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1. 北京航空航天大学 自动化科学与电气工程学院, 北京 100083;
2. 北京航空航天大学 飞行器控制一体化技术国防科技重点实验室, 北京 100083;
3. 北京电子工程总体研究所, 北京 100854;
4. 北京航空航天大学 大数据科学与脑机智能高精尖创新中心, 北京 100083

Time-varying output group formation tracking for heterogeneous multi-agent systems
TIAN Lei1,2, ZHAO Qilun3, DONG Xiwang1,2,4, LI Qingdong1,2, REN Zhang1,2,4
1. School of Automation Science and Electrical Engineering, Beihang University, Beijing 100083, China;
2. Science and Technology on Aircraft Control Laboratory, Beihang University, Beijing 100083, China;
3. Beijing Institute of Electronic System Engineering, Beijing 100854, China;
4. Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing 100083, China
Abstract: Air ground cooperative control is one of the hottest researches, and the differences between air ground agents' dynamic models represented by Quadrotor UAVs and unmanned vehicles bring great challenges. This paper investigates time-varying output group formation tracking control problems for heterogeneous high-order multi-agent systems with directed topologies. Agents in the structure of heterogeneous multi-agent systems' models given by this paper are classified into three types, including the virtual leader, the group leader, and the follower. The virtual leader can be used to control the trajectory of states associated with the whole heterogeneous multi-agent systems. Group leaders track the trajectory determined by the virtual leader and achieve the coordination among subgroups by communicating with each other. Followers track the states of group leaders and form the desired formation. Under the condition of directed topologies, based on neighboring information, observer theory and sliding mode control theory, a control protocol is constructed. Then the stability of heterogeneous multi-agent systems under the protocol is proven by using Lyapunov theory. Numerical simulations indicated that the approaches presented by this paper can be applied to the physical models represented by Quadrotor UAVs and Unmanned vehicles and have significant project application value.
Keywords: air ground cooperative control    heterogeneous high-order multi-agent systems    time-varying output group formation tracking    sliding mode control    observer theory

1 预备知识和问题描述 1.1 图论知识

1.2 问题描述

 $\left\{ {\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\dot x}}}_i}(t) = {\mathit{\boldsymbol{A}}_i}{\mathit{\boldsymbol{x}}_i}(t) + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{u}}_i}(t)}\\ {{\mathit{\boldsymbol{y}}_i}(t) = {\mathit{\boldsymbol{C}}_i}{\mathit{\boldsymbol{x}}_i}(t)} \end{array}} \right.$ （1）

 $\mathit{\boldsymbol{L}} = \left[ {\begin{array}{*{20}{c}} {\bf{0}}&{\bf{0}}&{\bf{0}}& \cdots &{\bf{0}}\\ {{\mathit{\boldsymbol{L}}_{{\rm{G,1}}}}}&{{\mathit{\boldsymbol{L}}_{{\rm{gro,1}}}}}&{\bf{0}}& \cdots &{\bf{0}}\\ {\bf{0}}&{{\mathit{\boldsymbol{L}}_{{\rm{G,2}}}}}&{{\mathit{\boldsymbol{L}}_{{\rm{gro,2}}}}}& \cdots &{\bf{0}}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {\bf{0}}&{{\mathit{\boldsymbol{L}}_{{\rm{G,1}} + M}}}&{\bf{0}}& \cdots &{{\mathit{\boldsymbol{L}}_{{\rm{gro,1}} + M}}} \end{array}} \right]$ （2）

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{E}}_g}{\mathit{\boldsymbol{A}}_1} = {\mathit{\boldsymbol{A}}_g}{\mathit{\boldsymbol{E}}_g} + {\mathit{\boldsymbol{B}}_g}{F_g}}\\ {{\bf{0}} = {\mathit{\boldsymbol{C}}_g}{\mathit{\boldsymbol{E}}_g} - {\mathit{\boldsymbol{C}}_1}} \end{array}} \right.$ （3）

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{E}}_j}{\mathit{\boldsymbol{A}}_{{{\bar g}_j}}} = {\mathit{\boldsymbol{A}}_j}{\mathit{\boldsymbol{E}}_j} + {\mathit{\boldsymbol{B}}_j}{F_j}}\\ {{\bf{0}} = {\mathit{\boldsymbol{C}}_j}{\mathit{\boldsymbol{E}}_j} - {\mathit{\boldsymbol{C}}_{{{\bar g}_j}}}} \end{array}} \right.$ （4）

 ${\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{H}}_g} - {\mathit{\boldsymbol{E}}_g}{\mathit{\boldsymbol{B}}_1} = {\bf{0}}$ （5）

 ${\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{H}}_j} - {\mathit{\boldsymbol{E}}_j}{\mathit{\boldsymbol{B}}_{{{\bar g}_j}}} = {\bf{0}}$ （6）

 $\left\{ {\begin{array}{*{20}{l}} {\mathop {{\rm{lim}}}\limits_{t \to \infty } ({\mathit{\boldsymbol{y}}_g}(t) - {\mathit{\boldsymbol{h}}_g}(t) - {\mathit{\boldsymbol{y}}_1}(t)) = {\bf{0}}}\\ {\mathop {{\rm{lim}}}\limits_{t \to \infty } ({\mathit{\boldsymbol{y}}_j}(t) - {\mathit{\boldsymbol{h}}_j}(t) - {\mathit{\boldsymbol{y}}_{{{\bar g}_j}}}(t)) = {\bf{0}}} \end{array}} \right.$ （7）

2 控制协议的设计和系统稳定性分析

1) 对于分组领导者g=2, 3, …, 1+M，设计如下控制协议：

 ${{\mathit{\boldsymbol{\dot {\hat x}}}}_{g,1}}(t) = {\mathit{\boldsymbol{A}}_1}{{\mathit{\boldsymbol{\hat x}}}_{g,1}}(t) - {\alpha _g}{\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{\zeta }}_{g,1}}(t) - {\beta _g}{\mathit{\boldsymbol{B}}_1}{{\mathit{\boldsymbol{\hat f}}}_g}(t)$ （8a）
 $\begin{array}{l} {\mathit{\boldsymbol{\zeta }}_{g,1}}(t) = {w_{g1}}({{\mathit{\boldsymbol{\hat x}}}_{g,1}}(t) - {\mathit{\boldsymbol{x}}_1}(t)) + \sum\limits_{k = 2}^{1 + M} {{w_{gk}}} \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({{\mathit{\boldsymbol{\hat x}}}_{g,1}}(t) - {{\mathit{\boldsymbol{\hat x}}}_{k,1}}(t)) \end{array}$ （8b）
 ${{\mathit{\boldsymbol{\hat f}}}_g}(t) = {\rm{sgn }}(\mathit{\boldsymbol{B}}_1^{\rm{T}}{\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{\zeta }}_g},1(t))$ （8c）
 $\mathit{\boldsymbol{\dot {\hat x}}}(t) = {\mathit{\boldsymbol{A}}_g}{{\mathit{\boldsymbol{\hat x}}}_g}(t) + {\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{u}}_g}(t) + {\mathit{\boldsymbol{S}}_g}({\mathit{\boldsymbol{C}}_g}{{\mathit{\boldsymbol{\hat x}}}_g}(t) - {\mathit{\boldsymbol{y}}_g}(t))$ （8d）
 $\begin{array}{l} \mathit{\boldsymbol{u}}(t) = \mathit{\boldsymbol{K}}_g^1({{\mathit{\boldsymbol{\hat x}}}_g}(t) - {{\mathit{\boldsymbol{\varpi }}} _g}(t)) + \mathit{\boldsymbol{K}}_g^2{{\mathit{\boldsymbol{\hat x}}}_{g,1}} - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\beta _g}{\mathit{\boldsymbol{H}}_g}{\mathit{\boldsymbol{f}}_g}(t) + {\mathit{\boldsymbol{v}}_g}(t) \end{array}$ （8e）
 ${\mathit{\boldsymbol{f}}_g}(t) = {\rm{sgn}} (\mathit{\boldsymbol{H}}_g^{\rm{T}}\mathit{\boldsymbol{B}}_g^{\rm{T}}\mathit{\boldsymbol{Q}}_g^{\rm{T}}({{\mathit{\boldsymbol{\hat x}}}_g}(t) - {{\mathit{\boldsymbol{\varpi }}} _g}(t) - {\mathit{\boldsymbol{E}}_g}{{\mathit{\boldsymbol{\hat x}}}_{g,1}}))$ （8f）
 $\mathit{\boldsymbol{v}}(t) = - {{\mathit{\boldsymbol{\hat B}}}_{g,1}}({\mathit{\boldsymbol{A}}_g}{{\mathit{\boldsymbol{\varpi }}} _g}(t) - {{\mathit{\boldsymbol{{\dot {\varpi }} }}}_g}(t))$ （8g）

2) 对于跟随者j=2+M, …, 1+M+N，控制协议设计为

 $\begin{array}{l} {{\mathit{\boldsymbol{\dot {\hat x}}}}_{j,{{\bar g}_j}}}(t) = {\mathit{\boldsymbol{A}}_{{{\bar g}_j}}}{{\mathit{\boldsymbol{\hat x}}}_{j,{{\bar g}_j}}}(t) - {\alpha _j}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\beta _j}{\mathit{\boldsymbol{B}}_{{{\bar g}_j}}}{{\mathit{\boldsymbol{\hat f}}}_j}(t) - {\eta _j}(t){\mathit{\boldsymbol{S}}_{{{\bar g}_j}}}{{\mathit{\boldsymbol{\bar f}}}_j}(t) \end{array}$ （9a）
 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}}(t) = {\mathit{\boldsymbol{w}}_{j{{\bar g}_j}}}({{\mathit{\boldsymbol{\hat x}}}_{j,{{\bar g}_j}}}(t) - {{\mathit{\boldsymbol{\hat x}}}_{{{\bar g}_j}}}(t)) + }\\ {\sum\limits_{k \in {N_j}} {{\mathit{\boldsymbol{w}}_{jk}}} ({{\mathit{\boldsymbol{\hat x}}}_{j,{{\bar g}_j}}}(t) - {{\mathit{\boldsymbol{\hat x}}}_{k,{{\bar g}_j}}}(t))} \end{array}$ （9b）
 ${{{\mathit{\boldsymbol{\hat f}}}_j}(t) = {\rm{sgn}} (\mathit{\boldsymbol{B}}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}}(t))}$ （9c）
 ${{{\mathit{\boldsymbol{\bar f}}}_j}(t) = {\rm{sgn}} (\mathit{\boldsymbol{S}}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}}(t))}$ （9d）
 ${{{\dot \eta }_j}(t) = w_{j{{\bar g}_j}}^\beta {{\left\| {\mathit{\boldsymbol{S}}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}}(t)} \right\|}_1}}$ （9e）
 ${{\mathit{\boldsymbol{\dot {\hat x}}}}_j}(t) = {\mathit{\boldsymbol{A}}_j}{{\mathit{\boldsymbol{\hat x}}}_j}(t) + {\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{u}}_j}(t) + {\mathit{\boldsymbol{S}}_j}({\mathit{\boldsymbol{C}}_j}{{\mathit{\boldsymbol{\hat x}}}_j}(t) - {\mathit{\boldsymbol{y}}_j}(t))$ （9f）
 $\begin{array}{l} \mathit{\boldsymbol{u}}(t) = \mathit{\boldsymbol{K}}_j^1({{\mathit{\boldsymbol{\hat x}}}_j}(t) - {{\mathit{\boldsymbol{\varpi }}} _j}(t)) + \mathit{\boldsymbol{K}}_j^2{{\mathit{\boldsymbol{\hat x}}}_{j,{{\bar g}_j}}} - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\beta _j}{\mathit{\boldsymbol{H}}_j}{\mathit{\boldsymbol{f}}_j}(t) + {\mathit{\boldsymbol{v}}_j}(t) \end{array}$ （9g）
 ${\mathit{\boldsymbol{f}}_j}(t) = {\rm{sgn}} (\mathit{\boldsymbol{H}}_j^{\rm{T}}\mathit{\boldsymbol{B}}_j^{\rm{T}}\mathit{\boldsymbol{Q}}_j^{\rm{T}}({{\mathit{\boldsymbol{\hat x}}}_j}(t) - {{\mathit{\boldsymbol{\varpi }}} _j}(t) - {\mathit{\boldsymbol{E}}_j}{{\mathit{\boldsymbol{\hat x}}}_{j,{{\bar g}_j}}}))$ （9h）
 ${\mathit{\boldsymbol{v}}_j}(t) = - {{\mathit{\boldsymbol{\hat B}}}_{j,1}}({\mathit{\boldsymbol{A}}_j}{{\mathit{\boldsymbol{\varpi }}} _j}(t) - {{{\mathit{\boldsymbol{\dot {\varpi }}}} }_j}(t))$ （9i）

