文章快速检索  
  高级检索
基于事件触发的异构多智能体最优包含控制
王东1,2, 王泽华1,2, 刘洋1,2, GUDongbing3, 王伟1,2     
1. 大连理工大学 工业装备智能控制与优化教育部重点实验室, 大连 116024;
2. 大连理工大学 控制科学与工程学院, 大连 116024;
3. 埃塞克斯大学 计算机科学与电子工程学院, 克彻斯特 CO4 3SQ
摘要: 针对异构多智能体系统的输出包含控制问题,提出一种基于边的事件触发最优控制协议,保证所有跟随者的输出能进入到由领航者的输出所形成的凸包中。同时,使系统达到最优性能,最小化控制代价。考虑到不是所有的跟随者都可获得领航者的信息,提出一种基于边的分布式事件触发观测器,估计领航者输出形成的凸包内点的轨迹。设计加权代价函数评价包含控制的性能,并将输出包含问题转化为最优状态反馈控制设计问题。利用贝尔曼方程和黎卡提方程,给出异构多智能体系统最优输出包含控制的参数设计。选择不同类型的机器人构成多智能体系统,验证算法的有效性。
关键词: 多智能体系统    最优包含控制    事件触发控制    贝尔曼方程    分布式观测器    
Event-triggered optimal containment control for heterogeneous multi-agent systems
WANG Dong1,2, WANG Zehua1,2, LIU Yang1,2, GU Dongbing3, WANG Wei1,2     
1. Key Laboratory of Intelligent Control and Optimization for Industrial Equipment, Ministry of Eduction, Dalian University of Technology, Dalian 116024, China;
2. School of Control Science and Engineering, Dalian University of Technology, Dalian 116024, China;
3. School of Computer Science and Electronic Engineering, University of Essex Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
Abstract: An edge-based event-triggered optimal control algorithm is proposed to solve the optimal output containment problem of general linear heterogeneous multi-agent systems. The proposed algorithm guarantees that the outputs of all followers enter into the convex hull formed by the outputs of leaders. At the same time, it can also render that the closed-loop system achieves the optimal performance and minimize the cost of control. Considering that the information of the leaders is unavailable to a part of followers, a distributed edge-based event-triggered observer is presented to estimate the trajectory for the interior point in the convex hull. In addition, a weighted performance function is designed to transform the output containment problem into the optimal state-feedback design problem. Based on the proposed function, the Bellman equation and algebraic Riccati equation are derived to solve the optimal output containment control problem of heterogeneous multi-agent systems. Multiple autonomous robots are employed as a group of multi-agent systems to verify the effectiveness of the proposed algorithm.
Keywords: multi-agent systems    optimal containment control    event-triggered control    Bellman equation    distributed observe    

近几十年来,由于多智能体系统在生物、物理和运输系统的广泛应用,人们越来越关注多智能体系统的协同控制[1]。协同控制问题的目标是跟踪参考信号或排除干扰。多智能体系统的基本问题是一致性,可以实现依赖状态的一致性[2-3]或者依赖输出的一致性[4-5]。根据领航者的有无,协同控制分为无领航者的一致性[6]、单个领航者跟踪控制[7]和带有多个领航者的包含控制[8]。多个领航者的控制目标是使得所有跟随者的状态或输出能够最终收敛到领航者对应信息组成的多边形中。包含控制是多智能体系统控制中具有挑战性问题之一,引起了人们的广泛关注。文献[9]中考虑了具有不同拓扑的单积分多智能体系统的包含控制问题。文献[10]研究了具有二阶动力学的多智能体系统的多领航者跟踪问题。然而,上述文献的一个共同局限是领航者和跟随者具有相同的动力学特性。但是在实际中,领航者可能自主地完成一些任务,而跟随者被动地跟随领航者执行任务,因此,它们的动力学可能是不同的。本文关注这种具有不同动力学的多智能体,即异构多智能体系统,研究其包含的控制问题。

每个智能体或多或少配备微处理器和通讯模块,通过与相邻的智能体通信实现邻居智能体的信息交流。自然地,智能体会受到带宽资源的限制,有必要考虑通信代价。与传统的周期性采样方法相比,事件触发控制作为一种有效减少不必要通信的技术手段,受到了越来越多研究者的关注。文献[11]中将事件触发策略应用于稳定单积分器控制问题。文献[12]设计了事件触发控制协议解决单积分多智能体系统的一致性问题。在文献[13]中,提出了一种基于组合测量的事件触发控制算法,智能体的控制仅在其自身的触发时刻进行更新,从而减少了一些不必要的通信并且降低了控制器更新的次数。此外,事件触发控制还可以应用于一些类似的场景以减轻通信负担。例如,文献[14]研究了一类离散时间复杂网络的状态估计问题,文献[15]研究了异构多智能体系统的输出一致性问题。本文将事件触发控制算法应用到异构多智能体系统的输出包含控制问题中。

值得提到的是,越来越多的研究者致力于将事件触发控制算法与最优性能的协同控制相结合。最早的尝试已在文献[16]中给出,并在文献[17]中进一步发展。上述文献提出了一种基于自适应动态规划的非线性连续时间系统事件触发方法。一些研究人员也在利用事件触发控制确保最佳性能。例如,文献[18]提出了一种基于事件采样神经动态规划的不确定连续时间系统的近似最优控制, 文献[19]提出了一种将事件触发控制与连续时间线性系统最优控制相结合的新控制策略。但是,上述的工作都是基于单个系统而言的。本文关注多智能体系统的最优包含控制。

本文的主要贡献包括3点:首先,将最优包含控制问题转化为最优一致问题, 引入贝尔曼方程,对控制策略进行评价,找出最优的控制策略; 其次,提出了基于边的事件触发观测器估计领航者组成的凸包里一个内点的状态; 最后,验证所提出的性能函数保证稳态最优,同时优化瞬态响应。

