Intelligent Perception and Recognition of Blast Furnace Anomalies via Burden Surface Video Image Analysis
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摘要: 智能感知、精准识别高炉(Blast furnace, BF)异常状态对高炉调控优化和稳定运行至关重要, 但高炉内部的黑箱状态致使传统检测方法难以直接感知并准确识别多种高炉异常状态. 新型工业内窥镜可获取大量料面视频图像, 为直接观测炉内运行状态提供了全新的手段. 基于此, 提出一种基于料面视频图像分析的高炉异常状态智能感知与识别方法. 首先, 提出基于多尺度纹理模糊C均值(Multi-scale texture fuzzy C-means, MST-FCM)聚类的高温煤气流区域提取方法, 准确获取煤气流图像, 并提取煤气流图像多元特征; 其次, 提出基于特征编码的高维特征降维方法, 结合自适应K-means++ 算法, 实现煤气流异常状态的粗粒度感知; 在此基础上, 通过改进雅可比−傅立叶矩(Jacobi-Fourier moments, JFM) 提取煤气流图像深层特征变化趋势, 进而提出细粒度煤气流异常状态感知方法; 最后, 基于煤气流异常状态感知结果, 结合料面视频图像, 提出多级残差通道注意力模块(Multi-level residual channel attention module, MRCAM), 建立高炉异常状态识别模型ResVGGNet, 实现高炉煤气流异常、塌料和悬料的精准在线识别. 实验结果表明, 所提方法能准确识别不同的高炉异常状态且识别速度快, 可为高炉平稳运行提供重要保障.
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关键词:
- 高炉 /
- 料面图像 /
- 高炉异常状态感知 /
- 高炉异常状态识别 /
- 多级残差通道注意力模块
Abstract: The intelligent perception and precise recognition of blast furnace (BF) anomalies are important for BF regulation, optimization and stable operation. However, the opaque nature of the internal workings of the BF makes it difficult for traditional detection methods to directly perceive and accurately recognize various BF anomalies. The novel industrial endoscope can capture a large number of BF burden surface video images, providing a new way for direct observation of the furnace＇s operational status. Based on this, an intelligent perception and precise recognition method for BF anomalies is proposed via burden surface video image analysis. Firstly, a method for extracting high-temperature gas flow regions based on multi-scale texture fuzzy C-means (MST-FCM) clustering is proposed to accurately obtain gas flow images and extract multi-features of gas flow images. Secondly, a high-dimensional feature dimensionality reduction method based on feature encoding is proposed, which is combined with the adaptive K-means++ algorithm to achieve coarse-grained perception of gas flow anomalies. On this basis, a fine-grained perception method for gas flow anomalies is proposed by refining Jacobi-Fourier moments (JFM) to extract the deep feature change trend of gas flow images. Finally, based on the perception results of gas flow anomalies, and combined with BF video images, a multi-level residual channel attention module (MRCAM) is put forward and the BF anomalies recognition model ResVGGNet is established. This model achieves precise and online recognition of gas flow anomalies, collapsing and hanging burden surface in the BF. Experimental results demonstrate that the proposed method can accurately recognize different BF anomalies with a fast recognition speed, providing crucial assurance for the smooth operation of the BF. -
近些年来, 由于多智能体系统的分布式协同控制在编队控制[1-2]、蜂拥[3-4]等多领域的应用, 现受到许多学者广泛关注. 目前为止, 多智能体系统的一致性研究已经由一阶[5]、二阶[6]逐步发展到高阶[7-8]. 一致性的基本思想是每个智能体通过自身和邻居信息来更新自身信息, 从而使得所有个体最终收敛于同一状态.
在实际的工程应用中, 智能体自身能量和通讯信道带宽往往都是有限的, 因此, 在设计控制协议时需要考虑智能体能量的损耗, 让其能有更长的运作时间. 由此, 将事件触发机制引入到多智能体系统具有很大意义. 文献[9]将事件触发策略引入多智能体系统的研究, 控制器不再连续更新控制输入, 而是依赖于与测量误差相关的事件触发函数, 当测量误差达到某一临界状态才更新控制输入. 文献[10]给出了一阶多智能体系统的事件触发控制协议, 设计了与智能体系统状态有关的触发条件. 文献[11]研究了在有向拓扑下, 带有扰动多智能体系统的均方一致性问题, 智能体最终收敛到系统初始状态的平均值, 并且进一步分析了切换拓扑的一致性. 在大多数已有的成果中, 对于触发条件的设计, 不仅与自身的触发时间有关, 还与其邻居的触发时间有关. 这样将会增加通讯负担和控制器的更新频率. 为了解决这个问题, 文献[12]提出了联合测量误差, 能减少智能体之间的通信次数. 为了进一步的减小通讯负担和控制器的更新频次, 文献[13]将事件触发机制引入到间歇控制, 给出了集中式和分布式两种事件触发控制策略.
值得注意的是, 大部分已有的基于事件触发控制策略只是基于渐近收敛. 然而, 在一些实际的工程应用中, 尤其在一些要求较高精度和较高收敛速度的控制问题中, 经常需要达到有限时间收敛. 因此, 基于事件触发的有限时间一致性问题有很大研究价值. 文献[14]研究了在无向拓扑下, 针对有领导者和无领导者两种情形, 通过将有限时间一致性控制器与事件触发相结合, 设计了两种控制协议, 然而, 并没有排除Zeno行为. 文献[15]在此基础上, 设计了新的事件触发条件, 给出了排除Zeno行为的证明和数值仿真. 文献[16]在文献[14]基础上, 研究了在有向拓扑下的有限时间一致性问题, 给出了两种事件触发条件. 尽管上述文献很好地解决了基于事件触发的有限时间一致性, 但是设置的收敛时间都与智能体的初始状态有关, 当系统初始状态很大时, 系统收敛时间会受较大影响. 为了排除这一影响, 文献[17]设计了两种控制协议: 1)通过引入符号函数来抑制外部扰动的固定时间一致性协议; 2)为消除前者符号函数所带来的抖振现象, 引入饱和函数, 并给出事件触发的条件.
上述文献大部分都是关于普通一致性问题, 文献[18]研究了比例一致性问题, 即各个智能体最终的状态能够趋于指定的比例, 而不是同一定值. 文献[19]研究了切换拓扑下带有通信时延的比例一致性问题. 文献[20]研究了一阶和二阶分组比例一致性问题, 设计了两种分布式控制协议. 文献[21]研究了带有外部扰动的比例一致性问题, 给出了基于渐近收敛、有限时间收敛和固定时间收敛三种控制策略.
本文研究了基于事件触发二阶多智能体系统的固定时间比例一致性问题, 提出了一种新的基于事件触发的比例一致性控制协议, 该控制协议包含基于状态信息和速度信息的分段式触发条件: 当智能体在追踪虚拟速度时, 采用与系统速度有关的触发条件; 当完成虚拟速度追踪后, 切换为基于状态信息的触发条件, 能有效的减小系统能量耗散及控制器更新频次. 基于Lyapunov稳定性理论、线性矩阵不等式和代数图论证明了所提事件触发控制策略能有效地实现二阶多智能体系统的固定时间比例一致性, 并且不存在Zeno行为. 相较于文献[14]、[16], 本文所给出的收敛时间不再依赖于系统的初始状态. 在文献[17]的基础上, 本文进一步拓展, 对二阶多智能体系统进行了研究, 同时多智能体不再收敛于同一状态, 而是按照既定的比例, 收敛到不同状态. 相较于文献[18]、[20], 本文采用事件触发的策略来设计控制协议, 能在达到比例一致性的同时有效节约系统资源.
1. 预备知识及问题描述
1.1 图论
$ N $ 个智能体可视为$ N $ 个节点, 可以用无向图$ G = $ $ (V,E,A) $ 表示,$V = \{ {v_1},\cdots,{v_N}\}$ 表示节点集合,$ E \subseteq $ $ V \times V $ 表示边集.$A = [{a_{ij}}] \in {{\bf R}^{n \times n}}$ 是具有元素$ {a_{ij}} $ 的加权矩阵, 其对角线元素$ {a_{ii}} = 0 .$ 如果$ ({v_i},{v_j}) \notin $ $ E $ ,$ {a_{ij}} = 0 $ , 否则$ {a_{ij}} > 0 . $ 若$ ({v_i},{v_j}) \in E = ({v_j},{v_i}) \in $ $ E $ ,$ {e_{ij}} = ({v_i},{v_j}) $ 表示第$ i $ 个智能体与第$ j $ 个智能体之间互相传输信息, 则图$ G $ 为无向图; 若$({v_i},{v_j}) \in $ $ E \ne ({v_j},{v_i}) \in E$ ,$ {e_{ij}} = ({v_i},{v_j}) $ 表示第$ i $ 个智能体向第$ j $ 个智能体传输信息, 则图$ G $ 为有向图, 从节点$ i $ 到节点$ j $ 的有向路径被称为有向边. 度矩阵$D \in $ $ {{{\bf R} ^{N \times N}}}$ 定义为$ D = {\rm diag}\{ {d_i}\} $ , 其中$ {d_i} = \sum\nolimits_{{v_j} \in V} {{a_{ij}}} . $ Laplacian矩阵$L \in {{{\bf R}^{N \times N}}}$ 被定义为$ L = [{l_{ij}}] $ ,$ L =$ $ D - A $ , 其中,$ {l_{ii}} = \sum\nolimits_{p \ne i}^n {a{}_{ip}} $ ,$ {l_{ij}} = - {a_{ij}},\forall i \ne j $ .1.2 相关引理
引理1[22].