 ${\mathit{\boldsymbol{Q}}_i}({\mathit{\boldsymbol{A}}_i} + {\mathit{\boldsymbol{B}}_i}\mathit{\boldsymbol{K}}_i^1) + {({\mathit{\boldsymbol{A}}_i} + {\mathit{\boldsymbol{B}}_i}\mathit{\boldsymbol{K}}_i^1)^{\rm{T}}}{\mathit{\boldsymbol{Q}}_i} = - 2{\mathit{\boldsymbol{I}}_{{n_i}}}$ （10）

 ${\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{A}}_1} + {\mathit{\boldsymbol{A}}_1}{\mathit{\boldsymbol{P}}_1} - \mathit{\boldsymbol{P}}_1^2 + {\mathit{\boldsymbol{I}}_{{n_1}}} = {\bf{0}}$ （11）

 ${\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{A}}_{{{\bar g}_j}}} + {\mathit{\boldsymbol{A}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}} - \mathit{\boldsymbol{P}}_{{{\bar g}_j}}^2 + {\mathit{\boldsymbol{I}}_{{n_{{{\bar g}_j}}}}} = {\bf{0}}$ （12）

 ${\mathit{\boldsymbol{\hat B}}_{i,2}}{\mathit{\boldsymbol{A}}_i}{{\mathit{\boldsymbol{\varpi }}} _i}(t) - {\mathit{\boldsymbol{\hat B}}_{i,2}}{{\mathit{\boldsymbol{\dot {\varpi }}}} _i}(t) = {\bf{0}}$ （13）

1) 对于分组领导者g=2, 3, …, 1+M，定义xg, 1(t)=$\hat{\boldsymbol{x}}$g, 1(t)-x1(t)为分组领导者g对虚拟领导者1的状态x1(t)的估计误差，根据式(1)和式(8a)可得

 $\begin{array}{l} {{\mathit{\boldsymbol{\dot {\bar x}}}}_{g,1}}(t) = {\mathit{\boldsymbol{A}}_1}{{\mathit{\boldsymbol{\bar x}}}_{g,1}}(t) - {\alpha _g}{\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{\zeta }}_{g,1}}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\beta _g}{\mathit{\boldsymbol{B}}_1}{{\mathit{\boldsymbol{\hat f}}}_g}(t) - {\mathit{\boldsymbol{B}}_1}{\mathit{\boldsymbol{u}}_1}(t) \end{array}$ （14）

 ${{{\mathit{\boldsymbol{\tilde x}}}_1}(t) = {{[\mathit{\boldsymbol{\bar x}}_{2,1}^{\rm{T}}(t),\mathit{\boldsymbol{\bar x}}_{3,1}^{\rm{T}}(t), \cdots ,\mathit{\boldsymbol{\bar x}}_{1 + M,1}^{\rm{T}}(t)]}^{\rm{T}}}}$
 ${{\mathit{\boldsymbol{\zeta }}_1}(t) = {{[\mathit{\boldsymbol{\zeta }}_{2,1}^{\rm{T}}(t),\mathit{\boldsymbol{\zeta }}_{3,1}^{\rm{T}}(t), \cdots ,\mathit{\boldsymbol{\zeta }}_{1 + M,1}^{\rm{T}}(t)]}^{\rm{T}}}}$
 ${{{\mathit{\boldsymbol{\tilde F}}}_1}(t) = [{{\mathit{\boldsymbol{\hat f}}}_2}(t),{{\mathit{\boldsymbol{\hat f}}}_3}(t), \cdots ,{{\mathit{\boldsymbol{\hat f}}}_{1 + M}}(t)]}$
 ${{{\mathit{\boldsymbol{\tilde \alpha }}}_1} = {\rm{diag}} \{ {\alpha _2},{\alpha _3}, \cdots ,{\alpha _{1 + M}}\} }$
 ${{{\mathit{\boldsymbol{\tilde \beta }}}_1} = {\rm{diag}} \{ {\beta _2},{\beta _3}, \cdots ,{\beta _{1 + M}}\} }$
 ${{{\mathit{\boldsymbol{\tilde D}}}_1} = {\rm{diag}} \{ {d_2},{d_3}, \cdots ,{d_{1 + M}}\} }$
 ${{{\mathit{\boldsymbol{\tilde u}}}_1}(t) = {{\bf{1}}_M} \otimes {\mathit{\boldsymbol{u}}_1}(t)}$

 $\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\dot {\tilde x}}}}_1}(t) = ({\mathit{\boldsymbol{I}}_M} \otimes {\mathit{\boldsymbol{A}}_1}){{\mathit{\boldsymbol{\tilde x}}}_1}(t) - ({{\mathit{\boldsymbol{\tilde \alpha }}}_1} \otimes {\mathit{\boldsymbol{P}}_1}){{\mathit{\boldsymbol{\tilde \zeta }}}_1}(t) - }\\ {({{\mathit{\boldsymbol{\tilde \beta }}}_1} \otimes {\mathit{\boldsymbol{B}}_1}){{\mathit{\boldsymbol{\tilde F}}}_1}(t) - ({\mathit{\boldsymbol{I}}_M} \otimes {\mathit{\boldsymbol{B}}_1}){{\mathit{\boldsymbol{\tilde u}}}_1}(t)} \end{array}$ （15）

 $\begin{array}{l} {{\mathit{\boldsymbol{\dot {\tilde \zeta} }}}_1}(t) = ({\mathit{\boldsymbol{I}}_M} \otimes {\mathit{\boldsymbol{A}}_1}){{\mathit{\boldsymbol{\tilde \zeta }}}_1}(t) - [({\mathit{\boldsymbol{L}}_{{\rm{gro,1}}}}{{\mathit{\boldsymbol{\tilde \alpha }}}_1}) \otimes {\mathit{\boldsymbol{P}}_1}] \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\mathit{\boldsymbol{\tilde \zeta }}}_1}(t) - [({\mathit{\boldsymbol{L}}_{{\rm{gro,1}}}}{{\mathit{\boldsymbol{\tilde \beta }}}_1}) \otimes {\mathit{\boldsymbol{B}}_1}]{{\mathit{\boldsymbol{\tilde F}}}_1}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({\mathit{\boldsymbol{L}}_{{\rm{gro,1}}}} \otimes {\mathit{\boldsymbol{B}}_1}){{\mathit{\boldsymbol{\tilde u}}}_1}(t) \end{array}$ （16）

 ${V_{{\rm{a}},1}}(t) = \sum\limits_{g = 2}^{1 + M} {{d_g}} \mathit{\boldsymbol{\zeta }}_{g,1}^{\rm{T}}(t){\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{\zeta }}_{g,1}}(t)$ （17）

Va, 1(t)求导可得

 $\begin{array}{l} {{\dot V}_{{\rm{a}},1}}(t) = \mathit{\boldsymbol{\tilde \zeta }}_1^{\rm{T}}(t)[{{\mathit{\boldsymbol{\tilde D}}}_1} \otimes ({\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{A}}_1} + \mathit{\boldsymbol{A}}_1^{\rm{T}}{\mathit{\boldsymbol{P}}_1})]{{\mathit{\boldsymbol{\tilde \zeta }}}_1}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{\tilde \zeta }}_1^{\rm{T}}(t)\{ [({{\mathit{\boldsymbol{\tilde D}}}_1}{\mathit{\boldsymbol{L}}_{{\rm{gro,1}}}} + \mathit{\boldsymbol{L}}_{{\rm{gro,1}}}^{\rm{T}},{{\mathit{\boldsymbol{\tilde D}}}_1}){{\mathit{\boldsymbol{\tilde \alpha }}}_1}] \otimes \mathit{\boldsymbol{P}}_1^2\} \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\mathit{\boldsymbol{\tilde \zeta }}}_1}(t) - 2\mathit{\boldsymbol{\tilde \zeta }}_1^{\rm{T}}(t)[({{\mathit{\boldsymbol{\tilde D}}}_1}{\mathit{\boldsymbol{L}}_{{\rm{gro,1}}}}{{\mathit{\boldsymbol{\tilde \beta }}}_1}) \otimes ({\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{B}}_1})] \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\mathit{\boldsymbol{\tilde F}}}_1}(t) - 2\mathit{\boldsymbol{\tilde \zeta }}_1^{\rm{T}}[({{\mathit{\boldsymbol{\tilde D}}}_1}{\mathit{\boldsymbol{L}}_{{\rm{gro,1}}}}) \otimes ({\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{B}}_1})]{{\mathit{\boldsymbol{\tilde u}}}_1}(t) \end{array}$ （18）

 $\mathit{\boldsymbol{\zeta }}_{g,1}^{\rm{T}}(t){\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{B}}_1}{\mathit{\boldsymbol{\hat f}}_g}(t) = {\left\| {\mathit{\boldsymbol{B}}_1^{\rm{T}}{\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{\zeta }}_g},1(t)} \right\|_1}$ （19）
 $\mathit{\boldsymbol{\zeta }}_{g,1}^{\rm{T}}(t){\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{B}}_1}{\mathit{\boldsymbol{\hat f}}_g}(t) \le {\left\| {\mathit{\boldsymbol{B}}_1^{\rm{T}}{\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{\zeta }}_g},1(t)} \right\|_1}$ （20）

