\overline \alpha } \right\|, g = - \frac{1}{2}\left( {\frac{{\overline \sigma \left( \mathit{\boldsymbol{P}} \right)\overline \sigma \left( \mathit{\boldsymbol{A}} \right)}}{{\underline \sigma \left( {\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}}} \right)}}{{\left\| \mathit{\boldsymbol{ \boldsymbol{\varLambda} }} \right\|}_F}\left\| {\mathit{\boldsymbol{\overline \alpha }} } \right\| + \overline \sigma \left( {{\mathit{\boldsymbol{P}}_1}} \right)} \right)$，其中，P1在式(8)已经定义，β＞0，Q在引理1已经说明，系数κ已在2.3.1节说明.
当m=1，2时，多智能体网络退化成2种特殊情形：一阶系统和双积分系统.
特别注意的是，RBF神经网络的权值自适应律式(13)和分布式控制协议式(15)适用于第i个智能体.
2.4 自适应RBF神经网络控制系统稳定性分析
定理1 考虑N个跟随者式(1)和一个领导者0式(2)组成多智能体系统，假设1~4皆成立，使用RBF神经网络的权值自适应律式(13)和分布式控制协议式(15)，得到如下结果:
1) 跟踪误差δ1, δ2, …, δM协同一致最终有界，这意味着在扩展网络G中，所有跟随者i可以趋近于领导者0的轨迹节点，实现追踪一致性.
2) 所有状态xi(t)是有界的.
证明：构建Lyapunov函数
$$ V(t) = {V_1}(t) + {V_2}(t) + {V_3}(t) $$ (18) 式中：${V_1}\left( t \right) = \frac{1}{2}{\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{Pr}};{V_2}\left( t \right) = \frac{1}{2}{\rm{tr}}\left\{ {{{\mathit{\boldsymbol{\widetilde W}}}^{\rm{T}}}{\mathit{\boldsymbol{F}}^{ - 1}}\mathit{\boldsymbol{\widetilde W}}} \right\};$${V_3}\left( t \right) = \frac{1}{2}{\rm{tr}}\left\{ {{\mathit{\boldsymbol{E}}_1}{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{E}}_1^{\rm{T}}} \right\};\mathit{\boldsymbol{P}} = {\mathit{\boldsymbol{P}}^{\rm{T}}} > 0;{\mathit{\boldsymbol{F}}^{ - 1}} = {\mathit{\boldsymbol{F}}^{ - T}} > 0$.
首先对V1(t)求导数，有
$$ \begin{array}{*{20}{c}} {{{\dot V}_1}(t) = {\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{P\dot r}} = }\\ {{\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{P}}[\mathit{\boldsymbol{\rho }} - (\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})(\mathit{\boldsymbol{f}} + \mathit{\boldsymbol{d}} + \mathit{\boldsymbol{u}} - {f_0}{\mathit{\boldsymbol{1}}_N})]} \end{array} $$ (19) 将式(16)代入式(19)，并考虑式(11)(12)和L=D-A，得
$$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} {{{\dot V}_1}(t) = {\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{P}}[\mathit{\boldsymbol{\rho }} - (\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})(\mathit{\boldsymbol{f}} + \mathit{\boldsymbol{d}} - {{(\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}})}^{ - 1}}\mathit{\boldsymbol{\rho }} - }\\ {\mathit{\boldsymbol{\hat f}} + k\mathit{\boldsymbol{r}} - {f_0}{\mathit{\boldsymbol{1}}_N})] = } \end{array}\\ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{P\rho }} - {\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{P}}(\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})(\mathit{\boldsymbol{\varepsilon }} + \mathit{\boldsymbol{d}} - {f_0}{\mathit{\boldsymbol{1}}_N}) - k{\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{P}}(\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})\mathit{\boldsymbol{r}} - }\\ {{\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{P}}[(\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}}) - \mathit{\boldsymbol{A}}] \cdot [{{(\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}})}^{ - 1}}\mathit{\boldsymbol{\rho }} + {{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}{\mathit{\boldsymbol{\phi }}} ] = } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{P}}(\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})(\mathit{\boldsymbol{\varepsilon }} + \mathit{\boldsymbol{d}} - {f_0}{\mathit{\boldsymbol{1}}_N}) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k{\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{P}}(\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})\mathit{\boldsymbol{r}} - {\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{P}}(\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}}){{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}{\mathit{\boldsymbol{\phi }}} + {\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{PA}}{{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}{\mathit{\boldsymbol{\phi }}} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{PA}}{(\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}})^{ - 1}}\mathit{\boldsymbol{\rho }} \end{array} $$ 根据xTy=tr{yxT}，其中，x, y∈${{\mathbb{R}}^{N}}$，并结合引理1，式(19)可化为
