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摘要: 作为多智能体对抗博弈问题的重要分支, 追逃博弈(Pursuit-evasion, PE)问题在控制和机器人领域得到了广泛应用, 受到众多研究者的密切关注. 追逃博弈问题主要聚焦于追逐者和逃跑者双方为实现各自目标而展开的动态博弈: 追逐者试图在最短时间内抓到逃跑者, 逃跑者的目标则是避免被捕获. 本文概述追逃博弈问题的相关研究进展, 从空间环境、信息获取等五个方面介绍追逃博弈问题的各类设定; 简述理论求解、数值求解等四种当下主流的追逃博弈问题求解方法. 通过对现有研究的总结和分析, 给出几点研究建议, 对未来追逃博弈问题的发展具有一定指导意义.Abstract: As an important branch of multi-agent adversarial games, pursuit-evasion (PE) games have found widespread applications in the fields of control and robotics, attracting considerable attention from researchers. PE games primarily focus on the dynamic games between pursuer and evader, each striving to achieve their respective objectives: The pursuer aims to capture the evader as quickly as possible, while the evader＇s goal is to avoid capture. This article provides an overview of the research progress in PE games, and introduces various settings of PE games across five key dimensions, including spatial environment, information acquisition, and so on. It briefly describes four mainstream methods for solving PE games, including theoretical approaches, numerical approaches, and so on. By summarizing and analyzing existing researches, this article offers several research suggestions, which are expected to provide significant guidance for future developments in PE games.
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Key words:
- Pursuit-evasion (PE) games /
- multi-agent /
- adversarial games /
- differential games
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多个独立的智能体通过信息交互以及多种方式组成多智能体系统[1−6], 旨在解决单个智能体无法解决的大规模复杂性任务. 多智能体系统能够完成在复杂网络下的信息传递, 已广泛应用在诸多重要的实际场景, 如航天器领域[7]、无人驾驶飞行器[8]和水下车辆系统[9]等. 特别地, 多智能体系统的跟踪控制策略[10−11]一直是进一步探索的热点研究方向. 例如, 文献[12]研究针对多智能体系统网络化预测PID的控制问题来达到输出信号一致性. 文献[13]针对带有非线性扰动的多智能体系统, 提出基于神经网络的自适应控制策略来实现共识控制目标. 然而, 智能体之间的信息交互严格依赖网络环境, 特别是存在大量智能体时将同时占用多个网络通信渠道, 不可避免地导致沉重的网络负担, 这一现象亟需解决.
近年来, 为缓解信息传递渠道上的通讯压力, 学者们提出事件触发策略[14−18]使控制器以非周期的方式更新. 文献[19]提出针对多智能体系统的事件触发分布式控制策略, 包括固定阈值策略、相对阈值策略和切换阈值策略. 根据这一概念, 相继研究出多种新颖的事件触发机制. 例如, 动态事件触发机制、状态触发机制和记忆事件触发机制等. 其中, 状态触发机制[20−21]引起广泛关注. 特别地, 文献[21]首次提出基于状态触发机制的非线性多智能体系统自适应一致性控制方案, 设计的事件触发机制首先将采样信号值与系统真实值构造出采样误差, 随后转换得到带有触发信号的同步误差, 并将之设计到控制输入信号中来缓解控制器−传感器渠道上的通讯压力. 值得注意的是, 现存文献中状态触发机制的阈值条件是动态变化的, 并且会采用根据分解的方式将产生的采样误差值的平方项化为常数项的方法判断下一次采样时刻. 需要进一步指出的是, 该阈值条件的设计方法严格依赖于稳定性条件, 从而限制阈值条件设计的灵活性. 但在实际系统的真实状况中, 状态信息通常不可测量, 因而会导致控制方案实施的准确性. 因此, 提出状态观测器来实现对原系统重构, 从而满足反馈控制的需要. 所以, 在大规模实际系统的控制运行中, 首先需要解决在未知状态不可测情况下系统通讯渠道负担重的问题, 并且在信息传递过程中多个通讯链路上的资源节约问题同样值得注意. 例如, 控制器−执行器环节、传感器−控制器环节、智能体与智能体之间的通讯渠道等. 因此, 如果实现同时在多个通讯渠道上节约通讯资源势必会大幅度降低整体控制系统通讯带宽占用率.
值得关注的是, 一些非线性因素[22−28]可能会导致系统性能下降或系统抖震现象发生. 模糊逻辑系统或神经网络是处理非线性项的常用近似工具. 然而在大多数情况下, 模糊逻辑系统或神经网络的逼近能力是有限的, 并且其逼近效果取决于模糊规则或神经网络节点的数量. 因此, 系统中非线性因素的存在会导致系统不稳定. 为解决上述情况并保证系统的实时性能, 文献[29]提出规定性能控制方法, 通过将跟踪误差约束在预定的范围内实现对系统瞬态性能和稳态性能的保证. 文献[30]针对非线性多智能体系统, 设计新颖的规定性能转换函数. 文献[31]利用误差转换方法和规定性能控制策略设计分布式自适应控制器来保证控制目标的有效实现. 因此可以看出, 规定性能控制方法在确保系统性能方面十分有效, 并且当发现性能指标无法满足时, 该机制可以及时采取相应的容错措施以提高系统的可靠性和安全性.
因此, 考虑到非线性多智能体系统中通讯资源负担重以及现存状态事件触发机制的阈值设计条件具有一定地局限性的双重问题, 本文展开基于混合双端事件触发机制的协同控制策略研究, 旨在提升现存状态事件触发机制阈值设计的灵活性且改善多通讯渠道的通讯压力状况, 进一步拓展多智能体系统一致性控制策略的多样性.
本文主要贡献如下:
1) 与现存结果相比[21], 所提出的状态触发机制的阈值条件可以在不使用杨氏不等式的情况下直接设计, 从而减少现有控制方案的缩放次数和保守性. 并且首次使用估计状态进行采样, 扩展了状态触发机制的应用范围. 基于新的状态触发机制与控制器触发机制, 构造新的双端分布式触发框架, 在较少的参数设计限制性下, 具有更小的通信压力.
2) 基于规定性能技术特性, 提出的自适应控制方案既减少系统不稳定情况发生的概率, 又保证多智能体一致性任务的精确度. 此外, 所设计的分布式观测器仅依赖于相对输出信息进行反馈调节来解决状态不可测问题, 具有良好的可扩展性和灵活性.
本文的组织结构安排如下. 第1节给出本文工作所需的预备知识; 第2节介绍模糊状态观测器的设计过程; 为获得预期的控制目标, 第3节提出自适应分布式控制器的设计方法、稳定性及芝诺行为分析; 第4节通过一个实际仿真例子证明所提出策略的有效性; 第5节给出本文的结论, 并对未来进行展望.
1. 预备知识
1.1 图论
针对非线性多智能体系统, 智能体之间的信息传输关系需要清晰描述. 考虑一个有向图来表明多个智能体之间的关系. 首先, 定义有向图为 $ \bar{{\cal{G}}}=({{\cal{V}}},\;\acute{{\cal{E}}},\;\bar{{\cal{A}}}) $, 其中, $ {{\cal{V}}}=\left( {1},\;\cdots,\;{N}\right) $ 为节点集合, $ \acute{{\cal{E}}}\subseteq{{\cal{V}}} \times {{\cal{V}}} $ 为边集合. 定义邻接矩阵为 $ {\bar{{\cal{A}}}}=[a_{h,\; \ell}] \in \bf{R}^{N\times N} $. 与此同时, 节点$ \ell $到节点 $ h $形成的边表示为 $ \left({{\cal{V}}}_{h},\;{{\cal{V}}}_{\ell}\right) \in \acute{{\cal{E}}} $. 当节点$ \ell $可以传输信息到节点$ h $, 即可以得到$ a_{h,\;\ell}>0 $. 否则, $ a_{h,\;\ell}=0 $. 此外, $ \check{{\cal{D}}} $ 表示度矩阵, 其中 $ \check{{\cal{D}}}=\hbox{diag}\left\{d_{1},\;\cdots,\;d_{N}\right\} $ 以及 $ d_{h} = {\sum\nolimits^N_{\ell=1}}a_{h,\;\ell} $. $ {{\cal{L}}} $ 表示拉普拉斯矩阵, 其定义为$ {{\cal{L}}}=\check{{\cal{D}}}-\bar{{\cal{A}}} $.
假设 1[32]. 将有向图$ {\cal{G}} $看作一个生成树, 需要保证任意一个节点到根节点都至少存在一条有效传输路径, 并将领导者节点$ {\bf{0}} $ 视为生成树的根.
引理1[33]. 定义$ {\cal{B}}=\hbox{diag}\left\{b_{h}\right\}\in \bf{R}^{N\times N} $, 并且使得$ b_{h}>0 $. 随后, 可以得到 $ {\cal{L}}+{\cal{B}} $是非奇异的.