 $\begin{array}{l} - 2\mathit{\boldsymbol{\tilde \zeta }}_1^{\rm{T}}(t)(({{\mathit{\boldsymbol{\tilde D}}}_1}{\mathit{\boldsymbol{L}}_{{\rm{gro,1}}}}{{\mathit{\boldsymbol{\tilde \beta }}}_1}) \otimes ({\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{B}}_1})){{\mathit{\boldsymbol{\tilde F}}}_1}(t) \le \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - 2\gamma \sum\limits_{g = 2}^{1 + M} {{d_g}} \mathit{\boldsymbol{\zeta }}_{1,g}^{\rm{T}}(t){\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{B}}_1}{\mathit{\boldsymbol{w}}_{g1}}{{\mathit{\boldsymbol{\hat f}}}_g}(t) - 2\gamma \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{g = 2}^{1 + M} {{d_g}} \mathit{\boldsymbol{\zeta }}_{1,g}^{\rm{T}}(t){\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{B}}_1}\sum\limits_{k = 2}^{1 + M} {{\mathit{\boldsymbol{w}}_{gk}}} ({{\mathit{\boldsymbol{\hat f}}}_g}(t) - {{\mathit{\boldsymbol{\hat f}}}_k}(t)) \le \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - 2\gamma \sum\limits_{g = 2}^{1 + M} {{d_g}} {w_{g1}}{\left\| {\mathit{\boldsymbol{B}}_1^{\rm{T}}{\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{\zeta }}_{g,1}}(t)} \right\|_1} \end{array}$ （21）

 $\begin{array}{*{20}{c}} { - 2\mathit{\boldsymbol{\tilde \zeta }}_1^{\rm{T}}(({{\mathit{\boldsymbol{\tilde D}}}_1}{\mathit{\boldsymbol{L}}_{{\rm{gro,1}}}}) \otimes ({\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{B}}_1})){{\mathit{\boldsymbol{\tilde u}}}_1}(t) = }\\ \begin{array}{l} {\kern 1pt} {\kern 1pt} - 2\sum\limits_{g = 2}^{1 + M} {{d_g}} {w_{g1}}\mathit{\boldsymbol{\zeta }}_{g,1}^{\rm{T}},{\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{B}}_1}{\mathit{\boldsymbol{u}}_1}(t) \le \\ 2\gamma \sum\limits_{g = 2}^{1 + M} {{d_g}} {w_{g1}}{\left\| {\mathit{\boldsymbol{B}}_1^{\rm{T}}{\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{\zeta }}_{g,1}}(t)} \right\|_1} \end{array} \end{array}$ （22）

 $\begin{array}{l} {{\dot V}_{{\rm{a}},1}}(t) \le \mathit{\boldsymbol{\tilde \zeta }}_1^{\rm{T}}(t)[{{\mathit{\boldsymbol{\tilde D}}}_1} \otimes ({\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{A}}_1} + \mathit{\boldsymbol{A}}_1^{\rm{T}}{\mathit{\boldsymbol{P}}_1})]{\mathit{\boldsymbol{\zeta }}_1}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{\tilde \zeta }}_1^{\rm{T}}(t)\{ [({{\mathit{\boldsymbol{\tilde D}}}_1}{\mathit{\boldsymbol{L}}_{{\rm{gro,1}}}} + \mathit{\boldsymbol{L}}_{{\rm{gro,1}}}^{\rm{T}},{{\mathit{\boldsymbol{\tilde D}}}_1}){{\mathit{\boldsymbol{\tilde \alpha }}}_1}] \otimes \mathit{\boldsymbol{P}}_1^2\} {{\mathit{\boldsymbol{\tilde \zeta }}}_1}(t) \end{array}$ （23）

 $\begin{array}{l} {{\dot V}_{{\rm{a}},1}}(t) \le \mathit{\boldsymbol{\tilde \zeta }}_1^{\rm{T}}(t)[{{\mathit{\boldsymbol{\tilde D}}}_1} \otimes ({\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{A}}_1} + \mathit{\boldsymbol{A}}_1^{\rm{T}}{\mathit{\boldsymbol{P}}_1})]{{\mathit{\boldsymbol{\tilde \zeta }}}_1}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\bar \lambda }_{\min }}\mathit{\boldsymbol{\tilde \zeta }}_1^{\rm{T}}(t)({{\mathit{\boldsymbol{\tilde \alpha }}}_1} \otimes \mathit{\boldsymbol{P}}_1^2){{\mathit{\boldsymbol{\tilde \zeta }}}_1}(t) \le \mathit{\boldsymbol{\tilde \zeta }}_1^{\rm{T}}(t)[{{\mathit{\boldsymbol{\tilde D}}}_1} \otimes \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({\mathit{\boldsymbol{P}}_1}{\mathit{\boldsymbol{A}}_1} + \mathit{\boldsymbol{A}}_1^{\rm{T}}{\mathit{\boldsymbol{P}}_1} - \mathit{\boldsymbol{P}}_1^2)]{{\mathit{\boldsymbol{\tilde \zeta }}}_1}(t) \le 0 \end{array}$ （24）

 ${\mathit{\boldsymbol{\dot {\bar x}}}_{g,g}}(t) = ({\mathit{\boldsymbol{A}}_g} + {\mathit{\boldsymbol{S}}_g}{\mathit{\boldsymbol{C}}_g}){\mathit{\boldsymbol{\bar x}}_{g,g}}(t)$ （25）

 $\begin{array}{l} {{\mathit{\boldsymbol{\dot x}}}_g}(t) = ({\mathit{\boldsymbol{A}}_g} + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1){\mathit{\boldsymbol{x}}_g}(t) + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1{{\mathit{\boldsymbol{\bar x}}}_{g,g}}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {{\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^2{\mathit{\boldsymbol{x}}_1}(t) + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^2{{\mathit{\boldsymbol{\bar x}}}_{g,1}}(t) - {\beta _g}{\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{H}}_g}{\mathit{\boldsymbol{f}}_g}(t) - }\\ {{\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1{{\mathit{\boldsymbol{\varpi }}} _g}(t) - {\mathit{\boldsymbol{B}}_g}{{\mathit{\boldsymbol{\hat B}}}_{g,1}}({\mathit{\boldsymbol{A}}_g}{\varpi _g}(t) - {{{\mathit{\boldsymbol{\dot {\varpi} }}} }_g}(t))} \end{array} \end{array}$ （26）

εg(t)=xg(t)-ϖg(t)-Egx1(t)，可得

 $\begin{array}{l} \begin{array}{*{20}{r}} {{{\mathit{\boldsymbol{\dot {\varepsilon} }}}_g}(t) = {{\mathit{\boldsymbol{\dot x}}}_g}(t) - {{{\mathit{\boldsymbol{\dot {\varpi} }}} }_g}(t) - {\mathit{\boldsymbol{E}}_g}{{\mathit{\boldsymbol{\dot x}}}_1}(t) = ({\mathit{\boldsymbol{A}}_g} + }\\ {{\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1){\mathit{\boldsymbol{\varepsilon }}_g}(t) + ({\mathit{\boldsymbol{A}}_g} + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1){\mathit{\boldsymbol{E}}_g}{\mathit{\boldsymbol{x}}_1}(t) - } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {{\mathit{\boldsymbol{E}}_g}({\mathit{\boldsymbol{A}}_1}{\mathit{\boldsymbol{x}}_1}(t) + {\mathit{\boldsymbol{B}}_1}{\mathit{\boldsymbol{u}}_1}(t)) + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1{{\mathit{\boldsymbol{\bar x}}}_{g,g}}(t) + }\\ {{\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^2{\mathit{\boldsymbol{x}}_1}(t) + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^2{{\mathit{\boldsymbol{\bar x}}}_{g,1}}(t) - {\beta _g}{\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{H}}_g}{\mathit{\boldsymbol{f}}_g}(t) - }\\ {{\mathit{\boldsymbol{B}}_g}{{\mathit{\boldsymbol{\hat B}}}_{g,1}}({\mathit{\boldsymbol{A}}_g}{{\mathit{\boldsymbol{\varpi }}} _g}(t) - {{{\mathit{\boldsymbol{\dot {\varpi} }}} }_g}(t)) - {{{\mathit{\boldsymbol{\dot {\varpi} }}} }_g}(t) + {\mathit{\boldsymbol{A}}_g}{{\mathit{\boldsymbol{\varpi }}} _g}(t)} \end{array} \end{array}$ （27）

δg(t)=-Bg$\hat{\boldsymbol{B}}$g, 1(Agϖg(t)$\dot{\boldsymbol{\varpi}}$g(t))$\dot{\boldsymbol{\varpi}}$g(t)+Agϖg(t)，由引理1与编队可行性条件(13)可得δg(t)=0，因此式(27)可写为

 $\begin{array}{l} {{\mathit{\boldsymbol{\dot \varepsilon }}}_g}(t) = ({\mathit{\boldsymbol{A}}_g} + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1){\mathit{\boldsymbol{\varepsilon }}_g}(t) + ({\mathit{\boldsymbol{A}}_g} + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1){\mathit{\boldsymbol{E}}_g}{\mathit{\boldsymbol{x}}_1}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{E}}_g}({\mathit{\boldsymbol{A}}_1}{\mathit{\boldsymbol{x}}_1}(t) + {\mathit{\boldsymbol{B}}_1}{\mathit{\boldsymbol{u}}_1}(t)) + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1{{\mathit{\boldsymbol{\bar x}}}_{g,g}}(t) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^2{\mathit{\boldsymbol{x}}_1}(t) + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^2{{\mathit{\boldsymbol{\bar x}}}_{g,1}}(t) - {\beta _g}{\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{H}}_g}{\mathit{\boldsymbol{f}}_g}(t) \end{array}$ （28）

 $\begin{array}{l} \begin{array}{*{20}{c}} {\mathit{\boldsymbol{\dot \varepsilon }}(t) = ({\mathit{\boldsymbol{A}}_g} + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1){\mathit{\boldsymbol{\varepsilon }}_g}(t) + ({\mathit{\boldsymbol{A}}_g} + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1){\mathit{\boldsymbol{E}}_g} \cdot }\\ {{\mathit{\boldsymbol{x}}_1}(t) - {\mathit{\boldsymbol{E}}_g}({\mathit{\boldsymbol{A}}_1}{\mathit{\boldsymbol{x}}_1}(t) + {\mathit{\boldsymbol{B}}_1}{\mathit{\boldsymbol{u}}_1}(t)) + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1 \cdot } \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} \begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\bar x}}}_{g,g}}(t) + {\mathit{\boldsymbol{B}}_g}({\mathit{\boldsymbol{F}}_g} - \mathit{\boldsymbol{K}}_g^1{\mathit{\boldsymbol{E}}_g}){\mathit{\boldsymbol{x}}_1}(t) + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^2 \cdot }\\ {{{\mathit{\boldsymbol{\bar x}}}_{g,1}}(t) - {\beta _g}{\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{H}}_g}{\mathit{\boldsymbol{f}}_g}(t) = ({\mathit{\boldsymbol{A}}_g} + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1) \cdot }\\ {{\mathit{\boldsymbol{\varepsilon }}_g}(t) - {\mathit{\boldsymbol{E}}_g}{\mathit{\boldsymbol{B}}_1}{u_1}(t) + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1{{\mathit{\boldsymbol{\bar x}}}_{g,g}}(t) + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^2 \cdot } \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}} {{{\mathit{\boldsymbol{\bar x}}}_{g,1}}(t) - {\beta _g}{\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{H}}_g}{\mathit{\boldsymbol{f}}_g}(t) = ({\mathit{\boldsymbol{A}}_g} + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1) \cdot }\\ {{\mathit{\boldsymbol{\varepsilon }}_g}(t) - {\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{H}}_g}{\mathit{\boldsymbol{u}}_1}(t) + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1{{\mathit{\boldsymbol{\bar x}}}_{g,g}}(t) + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^2 \cdot }\\ {{{\mathit{\boldsymbol{\bar x}}}_{g,1}}(t) - {\beta _g}{\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{H}}_g}{\mathit{\boldsymbol{f}}_g}(t)} \end{array} \end{array}$ （29）

