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基于带有噪声输入的稀疏高斯过程的人体姿态估计

夏嘉欣 陈曦 林金星 李伟鹏 吴奇

董滔, 李小丽, 赵大端. 基于事件触发的三阶离散多智能体系统一致性分析. 自动化学报, 2019, 45(7): 1366-1372. doi: 10.16383/j.aas.2017.c170406
引用本文: 夏嘉欣, 陈曦, 林金星, 李伟鹏, 吴奇. 基于带有噪声输入的稀疏高斯过程的人体姿态估计. 自动化学报, 2019, 45(4): 693-705. doi: 10.16383/j.aas.2018.c170397
DONG Tao, LI Xiao-Li, ZHAO Da-Duan. Event-triggered Consensus of Third-order Discrete-time Multi-agent Systems. ACTA AUTOMATICA SINICA, 2019, 45(7): 1366-1372. doi: 10.16383/j.aas.2017.c170406
Citation: XIA Jia-Xin, CHEN Xi, LIN Jin-Xing, LI Wei-Peng, WU Qi. Sparse Gaussian Process With Input Noise for Human Pose Estimation. ACTA AUTOMATICA SINICA, 2019, 45(4): 693-705. doi: 10.16383/j.aas.2018.c170397

基于带有噪声输入的稀疏高斯过程的人体姿态估计

doi: 10.16383/j.aas.2018.c170397
基金项目: 

国家自然科学基金 51705242

江苏省自然科学基金 BK20141430

上海浦江人才计划 15PJ1404300

国家自然科学基金 61473158

浙江大学CAD和CG国家重点实验室开放课题 A1713

国家自然科学基金 61671293

详细信息
    作者简介:

    夏嘉欣  上海交通大学电子信息与电气工程学院自动化系硕士研究生. 2015年获得上海交通大学学士学位.主要研究方向为图像处理与机器学习.E-mail: jessicax 1993@163.com

    陈曦  上海交通大学航空航天学院讲师.2014年获得皇家墨尔本理工大学航空工程专业博士学位.主要研究方向为故障预测与健康管理, 机器学习, 结构健康监测.E-mail:chenxi1@comac.cc

    林金星  南京邮电大学自动化学院副教授.主要研究方向为复杂系统智能建模与控制, 切换奇异系统.E-mail:jxlin2004@126.com

    李伟鹏  上海交通大学航空航天学院研究员.2008年获得哈尔滨工业大学硕士学位, 2011年获得东京大学航空航天工程博士学位.主要研究方向为湍流和气动噪声的数据挖掘.E-mail:liweipeng@sjtu.edu.cn

    通讯作者:

    吴奇  上海交通大学电子信息与电气工程学院自动化系副教授.2009年获得东南大学自动化学院控制工程与控制理论博士学位.主要研究方向为深度多层网络建模与学习算法, 机器学习与模式识别.本文通信作者.E-mail:wuqi7812@sjtu.edu.cn

Sparse Gaussian Process With Input Noise for Human Pose Estimation

Funds: 

National Natural Science Foundation of China 51705242

Natural Science Foundation of Jiangsu Province BK20141430

Shanghai Pujiang Program 15PJ1404300

National Natural Science Foundation of China 61473158

Open Project Program of the State Key Laboratory of CAD and CG, Zhejiang University A1713

National Natural Science Foundation of China 61671293

More Information
    Author Bio:

      Master student in the Department of Automation, School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University. She received her bachelor degree from Shanghai Jiao Tong University in 2015. Her research interest covers image processing and machine learning

     Lecturer at the School of Aeronautics and Astronautics, Shanghai Jiao Tong University. He received his Ph. D. degree in aerospace engineering from Royal Melbourne Institute of Technology University, Australia in 2014. His research interest covers prognosis and health management, machine learning, and structural health monitoring

     Associate professor at the School of Automation, Nanjing University of Posts and Telecommunications. His research interest covers intelligent modeling and control of complex systems, switched singular systems

     Professor at the School of Aeronautics and Astronautics, Shanghai Jiao Tong University. He received his master degree from Harbin Institute of Technology in 2008 and his Ph. D. degree in aerospace engineering from University of Tokyo, Japan in 2011. His research interest covers data mining for turbulence, drag reduction, and noise control

    Corresponding author: WU Qi  Associate professor in the Department of Automation, School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University. He received his Ph. D. degree in control theory and control engineering from the School of Automation, Southeast University in 2006. His research interest covers deep multi-layers network modeling and learning algorithm, machine learning, and pattern recognition. Corresponding author of this paper
  • 摘要: 高斯过程回归(Gaussian process regression,GPR)是一种广泛应用的回归方法,可以用于解决输入输出均为多元变量的人体姿态估计问题.计算复杂度是高斯过程回归的一个重要考虑因素,而常用的降低计算复杂度的方法为稀疏表示算法.在稀疏算法中,完全独立训练条件(Fully independent training conditional,FITC)法是一种较为先进的算法,多用于解决输入变量彼此之间完全独立的回归问题.另外,输入变量的噪声问题是高斯过程回归的另一个需要考虑的重要因素.对于测试的输入变量噪声,可以通过矩匹配的方法进行解决,而训练输入样本的噪声则可通过将其转换为输出噪声的方法进行解决,从而得到更高的计算精度.本文基于以上算法,提出一种基于噪声输入的稀疏高斯算法,同时将其应用于解决人体姿态估计问题.本文实验中的数据集来源于之前的众多研究人员,其输入为从视频序列中截取的图像或通过特征提取得到的图像信息,输出为三维的人体姿态.与其他算法相比,本文的算法在准确性,运行时间与算法稳定性方面均达到了令人满意的效果.
  • 近些年来, 由于多智能体协同控制在编队控制[1]、机器人网络[2]、群集行为[3]、移动传感器[4-5]等方面的广泛应用, 多智能体系统的协同控制问题受到了众多研究者的广泛关注.一致性问题是多智能体系统协同控制领域的一个关键问题, 其目的是通过与邻居之间的信息交换, 使所有智能体的状态达成一致.迄今为止, 对多智能体一致性的研究也已取得了丰硕的成果, 根据多智能体的动力学模型分类, 主要可以将其分为以下4种情形:一阶[6-9]、二阶[10-13]、三阶[14-15]、高阶[16-18].