$$ \begin{array}{l} {{\dot V}_1}(t) = - {\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{P}}(\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})(\mathit{\boldsymbol{\varepsilon }} + \mathit{\boldsymbol{d}} - {f_0}{\mathit{\boldsymbol{1}}_N}) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{2}k{\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{Qr}} - {\rm{tr}} \{ {{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}{\mathit{\boldsymbol{\phi }}} {\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{P}}(\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}})\} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{tr}} \{ {{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}{\mathit{\boldsymbol{\phi }}} {\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{PA}}\} + {\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{PA}}{(\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}})^{ - 1}}\mathit{\boldsymbol{\rho }} \end{array} $$ 因为$\mathop {\mathit{\boldsymbol{\widetilde W}}}\limits^. = \mathit{\boldsymbol{\dot W}} - \mathop {\mathit{\boldsymbol{\hat W}}}\limits^. = - \mathop {\mathit{\boldsymbol{\hat W}}}\limits^. $，对V2(t)求时间导数，将式(14)代入，可得
$$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\dot V}_2}(t) = \frac{1}{2} {\rm{tr}} \{ {{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}{\mathit{\boldsymbol{F}}^{ - 1}}\mathit{\boldsymbol{\tilde W}}\} = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{tr}} \{ {{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}{\mathit{\boldsymbol{F}}^{ - 1}}[\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{\phi }}} \mathit{\boldsymbol{rP}}(\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}}) + \kappa \mathit{\boldsymbol{F\hat W}}]\} = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{tr}} \{ {{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}{\mathit{\boldsymbol{\phi }}} \mathit{\boldsymbol{rP}}(\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}})\} + \kappa {\rm{tr}} \{ {{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}\mathit{\boldsymbol{\hat W}}\} = \\ {\rm{tr}} \{ {{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}{\mathit{\boldsymbol{\phi }}} \mathit{\boldsymbol{rP}}(\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}})\} + \kappa {\rm{tr}} \{ {{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}\mathit{\boldsymbol{W}}\} - \kappa {\rm{tr}} \{ {{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}\mathit{\boldsymbol{\tilde W}}\} \end{array} $$ 对V3(t)求时间导数得
$$ {\dot V_3}(t) = {\rm{tr}} \{ {\mathit{\boldsymbol{E}}_2}{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{E}}_1^{\rm{T}}\} $$ (20) 将式(7)代入式(20)，并考虑式(8)，得
$$ \begin{array}{*{20}{c}} {{{\dot V}_3}(t) = - \frac{\beta }{2} {\rm{tr}} \{ {\mathit{\boldsymbol{E}}_1}\mathit{\boldsymbol{E}}_1^{\rm{T}}\} + {\rm{tr}} \{ \mathit{\boldsymbol{r}}{\mathit{\boldsymbol{l}}^{\rm{T}}}{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{E}}_1^{\rm{T}}\} \le }\\ { - \frac{\beta }{2}\left\| {{\mathit{\boldsymbol{E}}_1}} \right\|_{\rm{F}}^2 + \bar \sigma ({\mathit{\boldsymbol{P}}_1})\left\| \mathit{\boldsymbol{r}} \right\|{{\left\| {{\mathit{\boldsymbol{E}}_1}} \right\|}_{\rm{F}}}} \end{array} $$ $$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} {\dot V(t) = {{\dot V}_1}(t) + {{\dot V}_2}(t) + {{\dot V}_3}(t) = }\\ { - {\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{P}}(\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})(\mathit{\boldsymbol{\varepsilon }} + \mathit{\boldsymbol{d}} - {f_0}{\mathit{\boldsymbol{1}}_N}) - \frac{1}{2}k{\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{Qr}} + } \end{array}\\ \begin{array}{*{20}{c}} { {\rm{tr}} \{ {{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}\phi {\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{PA}}\} + \kappa {\rm{tr }}\{ {{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}\mathit{\boldsymbol{W}}\} - \kappa {\rm{tr}} \{ {{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}\mathit{\boldsymbol{\tilde W}}\} + }\\ {{\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{PA}}{{(\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}})}^{ - 1}}\mathit{\boldsymbol{\rho }} - \frac{\beta }{2} {\rm{tr}} \{ {\mathit{\boldsymbol{E}}_1}\mathit{\boldsymbol{E}}_1^{\rm{T}}\} + {\rm{tr}} \{ \mathit{\boldsymbol{r}}{\mathit{\boldsymbol{l}}^{\rm{T}}}{\mathit{\boldsymbol{P}}_1}\mathit{\boldsymbol{E}}_1^{\rm{T}}\} = } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{P}}(\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})(\mathit{\boldsymbol{\varepsilon }} + \mathit{\boldsymbol{d}} - {f_0}{\mathit{\boldsymbol{1}}_N}) - \frac{1}{2}k{\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{Qr}} + }\\ {{\rm{tr}}\{ {{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}\phi {\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{PA}}\} + \kappa {\rm{tr}} \{ {{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}\mathit{\boldsymbol{W}}\} - \kappa {\rm{tr}} \{ {{\mathit{\boldsymbol{\tilde W}}}^{\rm{T}}}\mathit{\boldsymbol{\tilde W}}\} + } \end{array} \end{array} $$ 因此
$$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{r}}^{\rm{T}}}\mathit{\boldsymbol{PA}}{{(\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}})}^{ - 1}}({\mathit{\boldsymbol{E}}_1}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}^{\rm{T}}}\bar \alpha + \mathit{\boldsymbol{r}}{\mathit{\boldsymbol{l}}^{\rm{T}}}\mathit{\boldsymbol{\bar \alpha }}) - }\\ {\frac{\beta }{2} {\rm{tr}} \{ {\mathit{\boldsymbol{E}}_1}\mathit{\boldsymbol{E}}_1^{\rm{T}}\} + {\rm{tr}} \{ \mathit{\boldsymbol{r}}{\mathit{\boldsymbol{l}}^{\rm{T}}}{\mathit{\boldsymbol{P}}_{\rm{1}}}\mathit{\boldsymbol{E}}_1^{\rm{T}}\} \le } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {\bar \sigma (\mathit{\boldsymbol{P}})\bar \sigma (\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})\bar T\left\| \mathit{\boldsymbol{r}} \right\| - \frac{1}{2}k {\underline \sigma} (\mathit{\boldsymbol{Q}}){{\left\| \mathit{\boldsymbol{r}} \right\|}^2} + }\\ {\bar \phi \bar \sigma (\mathit{\boldsymbol{P}})\bar \sigma (\mathit{\boldsymbol{A}}){{\left\| {\mathit{\boldsymbol{\tilde W}}} \right\|}_{\rm{F}}}\left\| \mathit{\boldsymbol{r}} \right\| + \kappa \bar W{{\left\| {\mathit{\boldsymbol{\tilde W}}} \right\|}_{\rm{F}}} - } \end{array}\\ \kappa \left\| {\mathit{\boldsymbol{\tilde W}}} \right\|_{\rm{F}}^2 + \frac{{\bar \sigma (\mathit{\boldsymbol{P}})\bar \sigma (\mathit{\boldsymbol{A}})}}{{{\underline \sigma} (\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}})}}(\left\| \mathit{\boldsymbol{r}} \right\|{\left\| {{\mathit{\boldsymbol{E}}_1}} \right\|_{\rm{F}}}{\left\| \mathit{\boldsymbol{ \boldsymbol{\varLambda} }} \right\|_{\rm{F}}}\left\| {\mathit{\boldsymbol{\bar \alpha }}} \right\| + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} {{{\left\| \mathit{\boldsymbol{r}} \right\|}^2}\left\| \mathit{\boldsymbol{l}} \right\|\left\| {\mathit{\boldsymbol{\bar \alpha }}} \right\|) - }\\ {\frac{\beta }{2}\left\| {{\mathit{\boldsymbol{E}}_1}} \right\|_{\rm{F}}^2 + \bar \sigma ({P_1})\left\| \mathit{\boldsymbol{r}} \right\|{{\left\| {{\mathit{\boldsymbol{E}}_1}} \right\|}_{\rm{F}}}} \end{array} \end{array} $$ (21) 式中T=ε+d+F.
