文章快速检索 高级检索

1. 清华大学 自动化系, 北京 100084;
2. 北京信息科学与技术国家研究中心(BNRist), 北京 100084

Consensus control for multi-agent systems subject to actuator saturations
GAO Chen1, HE Xiao1,2
1. Department of Automation, Tsinghua University, Beijing 100084, China;
2. Beijing National Research Center for Information Science and Technology(BNRist), Beijing 100084, China
Abstract: Targeting at a class of discrete Multi-Agent Systems (MASs) subject to actuator saturations, a novel consensus controller based on Linear Matrix Inequalities (LMIs) is proposed and the existence of the solution for the LMI is examined. Due to the actuator saturation, the consensus control problem of MASs is related to the initial states of agents. Different controllers may tolerate different ranges of initial states. In order to quantitatively analyze the effects caused by the range of initial states, this paper introduces the concept of Domain of Attraction (DOA) to evaluate the controllers and, subsequently, the controller parameters are optimized to enlarge the DOA. Finally, simulation examples are provided to illustrate the effectiveness of the proposed controller.
Keywords: multi-agent system    actuator saturation    domain of attraction    consensus control    discrete system

1 问题描述

 ${\mathit{\boldsymbol{x}}_i}(k + 1) = \mathit{\boldsymbol{A}}{\mathit{\boldsymbol{x}}_i}(k) + \mathit{\boldsymbol{B}}\sigma ({\mathit{\boldsymbol{u}}_i}(k))$ （1）

 $\mathop {{\rm{lim}}}\limits_{k \to \infty } \left\| {{\mathit{\boldsymbol{x}}_i}(k) - {\mathit{\boldsymbol{x}}_j}(k)} \right\| = 0$ （2）

 $\mathit{\boldsymbol{D}} = \{ {\mathit{\boldsymbol{x}}_0}|\mathop {{\rm{lim}}}\limits_{k \to \infty } \psi (k,{\mathit{\boldsymbol{x}}_0}) = 0\}$ （3）

 ${\mathit{\boldsymbol{u}}_i}(k) = \mathit{\boldsymbol{K}}\sum\limits_{j = 1}^N {{a_{ij}}} ({\mathit{\boldsymbol{x}}_i}(k) - {\mathit{\boldsymbol{x}}_j}(k))$ （4）

x=[x1T, x2T, …, xNT]T，利用Kronecker积，可以得到闭环系统动态方程为

 $\begin{array}{l} \mathit{\boldsymbol{x}}(k + 1) = ({\mathit{\boldsymbol{I}}_N} \otimes \mathit{\boldsymbol{A}})\mathit{\boldsymbol{x}}(k) + ({\mathit{\boldsymbol{I}}_N} \otimes \mathit{\boldsymbol{B}}) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sigma ((\mathit{\boldsymbol{L}} \otimes \mathit{\boldsymbol{K}})\mathit{\boldsymbol{x}}(k)) \end{array}$ （5）

2 主要结果

2.1 一致性控制律设计

y=(TTIn)x=[y1T, y2T, …, yNT]T，其中，$\boldsymbol{y}_{i} \in {\bf R}^{n}(i \in \mathfrak{N})$，则式(5)可以转化为

 $\mathit{\boldsymbol{y}}(k + 1) = ({\mathit{\boldsymbol{I}}_N} \otimes \mathit{\boldsymbol{A}} + \mathit{\boldsymbol{ \boldsymbol{\varLambda} }} \otimes \mathit{\boldsymbol{BK}})\mathit{\boldsymbol{y}}(k)$ （6）

z=[y2T, y3T, …, yNT]T=(T1TIn)x，式(6)可以进一步转化为

 $\mathit{\boldsymbol{z}}(k + 1) = ({\mathit{\boldsymbol{I}}_{N - 1}} \otimes \mathit{\boldsymbol{A}} + \mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes \mathit{\boldsymbol{BK}})\mathit{\boldsymbol{z}}(k)$ （7）

z(k)=0n(N-1)时，x(k)满足：

 $\mathit{\boldsymbol{x}}(k) = (\mathit{\boldsymbol{T}} \otimes {\mathit{\boldsymbol{I}}_n})\mathit{\boldsymbol{y}}(k) = \sqrt {\frac{1}{N}} \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{y}}_1}(k)}\\ {{\mathit{\boldsymbol{y}}_1}(k)}\\ \vdots \\ {{\mathit{\boldsymbol{y}}_1}(k)} \end{array}} \right]$

 $\varepsilon (\mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes \mathit{\boldsymbol{P}},1) = \{ \mathit{\boldsymbol{z}} \in {{\bf{R}}^{(N - 1)n}}:{\mathit{\boldsymbol{z}}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes \mathit{\boldsymbol{P}})\mathit{\boldsymbol{z}} \le 1\}$ （8）
 $\wp (\mathit{\boldsymbol{H}}) = \{ \mathit{\boldsymbol{z}} \in {{\bf{R}}^{(N - 1)n}}:|{H_j}\mathit{\boldsymbol{z}}| \le 1\}$ （9）

