An Overview of Generalized KYP Lemma Based Methods for FiniteFrequency Analysis and Design
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摘要: 频域方法是控制理论与工程领域的一种基本研究手段，许多控制问题都可归结为有限频域性能指标的分析与综合问题.广义Kalman-Yakubovich-Popov（KYP）引理建立了频域方法（传递函数）与时域方法（状态空间）之间的一座桥梁，成为近年来系统与控制理论领域的研究热点之一.本文首先从信号和系统两个角度阐明有限频域分析与设计的背景和意义，并依次讨论三种主要研究方法（经典控制理论方法、频率加权法和广义性能指标法）各自的优缺点.然后简单介绍广义KYP引理的主体内容，并详细总结当前基于广义KYP引理的有限频域分析与设计的主要方向及研究进展.最后给出在使用广义KYP引理时很重要但容易忽视的几点注记，同时指明该领域目前存在并值得未来进一步研究的关键问题.
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关键词:
- 有限频域 /
- 广义Kalman-Yakubovich-Popov (KYP)引理 /
- 控制器设计 /
- 滤波 /
- 模型降阶
Abstract: Frequency-domain methods are a fundamental research approach in control theory and engineering. Many control problems can be viewed as analysis and design issues of finite frequency specifications. The generalized Kalman-Yakubovich-Popov (KYP) lemma, which bridges frequency-domain methods (transfer functions) and time-domain methods (state-space models), has been one of the hotspots in systems and control theory in recent years. In this paper, the background and significance of finite frequency analysis and design are first introduced from signal and system perspectives, respectively. Three main research methods (classical control theory, frequency-weighting strategy and generalized system specification based methodology) are discussed with respect to their individual advantages and disadvantages. The body of the generalized KYP lemma is then introduced briefly, which is followed by a detailed summary of main directions and recent progresses in finite frequency analysis and design based on the generalized KYP lemma. Finally, a few notes are presented, which are important but commonly overlooked in applying the generalized KYP lemma, and a few critical problems in the field are also pointed out, which are worth future investigation. -
近年来, 互联网数据呈现爆炸式增长, 特别是每年新增的图像数据以指数级发展, 如何从海量图像数据中快速检索出目标图像成为重要的研究课题.经典的海量图像检索技术主要包含两部分:高效的图像编码技术以及快速的最近邻搜索技术.目前, 国内外众多研究机构已经在海量图像检索相关领域取得了很多重要成果.
高效的编码方式:面对海量的图像数据, 视觉编码算法精度与效率的均衡是评价其性能的重要指标.词袋模型[1] (Bag of words, BOW)、Fisher向量[2] (Fisher vector, FV)、稀疏编码[3] (Sparse coding, SC)以及局部集聚向量描述[4] (Vector of locally aggregated descriptors, VLAD)等视觉编码框架的提出旨在解决海量图像检索过程中精度与效率均衡的问题.上述算法框架可用于百万量级规模的图像检索, 其中VLAD编码算法通过计算特征描述符与视觉单词的残差来表征图像, 具有良好的检索精度及速度, 特别适用于快速的海量图像检索应用.本文主要结合VLAD编码算法框架进行大规模海量图像检索.
然而, 传统的VLAD编码算法通过图像特征描述符空间的相似性计算图像相似度, 忽略了图像整体特征在几何空间的相关特性.虽然文献[5-6]提出了几何全校验重排算法, 利用了查询样本与候选样本的空间位置约束信息进行重排, 但算法较为耗时, 只能作用于少量检索结果.然而, 对于海量图像数据, 相关图像可能并不在少量检索结果中, 该几何重排算法精度难以满足要求.弱几何一致性校验[7] (Weak geometry consistency, WGC)的提出解决了几何重排校验无法适用于海量图像的问题, 该算法依据图像整体尺度及角度变化一致的假定筛选特征点, 通过直方图统计加速计算, 实现快速的几何重排.除上述几何重排序的研究工作外, 研究人员也展开对图像编码阶段融合几何信息的研究.文献[8]中, 作者沿用WGC几何变化一致的假定, 提出在特征集聚阶段按照特征角度划分分区, 特征点分区集聚的方法协变局部集聚向量描述符(Covariant vector of locally aggregated descriptors, CVLAD).然而由于图像存在旋转变化, 该算法需要通过穷举法估算查询图像与训练图像的旋转变化, 进而找到不同图像角度分区的对应关系, 算法较为耗时.文献[9]采用单项嵌入方式在特征集聚阶段将特征角度信息融入到图像整体表征中, 并以匹配核的形式重新解释了角度信息对于图像特征匹配的意义.文献[10]利用图像特征角度聚类的思想统一旋转图像的矢量表征, 并采用极坐标对特征角度进行参数化表征, 从而避免CVLAD中角度对应关系的穷举搜索计算.然而上述重排序、特征集聚阶段结合几何信息的工作, 仍然采用传统的K均值聚类算法训练视觉词典, 并未考虑几何信息对视觉词典的影响.因此, 特征编码阶段孤立了特征描述符与特征几何信息的关联性, 也即几何信息没有影响特征编码方式.
除特征点的几何信息外, 移动硬件传感器数据的融合是近年来海量图像检索领域研究的另一热点.文献[11]提出利用移动设备内置重力传感器信息构建重力对齐的特征描述符(Gravity aligned feature descriptors, GAFD). GAFD算法利用重力方向作为特征点主方向, 节约特征主方向计算时间, 同时提升相似结构特征的匹配性能.文献[12]中作者将GAFD算法用于VLAD编码框架, 其检索精度以及效率有明显提升.然而GAFD是以忽略特征原始梯度主方向为代价换取检索速度以及检索精度的少量提升.特征点梯度主方向对于海量图像检索也具有重要意义.
快速的最近邻检索:海量图像检索中, 为提高检索效率, 通常采用主成分分析(Principle component analysis, PCA)降低图像表达向量维度[13].然而单纯依靠PCA降维仍不能满足百万量级图像检索对于速度的需求.近年来, 国内外研究人员相继展开对海量图像最近邻搜索问题的研究.