    在实际应用中, 由于CPU处理速度和内存容量的限制, 智能体不能频繁地进行控制以及与其邻居交换信息.因此, 事件触发控制策略作为减少控制次数和通信负载的有效途径, 受到了越来越多的关注.到目前为止, 对事件触发控制机制的研究也取得了很多成果[19-23].Xiao等[19]基于事件触发控制策略, 解决了带有领航者的离散多智能体系统的跟踪问题.通过利用状态测量误差并且基于二阶离散多智能体系统动力学模型, Zhu等[20]提出了一种自触发的控制策略, 该策略使得所有智能体的状态均达到一致. Huang等[21]研究了基于事件触发策略的Lur$'$e网络的跟踪问题.针对不同的领航者-跟随者系统, Xu等[22]提出了3种不同类型的事件触发控制器, 包含分簇式控制器、集中式控制器和分布式控制器, 以此来解决对应的一致性问题.然而, 大多数现有的事件触发一致性成果集中于考虑一阶多智能体系统和二阶多智能体系统, 很少有成果研究三阶多智能体系统的事件触发控制问题, 特别是对于三阶离散多智能体系统, 成果更是少之又少.所以, 设计相应的事件触发控制协议来解决三阶离散多智能体系统的一致性问题已变得尤为重要.

    本文研究了基于事件触发控制机制的三阶离散多智能体系统的一致性问题, 文章主要有以下三点贡献:

    1) 利用位置、速度和加速度三者的测量误差, 设计了一种新颖的事件触发控制机制.

    2) 利用不等式技巧, 分析得到了保证智能体渐近收敛到一致状态的充分条件.与现有的事件触发文献[19-22]不同的是, 所得的一致性条件与通信拓扑的Laplacian矩阵特征值和系统的耦合强度有关.

    3) 给出了排除类Zeno行为的参数条件, 进而使得事件触发控制器不会每个迭代时刻都更新.

    智能体间的通信拓扑结构用一个有向加权图来表示, 记为.其中, $\vartheta = \left\{ {1, 2, \cdots, n} \right\}$表示顶点集, $\varsigma\subseteq\vartheta\times\vartheta$表示边集, 称作邻接矩阵, ${a_{ij}}$表示边$\left({j, i} \right) \in \varsigma $的权值.当$\left({j, i} \right) \in \varsigma $时, 有${a_{ij}} > 0$; 否则, 有${a_{ij}} = 0$. ${a_{ij}} > 0$表示智能体$i$能收到来自智能体$j$的信息, 反之则不成立.对任意一条边$j$, 节点$j$称为父节点, 节点$i$则称为子节点, 节点$i$是节点$j$的邻居节点.假设通信拓扑中不存在自环, 即对任意$i\in \vartheta $, 有${a_{ii}} = 0$.

    定义$L = \left({{l_{ij}}}\right)\in{\bf R}^{n\times n}$为图${\cal G}$的Laplacian矩阵, 其中元素满足${l_{ij}} = - {a_{ij}} \le 0, i \ne j$; ${l_{ii}} = \sum\nolimits_{j = 1, j \ne i}^n {{a_{ij}} \ge 0} $.智能体$i$的入度定义为${d_i} = \sum\nolimits_{j = 1}^n {{a_{ij}}} $, 因此可得到$L = D - \Delta $, 其中, .如果有向图中存在一个始于节点$i$, 止于节点$j$的形如的边序列, 那么称存在一条从$i$到$j$的有向路径.特别地, 如果图中存在一个根节点, 并且该节点到其他所有节点都有有向路径, 那么称此有向图存在一个有向生成树.另外, 如果有向图${\cal G}$存在一个有向生成树, 则Laplacian矩阵$L$有一个0特征值并且其他特征值均含有正实部.

    考虑多智能体系统由$n$个智能体组成, 其通信拓扑结构由有向加权图${\cal G}$表示, 其中每个智能体可看作图${\cal G}$中的一个节点, 每个智能体满足如下动力学方程:

    $ \begin{equation} \left\{ \begin{array}{l} {x_i}\left( {k + 1} \right) = {x_i}\left( k \right) + {v_i}\left( k \right)\\ {v_i}\left( {k + 1} \right) = {v_i}\left( k \right) + {z_i}\left( k \right)\\ {z_i}\left( {k + 1} \right) = {z_i}\left( k \right) + {u_i}\left( k \right) \end{array} \right. \end{equation} $

    (1)

    其中, ${x_i}\left(k \right) \in \bf R$表示位置状态, ${v_i}\left(k \right) \in \bf R$表示速度状态, ${z_i}\left(k \right) \in \bf R$表示加速度状态, ${u_i}\left(k \right) \in \bf R$表示控制输入.