 ${\varepsilon (\mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes \mathit{\boldsymbol{P}},1) \subset \wp ({\mathit{\boldsymbol{T}}_1}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes \mathit{\boldsymbol{K}})}$ （10a）
 ${\left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{W}}&{\mathit{\boldsymbol{W}}{\mathit{\boldsymbol{A}}^{\rm{T}}} + {\lambda _i}{\mathit{\boldsymbol{Y}}^{\rm{T}}}{\mathit{\boldsymbol{B}}^{\rm{T}}}}\\ {\mathit{\boldsymbol{AW}} + {\lambda _i}\mathit{\boldsymbol{BY}}}&\mathit{\boldsymbol{W}} \end{array}} \right] > 0}$ （10b）

 $V(k) = {\mathit{\boldsymbol{z}}^{\rm{T}}}(k)(\mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes \mathit{\boldsymbol{P}})\mathit{\boldsymbol{z}}(k)$ （11）

 $\begin{array}{l} \Delta V(k) = V(k + 1) - V(k) = {\mathit{\boldsymbol{z}}^{\rm{T}}}(k)(({\mathit{\boldsymbol{I}}_{N - 1}} \otimes \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{A}} + \mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes \mathit{\boldsymbol{BK}}{)^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes \mathit{\boldsymbol{P}}) \times ({\mathit{\boldsymbol{I}}_{N - 1}} \otimes \mathit{\boldsymbol{A}} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes \mathit{\boldsymbol{BK}}) - \mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes \mathit{\boldsymbol{P}})\mathit{\boldsymbol{z}}(k) = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{z}}^{\rm{T}}}(k){\rm{ diag}} ({\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2},{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_3}, \cdots ,{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_N})\mathit{\boldsymbol{z}}(k) \end{array}$ （12）

2.2 吸引域分析

 $\begin{array}{l} {\rm{max}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha \\ {\rm{s}}{\rm{.t}}{\rm{.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\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{c1}}){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \varepsilon (\mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes \mathit{\boldsymbol{P}},1) \subset \wp ({\mathit{\boldsymbol{T}}_1}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes \mathit{\boldsymbol{K}})\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\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{c2}}){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{W}}&{\mathit{\boldsymbol{W}}{\mathit{\boldsymbol{A}}^{\rm{T}}} + {\lambda _i}{\mathit{\boldsymbol{Y}}^{\rm{T}}}{\mathit{\boldsymbol{B}}^{\rm{T}}}}\\ {\mathit{\boldsymbol{AW}} + {\lambda _i}\mathit{\boldsymbol{BY}}}&\mathit{\boldsymbol{W}} \end{array}} \right] > 0\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\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{c3}}){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha \varepsilon (\mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes \mathit{\boldsymbol{R}},1) \subset \varepsilon (\mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes \mathit{\boldsymbol{P}},1) \end{array}$ （13）

 ${{\mathit{\boldsymbol{T}}_1}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes \mathit{\boldsymbol{K}} = ({\mathit{\boldsymbol{I}}_N} \otimes \mathit{\boldsymbol{K}})({\mathit{\boldsymbol{T}}_1}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes {\mathit{\boldsymbol{I}}_n}),}$
 ${\frac{1}{{{\lambda _N}}}{\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}^2} \otimes \mathit{\boldsymbol{P}} = \frac{1}{{{\lambda _N}}}(\mathit{\boldsymbol{ \boldsymbol{\varPhi} T}}_1^{\rm{T}} \otimes {\mathit{\boldsymbol{I}}_n})({\mathit{\boldsymbol{I}}_N} \otimes \mathit{\boldsymbol{P}})({\mathit{\boldsymbol{T}}_1}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes {\mathit{\boldsymbol{I}}_n}),}$

 $\left[ {\begin{array}{*{20}{l}} {\frac{1}{{{\lambda _N}}}}&{{{({\mathit{\boldsymbol{I}}_N} \otimes \mathit{\boldsymbol{K}})}_j}}\\ *&{({\mathit{\boldsymbol{I}}_N} \otimes \mathit{\boldsymbol{P}})} \end{array}} \right] \ge 0\;\:j = 1,2, \cdots ,mN$ （14）

 $\left[ {\begin{array}{*{20}{c}} {\frac{1}{{{\lambda _N}}}}&{{\mathit{\boldsymbol{K}}_j}}\\ *&\mathit{\boldsymbol{P}} \end{array}} \right] \ge 0\;\:j = 1,2, \cdots ,m$ （15）