1)无向图
$ G $ 的Laplacian矩阵$ L $ 为半正定, 有一个特征值为0. 如果无向图$ G $ 是连通的, 则除0以外的特征值均正定;2)无向图
$ G $ 的Laplacian矩阵$ L $ 的第二小特征值$ {\lambda _2}(L) $ 满足:$$ {\lambda _2}(L) = \mathop {\min }\limits_{||x|| \ne 0,\sum\limits_{i = 1}^N {{x_i} = 0} } \frac{{{x^{\rm{T}}}Lx}}{{||x|{|^2}}} > 0 $$ 当
$\sum_{i = 1}^n {x_i} = 0$ 时, 有:$${x^{\rm T}}Lx \ge {\lambda _2}(L){x^{\rm{T}}}x $;$ 3)对于任意
$x = {({x_1},{x_2},\cdots,{x_N})^{\rm{T}}} \in {{{\bf R}^N}}$ 有:$$ {x^{\rm{T}}}Lx = \frac{1}{2}\sum_{i = 1}^N {\sum_{j = 1}^N {{a_{ij}}{{({x_j} - {x_i})}^2}} } $.$ 引理2[23]. 如果存在一个连续的径向无界函数
$ V:{{{\bf R}^N}} \to {{{\bf R}_ + }} \cup \{ 0\} $ 满足:1)
$ V(x) = 0 \Leftrightarrow x = 0 $ ;2)系统任意的解
$ x(t) $ 满足:$$ {{D}^*}V(x(t)) \le - \alpha {V^p}(x(t)) - \beta ({V^q}(x(t)) $$ 其中,
$ \alpha ,\beta > 0 $ ,$ p = 1 - \dfrac{1}{{2\kappa }} $ ,$ q = 1 + \dfrac{1}{{2\kappa }} $ ,$ \kappa > 1 $ , 则系统在固定时间达到全局稳定, 且收敛时间$ T $ 满足$ T \le {T_{\max }} = \dfrac{{ {\text{π}} \kappa }}{{\sqrt {\alpha \beta } }} $ .引理3[24]. 假设
${w_1},{w_2},\cdots,{w_N} \ge 0$ ,$ 0 < p \le 1 $ ,$ q > 1 $ , 有:$$ \sum\limits_{i = 1}^N {w_i^p} \ge \left(\sum\limits_{i = 1}^N {w_i}\right){^p} ,\sum\limits_{i = 1}^N {w_i^q} \ge {N^{1 - q}}\left(\sum\limits_{i = 1}^N {w_i}\right){^q} $.$ 1.3 问题陈述
考虑到二阶多智能体系统由
$ N $ 个智能体组成, 智能体$ i $ 的动力学方程可写为:$$ \left\{ {\begin{aligned} &{{{\dot x}_i}(t) = {v_i}(t)}\\ &{{{\dot v}_i}(t) = {u_i}(t)} \end{aligned}} \right.,\quad i = 1,\cdots,N $$ (1) 上式中,
$ {x_i}(t) \in {\bf R} $ 表示为智能体$ i $ 的状态变量,$ {v_i}(t) \in {\bf R} $ 表示为智能体$ i $ 的速度变量,$ {u_i}(t) \in {\bf R} $ 表示为系统的控制输入,$x(t) = [{x_1}(t),{x_2}(t),\cdots,$ $ {x_N}(t)]^{\rm{T}} $ .定义1[20]. 对于给定的控制器
${u_i},i = 1,2,\cdots,N$ , 如果对于给定的任何初始值${x_i}(0),i = 1,2,\cdots,N$ , 存在一个与初始值有关的正数$ T $ 以及固定的常数$ {T_{\max }} > $ $ 0 $ ,$ T < {T_{\max }} $ , 于任意的$i,j = 1,2,\cdots,N$ 有:$$ \mathop {\lim }\limits_{t \to T} |{s_i}{x_i}(t) - {s_j}{x_j}(t)| = 0 $$ $$ \mathop {\lim }\limits_{t \to T} {v_i}(t) = 0 $$ $$ {s_i}{x_i}(t) = {s_j}{x_j}(t),v{}_i(t) = {v_j}(t),\quad\forall t \ge T $$ (2) 则称闭环系统达到固定时间比例一致性, 其中
$ {s_i}, $ $i = 1,2,\cdots,N$ , 为比例系数.2. 基于事件触发的固定时间比例一致性
受文献[7]、[25]的启发, 采用反推法来设计控制器, 引入虚拟速度:
$$ \begin{split} v_i^* =\;& - {c_1}{\rm sign}({s_i})sig{\left[ {\sum\limits_{j = 1}^N {{a_{ij}}({s_i}{x_i} - {s_j}{x_j})} } \right]^\alpha } - \\ &{c_2}{\rm sign}({s_i})sig{\left[ {\sum\limits_{j = 1}^N {{a_{ij}}({s_i}{x_i} - {s_j}{x_j})} } \right]^\beta } \end{split} $$ (3) 定义
$ sig{[m]^k} = {\rm sign}(m)|m{|^k} $ ,${\rm sign}(\cdot)$ 为符号函数. 其中,$i = 1,2,\cdots,N$ ,$ {c_1} > 0 $ ,$ {c_2} > 0 $ ,$ \alpha \in (0,1) $ ,$ \beta > 1 $ .定义速度跟踪误差:
$$ {\bar v_i} = {v_i} - v_i^* $$ (4) 令
$\bar v = {({\bar v_1},{\bar v_2},\cdots,{\bar v_N})^{\rm{T}}}$ , 式(3)、(4)求导得:$$ \begin{split} {{{\dot {\bar v}_i}}} =\;& {u_i} + {c_1}\alpha {\rm sign}({s_i})|\sum\limits_{j = 1}^N {{a_{ij}}} ({s_i}{x_i} - \\ &{s_j}{x_j}){|^{\alpha - 1}}\sum\limits_{j = 1}^N {{a_{ij}}({s_i}{v_i} - {s_j}{v_j})} + \\ &{c_{\rm{2}}}\beta {\rm sign}({s_i})|\sum\limits_{j = 1}^N {{a_{ij}}} ({s_i}{x_i} - \\ &{s_i}{x_j}){|^{\beta - 1}}\sum\limits_{j = 1}^N {{a_{ij}}({s_i}{v_i}} - {s_j}{v_j}) \end{split} $$ (5) 为设计智能体的事件触发策略, 对于每一个智能体
$ i $ 定义,$$ {\hat x_i}(t) = {s_i}{x_i}(t_i^k) $$ (6) $$ {\hat v_i}(t) = {s_i}{v_i}(t_i^k) $$ (7) 其中,
$ t_i^k $ 表示智能体$ i $ 第$ k $ 次事件触发时刻. 当$t \in $ $ [t_i^k, t_i^{k + 1}) $ 时, 定义:$$ \begin{split} \hat v_i^*(t) = \;&- {c_1}{\rm sign}({s_i})sig{\left[ {\sum\limits_{j = 1}^N {{a_{ij}}({{\hat x}_i}(t) - {{\hat x}_j}(t))} } \right]^\alpha } - \\ &{c_2}{\rm sign}({s_i})sig{\left[ {\sum\limits_{j = 1}^N {{a_{ij}}({{\hat x}_i}(t) - {{\hat x}_j}(t))} } \right]^\beta }\\[-15pt] \end{split} $$ (8) $$ {{\hat {\bar v}_i}}(t) = \frac{1}{{{s_i}}}{\hat v_i}(t) - \hat v_i^*(t) $$ (9) 经过以上分析, 给出基于事件触发的控制协议如下:
$$ \begin{split} &{u_i} = - {c_3}sig{\left[ {{{{\hat {\bar v}_i}}}} \right]^p} - {c_4} sig{\left[ {{{{\hat {\bar v}_i}}}} \right]^q} - \\ &\;\;\quad{c_1}\alpha {\rm sign}({s_i}){\left| {\sum\limits_{j = 1}^N {{a_{ij}}({{\hat x}_i} - {{\hat x}_j})} } \right|^{\alpha - 1}}\sum\limits_{j = 1}^N {{a_{ij}}({{\hat v}_i} - {{\hat v}_j})} - \\ &\;\;\quad{c_2}\beta {\rm sign}({s_i}){\left| {\sum\limits_{j = 1}^N {{a_{ij}}({{\hat x}_i} - {{\hat x}_j})} } \right|^{\beta - 1}}\sum\limits_{j = 1}^N {{a_{ij}}({{\hat v}_i} - {{\hat v}_j})} \end{split} $$ (10) 其中,
$ {c_3} $ 、$ {c_4} $ 为正常数, 且$ p \in (0,1),q > 1 $ . 定义$ x = $ ${({x_1},{x_2},\cdots,{x_N})^{\rm{T}}}$ . 为书写方便, 令$$ \begin{split} {\rho _i} = \;&{c_1}\alpha {\rm sign}({s_i})|\sum\limits_{j = 1}^N {{a_{ij}}({{\hat x}_i} - } \\ &{{\hat x}_j}){|^{\alpha - 1}}\sum\limits_{j = 1}^N {{a_{ij}}({{\hat v}_i}} - {{\hat v}_j}) + \\ &{c_2}\beta {\rm sign}({s_i})|\sum\limits_{j = 1}^N {{a_{ij}}({{\hat x}_i} - } \\ &{{\hat x}_j}){|^{\beta - 1}}\sum\limits_{j = 1}^N {{a_{ij}}({{\hat v}_i}} - {{\hat v}_j}) \end{split} $$ $$ \begin{split} {\varsigma _i} =\;& {c_1}\alpha {\rm sign}({s_i})|\sum\limits_{j = 1}^N {{a_{ij}}({s_i}{x_i} - } \\ & {s_j}{x_j}){|^{\alpha - 1}}\sum\limits_{j = 1}^N {{a_{ij}}({s_i}{v_i}} - {s_j}{v_j}) + \\ & {c_2}\beta {\rm sign}({s_i})|\sum\limits_{j = 1}^N {{a_{ij}}({s_i}{x_i} - } \\ & {s_j}{x_j}){|^{\beta - 1}}\sum\limits_{j = 1}^N {{a_{ij}}({s_i}{v_i} - {s_j}{v_j})} \end{split} $$ 当