    基于事件触发控制机制的控制器协议设计如下:

    $ \begin{equation} {u_i}\left( k \right) = \lambda {b_i}\left( {k_p^i} \right) + \eta {c_i}\left( {k_p^i} \right) + \gamma {g_i}\left( {k_p^i} \right), k \in \left[ {k_p^i, k_{p + 1}^i} \right) \end{equation} $

    (2)

    其中, $\lambda> 0$, $\eta> 0$, $\gamma> 0$表示耦合强度,

    $ \begin{align*}&{b_i}\left( k \right)= \sum\nolimits_{j \in {N_i}} {{a_{ij}}\left( {{x_j}\left( k \right) - {x_i}\left( k \right)} \right)} , \nonumber\\ &{c_i}\left( k \right)=\sum\nolimits_{j \in {N_i}} {{a_{ij}}\left( {{v_j}\left( k \right) - {v_i}\left( k \right)} \right)}, \nonumber\\ & {g_i}\left( k \right)=\sum\nolimits_{j \in {N_i}} {{a_{ij}}\left( {{z_j}\left( k \right) - {z_i}\left( k \right)} \right)} .\end{align*} $

    触发时刻序列定义为:

    $ \begin{equation} k_{p + 1}^i = \inf \left\{ {k:k > k_p^i, {E_i}\left( k \right) > 0} \right\} \end{equation} $

    (3)

    ${E_i}\left(k \right)$为触发函数, 具有以下形式:

    $ \begin{align} {E_i}\left( k \right)= & \left| {{e_{bi}}\left( k \right)} \right| + \left| {{e_{ci}}\left( k \right)} \right| + \left| {{e_{gi}}\left( k \right)} \right|- {\delta _2}{\beta ^k} - \nonumber\nonumber\\ &{\delta _1}\left| {{b_i}\left( {k_p^i} \right)} \right| - {\delta _1}\left| {{c_i}\left( {k_p^i} \right)} \right| - {\delta _1}\left| {{g_i}\left( {k_p^i} \right)} \right| \end{align} $

    (4)

    其中, ${\delta _1} > 0$, ${\delta _2} > 0$, $\beta > 0$, , ${e_{ci}}\left(k \right) = {c_i}\left({k_p^i} \right) - {c_i}\left(k \right)$, ${e_{gi}}\left(k \right) = {g_i}\left({k_p^i} \right) - {g_i}\left(k \right)$.

    令$\varepsilon _i\left(k\right)={x_i}\left(k\right)-{x_1}\left(k\right)$, ${\varphi _i}\left(k\right)={v_i}\left(k \right)-$ ${v_1}\left(k\right)$, ${\phi _i}(k) = {z_i}(k) - {z_1}\left(k \right)$, $i = 2, \cdots, n$. , $\cdots, {\varphi _n}\left(k \right)]^{\rm T}$, $\phi \left(k \right) = {\left[{{\phi _2}\left(k \right), \cdots, {\phi _n}\left(k \right)} \right]^{\rm T}}$. $\psi \left(k \right) = {\left[{{\varepsilon ^{\rm T}}\left(k \right), {\varphi ^{\rm T}}\left(k \right), {\phi ^{\rm T}}\left(k \right)} \right]^{\rm T}}$, , ${\bar e_b} = {\left[{{e_{b1}}\left(k \right), \cdots, {e_{b1}}\left(k \right)} \right]^{\rm T}}$, , ${e_{c1}}\left(k \right)]^{\rm T}$, , ${\bar e_g} = $ ${\left[{{e_{g1}}\left(k \right), \cdots, {e_{g1}}\left(k \right)} \right]^{\rm T}}$, $\tilde e\left(k \right) = [\tilde e_b^{\rm T}\left(k \right), \tilde e_c^{\rm T}\left(k \right), $ $\tilde e_g^{\rm T}\left(k \right)]^{\rm T}$, $\bar e\left(k \right) = [\bar e_b^{\rm T}\left(k \right), \bar e_c^T\left(k \right), \bar e_g^{\rm T}\left(k \right)]^{\rm T}$,

    $ \hat L = \left[ {\begin{array}{*{20}{c}} {{d_2} + {a_{12}}}&{{a_{13}} - {a_{23}}}& \cdots &{{a_{1n}} - {a_{2n}}}\\ {{a_{12}} - {a_{32}}}&{{d_3} + {a_{13}}}& \cdots &{{a_{1n}} - {a_{3n}}}\\ \vdots & \vdots & \ddots & \vdots \\ {{a_{12}} - {a_{n2}}}&{{a_{13}} - {a_{n3}}}& \cdots &{{d_n} + {a_{1n}}} \end{array}} \right] $

    再结合式(1)和式(2)可得到:

    $ \begin{equation} \psi \left( {k + 1} \right) = {Q_1}\psi \left( k \right) + {Q_2}\left( {\tilde e\left( k \right) - \bar e\left( k \right)} \right) \end{equation} $

    (5)

    其中, , .

    定义1.对于三阶离散时间多智能体系统(1), 当且仅当所有智能体的位置变量、速度变量、加速度变量满足以下条件时, 称系统(1)能够达到一致.

    $ \begin{align*} &{\lim _{k \to \infty }}\left\| {{x_j}\left( k \right) - {x_i}\left( k \right)} \right\| = 0 \nonumber\\ & {\lim _{k \to \infty }}\left\| {{v_j}\left( k \right) - {v_i}\left( k \right)} \right\| = 0 \nonumber\\ & {\lim _{k \to \infty }}\left\| {{z_j}\left( k \right) - {z_i}\left( k \right)} \right\| = 0 \\&\quad\qquad \forall i, j = 1, 2, \cdots , n \end{align*} $

    定义2.如果$k_{p + 1}^i - k_p^i > 1$, 则称触发时刻序列$\left\{ {k_p^i} \right\}$不存在类Zeno行为.

    假设1.假设有向图中存在一个有向生成树.