1.2 问题形成
1) 智能体的动态模型. 对于$ h=1,\;2,\;\cdots,\;M $和 $ g=1,\;2,\;\cdots,\;n-1 $, 定义第$ h $个智能体的动态模型为
$$ \begin{aligned} \left\{ \begin{aligned} &\dot{x}_{h,\;g}=x_{h,\;g+1}+\zeta_{h,\;g}(\bar{x}_{h,\;g})\\ &\dot{x}_{h,\;n}=u_{h}+\zeta_{h,\;n}(\bar{x}_{h,\;n})\\ &y_{h}=x_{h,\;1} \end{aligned} \right. \end{aligned} $$ (1) 其中, $ {\bar x_{h,\;g}} = [x_{h,\;1},\;x_{h,\;2},\;\cdots,\;x_{h,\;g}]^\text{T}\in {{\bf{R}}}^{g} $ 和 $ {\bar x_{h,\;n}}= [x_{h,\;1},\;x_{h,\;2},\;\cdots,\;x_{h,\;n}]^\text{T}\in {\bf{R}}^{n} $ 代表状态向量. $ u_{h} $和$ y_{h} $分别代表控制输入信号和输出信号. $ \zeta_{h,\;g}(\bar{x}_{h,\;g}) $和$ \zeta_{h,\;n}(\bar{x}_{h,\;n}) $表示未知的光滑非线性函数.
2)控制目标. 针对非线性多智能体系统, 设计一个带有规定性能机制的自适应混合双端事件触发跟踪控制策略来保证如下的两个控制目标:
a) 保证闭环内所有信号都是半全局一致最终有界的;
b) 使跟随者的输出轨迹和领导者的输出轨迹保持一致.
引理2[33]. 定义$ \bar{s}_{.1}=({s}_{1,\;1},\;{s}_{2,\;1},\;\cdots,\;{s}_{N,\;1})^\text{T} $, $ \bar{y}=(y_{1},\;y_{2},\;\cdots,\;y_{N})^\text{T} $ 和 $ \bar{y}_{r}=({y_{r},\;y_{r},\;\cdots,\; y_{r}})^\text{T} $. 三者满足如下关系:
$$ \begin{align} \begin{aligned} ||\bar{y}-\bar{y}_{r}||\le\frac{||\bar{s}_{.1}||}{{\bar{\sigma}}({\cal{L}}+{\cal{B}})} \end{aligned} \end{align} $$ (2) 其中, $ {\bar{\sigma}}({\cal{L}}+{\cal{B}}) $为矩阵$ {\cal{L}}+{\cal{B}} $的最小奇异值.
1.3 模糊逻辑系统
使用模糊逻辑系统理论近似严格反馈多智能体系统中存在的未知非线性函数. 考虑如下模糊逻辑系统:
$$ \begin{aligned} \tilde{y}(x)=\frac{{\sum\limits_{\flat=1}^r}\bar{y}_{\flat}{\prod\limits_{\hbar=1}^{\check{n}}}{\bar{\mu}_{F^{\flat}_{\hbar}}}(x_{\hbar})}{{\sum\limits_{\flat=1}^r}[\prod\limits_{\hbar=1}^{\check{n}}{\bar{\mu}_{F^{\flat}_{\hbar}}}(x_{\hbar})]} \end{aligned} $$ (3) 其中, $ \bar{y}_{\flat}=\max_{y\in{\bf{R}}}{\bar{\mu}_{G^{\flat}}}(\tilde{y}) $.
定义模糊基函数为
$$ \begin{aligned} \psi_{\hbar}(x)=\frac{{\prod\limits_{\hbar=1}^{\check{n}}}{\bar{\mu}_{F^{\flat}_{\hbar}}}(x_{\hbar})}{{\sum\limits_{\flat=1}^{r}}[\prod\limits_{\hbar=1}^{\check{n}}{\bar{\mu}_{F^{\flat}_{\hbar}}}(x_{\hbar})]} \end{aligned} $$ (4) 并且, 定义向量 $ \zeta^\text{T}=[\bar{y}_{1},\;\bar{y}_{2},\;\cdots,\;\bar{y}_{n}]= [\zeta_{1},\; \zeta_{2},\;\cdots, \zeta_{n}] $ 和 $ \psi(x)=[\psi_{1}(x),\;\cdots,\;\psi_{n}(x)]^\text{T} $. 随后, 进一步表示模糊逻辑系统为 $ \tilde{y}(x)=\zeta^\text{T}\psi(x) $.
引理3[33]. 对于$ \varepsilon>0 $, 表示模糊逻辑系统为
$$ \begin{align} \underset{x\in\varpi}\sup\lvert \zeta(x)-{\eta}^\text{T}\psi(x) \rvert \le\varepsilon \end{align} $$ (5) 其中, $ \zeta(x) $为定义在紧集$ \varpi $上的连续函数.
1.4 规定性能函数
定义规定性能机制[30]为
$$ \begin{align} \begin{aligned} -e_{h,\;1,\;{\mathrm{min}}}(t)<e_{h,\;1}(t)<e_{h,\;1,\;{\mathrm{max}}}(t) \end{aligned} \end{align} $$ (6) 接下来, 定义规定性能上下边界为
$$ \begin{aligned} e_{h,\;1,\;{\mathrm{min}}}(t)=\;&(e_{h,\;1,\;0,\;{\mathrm{min}}}-e_{h,\;1,\;\infty,\;{\mathrm{min}}})\text{e}^{-o_{h}t}\;+\\ & e_{h,\;1,\;\infty,\;{\mathrm{min}}}\nonumber\\ e_{h,\;1,\;{\mathrm{max}}}(t)=\;&(e_{h,\;1,\;0,\;{\mathrm{max}}}-e_{h,\;1,\;\infty,\;{\mathrm{max}}})\text{e}^{-o_{h}t}\;+\\ & e_{h,\;1,\;\infty,\;{\mathrm{max}}}\nonumber \end{aligned} $$ 其中, $ -e_{h,\;1,\;{\mathrm{min}}}\;(t) $表示设计的规定范围下界, $ e_{h,\;1,\;{\mathrm{max}}}(t) $ 表示设计的规定范围上界. $ o_{h} $ 是一个可设计的常数. $ e_{h,\;1,\;0,\;{\mathrm{min}}} $, $ e_{h,\;1,\;0,\;{\mathrm{max}}} $, $ e_{h,\;1,\;\infty,\;{\mathrm{min}}} $ 和 $ e_{h,\;1,\;\infty,\;{\mathrm{max}}} $ 是正的参数. 并且, 参数需要满足$ e_{h,\;1,\;0,\;{\mathrm{min}}} > e_{h,\;1,\;\infty,\;{\mathrm{min}}} $ 和 $ e_{h,\;1,\;0,\;{\mathrm{max}}} > e_{h,\;1,\;\infty,\;{\mathrm{max}}} $.
假设2[30]. 对于智能体$ h $, 初始同步误差必须满足受限不等式$ -e_{h,\;1,\;{\mathrm{min}}}(0) < e_{h,\;1}(0) < e_{h,\;1,\;{\mathrm{max}}}(0) $.
根据所考虑的规定性能方法, 得到误差转换机制为
$$ \begin{align} \begin{aligned} e_{h,\;1}=e_{h,\;1,\;{\mathrm{max}}}\Re_{h}(s_{h,\;1}) \end{aligned} \end{align} $$ (7) 其中, $ s_{h,\;1} $ 是转换后的误差. $ \Re_{h}(s_{h,\;1}) $ 表示误差转换函数, 其是光滑且严格单调递增的, 同时满足 $ \Re_{h}(s_{h,\;1})\in(-\kappa_{h},\;1) $ 且 $ \kappa_{h}={e_{h,\;1,\;{\mathrm{max}}}(t)}/{e_{h,\;1,\;{\mathrm{min}}}(t)} $.
转换函数的表达式为
$$ \begin{align} \begin{aligned} \Re_{h}(s_{h,\;1})=\frac{\text{e}^{s_{h,\;1}}-\text{e}^{-s_{h,\;1}}}{\text{e}^{s_{h,\;1}}+{\kappa}^{-1}_{h}\text{e}^{-s_{h,\;1}}} \end{aligned} \end{align} $$ (8) 将式(8)代入式(7), 可得
$$ \begin{align} s_{h,\;1}=\frac{1}{2}\ln\left(1+\frac{e_{h,\;1}}{e_{h,\;1,\;{\mathrm{min}}}}\right)-\frac{1}{2}\ln\left(1-\frac{e_{h,\;1}}{e_{h,\;1,\;{\mathrm{max}}}}\right) \end{align} $$ (9) 注1. 由于本文考虑状态触发机制, 该机制会导致系统出现阶跃现象或抖震现象. 为克服这一现象发生, 本文采用规定性能方法来约束系统的同步误差以减少系统性能下降的情况发生.
1.5 混合双端事件触发机制
针对非线性多智能体系统, 如何有效节省通信资源是十分重要的问题, 事件触发机制可以减少通信带宽的占用. 同时, 在设计事件触发机制时, 重要的是在设计相应的阈值条件时要考虑到通信资源和跟踪性能之间的平衡.