$ t \in [0,{T_1}] $ 时, 定义测量误差为:$$ \begin{split} {e_i} =\;& {c_3}sig{\left[ {{{{\hat {\bar v}_i}}}} \right]^p} + {c_4}sig{\left[ {{{{\hat {\bar v}_i}}}} \right]^q} + {\rho _i} - \\ &{c_3}sig{\left[ {{{\bar v}_i}} \right]^p} - {c_4}sig{\left[ {{{\bar v}_i}} \right]^q} - {\varsigma _i} \end{split} $$ (11) 令
$e = {({e_1},{e_2},\cdots,{e_N})^{\rm{T}}}$ .定理1. 假设多智能体系统的固定通信拓扑图
$ G $ 为无向图, 考虑到多智能体系统(1)在控制器(10)的作用下, 给出如下触发函数:$$ \begin{split} {f_i}(t) =\;& {\rm sign}\left( {\left| {{v_i} - v_i^*} \right|} \right)\left(\left\| {{e_i}} \right\| - {\mu _i}{c_4}{N^{\frac{{1 - q}}{{\rm{2}}}}}{\left\| {{{\bar v}_i}} \right\|^q}\right) + \\ &\left[ {1 - {\rm sign}\left( {\left| {{v_i} - v_i^*} \right|} \right)} \right]\Bigg(\left\| {{E_i}} \right\| - \\ &{\varepsilon _i}{c_2}{\left\| {\sum\limits_{j = 1}^N {{a_{ij}}\left( {{x_i} - {x_j}} \right)} } \right\|^\beta }\Bigg) \\[-15pt]\end{split} $$ (12) 其中,
$ {E_i} $ 定义为$ t \in ({T_1},T] $ 时的测量误差,$ {T_1} $ 表示智能体速度与虚拟速度达到一致的时间,$ T $ 表示多智能体系统收敛所需时间.$ {\mu _i} \in (0,1),{\varepsilon _i} \in (0,1) $ . 多智能体系统(1)在任意初始条件下均能实现固定时间比例一致性, 且收敛时间满足:$$ \begin{split} T =\;& {T_1} + {T_2} \le {T_{1\max }} + {T_{2\max }}=\\ &\frac{{2{\text{π}} }}{{(q - p)\sqrt {(1 - {\mu _i}){c_3}{c_4}{N^{{{(1 - q)} / 2}}}{2^{\frac{{p + q + 2}}{2}}}} }} + \\ &\frac{{2{\text{π}} }}{{(\beta - \alpha )\sqrt {(1 - {\varepsilon _i}){c_1}{c_2}{N^{{{(1 - \beta )} / 2}}}{{(2{\lambda _2}(L))}^{\frac{{\alpha + \beta + 2}}{2}}}} }} \end{split} $$ (13) $ {T_2} $ 表示智能体达到虚拟速度之后整个系统实现一致性的时间.证明. 由式(5)、(10)、(11)可得:
$$ {{{\dot {\bar v}_i}}} = - {c_3}sig{\left[ {{{\bar v}_i}} \right]^p} - {c_4}sig{\left[ {{{\bar v}_i}} \right]^q} - {e_i} $$ 当
$ t \in [0,{T_1}] $ 时, 选定Lyapunov函数为:$$ {V_1} = \frac{1}{2}{\bar v^{\rm{T}}}\bar v $$ 求导得
$$ \begin{split} {{\dot V}_1} =\;& \sum\limits_{i = 1}^N {{{\bar v}_i}} {{{\dot {\bar v}_i}}}=\\ &\sum\limits_{i = 1}^N {{{\bar v}_i}} ( - {c_3} sig{\left[ {{{\bar v}_i}} \right]^p} - {c_4}sig{\left[ {{{\bar v}_i}} \right]^q} - {e_i})=\\ &- {c_3}\sum\limits_{i = 1}^N {|{{\bar v}_i}{|^{p + 1}} - {c_4}\sum\limits_{i = 1}^N {|{{\bar v}_i}{|^{q + 1}}} } - \sum\limits_{i = 1}^N {{{\bar v}_i}{e_i}} =\\ &- {c_3}\sum\limits_{i = 1}^N {{{(\bar v_i^2)}^{\frac{{p + 1}}{2}}} - {c_4}\sum\limits_{i = 1}^N {{{(\bar v_i^2)}^{\frac{{q + 1}}{2}}}} } - \sum\limits_{i = 1}^N {{{\bar v}_i}{e_i}} \le\\ &- {c_3}{\left(\sum\limits_{i = 1}^N {\bar v_i^2} \right)^{\frac{{p + 1}}{2}}} - {c_4}{N^{\frac{{1 - q}}{2}}}{\left(\sum\limits_{i = 1}^N {\bar v_i^2} \right)^{\frac{{q + 1}}{2}}} - \\ &\sum\limits_{i = 1}^N {{{\bar v}_i}{e_i}}\\[-15pt] \end{split} $$ (14) 上式中
$$ \begin{split} \sum\limits_{i = 1}^N {{{\bar v}_i}{e_i}} =\;& {{\bar v}^{\rm{T}}}e \le \parallel \bar v\parallel \parallel e\parallel =\\ &\dfrac{{\parallel \bar v\parallel \parallel e\parallel {V_1}^{\frac{{1 + q}}{2}}}}{{{V_1}^{\dfrac{{1 + q}}{2}}}}=\\ &\dfrac{{{2^{\frac{{1 + q}}{2}}}\parallel \bar v\parallel \parallel e\parallel {V_1}^{\frac{{1 + q}}{2}}}}{{\parallel \bar v{\parallel ^{1 + q}}}}=\\& {2^{\dfrac{{1 + q}}{2}}}\parallel \bar v{\parallel ^{ - q}}\parallel e\parallel {V_1}^{\frac{{1 + q}}{2}} \end{split} $$ (15) 结合式(14)、(15)有:
$$ \begin{split} {{\dot V}_1} \le \;& - {c_3}{\left(\sum\limits_{i = 1}^N {\bar v_i^2} \right)^{\frac{{p + 1}}{2}}} - {c_4}{N^{\frac{{1 - q}}{2}}}{\left(\sum\limits_{i = 1}^N {\bar v_i^2} \right)^{\frac{{q + 1}}{2}}} + \\ &{2^{\frac{{1 + q}}{2}}}\parallel \bar v{\parallel ^{ - q}}\parallel e\parallel {V_1}^{\frac{{1 + q}}{2}}=\\ &- {2^{\frac{{p + 1}}{2}}}{c_3}V_1^{\frac{{p + 1}}{2}} - {2^{\frac{{q + 1}}{2}}}{N^{\frac{{1 - q}}{2}}}{c_4}V_1^{\frac{{q + 1}}{2}} + \\ &{2^{\frac{{1 + q}}{2}}}\parallel \bar v{\parallel ^{ - q}}\parallel e\parallel {V_1}^{\frac{{1 + q}}{2}}\\[-10pt] \end{split} $$ (16) 由引理2得, 事件触发条件为:
$$ \parallel {e_i}\parallel \le {\mu _i}{c_4}{N^{\frac{{1 - q}}{2}}}\parallel {\bar v_i}{\parallel ^q} $$ (17) 结合式(16)、(17)得:
$$ {\dot V_1} \le - {2^{\tfrac{{p + 1}}{2}}}{c_3}V_1^{\tfrac{{p + 1}}{2}} - (1 - {\mu _i}){2^{\tfrac{{q + 1}}{2}}}{c_4}{N^{\tfrac{{1 - q}}{2}}}V_1^{\tfrac{{q + 1}}{2}} $$ (18) 令
$ \dfrac{{p + 1}}{2} = 1 - \dfrac{1}{{2{\kappa _1}}} $ ,$ \dfrac{{q + 1}}{2} = 1 + \dfrac{1}{{2{\kappa _1}}} $ ,${\kappa _1} = $ $ \dfrac{2}{{q - p}}$ , 由引理2,在时间$ {T_1} $ 时, 虚拟速度的追踪误差$ {\bar v_i} $ 收敛于0, 意味着在固定时间内, 智能体系统能实现对虚拟速度$ v_i^* $ 的追踪, 且收敛时间$ {T_1} $ 满足:$$ \begin{split} {T_1} \le \;&{T_{1\max }}=\\ &\frac{2 \text{π} }{(q - p)\sqrt {(1 - {\mu _i}){c_3}{c_4}{N^{\frac{1 - q}{2}}2^{\frac{p + q + 2}{2}}} }} \end{split} $$ (19) 另一方面, 当
