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摘要: 多机协同围捕作为多机器人协同领域的一项重要分支, 着重研究多个机器人通过相互协作对动态可疑目标实现有效的追踪与围捕, 在军事侦查、紧急救援、协同探测等领域具有重要的研究意义与实际应用价值. 首先通过国内外科学引文数据库对多机协同围捕领域相关的文献进行全面检索, 深入剖析目前该领域前沿技术的发展现状与研究热点. 接下来从理论与技术层面分别针对多机协同围捕领域中的目标协同搜索、多机任务分配、协同围捕控制等方面进行全面总结, 重点阐述各研究内容常用方法与技术的工作原理、优缺点及适用范围等. 最后对该领域的发展现状进行总结, 并分析探讨目前尚未解决的难点, 对未来的发展方向提出展望.Abstract: As a significant branch of multi-robot coordination, multi-robot cooperative hunting mainly focuses on tracking and capturing dynamic suspicious targets effectively through cooperation. It has important significance and has been applied in various fields, such as military reconnaissance, emergency rescue, and collaborative detection. This paper conducts a comprehensive search of relevant literature in the field of multi-robot cooperative hunting through domestic and foreign scientific citation databases. It then thoroughly analyzes the current development status and research hotspots of frontier technologies in this field. Following this, the paper offers a thorough summary of research in the field of multi-robot cooperative hunting, covering theoretical and technical aspects, which concentrates on target cooperative search, multi-robot task allocation, and cooperative hunting control, etc.. The working principles, advantages, disadvantages, and application ranges of commonly used methods and technologies in each research aspect are introduced in detail. Finally, this paper summarizes the current state of development and unresolved challenges in this field, and suggests potential directions for future development.
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实际复杂工业过程如高炉炼铁过程、磨矿过程、造纸制浆过程、污水处理过程等, 通常涉及复杂的物理化学反应, 具有多变量、强耦合、非线性、大滞后等综合复杂动态特性, 利用传统的物理化学等机理方法时, 难以建立精确的数学模型[1-4]. 近年来, 随着大数据和人工智能技术的发展, 对于难以进行机理建模的工业过程, 甚至缺乏机理模型的复杂工业系统, 数据驱动建模常被看作是一种非常有效的建模方法[1, 4-5]. 目前, 数据驱动建模主要采用人工智能技术或多元统计分析技术来描述过程输入与输出之间的复杂未知动态关系, 在此基础上建立具有一定结构和适当模型参数的过程数据模型. 由于数据模型输出与实际过程输出之间存在一定的偏差, 为了最优化模型性能, 通常需要采用相关算法来优化关于建模误差的性能指标, 如均方根误差(Root mean square error, RMSE)、均方差(MSE)及平均绝对误差(Mean absolute error, MAE)等, 以此获得满意的数据模型参数[6].
在实际工业过程建模时, 单纯的RMSE、MSE、MAE等性能指标均以最小化一维的统计建模误差均值为目标, 并不能完全描述和刻画动态系统建模过程的随机性和不确定性[6]. 在复杂工业过程中, 外部不确定因素和随机动态干扰往往具有非高斯特性, 将其假设为高斯分布特征时, 难以获得满意的建模效果[7]. 另一方面, 复杂工业过程中, 其建模误差通常是一个未知的随机变量, 因而建模误差在时空尺度上的二维概率密度函数(Probability density function, PDF)形状分布能够包含动态系统建模误差的所有分布和统计信息. 因此, 误差PDF形状优化的思想受到了越来越广泛的关注, 并逐渐用于复杂工业过程的数据建模与控制中, 如以非高斯动态系统PDF形状为目标的随机分布控制已成为解决随机动态系统控制的非常有效的方法, 得到了广泛的应用[7-8]. 近年, 面向建模误差PDF形状优化的数据建模方法也逐渐引起重视. 文献[6]中, 作为有界随机分布系统建模与控制方法的扩展, Zhou等将输出PDF控制或随机分布控制思想引入到过程建模中. 通过优化建模误差PDF形状, 进而对模型参数求解, 使得实际建模误差PDF形状接近设定的期望PDF形状[6]. 该方法不仅可以获得较为满意的建模效果, 而且一定程度降低建模过程中的随机性和不确定性. 此外, 文献[9]通过优化建模误差PDF形状, 建立了选矿过程精矿品位的最小支持向量机模型. 而文献[10]通过优化建模误差PDF形状, 间接对模型参数进行调节, 建立了间歇过程的模糊神经网络模型.