在网络环境中进行信息交换时, 多个信息传输通道会同时进行数据传输. 基于这一考虑, 本文设计混合双端事件触发机制来同时释放控制器−执行器环节和传感器−控制器环节中通信渠道上的压力. 因此, 提出如下的混合双端事件触发机制:
$$ \left\{\begin{split} &\breve{\hat{x}}_{h,\;g}=\hat{x}_{h,\;g}(t^{x}_{h,\;k}),\;t\in[t^{x}_{h,\;k},\;t^{x}_{h,\;k+1})\\ &t^{x}_{h,\;k+1}=\inf\Bigr\{t>t^{x}_{h,\;k}:|\hat{x}_{h,\;g}(t)-\breve{\hat{x}}_{h,\;g}|\ge \\ &\qquad\qquad \nu_{h}+m_{h}\text{e}^{-b_{h}t}\Bigr\} \end{split}\right. $$ (10) 随后可得
$$ \left\{\begin{split} &\breve{u}_{h}(t)=u_{h}(t^{u}_{h,\;k}),\;t\in[t^{u}_{h,\;k},\;t^{u}_{h,\;k+1})\\ &t^{u}_{h,\;k+1}=\inf\Bigr\{t>t^{u}_{h,\;k}:|u_{h}(t)-\breve{u}_{h}(t)|\ge\\ &\qquad\qquad \rho_{h}+\mu_{h}\text{e}^{-\tau_{h}t}\Bigr\} \end{split} \right.$$ (11) 其中, $ t^{x}_{h,\;k} $ 表示系统状态的触发时刻. $ t^{u}_{h,\;k} $ 表示控制输入信号的触发时刻. $ \nu_{h} $, $ m_{h} $, $ b_{h} $, $ \rho_{h} $, $ \mu_{h} $ 和 $ \tau_{h} $ 是正的常数. 同时, 通常假设第一个触发发生在系统运行的初始时刻.
注2. 由于本文考虑未知不可测量状态问题, 提出的状态触发机制首次使用估计状态作为采样信号并构成触发误差, 拓宽了状态触发机制的应用范围. 并且, 设计的阈值条件会随着系统运行时间的变化而变化, 从而更好地平衡了系统性能和资源节约之间的关系.
2. 模糊状态观测器的设计
本节通过构造模糊状态观测器解决未知状态不可测量问题, 该观测器仅使用相对输出分布式误差信息进行反馈. 首先, 定义$ \zeta_{h}(x_{h},u_{h})=\zeta_{h,\;n}(\bar{x}_{h,\;n}) $ 且要求 $ |\zeta_{h}(x_{h},\;u_{h})|\le \bar{\zeta}_{h}(x_{h},\;u_{h}) $. 重新构造系统模型为
$$ \begin{aligned} \left\{ \begin{aligned} &\dot{x}_{h,\;g}=x_{h,\;g+1}+\zeta_{h,\;g}(\hat{\bar{x}}_{h,\;g})\\ &\dot{x}_{h,\;n}=u_{h}+\zeta_{h}(\hat{x}_{h},\;u_{h})\\ & y_{h}=x_{h,\;1} \end{aligned} \right. \end{aligned} $$ (12) 基于模糊逻辑系统理论, 考虑的多智能体系统包含非线性函数$ \zeta_{h,\;g}(\hat{\bar{x}}_{h,\;g}) $ 和 $ \zeta_{h}(\hat{x}_{h},\;u_{h}) $, 近似这两项可得:
$$ \begin{align} \begin{aligned} \hat{\zeta}_{h,\;g}(\hat{\bar{x}}_{h,\;g}|\eta_{h,\;g})=\eta^{*\text{T}}_{h,\;g}\psi_{h,\;g}(\hat{\bar{x}}_{h,\;g}) \end{aligned} \end{align} $$ (13) 其中, $ \hat{\bar{x}}_{h,\;g} $ 表示$ {\bar{x}}_{h,\;g} $的估计值.
$ \eta^{*}_{h,\;g} $ 是最优参数向量, 其可以表示为
$$ \begin{split} \eta^{*}_{h,\;g}=&\;\arg\underset{\eta_{h,\;g}\in\Omega_{h,\;g}}\min[\underset{(\bar{x}_{h,\;g},\;\hat{\bar{x}}_{h,\;g})\in U}\sup| \hat{\zeta}(\hat{\bar{x}}_{h,\;g}|\hat{\eta}_{h,\;g})\;-\\ &\zeta(\hat{\bar{x}}_{h,\;g})|] \\[-1pt]\end{split} $$ (14) 其中, $ U $ 和 $ \Omega_{h,\;g} $ 分别为 $ \hat{\bar{x}}_{h,\;g} $ 和 $ \eta_{h,\;g} $ 对应的紧集. $ \hat{\eta}_{h,\;g} $ 表示 $ \eta^{*}_{h,\;g} $ 的估计值. $ \varepsilon_{h,\;g} $ 表示模糊最小化近似误差且 $ \varepsilon_{h,\;g}=\zeta_{h,\;g}(\bar{x}_{h,\;g})-\hat{\zeta}_{h,\;g}(\hat{\bar{x}}_{h,\;g}|\eta_{h,\;g}) $. 接下来, 构建分布式状态观测器为
$$ \left\{\begin{aligned} &\dot{\hat x}_{h,\;g}=\hat{x}_{h,\;g+1}+\hat{\eta}^\text{T}_{h,\;g}\psi_{h,\;g}(\hat{\bar{x}}_{h,\;g})\;+\\ &\quad k^{*}_{h,\;g}\left(\sum_{\ell=1}^N a_{h,\;\ell}(y_{h}-y_{\ell})-\sum_{\ell=1}^N a_{h,\;\ell}(\hat{y}_{h}-y_{\ell})\right)\\ &\dot{\hat x}_{h,\;n}=u_{h}+\hat{\eta}^\text{T}_{h,\;n}\psi_{h,\;n}(\hat{\bar{x}}_{h,\;n})\;+\\ &\quad k^{*}_{h,\;n}\left(\sum_{\ell=1}^N a_{h,\;\ell}(y_{h}-y_{\ell})-\sum_{\ell=1}^N a_{h,\;\ell}(\hat{y}_{h}-y_{\ell})\right)\\ &\hat{y}_{h}=\hat{x}_{h,\;1} \\[-1pt] \end{aligned}\right. $$ (15) 其中, $ k^{*}_{h,\;g} $ 和 $ k^{*}_{h,\;n} $ 是正的常数. 此外, 相对输出误差表示为 $ \sum_{\ell=1}^N a_{h,\;\ell}(y_{h} - y_{\ell}) $. 定义 $ \Delta_{h} = \bar{x}_{h,\;n} - \hat{\bar{x}}_{h,\;n} $ 和 $ \hat{\bar{x}}_{h,\;n}=[\hat{x}_{h,\;1},\;\cdots,\;\hat{x}_{h,\;n}]^\text{T} $. 经过上述分析, 可得
$$ \begin{align} \begin{aligned} \dot{\Delta}_{h}=&\Xi_{h}\Delta_{h}+\varepsilon_{h}+\sum_{\eth=1}^{n} A_{h,\;\eth} \tilde{\eta}^\text{T}_{h,\;\eth}\psi_{h,\;\eth}(\hat{\bar{x}}_{h,\;\eth}) \end{aligned} \end{align} $$ (16) 其中, $ \varepsilon_{h} = [\varepsilon_{h,\,1},\,\cdots,\,\varepsilon_{h,\,n}]^\text{T} $, $ A_{h,\,\eth} = [0\cdots1\cdots0]_{n \times 1} $. 为了确保 $ \Xi_{h} $ 是一个严格的赫尔维兹矩阵, 选择向量 $ K_{h} $ 且 $ K_{h}=[k^{*}_{h,\;1},\;\cdots,\;k^{*}_{h,\;n}]^\text{T} $.
并且,
$$ \begin{align} \begin{aligned} A_{h}= \begin{bmatrix} -k^{*}_{h,\;1}\displaystyle\sum\limits_{\ell=1}^N a_{h,\;\ell}\\ -k^{*}_{h,\;2}\displaystyle\sum\limits_{\ell=2}^N a_{h,\;\ell}&H_{h,\;n-1}&\\ \vdots\\ -k^{*}_{h,\;n}\displaystyle\sum\limits_{\ell=n}^N a_{h,\;\ell}&...&0\nonumber \end{bmatrix} \end{aligned} \end{align} $$ 对于矩阵 $ Q_{h}=Q^\text{T}_{h}>0 $ 和矩阵 $ Y_{h}=Y^\text{T}_{h}>0 $, 满足如下关系
$$ \begin{align} \begin{aligned} A^\text{T}_{h}Y_{h}+Y_{h}A_{h}=-2Q_{h} \end{aligned} \end{align} $$ (17) 挑选如下Lyapunov函数 $ V_{h,\;0} $:
$$ \begin{align} \begin{aligned} V_{h,\;0}=\frac{1}{2}\tilde{x}^\text{T}_{h}Y_{h}\tilde{x}_{h} \end{aligned} \end{align} $$ (18) 基于上述分析, 计算$ V_{h,\;0} $的导数可以为
$$ \begin{split} \dot{V}_{h,\;0}=\;&-\Delta^\text{T}_{h}Q_{h}\Delta_{h}+\Delta^\text{T}_{h}Y_{h}\varepsilon_{h}\;+\\ &\Delta^\text{T}_{h}Y_{h}\sum_{\eth=1}^{n} A_{h,\;\eth} \tilde{\eta}^\text{T}_{h,\;\eth}\psi_{h,\;\eth} \end{split} $$ (19) 利用杨氏不等式, 可得
$$ \begin{align} \Delta^\text{T}_{h}Y_{h}\varepsilon_{h}\le\frac{1}{2}||Y_{h}||^{2}||\varepsilon^{*}_{h}||^{2}+\frac{1}{2}||\Delta_{h}||^{2} \end{align} $$ (20) $$ \begin{split} &e^\text{T}_{h}Y_{h}\sum_{\eth=1}^{n} A_{h,\;\eth} \tilde{\eta}^\text{T}_{h,\;\eth}\psi_{h,\;\eth}(\hat{\bar{x}}_{h,\;\eth})\le \\ &\qquad\frac{n}{2}||\Delta_{h}||^{2}+ \frac{1}{2}||Y_{h}||^{2}\sum_{\eth=1}^{n} A_{h,\;\eth} \tilde{\eta}^\text{T}_{h,\;\eth}\tilde{\eta}_{h,\;\eth} \end{split} $$ (21) 其中, $ \varepsilon^{*}_{h}=[\varepsilon^{*}_{h,\;1},\;\varepsilon^{*}_{h,\;2},\;\cdots,\;\varepsilon^{*}_{h,\;n}]^\text{T} $.