 ${V_{g,1}}(t) = \mathit{\boldsymbol{\varepsilon }}_g^{\rm{T}}(t){\mathit{\boldsymbol{Q}}_g}{\mathit{\boldsymbol{\varepsilon }}_g}(t)$ （30）

 $\begin{array}{l} {{\dot V}_{g,1}}(t) = \mathit{\boldsymbol{\varepsilon }}_g^{\rm{T}}(t)({\mathit{\boldsymbol{Q}}_g}({\mathit{\boldsymbol{A}}_g} + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1) = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {({\mathit{\boldsymbol{A}}_g} + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1)^{\rm{T}}}{\mathit{\boldsymbol{Q}}_g}){\mathit{\boldsymbol{\varepsilon }}_g}(t) + {\mathit{\boldsymbol{\xi }}_g}(t) \end{array}$ （31）

 $\begin{array}{l} {\mathit{\boldsymbol{\xi }}_g}(t) = - 2{\beta _g}\mathit{\boldsymbol{\varepsilon }}_g^{\rm{T}}(t){\mathit{\boldsymbol{Q}}_g}{\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{H}}_g}{\mathit{\boldsymbol{f}}_g}(t) - 2\mathit{\boldsymbol{\varepsilon }}_g^{\rm{T}}(t) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{Q}}_g}{\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{H}}_g}{\mathit{\boldsymbol{u}}_1}(t) + 2\mathit{\boldsymbol{\varepsilon }}_g^{\rm{T}}(t){\mathit{\boldsymbol{Q}}_g}({\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1{{\mathit{\boldsymbol{\bar x}}}_{g,g}}(t) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^2{{\mathit{\boldsymbol{\bar x}}}_{g,1}}(t)) \end{array}$

$\hat{\boldsymbol{\varepsilon}}_{g}$(t)=$\hat{\boldsymbol{\varepsilon}}_{g}$(t)-ϖg(t)-Eg$\hat{\boldsymbol{x}}$g, 1(t)，同时定义εg(t)=$\hat{\boldsymbol{\varepsilon}}_{g}$(t)-εg(t)，进而可以推出εg(t)=xg, g(t)-Egxg, 1(t)。根据假设1和算法1中的步骤2可知

 $\begin{array}{l} {\mathit{\boldsymbol{\xi }}_g}(t) = - 2{\beta _g}\mathit{\boldsymbol{\hat \varepsilon }}_g^{\rm{T}}(t){\mathit{\boldsymbol{Q}}_g}{\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{H}}_g}{\mathit{\boldsymbol{f}}_g}(t) + 2{\beta _g}\mathit{\boldsymbol{\bar \varepsilon }}_g^{\rm{T}}(t) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\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}}_g}{\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{H}}_g}{\mathit{\boldsymbol{f}}_g}(t) - \mathit{\boldsymbol{\hat \varepsilon }}_g^{\rm{T}}(t){\mathit{\boldsymbol{Q}}_g}{\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{H}}_g}{\mathit{\boldsymbol{u}}_1}(t) + 2\mathit{\boldsymbol{\bar \varepsilon }}_g^{\rm{T}}(t) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\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}} {{\mathit{\boldsymbol{Q}}_g}{\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{H}}_g}{\mathit{\boldsymbol{u}}_1}(t) + 2\mathit{\boldsymbol{\varepsilon }}_g^{\rm{T}}(t){\mathit{\boldsymbol{Q}}_g}({\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1{{\mathit{\boldsymbol{\bar x}}}_{g,g}}(t) + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^2 \cdot }\\ {{{\mathit{\boldsymbol{\bar x}}}_{g,1}}(t)) \le - 2({\beta _g} - \gamma ){{\left\| {\mathit{\boldsymbol{H}}_g^{\rm{T}}\mathit{\boldsymbol{B}}_g^{\rm{T}}\mathit{\boldsymbol{Q}}_g^{\rm{T}}{{\mathit{\boldsymbol{\hat \varepsilon }}}_g}(t)} \right\|}_1} + } \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} \begin{array}{*{20}{l}} {2({\beta _g} + \gamma ){{\left\| {\mathit{\boldsymbol{H}}_g^{\rm{T}}\mathit{\boldsymbol{B}}_g^{\rm{T}}\mathit{\boldsymbol{Q}}_g^{\rm{T}}{{\mathit{\boldsymbol{\bar \varepsilon }}}_g}(t)} \right\|}_1} + 2\mathit{\boldsymbol{\varepsilon }}_g^T(t){\mathit{\boldsymbol{Q}}_g} \cdot }\\ {({\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1{\mathit{\boldsymbol{x}}_{g,g}}(t) + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^2{{\mathit{\boldsymbol{\bar x}}}_{g,1}}(t)) \le 2({\beta _g} + \gamma ) \cdot } \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} \begin{array}{*{20}{l}} {{{\left\| {\mathit{\boldsymbol{H}}_g^{\rm{T}}\mathit{\boldsymbol{B}}_g^{\rm{T}}\mathit{\boldsymbol{Q}}_g^{\rm{T}}{{\mathit{\boldsymbol{\bar \varepsilon }}}_g}(t)} \right\|}_1} + 2\mathit{\boldsymbol{\varepsilon }}_g^{\rm{T}}(t){\mathit{\boldsymbol{Q}}_g}({\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^1 \cdot }\\ {{{\mathit{\boldsymbol{\bar x}}}_{g,g}}(t) + {\mathit{\boldsymbol{B}}_g}\mathit{\boldsymbol{K}}_g^2{{\mathit{\boldsymbol{\bar x}}}_{g,1}}(t))} \end{array} \end{array}$ （32）

σg(t)=Qg(BgKg1xg, g(t)+BgKg2xg, 1(t))，根据Young不等式可得

 $2\mathit{\boldsymbol{\varepsilon }}_g^{\rm{T}}(t){\mathit{\boldsymbol{\sigma }}_g}(t) \le \mathit{\boldsymbol{\varepsilon }}_g^{\rm{T}}(t){\mathit{\boldsymbol{\varepsilon }}_g}(t) + \mathit{\boldsymbol{\sigma }}_g^{\rm{T}}(t){\mathit{\boldsymbol{\sigma }}_g}(t)$ （33）

 $\begin{array}{*{20}{c}} {{{\dot V}_{g,1}}(t) \le - \frac{1}{{{\lambda _{{\rm{max}}}}({\mathit{\boldsymbol{Q}}_g})}}{V_{g,1}}(t) + 2({\beta _g} + \gamma ) \cdot }\\ {{{\left\| {\mathit{\boldsymbol{H}}_g^{\rm{T}}\mathit{\boldsymbol{B}}_g^{\rm{T}}\mathit{\boldsymbol{Q}}_g^{\rm{T}}{\mathit{\boldsymbol{\varepsilon }}_g}(t)} \right\|}_1} + \mathit{\boldsymbol{\sigma }}_g^{\rm{T}}(t){\mathit{\boldsymbol{\sigma }}_g}(t)} \end{array}$ （34）

2) 对于跟随者j=2+M, …, 1+M+N，定义xj,gj(t)=$\hat{\boldsymbol{x}}$j, gj(t)-$\hat{\boldsymbol{x}}$gj(t)为跟随者j对其分组领导者gj自身估计状态$\hat{\boldsymbol{x}}$gj(t)的估计误差。令ϕgj(t)=Cgj$\hat{\boldsymbol{x}}$gj(t)-ygj(t)=Cgjxgj, j(t)，由$\mathop {\lim }\limits_{t \to \infty }$xgj, gj(t)=0可知ϕg(t)是有界的，即||ϕg(t)1||≤γ，假定γ为未知的正实数。不失一般性，以分组gj∈{2, 3, …, 1+M}为例进行证明，假设该组有qgj个跟随者，令Ngj=$\sum\limits_{i = 1}^{{{\overline g }_j} - 1} {{q_i}}$，则跟随者j标号为2+Ngj, 3+Ngj, …, 1+qgj+Ngj，根据式(1)和式(9a)可得

 $\begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot {\bar x}}}(t) = {\mathit{\boldsymbol{A}}_{{{\bar g}_j}}}{{\mathit{\boldsymbol{\bar x}}}_{j,{{\bar g}_j}}}(t) - {\alpha _j}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}}(t) - {\beta _j}{\mathit{\boldsymbol{B}}_{{{\bar g}_j}}}}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\mathit{\boldsymbol{\hat f}}}_j}(t) - {\mathit{\boldsymbol{B}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{u}}_{{{\bar g}_j}}}(t) - {\eta _j}(t){\mathit{\boldsymbol{S}}_{{{\bar g}_j}}} \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\mathit{\boldsymbol{\bar f}}}_j}(t) - {\mathit{\boldsymbol{S}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{f}}_{{{\bar g}_j}}}(t)} \end{array}$ （35）