    假设$\kappa$是矩阵${Q_1}$的特征值, ${\mu _i}$是$L$的特征值, 则有如下等式成立:

    $ {\rm{det}}\left( {\kappa {I_{3n - 3}} - {Q_1}} \right)=\nonumber\\ \det \left(\! \!{\begin{array}{*{20}{c}} {\left( {\kappa - 1} \right){I_{n - 1}}}\!&\!{ - {I_{n - 1}}}\!&\!{{0_{n - 1}}}\\ {{0_{n - 1}}}\!&\!{\left( {\kappa - 1} \right){I_{n - 1}}}\!&\!{ - {I_{n - 1}}}\\ {\lambda {{\hat L}_{n - 1}}}\!&\!{\eta {{\hat L}_{n - 1}}}\!&\!{\left( {\kappa - 1} \right){I_{n - 1}} + \gamma {{\hat L}_{n - 1}}} \end{array}} \!\!\right)=\nonumber\\ \prod\limits_{i = 2}^n {\left[ {{{\left( {\kappa - 1} \right)}^3} + \left( {\lambda + \eta \left( {\kappa - 1} \right) + \gamma {{\left( {\kappa - 1} \right)}^2}} \right){\mu _i}} \right]} $

    $ \begin{align} {m_i}\left( \kappa \right)= &{\left( {\kappa - 1} \right)^3} + \nonumber\\&\left( {\lambda + \eta \left( {\kappa - 1} \right) + \gamma {{\left( {\kappa - 1} \right)}^2}} \right){\mu _i} = 0, \nonumber\\& \qquad\qquad\qquad\qquad\qquad i = 2, \cdots , n \end{align} $

    (6)

    则有如下引理:

    引理1[15].   如果矩阵$L$有一个0特征值且其他所有特征值均有正实部, 并且参数$\lambda $, $\eta $, $\gamma $满足下列条件:

    $ \left\{ \begin{array}{l} 3\lambda - 2\eta < 0\\ \left( {\gamma - \eta + \lambda } \right)\left( {\lambda - \eta } \right) < - \dfrac{{\lambda \Re \left( {{\mu _i}} \right)}}{{{{\left| {{\mu _i}} \right|}^2}}}\\ \left( {4\gamma + \lambda - 2\eta } \right)<\dfrac{{8\Re \left( {{\mu _i}} \right)}}{{{{\left| {{\mu _i}} \right|}^2}}} \end{array} \right. $

    那么, 方程(6)的所有根都在单位圆内, 这也就意味着矩阵${Q_1}$的谱半径小于1, 即$\rho \left({{Q_1}} \right) < 1$.其中, 表示特征值${\mu _i}$的实部.

    引理2[23].  如果, 那么存在$M \ge 1$和$0 < \alpha < 1$使得下式成立

    $ {\left\| {{Q_1}} \right\|^k} \le M{\alpha ^k}, \quad k \ge 0 $

    定理1.  对于三阶离散多智能体系统(1), 基于假设1, 如果式(2)中的耦合强度满足引理1中的条件, 触发函数(4)中的参数满足$0 < {\delta _1} < 1$, , $0 < \alpha < \beta < 1$, 则称系统(1)能够实现渐近一致.

    证明.令$\omega \left(k \right) = \tilde e\left(k \right) - \bar e\left(k \right)$, 式(5)能够被重新写成如下形式:

    $ \begin{equation} \psi \left( k \right) = Q_1^k\psi \left( 0 \right) + {Q_2}\sum\limits_{s = 0}^{k - 1} {Q_1^{k - 1 - s}\omega \left( s \right)} \end{equation} $

    (7)

    根据引理1和引理2可知, 存在$M \ge 1$和$0 < \alpha < 1$使得下式成立.

    $ \begin{align} \left\| {\psi \left( k \right)} \right\|\le & {\left\| {{Q_1}} \right\|^k}\left\| {\psi \left( 0 \right)} \right\| + \nonumber\\ & \left\| {{Q_2}} \right\|\sum\limits_{s = 0}^{k - 1} {{{\left\| {{Q_1}} \right\|}^{k - 1 - s}}\left\| {\omega \left( s \right)} \right\|}\le \nonumber\\ & M\left\| {\psi \left( 0 \right)} \right\|{\alpha ^k}+\nonumber\\ & M\left\| {{Q_2}} \right\|\sum\limits_{s = 0}^{k - 1} {{\alpha ^{k - 1 - s}}\left\| {\omega \left( s \right)} \right\|} \end{align} $

    (8)

    由触发条件可得:

    $ \begin{align} & \left| {{e_{bi}}\left( k \right)} \right| + \left| {{e_{ci}}\left( k \right)} \right| + \left| {{e_{gi}}\left( k \right)} \right|\le\nonumber\\ & \qquad{\delta _1}\left| {{b_i}\left( {k_p^i} \right)} \right| + {\delta _1}\left| {{c_i}\left( {k_p^i} \right)} \right| +\nonumber\\ &\qquad {\delta _1}\left| {{g_i}\left( {k_p^i} \right)} \right| + {\delta _2}{\beta ^k}\le\nonumber\\ &\qquad {\delta _1}\left\| L \right\| \cdot \left\| {\varepsilon \left( k \right)} \right\| + {\delta _1}\left\| L \right\| \cdot \left\| {\varphi \left( k \right)} \right\| + \nonumber\\ &\qquad{\delta _1}\left\| L \right\| \cdot \left\| {\phi \left( k \right)} \right\|+ {\delta _1}\left| {{e_{bi}} \left( k \right)} \right| + \nonumber\\ &\qquad{\delta _1}\left| {{e_{ci}} \left( k \right)} \right|+ {\delta _1}\left| {{e_{gi}}\left( k \right)} \right| + {\delta _2}{\beta ^k} \end{align} $

    (9)

    对上式移项可求解得:

    $ \begin{align} &\left| {{e_{bi}}\left( k \right)} \right| + \left| {{e_{ci}}\left( k \right)} \right| + \left| {{e_{gi}}\left( k \right)} \right|\le \nonumber\\ &\qquad\frac{{{\delta _1}\left\| L \right\| \cdot \left\| {\varepsilon \left( k \right)} \right\|}}{{1 - {\delta _1}}} + \frac{{{\delta _1}\left\| L \right\| \cdot \left\| {\varphi \left( k \right)} \right\|}}{{1 - {\delta _1}}}{\rm{ + }}\nonumber\\ &\qquad\frac{{{\delta _1}}}{{1 - {\delta _1}}}\left\| L \right\| \cdot \left\| {\phi \left( k \right)} \right\| + \frac{{{\delta _2}}}{{1 - {\delta _1}}}{\beta ^k} \end{align} $

    (10)

    又因为, 和, 可得出下列不等式:

    $ \begin{align} &\left| {{e_{bi}}\left( k \right)} \right| + \left| {{e_{ci}}\left( k \right)} \right| + \left| {{e_{gi}}\left( k \right)} \right|\le\nonumber\\ &\qquad \frac{{{\delta _1}\left\| L \right\|}}{{1 - {\delta _1}}} \cdot \left( {\left\| {\varepsilon \left( k \right)} \right\|{\rm{ + }}\left\| {\varphi \left( k \right)} \right\|{\rm{ + }}\left\| {\phi \left( k \right)} \right\|} \right) +\nonumber\\ &\qquad \frac{{{\delta _2}{\beta ^k}}}{{1 - {\delta _1}}}\le \frac{{3{\delta _1}}}{{1 - {\delta _1}}}\left\| L \right\| \cdot \left\| {\psi \left( k \right)} \right\| + \frac{{{\delta _2}}}{{1 - {\delta _1}}}{\beta ^k} \end{align} $

    (11)

    接着有如下不等式成立:

    $ \begin{align} \left\| {e\left( k \right)} \right\|\le \frac{{3\sqrt n {\delta _1}}}{{1 - {\delta _1}}}\left\| L \right\| \cdot \left\| {\psi \left( k \right)} \right\| + \frac{{\sqrt n {\delta _2}}}{{1 - {\delta _1}}}{\beta ^k} \end{align} $

    (12)

    其中, , ${e_b}(k) = \left[{{e_{b1}}(k), \cdots, {e_{bn}}(k)} \right]$, ${e_c}(k) = \left[{{e_{c1}}(k), \cdots, {e_{cn}}(k)} \right]$,

    注意到

    $ \begin{equation} \left\| {\tilde e( k )} \right\| + \left\| {\bar e( k )} \right\| \le \sqrt {6( {n - 1} )} \left\| {e( k )} \right\| \end{equation} $

    (13)

    于是有

    $ \begin{align} \left\| {\omega ( k )} \right\| &= \left\| {\tilde e( k ) - \bar e\left( k \right)} \right\| \le\nonumber\\ & \left\| {\tilde e\left( k \right)} \right\| + \left\| {\bar e\left( k \right)} \right\|\le\nonumber\\ & \frac{{3\sqrt {6n( {n - 1} )} {\delta _1}}}{{1 - {\delta _1}}}\left\| L \right\| \cdot \left\| {\psi \left( k \right)} \right\| +\nonumber\\ & \frac{{\sqrt {6n( {n - 1} )} {\delta _2}}}{{1 - {\delta _1}}}{\beta ^k} \end{align} $

    (14)

    把式(14)代入式(8)可得

    $ \begin{align} \left\| {\psi \left( k \right)} \right\| &\le M\left\| {\psi \left( 0 \right)} \right\|{\alpha ^k}+ \nonumber\\ &\frac{{M\left\| {{Q_2}} \right\|{\alpha ^{k - 1}} {\delta _1}3\sqrt {6n\left( {n - 1} \right)} \left\| L \right\|}}{{1 - {\delta _1}}}\times\nonumber\\ &\sum\limits_{s = 0}^{k - 1} {{\alpha ^{ - s}}\left\| {\psi \left( s \right)} \right\|} + M\left\| {{Q_2}} \right\|{\alpha ^{k - 1}}\times\nonumber\\ &\sum\limits_{s = 0}^{k - 1} {{\alpha ^{ - s}} \frac{{\sqrt {6n\left( {n - 1} \right)} {\delta _2}}} {{1 - {\delta _1}}}{\beta ^s}} \end{align} $

    (15)

    接下来的部分, 将证明下列不等式成立.

    $ \begin{equation} \left\| {\psi \left( k \right)} \right\| \le W{\beta ^k}.\end{equation} $

    (16)

    其中, $W = \max \left\{ {{\Theta _1}, {\Theta _2}} \right\}$,

    首先, 证明对任意的$\rho > 1$, 下列不等式成立.

    $ \begin{equation} \left\| {\psi \left( k \right)} \right\| < \rho W{\beta ^k} \end{equation} $

    (17)

    利用反证法, 先假设式(17)不成立, 则必将存在${k^ * } > 0$使得并且当$k \in \left({0, {k^ * }} \right)$时$\left\| {\psi \left(k \right)} \right\| < \rho W{\beta ^k}$成立.因此, 根据式(17)可得:

    $ \begin{align*} &\rho W{\beta ^{{k^ * }}} \le \left\| {\psi \left( {{k^ * }} \right)} \right\| \le\\ &\qquad M\left\| {\psi \left( 0 \right)} \right\|{\alpha ^{{k^ * }}} +\left\| {{Q_2}} \right\|{\alpha ^{{k^ * } - 1}}M\times \end{align*} $