1 研究基础及问题描述 1.1 图论

智能体之间的通信关系用图$ \mathcal{G}$表示,图中节点代表智能体,边代表智能体间的通信链路。图$ \mathcal{G}$由3个子集组成,分别是顶点集$ \mathcal{V}$、边集ε$ \mathcal{V}$×$ \mathcal{V}$和邻接矩阵$ \mathcal{A}$=[aij]N×N。智能体i的邻居用$ \mathcal{N}$i={j$ \mathcal{V}$|(j, i)∈ε}表示。图$ \mathcal{G}$的拉普拉斯矩阵为$ \mathcal{\mathcal{L}}$=$ \mathcal{\mathcal{D}}$-$ \mathcal{\mathcal{A}}$, $ \mathcal{\mathcal{D}}$,是每个节点的入度组成的对角阵。本文考虑N个智能体,假设前M(M≥1)个是跟随者,剩下的N-M个是领航者。跟随者和领航者的集合分别定义为$ \mathcal{\mathcal{F}}$={1, 2, …, M}和$ \mathcal{R}$={M+1, M+2, …, N}。考虑到领航者只向外发出信息, 对应于N个智能体的拉普拉斯矩阵为$ \mathcal{\mathcal{H}}$=$ \mathrm{\mathcal{H}=\mathcal{L}+}\sum\nolimits_{k=M+1}^{N}{{{\mathrm{ }\!\!\Delta\!\!\text{ }}_{_{k}}}}$, Δk是对角矩阵,元素为δ1k, δ2k, …, δMk。如果跟随者i可以收到领航者k的信息,那么δik=1,否则δik=0。

1.2 问题描述

跟随者的动力学系统为

$ \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)\;\:i \in \mathcal{F} }\\ {{\mathit{\boldsymbol{y}}_i}(t) = {\mathit{\boldsymbol{C}}_i}{\mathit{\boldsymbol{x}}_i}(t)} \end{array}} \right. $ (1)
 

式中:xi(t)∈Rniui(t)∈Rpiyi(t)∈Rq分别代表第i个跟随者的状态、控制输入和输出;AiBiCi是常数矩阵。

领航者的动力学系统为

$ \left\{ {\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\dot v}}}_k}(t) = \mathit{\boldsymbol{S}}{\mathit{\boldsymbol{v}}_k}(t)\;\:k \in \mathcal{L}}\\ {{\mathit{\boldsymbol{y}}_k}(t) = \mathit{\boldsymbol{F}}{\mathit{\boldsymbol{v}}_k}(t)} \end{array}} \right. $ (2)
 

式中:vk(t)和yk(t)是第k个领航者的状态和输出;SF是常数矩阵。

为方便研究,给出如下假设和定义。

假设1  对于每个跟随者,至少存在一条路径到达任一领航者。

定义1  对于具有多个领航者的多智能体系统,当t→∞时,跟随者进入到领航者形成的凸组合中,这种控制策略称之为包含控制。

本文的目标是考虑包括跟随者式(1)和领航者式(2),对每个跟随者设计分布式控制协议ui(t),使得,当t→∞时,包含控制误差

$ {\mathit{\boldsymbol{e}}_i}(t) = \sum\limits_{j \in {\mathcal{N}_i}} {{a_{ij}}} ({\mathit{\boldsymbol{y}}_i} - {\mathit{\boldsymbol{y}}_j}) + \sum\limits_{k = M + 1}^N {{\delta _{k,i}}} ({\mathit{\boldsymbol{y}}_i} - {\mathit{\boldsymbol{y}}_k}) $ (3)
 

收敛到原点,即limt→∞ei(t)=0。这意味着跟随者的输出能够最终收敛到由领航者的输出组成的凸包里,即实现多智能体包含控制。

2 基于边的事件触发观测器设计

由于领航者状态信息不完全可直接获得,设计观测器估计领航者凸包内点的状态。设计事件触发观测器为

$ \left\{ \begin{array}{l} {{\mathit{\boldsymbol{\dot z}}}_i}(t) = \mathit{\boldsymbol{S}}{\mathit{\boldsymbol{z}}_i}(t) + \mathit{\boldsymbol{T}}{\mathit{\boldsymbol{\mu }}_i}(t)\\ {\mathit{\boldsymbol{\mu }}_i}(t) = \mathit{\boldsymbol{uF}}\sum\limits_{k = M + 1}^N {{\delta _{k,i}}} {{\mathit{\boldsymbol{\bar z}}}_{k,i}}(t) + \mathit{\boldsymbol{uF}}\sum\limits_{j \in {N_i}} {{a_{ij}}} {{\mathit{\boldsymbol{\bar z}}}_{j,i}}(t) \end{array} \right. $ (4)
 

式中:zk, i(t)=zi(t)-vk(t); zj, i(t)=zi(t)-zj(t); k, i(t)和zj, i(t)是zk, i(t)和zj, i(t)的预测值。其动态为

$ \begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\dot {\bar z}}}}_{k,i}}(t) = \mathit{\boldsymbol{S}}{{\mathit{\boldsymbol{\bar z}}}_{k,i}}(t) + \mathit{\boldsymbol{F}}{\mathit{\boldsymbol{\mu }}_i}(t)}&{k \in \mathcal{R}} \end{array} $
$ \begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\dot {\bar z}}}}_{j,i}}(t) = \mathit{\boldsymbol{S}}{{\mathit{\boldsymbol{\bar z}}}_{j,i}}(t) + \mathit{\boldsymbol{F}}{\mathit{\boldsymbol{\mu }}_i}(t)}&{j \in \mathcal{L}} \end{array} $

且状态满足zk, i(tlk, i)=zk, i(tlk, i)和zj, i(tlj, i)=zj, i(tlj, i), tlj/k, i是事件触发时刻,并且定义初始时刻为t0j/k, i=0。

事件触发测量误差为

$ \left\{ {\begin{array}{*{20}{l}} {{ \epsilon _{k,i}}(t) = {{\mathit{\boldsymbol{\bar z}}}_{k,i}}(t) - {\mathit{\boldsymbol{z}}_{k,i}}(t)}&{t \in [t_l^{k,i},t_{l + 1}^{k,i})}\\ {{ \epsilon _{j,i}}(t) = {{\mathit{\boldsymbol{\bar z}}}_{j,i}}(t) - {\mathit{\boldsymbol{z}}_{j,i}}(t)}&{t \in [t_l^{j,i},t_{l + 1}^{j,i})} \end{array}} \right. $ (5)
 

对于跟随者i,定义观测器误差为

$ {\mathit{\boldsymbol{\xi }}_i}(t) = \sum\limits_{j \in {N_i}} {{a_{ij}}} ({\mathit{\boldsymbol{z}}_i} - {\mathit{\boldsymbol{z}}_j}) + \sum\limits_{k = M + 1}^N {{\delta _{k,i}}} ({\mathit{\boldsymbol{z}}_i} - {\mathit{\boldsymbol{v}}_k}) $

全局的观测器误差重写为

$ \mathit{\boldsymbol{\xi }}(t) = \sum\limits_{k = M + 1}^N {({\mathit{\boldsymbol{H}}_k} \otimes {\mathit{\boldsymbol{I}}_N})(\mathit{\boldsymbol{z}}(t) - {{\bf{1}}_n} \otimes {\mathit{\boldsymbol{v}}_k}(t))} $