 $\left[ {\begin{array}{*{20}{c}} {\frac{1}{{{\lambda _N}}}}&{{\mathit{\boldsymbol{Y}}_j}}\\ *&\mathit{\boldsymbol{W}} \end{array}} \right] \ge 0\;\:j = 1,2, \cdots ,m$ （16）

 $\begin{array}{l} {\rm{min}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma \\ {\rm{s}}{\rm{.t}}{\rm{.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({\rm{c1}}){\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ {\begin{array}{*{20}{c}} { - \frac{1}{{{\lambda _N}}}}&{ - {\mathit{\boldsymbol{Y}}_j}}\\ *&{ - \mathit{\boldsymbol{W}}} \end{array}} \right] \le 0\;\:j = 1,2, \cdots ,m\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({\rm{c2}}){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ {\begin{array}{*{20}{c}} { - \mathit{\boldsymbol{W}}}&*\\ { - \mathit{\boldsymbol{AW}} - {\lambda _i}\mathit{\boldsymbol{BY}}}&{ - \mathit{\boldsymbol{W}}} \end{array}} \right] < 0\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({\rm{c3}}){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{R}}^{ - 1}} \le \gamma \mathit{\boldsymbol{W}} \end{array}$ （17）

 $\begin{array}{l} \mathit{\boldsymbol{z}}{(0)^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varPhi} }} \otimes \mathit{\boldsymbol{P}})\mathit{\boldsymbol{z}}(0) = \mathit{\boldsymbol{x}}{(0)^{\rm{T}}}(\mathit{\boldsymbol{L}} \otimes \mathit{\boldsymbol{P}})\mathit{\boldsymbol{x}}(0) \le \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\lambda _{{\rm{max}}}}(\mathit{\boldsymbol{P}})\mathit{\boldsymbol{x}}{(0)^{\rm{T}}}(\mathit{\boldsymbol{L}} \otimes {\mathit{\boldsymbol{I}}_n})\mathit{\boldsymbol{x}}(0) = {\lambda _{{\rm{max}}}}(\mathit{\boldsymbol{P}}) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{{E_{ij}} \in \mathit{\boldsymbol{E}},i < j} {{{\left\| {{\mathit{\boldsymbol{x}}_i}(0) - {\mathit{\boldsymbol{x}}_j}(0)} \right\|}^2} \le } \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {N_G}{\lambda _{{\rm{max}}}}(\mathit{\boldsymbol{P}}){\rm{ma}}{{\rm{x}}_{i,j,{E_{ij}} \in \mathit{\boldsymbol{E}}}}\{ {\left\| {{\mathit{\boldsymbol{x}}_i}(0) - {\mathit{\boldsymbol{x}}_j}(0)} \right\|^2}\} \end{array}$ （18）

 ${\rm{ma}}{{\rm{x}}_{i,j,{E_{ij}} \in \mathit{\boldsymbol{E}}}}\{ \left\| {{\mathit{\boldsymbol{x}}_i}(0) - {\mathit{\boldsymbol{x}}_j}(0)} \right\|\} \le \sqrt {\frac{1}{{{N_G}{\lambda _{{\rm{max}}}}(\mathit{\boldsymbol{P}})}}}$ （19）

x(0)被包含在吸引域中，即任意两个互为邻居的个体之间初始状态差值的2-范数不超过$\sqrt{1 /\left(N_{G} \lambda_{\max }(\boldsymbol{P})\right)}$的多智能体系统，可以实现一致性。式(19)反映了在执行器存在饱和约束时，多智能体系统的一致性与系统的初值有关系，这是由于饱和环节的存在引入了非线性。

 $\begin{array}{l} \mathit{\boldsymbol{P}} - {\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{PA}} - {\lambda _i}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{PBK}} - {\lambda _i}{\mathit{\boldsymbol{K}}^{\rm{T}}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PA}} - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \lambda _i^2{\mathit{\boldsymbol{K}}^{\rm{T}}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PBK}} > 0 \end{array}$ （20）

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{P}} - {\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{PA}} + {c^2}\lambda _i^2{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{PB}}{{({\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PB}} + \mathit{\boldsymbol{I}})}^{ - 2}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PA}} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (2c{\lambda _i} - {c^2}\lambda _i^2){\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{PB}}{{({\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PB}} + \mathit{\boldsymbol{I}})}^{ - 1}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PA}} > 0} \end{array}$ （21）

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{P}} = {\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{PA}} - (2c{\lambda _N} - {c^2}\lambda _N^2){\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{PB}} \cdot }\\ {{{({\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PB}} + \mathit{\boldsymbol{I}})}^{ - 1}} \times {\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PA}} + \mathit{\boldsymbol{Q}}} \end{array}$ （22）