上述方法均是期望实际模型的建模误差PDF形状更好地跟踪期望的高斯分布形状, 以此建立具有最优参数的过程数据模型. 然而, 不管是常规建模方法的误差RMSE指标, 还是上述改进方法提到的建模误差PDF指标, 均仅仅体现过程模型输出与实际输出之间的误差大小情况, 难以衡量模型输出与实际动态过程输出之间拟合趋势是否一致. 实际上, 实际工业动态系统中, 过程输出变化趋势的估计和预测, 对于基于模型的预测控制、生产过程运行态势的把握与调控等诸多工程应用, 都具有十分重要的作用. 因此, 在动态系统建模时, 除了需要优化建模误差的PDF形状, 同时也需要考虑建模输出与样本数据之间拟合趋势最接近, 即曲线拟合动态变化趋势的相似度最大[11].
针对上述动态系统建模的实际需求和现有方法的不足, 本文以小波神经网络(Wavelet neural network, WNN)[12-13]数据建模为例, 提出一种新型的面向建模误差PDF形状与趋势拟合优度(相似度)多目标优化的动态系统数据建模方法. 所提方法不仅引入二维尺度的PDF指标来对动态建模误差在时间和空间进行全面刻画, 同时引入拟合优度(相似度)指标[11, 14]刻画动态系统数据建模的拟合趋势. 通过采用核密度估计(Kernel density estimation, KDE)[15-17]技术对实际建模误差PDF形状进行估计, 以及采用NSGA-II算法[18]对建模误差PDF形状的偏差以及拟合优度指标进行多目标优化, 从而建立具有最优模型参数的WNN模型. 数值仿真以及污水处理过程[19-20]数据验证表明所提方法的实际建模误差PDF能够更好地逼近设定的期望PDF, 并且模型输出与样本数据拟合趋势接近.
1. 建模思路与策略
1.1 WNN简介
小波神经网络是结合小波分析与神经网络的一种前馈型网络. WNN用小波函数代替传统Sigmoid函数作为激励函数, 通过仿射变换建立起小波变换与网络参数之间的连接, 能以任意精度对函数进行逼近[12-13]. 如图1所示, WNN通常采用三层网络结构, 其中:
$ {x_1,x_2,\cdots,x_M} $ 为WNN的输入变量,$ {y_1,y_2, \cdots,y_N} $ 为输出变量,${\omega _{{{I}},ji}}$ 和${\omega _{{{H}},lj}}$ 分别是输入层到隐含层的连接权值以及隐含层到输出层的连接权值,$ {\theta_H,_j} , {\theta_O,_l} $ 分别为隐含层和输出层节点阈值,$ {a_j} $ 和$ {b_j} $ 分别为小波基函数的伸缩因子和平移因子.WNN隐层激励函数
$ {\phi(t)} $ 通常采用如下的Morlet母小波函数[12]:$$ {\phi(t)} = \cos(1.75t)\exp\left(-{\frac{t^2}{2}}\right) $$ (1) 输出层激励函数
$ {f(t)} $ 则采用Sigmoid函数, 即:$$ f(t) = {(1 + {{\rm e}^{ - t}})^{ - 1}} $$ (2) 此外, 定义三层WNN的输入层、隐含层、输出层节点数分别为M, n和N, 则隐含层第j个节点的输入
$ {\rm{ne}}{{\rm{t}}_j} $ 和输出$ {z_j} $ 分别为:$$ \begin{split} &{ne{t_j} = \sum\limits_i {{\omega _{I,ji}}} {x_i} - {\theta _{H,j}},i = 1, \cdots ,M,}\\ &\qquad\qquad{j = 1,2, \cdots ,n} \end{split} $$ (3) $$ {z_j} = \varphi \left(\frac{{ne{t_j} - {b_j}}}{{{a_j}}}\right),\qquad j = 1, \cdots ,n $$ (4) 式中,
$ {b_j} $ 为小波基函数$ \varphi (t) $ 的平移因子,$ {a_j} $ 为小波基函数的伸缩因子,$ \varphi (t) $ 为Morlet母小波函数,$ {\theta _{H,\;j}} $ 为隐含层节点的阈值, 则WNN的最终输出为:$$ \begin{split} &{{{\hat y}_l} = f\left( {\sum\limits_j {{\omega _{H,lj}}} {z_j} - {\theta _{O,l}}} \right),}\\ &{j = 1, \cdots ,n,l = 1, \cdots ,N} \end{split} $$ (5) 式中,
$ {\theta _{O,\;l}} $ 为输出层节点的阈值, 函数$ f( \cdot ) $ 为输出层的激励函数.1.2 建模策略
对于常规WNN等现有多数建模方法, 通常采用如下均方根误差(RMSE)、均方差(MSE)、平均绝对误差(MAE)等单一的误差性能指标, 通过性能指标数值大小来评价建模精度.