当前, 主流的最近邻搜索分为两类.一类是线性搜索算法, 算法需要逐次计算查询向量到数据库中每个向量的距离.代表算法是局部敏感哈希(Locality sensitive hashing, LSH)[14], 将向量进行二值编码, 计算向量投影后的汉明距离. LSH主要依据是相似向量在哈希映射后仍以较高概率保持相似.随后, 一些改进算法如谱哈希(Spectral hashing, SH)[15]等算法优化了哈希函数, 谱哈希通过设计高效的特征向量图划分方法实现特征点的最佳哈希编码.另外文献[16]则利用最新的深度学习方法学习紧凑的哈希函数.相比LSH, 该类算法主要利用了样本的分布信息.另一类最近邻搜索算法是非线性搜索方法, 算法无需遍历数据库中的所有向量.该类算法的典型代表有K叉树(K-dimensional tree, K-D tree)[17]以及快速近似最近邻库(Fast library for approximate nearest neighbors, FLANN)[18].
乘积量化(Product quantization, PQ)[19]是近期提出的一种线性最近邻搜索算法.与LSH等二值编码方法不同, 该算法计算的是向量原始空间的L2距离, 通过对空间的分解优化量化精度, 成为当前海量图像检索中一种有效的压缩方式, 它常用于对图像表达向量PCA降维之后.乘积量化的目标函数可表示为
$ \begin{array}{l} {\rm{min}}\sum\limits_x {{{\left\| {\boldsymbol{x} - c\left( {i\left( \boldsymbol{x} \right)} \right)} \right\|}^2}} \\ {\rm{s}}.{\rm{t}}.c \in C = {C^1} \times \cdots \times {C^B} \end{array} $
(1) 查询向量x的最近邻码字c由B个较低维度的子码书分解构成.该方法能够产生大量有效码书并节约内存, 通过子码书量化器的引入能够加速最近邻搜索.然而由于PCA算法自身的问题, 造成奇异值较大的主成分集中于向量的前几维, PQ顺序均匀的子空间划分方法会导致子空间量化误差的不均衡, 显著影响检索精度.文献[20]提出的优化乘积量化(Optimized product quantization, OPQ)算法在一定程度上解决了奇异值分布不均的问题, 但该算法需要引入额外的正交投影矩阵.考虑到上述文献中出现的问题, 兼顾海量图像检索算法的性能与效率.本文设计了一种在特征编码及最近邻查询阶段融合重力信息的海量图像检索算法, 与传统VLAD编码算法相比, 本文主要贡献如下: 1)提出一种融合重力信息的角度编码方法来训练视觉词典, 充分利用相似图像的匹配特征在几何空间以及描述符空间的相关性. 2)提出一种基于角度信息的乘积量化方法用于最近邻检索, 将特征角度分区与乘积量化子分区相结合, 解决传统乘积量化方法中子空间量化误差不均衡的问题. 3)构建了一个包含GPS以及重力信息标注的北京地标建筑(Beijing landmark)数据库, 该数据库可作为后续移动视觉检索工作的基准数据库.本文第1节简单描述了融合重力信息的海量图像检索框架; 第2节介绍了在VLAD编码框架中融合重力信息的角度编码(Oriented coding)方法; 第3节提出基于角度乘积量化(Oriented coding)的最近邻检索并给出具体工作流程; 第4节对文中提出的算法进行实验分析; 最后, 总结全文并提出下一步工作计划.
1. 算法设计及框架
融合重力信息的海量图像检索框架如图 1所示, 离线阶段, 数据库图像根据重力信息以及特征角度信息进行分区VLAD量化并在每个子分区构建乘积量化子码书；在线查询阶段, 分区量化后的查询子向量与数据库子向量通过非对称计算得到两者的相似度度量, 具体流程如下:
图 1 融合重力信息和特征角度信息的海量图像检索框架Fig. 1 The framework of large-scale image retrieval based on a fusion of gravity aware orientation information离线阶段:
1)角度编码:数据库图像根据重力信息以及特征角度进行特征分区;
2) VLAD量化qc:对训练图像角度子分区进行码书映射及残差计算;
3)角度乘积量化qp:各角度分区VLAD子向量进行PCA降维以及子码书量化.
在线阶段:
1)角度编码:查询图像x根据重力信息以及特征角度进行特征分区;
2) VLAD量化qc:对查询图像角度子分区进行码书映射及残差计算;
3) PCA降维:各角度分区VLAD子向量降至同数据库图像相同维度;
4)非对称计算:计算查询子向量到训练图像子向量对应子码书中心的距离:
$ \sum\limits_{i = 1}^{i = B} {d\left( {{q_{ci}}\left( \boldsymbol{x} \right),{q_{pi}}\left( {{q_{ci}}\left( {{\boldsymbol{y}_i}} \right)} \right)} \right)} $
(2) 5)选择K近邻结果进行真实欧氏距离重排序, 排序后的结果即为检索结果.
本文基于文献[7-10]的工作基础之上, 将几何信息用于视觉词典的训练, 在特征编码阶段, 经重力旋转后具有相似角度的特征点被分区量化, 获得的图像向量表达融合了特征点的描述符以及角度两方面信息.角度乘积量化算法对每个角度子分区进行单独编码量化, 通过查询子向量与角度分区子码书的非对称计算取代查询向量与数据库向量的直接距离计算, 在保证查询精度的前提下大大降低了计算量.
2. 融合重力信息的角度编码算法
假定图像I提取n个尺度不变特征变换算法(Scale invariant feature transfarm, SIFT)[5]或者SURF (Speeded up robust features)[21]特征I={t1, t2, · · ·, tn}, 每个特征包含描述符Des、特征角度β以及尺度信息s, 即ti=(Des, β, s). BOW、VLAD等传统图像编码算法假定落在同一视觉单词上描述符相似的特征即为匹配特征:
$ \begin{array}{l} {\rm{if}}\left\| {De{s_1} - De{s_2}} \right\| < d\\ {\rm{then}}\;{t_1}\;{\rm{and}}\;{t_2}\;{\rm{are}}\;{\rm{matched}} \end{array} $
(3) 该假定忽略了每个特征点的上下文几何信息, 因此并不是一个充分条件.