式中:ξ(t)=[ξ1T(t), …, ξMT(t)]T, z(t)=[z1T(t), …, zMT(t)]T

对观测器误差求导可得

$ \mathit{\boldsymbol{\dot \xi }}(t) = ({\mathit{\boldsymbol{I}}_N} \otimes \mathit{\boldsymbol{S}})\mathit{\boldsymbol{\xi }}(t) + u\sum\limits_{k = M + 1}^N {({\mathit{\boldsymbol{H}}_k} \otimes \mathit{\boldsymbol{T}})} \mathit{\boldsymbol{\mu }}(t) $ (6)
 

式中:μ(t)=[μ1T(t), …, μMT(t)]T

定理1  在假设1成立的情况下,考虑多智能体系统(1)和(2),如果存在一个正整数u使得矩阵$ {{\mathit{\pmb{I}}}_{N}}\otimes \mathit{\pmb{S}}+u\sum\nolimits_{k=M+1}^{N}{({{\mathit{\pmb{H}}}_{k}}\otimes \mathit{\pmb{TF}}})$是赫尔维兹的,那么在如下的事件触发条件下

$ t_{l + 1}^{ji} = {\rm{inf}}\{ t > t_l^{ji}|\left\| {{ \epsilon _{ji}}(t)} \right\| > {\alpha _{ji}}{{\rm{e}}^{ - \beta t}}\} $ (7)
 

t→∞时,观测器状态ξi(t)能够估计到领航者凸包内某一内点的轨迹,也就是limt→∞ξi(t)=0。

证明  对测量误差ϵk, i(t)和ϵj, i(t)分别求导,可以得到

$ \left\{ {\begin{array}{*{20}{l}} {{{\dot \epsilon }_{k,i}}(t) = {{\mathit{\boldsymbol{\dot {\bar z}}}}_{k,i}}(t) - {{\mathit{\boldsymbol{\bar z}}}_{k,i}}(t) = \mathit{\boldsymbol{S}}{ \epsilon _{k,i}}(t)}\\ {{{\dot \epsilon }_{j,i}}(t) = {{\mathit{\boldsymbol{\dot {\bar z}}}}_{j,i}}(t) - {{\mathit{\boldsymbol{\bar z}}}_{j,i}}(t) = \mathit{\boldsymbol{S}}{ \epsilon _{j,i}}(t) + \mathit{\boldsymbol{F}}{\mathit{\boldsymbol{\mu }}_j}(t)} \end{array}} \right. $ (8)
 

从式(8)可知, 测量误差ϵk, i(t)是一个自治系统,并且在每个触发时刻都有ϵk, i(tkij)=0。这意味着zk, i(t)≡zk, i(t)。而测量误差ϵj, i(t)的解为ϵj, i(t)=$ \int_{t_k^{ij}}^t {\exp (\mathit{\boldsymbol{S}}(t} - \tau ))\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{\mu }}_j}(\tau ){\rm{d}}\tau $。因此,观测器误差系统(6)改写为

$ \begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot \xi }}(t) = \left( {{\mathit{\boldsymbol{I}}_N} \otimes \mathit{\boldsymbol{S}} + u\sum\limits_{k = M + 1}^N ( {\mathit{\boldsymbol{H}}_k} \otimes \mathit{\boldsymbol{TF}})} \right)\mathit{\boldsymbol{\xi }}(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} {\kern 1pt} {\kern 1pt} ({\mathit{\boldsymbol{I}}_N} \otimes \mathit{\boldsymbol{TF}}){\rm{d}}(t)} \end{array} $ (9)
 

式中:ϵ(t)=[ϵ1(t), …, ϵM(t)]T; ϵi(t)=$ \sum\nolimits_{j\in {{N}_{i}}}{{{\alpha }_{ji}}}$ ϵji(t)。

从式(9)可以得到ξ(t)的解为

$ \mathit{\boldsymbol{\xi }}(t) = {{\rm{e}}^{\mathit{\boldsymbol{\bar M}}t}}\mathit{\boldsymbol{\xi }}(0) + \int_0^t {{{\rm{e}}^{\mathit{\boldsymbol{\bar M}}(t - \tau )}}} ({\mathit{\boldsymbol{I}}_N} \otimes \mathit{\boldsymbol{TF}}) \epsilon (\tau ){\rm{d}}\tau $ (10)
 

式中:$ \mathit{\boldsymbol{\bar M = }}{\mathit{\boldsymbol{I}}_N} \otimes \mathit{\boldsymbol{S + }}u\sum\nolimits_{k = M + 1}^N {({\mathit{\boldsymbol{H}}_k} \otimes \mathit{\boldsymbol{TF}}} )$

对式(10)取范数, 并且利用$ \left\| {{{\rm{e}}^{\mathit{\boldsymbol{\bar M}}t}}} \right\| \le \varsigma {{\rm{e}}^{ - \rho t}}$,得

$ \begin{array}{l} \left\| {\mathit{\boldsymbol{\xi }}(t)} \right\| \le \varsigma {{\rm{e}}^{ - \rho t}}\left\| {\mathit{\boldsymbol{\xi }}(0)} \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} \varsigma \left\| {\int_0^t {{{\rm{e}}^{ - \rho (t - \tau )}}} ({\mathit{\boldsymbol{I}}_N} \otimes \mathit{\boldsymbol{TF}}) \epsilon (\tau ){\rm{d}}\tau } \right\| \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} \varsigma {{\rm{e}}^{ - \rho t}}\left\| {\mathit{\boldsymbol{\xi }}(0)} \right\| + \left\| {\mathit{\boldsymbol{TF}}} \right\|\int_0^t {{{\rm{e}}^{ - \rho (t - \tau )}}} \left\| { \epsilon (\tau )} \right\|{\rm{d}}\tau \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} \varsigma {{\rm{e}}^{ - \rho t}}\left\| {\mathit{\boldsymbol{\xi }}(0)} \right\| + \varsigma \left\| {\mathit{\boldsymbol{TF}}} \right\|\sum\limits_i^M {\int_0^t {{{\rm{e}}^{M(t - \tau )}}} } \left\| {{ \epsilon _i}(\tau )} \right\|{\rm{d}}\tau \end{array} $