$ t \in [0,{T_1}] $ 时, 由式(18)及引理2知$ {\bar v_i} $ 有界, 再由式(1)、(3)、(4)得:$$ \begin{split} {{\dot x}_i} =\;& - {c_1}{\rm sign}({s_i})sig{\left[ {\sum\limits_{j = 1}^N {{a_{ij}}({s_i}{x_i} - {s_j}{x_j})} } \right]^\alpha } - \\ &{c_2}{\rm sign}({s_i})sig{\left[ {\sum\limits_{j = 1}^N {{a_{ij}}({s_i}{x_i} - {s_j}{x_j})} } \right]^\beta } + {{\bar v}_i} \end{split} $$ (20) 根据式(20),由于
$ {\bar v_i} $ 有界, 当$ t \in [0,{T_1}] $ , 要初始状态有界, 则$ {x_i} $ 有界. 同时, 由于$ {x_i} $ 有界, 由式(3)、(4)不难得出$ {v_i} $ 有界.注意到, 当
$ {\bar v_i} = 0 $ 时, 由式(4)得:$$ \begin{split} {v_i} = \;&- {c_1}{\rm sign}({s_i})sig{\left[ {\sum\limits_{j = 1}^N {{a_{ij}}({s_i}{x_i} - {s_j}{x_j})} } \right]^\alpha } - \\ &{c_2}{\rm sign}({s_i})sig{\left[ {\sum\limits_{j = 1}^N {{a_{ij}}({s_i}{x_i} - {s_j}{x_j})} } \right]^\beta } \end{split} $$ (21) 考虑到控制器
$ {u_i} $ 是不连续更新控制输入的, 故有:$$ \begin{split} {{\dot x}_i} = \;&- {c_1}{\rm sign}({s_i})sig{\left[ {\sum\limits_{j = 1}^N {{a_{ij}}({{\hat x}_i} - {{\hat x}_j})} } \right]^\alpha } - \\ &{c_2}{\rm sign}({s_i})sig{\left[ {\sum\limits_{j = 1}^N {{a_{ij}}({{\hat x}_i} - {{\hat x}_j})} } \right]^\beta } \end{split} $$ (22) 当
$ t \in ({T_1},T] $ 时, 定义测量误差:$$ \begin{split} {E_i} =\;& {c_1}{\rm sign}({s_i})sig{\left[ {\sum\limits_{j = 1}^N {{a_{ij}}({{\hat x}_i} - {{\hat x}_j})} } \right]^\alpha } + \\ &{c_2}{\rm sign}({s_i})sig{\left[ {\sum\limits_{j = 1}^N {{a_{ij}}({{\hat x}_i} - {{\hat x}_j})} } \right]^\beta } - \\ &{c_1}{\rm sign}({s_i})sig{\left[ {\sum\limits_{j = 1}^N {{a_{ij}}({s_i}{x_i} - {s_j}{x_j})} } \right]^\alpha } - \\ &{c_2}{\rm sign}({s_i})sig{\left[ {\sum\limits_{j = 1}^N {{a_{ij}}({s_i}{x_i} - {s_j}{x_j})} } \right]^\beta } \end{split} $$ (23) 选取Lyapunov函数为:
$$ {V_2} = \frac{1}{2}{x^{\rm{T}}}{S^{\rm{T}}}LSx $$ 其中,
$S = {\rm diag}\{{s_i}{\rm sign}({s_i})\}$ , 导有:$$ {\dot V_2} = {x^{\rm{T}}}{S^{\rm{T}}}LS\dot x = \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_{ij}}({s_i}{x_i} - {s_j}{x_j}){{\dot x}_i}} } $$ 令
${y_i} = \sum\nolimits_{j = 1}^N {{a_{ij}}({s_i}{x_i} - {s_j}{x_j})},$ $y = ({y_1},{y_2},\cdots, $ $ {y_N})^{\rm{T}}$ .$$ \begin{split} {{\dot V}_2} =\;& \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_{ij}}({s_i}{x_i} - {s_j}{x_j})( - {c_1}{\rm sign}({s_i})sig{{\left[ y \right]}^\alpha }} } - \\ &{c_2}{\rm sign}({s_i})sig{\left[ y \right]^\beta } - {E_i}) \le \\ &- {c_1}\sum\limits_{i = 1}^N {\parallel {y_i}{\parallel ^{\alpha + 1}}} - {c_2}\sum\limits_{i = 1}^N {\parallel {y_i}{\parallel ^{\beta + 1}}} + \\ &\sum\limits_{i = 1}^N {\parallel {E_i}\parallel \parallel {y_i}\parallel }\\[-15pt] \end{split} $$ (24) 于是, 可以得到事件触发条件:
$$ \parallel {E_i}\parallel \le {\varepsilon _i}{c_2}\parallel {y_i}{\parallel ^\beta } $$ (25) 由引理3, 式(24)可以写成:
$$ {\dot V_2} \le - {c_1}{\left[ {\sum\limits_{i = 1}^N {y_i^2} } \right]^{\frac{{\alpha + 1}}{2}}} - (1 - {\varepsilon _i}){c_2}{N^{\frac{{1 - \beta }}{2}}}{\left[ {\sum\limits_{i = 1}^N {y_i^2} } \right]^{\frac{{\beta + 1}}{2}}} $$ (26) 由引理1,
$$ \begin{split} \sum\limits_{i = 1}^N {y_i^2} =\;& {({L^{\frac{1}{2}}}Sx)^{\rm T}}L({L^{\frac{1}{2}}}Sx)\ge\\ &{\lambda _2}(L){x^{\rm T}}SLSx = 2{\lambda _2}(L){V_2} \end{split} $$ 于是式子(26)可以写成:
$$ \begin{split} {{\dot V}_2} \le\;& - {c_1}{\left[ {2{\lambda _2}(L){V_2}} \right]^{\frac{{\alpha + 1}}{2}}} - \\ &(1 - {\varepsilon _i}){c_2}{N^{\frac{{1 - \beta }}{2}}}{\left[ {2{\lambda _2}(L){V_2}} \right]^{\frac{{\beta + 1}}{2}}} \end{split} $$ (27) 其中,
$ {\lambda _2}(L) $ 为矩阵$ L $ 的第二小特征值. 令$\dfrac{{\alpha + 1}}{2} = 1 -$ $ \dfrac{1}{{2{\kappa _2}}} $ ,$ \dfrac{{\beta + 1}}{2} = 1 + \dfrac{1}{{2{\kappa _2}}} $ ,$ {\kappa _2} = \dfrac{2}{{\beta - \alpha }} $ , 由引理2, 当$ {\bar v_i} $ 收敛后,$ {x_i} $ 可以实现固定时间一致性, 且收敛时间$ {T_2} $ 满足:$$ \begin{split} {T_2} \le\;& {T_{2\max }}=\\ &\frac{{2 {\text{π}} }}{{(\beta - \alpha )\sqrt {(1 - {\varepsilon _i}){c_1}{c_2}{N^{{{(1 - \beta )} / 2}}}{{(2{\lambda _2}(L))}^{\frac{{\alpha + \beta + 2}}{2}}}} }} \end{split} $$ (28) 结合式(19)、(28),可得多智能体系统(1)在控制输入(10)及触发条件(12)的作用下, 可以实现固定时间一致性, 且收敛时间T满足:
$$ \begin{split} T =\;& {T_1} + {T_2} \le {T_{1\max }} + {T_{2\max }} = \\ &\dfrac{{2 {\text{π}} }}{{(q - p)\sqrt {(1 - {\mu _i}){c_3}{c_4}{N^{{{(1 - q)} / 2}}}{2^{\frac{{p + q + 2}}{2}}}} }} + \\ &\dfrac{{2 {\text{π}} }}{{(\beta - \alpha )\sqrt {(1 - {\varepsilon _i}){c_1}{c_2}{N^{{{(1 - \beta )} / 2}}}{{(2{\lambda _2}(L))}^{\frac{{\alpha + \beta + 2}}{2}}}} }} \end{split} $$ 考虑到时间