文献[7]指出, 图像整体的主方向旋转以及特征尺度缩放是一致的, 即相似图像的匹配特征角度变化β1 -β2=Δβ以及尺度缩放s1/s2=Δs是定值, 它们满足数学上的线性相关性, 角度以及尺度等几何信息的利用有助于提高图像匹配性能.将几何信息用于BOW、FV、VLAD以及SC等传统图像编码框架, 匹配特征同时满足描述符的相似性以及几何信息的线性相关性, 增加了特征编码的准确性.
$ \begin{array}{l} {\rm{if}}\left\| {De{s_1} - De{s_2}} \right\| < d\\ {\beta _1} - {\beta _2}{\rm{ = }}\Delta \beta \;{\rm{and}}\;\frac{{{s_1}}}{{{s_2}}} = \Delta s\\ {\rm{then}}\;{t_1}\;{\rm{and}}\;{t_2}\;{\rm{are}}\;{\rm{matched}} \end{array} $
(4) 重力传感器已成为当前移动智能硬件的标配, 重力传感器能够记录相机姿态的旋转.由于图像旋转变化的整体性, 特征点主方向的旋转变化Δβ也能够通过重力传感器测得[22].因此, 在词典训练过程中可以充分利用匹配特征主方向的线性相关性.
在本节中, 我们提出一种融合重力信息的角度编码方法(Oriented coding)来训练具有几何信息的视觉词典, 并将其应用于VLAD检索框架中.
2.1 融合重力信息的角度编码
本节主要介绍融合重力信息的角度编码(Oriented coding)算法, 该方法由重力自适应的特征主方向计算以及角度量化两部分构成.
1)重力自适应的特征主方向
图 2反映了在拍摄地标建筑过程中可能发生的相机旋转.地标建筑垂直方向始终与重力方向平行, 因此可以通过图像坐标系中重力方向的变化估算地标建筑的旋转.图 2 (a)中重力方向垂直向下, 设此为基准位置, 图 2 (b)中重力方向相较于图像坐标轴的旋转角度θ可由重力传感器信息Gi测得, 需要特别注意的是图像坐标系与重力坐标系三轴相平行, 但其X轴与Y轴坐标刚相反, 即重力Y轴为图像坐标X轴.
图 2 不同拍摄角度的地标建筑及对应重力信息Fig. 2 The landmark building with different viewing angles and corresponding gravity information若图 2 (b)中重力传感器信息为Gi=[gx(i), gy(i), gz(i)], 则此时地标建筑旋转角度:
$ \theta {\rm{ = }}\left\{ \begin{array}{l} \frac{{gy\left( i \right)}}{{\left| {gy\left( i \right)} \right|}} \times \frac{\pi }{2} - \arctan \frac{{gx\left( i \right)}}{{gy\left( i \right)}},\;\;\;\;gy\left( i \right) \ne 0\\ \frac{\pi }{2} - \frac{{gx\left( i \right)}}{{\left| {gx\left( i \right)} \right|}} \times \frac{\pi }{2},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;gy\left( i \right) = 0 \end{array} \right. $
(5) 将地标建筑旋转θ, 即旋转至与图像坐标轴平行位置, 所得图像同图 2 (a)相似.由于图像特征旋转的整体性, 则特征主方向β的旋转为β + θ.经重力信息旋转后, 两张图像对应的匹配特征(t1, t2)角度之间的相互关系可以表示为
$ \beta \left( {{t_1}} \right) + \theta \left( {{t_1}} \right) = \beta \left( {{t_2}} \right) + \theta \left( {{t_2}} \right) $
(6) θ(t1)与θ(t2)分别是(t1, t2)所在图像的重力方向旋转角度.如果不考虑重力计算旋转角的误差, 则重力旋转后相似图像匹配特征具有相同的主方向.
2)角度量化考虑到重力传感器精度以及匹配特征的角度计算误差, 本文将旋转后的主方向映射至B个分区, 并建立特征描述符到角度分区的分区索引, O是映射后的分区索引:
$ O = \left\lfloor {B \times \frac{{\beta + \theta }}{{2{\rm{\pi }}}}} \right\rfloor ,\;\;\;\beta \in \left[ {0,2{\rm{\pi }}} \right) $
(7) 通过角度分区映射, 在特征编码阶段, 角度相关的特征点落入相同分区中.如图 3所示, 经过重力信息旋转后, 相似图像的主方向角度分布直方图具有高度相似性.本文在编码过程中综合利用了匹配特征描述符的相似性以及特征主方向的相关性, 训练的视觉词典在几何空间以及描述符空间具有双重区分力.
相比较CVLAD等算法将角度信息用在特征集聚阶段, 文中提出的算法在保持相同维度情况下将角度信息融入到视觉单词中, 增加了视觉单词的区分度, 本文将在后续实验中验证该算法有效性.
2.2 基于VLAD算法框架的角度编码
下面将上节提出的Oriented coding方法应用于VLAD检索算法框架中.不失一般性, 本文以文献[4]提出的VLAD算法为例介绍编码过程, 这里没有采用软量化策略[23], 因此编码主要包含两部分:角度编码OC(t)=j, j=1, · · ·, B以及最近邻视觉单词查找NN(t, Cj)=cj, k, k=1, · · ·, K.其中t为输入特征, B, K分别为角度分区数及视觉单词数, Cj代表主方向映射为第j角度分区的视觉词典, cj, k是Cj中第k视觉单词. xj, k是落在视觉单词cj, k上的所有特征的集聚残差向量和, 也即为图像向量表达x的第j角度分区中第k子向量:
$ \begin{array}{l} {\boldsymbol{x}_{j,k}} = \sum {\left( {t - {c_{j,k}}} \right)} \\ {\rm{s}}{\rm{.t}}{\rm{.}}t \in \left\{ {\left. t \right|OC\left( t \right) = j \cap NN\left( {t,{\boldsymbol{C}_j}} \right) = {c_{j,k}}} \right\} \end{array} $
(8) 通过在特征编码过程中融合特征几何信息, 算法充分利用了匹配特征在描述符以及几何空间的相关性, 并且有助于消除Burstiness[24]现象带来的负面影响.
3. 基于角度乘积量化的最近邻检索
图像向量表达维度与海量图像检索速度直接相关.海量图像在线检索时间主要消耗在特征编码以及最近邻检索阶段.假定数据库包含N张图像, 特征描述维度为d, VLAD编码视觉单词个数为K, 角度分区数为B, 则VLAD、CVLAD以及本文提出的Oriented coding算法特征编码时间几乎相同, 而最近邻检索时间分别为O(N × K × d)、O(N × B × B × K × d)以及O(N × B × K × d).