根据ϵi(t)的定义, 可得

$ \begin{array}{*{20}{c}} {\left\| {\mathit{\boldsymbol{\xi }}(t)} \right\| \le \varsigma \left\| {\mathit{\boldsymbol{TF}}} \right\|\sum\limits_{i = 1}^M {\int_0^t {{{\rm{e}}^{ - \rho (t - \tau )}}} } \left\| {\sum\limits_{j \in {\mathcal{N} _i}} {{a_{ij}}} } \right\| \cdot }\\ {\left\| {{ \epsilon _{ji}}(\tau )} \right\|{\rm{d}}\tau + \varsigma {{\rm{e}}^{ - \rho t}}\left\| {\mathit{\boldsymbol{\xi }}(0)} \right\|} \end{array} $

根据事件触发条件(7)可知:

$ \begin{array}{l} \left\| {\mathit{\boldsymbol{\xi }}(t)} \right\| \le \varsigma \left\| {\mathit{\boldsymbol{TF}}} \right\|\sum\limits_{i = 1}^M {{\kappa _i}} {(\beta - \rho )^{ - 1}}({{\rm{e}}^{ - \beta t}} - {{\rm{e}}^{ - \rho 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} \varsigma {{\rm{e}}^{ - \rho t}}\left\| {\mathit{\boldsymbol{\xi }}(0)} \right\| \end{array} $

式中:$ {{\kappa }_{i}}=\sum\nolimits_{j\in {{\mathcal{N}}_{i}}}{{{\alpha }_{ji}}}$

从上述结果可知, 矩阵M是赫尔维兹的, 即,当t→∞时, 状态ξ(t)可以收敛到原点。这意味着观测器能估计到领航者凸包内某一内点的轨迹。证毕。

接下来,验证事件触发间隔存在一个正的下界,即,所提出的基于边的事件触发观测器能够避免Zeno现象。

ϵj, i的定义,可知:

$ \left\| {{ \epsilon _{j,i}}(t)} \right\| \le \int_{t_l^{ij}}^t {{\rm{exp}}} \left\| \mathit{\boldsymbol{S}} \right\|(t - \tau ))\left\| {\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{\mu }}_j}(\tau )} \right\|{\rm{d}}\tau $ (11)
 

根据μj定义, 可得

$ \begin{array}{*{20}{c}} {\left\| {\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{\mu }}_j}(t)} \right\| \le \left\| {\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{\xi }}_j}(t)} \right\| + \left\| {\mathit{\boldsymbol{F}}\sum\limits_{j \in {N_i}} {{a_{ij}}} { \epsilon _{j,i}}(t)} \right\| \le }\\ {\left\| {\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{\xi }}_j}(t)} \right\| + \sum\limits_{j \in {N_i}} {\left\| {{\alpha _{ji}}} \right\|} {{\rm{e}}^{ - \beta t}} \le {\sigma _i}{{\rm{e}}^{ - \beta t}}} \end{array} $

式中:$ {{\sigma }_{i}}=\left\| \xi (0) \right\|+{{\kappa }_{i}}$

结合式(11)可知:

$ \left\| {{ \epsilon _{j,i}}(t)} \right\| \le \frac{{{{\rm{e}}^{ - \beta t}}}}{{\left\| \mathit{\boldsymbol{S}} \right\| + \beta }}({{\rm{e}}^{(\left\| \mathit{\boldsymbol{S}} \right\| + \beta )(t - t_k^{ii})}} - 1) $

由于$ \left\| {{\mathrm{ϵ}}_{j, i}}(t) \right\|$在区间[tlji, tl+1ji)增长,且包络线是αjie-βt,可解出事件触发的时间间隔为

$ t_{l + 1}^{ji} - t_l^{ji} \ge {\rm{ln}}\left( {\frac{{{\alpha _{ji}}(\left\| \mathit{\boldsymbol{S}} \right\| + \beta ) + 1}}{{{\sigma _i}}}} \right)/(\left\| \mathit{\boldsymbol{S}} \right\| + \beta ) > 0 $ (12)
 

由式(12)可知,事件触发间隔存在一个正的下界, 也就是说,所设计的观测器能避免Zeno现象。

注解1  在文献[20]中,作者设计了一种基于点的事件触发控制协议,这种通信协议依赖于邻居信息。而本文设计的基于边的控制协议不依赖于邻居信息。且与传统的基于邻居的通信方式相比,基于边的通信方式是一种更为分布式的通信方式。

3 最优控制器设计

本节设计最优包含控制器,将包含控制问题转化成一致性控制问题。整合跟随者和观测器的动态方程, 得到增广系统:

$ {\mathit{\boldsymbol{\dot \eta }}_i}(t) = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{A}}_i}}&{\bf{0}}\\ {\bf{0}}&\mathit{\boldsymbol{S}} \end{array}} \right]{\mathit{\boldsymbol{\eta }}_i}(t) + \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{B}}_i}}\\ {\bf{0}} \end{array}} \right]{\mathit{\boldsymbol{u}}_i}(t) + \left[ {\begin{array}{*{20}{l}} {\bf{0}}\\ \mathit{\boldsymbol{T}} \end{array}} \right]{\mathit{\boldsymbol{\mu }}_i}(t) $ (13)
 

式中:ηi(t)=[xiT(t) ziT(t)]T

引入辅助变量:

$ {\mathit{\boldsymbol{\chi }}_i}(t) = ({\mathit{\boldsymbol{I}}_n} - {\mathit{\boldsymbol{I}}_n}){\mathit{\boldsymbol{\eta }}_i}(t) $ (14)
 

设计控制器

$ \begin{array}{l} {\mathit{\boldsymbol{u}}_i}(t) = {\mathit{\boldsymbol{K}}_{i1}}{\mathit{\boldsymbol{x}}_i}(t) + {\mathit{\boldsymbol{K}}_{i2}}{\mathit{\boldsymbol{z}}_i}(t) + {\mathit{\boldsymbol{c}}_i}(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{K}}_{i1}}\;\:{\mathit{\boldsymbol{K}}_{i2}}]{\mathit{\boldsymbol{\eta }}_i}(t) + {\mathit{\boldsymbol{c}}_i}(t) = {\mathit{\boldsymbol{K}}_i}{\mathit{\boldsymbol{\eta }}_i}(t) + {\mathit{\boldsymbol{c}}_i}(t) \end{array} $ (15)
 

变换后系统的性能评价函数定义为

$ \begin{array}{l} V_i^*(t) = \int_t^\infty {{{\rm{e}}^{ - {\gamma _i}(\tau t)}}} \mathit{\boldsymbol{\chi }}_i^{\rm{T}}(\tau ){\mathit{\boldsymbol{Q}}_i}{\mathit{\boldsymbol{\chi }}_i}(\tau ){\rm{d}}\tau + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \int_t^\infty {{{\rm{e}}^{ - {\gamma _i}(\tau - t)}}} \mathit{\boldsymbol{u}}_i^{\rm{T}}(\tau ){\mathit{\boldsymbol{R}}_i}{\mathit{\boldsymbol{u}}_i}(\tau ){\rm{d}}\tau \end{array} $ (16)
 