随后, 可得
$$ \begin{split} \dot{V}_{h,\;0}\le\;&\frac{1+n}{2}||\Delta_{h}||^{2}-\Delta^\text{T}_{h}Q_{h}\Delta_{h}+\frac{1}{2}||Y_{h}||^{2}||\varepsilon^{*}_{h}||^{2}\;+\\ &\frac{1}{2}||Y_{h}||^{2}\sum_{\eth=1}^{n} A_{h,\;\eth} \tilde{\eta}^\text{T}_{h,\;\eth}\tilde{\eta}_{h,\;\eth}\le\frac{1}{2}||Y_{h}||^{2}||\tilde{\eta}^{2}_{h,\;n}\;+\\ & \frac{1}{2}||Y_{h}||^{2}||\varepsilon^{*}_{h}||^{2}+ \xi_{0}||\Delta_{h}||^{2}\\[-1pt] \end{split} $$ (22) 其中, $ \xi_{0} = \min\left\{\tau_{{\mathrm{min}}}(Q_{h}) - \frac{1+n}{2}\right\} $ 和 $ \xi_{0} > 0 $. $ \tau_{{\mathrm{min}}}(Q_{h}) $ 是矩阵 $ Q_{h} $ 的最小特征值.
注3. 本文设计的模糊状态观测器仅依赖于智能体的相对输出信息进行反馈, 表明仅使用部分的分布式信息就可以解决未知不可测状态问题. 此外, 设计的观测器可识别严格反馈多智能体系统中的未知非线性函数.
3. 主要内容
3.1 自适应分布式控制器设计
本节给出自适应控制器的设计过程且解决非线性多智能体系统的自适应模糊跟踪控制问题. 定义局部的同步误差 $ e_{h,\;1} $ 和 $ e_{h,\;g} $ ($ g\;=\; 2,\;\cdots,\;n $) 为
$$ \left\{\begin{split} &e_{h,\;1}=\sum_{\ell=1}^Na_{h,\;\ell}(y_{h}-y_{\ell})+b_{h}(y_{h}-y_{r})\\ &e_{h,\;g}=\hat{x}_{h,\;g}-\alpha_{hf,\;g-1} \end{split}\right.$$ (23) 其中, $ \alpha_{hf,\;g-1} $ 表示滤波后的虚拟控制器.
在传统的反步法框架下, 为避免“复杂性爆炸”问题, 引入了一阶滤波器:
$$ \left\{\begin{split} &\Phi_{h,\;g-1}\dot{\alpha}_{hf,\;g-1}+{\alpha}_{hf,\;g-1}={\alpha}_{h,\;g-1}\\ &{\alpha}_{hf,\;g-1}(0)={\alpha}_{h,\;g-1}(0) \end{split}\right. $$ (24) 其中, $ \alpha_{h,\;g - 1} $ 是虚拟控制信号. 虚拟控制信号$ \alpha_{h,\;g - 1} $ 通过一阶滤波器 $ \Phi_{h,\;g-1}>0 $ 会产生一个新的信号 $ {\alpha}_{hf,\;g-1} $.
随后, 表示一阶滤波器的误差为
$$ \begin{align} \begin{aligned} \vartheta_{h,\;g-1}={\alpha}_{hf,\;g-1}-{\alpha}_{h,\;g-1} \end{aligned} \end{align} $$ (25) 基于式 (9), 计算 $ s_{h,\;1} $ 的导数为
$$ \begin{align} \begin{aligned} \dot{s}_{h,\;1}=\rho_{h}\dot{e}_{h,\;1}-\phi_{h}e_{h,\;1} \end{aligned} \end{align} $$ (26) 其中,
$$ \begin{align} \begin{aligned} \rho_{h}=\;&\frac{1}{2}\bigg(\frac{1}{e_{h,\;1,\;{\mathrm{min}}}+e_{h,\;1}}+\frac{1}{e_{h,\;1,\;{\mathrm{max}}}-e_{h,\;1}}\bigg)\\ \phi_{h}=\;&\frac{1}{2}\bigg(\frac{\dot{e}_{h,\;1,\;{\mathrm{min}}}}{e_{h,\;1,\;{\mathrm{min}}}(e_{h,\;1,\;{\mathrm{min}}}+e_{h,\;1})}\;+\\ &\frac{\dot{e}_{h,\;1,\;{\mathrm{max}}}}{e_{h,\;1,\;{\mathrm{max}}}(e_{h,\;1,\;{\mathrm{max}}}-e_{h,\;1})}\bigg)\nonumber \end{aligned} \end{align} $$ 步骤 1. 选择如下的Lyapunov函数:
$$ \begin{align} \begin{aligned} V_{h,\;1}=\frac{1}{2}s^{2}_{h,\;1}+\frac{1}{2}\eta^\text{T}_{h,\;1}\eta_{h,\;1}+\frac{1}{2}\vartheta^{2}_{h,\;1} \end{aligned} \end{align} $$ (27) 计算$ V_{h,\;1} $的导数为
$$ \begin{split} \dot{V}_{h,\;1}&=s_{h,\;1}\dot{s}_{h,\;1}-\tilde{\eta}_{h,\;1}\dot{\hat{\eta}}_{h,\;1}+\vartheta_{h,\;1}\dot{\vartheta}_{h,\;1}=\\ &s_{h,\;1}(\rho_{h}\dot{e}_{h,\;1}-\phi_{h}e_{h,\;1})-\tilde{\eta}_{h,\;1}\dot{\hat{\eta}}_{h,\;1}+\vartheta_{h,\;1}\dot{\vartheta}_{h,\;1}=\\ &s_{h,\;1}\rho_{h}\left(\sum_{\ell=1}^Na_{h,\;\ell}(\dot{y}_{h}-\dot{y}_{\ell})+b_{h}(\dot{y}_{h}-\dot{y}_{r})\right)-\\ &\phi_{h}e_{h,\;1}s_{h,\;1}+\vartheta_{h,\;1}\dot{\vartheta}_{h,\;1}-\tilde{\eta}_{h,\;1}\dot{\hat{\eta}}_{h,\;1}\\[-1pt] \end{split} $$ (28) 根据式 (15), 进一步可得
$$ \begin{split} \dot{V}_{h,\;1}=\;&s_{h,\;1}\rho_{h}\big((b_{h}+d_{h})(\hat{x}_{h,\;2}+\zeta_{h,\;1})\;-\\ & d_{h}(\hat{x}_{\ell,\;2}+\zeta_{\ell,\;1})-b_{h}\dot{y}_{r}\big)-\phi_{h}e_{h,\;1}s_{h,\;1}\;-\\ &\tilde{\eta}_{h,\;1}\dot{\hat{\eta}}_{h,\;1}+\vartheta_{h,\;1}\dot{\vartheta}_{h,\;1}=s_{h,\;1}\rho_{h}\big((b_{h}\;+\\ & d_{h})(e_{h,\;2}+\alpha_{h,\;1}+\vartheta_{h,\;1}+\zeta_{h,\;1})\;-\\ & d_{h}(\hat{x}_{\ell,\;2}+\zeta_{\ell,\;1})-b_{h}\dot{y}_{r}-\phi_{h}e_{h,\;1}\rho^{-1}_{h}\big)\;-\\ & \tilde{\eta}_{h,\;1}\dot{\hat{\eta}}_{h,\;1}+\vartheta_{h,\;1}\dot{\vartheta}_{h,\;1} \end{split} $$ (29) 其中, $ e_{h,\;2}=\hat{x}_{h,\;2}-\alpha_{hf,\;1} $ 且 $ \vartheta_{h,\;1}={\alpha}_{hf,\;1}-{\alpha}_{h,\;1} $. $ \sum_{j=1}^Na_{h,\;\ell} $ 可以由 $ d_{h} $ 来表示.