海量图像检索中, 图像数目N可达百万量级以上, 即使通过PCA将图像整体表征矢量降至D维, 最近邻检索时间复杂度由O(N × B × K × d)降至O(N × D), 速度也通常难以满足快速查询的需求.因此, 本文在传统PCA降维基础上提出基于角度分区的乘积量化方法对检索向量进一步压缩.首先, 通过角度编码算法将图像向量表达分解成若干个子分区, 将每个子分区进行PCA降维; 然后, 在每个分解子分区上进行K均值聚类生成子码书量化器, 构建角度乘积量化方法.该方法采用角度子分区作为乘积量化子分区, 在不额外引入投影矩阵的情况下解决了特征值分布不均的问题.
假定数据库向量yi=[yi, 1, · · ·, yi, B]由B个角度子分区构成, 其中i=1, · · ·, N, N为数据库图像数目, x=[x1, · · ·, xB]为查询向量.
离线数据库构建阶段, 首先对角度子分区进行PCA降维:
$ {\boldsymbol{y}_{i,j}} = {P_j}{\boldsymbol{y}_{i,j}},\;\;\;\;j = 1, \cdots ,B $
(9) 然后对降维后的子分区分别进行K均值聚类, 构建B个子分区码书Cpj, j=1, · · ·, B.每个子分区码书包含k*个视觉单词Cpj=[cpj1, · · ·, cpjk*], 总共B × k*个子码书, 分区子码书之间采用笛卡尔积形式组合, 构成(k*)B个大码书.最后对数据库向量进行分区编码, qpj为第j分区量化器, 量化结果为子码书中心索引标号:
$ \begin{array}{l} {q_p}\left( {{\boldsymbol{y}_i}} \right) = \left[ {{q_{p1}}\left( {{\boldsymbol{y}_{i,1}}} \right), \cdots ,{q_{pB}}\left( {{\boldsymbol{y}_{i,B}}} \right)} \right]\\ {q_{pj}}\left( {{\boldsymbol{y}_{i,j}}} \right) = k\\ {\rm{s}}{\rm{.t}}{\rm{.}}j = 1, \cdots ,B,\;\;\;\;\;k = 1, \cdots ,k* \end{array} $
(10) 在线查询阶段, 首先将查询子向量进行PCA降维, 并计算与所在子分区每个码书中心的欧氏距离, 生成B × k*查找表:
$ \begin{array}{l} d\left( {{x_j},{\boldsymbol{C}_{pj}}} \right) = \left[ {d\left( {{x_j},c_{pj}^1} \right), \cdots ,d\left( {{x_j},c_{pj}^{k*}} \right)} \right]\\ d\left( {{x_j},c_{pj}^k} \right) = \left\| {{x_j} - c_{pj}^k} \right\|_2^2\\ {\rm{s}}{\rm{.t}}{\rm{.}}j = 1, \cdots ,B,\;\;\;\;\;k = 1, \cdots ,k* \end{array} $
(11) 然后针对每一数据库向量, 根据离线存储的子码书中心映射索引标号qpj(yi, j)在B × k*查找表中寻找非对称距离:
$ d\left( {{x_j},{y_{i,j}}} \right) = d\left( {{x_j},C_{pj}^{{q_{pj}}\left( {{y_{i,j}}} \right)}} \right) $
(12) 其中, cpjqpj(yi, j)代表第j分区第qpj(yi, j)个码书中心.则查询向量x与数据库向量yi的距离为所有子分区的非对称距离和:
$ d\left( {x,{y_i}} \right) = \sum\limits_{j = 1}^{j = B} {d\left( {{x_j},{y_{i,j}}} \right)} $
(12) 根据非对称计算返回Top K的最近邻距离并进行欧氏距离重排, 最后输出结果即为检索结果.
算法流程如下:
算法1.角度乘积量化
输入.数据库向量以及查询向量.
输出.查询向量的K最近邻.
步骤1. PCA降维后的数据库向量子分区分别进行K均值聚类构建子分区码书, 并进行分区编码;
步骤2.查询子向量PCA降维, 并依次计算与子分区码书中心的距离, 形成B × k*查找表;
步骤3.通过查找表以及数据库子向量的码书映射情况计算非对称距离;
步骤4.返回Top K的最近邻距离进行欧氏距离重排.
若PCA降维后图像整体表征矢量为D维, 则其线性搜索时间复杂度为O(N × D), 而文中提出的角度乘积量化算法的时间复杂度为O(N × B + k* × D), 其中k*为子分区量化器大小, 一般k* « N, 因此角度乘积量化算法大大降低了图像在线查询时间.
4. 实验结果及分析
4.1 测试数据库
为验证文中提出的Oriented coding方法以及Oriented PQ方法, 本文在自己构建的北京地标建筑(Beijing landmark)图像数据库、Holidays数据库[4]以及SUN397数据库[25]中进行测试. Beijing landmark由课题组使用智能手机在北京周边搜集构建.图像采用自己编写的Android平台拍照应用程序搜集, 在拍照的同时记录GPS以及重力传感器信息.该图像数据库总共包含4 000张地标图像, 每5张图像为一个地标建筑, 分别包含尺度、视角以及旋转等不同变化.数据库中的部分图片如图 4所示. Beijing landmark提供公开下载, 可供后续测试研究(http://pan.baidu.com/s/1c04gdlI).由于每个场景图像数目相同, 本文选择与Zurich以及UKB等标准图像库相类似的评价标准, 取检索到前5张图像中相关图像数目的平均值作为图像检索精度的评价标准, 即为Recall@Top5. Holidays数据库[4]包含1 491张数据库图像以及500张查询图片.本文将图像降采样至最大786 432像素, 采用mAP作为衡量检索精度的标准.考虑到无法获取Holidays数据库的重力信息, 与文献[10]相类似, 本文实验将Holidays数据库图像做人工旋转校正. SUN397包含130 519张各类建筑图像, 本文随机选择其中96 000张图像赋以随机重力信息作为干扰图像, 同Beijing landmark相混合成100 KB大小数据库, 作为海量检索图像数据库.