式中: QiRi是正定矩阵;γi是加权系数,保证性能函数有界性。

注解2  在文献[19]中, 基于线性二次型的性能函数要求领航者动力学必须是渐近稳定的。本文引入加权系数克服这个要求。

定理2  考虑满足假设1的多智能体系统(1)和(2),假设(Ai, Bi)是可镇定的且(Ai, Ci)是可观测的。如果存在正定矩阵Pi满足:

$ \begin{array}{*{20}{l}} {{\bf{0}} = \mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_i^{\rm{T}}\mathit{\boldsymbol{P}}_i^* + \mathit{\boldsymbol{P}}_i^*{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_i} - {\gamma _i}\mathit{\boldsymbol{P}}_i^* - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{P}}_i^*{\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_i}\mathit{\boldsymbol{R}}_i^{ - 1}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_i^{\rm{T}}\mathit{\boldsymbol{P}}_i^* + \mathit{\boldsymbol{C}}_i^{\rm{T}}{\mathit{\boldsymbol{Q}}_i}{\mathit{\boldsymbol{C}}_i}} \end{array} $ (17a)
 
$ \begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot M}}_i^* = - (\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_i^{\rm{T}} - \mathit{\boldsymbol{P}}_i^*{\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_i}\mathit{\boldsymbol{R}}_i^{ - 1}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_i^{\rm{T}})\mathit{\boldsymbol{M}}_i^* - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{P}}_i^*{\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_i}{\mathit{\boldsymbol{\mu }}_i} + {\gamma _i}\mathit{\boldsymbol{M}}_i^*} \end{array} $ (17b)
 
$ \begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot N}}_i^* = - 2{{(\mathit{\boldsymbol{M}}_i^*)}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_i}{\mathit{\boldsymbol{\mu }}_i} + {{(\mathit{\boldsymbol{M}}_i^*)}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_i}\mathit{\boldsymbol{R}}_i^{ - 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{ \boldsymbol{\varPhi} }}_i^{\rm{T}}\mathit{\boldsymbol{M}}_i^* + {\gamma _i}\mathit{\boldsymbol{N}}_i^*} \end{array} $ (17c)
 

那么:

1) 包含误差χi(t)能渐近收敛到原点,即实现包含控制。

2) 最优控制器设计为ui*(t)=-Ri-1·(Pi*ηi(t)+Mi*), 且性能函数的最小值为

$ \begin{array}{*{20}{c}} {{J_i}({\mathit{\boldsymbol{x}}_i}(0),\mathit{\boldsymbol{u}}_i^*) = \frac{1}{2}(\mathit{\boldsymbol{\eta }}_i^{\rm{T}}(0)\mathit{\boldsymbol{P}}_i^*{\mathit{\boldsymbol{\eta }}_i}(0) + }\\ {{\mathit{\boldsymbol{N}}_i}(0)) + \mathit{\boldsymbol{\eta }}_i^{\rm{T}}(0){\mathit{\boldsymbol{M}}_i}(0)} \end{array} $

证明  构造候选李雅普诺夫函数:

$ {V_i}(t) = \frac{1}{2}(\mathit{\boldsymbol{\eta }}_i^{\rm{T}}(t){\mathit{\boldsymbol{P}}_i}{\mathit{\boldsymbol{\eta }}_i}(t) + 2\mathit{\boldsymbol{\eta }}_i^{\rm{T}}(t){\mathit{\boldsymbol{M}}_i}(t) + {\mathit{\boldsymbol{N}}_i}(t)) $ (18)
 

式中:

$ {\mathit{\boldsymbol{M}}_i}(t) = \int_0^\infty {{{\rm{e}}^{ - \gamma t}}} (\int_0^\tau {{{\rm{e}}^{{{\mathit{\boldsymbol{\bar A}}}_{{i^\tau }}}}}} \mathit{\boldsymbol{\bar T}}{\mathit{\boldsymbol{\mu }}_i}(\tau ){\rm{d}}\tau ){\rm{d}}t $
$ {\mathit{\boldsymbol{N}}_i}(t) = \mathit{\boldsymbol{M}}_i^{\rm{T}}(t){\mathit{\boldsymbol{M}}_i}(t) $
$ {{\mathit{\boldsymbol{\bar A}}}_i} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{A}}_i} + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{K}}_i}}&{\bf{0}}\\ {\bf{0}}&\mathit{\boldsymbol{S}} \end{array}} \right],\mathit{\boldsymbol{\bar T}} = \left[ {\begin{array}{*{20}{l}} {\bf{0}}\\ \mathit{\boldsymbol{T}} \end{array}} \right] $

沿着增广系统(13)的轨迹对李雅普诺夫函数(18)求导,得到如下哈密顿-雅克比-贝尔曼方程:

$ \begin{array}{l} H({\mathit{\boldsymbol{y}}_i}(t),{\mathit{\boldsymbol{u}}_i}(t)) = \frac{{{\rm{d}}{V_i}(t)}}{{{\rm{d}}t}} - {\gamma _i}{V_i}(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} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{2}\mathit{\boldsymbol{y}}_i^{\rm{T}}(t){\mathit{\boldsymbol{Q}}_i}{\mathit{\boldsymbol{y}}_i}(t) + \mathit{\boldsymbol{u}}_i^{\rm{T}}(t){\mathit{\boldsymbol{R}}_i}{\mathit{\boldsymbol{u}}_i}(t) = \\ \begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{\eta }}_i^{\rm{T}}(t){\mathit{\boldsymbol{P}}_i}{\mathit{\boldsymbol{\eta }}_i}(t) + 2\mathit{\boldsymbol{\eta }}_i^{\rm{T}}(t){\mathit{\boldsymbol{M}}_i}(t) + 2\mathit{\boldsymbol{M}}_i^{\rm{T}}(t){{\dot \eta }_i}(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} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\dot N}_i}(t) - {\gamma _i}(\mathit{\boldsymbol{\dot \eta }}_i^{\rm{T}}(t){\mathit{\boldsymbol{P}}_i}{\mathit{\boldsymbol{\eta }}_i}(t) + 2\mathit{\boldsymbol{\eta }}_i^{\rm{T}}(t){\mathit{\boldsymbol{M}}_i}(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} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{N}}_i}(t)) + \frac{1}{2}\mathit{\boldsymbol{y}}_i^{\rm{T}}(t){\mathit{\boldsymbol{Q}}_i}{\mathit{\boldsymbol{y}}_i}(t) + \mathit{\boldsymbol{u}}_i^{\rm{T}}(t){\mathit{\boldsymbol{R}}_i}{\mathit{\boldsymbol{u}}_i}(t)} \end{array} \end{array} $ (19)
 