随后, 根据引理3, 可得
$$ \begin{split} &\bar{F}_{h,\;1}({x}_{h,\;1},\;{x}_{\ell,\;1})=(b_{h}+d_{h})f_{h,\;1}-d_{h}(\hat{x}_{\ell,\;2}\;+\\ &\quad f_{\ell,\;1})\nonumber-\phi_{h}e_{h,\;1}\rho^{-1}_{h}\nonumber={\eta}^{\mathrm{T}}_{h,\;1}\psi_{h,\;1}+\varepsilon_{h,\;1} \end{split} $$ 式 (29) 可进一步表示为
$$ \begin{split} \dot{V}_{h,\;1}=\;&s_{h,\;1}\rho_{h}\big((b_{h}+d_{h})(e_{h,\;2}+\alpha_{h,\;1}+\vartheta_{h,\;1})\nonumber\;-\\ & b_{h}\dot{y}_{r}+\bar{F}_{h,\;1}\big)-\tilde{\eta}_{h,\;1}\dot{\hat{\eta}}_{h,\;1}+\vartheta_{h,\;1}\dot{\vartheta}_{h,\;1}\nonumber=\\ & s_{h,\;1}\rho_{h}\big((b_{h}+d_{h})(e_{h,\;2}+\alpha_{h,\;1}+\vartheta_{h,\;1})\nonumber\;+\\ & {\eta}^{\mathrm{T}}_{h,\;1}\psi_{h,\;1}+\varepsilon_{h,\;1}-b_{h}\dot{y}_{r})\big)\nonumber\;-\\ &\tilde{\eta}_{h,\;1}\dot{\hat{\eta}}_{h,\;1}+\vartheta_{h,\;1}\dot{\vartheta}_{h,\;1} \end{split} $$ 基于杨氏不等式, 可得
$$ \begin{align} \left\{\begin{aligned} &(b_{h}+d_{h})s_{h,\;1}e_{h,\;2}\le\frac{(b_{h}+d_{h})^{2}}{2}s^{2}_{h,\;1}+\frac{1}{2}e^{2}_{h,\;2}\nonumber\\&(b_{h}+d_{h})s_{h,\;1}\vartheta_{h,\;1}\le\frac{(b_{h}+d_{h})^{2}}{2}s^{2}_{h,\;1}+\frac{1}{2}\vartheta^{2}_{h,\;1}\nonumber\\ &s_{h,\;1}\varepsilon_{h,\;1}\le\frac{1}{2}s^{2}_{h,\;1}+\frac{1}{2}\varepsilon^{2}_{h,\;1} \end{aligned}\right. \end{align} $$ 设计虚拟控制器为
$$ \begin{split} \alpha_{h,\;1}=\; &\rho^{-1}_{h}\bigg(-(b_{h}+d_{h})s_{h,\;1}-\frac{1}{2(b_{h}+d_{h})}s_{h,\;1}\;-\\ & c_{h,\;1}s_{h,\;1}+\frac{b_{h}}{b_{h}+d_{h}}\dot{y}_{r}-\frac{1}{b_{h}+d_{h}}\hat{\eta}_{h,\;1}\psi_{h,\;1}\bigg) \end{split} $$ (30) 其中, $ c_{h,\;1} $ 是设计参数. $ \hat{\eta}_{h,\;1} $ 是 $ {\eta}_{h,\;1} $的估计值.
在触发时刻, 系统会更新自适应律, 并且在触发间隔区间中保持不变. 因此, 设计自适应律为
$$ \left\{\begin{aligned} \hat{\eta}^+_{h,\;1}&=-N_{h,\;1}\hat{\eta}_{h,\;1}+s_{h,\;1}\psi_{h,\;1},&& t=t_{h,\;k}\\ \dot{\hat{\eta}}_{h,\;1}&=0,&&t\in[t_{h,\;k},\;t_{h,\;k+1}) \end{aligned} \right. $$ (31) 其中, $ N_{h,\;1} $ 是设计参数.
基于上述分析, 可知
$$ \begin{split} \dot{V}_{h,\;1}&\le-{c}_{h,\;1}s^{2}_{h,\;1}-\frac{1}{2}N_{h,\;1}\tilde{\eta}^{2}_{h,\;1}+\frac{1}{2}N_{h,\;1}{\eta}^{2}_{h,\;1}\;+\nonumber\\ &\quad\frac{1}{2}\rho_{h}\vartheta^{2}_{h,\;1}+\frac{1}{2}\varepsilon^{2}_{h,\;1}\rho_{h}+\frac{1}{2}\rho_{h}e^{2}_{h,\;2}+\vartheta_{h,\;1}\dot{\vartheta}_{h,\;1} \end{split} $$ 随后, 可得
$$ \begin{align} \begin{aligned} \dot{\vartheta}_{h,\;1}=-\frac{1}{\upsilon_{h,\;1}}{\vartheta}_{h,\;1}-\dot{\alpha}_{h,\;1} \end{aligned} \end{align} $$ (32) 其中, $ \upsilon_{h,\;1} $ 是设计常数.
根据上述分析, $ \dot{V}_{h,\;1} $ 满足如下不等式:
$$ \begin{split} \dot{V}_{h,\;1}\le &-c_{h,\;1}s^{2}_{h,\;1}-\frac{1}{2}N_{h,\;1}\tilde{\eta}^{2}_{h,\;1}+\frac{1}{2}N_{h,\;1}{\eta}^{2}_{h,\;1}\;+\\ &\frac{1}{2}\varepsilon^{2}_{h,\;1}\rho_{h}+\frac{1}{2}\rho_{h}e^{2}_{h,\;2}-\frac{1}{\upsilon_{h,\;1}}{\vartheta}^{2}_{h,\;1}\;+\\ &\frac{1}{2}\dot{\alpha}^{2}_{h,\;1}+\frac{1}{2}\rho_{h}\vartheta^{2}_{h,\;1}\\[-1pt] \end{split} $$ (33) 步骤g. $(g=2, \cdots, n-1) $选择如下的Lyapunov函数:
$$ \begin{align} \begin{aligned} V_{h,\;g}=\frac{1}{2}e^{2}_{h,\;g}+\frac{1}{2}\eta^\text{T}_{h,\;g}\eta_{h,\;g}+\frac{1}{2}\vartheta^{2}_{h,\;g} \end{aligned} \end{align} $$ (34) 计算$ V_{h,\;g} $的导数为
$$ \begin{split} \dot{V}_{h,\;g}=\;&e_{h,\;g}\dot{e}_{h,\;g}-\tilde{\eta}_{h,\;g}\dot{\hat{\eta}}_{h,\;g}+\vartheta_{h,\;g}\dot{\vartheta}_{h,\;g}=\\ &e_{h,\;g}(\dot{\hat{x}}_{h,\;g}-\dot{\alpha}_{hf,\;g-1})\;-\\ & \tilde{\eta}_{h,\;g}\dot{\hat{\eta}}_{h,\;g}+\vartheta_{h,\;g}\dot{\vartheta}_{h,\;g} \end{split} $$ (35) 基于式 (15), 式 (35)重新表示为
$$ \begin{split} \dot{V}_{h,\;g}=\;&e_{h,\;g}\big(\hat{x}_{h,\;g+1}+\eta_{h,\;g}\psi_{h,\;g}+k^{*}_{h,\;g}\big(d_{h}(y_{h}-y_{\ell})\nonumber\;-\\ & d_{h}(\hat{y}_{h}-y_{\ell}) \big) -\dot{\alpha}_{hf,\;g-1} \big) -\tilde{\eta}_{h,\;g}\dot{\hat{\eta}}_{h,\;g}\nonumber + \vartheta_{h,\;g}\dot{\vartheta}_{h,\;g} \end{split} $$ 其中, $ \hat{x}_{h,\;g+1}=e_{h,\;g+1}+\alpha_{hf,\;g} $.
随后, 可得
$$\begin{aligned} \dot{V}_{h,\;g}=\;&e_{h,\;g}\big(e_{h,\;g+1}+\alpha_{hf,\;g}+\eta_{h,\;g}\psi_{h,\;g}\;+\\ & k^{*}_{h,\;g}\big(d_{h}(y_{h}\nonumber-y_{\ell})-d_{h}(\hat{y}_{h}-y_{\ell})\big)-\dot{\alpha}_{hf,\;g-1}\big)\;-\\ & \tilde{\eta}_{h,\;g}\dot{\hat{\eta}}_{h,\;g}\nonumber+\vartheta_{h,\;g}\dot{\vartheta}_{h,\;g}\nonumber=e_{h,\;g}\big(e_{h,\;g+1}+\alpha_{h,\;g}\;+\\ & {\vartheta}_{h,\;g}+\eta_{h,\;g}\psi_{h,\;g}\nonumber+k^{*}_{h,\;g}\big(d_{h}(y_{h}-y_{\ell})\;-\\ &d_{h}(\hat{y}_{h} - y_{\ell}) \big)\nonumber-\dot{\alpha}_{hf,\;g-1}\big) -\tilde{\eta}_{h,\;g}\dot{\hat{\eta}}_{h,\;g} +\vartheta_{h,\;g}\dot{\vartheta}_{h,\;g} \end{aligned} $$ 基于杨氏不等式, 可知
$$ \begin{align} \begin{aligned} e_{h,\;g}\vartheta_{h,\;g}\le\frac{1}{2}e^{2}_{h,\;g}+\frac{1}{2}\vartheta^{2}_{h,\;g} \end{aligned} \end{align} $$ (36) 设计虚拟控制器为
$$ \begin{split} \alpha_{h,\;g}=\; &-k^{*}_{h,\;g}\big(d_{h}(y_{h}-y_{\ell})-d_{h}(\hat{y}_{h}-y_{\ell})\big)\;-\\ &\hat{\eta}_{h,\;g}\psi_{h,\;g}-\left(\frac{1}{2}+c_{h,\;g}\right)e_{h,\;g}+\dot{\alpha}_{hf,\;g-1} \end{split} $$ (37) 其中, $ c_{h,\;g} $ 是设计常数.