4.2 参数设定
本文实验主要测试文中提出的融合重力信息的Oriented coding方法以及Oriented PQ方法对于海量图像检索性能的影响.考虑到移动端的计算性能以及快速查询要求, 文中采用d=64维SURF特征作为主要特征提取算法, K均值聚类方法作为描述符聚类的主要方法.其中K均值聚类的聚类中心个数设定为K, 特征点进行角度映射的分区数目设置为B.文中将在第4.3节重点讨论这些参数对于检索性能的影响.
4.3 检索性能
本节主要测试融合重力信息的Oriented coding方法对于海量图像检索精度的影响, 并与VLAD算法[4], 改进的VLAD+算法[26]以及CVLAD[8]进行性能对比.测试数据库为Beijing landmark数据库以及Holidays数据库.
图 5所示为Beijing landmark数据库下VLAD、VLAD+、CVLAD以及Oriented coding方法检索精度对比.可以看出, 在角度分区B=4, 6, 8的情况下, 文中提出的Oriented coding方法均比VLAD、VLAD+以及CVLAD有较大的精度提升.其中CVLAD检索精度随分区数目增长而提高, 这是由于随着角度分区数目的增加, CVLAD的穷举搜索算法能够更准确地预估图像旋转变化, 因此其检索精度随之提高.但是, 这同样带来了内存以及速度的牺牲.
与CVLAD不同, 图 5中Oriented coding方法检索精度B=6 > 8 > 4, 图 6具体给出K=16情况下Oriented coding方法随角度分区数目B变化的检索精度变化曲线, 其中B=1代表不分区, 即传统VLAD方法.可以看出, Oriented coding方法在分区数目达到B=6后, 检索精度开始有所下降, 这是由于本文采用重力传感器XY轴计算图像旋转角度, 对图像拍摄时俯仰变化比较敏感.过细角度分区划分会影响到俯仰角较大拍摄图像的特征角度分区, 容易出现错分现象, 导致检索精度下降.因此后续实验中, 无特别说明, 角度分区数目主要设置为B=6进行实验.
图 6 Oriented coding检索精度与分区数目关系Fig. 6 The relationship of oriented coding retrieval accuracy and partition number考虑到本文Oriented coding方法相比VLAD、VLAD+以及CVLAD多利用了图像的重力信息, 为公平对比, 本文将重力信息赋予CVLAD算法, 即假定CVLAD也通过重力获取旋转角, 将特征角度用于特征集聚阶段, 无需穷举搜索计算图像旋转角度.本文将CVLAD的重力版本定义为gCVLAD. Oriented coding与gCVLAD方法对比结果如图 7.可以看出本文的方法在相同角度分区数目情况下均优于gCVLAD方法, gCVLAD与Oriented coding相同的是, B=6检索精度最高.
图 7 Oriented coding与重力版本CVLAD方法检索精度对比Fig. 7 Comparison of retrieval accuracy with oriented coding and gCVLAD由于重力信息存在计算误差, 下面分别在Rotated Holidays数据库以及原始Holidays数据库中进行对比测试, 这里仅考虑角度编码影响, 剔除重力信息影响.实验中码书大小设置为变化的K, CVLAD以及Oriented coding中角度分区数目B=6.在Rotated Holidays数据库测试中, 由于校正数据库中相似图像匹配特征主方向相同, CVLAD方法无需穷举搜索图像主方向变化.
表 1所示为Holidays数据库下各方法检索精度对比.可以看出, CVLAD以及Oriented coding方法在校正图像数据库下均具有良好的检索结果, 其中本文提出的Oriented coding方法优于CVLAD, 而VLAD以及VLAD+提升不大, 这验证了正确的角度编码对于检索精度提升的效果.同时, CVLAD和Oriented coding相比无几何信息的VLAD和VLAD+, 随着码书大小K的增加, 检索精度的提升相对较小.
表 1 Holidays数据库检索精度(mAP)Table 1 The retrieval accuracy of Holidays dataset (mAP)码书大小 K=8 K=16 K=32 K=64 Holidays Rotated Holidays Rotated Holidays Rotated Holidays Rotated VLAD 0.512 0.515 0.534 0.542 0.551 0.559 0.579 0.587 VLAD+ 0.560 0.564 0.581 0.586 0.597 0.605 0.613 0.622 CVLAD 0.658 0.687 0.663 0.694 0.683 0.709 0.697 0.719 Oriented coding \ 0.709 \ 0.716 \ 0.728 \ 0.736 在时间消耗方面, CVLAD采用穷举遍历法计算图像的旋转, 时间代价为O(N × B × B × K × d).而本文的融合重力信息的Oriented coding方法仅需O(N × B × K × d)时间复杂度, 在B=6的情况下时间节约接近80 %.
4.4 特征降维
为保证算法性能对比的公平性, 本实验在VLAD、VLAD+、CVLAD以及本文提出的Oriented coding等方法编码后采用PCA降低到相同维度, 然后进行检索精度对比.假定上述编码算法采用的特征描述维度为d, 编码视觉单词个数为K, 角度分区数为B, 则VLAD以及VLAD+图像表达向量维度为K×d, CVLAD以及文中提出的Oriented coding向量维度为B × K × d.本实验对上述编码方法通过PCA将维度降低至同一维度, 比较其检索精度.图 8为分区数目B=6, 词典数目K=16情况下各算法的检索精度比较.实验结果表明, 文中提出的结合重力以及几何信息的Oriented coding编码算法在相同维度下有较大精度优势.而CVLAD在相同维度下相比VLAD+并没有明显的精度优势.另外上述检索算法随着维度增加到一定程度, 检索精度并未继续增长, 甚至部分情况下会发生精度下降的情况.
图 8 PCA降维后Oriented coding检索算法精度Fig. 8 The retrieval accuracy of oriented coding after PCA4.5 基于角度乘积量化的海量图像检索方法
本节主要测试文中提出的Oriented coding最近邻搜索算法的性能, 并将Oriented PQ与Oriented coding相结合, 应用于海量图像检索中.测试数据库由SUN397与Beijing landmark混合构成, 数据库大小为100 KB. PCA降维后图像向量表达维度D=128, 乘积量化子分区数B=8, 子分区聚类词典数目k*=256.