$$ \left\{ \begin{array}{l} {J_{{\rm{RMSE}}}} = \sqrt {\dfrac{1}{m}\displaystyle\sum\limits_{l = 1}^m {{{({y_l} - {{\hat y}_l})}^2}} } \\ {J_{{\rm{MSE}}}} = \dfrac{1}{m}\displaystyle\sum\limits_{l = 1}^m {{{({y_l} - {{\hat y}_l})}^2}} \\ {J_{{\rm{MAE}}}} = \dfrac{1}{m}\displaystyle\sum\limits_{l = 1}^m {\left| {{y_l} - {{\hat y}_l}} \right|} \end{array} \right. $$ (6) 然而, 式(6)所示传统性能指标是从建模误差的均值角度评价模型精度, 并不能全面描述动态系统建模误差在时空尺度上的随机特性. 此外, 对于时序相关动态系统建模, 运行数据拟合趋势的估计对于建模效果有很大影响, 并且更有实际意义. 而式(6)所示常规建模性能评价指标仅希望建模输出与实际数据之间偏差最小, 却难以描述动态系统的拟合趋势好坏.
为了解决上述问题, 本文以WNN智能建模为基础, 通过引入建模误差概率密度函数(PDF)指标从时空二维角度对建模误差进行全面刻画, 以及引入拟合优度指标对动态系统数据建模的拟合趋势进行相似性评估, 从而提出图2所示的面向建模误差PDF形状与趋势拟合优度的动态系统优化建模方法, 具体如下:
图 2 面向建模误差PDF形状与趋势拟合优度的优化建模策略Fig. 2 Optimized modeling strategy towards modeling error PDF shape and goodness of fit1)首先, 构建动态系统数据建模的实际建模误差PDF与期望建模误差PDF的偏差平方积分作为多目标优化计算的第一个评价指标, 如下所示:
$$ {J_1} = \int_{ - \infty }^{ + \infty } {{{(\Gamma (e) - {\Gamma _{\rm target}}(e))}^2}{\rm d}e} $$ (7) 式中,
$ \Gamma(e) $ 和${\Gamma _{\rm target}}(e)$ 分别为实际建模误差PDF和期望建模误差PDF. 本文实际建模误差PDF是采用核密度估计技术对所建立数据模型的建模误差序列进行求解获得, 而期望建模误差PDF是设置的一个较为理想的(即均值为0、方差尽量小)高斯分布形状的PDF, e为建模误差PDF的自变量.2)其次, 引入式(8)所示的拟合优度指标
$ {\rho _{AB}} $ [11, 14]对动态系统数据建模的动态拟合趋势进行相似性评估, 然后构建式(9)所示关于拟合优度的性能指标作为第二个评价指标.$$ {\rho _{AB}} = \frac{{\sum\limits_m {\sum\limits_n {({A_{mn}} - \bar A)} } ({B_{mn}} - \bar B)}}{{\sqrt {\left( {{{\sum\limits_m {\sum\limits_n {({A_{mn}} - \bar A)} } }^2}} \right)\left( {{{\sum\limits_m {\sum\limits_n {({B_{mn}} - \bar B)} } }^2}} \right)} }} $$ (8) $$ {J_2} = \frac{1}{\rho_{AB}} $$ (9) 式中,
$ A,B $ 为两个数据矩阵,$ \bar A,\bar B $ 分别为数据矩阵$ A,B $ 的均值. 事实上,$ {\rho _{AB}} $ 是衡量数据矩阵A和B之间近似程度的量,$ \left| {{\rho _{AB}}} \right| \to 1 $ 表示数据矩阵A和B之间相关性很强, 而$ \left| {{\rho _{AB}}} \right| \to 0 $ 意味着数据矩阵A和B之间相关性较弱. 由于本文要衡量建模输出与实际输出的时序相关数据之间的动态拟合趋势, 所以本文式(8)中A和B分别表示小波神经网络模型输出和实际过程输出所构成的时序相关数据矩阵.3)最后, 分别将式(8)和式(9)作为数据建模的综合性能评价指标的适应度函数, 采用运算速度快、解集收敛性好的NSGA-II算法[11]来获得WNN模型的最优参数集
$[{\omega _{{{I}},ji}},{\omega _{{{H}},lj}},{a_j},{b_j},{\theta _{{{H,}}j}},{\theta _{{{O,}}l}}]$ .2. 建模算法