根据2.3.2节对γ、h和g的定义，式(21)可化为
$$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \dot V(t) \le - \left( {\frac{1}{2}k\sigma (\mathit{\boldsymbol{Q}}) - \frac{{\bar \sigma (\mathit{\boldsymbol{P}})\bar \sigma (\mathit{\boldsymbol{A}})}}{{\sigma (\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}})}}\left\| {\mathit{\boldsymbol{\bar \alpha }}} \right\|} \right) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} {{{\left\| \mathit{\boldsymbol{r}} \right\|}^2} - \kappa \left\| {\mathit{\boldsymbol{\tilde W}}} \right\|_{\rm{F}}^2 - \frac{\beta }{2}\left\| {{\mathit{\boldsymbol{E}}_1}} \right\|_{\rm{F}}^2 + \bar {\mathit{\boldsymbol{\phi }}} \bar \sigma (\mathit{\boldsymbol{P}})\bar \sigma (A) \cdot }\\ {{{\left\| {\mathit{\boldsymbol{\tilde W}}} \right\|}_{\rm{F}}}\left\| \mathit{\boldsymbol{r}} \right\| + \left( {\frac{{\bar \sigma (\mathit{\boldsymbol{P}})\bar \sigma (\mathit{\boldsymbol{A}})}}{{\bar \sigma (\mathit{\boldsymbol{D}} + \mathit{\boldsymbol{B}})}}{{\left\| \mathit{\boldsymbol{ \boldsymbol{\varLambda} }} \right\|}_{\rm{F}}}\left\| {\mathit{\boldsymbol{\bar \alpha }}} \right\| + } \right.} \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{c}} {\left. {\bar \sigma ({\mathit{\boldsymbol{P}}_1})} \right)\left\| \mathit{\boldsymbol{r}} \right\|{{\left\| {{\mathit{\boldsymbol{E}}_1}} \right\|}_{\rm{F}}} + }\\ {\bar \sigma (\mathit{\boldsymbol{P}})\bar \sigma (\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})\bar T\left\| \mathit{\boldsymbol{r}} \right\| + \kappa \bar W{{\left\| {\mathit{\boldsymbol{\tilde W}}} \right\|}_{\rm{F}}} = }\\ { - (\frac{1}{2}k {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\sigma }} (Q) - h){{\left\| \mathit{\boldsymbol{r}} \right\|}^2} - \kappa \left\| {\mathit{\boldsymbol{\tilde W}}} \right\|_F^2 - } \end{array}\\ \begin{array}{*{20}{c}} {\frac{\beta }{2}\left\| {{\mathit{\boldsymbol{E}}_1}} \right\|_{\rm{F}}^2 - 2\gamma {{\left\| {\mathit{\boldsymbol{\tilde W}}} \right\|}_{\rm{F}}}\left\| \mathit{\boldsymbol{r}} \right\| - 2g\left\| \mathit{\boldsymbol{r}} \right\|{{\left\| {{\mathit{\boldsymbol{E}}_1}} \right\|}_{\rm{F}}} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \bar \sigma (P)\bar \sigma (L + B)\bar T\left\| \mathit{\boldsymbol{r}} \right\| + \kappa \bar W{{\left\| {\mathit{\boldsymbol{\tilde W}}} \right\|}_{\rm{F}}} = } \end{array}\\ - {\left[ {\begin{array}{*{20}{c}} {{{\left\| {{\mathit{\boldsymbol{E}}_1}} \right\|}_{\rm{F}}}}\\ {{{\left\| {\mathit{\boldsymbol{\tilde W}}} \right\|}_{\rm{F}}}}\\ {\left\| \mathit{\boldsymbol{r}} \right\|} \end{array}} \right]^{\rm{T}}}\left[ {\begin{array}{*{20}{c}} {\frac{\beta }{2}}&0&g\\ 0&\kappa &\gamma \\ g&\gamma &{\frac{1}{2}k\bar \sigma (Q) - h} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{\left\| {{\mathit{\boldsymbol{E}}_1}} \right\|}_{\rm{F}}}}\\ {{{\left\| {\mathit{\boldsymbol{\tilde W}}} \right\|}_{\rm{F}}}}\\ {\left\| \mathit{\boldsymbol{r}} \right\|} \end{array}} \right] + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ {\begin{array}{*{20}{c}} 0&{\kappa \bar W}&{\bar \sigma (\mathit{\boldsymbol{P}})\bar \sigma (\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})\bar T} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{\left\| {{\mathit{\boldsymbol{E}}_1}} \right\|}_{\rm{F}}}}\\ {{{\left\| \mathit{\boldsymbol{W}} \right\|}_{\rm{F}}}}\\ {\left\| \mathit{\boldsymbol{r}} \right\|} \end{array}} \right] \end{array} $$ 令$\mathit{\boldsymbol{z = }}{\left[ {{{\left\| {{\mathit{\boldsymbol{E}}_1}} \right\|}_{\rm{F}}}\;\;\;{{\left\| {\mathit{\boldsymbol{\tilde W}}} \right\|}_{\rm{F}}}\;\;\;\left\| \mathit{\boldsymbol{r}} \right\|} \right]^{\rm{T}}}, \mathit{\boldsymbol{ \boldsymbol{\varXi} = }}\left[ {\begin{array}{*{20}{c}} {\frac{\beta }{2}}&0&g\\ 0&\kappa &\gamma \\ g&\gamma &{\frac{1}{2}k\underline \sigma \left( Q \right) - h} \end{array}} \right], \mathit{\boldsymbol{\omega }} = {\left[ {0\;\;\kappa \overline W \;\;\overline \sigma \left( \mathit{\boldsymbol{P}} \right)\overline \sigma \left( {\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}}} \right)\overline T } \right]^{\rm{T}}}$，可得
$$ \dot V(t) \le - {\mathit{\boldsymbol{z}}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varXi} z}} + {\mathit{\boldsymbol{\omega }}^{\rm{T}}}\mathit{\boldsymbol{z}} = - {V_z}(\mathit{\boldsymbol{z}}) $$ (22) 系统实现渐近稳定的条件是Vz(z)为正定函数，即满足下面2个条件：
1) 矩阵Ξ是正定的；
2) $\left\| \mathit{\boldsymbol{z}} \right\| > \frac{{\left\| \mathit{\boldsymbol{\omega }} \right\|}}{{\overline \sigma \left( \mathit{\boldsymbol{ \boldsymbol{\varXi} }} \right)}}$.
为了检验矩阵Ξ的正定性，可对各顺序主子行列式校验其是否大于0，即
$$ \begin{array}{*{20}{l}} {\beta > 0}\\ {\beta \kappa > 0}\\ {\kappa \left[ {\beta (\frac{1}{2}k {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\sigma }} (Q) - h) - 2{g^2}} \right] - \beta {\gamma ^2} > 0} \end{array} $$ 解上述不等式，可得式(17)的条件.
易知‖ω‖1>‖ω‖，如果‖z‖≥Bd，上述条件2成立，可求得
$$ Bd = \frac{{{{\left\| \mathit{\boldsymbol{\omega }} \right\|}_1}}}{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\sigma }} (\mathit{\boldsymbol{ {\varXi} }})}} = \frac{{\bar \sigma (\mathit{\boldsymbol{P}})\bar \sigma (\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{B}})\bar T + \kappa \bar W}}{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\sigma }}(\mathit{\boldsymbol{ \boldsymbol{\varXi} }})}} $$ 因此，条件1和2皆满足，最终得到
$$ \dot V(t) \le - {V_z}(\mathit{\boldsymbol{z}}),\forall \left\| \mathit{\boldsymbol{z}} \right\| \ge Bd $$ Vz(z)是连续正定函数.