设计自适应律为
$$ \begin{align} \left\{ \begin{aligned} \hat{\eta}^+_{h,\;g}&=-N_{h,\;g}\hat{\eta}_{h,\;g}+e_{h,\;g}\psi_{h,\;g},&& t=t_{h,\;k}\\ \dot{\hat{\eta}}_{h,\;g}&=0,&& t\in[t_{h,\;k},\;t_{h,\;k+1}) \end{aligned} \right. \end{align} $$ (38) 并且, 可知
$$ \begin{align} \begin{aligned} \dot{\vartheta}_{h,\;g}=-\frac{1}{\upsilon_{h,\;g}}{\vartheta}_{h,\;g}-\dot{\alpha}_{h,\;g} \end{aligned} \end{align} $$ (39) 其中, $ \upsilon_{h,\;g} $ 是一个设计常数.
接下来, $ \dot{V}_{h,\;g} $ 满足如下不等式:
$$ \begin{split} \dot{V}_{h,\;g}\le&-{c}_{h,\;g}e^{2}_{h,\;g}-\frac{1}{2}N_{h,\;g}\tilde{\eta}^{2}_{h,\;g}+\frac{1}{2}N_{h,\;g}{\eta}^{2}_{h,\;g}\;+\\ &\frac{1}{2}\vartheta^{2}_{h,\;g}-\frac{1}{\upsilon_{h,\;g}}{\vartheta}^{2}_{h,\;g}+\frac{1}{2}\dot{\alpha}^{2}_{h,\;g} \\[-1pt]\end{split} $$ (40) 步骤${\boldsymbol{n}} $. 当应用一阶滤波器, 可得
$$\left\{\begin{split} &\Phi_{h,\;n-1}\dot{\alpha}_{hf,\;n-1}+{\alpha}_{hf,\;n-1}={\alpha}_{h,\;n-1}\\ &{\alpha}_{hf,\;n-1}(0)={\alpha}_{h,\;n-1}(0) \end{split}\right. $$ (41) 其中, $ \alpha_{h,\;n-1} $ 是虚拟控制信号. 虚拟控制信号$ \alpha_{h,\;n-1} $ 通过一阶滤波器 $ \Phi_{h,\;n-1}>0 $ 会产生一个新的信号 $ {\alpha}_{hf,\;n-1} $.
挑选如下的Lyapunov函数:
$$ \begin{align} \begin{aligned} V_{h,\;n}=\frac{1}{2}e^{2}_{h,\;n}+\frac{1}{2}\eta^\text{T}_{h,\;n}\eta_{h,\;n} \end{aligned} \end{align} $$ (42) 计算$ V_{h,\;n} $的导数为
$$ \begin{split} \label{lin} \begin{aligned} \dot{V}_{h,\;n}=\;&e_{h,\;n}\dot{e}_{h,\;n}-\tilde{\eta}_{h,\;n}\dot{\hat{\eta}}_{h,\;n}\nonumber=\\ &e_{h,\;n}(\dot{\hat{x}}_{h,\;n}-\dot{\alpha}_{hf,\;n-1})-\tilde{\eta}_{h,\;n}\dot{\hat{\eta}}_{h,\;n} \end{aligned} \end{split} $$ 随后, 可得
$$ \begin{split} \dot{V}_{h,\;n}=\;&e_{h,\;n}\Bigg(u_{h}+{\eta}_{h,\;n}\psi_{h,\;n}(\hat{\bar{x}}_{h,\;n})\;+\\ &k^{*}_{h,\;n}\Biggr(\sum_{\ell=1}^N a_{h,\;\ell}(y_{h}-y_{\ell})-\sum_{\ell=1}^N a_{h,\;\ell}(\hat{y}_{h}\nonumber\;-\\ &y_{\ell})\Biggr)-\dot{\alpha}_{hf,\;n-1}\Bigg)-\tilde{\eta}_{h,\;n}\dot{\hat{\eta}}_{h,\;n}\nonumber=\\ &e_{h,\;n}\Bigg(\breve{u}_{h}+(u_{h}-\breve{u}_{h})+\eta^\text{T}_{h,\;n}\psi_{h,\;n}(\hat{\bar{x}}_{h,\;n})\;+\\ & k^{*}_{h,\;n}\Biggr(\sum_{\ell=1}^N a_{h,\;\ell}(y_{h}-y_{\ell})-\sum_{j=1}^N a_{h,\;\ell}(\hat{y}_{h}\nonumber\;-\\ & y_{\ell})\Biggr)-\dot{\alpha}_{hf,\;n-1}\Bigg)-\tilde{\eta}_{h,\;n}\dot{\hat{\eta}}_{h,\;n} \end{split} $$ 设计自适应事件触发控制器 $ \breve{u}_{h} $ 为
$$ \begin{split} \breve{u}_{h}=\;&-c_{h,\;n}\breve{e}_{h,\;n}-\hat{\eta}_{h,\;n}\psi_{h,\;n}\;+\\ &\frac{\breve{\alpha}_{h,\;n-1}-\breve{\alpha}_{hf,\;n-1}}{\Phi_{h,\;n-1}}- k^{*}_{h,\;n}\Biggr(\sum_{\ell=1}^Na_{h,\;\ell}(y_{h}-y_{\ell})\;-\\ &\sum_{\ell=1}^N a_{h,\;\ell}(\hat{y}_{h}-y_{\ell})\Biggr)\\[-1pt]\end{split} $$ (43) 其中,
$$ \begin{split} &\breve{e}_{h,\;n}(t)=\breve{\hat{x}}_{h,\;n}(t)-\breve{\alpha}_{hf,\;n-1}(t)\nonumber\\ &\breve{\alpha}_{hf,\;n-1}(t)={\alpha}_{hf,\;n-1}(t^{\alpha_{f}}_{h,\;k}),\quad t\in[t^{\alpha_{f}}_{h,\;k},\;t^{\alpha_{f}}_{h,\;k+1})\nonumber \end{split} $$ $$ \begin{split} t^{k+1}_{h,\;\alpha_{f}}=\;&\;\inf\Bigr\{t>t^{k}_{h,\;\alpha_{f}}:|{\dot{\alpha}}_{hf,\;n-1}(t)\;-\\ &{\dot{\breve{\alpha}}}_{hf,\;n-1}(t)|\ge\Theta_{h,\;\alpha_{f}}\Bigr\} \end{split} $$ (44) $ \breve{\alpha}_{h,\;n-1} $ 是 $ \alpha_{h,\;n-1} $ 触发后的信号. $ \breve{\alpha}_{hf,\;n-1} $ 是 $ \alpha_{hf,\;n-1} $ 触发后的信号. 在本文考虑的状态触发机制中, $ {\alpha}_{hf,\;n-1} $ 和 $ {\alpha}_{h,\;n-1} $ 均依赖于系统的状态值.
选择自适应律为
$$ \left\{\begin{split}& \hat{\eta}^+_{h,\;n}=-N_{h,\;n}\hat{\eta}_{h,\;n}+e_{h,\;n}\psi_{h,\;n}, \;\; t=t_{h,\;k}\\ &\dot{\hat{\eta}}_{h,\;n}=0,\qquad\qquad\qquad\qquad\qquad\; t\in[t_{h,\;k},\;t_{h,\;k+1}) \end{split} \right.$$ (45) 其中, $ N_{h,\;n} $ 是设计参数.
随后, 可知
$$ \begin{split} \dot{V}_{h,\;n}=\;&e_{h,\;n}\big((u_{h}-\breve{u}_{h})+{c}_{h,\;n}(e_{h,\;n}-\breve{e}_{h,\;n})\;+\\ & \tilde{\eta}_{h,\;n}\psi_{h,\;n}\nonumber+\frac{\breve{\alpha}_{h,\;n-1}-\breve{\alpha}_{hf,\;n-1}}{\Pi_{h,\;n-1}}\;-\\&\frac{{\alpha}_{h,\;n-1}-{\alpha}_{hf,\;n-1}}{\Pi_{h,\;n-1}}\nonumber\;-\\&{c}_{h,\;n}e_{h,\;n}\big)-\tilde{\eta}_{h,\;n}\dot{\hat{\eta}}_{h,\;n} \end{split} $$ 引理4[34]. 触发误差的上界可以表示为
$$ \begin{split} &|e_{h,\;n}-\breve{e}_{h,\;n}|\le\bar{\Theta}_{h}\nonumber\\ &\qquad\quad\left|\frac{\breve{\alpha}_{h,\;n-1}\breve{\alpha}_{hf,\;n-1}}{\Phi_{h,\;n-1}}-\frac{{\alpha}_{h,\;n-1}{\alpha}_{hf,\;n-1}}{\Phi_{h,\;n-1}}\right|\le \\ &\qquad\quad \frac{\Theta_{h,\;\alpha_{f}}+\Theta_{hf,\;n-1}}{\Phi_{h,\;n-1}} \end{split} $$ 接下来, 可得
$$ \begin{split} \dot{V}_{h,\;n}\le\;&-c_{h,\;n}e^{2}_{h,\;n}-\frac{1}{2}N_{h,\;n}\tilde{\eta}^{2}_{h,\;n}\;+\\ &\frac{1}{2}N_{h,\;n}{\eta}^{2}_{h,\;n}+\varsigma_{h,\;n} \end{split} $$ (46) 其中, $ \varsigma_{h,\;n}=\bar{\Theta}^{2}_{h}+\frac{{(\Theta_{h,\;\alpha_{f}}+\Theta_{hf,\;n-1})}^{2}}{\Phi^{2}_{h,\;n-1}}+p^{2}_{h}+\rho^{2}_{h} $.