如图 9所示, 文中提出的Oriented PQ方法相比传统的PQ能够提升一定的检索精度, 在100 KB数据库中, 检索精度提升3 %以上, 并且检索精度更加接近直接线性查找算法, 误差在1 %以内.
图 9 海量图像最近邻检索方法精度对比Fig. 9 Comparison of retrieval accuracy with different ANN methods表 2所示为海量检索过程中的时间开销.从表 2可见, PQ方法相比PCA降维有较大的检索速度提升.但在较小的10 KB数据库中, 由于子分区中心距离的计算将消耗一定时间O(k* × D), PQ速度提升并不明显.而子分区距离中心计算时间不随图像数目发生变化, 在100 KB数据库中, 影响检索速度的主要因素是特征维度以及图片数目, 此时O(N × B + k* × D) < O(N × D), 随着图片数目的递增, 文中提出的检索算法时间消耗接近于PCA降维的B/D.由于影响PQ速度的关键因素子分区数目相同, 文中提出的Oriented PQ方法与PQ有较为接近的检索速度.
表 2 海量检索时间消耗(ms)Table 2 Time consuming of image retrieval (ms)数据库大小 10 KB 100 KB PCA 62.1 671.3 PQ 22.7 104.2 Oriented PQ 24.3 108.5 4.6 与其他检索算法框架相结合
稀疏编码是另外一种比较主流的图像检索算法[27-29], 其检索精度与VLAD算法相当, 因此, 本实验测试Oriented coding算法与稀疏编码框架结合的检索精度, 实验结果如图 10所示, 其中“B bins SC with max pooling”代表角度分区数目为B的稀疏编码方法, 每个分区取稀疏系数的最大值, B=1时即为传统的稀疏编码方法.实验结果表明, Oriented coding方法应用于稀疏编码同样能够提升检索精度.然而稀疏编码特征编码阶段较为耗时, 当前还无法应用于实时性要求较高的移动视觉检索系统.
图 10 基于稀疏编码框架的Oriented coding方法检索精度Fig. 10 The retrieval accuracy of oriented coding based on sparse coding framework5. 总结和展望
本文研究了海量图像检索过程中利用重力传感器以及特征点角度信息的问题, 提出了一种融合重力信息的Oriented coding以及Oriented PQ快速检索算法.本文搜集了GPS以及重力信息标注的Beijing landmark图像数据库, 可用于后续算法研究.文中提出的融合重力信息的Oriented coding算法综合利用了匹配特征在描述符空间以及几何空间的相关性, Oriented PQ算法在100 KB图像检索中保持较高的检索速度及精度.同时本文提出的Oriented coding算法还存在一些问题, 例如不同角度之间视觉单词的相关性如何消除, 特征尺度信息如何利用, 这些问题的研究有助于进一步降低图像表达向量维度, 本文会在后续研究中继续深入探索这些问题.
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表 1 集合$\Omega $与$\Lambda $以及矩阵$\Phi $和$\Psi $的取值
Table 1 The values of sets$\Omega $and$\Lambda $and matrices$\Phi $and$\Psi $
$\Lambda $ $\Phi $ $\Omega $ $\Psi $ 连续系统 $\left[ \begin{matrix} 0 & 1 \\ 1 & 0 \\ \end{matrix} \right]$ 低频 $\left[ \begin{matrix} -1 & 0 \\ 0 & \omega _{l}^{2} \\ \end{matrix} \right]$ 中频 $\left[ \begin{matrix} -1 & \text{j}{{\omega }_{c}} \\ -\text{j}{{\omega }_{c}} & -{{\omega }_{1}}{{\omega }_{2}} \\ \end{matrix} \right]$ 高频 $\left[ \begin{matrix} 1 & 0 \\ 0 & -\omega _{h}^{2} \\ \end{matrix} \right]$ 离散系统 $\left[ \begin{matrix} 1 & 0 \\ 0 & -1 \\ \end{matrix} \right]$ 低频 $\left[ \begin{matrix} 0 & 1 \\ 1 & -2\cos {{\omega }_{r}} \\ \end{matrix} \right]$ 中频 $\left[ \begin{matrix} 0 & {{\text{e}}^{\text{j}\omega c}} \\ {{\text{e}}^{-\text{j}\omega \text{c}}} & -2\cos {{\omega }_{r}} \\ \end{matrix} \right]$ 高频 $\left[ \begin{matrix} 0 & -1 \\ -1 & 2\cos {{\omega }_{h}} \\ \end{matrix} \right]$ 
下载: 导出CSV
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[1] Du C L, Xie L H, Guo G X, Teoh J N. A generalized KYP lemma based approach for disturbance rejection in data storage systems. Automatica, 2007, 43(12): 2112-2118 doi: 10.1016/j.automatica.2007.04.023 [2] Chen Y, Zhang W L, Gao H J. Finite frequency H∞ control for building under earthquake excitation. Mechatronics, 2010, 20(1): 128-142 doi: 10.1016/j.mechatronics.2009.11.001 [3] Lim J S, Ryoo J R, Lee Y I, Son S Y. Design of a fixed-order controller for the track-following control of optical disc drives. IEEE Transactions on Control Systems Technology, 2011, 20(1): 205-213 http://cn.bing.com/academic/profile?id=2078120591&encoded=0&v=paper_preview&mkt=zh-cn [4] Paszke W, Rogers E, Galkowski K. On the design of ILC schemes for finite frequency range tracking specifications. In: Proceedings of the 49th IEEE Conference on Decision and Control. Atlanta, GA: IEEE, 2010. 6979-6984 [5] Pipeleers G, Swevers J. Optimal feedforward controller design for periodic inputs. International Journal of Control, 2010, 83(5): 1044-1053 doi: 10.1080/00207170903552067 [6] Hatada K, Hirata K. Energy-efficient power assist control for periodic motions. In: Proceedings of the SICE Annual Conference 2010. Taipei, China: IEEE, 2010. 2004-2009 [7] Pipeleers G, Demeulenaere B, De Schutter J, Swevers J. Generalised repetitive control: relaxing the period-delay-based structure. IET Control Theory and Applications, 2009, 3(11): 1528-1536 doi: 10.1049/iet-cta.2008.0499 [8] Pipeleers G, Demeulenaere B, Al-Bender F, De Schutter J, Swevers J. Optimal performance tradeoffs in repetitive control: experimental validation on an active air bearing setup. IEEE Transactions on Control Systems Technology, 2009, 17(4): 970-989 doi: 10.1109/TCST.2009.2014358 [9] Zhou K M, Doyle J C, Glover K. Robust and Optimal Control. New Jersey: Prentice-Hall, 1996. [10] Sun W C, Gao H J, Kaynak O. Finite frequency H∞ control for vehicle active suspension systems. IEEE Transactions on Control Systems Technology, 2011, 19(2): 416-422 doi: 10.1109/TCST.2010.2042296 [11] 孙维超. 汽车悬架系统的主动振动控制[博士学位论文], 哈尔滨工业大学, 中国, 2013Sun Wei-Chao. Active Vibration Control for Vehicle Suspension Systems [Ph.D. dissertation], Harbin Institute of Technology, China, 2013 [12] Costa-Castello R, Wang D W, Grino R. A passive repetitive controller for discrete-time finite-frequency positive-real systems. IEEE Transactions on Automatic Control, 2009, 54(4): 800-804 doi: 10.1109/TAC.2008.2009594 [13] Wongsura S, Liu L, Hara S. Conditions for mixed small gain and positive real property for LTI systems. In: Proceedings of the SICE Annual Conference 2010. Taipei, China: IEEE, 2010. 