根据二次正定函数(18)，可以得到不等式
$$ \begin{array}{*{20}{l}} {\frac{1}{2}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\sigma }} (\mathit{\boldsymbol{P}}){{\left\| \mathit{\boldsymbol{r}} \right\|}^2} + \frac{1}{2}\sigma ({\mathit{\boldsymbol{F}}^{ - 1}}){{\left\| {\mathit{\boldsymbol{\tilde W}}} \right\|}^2} + }\\ {\frac{1}{2}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\sigma }} ({\mathit{\boldsymbol{P}}_1}){{\left\| {{\mathit{\boldsymbol{E}}_1}} \right\|}^2} \le V \le \frac{1}{2}\bar \sigma (\mathit{\boldsymbol{P}}){{\left\| \mathit{\boldsymbol{r}} \right\|}^2} + }\\ {\frac{1}{2}\bar \sigma ({\mathit{\boldsymbol{F}}^{ - 1}}){{\left\| {\mathit{\boldsymbol{\tilde W}}} \right\|}^2} + \frac{1}{2}\bar \sigma ({\mathit{\boldsymbol{P}}_1}){{\left\| {{\mathit{\boldsymbol{E}}_1}} \right\|}^2}} \end{array} $$ (23) 令${\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1} = {\rm{diag}}\left( {\frac{1}{2}\underline \sigma \left( \mathit{\boldsymbol{P}} \right)\;\;\;\frac{1}{2}\underline \sigma \left( {{\mathit{\boldsymbol{F}}^{ - 1}}} \right)\;\;\;\frac{1}{2}\underline \sigma \left( {{\mathit{\boldsymbol{P}}_{\rm{1}}}} \right)} \right), {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2} = {\rm{diag}}\left( {\frac{1}{2}\overline \sigma \left( \mathit{\boldsymbol{P}} \right)\;\;\;\frac{1}{2}\overline \sigma \left( {{\mathit{\boldsymbol{F}}^{ - 1}}} \right)\;\;\;\frac{1}{2}\overline \sigma \left( {{\mathit{\boldsymbol{P}}_{\rm{1}}}} \right)} \right)$则式(23)可以等价于
$$ {\underline \sigma} ({\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1}){\left\| \mathit{\boldsymbol{z}} \right\|^2} \le V \le \bar \sigma ({\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2}){\left\| \mathit{\boldsymbol{z}} \right\|^2} $$ 根据文献[18]定理4.10，存在有限时间T0，得到‖z(t)‖是有界的，满足关系
$$ \left\| \mathit{\boldsymbol{z}} \right\| \le \sqrt {\bar \sigma ({\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2})/{\underline \sigma} ({\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1})} Bd,\forall t \ge {t_0} + {T_0} $$ 令$\mu = \mathop {\min }\limits_{\left\| \mathit{\boldsymbol{z}} \right\| \ge Bd} {V_z}\left( \mathit{\boldsymbol{z}} \right)$，对式(22)两边积分，可以求出有限收敛时间
$$ {T_0} = \frac{{V({t_0}) - \bar \sigma ({\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2})B{d^2}}}{\mu } $$ 通过向量z可知，r(t)是最终有界的，进一步得到ri(t)有界.通过引理3，ei(t)是最终有界的，意味着ei1, ei2, …, eiM是最终有界的.通过引理2，跟随误差δm是协同一致最终有界的，则跟随者状态xim(t)收敛到领导者x0m(t)的邻域内.
3. 数值仿真
3.1 同构多智能体系统的一致性
在仿真实例中，考虑4个跟随者与1个领导者0组成的有向网络拓扑G，如图 1所示，满足假设1.为了简化仿真设计，假设非线性网络中互相有通信的边的权重及牵制增益均为1.
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图 1 由4个追随智能体和1个领导智能体组成的有向图Figure 1. Directed topology with four followers and one leader4个跟随者具有如下的三阶不确定非线性多智能体系统：
$$ \left\{ {\begin{array}{*{20}{l}} {{{\dot x}_{i1}}(t) = {x_{i2}}(t)}\\ {{{\dot x}_{i2}}(t) = {x_{i3}}(t)}\\ {{{\dot x}_{i3}}(t) = {f_i}(t,{x_i}) + {d_i}(t) + {u_i}(t)} \end{array}} \right. $$ 假设非线性函数fi的数学表达式与领导者相同，即
$$ \begin{array}{*{20}{c}} {f(t,{x_i}(t)) = }\\ { - {\rm{sin}}{\kern 1pt} {\kern 1pt} {x_{i1}} - 0.25{x_{i2}} + 1.5{\rm{cos}}(2.5t)} \end{array} $$ 领导者具有式(2)的动力学方程，其中，f0=-sin x01-0.25x02+1.5cos(2.5t).