注 4. 针对控制器−执行器环节和传感器−控制器环节, 本文设计混合双端事件触发机制, 可同时缓解双信道的通讯负担. 首先, 考虑在传感器−控制器环节上设置事件触发机制. 因为每次传输闭环系统的信息时, 输入信号都是根据输出反馈结果设置的, 所以输出信号的有效更新和更新次数是需要考虑的重要环节. 另外, 考虑控制器−执行器环节的资源节约. 通过在输入信号中设置触发项并进一步设置事件采样所需的条件, 实现节省通信资源的目的. 在多智能体系统或分布式系统中, 通信资源通常是有限的. 双端事件触发机制需要在有限的通信条件下, 确保信息的及时传输和系统的协调运行. 因此, 可以通过调整参数$ \nu_{h} $, $ m_{h} $, $ b_{h} $, $ \rho_{h} $, $ \mu_{h} $, $ \tau_{h} $ 和 $ \Theta_{h,\;\alpha_{f}} $实现对系统实时性能和通讯带宽占用率的有效平衡.
3.2 稳定性分析
定理1. 在假设1和假设 2下, 针对非线性多智能体系统 (1), 考虑混合双端事件触发机制 (10), (11), (44) 和模糊状态观测器(15), 设计自适应律 (31), (38) 和 (45), 虚拟控制器 (30), (37) 和分布式控制器 (43)可以使得闭环系统内的所有信号是半全局一致最终有界的.
此外, 对于 $ \forall \chi>0 $, 设计的参数满足
$$ \begin{align} \begin{aligned} \underset{t\to\infty}\lim||y-{y}_{r}||\leq\chi \end{aligned} \end{align} $$ (47) 证明. 为证明整体闭环系统的稳定性, 选择总Lyapunov函数为
$$ \begin{align} \begin{aligned} V_{h}=\sum_{m=1}^n V_{h,\;m}+V_{h,\;0} \end{aligned} \end{align} $$ (48) 根据式 (22), (33), (40) 和 (46), 可得
$$ \begin{split} \dot{V}_{h}\le\;&\xi_{0}||\Delta_{h}||^{2} + \frac{1}{2}||Y_{h}||^{2}||\tilde{\eta}^\text{T}_{h,\;n}\tilde{\eta}_{h,\;n} + \frac{1}{2}||Y_{h}||^{2}||\varepsilon^{*}_{h}||^{2}\;+\\ &\frac{1}{2}||Y_{h}||^{2}\sum_{m=1}^{n} A_{h,\;m} \tilde{\eta}^\text{T}_{h,\;m}\tilde{\eta}_{h,\;m}+\frac{1}{2}\varepsilon^{2}_{h,\;1}\rho_{h}\;-\\ &\sum_{m=1}^{n}\frac{1}{2}N_{h,\;m}\tilde{\eta}^{2}_{h,\;m}+\sum_{m=1}^{n}\frac{1}{2}N_{h,\;m}{\eta}^{2}_{h,\;m}\;+\\ &\sum_{m=1}^{n-1}\frac{1}{2}\dot{\alpha}^{2}_{h,\;m}-\sum_{m=2}^{n-1}\left(\frac{1}{\upsilon_{h,\;1}}-\frac{1}{2}\right){\vartheta}^{2}_{h,\;m}\;-\\ & {c}_{h,\;1}s^{2}_{h,\;1}-\sum_{m=2}^{n-1} {c}_{h,\;m}e^{2}_{h,\;m}-{c}_{h,\;n}e^{2}_{h,\;n}+\varsigma_{h,\;n}\;-\\ & \left(\frac{1}{\upsilon_{h,\;1}}-\frac{1}{2}\rho_{h}\right){\vartheta}^{2}_{h,\;1}\le-\partial_{h}V_{h}+\omega_{h}\\[-1pt] \end{split} $$ (49) 其中,
$$ \begin{split} \partial_{h}=\;&\;\min\bigg\{ \bar{c}_{h,\;1},\; \bar{c}_{h,\;m},\; {c}_{h,\;n},\; \frac{N_{h,\;m}}{2},\;\\& \left(\frac{1}{\upsilon_{h,\;1}}-\frac{1}{2}\right),\; \left(\frac{1}{\upsilon_{h,\;1}}-\frac{\rho_{h}}{2}\right) \bigg\} \end{split} $$ 且
$$ \begin{split} \omega_{h}=\;&\;\sum_{m=1}^{n}\frac{1}{2}N_{h,\;m}{\eta}^{2}_{h,\;m}+\frac{1}{2}\varepsilon^{2}_{h,\;1}\rho_{h}\;+\\ &\varsigma_{h,\;n}+\sum_{m=1}^{n-1}\frac{1}{2}\dot{\alpha}^{2}_{h,\;m}\nonumber \end{split} $$ 在不等式(49)两边同时乘以 $ \text{e}^{\partial_{h}t} $, 并在定义域 $ [0,\;t] $ 上同时积分, 可得
$$ \begin{align} \begin{aligned} 0\le V(t)\le \text{e}^{-\partial_{h}t}V(0)+\frac{\omega_{h}}{\partial_{h}}(1-\text{e}^{-\partial_{h}t}) \end{aligned} \end{align} $$ (50) 根据 $ V $ 的定义和式(50), 可知
$$ \begin{align} \begin{aligned} ||s_{.1}||^2\le2\text{e}^{-\partial_{h}t}V_{0}+\frac{\omega_{h}}{\partial_{h}}(1-\text{e}^{-\partial_{h}t}) \end{aligned} \end{align} $$ (51) 针对 $ \forall\chi>0 $, 基于 $ \partial_{h} $ 和 $ \omega_{h} $ 的定义, 选择合适的参数, 可得如下关系式
$$ \begin{align} \frac{\omega_{h}}{\partial_{h}}\le\frac{\chi^{2}}{2}(\sigma({\cal{L}}+{\cal{B}}))^2 \end{align} $$ (52) 并且, 根据引理2, 当 $ t\to\infty $ 时, 不等式 (47) 成立.
□ 3.3 芝诺行为分析
本节需要证明提出的事件触发机制的间隔时间是有下界的, 即同时排除芝诺行为发生的可能性. 对于 $ \varkappa=1,\;\cdots,\;n $, 定义
$$\left\{\begin{split} &\sigma_{h,\;\varkappa}(t)=\hat{x}_{h,\;\varkappa}(t)-\breve{\hat{x}}_{h,\;\varkappa}(t),\; \quad t\in[t^{x}_{h,\;k},\;t^{x}_{h,\;k+1})\nonumber\\ &\sigma_{h,\;u}(t)=u_{h}(t)-\breve{u}_{h}(t),\; \quad\quad\quad t\in[t^{u}_{h,\;k},\;t^{u}_{h,\;k+1})\nonumber\\ &\sigma_{hf,\;\varkappa-1}(t)={\dot{\alpha}}_{hf,\;\varkappa-1}(t)-{\dot{\breve{\alpha}}}_{hf,\;\varkappa-1}(t),\;\\ &\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad t\in[t^{\alpha_{f}}_{h,\;k},\;t^{\alpha_{f}}_{h,\;k+1})\nonumber \end{split}\right. $$ 计算以上变量的导数值为
$$ \begin{align} \begin{aligned} &\frac{{\rm{d}}}{{\rm{d}}t}|\sigma_{h,\;\varkappa}|=\frac{{\rm{d}}}{{\rm{d}}t}(\sigma_{h,\;\varkappa}*\sigma_{h,\;\varkappa})^{\frac{1}{2}}\nonumber=\\ &\qquad\hbox{sign}(\hat{x}_{h,\;n}-\breve{\hat{x}}_{h,\;n})\dot{\hat{x}}_{h,\;n}\le|\dot{\hat{x}}_{h,\;n}|\nonumber\\& \frac{{\rm{d}}}{{\rm{d}}t}|\sigma_{h,\;u}|=\frac{{\rm{d}}}{{\rm{d}}t}(\sigma_{h,\;u}*\sigma_{h,\;u})^{\frac{1}{2}}\nonumber=\\ &\qquad\hbox{sign}(u_{h}-\breve{u}_{h})\dot{u}_{h}\le|\dot{u}_{h}|\nonumber\\ &\frac{{\rm{d}}}{{\rm{d}}t}|\sigma_{hf,\;\varkappa-1}|=\frac{{\rm{d}}}{{\rm{d}}t}(\sigma_{hf,\;\varkappa-1}*\sigma_{hf,\;\varkappa-1})^{\frac{1}{2}}\nonumber=\\ &\qquad\hbox{sign}(\dot{\alpha}_{hf,\;g-1}-\dot{\breve{\alpha}}_{hf,\;g-1})\ddot{\alpha}_{hf,\;g-1}\nonumber \le\\ &\qquad|\ddot{\alpha}_{hf,\;g-1}| \end{aligned} \end{align} $$ 其中, $ \dot{u}_{h} $和$ \ddot{\alpha}_{hf,\;g-1} $与$ \hat{x}_{h,\;g} $, $ e_{h,\;g} $, $ \tilde{\eta}_{h,\;g} $和$ \dot{\alpha}_{h,\;g} $信号有关.