$ {T_1} $ 是不确定的, 对两个事件触发条件进行合并. 由式(29)可以看出, 当$ t \in [0,{T_1}] $ 时, 智能体处于追踪虚拟速度的状态,$ {\rm sign}\left( {\left| {{v_i} - v_i^*} \right|} \right)=$ $ 1 $ , 此时事件触发条件为式(17);当$ t \in ({T_1},T] $ 时, 触发条件为式(25).$$ \begin{split} {f_i}(t) =\;& {\rm sign}\left( {\left| {{v_i} - v_i^*} \right|} \right)\left( {\left\| {{e_i}} \right\| - {\mu _i}{c_4}{N^{\frac{{1 - q}}{2}}}{{\left\| {{{\bar v}_i}} \right\|}^q}} \right) + \\ &\left[ {1 - {\rm sign}\left( {\left| {{v_i} - v_i^*} \right|} \right)} \right]\left( {\left\| {{E_i}} \right\| - {\varepsilon _i}{c_2}{{\left\| {{y_i}} \right\|}^\beta }} \right) \end{split} $$ (29) 定理2. 假设固定通信拓扑图
$ G $ 是无向连通的, 考虑多智能体系统(1)在控制器(10)和触发条件(12)的作用下, 系统能实现一致且不存在Zeno行为.证明. 当
$ t \in [0,{T_1}] $ 时, 定义$ \gamma = \parallel e\parallel /\parallel {\bar v^q}\parallel $ , 每个时间段$ [t_k^i,t_{k + 1}^i)) $ 内有:$$ \begin{split} \dot \gamma =\;& \dfrac{{{{(e)}^{\rm{T}}}(e)'}}{{\parallel e\parallel \parallel {{\bar v}^q}\parallel }} - \dfrac{{\parallel e\parallel }}{{\parallel {{\bar v}^q}\parallel }}\frac{{{{({{\bar v}^q})}^{\rm{T}}}({{\bar v}^q})'}}{{\parallel {{\bar v}^q}{\parallel ^2}}}=\\ &- \dfrac{{{{(e)}^{\rm{T}}}(U)'}}{{\parallel e\parallel \parallel {{\bar v}^q}\parallel }} - \dfrac{{\parallel e\parallel }}{{\parallel {{\bar v}^q}\parallel }}\frac{{{{({{\bar v}^q})}^{\rm{T}}}({{\bar v}^q})'}}{{\parallel {{\bar v}^q}{\parallel ^2}}} \end{split} $$ 其中,
$ U $ 表示控制器(10)不采用事件触发机制时的控制协议. 定理1中给出了事件触发条件, 并在其后证明了在控制器(10)的作用下多智能体系统的稳定性. 控制器$ U $ 实时更新控制输入, 也就是说, 控制器$ U $ 比控制器(10)更为保守. 不难证明, 在控制器$ U $ 的作用下, 依然能实现多智能体系统的固定时间一致性, 因此$ U' $ 必定是有界的. 假定$ U' $ 绝对值的最大值为$ {G_{\max }} $ ,则有:$$ \begin{split} \dot \gamma \le \;&\dfrac{{{G_{\max }}}}{{\parallel {{\bar v}^q}\parallel }} + \gamma \dfrac{{\parallel ({{\bar v}^q})'\parallel }}{{\parallel {{\bar v}^q}\parallel }}\le\\ &(1 + \gamma )\dfrac{{{G_{\max }}}}{{\parallel {{\bar v}^q}\parallel }} + (1 + \gamma )\dfrac{{\parallel ({{\bar v}^q})'\parallel }}{{\parallel {{\bar v}^q}\parallel }}=\\ &(1 + \gamma )\left( {\dfrac{{{G_{\max }}}}{{\parallel {{\bar v}^q}\parallel }} + \dfrac{{\parallel ({{\bar v}^q})'\parallel }}{{\parallel {{\bar v}^q}\parallel }}} \right)=\\ &(1 + \gamma )\left( {\dfrac{{{G_{\max }}}}{{\parallel {{\bar v}^q}\parallel }} + \dfrac{{q\parallel {\bar v}{\parallel ^{q - 1}}\parallel \dot {\bar v}\parallel }}{{\parallel {{\bar v}^q}\parallel }}} \right)=\\ &(1 + \gamma )\left( {\dfrac{{{G_{\max }}}}{{\parallel {{\bar v}^q}\parallel }} + } \right.\\ &\left. {q\parallel \bar v\parallel ^{q - 1}\dfrac{{\parallel e\parallel + {c_3}\parallel {\bar v}{\parallel ^p} + {c_4}\parallel {\bar v}\parallel ^q}}{{\parallel {{\bar v}^q}\parallel }}} \right) \le\\ &q\parallel {\bar v}\parallel ^{q - 1}(1 + \gamma )\left( {\dfrac{{{G_{\max }}}}{{q{N^{1 - q}}\parallel {\bar v}{\parallel ^{2q - 1}}}} + } \right.\\ &\left. {\dfrac{{{c_3}}}{{{N^{1 - q}}}}\parallel {\bar v}\parallel ^{p - q} + \dfrac{{{c_4}}}{{{N^{1 - q}}}} + \gamma } \right) \le\\ &q\parallel {\bar v}\parallel ^{q - 1}\left( {1 + \dfrac{{{G_{\max }}}}{{q{N^{1 - q}}\parallel \bar v{\parallel ^{2q - 1}}}} + } \right.\\ &{\left. {\dfrac{{{c_3}}}{{{N^{1 - q}}}}\parallel {\bar v}\parallel ^{p - q} + \dfrac{{{c_4}}}{{{N^{1 - q}}}} + \gamma } \right)^2} \end{split} $$ 考虑到
$\parallel \bar v\parallel = \sqrt {{{\bar v}^T}\bar v} = \sqrt {2{V_1}} \le \sqrt {2{V_1}(0)}$ , 因此$ 1 + $ $ \dfrac{{{G_{\max }}}}{{q{N^{1 - q}}\parallel \bar v{\parallel ^{2q - 1}}}} + \dfrac{{{c_3}}}{{{N^{1 - q}}}}\parallel \bar v{\parallel ^{p - q}} + \dfrac{{{c_4}}}{{{N^{1 - q}}}} $ 必定存在最大值, 假定其最大值为$ {\omega _2} $ , 上式可以写成:$$ \dot \gamma \le {\omega _1}{({\omega _2} + \gamma )^2} $$ (30) 其中,
$ {\omega _1} = q{(2{V_1}(0))^{{{(q - 1)} / 2}}} $ , 此$ {\dot \gamma _i}(t) $ 满足:$$ {\dot \gamma _i}(t) \le {\phi _i}(t,\phi _0^i) $$ (31) 其中,
$ {\phi _i}(t,\phi _0^i) $ 为下式的解:$$ {\dot \phi _i} = {\omega _1}{({\omega _2} + {\phi _i})^2},{\phi _i}(0,\phi _0^i) = \phi _0^i $$ (32) 上述等式的解为:
$$ {\phi _i}({\tau _i},0) = \frac{{{\tau _i}{\omega _1}\omega _2^2}}{{1 - {\tau _i}{\omega _1}{\omega _2}}} $$ (33) 由事件触发条件(17)得:
$$ \frac{{\parallel e\parallel }}{{\parallel \bar v\parallel ^q}} \le {\mu _{\min }}{c_4}{N^{\frac{{1 - q}}{2}}} $$ (34) 其中,
${\mu _{\min }} = \min \left\{ {{\mu _1},{\mu _2},\cdots,{\mu _N}} \right\}$ . 由引理3得:$$ \frac{{\parallel e\parallel }}{{\parallel {{\bar v}^q}\parallel }} \le \frac{1}{{{N^{1 - q}}}}\frac{{\parallel e\parallel }}{{\parallel \bar v{\parallel ^q}}} $$ (35) 于是式(33)可以写成:
$$ \frac{{\parallel e\parallel }}{{\parallel {{\bar v}^q}\parallel }} \le {\mu _{\min }}{c_4}{N^{\frac{{q - 1}}{2}}} $$ (36) 于是在时间间隔
$ [t_k^i,t_{k + 1}^i) $ 内, 有:$$ {\phi _i}({\tau _i},0) = {\mu _{\min }}{c_4}{N^{\frac{{q - 1}}{2}}} $$ (37) 结合等式(32)、(36),可得最小触发时间间隔:
$$ {\tau _i} = \frac{{{\mu _{\min }}{c_4}{N^{\frac{{q - 1}}{2}}}}}{{{\omega _1}\omega _2^2 + {\mu _i}{N^{\frac{{q - 1}}{2}}}{c_4}{\omega _1}{\omega _2}}} $$ (38) 当