 $\begin{array}{l} {{\mathit{\boldsymbol{\tilde x}}}_{{{\bar g}_j}}}(t) = [\mathit{\boldsymbol{\bar x}}_{2 + {{\bar N}_{{{\bar g}_j}}},{{\bar g}_j}}^{\rm{T}}(t),\mathit{\boldsymbol{\bar x}}_{3 + {{\bar N}_{{{\bar g}_j}}},{{\bar g}_j}}^{\rm{T}}(t), \cdots ,\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{\bar x}}_{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{{{\bar g}_j}}},{{\bar g}_j}}^{\rm{T}}(t){]^{\rm{T}}} \end{array}$
 $\begin{array}{l} {{\mathit{\boldsymbol{\tilde \zeta }}}_{{{\bar g}_j}}}(t) = [\mathit{\boldsymbol{\zeta }}_{2 + {{\bar N}_{{{\bar g}_j}}},{{\bar g}_j}}^{\rm{T}}(t),\mathit{\boldsymbol{\zeta }}_{3 + {{\bar N}_{{{\bar g}_j}}},{{\bar g}_j}}^{\rm{T}}(t), \cdots ,\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{\zeta }}_{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{{{\bar g}_j}}},{{\bar g}_j}}^{\rm{T}}(t){]^{\rm{T}}} \end{array}$
 ${\mathit{\boldsymbol{\tilde F}}_{{{\bar g}_j}}}(t) = {[\mathit{\boldsymbol{\hat f}}_{2 + {{\bar N}_{{{\bar g}_j}}}}^{\rm{T}}(t),\mathit{\boldsymbol{\hat f}}_{3 + {{\bar N}_{{{\bar g}_j}}}}^{\rm{T}}(t), \cdots ,\mathit{\boldsymbol{\hat f}}_{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{{{\bar g}_j}}}}^{\rm{T}}(t)]^{\rm{T}}}$
 ${\mathit{\boldsymbol{\bar F}}_{{{\bar g}_j}}}(t) = {[\mathit{\boldsymbol{\bar f}}_{2 + {{\bar N}_{{{\bar g}_j}}}}^{\rm{T}}(t),\mathit{\boldsymbol{\bar f}}_{3 + {{\bar N}_{{{\bar g}_j}}}}^{\rm{T}}(t), \cdots ,\mathit{\boldsymbol{\bar f}}_{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{{{\bar g}_j}}}}^{\rm{T}}(t)]^{\rm{T}}}$
 ${\mathit{\boldsymbol{\bar {\varGamma }}}_{{{\bar g}_j}}}(t) = {[\eta _{2 + {{\bar N}_{{{\bar g}_j}}}}^{\rm{T}}(t),\eta _{3 + {{\bar N}_{{{\bar g}_j}}}}^{\rm{T}}(t), \cdots ,\eta _{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{{{\bar g}_j}}}}^{\rm{T}}(t)]^{\rm{T}}}$
 ${\mathit{\boldsymbol{\tilde \alpha }}_{{{\bar g}_j}}} = {\rm{diag}} \{ {\alpha _{2 + {{\bar N}_{{{\bar g}_j}}}}},{\alpha _{3 + {{\bar N}_{{{\bar g}_j}}}}}, \cdots ,{\alpha _{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{{{\bar g}_j}}}}}\}$
 ${\mathit{\boldsymbol{\tilde \beta }}_{{{\bar g}_j}}} = {\rm{diag}} \{ {\beta _{2 + {{\bar N}_{{{\bar g}_j}}}}},{\beta _{3 + {{\bar N}_{{{\bar g}_j}}}}}, \cdots ,{\beta _{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{{{\bar g}_j}}}}}\}$
 ${\mathit{\boldsymbol{\tilde D}}_{{{\bar g}_j}}} = {\rm{diag}} \{ {d_{2 + {{\bar N}_{{{\bar g}_j}}}}},{d_{3 + {{\bar N}_{{{\bar g}_j}}}}}, \cdots ,{d_{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{{{\bar g}_j}}}}}\}$
 ${\mathit{\boldsymbol{\bar \varPhi }}_{{{\bar g}_j}}}(t) = {{\bf{1}}_{{q_{{{\bar g}_j}}}}} \otimes {{\mathit{\boldsymbol{\phi }}} _{{{\bar g}_j}}}(t),{\mathit{\boldsymbol{\tilde u}}_{{{\bar g}_j}}}(t) = {{\bf{1}}_{{q_{{{\bar g}_j}}}}} \otimes {\mathit{\boldsymbol{u}}_{{{\bar g}_j}}}(t)$

 $\begin{array}{l} \begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\dot {\tilde x}}}}_{{{\bar g}_j}}}(t) = ({\mathit{\boldsymbol{I}}_{{q_{{{\bar g}_j}}}}} \otimes {\mathit{\boldsymbol{A}}_{{{\bar g}_j}}}){{\mathit{\boldsymbol{\tilde x}}}_{{{\bar g}_j}}}(t) - ({{\mathit{\boldsymbol{\tilde \alpha }}}_{{{\bar g}_j}}} \otimes {\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}){{\mathit{\boldsymbol{\tilde \zeta }}}_{{{\bar g}_j}}}(t) - }\\ {({{\mathit{\boldsymbol{\tilde \beta }}}_{{{\bar g}_j}}} \otimes {\mathit{\boldsymbol{B}}_{{{\bar g}_j}}}){{\mathit{\boldsymbol{\tilde F}}}_{{{\bar g}_j}}}(t) - ({\mathit{\boldsymbol{I}}_{{q_{{{\bar g}_j}}}}} \otimes {\mathit{\boldsymbol{B}}_{{{\bar g}_j}}}){{\mathit{\boldsymbol{\tilde u}}}_{{{\bar g}_j}}}(t) - } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({{\mathit{\boldsymbol{\bar \varGamma }}}_{{{\bar g}_j}}}(t) \otimes {\mathit{\boldsymbol{S}}_{{{\bar g}_j}}}){{\mathit{\boldsymbol{\bar F}}}_{{{\bar g}_j}}}(t) - ({\mathit{\boldsymbol{I}}_{{q_{{{\bar g}_j}}}}} \otimes {\mathit{\boldsymbol{S}}_{{{\bar g}_j}}}){{\mathit{\boldsymbol{\bar \varPhi }}}_{{{\bar g}_j}}}(t) \end{array}$ （36）

 $\begin{array}{l} {{\mathit{\boldsymbol{\dot {\tilde \zeta} }}}_{{{\bar g}_j}}}(t) = ({\mathit{\boldsymbol{I}}_{{q_{{{\bar g}_j}}}}} \otimes {\mathit{\boldsymbol{A}}_{{{\bar g}_j}}} - {\mathit{\boldsymbol{L}}_{{\rm{gro}},{{\bar g}_j}}}{{\mathit{\boldsymbol{\tilde \alpha }}}_{{{\bar g}_j}}} \otimes {\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}){{\mathit{\boldsymbol{\tilde \zeta }}}_{{{\bar g}_j}}}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} [({\mathit{\boldsymbol{L}}_{{\rm{gro}},{{\bar g}_j}}}{{\tilde \beta }_{{{\bar g}_j}}}) \otimes {\mathit{\boldsymbol{B}}_{{{\bar g}_j}}}]{{\mathit{\boldsymbol{\tilde F}}}_{{{\bar g}_j}}}(t) - ({\mathit{\boldsymbol{L}}_{{\rm{gro}},{{\bar g}_j}}} \otimes {\mathit{\boldsymbol{B}}_{{{\bar g}_j}}}) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\tilde u}}}_{{{\bar g}_j}}}(t) - [({\mathit{\boldsymbol{L}}_{{\rm{gro}},{{\bar g}_j}}}{{\mathit{\boldsymbol{\bar \varGamma }}}_{{{\bar g}_j}}}(t)] \otimes {\mathit{\boldsymbol{S}}_{{{\bar g}_j}}}){{\mathit{\boldsymbol{\bar F}}}_{{{\bar g}_j}}}(t) - }\\ {({\mathit{\boldsymbol{L}}_{{\rm{gro}},{{\bar g}_j}}} \otimes {\mathit{\boldsymbol{S}}_{{{\bar g}_j}}}){{\mathit{\boldsymbol{\bar \varPhi }}}_{{{\bar g}_j}}}(t)} \end{array} \end{array}$ （37）

 $\begin{array}{l} {V_{{\rm{b}},{{\bar g}_j}}}(t) = \mathop \sum \limits_{j = 2 + {{\bar N}_{{{\bar g}_j}}}}^{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{{{\bar g}_j}}}} {d_j}\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}^{\rm{T}}(t){\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}}(t) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathop \sum \limits_{j = 2 + {{\bar N}_{{{\bar g}_j}}}}^{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{{{\bar g}_j}}}} {d_j}{({\eta _j}(t) - \bar \eta )^2} \end{array}$ （38）

 $\begin{array}{l} {{\dot V}_{{\rm{b}},{{\bar g}_j}}}(t) = \mathit{\boldsymbol{\tilde \zeta }}_{{{\bar g}_j}}^{\rm{T}}(t)[{{\mathit{\boldsymbol{\tilde D}}}_{{{\bar g}_j}}} \otimes ({\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{A}}_{{{\bar g}_j}}} + \mathit{\boldsymbol{A}}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}})] \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\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{\tilde \zeta }}}_{{{\bar g}_j}}}(t) - \mathit{\boldsymbol{\tilde \zeta }}_{{{\bar g}_j}}^{\rm{T}}(t)\{ [({{\mathit{\boldsymbol{\tilde D}}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{L}}_{{\rm{gro}},{{\bar g}_j}}} + \mathit{\boldsymbol{L}}_{{\rm{gro}},{{\bar g}_j}}^{\rm{T}}{{\tilde D}_{{{\bar g}_j}}}) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\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{\tilde \alpha }}}_{{{\bar g}_j}}}] \otimes \mathit{\boldsymbol{P}}_{{{\bar g}_j}}^2\} {{\mathit{\boldsymbol{\tilde \zeta }}}_{{{\bar g}_j}}}(t) - 2\mathit{\boldsymbol{\tilde \zeta }}_{{{\bar g}_j}}^{\rm{T}}(t)[({{\mathit{\boldsymbol{\tilde D}}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{L}}_{{\rm{gro}},{{\bar g}_j}}} \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\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}} {{{\mathit{\boldsymbol{\tilde \beta }}}_{{{\bar g}_j}}}) \otimes ({\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{B}}_{{{\bar g}_j}}})]{{\mathit{\boldsymbol{\tilde F}}}_{{{\bar g}_j}}}(t) - 2\mathit{\boldsymbol{\tilde \zeta }}_{\bar g}^{\rm{T}}[({{\mathit{\boldsymbol{\tilde D}}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{L}}_{{\rm{gro}},{{\bar g}_j}}}) \otimes }\\ {({\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{B}}_{{{\bar g}_j}}})]{{\mathit{\boldsymbol{\tilde u}}}_{{{\bar g}_j}}}(t) - 2\mathit{\boldsymbol{\tilde \zeta }}_{{{\bar g}_j}}^{\rm{T}}(t)[({{\mathit{\boldsymbol{\tilde D}}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{L}}_{gro,{{\bar g}_j}}} \cdot } \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}} {{{\mathit{\boldsymbol{\bar \varGamma }}}_{{{\bar g}_j}}}(t)) \otimes ({\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{S}}_{{{\bar g}_j}}})]{{\mathit{\boldsymbol{\bar F}}}_{{{\bar g}_j}}}(t) - 2\mathit{\boldsymbol{\tilde \zeta }}_{{{\bar g}_j}}^{\rm{T}} \cdot }\\ {[({{\mathit{\boldsymbol{\tilde D}}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{L}}_{{\rm{gro}},{{\bar g}_j}}}) \otimes ({\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{S}}_{{{\bar g}_j}}})]{{\mathit{\boldsymbol{\bar \varPhi }}}_{{{\bar g}_j}}}(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} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2\sum\limits_{j = 2 + {{\bar N}_{{{\bar g}_j}}}}^{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{\bar g}}_j} {{d_j}({\eta _j}(t) - \bar \eta )w_{j{{\bar g}_j}}^\beta {{\left\| {\mathit{\boldsymbol{S}}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}}(t)} \right\|}_1}} \end{array}$ （39）