利用平稳性条件H(yi(t), ui(t))/u=0,可以得到最优控制器ui*(t)=-Ri-1(Pi*ηi(t)+Mi*(t))。进而,可以得到

$ \begin{array}{l} {{\dot V}_i}(t) = \mathit{\boldsymbol{\eta }}_i^{\rm{T}}(t){\mathit{\boldsymbol{P}}_i}{{\mathit{\boldsymbol{\dot \eta }}}_i}(t) + 2\mathit{\boldsymbol{\eta }}_i^{\rm{T}}(t){{\mathit{\boldsymbol{\dot M}}}_i}(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} 2\mathit{\boldsymbol{M}}_i^{\rm{T}}(t){{\mathit{\boldsymbol{\dot \eta }}}_i}(t) + {{\mathit{\boldsymbol{\dot N}}}_i}(t) \end{array} $ (20)
 

根据Ni(t)和Mi(t)的定义,当t→∞时可以得到Ni*(∞)=Mi*(∞)=0。此外,根据事件触发条件可知,limt→∞ϵij(t)=0。进而,得到李雅普诺夫函数Vi(t)→1/2ηiT(t)Piηi(t),并且其导数$ {{{\dot{V}}}_{i}}(t)$满足:

$ {\dot V_i}(t) \to \mathit{\boldsymbol{\eta }}_i^{\rm{T}}(t)({\mathit{\boldsymbol{P}}_i}{\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_i} + \mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_i^{\rm{T}}{\mathit{\boldsymbol{P}}_i}){\mathit{\boldsymbol{\eta }}_i}(t) \le 0 $

根据拉塞尔不变集原理,可知当$ {{{\dot{V}}}_{i}}(t)$=0时,ηi(t)可以收敛到最大的不变集。根据式(17)的黎卡提方程,如果Pi*ηi(t)=0则意味着$ {{{\dot{V}}}_{i}}(t)$=0,因此,包含误差渐近地收敛到原点。进而,得到稳态的性能函数:

$ \begin{array}{l} {J_i}({\mathit{\boldsymbol{x}}_i}(0),\mathit{\boldsymbol{u}}_i^*) = \int_0 {{{\rm{e}}^{ - {\gamma _i}\tau }}} \mathit{\boldsymbol{\chi }}_i^{\rm{T}}(\tau ){\mathit{\boldsymbol{Q}}_i}{\mathit{\boldsymbol{\chi }}_i}(\tau ){\rm{d}}\tau + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \int_0^\infty {{{\rm{e}}^{ - {\gamma _i}\tau }}} \mathit{\boldsymbol{u}}_i^{\rm{T}}(\tau ){\mathit{\boldsymbol{R}}_i}{\mathit{\boldsymbol{u}}_i}(\tau ){\rm{d}}\tau + \int_0^\infty {{{\bar V}_i}} (\tau ){\rm{d}}\tau + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\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 V}_i}(0) - {{\bar V}_i}(\infty ) \end{array} $ (21)
 

式中:Vi(t)=eγitVi(t)。因此,可知Vi(∞)=0, Vi(0)=Vi(0)。根据黎卡提方程(17)可得

$ \begin{array}{l} {J_i}({\mathit{\boldsymbol{x}}_i}(0),\mathit{\boldsymbol{u}}_i^*) = \frac{1}{2}\mathit{\boldsymbol{\eta }}_i^{\rm{T}}(0)\mathit{\boldsymbol{P}}_i^*{\mathit{\boldsymbol{\eta }}_i}(0) + {\mathit{\boldsymbol{N}}_i}(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} \mathit{\boldsymbol{\eta }}_i^{\rm{T}}(0){\mathit{\boldsymbol{M}}_i}(0) \end{array} $

证毕

注解3  文献[21-22]研究了异构多智能体的包含控制问题,但是领航者和跟随者的维数是相同的。本文中所设计的最优控制策略适用于具有不同维数的异构多智能体系统。此外,在文献[22]中,领航者是一个虚拟的动态轨迹,这将无法实现最优包含控制。

4 仿真分析

采用文献[23]中机器人模型作为多智能体系统,验证分布式最优包含控制协议的有效性。选取8个机器人构成多智能体系统,4个设定为领航者,4个作为跟随者, 其方程为

$ {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_i}{\mathit{\boldsymbol{\dot x}}_i} + {\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}_i}{\mathit{\boldsymbol{x}}_i} = {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_i}{\mathit{\boldsymbol{\delta }}_{iS}} $ (22)
 

式中:xiT=[ωiqiθizi]; Ψi=[Z Mis0 0];

$ {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_i} = \left[ {\begin{array}{*{20}{c}} {{m_i} - {{\bar z}_{i{{\dot {\bar w}}_i}}}}&{{{\bar Z}_{i{{\dot {\bar q}}_i}}}}&0&0\\ { - {{\bar M}_{i{{\dot {\bar w}}_i}}}}&{{I_{iy}} - {{\bar M}_{i{{\dot {\bar w}}_i}}}}&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{array}} \right] $
$ {\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}_i} = \left[ {\begin{array}{*{20}{c}} { - {{\bar z}_{i{{\bar w}_i}}}}&{{m_i} - {{\bar u}_{i0}}{{\bar Z}_{i{{\bar q}_i}}}}&0&0\\ { - {M_{i{{\bar w}_i}}}}&{ - {M_{i{{\bar q}_i}}}}&{{M_{i{{\bar \theta }_i}}}}&0\\ 0&{ - 1}&0&0\\ { - 1}&0&{{{\bar u}_{i0}}}&0 \end{array}} \right] $

写成线性系统的典型形式:

$ {\mathit{\boldsymbol{\dot x}}_i} = {\mathit{\boldsymbol{A}}_i}{\mathit{\boldsymbol{x}}_i} + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{u}}_i} $