    $ \begin{align*} &\qquad\sum\limits_{s = 0}^{{k^ * } - 1} {\alpha ^{ - s}}\left[ {\frac{{3\sqrt {6n\left( {n - 1} \right)} {\delta _1}\left\| L \right\| \cdot \left\| {\psi \left( s \right)} \right\|}}{{1 - {\delta _1}}}} \right]+ \\ &\qquad M\left\| {{Q_2}} \right\|{\alpha ^{{k^ * } - 1}} \sum\limits_{s = 0}^{{k^ * } - 1} {{\alpha ^{ - s}} \left[ {\frac{{\sqrt {6n\left( {n - 1} \right)} {\delta _2}}}{{1 - {\delta _1}}}{\beta ^s}} \right]} < \\ &\qquad \rho M\left\| {\psi \left( 0 \right)} \right\|{\alpha ^{{k^ * }}} + \rho M\left\| {{Q_2}} \right\|{\alpha ^{{k^ * } - 1}}\times\\ &\qquad \sum\limits_{s = 0}^{{k^ * } - 1} {{\alpha ^{ - s}} \left[ {\frac{{3\sqrt {6n\left( {n - 1} \right)} {\delta _1}\left\| L \right\| \cdot W{\beta ^s}}} {{1 - {\delta _1}}}} \right]} +\\ &\qquad\rho M\left\| {{Q_2}} \right\|{\alpha ^{{k^ * } - 1}} \sum\limits_{s = 0}^{{k^ * } - 1} {{\alpha ^{ - s}} \left[ {\frac{{\sqrt {6n\left( {n - 1} \right)} {\delta _2}{\beta ^s}}}{{1 - {\delta _1}}}} \right]=} \\ &\qquad \rho M\left\| {\psi \left( 0 \right)} \right\|{\alpha ^{{k^ * }}}- \nonumber\\ &\qquad \rho \frac{{M\left\| {{Q_2}} \right\|\sqrt {6n\left( {n - 1} \right)} \left( {3{\delta _1}\left\| L \right\|W + {\delta _2}} \right)}}{{\left( {\beta - \alpha } \right)\left( {1 - {\delta _1}} \right)}}{\alpha ^{{k^ * }}}+\nonumber\\ &\qquad \rho \frac{{M\left\| {{Q_2}} \right\|\sqrt {6n\left( {n - 1} \right)} \left( {3{\delta _1}\left\| L \right\|W + {\delta _2}} \right)}}{{\left( {\beta - \alpha } \right)\left( {1 - {\delta _1}} \right)}}{\beta ^{{k^ * }}} \end{align*} $

    1) 当$W = M\left\| {\psi \left(0 \right)} \right\|$时, 则有

    $ \begin{equation*} \begin{aligned} &M\left\| {\psi \left( 0 \right)} \right\| - \nonumber\\ &\qquad \frac{{M\left\| {{Q_2}} \right\|\sqrt {6n\left( {n - 1} \right)} \left( {3{\delta _1}\left\| L \right\|W + {\delta _2}} \right)}}{{\left( {\beta - \alpha } \right)\left( {1 - {\delta _1}} \right)}} \ge 0 \end{aligned} \end{equation*} $

    所以可得到

    $ \begin{equation} \rho W{\beta ^{{k^ * }}} \le \left\| {\psi \left( {{k^ * }} \right)} \right\| \le \rho M\left\| {\psi \left( 0 \right)} \right\|{\beta ^{{k^ * }}}=\rho W{\beta ^{{k^ * }}} \end{equation} $

    (18)

    2) 当时, 则有

    $ \begin{equation*} \begin{aligned} &M\left\| {\psi \left( 0 \right)} \right\|- \nonumber\\ &\qquad\frac{{M\left\| {{Q_2}} \right\|\sqrt {6n\left( {n - 1} \right)} \left( {3{\delta _1}\left\| L \right\|W + {\delta _2}} \right)}}{{\left( {\beta - \alpha } \right)\left( {1 - {\delta _1}} \right)}} < 0 \end{aligned} \end{equation*} $

    所以有

    $ \begin{align} &\rho W{\beta ^{{k^ * }}} \le \left\| {\psi \left( {{k^ * }} \right)} \right\|\le\nonumber\\ & \frac{{\rho {\delta _2}M\left\| {{Q_2}} \right\|\sqrt {6n\left( {n - 1} \right)} {\beta ^{{k^ * }}}}}{{\left( {\beta - \alpha } \right)\left( {1 - {\delta _1}} \right) - 3{\delta _1}M\left\| {{Q_2}} \right\|\left\| L \right\|\sqrt {6n\left( {n - 1} \right)} }}=\nonumber\\ &\rho W{\beta ^{{k^ * }}} \end{align} $

    (19)

    根据以上结果, 式(18)和式(19)都与假设相矛盾.这说明原命题成立, 即对任意的$\rho > 1$, 式(17)成立.易知, 如果$\rho \to 1$, 则式(16)成立.根据式(16)可知, 当$k \to + \infty $时, 有, 则系统(5)是收敛的.由$\psi \left(k \right)$的定义可知, 系统(1)能够实现渐近一致.

    定理2.  对于系统(1), 如果定理1中的条件成立, 并且控制器(2)中的设计参数满足如下条件,

    $ {\delta _1} \in \left( {\frac{{\left( {\beta - \alpha } \right)}}{{\left( {\beta - \alpha } \right) + 3\sqrt {6n\left( {n - 1} \right)} M\left\| {{Q_{\rm{2}}}} \right\|\left\| L \right\|}}, 1} \right)\\ {\delta _2} > \frac{{\left\| L \right\|\left\| {\psi \left( 0 \right)} \right\|M\left( {1 + \beta } \right)}}{\beta } $

    那么触发序列中的类Zeno行为将被排除.