所提方法的具体建模算法包括如下几个过程: 首先,采用式
$(1) \sim (5)$ 所示算法构建初始的WNN数据模型, 通过比较WNN数据模型输出与过程输出或者相应的实际值, 可以得到特定时间内的建模误差序列. 然后, 采用第3.1节的核密度估计技术对实际建模误差PDF进行计算. 最后, 采用NSGA-II算法优化式(8)和式(9)所示的多目标性能指标, 获得同时具有较好建模误差PDF形状与拟合优度值的多组WNN模型参数集解.2.1 建模误差PDF核密度估计
核密度估计(Kernel density estimation, KDE)是由Parzen提出的一种非参数估计方法[15-17], 用于求解给定随机变量数据集合分布的概率密度函数. 假设
$ {x_i} \in R,i = 1, \cdots,n $ 为独立同分布的随机变量数据集, 其所服从的分布密度函数为$f(x),x \in {\bf R},$ 则$ f(x) $ 的核密度估计$ {\hat f_h}(x) $ 定义如下:$$ {\hat f_h}(x) = \frac{1}{{n{h_p}}}\sum\limits_{i = 1}^n {\phi \left(\frac{{{x_i} - x}}{{{h_p}}}\right), \quad x \in {\bf R}} $$ (10) 式中, 窗宽
$ {h_p} $ 是一个给定的正数,$ \phi (x) $ 为核函数, n为样本数.对于所提建模方法, 采用KDE对建模误差PDF进行估计, 可以得到估计的建模误差概率密度函数
$ {\Gamma _e} $ 为:$$ {\Gamma _e} = \frac{1}{K}\sum\limits_{k = 1}^K {\frac{1}{{{h_p}}}\phi \left( - \frac{{e - e(k)}}{{{h_p}}}\right)} $$ (11) 式中, K为建模误差样本数目, 通过设置期望的建模误差PDF, 可以构造式(7)所示的性能指标. 式(11)所示基于KDE的WNN实际建模误差PDF估计求解步骤如下:
1)选择核函数: 在估计随机变量未知概率密度函数时, 常用的核函数有高斯核函数、矩形窗核函数、Epanechnikov核函数等. 核函数的不同选择在KDE中不敏感, 当样本数据很大时, 对核函数密度估计的结果影响不大. 本文选取高斯核函数, 其表式如下:
$$ \phi (x) = \frac{1}{{\sqrt {2\pi } }}{{\rm e}^{ - \frac{{{{\left\| x \right\|}^2}}}{2}}} $$ (12) 2)选择窗宽: 窗宽
$ {h_p} $ 的选择对核函数的密度估计起着局部光滑的作用, 如果$ {h_p} $ 过大会使模型误差PDF形状很光滑, 使其主要部分的某些特征(如多峰性)被掩盖起来, 从而增加估计量的偏差; 而若$ {h_p} $ 过小, 则整个密度函数表现粗糙. 本文基于正态参照规则方法[17]进行窗宽选择, 假设建模误差服从正态分布, 则窗宽$ {h_p} $ 设置为$ {h_p} = 1.06\sigma {K^{ - 1/5}} $ 其中$ \sigma $ 由$ \min \{ s,{Q / {1.34}}\} $ 估计, s表示样本标准差, Q为四分位数间距.3)求解模型误差PDF: 根据步骤1)和2)选择合适窗函数和窗宽参数, 然后代入式(11), 可以得到WNN建模误差PDF函数的估计值为:
$$ \begin{split} {\Gamma _e} =& \left( {x(k),y(k),{\theta _{WNN}},\varphi (e),{h_p}} \right)=\\ & \frac{1}{K}\sum\limits_{k = 1}^K {\frac{1}{{{h_p}}}\varphi \left( - \frac{{e - \Theta \left( {x(k),y(k),{\theta _{WNN}}} \right)}}{{{h_p}}}\right)} \end{split} $$ (13) 2.2 模型参数多目标优化求解
式(1) ~ (5)所示基本WNN数据模型的参数主要包括: 输入层连接权值
$ {\omega _{I,ji}} $ 、隐含层连接权值$ {\omega _{H,lj}} $ 、隐含层阈值$ {\theta _{H,j}} $ 、输出层阈值$ {\theta _{O,l}} $ 、小波基函数的伸缩因子$ {a_j} $ 以及平移因子$ {b_j} $ . 这些参数的取值直接决定了WNN数据模型的性能, 因而基于前述构建的多目标建模性能指标, 采用NSGA-II算法对模型参数进行优化, 步骤如下:1)网络参数的编码. 将WNN模型参数集