$ t \in ({T_1},T] $ 时, 定义$ \ell = {{{\parallel E\parallel } / {\parallel y}}^\beta }\parallel $ , 每个时间段$ [t_h^i,t_{h + 1}^i) $ 内有:$$ \begin{split} \dot \ell =\;& \dfrac{{{{(E)}^{\rm{T}}}(E)'}}{{\parallel E\parallel \parallel {y^\beta }\parallel }} - \dfrac{{\parallel E\parallel }}{{\parallel {y^\beta }\parallel }}\dfrac{{{{({y^\beta })}^{\rm{T}}}({y^\beta })'}}{{\parallel {y^\beta }{\parallel ^2}}}\le\\ & \dfrac{1}{{\parallel {y^\beta }\parallel }}\left[ {{c_1}({y^\alpha })' + {c_2}({y^\beta })'} \right] + \dfrac{{\parallel E\parallel ({y^\beta })'}}{{\parallel {y^\beta }{\parallel ^2}}}=\\ &\dfrac{{{c_1}({y^\alpha })' + {c_2}({y^\beta })'}}{{\parallel {y^\beta }\parallel }} + \ell \dfrac{{({y^\beta })'}}{{\parallel {y^\beta }\parallel }}\le \\ &(1 + \ell )\dfrac{{{ c_1}({y^\alpha })' + (1 + {c_2})({y^\beta })'}}{{\parallel {y^\beta }\parallel }}=\\ &(1 + \ell )\dfrac{{{\alpha c_1}{y^{\alpha - 1}}y' + \beta(1 + {c_2}){y^{\beta - 1}}y'}}{{\parallel {y^\beta }\parallel }}\le\\ & \beta{\rm sign} (y')(1 + \ell )\left[ {{c_1}\parallel {y^{\alpha - 1}}\parallel + (1 + {c_2})\parallel {y^{\beta - 1}}\parallel } \right]\times\\ &\dfrac{{y'}}{{\parallel {y^\beta }\parallel }}\le\beta(1 + \ell )\parallel L\parallel \parallel S\parallel \bigg[ {\dfrac{{{c_1}}}{{\parallel y{\parallel ^{1 - \alpha }}}} + } \\ & {(1 + {c_2}){N^{2 - \beta }}\parallel y{\parallel ^{\beta - 1}}} \bigg]\bigg( {\ell + \dfrac{{{c_1}}}{{{N^{1 - \beta }}\parallel y{\parallel ^{\beta - \alpha }}}} + } \\ & {\dfrac{{{c_2}}}{{{N^{1 - \beta }}}}} \bigg)\le\beta\parallel L\parallel \parallel S\parallel \bigg[ {\dfrac{{{c_1}}}{{\parallel y{\parallel ^{1 - \alpha }}}} + } \\ & {(1 + {c_2}){N^{2 - \beta }}\parallel y{\parallel ^{\beta - 1}}} \bigg]\bigg( {\ell + \dfrac{{{c_1}}}{{{N^{1 - \beta }}\parallel y{\parallel ^{\beta - \alpha }}}} + } \\ &{ {\dfrac{{{c_2}}}{{{N^{1 - \beta }}}} + 1} \bigg)^2} \\[-15pt]\end{split} $$ (39) 由引理1,
$ \sum\nolimits_{i = 1}^N {y_i^2} = {({L^{{1/ 2}}}Sx)^{\rm{T}}}L({L^{{1 / 2}}}Sx) \le \\ {\lambda _{\max }}(L){x^{\rm{T}}}SLSx \le 2{\lambda _{\max }}(L){V_2} \le 2{\lambda _{\max }}(L){V_2}(0) $ 于是不难得到:
$$ \parallel y\parallel \le \sqrt {2{\lambda _{\max }}(L){V_2}(0)} $$ (40) 因此, 式子
$$ \beta\parallel L\parallel \parallel S\parallel \left[ {\frac{{{c_1}}}{{\parallel y{\parallel ^{1 - \alpha }}}} + (1 + {c_2}){N^{2 - \beta }}\parallel y{\parallel ^{\beta - 1}}} \right] $$ 以及
$ \dfrac{{{c_1}}}{{{N^{1 - \beta }}\parallel y{\parallel ^{\beta - \alpha }}}} + \dfrac{{{c_2}}}{{{N^{1 - \beta }}}} + 1 $ 必定存在最大值, 分别假定其最大值为$ {\chi _1},{\chi _2} $ , 结合式(39)、(40),可以得到:$$ \dot \ell \le {\chi _1}{(\ell + {\chi _2})^2} $$ (41) 以下证明类似于
$ t \in [0,{T_1}] $ 的情况, 可得最小触发时间间隔为:$$ {\tau '_i} = \frac{{{\varepsilon _{\min }}{c_2}}}{{{N^{1 - \beta }}{\chi _1}\chi _2^2 + {\varepsilon _i}{c_2}{\chi _1}{\chi _2}}} $$ (42) 其中,
${\varepsilon _{\min }} = \min \left\{ {{\varepsilon _1},{\varepsilon _2},\cdots,{\varepsilon _N}} \right\}$ . 综合上述论证, 在两个时间段内, 事件触发间隔都存在正下界. □3. 实例仿真
实例1. 考虑到多智能体系统由5个智能体组成, 5个智能体互连构成的连通拓扑图如图1所示.
由通信拓扑图不难得到Laplacian矩阵
$ L $ :$$ L = \left[ {\begin{array}{*{20}{c}} 2&{ - 1}&{ - 1}&0&0\\ { - 1}&3&{ - 1}&0&{ - 1}\\ { - 1}&{ - 1}&3&{ - 1}&0\\ 0&0&{ - 1}&2&{ - 1}\\ 0&{ - 1}&0&{ - 1}&2 \end{array}} \right] $$ 其中,
$ {\lambda _2}(L) = 1.38. $ 选定初始的状态为$ x(0) =[ - 0.5,$ $ - 0.3,0.1,0.2, - 0.1], $ 初始速度为$v(0) = [ - 0.2,0.1,$ $- 0.3, 0.2, - 0.1].$ 设定控制增益分别为:$ {c_1} = 0.22, $ ${c_2} = $ $ 1.2,$ $ {c_3} = 0.85 $ ,$ {c_4} = 0.4 $ . 比例参数设置为:$ {s_1} = - 1.3, $ $ {s_2} = - 1.3, $ $ {s_3} = 0.3, $ $ {s_4} = 0.3, $ $ {s_5} = 1. $ 其他需要设定的参数分别为:$ \alpha = 0.6, $ $ \beta = 1.8, $ $ p = 0.8, $ $ q = 1.1, $ $ {\mu _i} = 0.5, $ $ {\varepsilon _i} = 0.95. $ 由等式(19)不难得出${T_{1\max }} = $ $ 26.9\;{\rm{s}},$ 由等式(26)得$ {T_{2\max }} = 20.9\;{\rm{s}} . $ 系统总的收敛时间满足:$ T = {T_1} + {T_2} \le {T_{1\max }} +{T_{2\max }} =$ $ 47.8\;{\rm{s}}. $ 图2表明每个智能体最后收敛到不同状态. 由图2和图3知, 系统总体的收敛时间在10 s左右, 显然小于47.8 s. 由图4知, 当各个智能体追踪虚拟速度的误差趋近于零时, 所需要的时间小于5 s, 显然小于
${T_{1\max }}.$ 图5和图6表示智能体1在事件触发控制协议(10)及事件触发函数(12)下, 其误差范数的演化过程. 图5表示的是在
$ t \in [0,{T_1}] $ , 用基于速度信息的事件触发条件时, 智能体1误差范数的演化过程. 图6表示在$ t \in ({T_1},T] $ , 当多智能体系统完成虚拟速度追踪时, 切换为基于状态信息的事件触发条件, 智能体1的误差范数演化过程.
图 5 智能体1在触发条件(17)下的测量误差及阈值变化趋势Fig. 5 The evolution of the error norm and the threshold of agent 1 with trigger function (17)
图 6 智能体1在触发条件(25)下的测量误差及阈值变化趋势Fig. 6 The evolution of the error norm and the threshold of agent 1 with trigger function (25)图7中
$ i = 1,2, \cdots ,5 $ 为在控制策略(10)下, 各个智能体触发间隔;$ i = 6 $ 为在时间触发下, 每个智能体触发间隔. 图7表明本文所提出的事件触发控制策略在减小系统的能量耗散和控制器的更新频次的优越性.
图 7 各智能体在控制策略(10)下的触发间隔及在时间触发控制策略下的触发间隔Fig. 7 The triggered interval of each agent undercontrol scheme (10) and the trigger interval underthe time trigger control strategy图8给出了智能体1在控制协议(10)下, 与其它智能体之间的状态误差. 当系统达到稳定状态时, 各个智能体收敛于不同的值, 且满足既定的比例关系.