    证明.  易知排除类Zeno行为的关键是要证明不等式$k_{p + 1}^i - k_p^i > 1$成立.根据事件触发机制可知, 下一个触发时刻将会发生在触发函数(4)大于0时.进而可得到如下不等式

    $ \begin{align} &\left| {{e_{bi}}\left( {k_{p + 1}^i} \right)} \right| + \left| {{e_{ci}}\left( {k_{p + 1}^i} \right)} \right| + \left| {{e_{gi}}\left( {k_{p + 1}^i} \right)} \right|\ge\nonumber\\ &\qquad{\delta _1}\left| {{b_i}\left( {k_p^i} \right)} \right| + {\delta _1}\left| {{c_i}\left( {k_p^i} \right)} \right| +\nonumber\\ &\qquad {\delta _1}\left| {{g_i}\left( {k_p^i} \right)} \right| + {\delta _2}{\beta ^{k_{p + 1}^i}} \end{align} $

    (20)

    定义, .结合式(20), 可得到下式

    $ \begin{equation} {G_i}\left( {k_{p + 1}^i} \right) \ge {\delta _1}{H_i}\left( {k_p^i} \right) + {\delta _2}{\beta ^{k_{p + 1}^i}} \end{equation} $

    (21)

    结合式(16)和式(21)可得

    $ \begin{align} {\delta _2}{\beta ^{k_{p + 1}^i}} &\le {G_i}\left( {k_{p + 1}^i} \right) - {\delta _1}{H_i}\left( {k_p^i} \right)\le\nonumber\\ & \left\| L \right\|\left( {\left\| {\psi \left( {k_p^i} \right)} \right\| + \left\| {\psi \left( {k_{p + 1}^i} \right)} \right\|} \right)\le\nonumber\\ & W\left\| L \right\|\left( {{\beta ^{k_p^i}} + {\beta ^{k_{p + 1}^i}}} \right) \end{align} $

    (22)

    求解上式得

    $ \begin{equation} \left( {{\delta _2} - \left\| L \right\|W} \right){\beta ^{k_{p + 1}^i}} \le \left\| L \right\|W{\beta ^{k_p^i}} \end{equation} $

    (23)

    根据式(23)可得

    $ \begin{equation} k_{p + 1}^i - k_p^i > \dfrac{{\ln \dfrac{{W\left\| L \right\|}}{{{\delta _2} - W\left\| L \right\|}}} } {\ln \beta } \end{equation} $

    (24)

    基于(24)易知当时, 有如下不等式成立

    $ \begin{equation} \dfrac{{\ln \dfrac{{W\left\| L \right\|}}{{{\delta _2} - W\left\| L \right\|}}}} {\ln \beta } > 1 \end{equation} $

    (25)

    此外, 因为$W = M\left\| {\psi \left(0 \right)} \right\|$以及

    $ \begin{equation} {\delta _1} > \frac{{\left( {\beta - \alpha } \right)}}{{\left( {\beta - \alpha } \right) + 3\sqrt {6n\left( {n - 1} \right)} M\left\| {{Q_{\rm{2}}}} \right\|\left\| L \right\|}} \end{equation} $

    (26)

    又可以得出

    $ \begin{equation} {\delta _2} > \frac{{\left\| L \right\|\left\| {\psi \left( 0 \right)} \right\|M\left( {1 + \beta } \right)}}{\beta } = \frac{{\left\| L \right\|W\left( {1 + \beta } \right)}}{\beta } \end{equation} $

    (27)

    该式意味着式(25)成立, 又结合式(24)易知$k_{p + 1}^i - k_p^i > 1$, 即排除类Zeno行为的条件得已满足.

    注2.类Zeno行为广泛存在于基于事件触发控制机制的离散系统中.然而, 当前极少有文献研究如何排除类Zeno行为, 尤其是对于三阶多智能体动态模型.定理2给出了排除三阶离散多智能体系统的类Zeno行为的参数条件.

    本部分将利用一个仿真实验来验证本文所提算法及理论的正确性和有效性.假设三阶离散多智能体系统(1)包含6个智能体, 且有向加权通信拓扑结构如图 1所示, 权重取值为0或1, 可以明显地看出该图包含有向生成树(满足假设1).

    图 1  6个智能体通信拓扑结构图
    Fig. 1  The communication topology with six agents

    通过简单的计算可得, ${\mu _1} = 0$, ${\mu _2} = 0.6852$, ${\mu _3} = 1.5825 + 0.3865$i, ${\mu _4} = 1.5825 - 0.3865$i, ${\mu _5} = 3.2138$, ${\mu _6} = 3.9360$.令$M = 1$, 结合定理1和定理2可得到$0.035 < {\delta _1} < 1$, ${\delta _2} > 44.0025$, $0 < \alpha < \beta < 1$.令${\delta _1} = 0.2$, ${\delta _2} = 200$, $\alpha = 0.6$, $\beta = 0.9$, $\lambda = 0.02$, $\eta = 0.3$, $\gamma = 0.5$, 不难验证满足引理1的条件并且计算可知$\rho \left({{Q_1}} \right) = 0.9958 < 1$.三阶离散多智能体系统(1)的一致性结果如图 2~图 6所示.根据定理1可知, 基于控制器(2)和事件触发函数(4)的系统(1)能实现一致.从图 2~图 6可以看出, 仿真结果与理论分析符合.