$ {\theta _{WNN}} = $ $[{\omega _{I,ji}},{\omega _{H,lj}},{a_j},{b_j},{\theta _{H,j}},{\theta _{O,l}}] $ 与每条染色体相对应, 即对WNN模型参数进行如下形式的编码:$$ \begin{split} R =\;& [{\omega _{I,11}}, \cdots ,{\omega _{I,1n}}, \cdots ,{\omega _{I,Mn}},\\ &{\omega _{H,11}}, \cdots ,{\omega _{H,1m}}, \cdots ,{\omega _{H,nN}},\\ &{a_1}, \cdots ,{a_n},{b_1}, \cdots ,{b_n},{\theta _{H,1}}, \cdots ,\\ &{\theta _{H,n}},{\theta _{O,1}}, \cdots ,{\theta _{O,n}}] \end{split} $$ 式中, 染色体基因数为S = (M + 3)n + (n + 1)N,
$P = {[{S_1},{S_2}, \cdots ,{S_i}, \cdots ,{S_Q}]^{\rm T}}$ 表示包含Q条染色体的初始种群.2)个体适应度计算. 每条染色体中的各个基因分别代表WNN的各个参数, 将第t代种群中第h条染色体上的各个基因代入下式的第t代、第h个个体多适应度函数中:
$$ F_{C1}^{t,h} = \int_{ - \infty }^{ + \infty } {{{(\Gamma (e) - {\Gamma _{\rm target}}(e))}^2}{\rm d}e} $$ (14) $$ F_{C2}^{t,h} = \frac{1}{\rho _{{{AB}}}} $$ (15) 3)选择算子. 根据非支配排序结果, 选择非支配排序中支配层较低的个体. 如果有多个个体在同一支配层, 从种群多样性角度考虑, 选择拥挤度距离较大的个体.
4) 模拟二进制交叉. 基于实数编码, 交叉后代为父代的线性组合, 即
$$ \begin{split}& G_{1,i}^{t + 1} = 0.5\left[ {(1 - {\beta _k}(\varepsilon ))G_{1,i}^t + (1 + {\beta _k}(\varepsilon ))G_{2,i}^t} \right]\\ &G_{2,i}^{t + 1} = 0.5\left[ {(1 + {\beta _k}(\varepsilon ))G_{1,i}^t + (1 - {\beta _k}(\varepsilon ))G_{2,i}^t} \right] \end{split} $$ 式中,
$ \varepsilon $ 为在(0,1)内服从均匀分布的随机数. 当$ \varepsilon > $ $ 0.5 $ 时$ {\beta _k}(\varepsilon ) = {({[2(1 - \varepsilon )]^{{{({\eta _c} + 1)}^{ - 1}}}})^{ - 1}} ,$ 当$ \varepsilon \le 0.5$ 时,$ {\beta _k}(\varepsilon ) = {(2\varepsilon )^{{{({\eta _c} + 1)}^{ - 1}}}} , {\eta _c} $ 为交叉分布指数,$ i = 1,2 $ 为目标函数的个数.5) 多项式变异. 二进制交叉后, 进行多项式变异, 变异后的个体为:
$$ G_i^{t + 1} = G_i^{t + 1} + ({B^U} - {B^L}){\delta _k} $$ 式中,
$ {B^U},{B^L} $ 分别为优化变量的上、下限,$ {\delta _k} $ 为变异参数. 当$ {r_k} > 0.5 $ 时,$ {\delta _k} = {(2{r_k})^{{{({\eta _m} + 1)}^{ - 1}}}} ,$ 而当$ {r_k} \le $ $ 0.5 $ 时,$ {\delta _k} = {(1 - [2(1 - {r_k})])^{{{({\eta _m} + 1)}^{ - 1}}}} , {r_k} $ 为在(0,1)服从均匀分布的随机数,$ {\eta _m} $ 为变异分布指数.采用NSGA-II算法优化WNN模型参数时, 每个待优化的参数对应染色体上的一个基因. 在遗传算法中, 适应度函数的选择决定着遗传优化的精度和收敛速度. 描述个体性能的指标主要通过适应度函数值体现, 依据适应度值的大小对个体进行优胜劣汰. 本文多目标适应度函数为实际建模误差PDF与期望建模误差PDF之间的二维偏差平方和以及趋势拟合优度的倒数, 并通过基因之间的选择、二进制交叉、变异产生最优个体即最优模型参数.