式中:Ai=Θi-1Δi; Bi=Θi-1Ψi; ui=δiS

每个机器人的参数如表 1所示。

表 1 机器人的各项参数取值 Table 1 Parameters value of robots
参数 跟随者1, 2 跟随者3, 4
mi 4.154 8 3.57
Iiy 0.573 2 0.615 2
Miωi 0.003 14 0.003 46
$ {{M}_{i{{{\dot{\bar{\omega }}}}_{i}}}}$ -0.000 825 -0.000 134
Miqi -0.001 117 -0.002 24
Miθi 1.125 2.345 6
Ziqi -0.002 38 -0.001 25
ui0 1 2
z -0.008 73 -0.005 68

表 1中参数代入到动态方程中,并进行一系列计算, 可以得到跟随者的系统参数为

$ \begin{array}{l} {\mathit{\boldsymbol{A}}_1} = {\mathit{\boldsymbol{A}}_2} = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ {\begin{array}{*{20}{c}} { - 0.002{\kern 1pt} {\kern 1pt} {\kern 1pt} 1}&{ - 0.999{\kern 1pt} {\kern 1pt} {\kern 1pt} 2}&0&0\\ {0.005}&{ - 0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 8}&{ - 0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 8}&0\\ 0&1&0&0\\ 1&0&0&{ - 1} \end{array}} \right] \end{array} $
$ {\mathit{\boldsymbol{A}}_3} = {\mathit{\boldsymbol{A}}_4} = \left[ {\begin{array}{*{20}{c}} { - 1.7}&{0.63}&0&0\\ { - 16.09}&{73.19}&0&0\\ 1&0&{ - 3.6}&0\\ 0&1&0&0 \end{array}} \right] $
$ {{\mathit{\boldsymbol{B}}_1} = {\mathit{\boldsymbol{B}}_2} = \left[ {\begin{array}{*{20}{l}} {0.10}&{3.52}&0&0 \end{array}} \right]} $
$ {{\mathit{\boldsymbol{C}}_1} = {\mathit{\boldsymbol{C}}_2} = \left[ {\begin{array}{*{20}{c}} { - 0.1}&{1.2}&0&0\\ {0.4}&{1.4}&0&0 \end{array}} \right]} $
$ {{\mathit{\boldsymbol{B}}_3} = {\mathit{\boldsymbol{B}}_4} = \left[ {\begin{array}{*{20}{l}} {0.081{\kern 1pt} {\kern 1pt} {\kern 1pt} 6}&0&0&0 \end{array}} \right]} $
$ {{\mathit{\boldsymbol{C}}_3} = {\mathit{\boldsymbol{C}}_4} = \left[ {\begin{array}{*{20}{c}} { - 0.5}&{1.3}&0&0\\ {0.2}&{1.4}&0&0 \end{array}} \right]} $

而领航者的系统参数为

$ \mathit{\boldsymbol{S}} = \left[ {\begin{array}{*{20}{c}} 1&{ - 3}&0&0\\ 1&{ - 1}&0&0\\ 0&0&0&1\\ 0&0&{ - 1}&0 \end{array}} \right],\mathit{\boldsymbol{T}} = \left[ {\begin{array}{*{20}{c}} 1&0\\ 0&1\\ 0&0\\ 0&0 \end{array}} \right],\mathit{\boldsymbol{R}} = {\mathit{\boldsymbol{T}}^{\rm{T}}} $

智能体之间的通信图如图 1所示,节点1、2、3、4代表领航者,其他代表跟随者。

图 1 通信拓扑图 Fig. 1 Communication topology

利用极点配置方法,可以得到观测器反馈增益矩阵为

$ \begin{array}{l} \mathit{\boldsymbol{F}} = \\ \left[ {\begin{array}{*{20}{c}} { - 0.507{\kern 1pt} {\kern 1pt} 9}&{ - 0.163{\kern 1pt} {\kern 1pt} 2}&{0.102{\kern 1pt} {\kern 1pt} 1}&{ - 0.057{\kern 1pt} {\kern 1pt} 3}\\ { - 0.220{\kern 1pt} {\kern 1pt} 5}&{ - 0.219{\kern 1pt} {\kern 1pt} 1}&{ - 0.067{\kern 1pt} {\kern 1pt} 3}&{ - 0.481{\kern 1pt} {\kern 1pt} 4} \end{array}} \right] \end{array} $

选择性能函数(16)中的权重矩阵和参数为

$ {{Q_1} = {Q_2} = 90,{R_1} = {R_2} = 0.09,{\gamma _1} = {\gamma _2} = 0.04} $
$ {{Q_3} = {Q_4} = 80,{R_3} = {R_4} = 0.08,{\gamma _3} = {\gamma _4} = 0.03} $

根据定理2中的黎卡提方程(17),可以获得增益矩阵

$ \begin{array}{*{20}{l}} {{\mathit{\boldsymbol{K}}_{11}} = {\mathit{\boldsymbol{K}}_{21}} = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - [34.480{\kern 1pt} {\kern 1pt} {\kern 1pt} 4\;\:30.421{\kern 1pt} {\kern 1pt} {\kern 1pt} 4\;\:1.778{\kern 1pt} {\kern 1pt} {\kern 1pt} 0\;\:9.207{\kern 1pt} {\kern 1pt} {\kern 1pt} 7]} \end{array} $
$ \begin{array}{*{20}{l}} {{\mathit{\boldsymbol{K}}_{31}} = {\mathit{\boldsymbol{K}}_{41}} = }\\ {\;\:[\begin{array}{*{20}{c}} {{\rm{14}}{\rm{.710}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{5}}}&{ - 59.242}&{0.747{\kern 1pt} {\kern 1pt} {\kern 1pt} 8}&{ - 30.159{\kern 1pt} {\kern 1pt} {\kern 1pt} 5} \end{array}]} \end{array} $

在这种情况下, 计算得到性能成本为Jx(t)=17.31。每个跟随者的输出包含误差的运动轨迹如图 2图 3所示。可以观察到,输出包含误差最终衰减为零。这意味者跟随者进入到领航者所形成的凸包中,由定义1可知,实现了包含控制。又从图 4可以看出,性能函数收敛到一个有限值,并不再增加,达到最大值,即性能最优。从这些结果可以看出,在基于边的事件触发观测器和最优控制协议下,实现了输出最优包含控制。

图 2 包含误差分量χi1轨迹 Fig. 2 Trajectories of containment error χi1
图 3 包含误差分量χi2轨迹 Fig. 3 Trajectories of containment error χi2
图 4 性能函数Jx(t)轨迹 Fig. 4 Trajectory of performance function Jx(t)
5 结论