 $\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}^{\rm{T}}(t){\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{B}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{\hat f}}_j}(t) = {\left\| {\mathit{\boldsymbol{B}}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}}(t)} \right\|_1}$ （40a）
 $\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}^{\rm{T}}(t){\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{B}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{\hat f}}_k}(t) \le {\left\| {\mathit{\boldsymbol{B}}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}}(t)} \right\|_1}$ （40b）
 $\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}^{\rm{T}}(t){\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{S}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{\hat f}}_j}(t) = {\left\| {\mathit{\boldsymbol{S}}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}}(t)} \right\|_1}$ （41a）
 $\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}^{\rm{T}}(t){\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{S}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{\hat f}}_k}(t) \le {\left\| {\mathit{\boldsymbol{S}}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}}(t)} \right\|_1}$ （41b）

 $\begin{array}{l} - 2\mathit{\boldsymbol{\tilde \zeta }}_{{{\bar g}_j}}^{\rm{T}}(t)(({{\mathit{\boldsymbol{\tilde D}}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{L}}_{{\rm{gro}},{{\bar g}_j}}}{{\tilde \beta }_{{{\bar g}_j}}}) \otimes ({\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{B}}_{{{\bar g}_j}}})){{\mathit{\boldsymbol{\tilde F}}}_{{{\bar g}_j}}}(t) \le \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - 2\gamma \sum\limits_{j = 2 + {{\bar N}_{{{\bar g}_j}}}}^{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{{{\bar g}_j}}}} {{d_j}{w_{j{{\bar g}_j}}}{{\left\| {\mathit{\boldsymbol{B}}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{\zeta }}_{j,{{\bar g}_j}}}(t)} \right\|}_1}} \end{array}$ （42a）
 $\begin{array}{l} - 2\mathit{\boldsymbol{\tilde \zeta }}_{{{\bar g}_j}}^{\rm{T}}(t)(({{\mathit{\boldsymbol{\tilde D}}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{L}}_{{\rm{gro}},{{\bar g}_j}}}) \otimes ({\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{B}}_{{{\bar g}_j}}})){{\mathit{\boldsymbol{\tilde u}}}_{{{\bar g}_j}}}(t) = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - 2\sum\limits_{j = 2 + {{\bar N}_{{{\bar g}_j}}}}^{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{{{\bar g}_j}}}} {{d_j}{w_{j{{\bar g}_j}}}\mathit{\boldsymbol{\tilde \zeta }}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_j}{\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{u}}_{{{\bar g}_j}}}(t)} \le \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2\gamma \sum\limits_{j = 2 + {{\bar N}_{{{\bar g}_j}}}}^{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{{{\bar g}_j}}}} {{d_j}{w_{j{{\bar g}_j}}}{{\left\| {\mathit{\boldsymbol{B}}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{{\mathit{\boldsymbol{\tilde \zeta }}}_{j,{{\bar g}_j}}}(t)} \right\|}_1}} \end{array}$ （42b）
 $\begin{array}{l} - 2\mathit{\boldsymbol{\tilde \zeta }}_{{{\bar g}_j}}^{\rm{T}}(t)[({{\mathit{\boldsymbol{\tilde D}}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{L}}_{{\rm{gro}},{{\bar g}_j}}}{{\mathit{\boldsymbol{\bar \varGamma }}}_{{{\bar g}_j}}}(t)) \otimes ({\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{S}}_{{{\bar g}_j}}})] \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\mathit{\boldsymbol{\bar F}}}_{{{\bar g}_j}}}(t) \le - 2\sum\limits_{j = 2 + {{\bar N}_{{{\bar g}_j}}}}^{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{{{\bar g}_j}}}} {{d_j}{\eta _j}(t){w_{j{{\bar g}_j}}}{{\left\| {\mathit{\boldsymbol{S}}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{{\mathit{\boldsymbol{\tilde \zeta }}}_{j,{{\bar g}_j}}}(t)} \right\|}_1}} \end{array}$ （43a）
 $\begin{array}{l} - 2\mathit{\boldsymbol{\tilde \zeta }}_{{{\bar g}_j}}^{\rm{T}}(t)(({{\mathit{\boldsymbol{\tilde D}}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{L}}_{{\rm{gro}},{{\bar g}_j}}}) \otimes ({\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{S}}_{{{\bar g}_j}}})){{\mathit{\boldsymbol{\bar \varPhi }}}_{{{\bar g}_j}}}(t) = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - 2\sum\limits_{j = 2 + {{\bar N}_{{{\bar g}_j}}}}^{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{{{\bar g}_j}}}} {{d_j}{w_{j{{\bar g}_j}}}\mathit{\boldsymbol{\tilde \zeta }}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_j}{\mathit{\boldsymbol{S}}_j}{{\mathit{\boldsymbol{\bar \varPhi }}}_{{{\bar g}_j}}}(t)} \le \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2\gamma \sum\limits_{j = 2 + {{\bar N}_{{{\bar g}_j}}}}^{1 + {q_{{{\bar g}_j}}} + {{\bar N}_{{{\bar g}_j}}}} {{d_j}{w_{j{{\bar g}_j}}}{{\left\| {\mathit{\boldsymbol{S}}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{{\mathit{\boldsymbol{\tilde \zeta }}}_{j,{{\bar g}_j}}}(t)} \right\|}_1}} \end{array}$ （43b）

 $\begin{array}{l} {{\dot V}_{{\rm{b}},{{\bar g}_j}}}(t) \le \mathit{\boldsymbol{\tilde \zeta }}_{{{\bar g}_j}}^{\rm{T}}(t)[{{\mathit{\boldsymbol{\tilde D}}}_{{{\bar g}_j}}} \otimes ({\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{A}}_{{{\bar g}_j}}} + \mathit{\boldsymbol{A}}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}})] \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\mathit{\boldsymbol{\tilde \zeta }}}_{{{\bar g}_j}}}(t) - \mathit{\boldsymbol{\tilde \zeta }}_{{{\bar g}_j}}^{\rm{T}}(t)\{ [({{\mathit{\boldsymbol{\tilde D}}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{L}}_{{\rm{gro}},{{\bar g}_j}}} + \mathit{\boldsymbol{L}}_{{\rm{gro}},{{\bar g}_j}}^{\rm{T}}{{\mathit{\boldsymbol{\tilde D}}}_{{{\bar g}_j}}}) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\tilde \alpha }_{{{\bar g}_j}}}] \otimes \mathit{\boldsymbol{P}}_{{{\bar g}_j}}^2\} {{\mathit{\boldsymbol{\tilde \zeta }}}_{{{\bar g}_j}}}(t) \end{array}$ （44）

 $\begin{array}{l} {{\dot V}_{{\rm{b}},{{\bar g}_j}}} \le \mathit{\boldsymbol{\tilde \zeta }}_{{{\bar g}_j}}^T(t)[{{\mathit{\boldsymbol{\tilde D}}}_{{{\bar g}_j}}} \otimes ({\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{A}}_{{{\bar g}_j}}} + \mathit{\boldsymbol{A}}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}})]{{\mathit{\boldsymbol{\tilde \zeta }}}_{{{\bar g}_j}}}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\bar \lambda }_{{\rm{min}}}}\mathit{\boldsymbol{\tilde \zeta }}_{{{\bar g}_j}}^{\rm{T}}(t)({{\mathit{\boldsymbol{\tilde \alpha }}}_{{{\bar g}_j}}} \otimes \mathit{\boldsymbol{P}}_{{{\bar g}_j}}^2){{\mathit{\boldsymbol{\tilde \zeta }}}_{{{\bar g}_j}}}(t) \le \mathit{\boldsymbol{\tilde \zeta }}_{{{\bar g}_j}}^{\rm{T}}(t)[{{\mathit{\boldsymbol{\tilde D}}}_{{{\bar g}_j}}} \otimes \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({\mathit{\boldsymbol{P}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{A}}_{{{\bar g}_j}}} + \mathit{\boldsymbol{A}}_{{{\bar g}_j}}^{\rm{T}}{\mathit{\boldsymbol{P}}_{{{\bar g}_j}}} - \mathit{\boldsymbol{P}}_{{{\bar g}_j}}^2)]{{\mathit{\boldsymbol{\tilde \zeta }}}_{{{\bar g}_j}}}(t) \le 0 \end{array}$ （45）

 ${\mathit{\boldsymbol{\dot {\bar x}}}_{j,j}}(t) = ({\mathit{\boldsymbol{A}}_j} + {\mathit{\boldsymbol{S}}_j}{\mathit{\boldsymbol{C}}_j}){\mathit{\boldsymbol{\bar x}}_{j,j}}(t)$ （46）

 $\begin{array}{l} {{\mathit{\boldsymbol{\dot x}}}_j}(t) = ({\mathit{\boldsymbol{A}}_j} + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1){\mathit{\boldsymbol{x}}_j}(t) + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1{{\mathit{\boldsymbol{\bar x}}}_{j,j}}(t) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^2{{\mathit{\boldsymbol{\hat x}}}_{{{\bar g}_j}}}(t) + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^2{{\mathit{\boldsymbol{\bar x}}}_{j,{{\bar g}_j}}}(t) - {\beta _j}{\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{H}}_j}{\mathit{\boldsymbol{f}}_j}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{B}}_j}K_j^1{{\mathit{\boldsymbol{\varpi }}} _j}(t) - {\mathit{\boldsymbol{B}}_j}{{\mathit{\boldsymbol{\hat B}}}_{j,1}}({\mathit{\boldsymbol{A}}_j}{{\mathit{\boldsymbol{\varpi }}} _j}(t) - {{{\mathit{\boldsymbol{\dot \varpi }}} }_j}(t)) \end{array}$ （47）

εj(t)=xj(t)-ϖj(t)-Ejxgj(t)，可得

 $\begin{array}{l} {{\mathit{\boldsymbol{\dot \varepsilon }}}_j}(t) = {{\mathit{\boldsymbol{\dot x}}}_j}(t) - {{{\mathit{\boldsymbol{\dot \varpi }}} }_j}(t) - {\mathit{\boldsymbol{E}}_j}{{\mathit{\boldsymbol{\dot x}}}_{{{\bar g}_j}}}(t) = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\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{A}}_j} + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1){\mathit{\boldsymbol{\varepsilon }}_j}(t) + ({\mathit{\boldsymbol{A}}_j} + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1){\mathit{\boldsymbol{E}}_j}{\mathit{\boldsymbol{x}}_{{{\bar g}_j}}}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\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}} {{\mathit{\boldsymbol{E}}_j}({\mathit{\boldsymbol{A}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{x}}_{{{\bar g}_j}}}(t) + {\mathit{\boldsymbol{B}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{u}}_{{{\bar g}_j}}}(t)) + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1{{\bar x}_{j,j}}(t) + }\\ {{\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^2{{\mathit{\boldsymbol{\hat x}}}_{{{\bar g}_j}}}(t) + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^2{{\mathit{\boldsymbol{\bar x}}}_{j,{{\bar g}_j}}}(t) - {\beta _j}{\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{H}}_j}{\mathit{\boldsymbol{f}}_j}(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} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{B}}_j}{{\mathit{\boldsymbol{\hat B}}}_{j,1}}({\mathit{\boldsymbol{A}}_j}{{\mathit{\boldsymbol{\varpi }}} _j}(t) - {{{\mathit{\boldsymbol{\dot \varpi }}} }_j}(t)) - {{{\mathit{\boldsymbol{\dot \varpi }}} }_j}(t) + {\mathit{\boldsymbol{A}}_j}{\mathit{\boldsymbol{\varpi }}} j(t) \end{array}$ （48）