3. 数值仿真及工业数据验证
3.1 数值仿真
为了验证所提方法的有效性和优越性, 首先使用下述两输入一输出非线性动态系统进行数值验证:
$$ y(k + 1) = {u^3}(k) + \frac{{y(k)}}{{1 + {y^2}(k)}} + \omega (k) $$ (16) 式中,
$ y(0) = 0.1, u(k) $ 为在区间(0, 1)内服从均匀分布的随机序列,$ \omega (k) $ 为通过参数$ \sigma $ 描述的、服从瑞利分布的非高斯随机干扰序列. 针对以上非线性系统, 利用提出的建模方法进行建模, 所要建立的WNN数据模型可以表示为:$$ \tilde y(k + 1) = {f_{WNN}}(y(k),u(k),\;\omega (k),{\theta _{WNN}}) $$ 假设
$ \omega $ 为随机产生、服从瑞利分布且参数为0.2的非高斯干扰. WNN隐层节点数选择为6, 迭代优化步长r为0.003. 采用NSGA-II算法对WNN模型参数进行寻优时, 交叉分布指数${\eta _c} = 20 ,$ 变异分布指数$ {\eta _m}{\rm{ = }}20, $ 优化变量的上限与下限分别设定为1和−1, 交叉率和突变率分别设为0.9和0.1.建模后, 得到60组Pareto前沿解进化过程如图3所示, 而图4为60组多目标优化解对应的拟合优度变化曲线, 这里将所提方法与常规WNN方法以及近年文献[6]中提出的面向建模误差PDF优化的WNN方法进行比较. 由于文献[6]是采用梯度下降方法来优化WNN模型的建模误差PDF, 因而本文将其称为GD-WNN. 从图4可以看出, 采用所提建模方法可以获得具有较大动态变化趋势拟合优度的一组解集, 这些解对应的拟合优度均远好于常规WNN方法以及GD-WNN方法. 所提方法得到的解对应的拟合优度指标最高达到0.96, 而文献[6]中方法的拟合优度指标仅为0.83, 以及常规WNN方法的拟合优度指标甚至仅为0.75. 并且图4还可以看出所提方法有39组解的拟合优化度指标好于文献[6]中方法得到的拟合优度值. 图5是图3中1到60号解对应的建模误差PDF曲线变化图, 可以看出60号解对应模型的建模误差PDF最好. 图6为多目标优化后30号解与60号解的建模误差PDF曲线与其他两种现有方法的建模误差PDF曲线的对比图, 本文设置的期望PDF为均值为0方差为0.25的高斯型概率密度函数.
图 4 不同优化解对应的拟合优度值变化曲线Fig. 4 Change curve of goodness of fit corresponding to different optimization solutions
图 5 不同优化解对应的建模误差PDF变化曲面Fig. 5 PDF changing surface corresponding to different optimization solutions可以看出, 本文方法获得的非最优30号解对应的建模误差PDF曲线也要远好于常规WNN方法和GD-WNN方法. 所提方法得到的建模误差PDF较高且较窄, 与其他方法相比方差更小, 即模型的随机性和不确定性更小, 这也表明所提方法的有效性和优越性. 图7和图8分别是本文方法中非最优的30号解对应的建模效果和新样本测试效果, 可以看出所提方法得到非最优解对应的模型不论是建模和新样本测试均好于其他两种方法.