625-629 [14] Yang H J, Xia Y Q, Shi P, Fu M Y. Stability analysis for high frequency networked control systems. IEEE Transactions on Automatic Control, 2012, 57(10): 2694-2700 doi: 10.1109/TAC.2012.2190194 [15] Iwasaki T, Hara S, Yamauchi H. Dynamical system design from a control perspective: finite frequency positive-realness approach. IEEE Transactions on Automatic Control, 2003, 48(8): 1337-1354 doi: 10.1109/TAC.2003.815013 [16] Forbs J R. Extensions of Input-Output Stability Theory and the Control of Aerospace Systems [Ph.D. dissertation], University of Toronto, Canada, 2011 [17] Wu S P, Boyd S, Vandenberghe L. FIR filter design via spectral factorization and convex optimization. Applied and Computational Control, Signals, and Circuits. New York: Springer, 1999. 215-245 http://cn.bing.com/academic/profile?id=1591179352&encoded=0&v=paper_preview&mkt=zh-cn [18] Lei C U, Wong N. IIR approximation of FIR filters via discrete-time hybrid-domain vector fitting. IEEE Signal Processing Letters, 2009, 16(6): 533-537 doi: 10.1109/LSP.2009.2017478 [19] Apkarian P, Noll D. Nonsmooth optimization for multiband frequency domain control design. Automatica, 2007, 43(4): 724-731 doi: 10.1016/j.automatica.2006.08.031 [20] SimÖes A M, Apkarian P, Noll D. Nonsmooth multi-objective synthesis with applications. Control Engineering Practice, 2009, 17(11): 1338-1348 doi: 10.1016/j.conengprac.2009.06.010 [21] 裴润, 宋申民. 自动控制原理. 哈尔滨: 哈尔滨工业大学出版社, 2006.Pei Run, Song Shen-Min. Principles of Automatic Control. Harbin: Harbin Institute of Technology Press, 2006. [22] 郑大钟. 线性系统理论. 北京: 清华大学出版社, 2002.Zheng Da-Zhong. Linear System Theory. Beijing: Tsinghua University Press, 2002. [23] Anderson B D O, Moore J B. Optimal Control: Linear Quadratic Methods. Englewood Cliffs, New Jersey: Prentice-Hall, 1989. [24] Enns D F. Model reduction with balanced realizations: an error bound and a frequency weighted generalization. In: Proceedings of the 23rd Conference on Decision and Control. Las Vegas, NV, USA: IEEE, 1984. 127-132 [25] Anderson B D O. Weighted Hankel-norm approximation: calculation of bounds. Systems & Control Letters, 1986, 7(4): 247-255 http://cn.bing.com/academic/profile?id=2031798783&encoded=0&v=paper_preview&mkt=zh-cn [26] Chow Y L, Hu Y B, Li X W, Kominek A, Lam J. Mixed additive/multiplicative H∞ model reduction. Journal of Dynamic Systems, Measurement, and Control, 2013, 135(5): 051005, DOI: 10.1115/1.4024111 [27] Zhou K M. Frequency-weighted L∞ norm and optimal Hankel norm model reduction. IEEE Transactions on Automatic Control, 1995, 40(10): 1687-1699 doi: 10.1109/9.467681 [28] Luo H, Lu W S, Antoniou A. A weighted balanced approximation for 2-D discrete systems and its application to model reduction. IEEE Transactions on Circuits and Systems—I: Fundamental Theory and Applications, 1995, 42(8): 419-429 doi: 10.1109/81.404046 [29] Wang G, Sreeram V, Liu W Q. A new frequency-weighted balanced truncation method and an error bound. IEEE Transactions on Automatic Control, 1999, 44(9): 1734-1737 doi: 10.1109/9.788542 [30] Ghafoor A, Wang J, Sreeram V. Frequency-weighted model reduction method with error bounds for 2-D separable denominator discrete systems. In: Proceedings of the 2005 IEEE International Symposium on, Mediterrean Conference on Control and Automation, Intelligent Control. Limassol, Cyprus: IEEE, 2005. 525-530 [31] Ghafoor A, Sreera V, Treasure R. Frequency weighted model reduction technique retaining Hankel singular values. Asian Journal of Control, 2007, 9(1): 50-56 http://cn.bing.com/academic/profile?id=2084554706&encoded=0&v=paper_preview&mkt=zh-cn [32] Ghafoor A, Sreeram V. A survey/review of frequency-weighted balanced model reduction techniques. Journal of Dynamic Systems, Measurement, and Control, 2008, 130(6): 758-767 http://cn.bing.com/academic/profile?id=1973799947&encoded=0&v=paper_preview&mkt=zh-cn [33] Liu Y, Anderson B D O. Frequency weighted controller reduction methods and loop transfer recovery. Automatica, 1990, 26(3): 487-497 doi: 10.1016/0005-1098(90)90020-I [34] Houlis P, Sreeram V. A parametrized controller reduction technique via a new frequency weighted model reduction formulation. IEEE Transactions on Automatic Control, 2009, 54(5): 1087-1093 doi: 10.1109/TAC.2008.2010993 [35] Gawronski W, Juang J N. Model reduction in limited time and frequency intervals. International Journal of Systems Science, 1990, 21(2): 349-376 doi: 10.1080/00207729008910366 [36] Rotea M A. The generalized H2 