对于网络G中第i个智能体，假设其包含外界扰动和传感器噪声等不确定性项统一建模为0.01sin t.神经网络隐含层节点数为7个，网络的初始值为0，高斯函数参数设置为
$$ \mathit{\boldsymbol{c}} = \left[ {\begin{array}{*{20}{c}} { - 4.5}&{ - 3}&{ - 1.5}&0&{1.5}&3&{4.5}\\ { - 4.5}&{ - 3}&{ - 1.5}&0&{1.5}&3&{4.5}\\ { - 4.5}&{ - 3}&{ - 1.5}&0&{1.5}&3&{4.5} \end{array}} \right] $$ 和${\hat b_j}$=2.4，采用RBF神经网络的权值自适应律式(13)和分布式控制协议式(15)，参数α1=100，α2=20，k=600, κ=0.01，Fi=2 000I7.
多智能体系统初始状态信息为
$$ \begin{array}{l} \begin{array}{*{20}{l}} {{\mathit{\boldsymbol{x}}_1} = {{\left[ {\begin{array}{*{20}{l}} {2.8}&{1.2}&{2.3} \end{array}} \right]}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{x}}_2} = {{\left[ {\begin{array}{*{20}{l}} {2.4}&{2.1}&{1.1} \end{array}} \right]}^{\rm{T}}}} \end{array}\\ \begin{array}{*{20}{l}} {{\mathit{\boldsymbol{x}}_3} = {{\left[ {\begin{array}{*{20}{l}} { - 1.5}&{ - 1.3}&{ - 1.4} \end{array}} \right]}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{x}}_4} = {{\left[ {\begin{array}{*{20}{r}} { - 1.5}&{ - 2.1}&{ - 2.3} \end{array}} \right]}^{\rm{T}}}} \end{array}\\ {\mathit{\boldsymbol{x}}_0} = {\left[ {\begin{array}{*{20}{c}} { - 2.5}&{1.5}&{ - 0.5} \end{array}} \right]^{\rm{T}}} \end{array} $$ 三阶系统的所有智能体的各阶状态轨迹如图 2~4所示，4个同构跟随者的位置状态都渐渐逼近领导者0的位置状态，即实现领导跟随一致性.
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图 2 同构系统xi1(i=0, 1, …, 4)轨迹图Figure 2. Homogeneous systems state trajectories of xi1(i=0, 1, …, 4)![]()
图 3 同构系统xi2(i=0, 1, …, 4)轨迹图Figure 3. Homogeneous systems state trajectories of xi2(i=0, 1, …, 4)![]()
图 4 同构系统xi3(i=0, 1, …, 4)轨迹图Figure 4. Homogeneous systems state trajectories of xi3(i=0, 1, …, 4)3.2 异构多智能体系统的一致性
为了说明所提协议能够应用于异构多智能体系统，考虑如下4个智能体：
$$ \begin{array}{*{20}{l}} {{{\dot x}_{i1}}(t) = {x_{i2}}(t)}\\ {{{\dot x}_{i2}}(t) = {x_{i3}}(t)}\\ {{{\dot x}_{i3}}(t) = {f_i}(t,{x_i}) + {d_i}(t) + {u_i}(t)} \end{array} $$ 假设非线性函数fi的数学表达式为
$$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {f_1}(t,{x_1}(t)) = - {\rm{sin}}{\kern 1pt} {\kern 1pt} {x_{11}} - 0.2{x_{12}} + 2{\rm{cos}}(2.5t);\\ {f_2}(t,{x_2}(t)) = - 1.2{\rm{sin}}{\kern 1pt} {\kern 1pt} {x_{21}} - 0.25{x_{22}};{f_3}(t,{x_3}(t)) = \\ - 0.8{\rm{sin}}{\kern 1pt} {\kern 1pt} {x_{31}} - 0.25{x_{32}} + {\rm{cos}}(3t);{f_4}(t,{x_4}(t)) = {x_{41}} - \\ 0.25{\rm{sin}}{\kern 1pt} {\kern 1pt} {x_{42}} \end{array} $$ 控制器和神经网络的参数、网络拓扑结构、系统的初始状态、不确定项di(t)、领导者的系统模型皆与3.1节相同.三阶系统所有智能体的各阶状态轨迹如图 5~7所示，4个异构跟随者的位置状态都渐渐逼近领导者0的位置状态，实现了状态的一致性.