图 7 所提方法30号优化解对应的建模效果Fig. 7 Modeling result corresponding to the 30th optimization solution of the proposed method
图 8 所提方法30号优化解对应的测试效果Fig. 8 Testing result corresponding to the 30th optimization solution of the proposed method3.2 活性污泥污水处理过程生化反应池出水COD含量建模
目前, 城市污水处理广泛采用活性污泥法[3, 19-22]. 图9为典型活性污泥污水处理的工艺流程图, 主要包括三个级别的处理过程. 一级处理主要进行物理反应, 除去原生污水中的悬浮固体. 二级处理过程包括曝气池及二沉池处理两个部分. 曝气池是污水处理的核心部分, 主要进行微生物自身的代谢活动, 从而达到对污水中有机污染物如氮、磷的去除以及有氧生物的降解.
图 9 典型活性污泥法污水处理过程工艺流程图Fig. 9 Flow chart of a typical activated sludge wastewater treatment process经过曝气池处理后的污水流入二沉池进行固液分离, 上层是澄清的液体, 下层的污泥一部分回流至曝气池, 以维持曝气池内的污泥浓度, 另一部分污泥排出系统. 经过二级处理后的污水进入三级处理过程, 通过加入药剂进而得到达标的出水. 判断水质是否达标主要通过水质参数进行衡量. 在众多水质参数指标中, 出水COD含量不仅代表污水中含有的有机物的量, 同时还包括污水中还原性无机物被氧化时所消耗的氧气量. COD数值越小, 在氧化过程中氧气的消耗量就越少, 即水体中有机物的量越少. 因此, 该指标能够反映有机污染物受纳的程度, 是非常重要的出水水质指标. 虽然目前有许多COD含量的在线检测仪, 但是都存在检测周期长、价格昂贵的问题. 所以, 通过基于数据的智能建模技术来预测曝气池出水COD含量对于判断出水水质是否达标具有重要意义.
影响污水出水COD含量的参数较多, 包括: 入水流量(Q)、入水化学需氧量(COD)、溶解氧浓度(DO)、污泥浓度(MLSS)、悬浮固体浓度(SS)、出水PH值等. 为此, 根据过程机理分析, 确定出水COD预测建模的输入变量为: 入水COD、入水流量(Q)和污泥浓度(MLSS). 为消除变量间的量纲影响, 建模所用训练与测试数据都归一化处理. 设定WNN隐层节点数为6, 迭代优化步长为0.003, 得到的100组多目标优化Pareto前沿解如图10所示. 图11和图12分别为100组多目标优化解对应的拟合优度变化曲线和建模误差PDF变化曲线. 同样, 这里也将所提方法与常规WNN方法以及文献[6]中的GD-WNN方法进行比较. 可以看出, 采用所提建模方法可以获得具有较大动态趋势拟合优度的70余组解, 这些解对应的拟合优度均远好于常规WNN方法以及文献[6]中的GD-WNN方法, 并且所提方法所得解的最优拟合优度已非常接近1. 从图12所有多目标优化解的建模误差PDF变化曲线可以看出, 从第1号解到第100号解, 建模误差PDF的形状越来越窄而尖, 并且越来越接近设定的理想PDF形状, 即模型的随机性和不确定性很小.
图 10 COD含量建模Pareto前沿进化过程Fig. 10 Pareto front evolution process of COD content modeling
图 11 不同优化解对应的拟合优度值变化曲线Fig. 11 Change curve of goodness of fit corresponding to different optimization solutions
图 12 不同优化解对应的COD含量建模误差PDF变化曲面Fig. 12 PDF changing surface of COD content modeling error corresponding to different optimization solutions图13为50号解与100号解的建模误差PDF曲线与其它方法的建模误差PDF曲线对比图, 这里设置的期望PDF为均值为0方差为0.3的高斯型概率密度函数. 图中可以看出, 本文方法获得的非最优50号解对应的建模误差PDF形状为窄而高的形状, 仍然远好于常规WNN方法和GD-WNN方法. 图14和图15分别是本文方法所得非最优50号解的建模效果以及对新样本的测试效果, 虽然是选取的本文方法的非最优解, 但是从图中可以看出非最优解的建模和新样本测试效果均好于其他两种对比方法.