δj(t)=-Bj$\hat{\boldsymbol{B}}$j, 1(Ajϖj(t)-$\dot{\boldsymbol{\varpi}}$j(t))-$\dot{\boldsymbol{\varpi}}$j(t)+Ajϖj(t)，由引理1与编队可行性条件式(13)可知δj(t)=0，因此式(48)可写为

 $\begin{array}{l} {{\mathit{\boldsymbol{\dot \varepsilon }}}_j}(t) = ({\mathit{\boldsymbol{A}}_j} + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1){\mathit{\boldsymbol{\varepsilon }}_j}(t) + ({\mathit{\boldsymbol{A}}_j} + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1){\mathit{\boldsymbol{E}}_j} \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{x}}_{{{\bar g}_j}}}(t) - {\mathit{\boldsymbol{E}}_j}({\mathit{\boldsymbol{A}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{x}}_{{{\bar g}_j}}}(t) + {\mathit{\boldsymbol{B}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{u}}_{{{\bar g}_j}}}(t)) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1{{\mathit{\boldsymbol{\bar x}}}_{j,j}}(t) + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^2{{\mathit{\boldsymbol{\hat x}}}_{{{\bar g}_j}}}(t) + {\mathit{\boldsymbol{B}}_j}K_j^2{{\mathit{\boldsymbol{\bar x}}}_{j,{{\bar g}_j}}}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\beta _j}{\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{H}}_j}{\mathit{\boldsymbol{f}}_j}(t) \end{array}$ （49）

 $\begin{array}{l} {{\mathit{\boldsymbol{\dot \varepsilon }}}_j}(t) = ({\mathit{\boldsymbol{A}}_j} + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1){\mathit{\boldsymbol{\varepsilon }}_j}(t) + ({\mathit{\boldsymbol{A}}_j} + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1){\mathit{\boldsymbol{E}}_j}{\mathit{\boldsymbol{x}}_{{{\bar g}_j}}}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\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{E}}_j}({\mathit{\boldsymbol{A}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{x}}_{{{\bar g}_j}}}(t) + {\mathit{\boldsymbol{B}}_{{{\bar g}_j}}}{\mathit{\boldsymbol{u}}_{{{\bar g}_j}}}(t)) + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1{{\mathit{\boldsymbol{\bar x}}}_{j,j}}(t) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\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}} {{\mathit{\boldsymbol{B}}_j}({\mathit{\boldsymbol{F}}_j} - \mathit{\boldsymbol{K}}_j^1{\mathit{\boldsymbol{E}}_j}){{\mathit{\boldsymbol{\hat x}}}_{{{\bar g}_j}}}(t) + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^2{{\mathit{\boldsymbol{\bar x}}}_{j.{{\bar g}_j}}}(t) - }\\ {{\beta _j}{\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{H}}_j}{\mathit{\boldsymbol{f}}_j}(t) = ({\mathit{\boldsymbol{A}}_j} + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1){\mathit{\boldsymbol{\varepsilon }}_j}(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} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {{\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{H}}_j}{\mathit{\boldsymbol{u}}_{{{\bar g}_j}}}(t) - {\beta _j}{\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{H}}_j}{\mathit{\boldsymbol{f}}_j}(t){\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1{{\mathit{\boldsymbol{\bar x}}}_{j,j}}(t) + }\\ {{\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^2{{\mathit{\boldsymbol{\bar x}}}_{j.{{\bar g}_j}}}(t) + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^2{{\mathit{\boldsymbol{\bar x}}}_{{{\bar g}_j} \cdot {{\mathit{\boldsymbol{\bar g}}}_j}}}(t)} \end{array} \end{array}$ （50）

 ${V_{j,{{\bar g}_j}}}(t) = \mathit{\boldsymbol{\varepsilon }}_j^{\rm{T}}(t){\mathit{\boldsymbol{Q}}_j}{\mathit{\boldsymbol{\varepsilon }}_j}(t)$ （51）

 $\begin{array}{l} {{\dot V}_{j,{{\bar g}_j}}}(t) = \mathit{\boldsymbol{\varepsilon }}_j^{\rm{T}}(t)({\mathit{\boldsymbol{Q}}_j}({\mathit{\boldsymbol{A}}_j} + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1) + ({\mathit{\boldsymbol{A}}_j} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1{)^{\rm{T}}}{\mathit{\boldsymbol{Q}}_j}){\mathit{\boldsymbol{\varepsilon }}_j}(t) + {\xi _j}(t) \end{array}$ （52）

 $\begin{array}{l} {\xi _j}(t) = - 2{\beta _j}\mathit{\boldsymbol{\varepsilon }}_j^{\rm{T}}(t){\mathit{\boldsymbol{Q}}_j}{\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{H}}_j}{\mathit{\boldsymbol{f}}_j}(t) - 2\mathit{\boldsymbol{\varepsilon }}_j^{\rm{T}}(t) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{Q}}_j}{\mathit{\boldsymbol{B}}_j}{\mathit{\boldsymbol{H}}_j}{\mathit{\boldsymbol{u}}_{{{\bar g}_j}}}(t) + 2\mathit{\boldsymbol{\varepsilon }}_j^{\rm{T}}(t){\mathit{\boldsymbol{Q}}_j}({\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1{{\mathit{\boldsymbol{\bar x}}}_{j,j}}(t) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^2{{\mathit{\boldsymbol{\bar x}}}_{j,{{\bar g}_j}}}(t) + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^2{{\mathit{\boldsymbol{\bar x}}}_{{{\bar g}_j},{{\bar g}_j}}}(t)) \end{array}$

$\hat{\boldsymbol{\varepsilon}}$j(t)=$\hat{\boldsymbol{x}}$j(t)-ϖj(t)-Ej$\hat{{x}}$j, gj(t)，同时定义εj(t)=$\hat{\boldsymbol{\varepsilon}}$j(t)-εj(t)，进而可以推出εj(t)=xj, j(t)-Ej(xj, gj(t)+xgj, gj(t))。根据假设1和算法1中的步骤2可得

 $\begin{array}{l} {\xi _j}(t) \le 2({\beta _j} + \gamma )\left\| {\mathit{\boldsymbol{H}}_j^{\rm{T}}\mathit{\boldsymbol{B}}_j^{\rm{T}}\mathit{\boldsymbol{Q}}_j^{\rm{T}}{{\mathit{\boldsymbol{\bar \varepsilon }}}_j}(t)} \right\| + 2\mathit{\boldsymbol{\varepsilon }}_j^{\rm{T}}(t) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{Q}}_j}({\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^1{{\mathit{\boldsymbol{\bar x}}}_{j,j}}(t) + {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^2{{\mathit{\boldsymbol{\bar x}}}_{j,{{\bar g}_j}}}(t) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{B}}_j}\mathit{\boldsymbol{K}}_j^2{{\mathit{\boldsymbol{\bar x}}}_{{{\bar g}_j},{{\bar g}_j}}}(t)) \end{array}$ （53）

σj(t)=Qj(BjKj1xj, j(t)+BjKj2xj, gj(t)+BjKj2xgj, gj(t))，根据Young不等式可得

 $2\mathit{\boldsymbol{\varepsilon }}_j^{\rm{T}}(t){\mathit{\boldsymbol{\sigma }}_j}(t) \le \mathit{\boldsymbol{\varepsilon }}_j^{\rm{T}}(t){\mathit{\boldsymbol{\varepsilon }}_j}(t) + \mathit{\boldsymbol{\sigma }}_j^{\rm{T}}(t){\mathit{\boldsymbol{\sigma }}_j}(t)$ （54）

 $\begin{array}{*{20}{c}} {{{\dot V}_{j,{{\bar g}_j}}}(t) \le - \frac{1}{{{\lambda _{{\rm{max}}}}({\mathit{\boldsymbol{Q}}_j})}}{V_{j,{{\bar g}_j}}}(t) + 2({\beta _j} + \gamma ) \cdot }\\ {\left\| {\mathit{\boldsymbol{H}}_j^{\rm{T}}\mathit{\boldsymbol{B}}_j^{\rm{T}}\mathit{\boldsymbol{Q}}_j^{\rm{T}}{{\mathit{\boldsymbol{\bar \varepsilon }}}_j}(t)} \right\| + \mathit{\boldsymbol{\sigma }}_j^{\rm{T}}(t){\mathit{\boldsymbol{\sigma }}_j}(t)} \end{array}$ （55）

3 数值仿真

 ${\mathit{\boldsymbol{A}}_{\rm{F}}} = \left[ {\begin{array}{*{20}{l}} 0&1\\ 0&0 \end{array}} \right],{\mathit{\boldsymbol{B}}_{\rm{F}}} = \left[ {\begin{array}{*{20}{l}} 0\\ 1 \end{array}} \right],{\mathit{\boldsymbol{C}}_{\rm{F}}} = [1,0]$

 图 1 异构多智能体系统模型架构示意图 Fig. 1 Model structure of heterogeneous multi-agent systems
 图 2 异构多智能体系统通信拓扑关系 Fig. 2 Interaction topologies of heterogeneous multi-agent systems
 ${\mathit{\boldsymbol{\varpi }}} _i^X(t) = \left\{ {\begin{array}{*{20}{l}} {8{\rm{sin}}\left( { - \frac{\pi }{2}} \right)}&{i = 2}\\ {8{{\left[ {{\rm{sin}}\left( {\frac{{4i + 1}}{6}\pi } \right),0} \right]}^{\rm{T}}}}&{i = 3,4}\\ {4{\rm{sin}}\left( {\frac{{i\pi }}{2} + t} \right)}&{i = 5,6,7,8}\\ {4\left[ {{\rm{sin}}\left( {\frac{{2i - 1}}{4}\pi + t} \right)} \right.,}&{}\\ {{{\left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{cos}}\left( {\frac{{2i - 1}}{4}\pi + t} \right)} \right]}^{\rm{T}}}}&{i = 9,10, \cdots ,16} \end{array}} \right.$
 $\begin{array}{l} {\mathit{\boldsymbol{\varpi }}} _i^Y(t) = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ {\begin{array}{*{20}{l}} {8{\rm{cos}}\left( { - \frac{\pi }{2}} \right)}&{i = 2}\\ {8{{\left[ {{\rm{cos}}\left( {\frac{{4i + 1}}{6}\pi } \right),0} \right]}^{\rm{T}}}}&{i = 3,4}\\ {4{\rm{cos}}\left( {\frac{{i\pi }}{2} + t} \right)}&{i = 5,6,7,8}\\ {4\left[ {{\rm{cos}}\left( {\frac{{2i - 1}}{4}\pi + t} \right)} \right.,}&{}\\ {{{\left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {\rm{sin}}\left( {\frac{{2i - 1}}{4}\pi + t} \right)} \right]}^{\rm{T}}}}&{i = 9,10, \cdots ,16} \end{array}} \right. \end{array}$
 $\varpi _i^Z(t) = \left\{ {\begin{array}{*{20}{l}} 0&{i = 2}\\ {{{[5,0]}^{\rm{T}}}}&{i = 3,4}\\ 0&{i = 5,6,7,8}\\ {{{[0,0]}^{\rm{T}}}}&{i = 9,10, \cdots ,16} \end{array}} \right.$