    图 2  三阶离散多智能体系统的位置轨迹图
    Fig. 2  The trajectories of position in third-order discrete-time multi-agent systems
    图 3  三阶离散多智能体系统的速度轨迹图
    Fig. 3  The trajectories of speed in third-order discrete-time multi-agent systems
    图 4  三阶离散多智能体系统的加速度轨迹图
    Fig. 4  The trajectories of acceleration in third-order discrete-time multi-agent systems
    图 5  三阶离散多智能体系统的控制轨迹图
    Fig. 5  The trajectories of control in third-order discrete-time multi-agent systems
    图 6  100次迭代内所有智能体的触发时刻
    Fig. 6  Triggering instants of all agents within 100 iterations

    图 2~图 4分别表征了系统(1)中所有智能体的位置、速度和加速度的轨迹, 从图中可以看出以上3个变量确实达到了一致.图 5展示了控制输入的轨迹.为了更清楚地体现事件触发机制的优点, 图 6给出了0$ \sim $100次迭代内的各智能体的触发时刻轨迹.从图 6可以看出, 本文设计的事件触发协议确实达到了减少更新次数, 节省资源的目的.

    针对三阶离散多智能体系统的一致性问题, 构造了一个新颖的事件触发一致性协议, 分析得到了在通信拓扑为有向加权图且包含生成树的条件下, 系统中所有智能体的位置状态、速度状态和加速度状态渐近收敛到一致状态的充分条件.同时, 该条件指出了通信拓扑的Laplacian矩阵特征值和系统的耦合强度对系统一致性的影响.另外, 给出了排除类Zeno行为的参数条件.仿真实验结果也验证了上述结论的正确性.将文中获得的结论扩展到拓扑结构随时间变化的更高阶多智能体网络是极有意义的.这将是未来研究的一个具有挑战性的课题.


  • 本文责任编委 黄庆明
  • 图  1  GP, FITC, NIGP和SGPIN算法预测结果

    Fig.  1  Predicting results of GP, FITC, NIGP and SGPIN

    图  2  TGP, TGPKNN与SGPIN算法的误差比较

    Fig.  2  Error comparison of TGP, TGPKNN and SGPIN

    图  3  GP, KTA, HSICKNN与SGPIN算法的误差比较

    Fig.  3  Error comparison of GP, KTA, HSICKNN and SGPIN

    表  1  GP, FITC, NIGP和SGPIN算法比较

    Table  1  Comparison of GP, FITC, NIGP and SGPIN

    算法训练点个数MSE ($10^{-3}$)运行时间(s)
    GP20031.13261.876034
    FITC80018.62790.062001
    NIGP20018.627913.630882
    SGPIN8008.62650.003087
    20018.49460.002612
    下载: 导出CSV

    表  2  实验数据集

    Table  2  Experimental set

    特征动作个体1个体2个体3总数
    HoGWalking1 1768768952 947
    Jogging4397958312 065
    Throw/Catch21780601 023
    Gestures8016812141 696
    Box5024649331 889
    Total3 1353 6222 8739 630
    下载: 导出CSV

    表  3  基于HumanEva-I数据集HoG特征的不同算法的平均误差

    Table  3  Evaluation of average error of difierent algorithms based on HoG feature of HumanEva-I

    研究个体动作样本数GPTGPTGPKNNKTAHSICKNNSGPIN
    S1Walking1 176398.5823197.1179193.9949213.5265218.6241161.2112
    Jogging439383.7747212.3234212.2018188.6683196.0839154.5919
    Throw/Catch217414.5873174.2834///100.7592
    Gestures801415.310698.6237102.552092.1541156.6464 20.1770
    Box502426.6358162.6801163.3203118.0500149.5003 82.3949
    S2Walking876398.5817197.1496195.5694206.7040211.9735160.4342
    Jogging795405.1201213.0572207.2430227.3562231.1777176.1768
    Throw/Catch806421.5898210.1543199.3265173.2717189.7417 92.6742
    Gestures681410.0671201.1053201.7576153.9103173.0548 63.2473
    Box464421.3947171.6007109.1912137.1031159.5833 98.3920
    S3Walking895412.0019219.2579214.8589236.1566239.6487177.3461
    Jogging831441.7053211.1343206.1400233.5746236.5287184.2251
    Throw/Catch0//////
    Gestures214473.7616159.7482/// 40.3100
    Box933483.6534214.1621207.7578186.5170195.9815120.6541
    总数9 630284.0985160.1196162.0768//155.3066
    下载: 导出CSV

    表  4  基于HumanEva-I数据集HoG特征的不同算法的运行时间

    Table  4  Evaluation of runtime of difierent algorithms based on HoG feature of HumanEva-I

    研究个体动作样本数GPTGPTGPKNNKTAHSICKNNSGPIN
    S1Walking1 176 0.1126.7724.6728.1627.8718.02
    Jogging439 0.038.4710.4310.1810.2621.65
    Throw/Catch217 0.013.77///22.44
    Gestures801 0.0727.1527.3118.6419.4219.78
    Box502 0.0310.1111.1911.7511.8421.90
    S2Walking876 0.0820.8625.8320.0320.2622.04
    Jogging795 0.0718.0617.8617.6417.7423.32
    Throw/Catch806 0.0218.5626.5920.1320.0221.69
    Gestures681 0.0414.3215.5215.9116.6418.38
    Box464 0.039.0210.3510.7111.4323.69
    S3Walking895 0.0922.7822.6320.7520.9521.13
    Jogging831 0.0821.8320.1318.5119.0120.36
    Throw/Catch0//////
    Gestures214 0.046.13///22.62
    Box933 0.1023.7023.6722.6823.5721.69
    总数9 630 11192844249149541
    下载: 导出CSV

    表  5  个体3行走姿态的预测误差

    Table  5  Predicting errors of subject 3 walking

    GPTGPTGPKNNHSICKNNKTASGPIN
    1412.0219.1214.8236.3239.6177.6
    2412.0218.0214.9232.5235.7176.9
    3412.0220.2214.6237.1240.9177.7
    4412.0220.4215.6241.6244.8177.5
    5412.0218.7214.5233.4237.3177.0
    方差0.001.040.1812.8912.380.14
    下载: 导出CSV
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