针对异构多智能体系统领航者信息不完全可测的包含控制问题,设计基于观测器的最优控制器,实现输出包含控制,达到最优性能。

1) 所设计的事件触发观测器实现了对领航者组成的凸包某一内点的估计,并节约了通信成本。

2) 所设计的最优控制协议保证了跟随者的输出能够进入领航者输出组成的凸包内,同时也保证了系统的性能达到最优。

参考文献
[1] DUSAN T. Transport modeling by multi-agent systems:a swarm intelligence approach[J]. Transportation Planning and Technology, 2003, 26(4): 289-312.
Click to display the text
[2] ZHAO Q L, DONG X W, LIANG Z X, et al. Distributed cooperative guidance for multiple missiles with fixed and switching communication topologies[J]. Chinese Journal of Aeronautics, 2017, 30(4): 1570-1581.
Click to display the text
[3] 陈旿, 张鑫, 金鑫, 等. 一种多智能体协同信息一致性算法[J]. 航空学报, 2017, 38(12): 321222.
CHEN W, ZHANG X, JIN X, et al. A cooperative information consensus algorithm for multi-agent system[J]. Acta Aeronautics et Astronautica Sinica, 2017, 38(12): 321222. (in Chinese)
Cited By in Cnki | Click to display the text
[4] KIM H, SHIM H, SEO J H. Output consensus of heterogeneous uncertain linear multi-agent systems[J]. IEEE Transactions on Automatic Control, 2010, 56(1): 200-206.
Click to display the text
[5] 周绍磊, 祁亚辉, 张雷, 等. 切换拓扑下无人机集群系统时变编队控制[J]. 航空学报, 2017, 38(4): 320452.
ZHOU S L, QI Y H, ZHANG L, et al. Time-varying formation control of UAV swarm systems with switching topologies[J]. Acta Aeronautics et Astronautica Sinica, 2017, 38(4): 320452. (in Chinese)
Cited By in Cnki (14) | Click to display the text
[6] JIANG F C, WANG L, JIA Y M. Consensus in leaderless networks of high-order-integrator agents[C]//American Control Conference, 2009: 4458-4463.
[7] MEI J, ZHANG H B, MA G F. Adaptive coordinated tracking for networked Euler-Lagrange systems under a directed graph[J]. Acta Automatica Sinica, 2011, 37(5): 596-603.
Click to display the text
[8] DONG X W, SHI Z Y, LU G, et al. Formation-containment analysis and design for high-order linear time-invariant swarm systems[J]. International Journal of Robust and Nonlinear Control, 2015, 25(17): 3439-3456.
Click to display the text
[9] NOTARSTEFANO G, EGERSTEDT M, HAQUE M. Containment in leader-follower networks with switching communication topologies[J]. Automatica, 2011, 47(5): 1035-1040.
Click to display the text
[10] LI J, GUAN Z H, LIAO R Q, et al. Impulsive containment control for second-order networked multi-agent systems with sampled information[J]. Nonlinear Analysis:Hybrid Systems, 2014, 12: 93-103.
Click to display the text
[11] TABUADA P. Event-triggered real-time scheduling of stabilizing control tasks[J]. IEEE Transactions on Automatic Control, 2007, 52(9): 1680-1685.
Click to display the text
[12] DIMAROGONAS D V, FRAZZOLI E, JOHANSSON K H. Distributed event-triggered control for multi-agent systems[J]. IEEE Transactions on Automatic Control, 2011, 57(5): 1291-1297.
Click to display the text
[13] FAN Y, FENG G, WANG Y, et al. Distributed event-triggered control of multi-agent systems with combinational measurements[J]. Automatica, 2013, 49(2): 671-675.
Click to display the text
[14] WANG L C, WANG Z D, HUANG T W, et al. An event-triggered approach to state estimation for a class of complex networks with mixed time delays and nonlinearities[J]. IEEE Transactions on Cybernetics, 2015, 46(11): 2497-2508.
Click to display the text
[15] HU W F, LIU L, FENG G. Output consensus of heterogeneous linear multi-agent systems by distributed event-triggered/self-triggered strategy[J]. IEEE Transactions on Cybernetics, 2016, 47(8): 1914-1924.
Click to display the text
[16] MOLIN A, HIRCHE S. Suboptimal event-triggered control for networked control systems[J]. ZAMM-Journal of Applied Mathematics and Mechanics, 2014, 94(4): 277-289.
Click to display the text
[17] WANG J, MOUNIER H, CELA A, et al. Event driven intelligent PID controllers with applications to motion control[J]. IFAC Proceedings Volumes, 2011, 44(1): 10080-10085.
Click to display the text
[18] SAHOO A, XU H, JAGANNATHAN S. Approximate optimal control of affine nonlinear continuous-time systems using event-sampled neuro-dynamic programming[J]. IEEE Transactions on Neural Networks and Learning Systems, 2016, 28(3): 639-652.
Click to display the text
[19] VAMVOUDAKIS K G, FERRAZ H. Model-free event-triggered control algorithm for continuous-time linear systems with optimal performance[J]. Automatica, 2018, 87: 412-420.
Click to display the text
[20] LIU K E, JI Z J, XIE G M, et al. Event-based broadcasting containment control for multi-agent systems under directed topology[J]. International Journal of Control, 2016, 89(11): 2360-2370.
Click to display the text
[21] LEWIS F L, ZHANG H W, HENGSTER M K, et al. Cooperative control of multi-agent systems:optimal and adaptive design approaches[M]. Springer Science & Business Media, 2013.
[22] HAGHSHENAS H, BADAMCHIZADEH M A, BARADARANNIA M. Containment control of heterogeneous linear multi-agent systems[J]. Automatica, 2015, 54: 210-216.
Click to display the text
[23] NARASIMHAN M, SINGH S N. Adaptive optimal control of an autonomous underwater vehicle in the dive plane using dorsal fins[J]. Ocean Engineering, 2006, 33(3-4): 404-416.
Click to display the text
http://dx.doi.org/10.7527/S1000-6893.2019.23775
中国航空学会和北京航空航天大学主办。
0

文章信息

王东, 王泽华, 刘洋, GUDongbing, 王伟
WANG Dong, WANG Zehua, LIU Yang, GU Dongbing, WANG Wei
基于事件触发的异构多智能体最优包含控制
Event-triggered optimal containment control for heterogeneous multi-agent systems
航空学报, 2020, 41(S1): 723775.
Acta Aeronautica et Astronautica Sinica, 2020, 41(S1): 723775.
http://dx.doi.org/10.7527/S1000-6893.2019.23775

文章历史

收稿日期: 2019-12-13
退修日期: 2019-12-26
录用日期: 2019-12-30
网络出版时间: 2020-01-13 16:57

相关文章

工作空间