以上变量满足如下的关系式: $ |\dot{\hat{x}}_{h,\;m}|\;\le\;\digamma_{h} $, $ |\dot{u}_{h}|\le\digamma_{\breve{u}_{h}} $ 和 $ |\ddot{\digamma}_{h,\;\alpha_{f}}|\le\digamma_{h,\;\alpha_{f}} $. 值得注意的是, 在 $ t^{x}_{h,\;k} $, $ t^{u}_{h,\;k} $ 和 $ t^{\alpha_{f}}_{h,\;k} $ 时刻, 三个变量满足 $ \sigma_{h,\,\varkappa}(t^{x}_{h,\,k}) = 0 $, $ \sigma_{h,\;\varkappa}(t^{u}_{h,\;k})=0 $ 和 $ \sigma_{hf,\;\varkappa-1}(t^{\alpha_{f}}_{h,\;k})=0 $.
随后, 可得
$$\begin{split} &\lim_{t\rightarrow t^{x}_{h,\;k}}\sigma_{h,\;\varkappa}(t)=\nu_{h}+ m_{h}\text{e}^{-b_{h}t^{x}_{h,\;k}} \\ &\lim_{t\rightarrow t^{u}_{h,\;k}} \sigma_{h,\;\varkappa}(t)=\rho_{h}+\mu_{h}\text{e}^{-\tau_{h}t^{u}_{h,\;k}}\\ &\lim_{t\rightarrow t^{\alpha_{f}}_{h,\;k}} \sigma_{h,\;\varkappa}(t)=\Theta_{h,\;\alpha_{f}} \end{split} $$ 因此, 得到本文提出的事件触发机制间隔的最小界限值为
$$ \begin{align} t^{x}_{h,\;k+1}-t^{x}_{h,\;k}\ge\frac{\nu_{h}+m_{h}\text{e}^{-b_{h}t^{x}_{h,\;k}}}{\digamma_{h}} \end{align} $$ (53) $$ \begin{align} t^{u}_{h,\;k+1}-t^{u}_{h,\;k}\ge\frac{\rho_{h}+\mu_{h}\text{e}^{-\tau_{h}t^{u}_{h,\;k}}}{\digamma_{\breve{u}_{h}}} \end{align} $$ (54) $$ \begin{align} t^{\alpha_{f}}_{h,\;k+1}-t^{\alpha_{f}}_{h,\;k}\ge\frac{\Theta_{h,\;\alpha_{f}}}{\digamma_{h,\;\alpha_{f}}} \end{align} $$ (55) 通过以上分析可以得出, 本文提出的事件触发机制不会发生芝诺行为.
4. 仿真结果
一些仿真结果验证了本文控制方案的有效性.
图1是本文所考虑的通讯拓扑结构, 其表明4个跟随者与1个领导者之间的信息传输关系. 基于图1, 邻接矩阵 $ \bar{{\cal{A}}} $ 和拉普拉斯矩阵 $ {\cal{L}} $ 表示如下:
$$ \begin{align} \begin{aligned} \bar{{\cal{A}}}= \begin{bmatrix} 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0\\ 0&1&1&0\nonumber \end{bmatrix} ,\; {\cal{L}}= \begin{bmatrix} \;\;\,1&\;\;\;\,0&\;\;\,0&-1\\ -1&\;\;1&\;\;\,0&\;\;0\\ \,\;\;0&-1&\;\;\,1&\;\;0\\ \;\;\,0&-1&-1&\;\;2\nonumber \end{bmatrix} \end{aligned} \end{align} $$ 对于 $ h=1,\;2,\;3,\;4 $, 挑选一组强阻尼系统, 其动态模型为
$$ \begin{align} \begin{aligned} \Lambda_{h}=\frac{\pi_{h}}{M_{h}L^{2}_{h}}\Lambda_{h}-\frac{v_{h}\bar{g}}{L_{h}}\sin(\Lambda_{h})+u_{h} \end{aligned} \end{align} $$ (56) 其中, $ L_{h} $ 是摆长. $ \Lambda_{h} $ 是从垂直向下的位置逆时针测量的杆的角度. $ u_{h} $ 表示传动转矩. $ \bar{g} $ 为重力加速度. $ v_{h} $ 定义为恢复转矩系数. $ \pi $ 为阻尼系数. $ M_{h} $ 为钟摆的质量.
针对$ h = 1,\;2,\;3,\;4 $, 设$ x_{h,\;1} = \Lambda_{h} $ 和 $ {x}_{h,\;2}=\dot{\Lambda}_{h} $, 其动力学方程可转化为
$$ \left\{\begin{aligned} &\dot{x}_{h,\;1}=x_{h,\;2}\nonumber\\ &\dot{x}_{h,\;2}=u_{h}-\frac{\pi_{h}}{M_{h}L^{2}_{h}}x_{h,\;2}-\frac{v_{h}\bar{g}}{L_{h}}\sin(x_{h,\;1})\nonumber\\ & y_{h}=x_{h,\;1} \end{aligned} \right. $$ 强阻尼系统的初始值矩阵选择为 $ x_{h}(0) = [0.1,\; 0.1\;]^\text{T} $. 自适应参数的初始值为 $ \hat{\eta}\;_{h,\;1}\; (0) \;=\; 0.1 $ 和 $ \hat{\eta}\;_{h,\,2} (0) = 0.1 $. 挑选参数值分别为 $ e_{h,\;1,\;0,\;\min}=0.5 $, $ e_{h,\;1,\;\infty,\;\max}=0.5 $, $ e_{h,\;1,\;\infty,\;\min}=0.3 $, $ e_{h,\;1,\;0,\;\max}=0.5 $, $ o_{h}=-1 $, $ c_{h,\;1}=c_{h,\;2}=50 $, $ \phi_{h,\;2}=2 $, $ N_{h,\;1}= N_{h,\;2}= 0.1 $, $ p_{h}=0.1 $, $ m_{h}=0.1 $, $ b_{h}=0.1 $, $ \rho_{h}=10 $, $ \mu_{h}=2 $, $ \tau_{h}=2, $ $ M_{h}=1, $ $ L_{h}=1, $ $ \bar{g}=9.8 $, $ v_{h}=\frac{1}{9.8} $ 和 $ \pi=-0.25 $.
图2表明4个跟随者的输出轨迹与既定的领导者轨迹是一致的, 设计的控制算法可使强阻尼系统的输出稳定于给定的参考信号. 基于此, 可以看出考虑的控制目标得到实现, 控制策略有效地实现分布式强阻尼系统的一致性目标.
图 2 4个跟随者和1个领导者的输出轨迹Fig. 2 The output trajectories of the four followers and one leader在图3中, 将一致性误差的值域限制在预设的范围中, 误差输出轨迹小于设计的规定性能预设边界$ (-0.5,\;0.3) $, 表明良好的规定性能效果. 图4为在双端触发框架下的控制输入曲线, 从图4中可以看出输入信号$ u_{h} $是有界的.
图5和图6凸显了模糊逻辑系统权重参数的变化过程, 证明本文所有自适应律参数是有界的, 并根据上文展示的控制性能, 说明本文控制框架针对实际系统具备有效性与适用性. $ \hat{\eta}_{h,\;g} $ 表示模糊逻辑系统的权重, 且用来调整模糊逻辑系统对非线性函数$ \zeta_{h,\;g}(\bar{x}_{h,\;g}) $ 的逼近效果.
图 5 自适应律参数$\hat{\eta}_{h,\;1}$的轨迹Fig. 5 The trajectories of the adaptive law parameters $\hat{\eta}_{h,\;1}$
图 6 自适应律参数$\hat{\eta}_{h,\;2}$的轨迹Fig. 6 The trajectories of the adaptive law parameters $\hat{\eta}_{h,\;2}$图7为观测误差的数值变化轨迹. 同时, 依据前文的设计过程, 可知考虑的观测器仅使用相对输出信息进行反馈, 能够大幅度提高观测器的实用性. 图8展示了所提出事件触发机制的触发间隔. 同时, 也说明所提出的混合双端事件触发机制具有可节省控制器环节通信资源的优势. 以智能体1为例, 正常迭代次数为
3000 次, 通过本文事件触发的设计后, 控制器的更新次数为908 次, 节省了69.7%的通讯资源. 针对多智能体系统, 智能体之间在通讯网络下进行信息传输, 当智能体数量增多时, 必会造成一定程度的通讯压力, 这足以可见本文设计事件触发机制的重要性. 并且, 针对多渠道通讯网络, 本文同时降低了控制器−执行器环节和传感器−控制器环节的通讯负担.
图 7 观测误差$\Delta_{h,\;2} $的变化情况Fig. 7 The changes in observation errors $\Delta_{h,\;2}$5. 结束语
本文研究了双端事件触发自适应模糊跟踪控制问题. 针对控制器−执行器和传感器−控制器环节, 提出基于状态触发机制和控制器触发机制的混合双端分布式事件触发机制, 并且设计一种改进的状态触发机制, 首次将估计的状态信号作为触发信号来达到节约通讯资源的目的. 最终, 一些仿真结果证明了提出控制方案的有效性. 在未来的研究工作中, 我们将致力于探索电力系统控制需求, 并将多种事件触发控制策略融合实际系统的需要, 以满足智能化、高效化与绿色化的能源转型目标.
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