 ${{\alpha _i} = 10\quad i = 2,3, \cdots ,16}$
 ${{\beta _i} = 10\quad i = 2,3, \cdots ,16}$
 ${{\mathit{\boldsymbol{E}}_2} = [1,0],{\mathit{\boldsymbol{E}}_i} = 1\quad i = 5,6,7,8}$
 ${\mathit{\boldsymbol{E}}_i} = {\mathit{\boldsymbol{I}}_2}\quad i = 3,4,9,10, \cdots ,16$
 ${\mathit{\boldsymbol{F}}_2} = \left[ {0,1} \right],{\mathit{\boldsymbol{F}}_i} = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = 5,6,7,8$
 ${\mathit{\boldsymbol{F}}_i} = [0,0]\quad i = 3,4,9,10, \cdots ,16$
 ${{\mathit{\boldsymbol{H}}_2} = 0,{\mathit{\boldsymbol{H}}_i} = 1\quad i = 3,4, \cdots ,16}$
 ${{\mathit{\boldsymbol{S}}_i} = - 1\quad i = 2,5,6,7,8}$
 ${{\mathit{\boldsymbol{S}}_i} = {{[ - 2, - 1]}^{\rm{T}}}\quad i = 3,4,9,10, \cdots ,16}$
 ${\mathit{\boldsymbol{K}}_i^1 = - 2(i = 2,5,6,7,8)}$
 ${\mathit{\boldsymbol{K}}_i^1 = [ - 2, - 4]\quad i = 3,4,9,10, \cdots ,16}$
 ${\mathit{\boldsymbol{K}}_2^2 = [2,1],\mathit{\boldsymbol{K}}_i^2 = - 2\quad i = 5,6,7,8}$
 ${\mathit{\boldsymbol{K}}_i^2 = [2,4]\quad i = 3,4,9,10, \cdots ,16}$
 ${\mathit{\boldsymbol{Q}}_i} = \left[ {\begin{array}{*{20}{c}} {2.75}&{0.5}\\ {0.5}&{0.375} \end{array}} \right]\quad i = 3,4,9,10, \cdots ,16$
 ${\mathit{\boldsymbol{Q}}_i} = \frac{1}{2}\quad i = 2,5,6,7,8$
 ${\mathit{\boldsymbol{P}}_i} = \left[ {\begin{array}{*{20}{l}} {0.910}&2&{0.4142}\\ {0.414}&2&{1.287} \end{array}} \right]\quad i = 1,3,4$
 ${\mathit{\boldsymbol{P}}_2} = 1$

 图 3 异构多智能体系统的搜索轨迹与各智能体部分时刻的位置信息 Fig. 3 Trajectory and position at certain time of heterogeneous multi-agent systems
 图 4 异构多智能体系统在惯性空间各分量的分组输出时变编队跟踪误差 Fig. 4 Tracking errors of time-varying output group formation of heterogeneous multi-agent systems in inertial space
 $\begin{array}{l} {e_i}{(t)^ \buildrel \Delta \over = }\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ {\begin{array}{*{20}{l}} {{{\left\| {{\mathit{\boldsymbol{y}}_i}(t) - {\mathit{\boldsymbol{h}}_i}(t) - {\mathit{\boldsymbol{y}}_1}(t)} \right\|}_2}}&{i = 2,3,4}\\ {{{\left\| {{\mathit{\boldsymbol{y}}_i}(t) - {\mathit{\boldsymbol{h}}_i}(t) - {\mathit{\boldsymbol{y}}_2}(t)} \right\|}_2}}&{i = 5,6,7,8}\\ {{{\left\| {{\mathit{\boldsymbol{y}}_i}(t) - {\mathit{\boldsymbol{h}}_i}(t) - {\mathit{\boldsymbol{y}}_3}(t)} \right\|}_2}}&{i = 9,10,11,12}\\ {{{\left\| {{\mathit{\boldsymbol{y}}_i}(t) - {\mathit{\boldsymbol{h}}_i}(t) - {\mathit{\boldsymbol{y}}_4}(t)} \right\|}_2}}&{i = 13,14,15,16} \end{array}} \right. \end{array}$

4 结论

 [1] 林来兴. 小卫星编队飞行及其轨道构成[J]. 中国空间科学技术, 2001, 21(1): 23-28. LIN L X. Formation flying of small satellite and its orbital configuration[J]. Chinese Space Science and Technology, 2001, 21(1): 23-28. (in Chinese) Cited By in Cnki (202) | Click to display the text [2] RAY R J, COBLEIGH B R, VACHON M J. Flight test techniques used to evaluate performance benefits during formation flight[R]. Washington, D.C.: NASA Dryden Research Center, 2002. [3] 宗群, 王丹丹, 邵士凯, 等. 多无人机协同编队飞行控制研究现状及发展[J]. 哈尔滨工业大学学报, 2017, 49(3): 1-14. ZONG Q, WANG D D, SHAO S K, et al. Research status and development of muli-UAV coordinated formation flight control[J]. Journal of Harbin Institute of Technology, 2017, 49(3): 1-14. (in Chinese) Cited By in Cnki (19) | Click to display the text [4] OH K K, PARK M C, AHN H S. A survey of multi-agent formation control[J]. Automatica, 2015, 53: 424-440. Click to display the text [5] REN W. Consensus strategies for cooperative control of vehicle formations[J]. IET Control Theory & Applications, 2007, 1(2): 505-512. Click to display the text [6] DONG X W, HU G Q. Time-varying formation control for general linear multi-agent systems with switching directed topologies[J]. Automatica, 2016, 73: 47-55. Click to display the text [7] HE L L, ZHANG J Q, HOU Y Q, et al. Time-varying formation control for second-order discrete-time multi-agent systems with switching topologies and nonuniform communication delays[J]. IEEE Access, 2019, 7: 65379-65389. Click to display the text [8] HE L L, ZHANG J Q, HOU Y Q, et al. Time-varying formation control for second-order discrete-time multi-agent systems with directed topologies and nonuniform communication delays[J]. IEEE Access, 2019, 7: 33517-33527. Click to display the text [9] HUA Y Z, DONG X W, LI Q D, et al. Distributed time-varying formation robust tracking for general linear multiagent systems with parameter uncertainties and external disturbances[J]. IEEE Transactions on Cybernetics, 2017, 47(8): 1959-1969. Click to display the text [10] YU J L, DONG X W, LI Q D, et al. Robust H∞ guaranteed cost time-varying formation tracking for high-order multiagent systems with time-varying delays[J]. IEEE Transactions on Systems, Man, and Cybernetics:Systems, 2020, 50(4): 1465-1475. Click to display the text [11] YU J L, DONG X W, LI Q D, et al. Practical time-varying formation tracking for multiple non-holonomic mobile robot systems based on the distributed extended state observers[J]. IET Control Theory & Applications, 2018, 12(12): 1737-1747. Click to display the text [12] DONG X W, LI Q D, ZHAO Q L, et al. Time-varying group formation analysis and design for second-order multi-agent systems with directed topologies[J]. Neurocomputing, 2016, 205: 367-374. Click to display the text [13] LI Y F, DONG X W, LI Q D, et al. Time-varying group formation control for second-order multi-agent systems with switching directed topologies[C]//Proceedings of 201736th Chinese Control Conference, 2017: 8530-8535. [14] 韩亮, 任章, 董希旺, 等. 多无人机协同控制方法及应用研究[J]. 导航定位与授时, 2018, 25(4): 5-11. HAN L, REN Z, DONG X W, et al. Research on cooperative control method and application for multiple unmanned aerial vehicles[J]. Navigation Positioning & Timing, 2018, 25(4): 5-11. (in Chinese) Cited By in Cnki (2) | Click to display the text [15] LI Z, CHEN M Z Q, DING Z. Distributed adaptive controllers for cooperative output regulation of heterogeneous agents over directed graphs[J]. Automatica, 2016, 68: 179-183. Click to display the text [16] ADIB Y F, LEWIS F L, SU R. Output regulation of linear heterogeneous multi-agent systems via output and state feedback[J]. Automatica, 2016, 67: 157-164. Click to display the text [17] LIU X K, WANG Y W, XIAO J W, et al. Distributed hierarchical control design of coupled heterogeneous linear systems under switching networks[J]. International Journal of Robust and Nonlinear Control, 2016, 27(8): 1242-1259. Click to display the text [18] LI Z, LIU X, REN W, et al. Distributed tracking control for linear multiagent systems with a leader of bounded unknown input[J]. IEEE Transactions on Automatic Control, 2013, 58(2): 518-523. Click to display the text [19] TANG Y, HONG Y, WANG X. Distributed output regulation for a class of nonlinear multi-agent systems with unknown-input leaders[J]. Automatica, 2015, 62: 154-160. Click to display the text [20] LYU Y, LI Z, DUAN Z, et al. Distributed adaptive output feedback consensus protocols for linear systems on directed graphs with a leader of bounded input[J]. Automatica, 2016, 74: 308-314. Click to display the text [21] QU Z H. Cooperative control of dynamical systems:Applications to autonomous vehicles[M]. London: Springer-Verlag, 2009. [22] 田磊, 王蒙一, 赵启伦, 等. 拓扑切换的集群系统分布式分组时变编队跟踪控制[J]. 中国科学:信息科学, 2020, 50(3): 408-423. TIAN L, WANG M Y, ZHAO Q L, et al. Distributed time-varying group formation tracking for cluster systems with switching interaction topologies[J]. Scientia Sinica Informations, 2020, 50(3): 408-423. (in Chinese) Cited By in Cnki | Click to display the text [23] WANG Z M, YANG C, DONG X W, et al. Time-varying formation control for mobile robots: Algorithms and experiments[C]//Proceedings of 201743rd IEEE Industrial Electronics Society Conference. Piscataway: IEEE Press, 2017: 7239-7244.
http://dx.doi.org/10.7527/S1000-6893.2019.23727

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#### 文章信息

TIAN Lei, ZHAO Qilun, DONG Xiwang, LI Qingdong, REN Zhang

Time-varying output group formation tracking for heterogeneous multi-agent systems

Acta Aeronautica et Astronautica Sinica, 2020, 41(7): 323727.
http://dx.doi.org/10.7527/S1000-6893.2019.23727