实例2. 为了证明本文给出的结果能够适应更为复杂的多智能体系统, 选用12个智能体组成的复杂多智能体系统, 12个智能体互连构成的连通拓扑图如图9所示. 其中, 取多智能体系统的初始状态、初始速度分别为
$x(0) = [ - 1,2, - 3,4, - 5,6,7, - 8, $ $ 9,10, - 11, - 7], $ $ v(0) = [ - 2, - 1,3,4, - 2, - 6,7,8,9, - 3,$ $ 3.5] $ . 设定控制增益分别为:$ {c_1} = 0.2 $ ,$ {c_2} = 0.3 $ ,${c_3} = $ $ 1.2 $ ,$ {c_4} = 0.4 $ .$ {\lambda _2}(L) = 0.558 $ . 其他需要设定的参数分别为:$ \alpha = 0.4 $ ,$ \beta = 1.8 $ ,$ p = 0.8 $ ,$ q = 1.1 $ ,$ {\mu _i} = 0.5 $ ,$ {\varepsilon _i} = 0.66 $ .由等式(19)不难得出
$ {T_{1\max }} = 32.7\;{\rm{s}} $ , 由等式(26)得$ {T_{2\max }} = 26.2\;{\rm{s}} . $ 系统总的收敛时间满足:$ T = {T_1} + {T_2} \le {T_{1\max }} + {T_{2\max }} = 58.9\;{\rm{s}} $ . 图10表明在控制策略(10)下, 多智能体系统能收敛至5个不同的子群. 图11表示各个智能体速度状态轨迹图. 而图12则表明智能体追踪虚拟速度的误差. 从图中可以看出, 智能体大约在4 s左右实现虚拟速度的追踪, 系统总体的收敛时间在10 s左右, 显然小于58.9 s.4. 结论
本文研究了基于事件触发二阶多智能体系统的固定时间比例一致性. 为使得系统状态收敛到不同状态, 设计了一种基于事件触发的固定时间非线性比例一致控制策略, 该控制协议包含一种基于状态信息和速度信息的分段式触发条件: 当多智能体系统处于追踪虚拟速度的状态时, 采用基于速度信息的事件触发条件; 当完成虚拟速度的追踪时, 采用基于状态信息的事件触发条件, 能有效的减小系统的能量耗散和控制器的更新频次. 利用Lyapunov稳定性理论、线性矩阵不等式和代数图论, 证明在该控制策略下, 二阶多智能体系统能实现固定时间比例一致性, 且不存在Zeno行为.
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图 1 新型工业内窥镜安装示意图((a) 新型工业内窥镜实际安装位置; (b) 新型工业内窥镜成像区域示意图)
Fig. 1 The schematic diagram of installation of the novel industrial endoscope ((a) Actual installation position of the novel industrial endoscope; (b) Schematic diagram of imaging area of the novel industrial endoscope)
图 2 不同高炉运行状态下的高炉料面图像
Fig. 2 BF burden surface images under different BF operation statuses
图 3 不同炉况下不同方法获取高温煤气流图像对比结果 ((a) 稳定1; (b) 稳定2; (c) 煤气流状态异常; (d) 悬料1; (e) 悬料2; (f) 高料位)
Fig. 3 Comparison results of high temperature gas flow images acquired by different methods under different BF conditions ((a) Stable 1; (b) Stable 2; (c) Abnormal gas flow status; (d) Hanging 1; (e) Hanging 2; (f) High stockline)
图 4 基于料面视频图像分析的高炉异常状态智能感知与识别框图
Fig. 4 Block diagram of intelligent BF anomalies perception and recognition via burden surface video image analysis
图 9 不同状态下高温煤气流图像多元特征降维结果((a) 正常状态; (b) 异常状态)
Fig. 9 The multi-feature results of high-temperature gas flow images from different BF statuses after dimensionality reduction ((a) Normal status; (b) Abnormal status)
图 10 不同状态下高温煤气流图像多元特征聚类结果((a) 正常状态; (b) 异常状态)
Fig. 10 The multi-feature results of high-temperature gas flow images from different BF statuses after clustering ((a) Normal status; (b) Abnormal status)
图 11 基于HGJM趋势变化的煤气流异常状态感知结果
Fig. 11 The perception results of gas flow anomaly based on HGJM trend change
图 12 不同分类网络模型训练与测试结果((a) 训练精度; (b) 训练损失; (c) 测试精度; (d) 测试损失)
Fig. 12 Training and test results of different classification network models ((a) Training accuracy; (b) Training loss; (c) Test accuracy; (d) Test loss)
表 1 高炉料面图像数据集
Table 1 Dataset of BF burden surface images
高炉状态 塌料 煤气流异常 悬料 正常 训练集 640 1920 960 1920 测试集 160 480 240 480 
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表 2 不同分类网络在高炉料面图像数据集下的识别结果
Table 2 Recognition results of different classification networks under BF burden surface image dataset
网络名称 异常状态检测率↑ 误报率↓ 速度(帧/s)↑ 塌料 煤气流异常 悬料 正常状态 ResNet18 100.00% 98.54% 99.58% 0.42% 42.94 VGG11 100.00% 98.33% 99.58% 2.29% 35.29 ViT16 100.00% 99.38% $\underline{98.75\%}$ 1.67% 23.32 SwinT-t 100.00% $\underline{93.96\%}$ 99.58% $\underline{3.96\%}$ $\underline{8.98}$ ResVGGNet 100.00% 99.30% 99.58% 0.21% 60.26 注: ↑ 表示指标越大越好, ↓ 表示指标越小越好, 粗体表示指标最优, 下划线表示指标最差.
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表 3 不同高炉异常状态识别方法对比
Table 3 Comparison among different BF anomaly recognition methods
类型 方法名称 悬料状态 正常状态 检测率↑ 误报率↓ 多元统计分析 CA $\underline{71.20\%}$ 4.60% MWPCA 96.45% 3.76% SFICVA 89.50% 3.00% L-DBKSSA 100.00% 1.24% A-DiASSA 92.80% 1.40% 深度学习 DSKL-SVM 100.00% $\underline{17.00\%}$ SD-DAE 93.77% 10.40% 料面图像法 所提方法 99.58% 0.21% 注: ↑ 表示指标越大越好, ↓ 表示指标越小越好, 粗体表示指标最优, 斜体表示指标第二优, 下划线表示指标最差.
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[1] Liu J S, Jiang Z H, Gui W H, Chen Z W. A novel particle size detection system based on RGB-laser fusion segmentation with feature dual-recalibration for blast furnace materials. IEEE Transactions on Industrial Electronics, 2023, 70(10): 10690−10699 doi: 10.1109/TIE.2022.3219054 [2] Zhu J L, Gui W H, Chen Z P, Jiang Z H. A novel non-contact and real-time blast furnace stockline detection method based on burden surface video streams. IEEE Transactions on Instrumentation and Measurement, 2023, 72: Article No. 4502213 [3] 周平, 刘记平, 梁梦圆, 张瑞垚. 基于KPLS 鲁棒重构误差的高炉燃料比监测与异常识别. 自动化学报, 2021, 47(7): 1661−1671Zhou Ping, Liu Ji-Ping, Liang Meng-Yuan, Zhang Rui-Yao. KPLS robust reconstruction error-based monitoring and anomaly identification of fuel ratio in blast furnace ironmaking. Acta Automatica Sinica, 2021, 47(7): 1661−1671 [4] Jiang K, Jiang Z H, Xie Y F, Pan D, Gui W H. Abnormality monitoring in the blast furnace ironmaking process based on stacked dynamic target-driven denoising autoencoders. IEEE Transactions on Industrial Informatics, 2022, 18(3): 1854−1863 doi: 10.1109/TII.2021.3084911 [5] Takatani K, Iwanaga Y. Fundamental study on thermal degradation of coke during rapid heating. Tetsu-to-Hagane, 1986, 72(2): 189−194 doi: 10.2355/tetsutohagane1955.72.2_189 [6] 李骏峰. 高炉炉体热流强度与炉况顺行的研究 [硕士学位论文], 昆明理工大学, 中国, 2014.Li Jun-Feng. Study on Heat Flux Intensity of Blast Furnace Body and Smooth Operation of Blast Furnace Condition [Master thesis], Kunming University of Science and Technology, China, 2014. [7] Zhou P, Li H L, Shi P Y, Zhou C Q. Simulation of the transfer process in the blast furnace shaft with layered burden. Applied Thermal Engineering, 2016, 95: 296−302 doi: 10.1016/j.applthermaleng.2015.11.004 [8] 陈建华, 徐红阳. “高炉专家系统”应用现状和发展趋势. 现代冶金, 2012, 40(3): 6−10Chen Jian-Hua, Xu Hong-Yang. Application status and development trend of “blast furnace expert system”. Modern Metallurgy, 2012, 40(3): 6−10 [9] Dong Y N, Qin S J. A novel dynamic PCA algorithm for dynamic data modeling and process monitoring. Journal of Process Control, 2018, 67: 1−11 doi: 10.1016/j.jprocont.2017.05.002 [10] Zhou P, Zhang R Y, Xie J, Liu J P, Wang H, Chai T Y. Data-driven monitoring and diagnosing of abnormal furnace conditions in blast furnace ironmaking: An integrated PCA-ICA method. IEEE Transactions on Industrial Electronics, 2021, 68(1): 622−631 doi: 10.1109/TIE.2020.2967708 [11] Chen Q, Kruger U, Leung A Y T. Cointegration testing method for monitoring nonstationary processes. Industrial and Engineering Chemistry Research, 2009, 48(7): 3533−3543 [12] Lou S W, Wu P, Yang C J, Xu Y H. Structured fault information-aided canonical variate analysis model for dynamic process monitoring. Journal of Process Control, 2023, 124: 54−69 doi: 10.1016/j.jprocont.2023.01.011 [13] 赵明. 