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图 5 异构系统xi1(i=0, 1, …, 4)轨迹图Figure 5. Heterogeneous systems state trajectories of xi1(i=0, 1, …, 4)![]()
图 6 异构系统xi2(i=0, 1, …, 4)轨迹图Figure 6. Heterogeneous systems state trajectories of xi2(i=0, 1, …, 4)![]()
图 7 异构系统xi3(i=0, 1, …, 4)轨迹图Figure 7. Heterogeneous systems state trajectories of xi3(i=0, 1, …, 4)上述仿真结果表明，在设计控制协议作用下，系统跟随者状态轨迹在有限时间内到达切换流形，从而所有智能体的各阶状态都能相应地收敛到一起, 即
$$ \begin{array}{l} \mathop {{\rm{lim}}}\limits_{t \to \infty } \left\| {{\mathit{\boldsymbol{x}}_i}(t) - {x_0}(t)} \right\| = 0\\ \mathop {{\rm{lim}}}\limits_{t \to \infty } \left\| {{x_i}(t) - {x_j}(t)} \right\| = 0,\forall i,j = 1,2, \cdots ,N \end{array} $$ 从而验证了定理的正确性.通过同质和异质多智能体系统仿真比较，验证了所提出的自适应一致性协议不仅适用于同质多智能体系统，也可应用于异质多智能体系统.
另外，从状态轨迹图可知，在不改变领导智能体的初始条件下，利用牵制控制技术，将领导者的状态信息传递给跟随者，而异质跟随智能体通过控制输入，能够逼近同质跟随智能体的状态.
注3 针对系统中存在外界扰动和不确定项问题，目前主要有鲁棒控制方法和状态观测器方法.采用鲁棒控制对外界扰动有一定抑制作用，但不能有效地消除外界扰动对一致性的影响，如文献[19].目前较为常用的方法是采用状态观测器，对多智能系统不确定项进行估计来补偿未知项，实现多智能体的一致性，而且设计的是针对线性系统[20].本文针对高阶非线性不确定多智能体系统展开研究，对系统中非线性部分采用RBF神经网络进行逼近，设计滑模控制器对外界扰动进行了补偿，使多智能体实现镇定，从而实现了领导者跟随者一致性问题.
4. 结论
1) 为克服文献[8]需要已知非线性项和不确定项上界，本文引入RBF神经网络对非线性项进行逼近，同时抑制系统抖振，提高系统的鲁棒性能.
2) 理论结果表明，Brunovsky型高阶非线性智能体系统达到一致性所需要的有限时间不仅取决于所设计算法的相关控制参数和信息拓扑结构，而且取决于多智能体的初始状态.
3) 仿真结果证明了所提出的分布式RBF神经网络逼近的自适应滑模控制算法不仅可以有效处理未知甚至异构的非线性动态，而且具有良好的抗干扰能力，保证了系统跟踪误差收敛.
4) 在弱连通的条件下，使用牵制控制技术来定位网络G中的相关节点，实现领导智能体和跟随智能体之间信息交互，有效地解决了Brunovsky型高阶非线性智能体系统的协同跟踪控制问题.
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图 1 由4个追随智能体和1个领导智能体组成的有向图
Figure 1. Directed topology with four followers and one leader
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图 2 同构系统xi1(i=0, 1, …, 4)轨迹图
Figure 2. Homogeneous systems state trajectories of xi1(i=0, 1, …, 4)
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图 3 同构系统xi2(i=0, 1, …, 4)轨迹图
Figure 3. Homogeneous systems state trajectories of xi2(i=0, 1, …, 4)
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图 4 同构系统xi3(i=0, 1, …, 4)轨迹图
Figure 4. Homogeneous systems state trajectories of xi3(i=0, 1, …, 4)
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图 5 异构系统xi1(i=0, 1, …, 4)轨迹图
Figure 5. Heterogeneous systems state trajectories of xi1(i=0, 1, …, 4)
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图 6 异构系统xi2(i=0, 1, …, 4)轨迹图
Figure 6. Heterogeneous systems state trajectories of xi2(i=0, 1, …, 4)
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