图 13 不同方法COD含量建模误差PDF比较Fig. 13 PDF comparison of COD content modeling error with different methods
图 14 所提方法50号优化解对应的COD含量建模效果Fig. 14 Modeling result of COD content corresponding to the 50th optimization solution of the proposed method
图 15 所提方法50号优化解对应的COD含量测试效果Fig. 15 Testing result of COD content corresponding to the 50th optimization solution of the proposed method4. 结语
基于误差最小的数据驱动工业系统建模时, 通常基于单一的RMSE等一维性能指标. 但是RMSE等时间维度的一维性能指标并不能充分体现动态系统建模的随机性和不确定性. 同时, 对于传统动态系统建模方法, 并没有考虑模型输出和动态系统实际输出之间的拟合趋势. 为此, 本文基于数据驱动小波神经网络智能建模、多目标参数优化以及核密度估计技术, 提出综合考虑建模误差PDF形状与趋势拟合优度的动态系统优化建模方法. 其中多目标参数优化的性能指标分别为实际建模误差PDF与期望建模误差PDF之间二维偏差平方、趋势拟合优度. 仿真实验以及污水处理过程数据验证表明: 相比于对比的两种现有建模方法, 所提方法不仅具有更好的建模精度和泛化能力, 还可控制建模误差的空间分布状态, 使得所提方法的建模误差PDF比传统建模方法的建模误差PDF更高、更窄, 即模型中含有的随机性和不确定性更小. 此外, 所提方法还可以获得一大类具有建模误差PDF形状接近期望分布形状, 且模型输出与实际输出的趋势拟合优度值较大的数据模型参数解, 因而具有更好的实用性.
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表 2 围捕机器人常见运动学与动力学约束
Table 2 Common kinematic and dynamic constraints of hunting robots
运动学与动力学约束 约束描述 最小、最大运动速度约束 围捕机器人速度须介于最小速度和最大速度之间 最小、最大运动加速度约束 围捕机器人加速度须介于最小加速度和最大加速度之间 最小步长约束 围捕机器人轨迹从当前状态到改变行进方向的下一状态之间的直线运动距离须大于最小步长 最小转弯半径约束 围捕机器人轨迹的转弯半径须大于最小转弯半径 最大航偏角约束 围捕机器人运动过程中的航偏角须小于最大航偏角 
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表 3 多机器人协同围捕任务分配方法总结
Table 3 Summary of multi-robot cooperative hunting task allocation methods
任务分配算法 架构特点 优点 缺点 贪婪算法[91] 集中式 实时性高, 简单易实现 局部最优解, “死锁”分配 匈牙利算法[74−75] 集中式 效率高, 局部最优解, 简单易实现 不适合大规模任务分配场景, 只适合一对一分配 遗传算法[81] 集中式 全局搜索能力, 适应性和鲁棒性, 并行性 计算成本高, 难以收敛, 参数设置敏感, 编码方式选择困难 粒子群算法[88, 90] 集中式 快速收敛, 简单易实现, 适应性强 过早收敛, 参数敏感 契约合同网络方法[95−96] 分布式 分布式协作, 动态灵活性, 适应性 高复杂性和通讯开销大, 信任建立和信息共享困难 拍卖算法[106−107] 分布式 分布式协作, 灵活性, 任务动态调整 竞争激烈可能导致效率下降, 存在信息不对称, 对于大规模问题不再适用
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表 4 生物启发式神经网络方法分析
Table 4 Biologically inspired neural network method analysis
领域 内容 优点 缺点 路径规划[125−126, 128] 解决机器人路径规划、防碰撞问题 实时性好 仅适用于二维平面 多机围捕[137] 首次将围捕问题与生物神经网络模型对应 兼顾围捕任务目标搜索、任务分配过程 模型设计复杂 路径规划[129] 引入假想非障碍物相邻点, 考虑转角因素 解决路径错判问题 计算效率低 路径规划[130] 模型考虑相邻神经元权值影响 使模型更具网络特性 增加碰撞检测计算节点 路径规划[131−132] 决策项考虑洋流影响 使方法更贴合实际环境 缺乏高效任务分配方法 多机围捕[133−134] 决策项考虑洋流影响 使方法更贴合实际环境 规划路径可能并非最优解 多机围捕[135] 设计基于协商思想的任务分配方法 增加围捕任务效率 仅适用于低维环境 多机围捕[136] 将方法拓展至三维环境 增强方法的可拓展性 奖励函数设计复杂、计算效率低
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