基于神经网络的高炉炉况诊断与预报研究 [硕士学位论文], 东北大学, 中国, 2010.Zhao Ming. Study on Diagnose and Forecast of Blast Furnace Condition Based on Neural Network [Master thesis], Northeastern University, China, 2010. [14] Gao C H, Jian L, Luo S H. Modeling of the thermal state change of blast furnace hearth with support vector machines. IEEE Transactions on Industrial Electronics, 2012, 59(2): 1134−1145 [15] Luo S H, Dai Z A, Chen T X, Chen H Y, Jian L. A weighted SVM ensemble predictor based on AdaBoost for blast furnace ironmaking process. Applied Intelligence, 2020, 50: 1997−2008 doi: 10.1007/s10489-020-01662-y [16] Li J P, Hua C C, Yang Y N, Guan X P. Bayesian block structure sparse based T-S fuzzy modeling for dynamic prediction of hot metal silicon content in the blast furnace. IEEE Transactions on Industrial Electronics, 2018, 65(6): 4933−4942 doi: 10.1109/TIE.2017.2772141 [17] 赵辉, 赵德涛, 岳有军, 王红君. 基于深度置信网络的高炉炉况故障分类方法的研究. 铸造技术, 2018, 39(5): 1028−1032Zhao Hui, Zhao De-Tao, Yue You-Jun, Wang Hong-Jun. Research on fault classification of blast furnace condition based on deep belief network. Foundry Technology, 2018, 39(5): 1028−1032 [18] Lou S W, Yang C J, Wu P, Kong L Y, Xu Y H. Fault diagnosis of blast furnace iron-making process with a novel deep stationary kernel learning support vector machine approach. IEEE Transactions on Instrumentation and Measurement, 2022, 71: Article No. 3521913 [19] Han Y H, Li Q, Wang C, Zhao Q. A novel knowledge enhanced graph neural networks for fault diagnosis with application to blast furnace process safety. Process Safety and Environmental Protection, 2022, 166: 143−157 doi: 10.1016/j.psep.2022.08.014 [20] Lou S W, Yang C J, Wu P. A local dynamic broad kernel stationary subspace analysis for monitoring blast furnace ironmaking process. IEEE Transactions on Industrial Informatics, 2023, 19(4): 5945−5955 [21] Lou S W, Yang C J, Zhu X Z, Zhang H W, Wu P. Adaptive dynamic inferential analytic stationary subspace analysis: A novel method for fault detection in blast furnace ironmaking process. Information Sciences, 2023, 642: Article No. 119176 doi: 10.1016/j.ins.2023.119176 [22] Shi L, Wen Y B, Zhao G S, Yu T. Recognition of blast furnace gas flow center distribution based on infrared image processing. Journal of Iron and Steel Research International, 2016, 23(3): 203−209 doi: 10.1016/S1006-706X(16)30035-8 [23] Pan D, Jiang Z H, Chen Z P, Jiang K, Gui W H. Compensation method for molten iron temperature measurement based on heterogeneous features of infrared thermal images. IEEE Transactions on Industrial Informatics, 2020, 16(11): 7056−7066 doi: 10.1109/TII.2020.2972332 [24] He L, Jiang Z H, Xie Y F, Chen Z P, Gui W H. Velocity measurement of blast furnace molten iron based on mixed morphological features of boundary pixel sets. IEEE Transactions on Instrumentation and Measurement, 2021, 70: Article No. 5018312 [25] He L, Jiang Z H, Xie Y F, Gui W H, Chen Z P. Mass flow measurement of molten iron from blast furnace, based on trusted region stacking using single high-speed camera. IEEE Transactions on Instrumentation and Measurement, 2021, 70: Article No. 5013011 [26] Li Q, Wang Z J, Wang S, Li M M, Lei H, Zou Z S. A deep learning-based diagnosis model driven by tuyere images big data for iron-making blast furnaces. Steel Research International, 2022, 93(8): Article No. 2100826 doi: 10.1002/srin.202100826 [27] Yi Z H, Jiang Z H, Huang J C, Chen X F, Gui W H. Optimization method of the installation direction of industrial endoscopes for increasing the imaged burden surface area in blast furnaces. IEEE Transactions on Industrial Informatics, 2022, 18(11): 7729−7740 doi: 10.1109/TII.2022.3151747 [28] Xu T X, Chen Z P, Jiang Z H, Huang J C, Gui W H. A real-time 3D measurement system for the blast furnace burden surface using high-temperature industrial endoscope. Sensors, 2020, 20(3): Article No. 869 doi: 10.3390/s20030869 [29] 赵志坚, 佘雪峰, 赵奕喆, 刘燕军, 王艾军, 李丽红, 等. 高炉料面结构对煤气流分布影响的模拟. 材料与冶金学报, 2023, 22(1): 38−44Zhao Zhi-Jian, She Xue-Feng, Zhao Yi-Zhe, Liu Yan-Jun, Wang Ai-Jun, Li Li-Long, et al. Numerical simulation study on the influence of burden profile on gas distribution in blast furnace. Journal of Materials and Metallurgy, 2023, 22(1): 38−44 [30] 郭恒光, 朱默, 赵亮. 基于多尺度方向分解的纹理特征提取方法研究. 计算技术与自动化, 2021, 40(4): 102−107Guo Heng-Guang, Zhu Mo, Zhao Liang. Texture feature extraction method based on multiscale orientation decomposition. Computing Technology and Automation, 2021, 40(4): 102−107 [31] Guo W H, Xu G X, Liu B D, Wang Y J. Hyperspectral image classification using CNN-enhanced multi-level Haar wavelet features fusion network. IEEE Geoscience and Remote Sensing Letters, 2022, 19: Article No. 6008805 [32] Shi Z H, Wu D R, Guo C F, Zhao C M, Cui Y Q, Wang F Y. FCM-RDpA: TSK fuzzy regression model construction using fuzzy C-means clustering, regularization, DropRule, and Powerball AdaBelief. Information Sciences, 2021, 574: 490−504 doi: 10.1016/j.ins.2021.05.084 [33] Kirillov A, Mintun E, Ravi N, Mao H Z, Rolland C, Gustafson L, et al. Segment anything. arXiv preprint arXiv: 2304.02643, 2023. [34] 朱叶, 申铉京, 陈海鹏. 基于彩色LBP的隐蔽性复制−粘贴篡改盲鉴别算法. 自动化学报, 2017, 43(3): 390−397Zhu Ye, Shen Xuan-Jing, Chen Hai-Peng. Covert copy-move forgery detection based on color LBP. Acta Automatica Sinica, 2017, 43(3): 390−397 [35] Hosny K M, Darwish M M, Aboelenen T. Novel fractional-order generic Jacobi-Fourier moments for image analysis. Signal Processing, 2020, 172: Article No. 107545 [36] 廖陆峰, 李思坤, 王向朝. 基于预训练VGG11模型的光刻坏点检测方法. 光学学报, 2023, 43(3): 140−149Liao Lu-Feng, Li Si-Kun, Wang Xiang-Chao. Bad spot detection method of lithography based on pre-trained VGG11 model. Journal of Optics, 2023, 43(3): 140−149 [37] Kakarwal S, Paithane P. Automatic pancreas segmentation using ResNet-18 deep learning approach. System Research and Information Technologies, DOI: https://doi.org/10.20535/SRIT.2308-8893.2022.2.08 [38] Zhang T, Qi G J, Xiao B, Wang J D. Interleaved group convolutions. In: Proceedings of the IEEE International Conference on Computer Vision (ICCV). Venice, Italy: IEEE, 2017. 4383−4392 [39] Mao X F, Qi G G, Chen Y F, Li X D, Duan R J, Ye S K, et al. Towards robust vision Transformer. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). New Orleans, USA: IEEE, 2022. 12032−12041 [40] Liu Z, Lin Y T, Cao Y, Hu H, Wei Y X, Zhang Z, et al. Swin Transformer: Hierarchical vision Transformer using shifted windows. In: Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV). Montreal, Canada: IEEE, 2021. 9992−10002 [41] Wang X, Kruger U, Irwin G W. Process monitoring approach using fast moving window PCA. Industrial and Engineering Chemistry Research, 2005, 44(15): 5691−5702 doi: 10.1021/ie048873f -
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