1) 如果A的所有特征根都在单位圆内或单位圆上，则对于0 < c < 2/λN，式(22)均存在唯一解。

2) 如果A至少有一个特征根(记为λiu(A))位于单位圆外，并且rank(B)=1时，则对于$2 c \lambda_{N}-c^{2} \lambda_{N}^{2}>1-1 /\left|\prod_{i}\lambda_{i}^{u}(\boldsymbol{A})\right|^{2}$，式(22)存在唯一解。

3) 式(22)的唯一解满足P=limk→∞P(k)，其中P(k)=ATP(k-1)A-(2i-c2λi2)ATP(k-1)B(BTP(k-1)B+I)-1BTP(k-1)A+Q

3 仿真分析

1) 多智能体系统1：二阶积分系统

 $\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} 1&{0.01}\\ 0&1 \end{array}} \right]}\\ {\mathit{\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} {0.0001}\\ {0.01} \end{array}} \right]} \end{array}} \right.$

 $\mathit{\boldsymbol{Y}} = [ - 129.552{\kern 1pt} {\kern 1pt} {\kern 1pt} 9, - 10.044{\kern 1pt} {\kern 1pt} {\kern 1pt} 2],$
 $\mathit{\boldsymbol{P}} = \left[ {\begin{array}{*{20}{c}} {0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 009{\kern 1pt} {\kern 1pt} {\kern 1pt} 6}&{0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 028}\\ {0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 028}&{0.001{\kern 1pt} {\kern 1pt} {\kern 1pt} 6} \end{array}} \right],\gamma = 0.001{\kern 1pt} {\kern 1pt} {\kern 1pt} 9,$
 $\mathit{\boldsymbol{K}} = [ - 0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 54,\;\: - 0.019{\kern 1pt} {\kern 1pt} {\kern 1pt} 6],\alpha = 22.805{\kern 1pt} {\kern 1pt} {\kern 1pt} 1。$
 图 1 多智能体系统拓扑图 Fig. 1 Topology of multi-agent system

 图 2 多智能体系统1的x1j-xij变化曲线(i=2, 3, 4, 5, j=1, 2, γ=0.001 9) Fig. 2 Variation of x1j-xij in MAS 1 (i=2, 3, 4, 5, j=1, 2, γ=0.001 9)
 图 3 多智能体系统1的输入曲线(γ=0.001 9) Fig. 3 Variation of input variable in the MAS 1 (γ=0.001 9)
 图 4 多智能体系统1的x1j-xij变化曲线(γ=4) Fig. 4 Variation of x1j-xij in MAS 1 (γ=4)
 图 5 多智能体系统1的输入曲线(γ=4) Fig. 5 Variation of input variable in MAS 1 (γ=4)

2) 多智能体系统2：不稳定系统

 $\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} 1&{0.1}\\ {0.15}&{0.5} \end{array}} \right]}\\ {\mathit{\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} {0.2}\\ {0.25} \end{array}} \right]} \end{array}} \right.$

 $\mathit{\boldsymbol{Y}} = [ - 1.495{\kern 1pt} {\kern 1pt} {\kern 1pt} 7, - 0.336{\kern 1pt} {\kern 1pt} {\kern 1pt} 3],$
 $\mathit{\boldsymbol{P}} = \left[ {\begin{array}{*{20}{c}} {0.068{\kern 1pt} {\kern 1pt} 4}&{0.011{\kern 1pt} {\kern 1pt} 6}\\ {0.011{\kern 1pt} {\kern 1pt} 6}&{0.008{\kern 1pt} {\kern 1pt} 0} \end{array}} \right],$
 $\mathit{\boldsymbol{K}} = \left[ {\begin{array}{*{20}{c}} { - 0.106{\kern 1pt} {\kern 1pt} 2,}&{ - 0.020{\kern 1pt} {\kern 1pt} 1} \end{array}} \right]$

 图 6 多智能体系统2的x1j-xij变化曲线 Fig. 6 Variation of x1j-xij in MAS 2
 图 7 多智能体系统2的输入曲线 Fig. 7 Variation of input variable in MAS 2

3) 多智能体系统3：有向图

 $\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{l}} {1.006}&{ - 0.008}\\ {0.006}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1.006} \end{array}} \right]}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{B}} = \begin{array}{*{20}{c}} {0.019{\kern 1pt} {\kern 1pt} 9}\\ {0.040{\kern 1pt} {\kern 1pt} 2} \end{array}} \end{array}} \right.$

 图 8 多智能体系统3的x1j-xij变化曲线 Fig. 8 Variation of x1j-xij in MAS 3
 图 9 多智能体系统3的输入曲线 Fig. 9 Variation of input variable in MAS 3
4 结论

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http://dx.doi.org/10.7527/S1000-6893.2019.23760

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文章信息

GAO Chen, HE Xiao

Consensus control for multi-agent systems subject to actuator saturations

Acta Aeronautica et Astronautica Sinica, 2020, 41(S1): 723760.
http://dx.doi.org/10.7527/S1000-6893.2019.23760