control problem. Automatica, 1993, 29(2): 373-385 doi: 10.1016/0005-1098(93)90130-L [37] Li X W, Gao H J. A delay-dependent approach to robust generalized H2 filtering for uncertain continuous-time systems with interval delay. Signal Processing, 2011, 91(10): 2371-2378 doi: 10.1016/j.sigpro.2011.04.032 [38] Gugercin S, Antoulas A C. A survey of model reduction by balanced truncation and some new results. International Journal of Control, 2004, 77(8): 748-766 doi: 10.1080/00207170410001713448 [39] Ghafoor A, Sreeram V. Model reduction via limited frequency interval gramians. IEEE Transactions on Circuits and Systems—I: Regular Papers, 2008, 55(9): 2806-2812 doi: 10.1109/TCSI.2008.920092 [40] Sahlan S, Ghafoor A, Sreeram V. A new method for the model reduction technique via a limited frequency interval impulse response gramian. Mathematical and Computer Modelling, 2012, 55(3-4): 1034-1040 doi: 10.1016/j.mcm.2011.09.028 [41] Petersson D, LÖfberg J. Model Reduction Using a Frequency-Limited H2-Cost, Technical Report LiTH-ISY-R-3045, Division of Automatic Control, LinkÖpings University, Sweden, 2012. http://www.control.isy.liu.se/publications/?lname=norrlof&fname=&title=&year=&type=any&number=LiTH&keywords=&go=Search&output=html [42] Iwasaki T, Meinsma G, Fu M Y. Generalized S-procedure and finite frequency KYP lemma. Mathematical Problems in Engineering, 2000, 6(2-3): 305-320 doi: 10.1155/S1024123X00001368 [43] Iwasaki T, Hara S. Generalized KYP lemma: unified frequency domain inequalities with design applications. IEEE Transactions on Automatic Control, 2005, 50(1): 41-59 doi: 10.1109/TAC.2004.840475 [44] Rantzer A. On the Kalman-Yakubovich-Popov lemma. Systems & Control Letters, 1996, 28(1): 7-10 http://cn.bing.com/academic/profile?id=2077418568&encoded=0&v=paper_preview&mkt=zh-cn [45] Iwasaki T, Hara S. Generalization of Kalman-Yakubovic-Popov lemma for restricted frequency inequalities. In: Proceedings of the 2003 American Control Conference. Denver, Colorado, USA: IEEE, 2003. 3828-3833 [46] Nesterov Y, Nemirovskii A. Interior-Point Polynomial Algorithms in Convex Programming. Philadelphia, PA: SIAM, 1994. [47] Boyd S, El Ghaoui L, Feron E, Balakrishnan V. Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [48] 高会军. 基于参数依赖Lyapunov函数的不确定动态系统的分析与综合[博士学位论文], 哈尔滨工业大学, 中国, 2005Gao Hui-Jun. Analysis and Synthesis of Uncertain Dynamic Systems Based on Parameter-Dependent Lyapunov Function [Ph.D. dissertation], Harbin Institute of Technology, China, 2005 [49] 俞立. 鲁棒控制--线性矩阵不等式处理方法. 北京: 清华大学出版社, 2002.Yu Li. Robust Control-Linear Matrix Inequality Method. Beijing: Tsinghua University Press, 2002. [50] Toh K C, Todd M J, TÜtÜncÜ R H. SDPT3—a matlab software package for semidefinite programming, version 1.3. Optimization Methods and Software, 1999, 11(1-4): 545-581 doi: 10.1080/10556789908805762 [51] Sturm J F. Using SeDuMi 1.02, a matlab toolbox for optimization over symmetric cones. Optimization Methods and Software, 1999, 11(1-4): 625-653 doi: 10.1080/10556789908805766 [52] Skelton R E, Iwasaki T, Grigoriadis K M. A Unified Algebraic Approach to Linear Control Design. London and Bristol, PA: Taylor & Francis, 1998. [53] Hara S, Iwasaki T. From generalized KYP lemma to engineering applications. In: Proceedings of the 42nd IEEE Conference on Decision and Control. Maui, Hawaii, USA: IEEE, 2003. 792-797 [54] Iwasaki T, Hara S. Robust control synthesis with general frequency domain specifications: static gain feedback case. In: Proceedings of the 2004 American Control Conference. Boston, MA: IEEE, 2004. 4613-4618 [55] Iwasaki T, Hara S. Feedback control synthesis of multiple frequency domain specifications via generalized KYP lemma. International Journal of Robust & Nonlinear Control, 2007, 17(5-6): 415-434 http://cn.bing.com/academic/profile?id=2004080117&encoded=0&v=paper_preview&mkt=zh-cn [56] Hara S, Iwasaki T. Sum-of-squares decomposition via generalized KYP lemma. IEEE Transactions on Automatic Control, 2009, 54(5): 1025-1029 doi: 10.1109/TAC.2009.2017149 [57] Hara S, Iwasaki T, Shiokata D. Robust PID control using generalized KYP synthesis: direct open-loop shaping in multiple frequency ranges. IEEE Control Systems Magazine, 2006, 26(1): 80-91 doi: 10.1109/MCS.2006.1580156 [58] Shiokata D, Hara S, Iwasaki T. From Nyquist/Bode to GKYP design: design algorithms with CACSD tools. In: Proceedings of the SICE 2004 Annual Conference. Sapporo: SICE, 2004. 1780-1785 [59] El Ghaoui L, Oustry F, AitRami M. A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Transactions on Automatic Control, 1997, 42(8): 1171-1176 doi: 10.1109/9.618250 [60] Li X W, Gao H J. A heuristic approach to static output-feedback controller synthesis with restricted frequency-domain specifications. IEEE Transactions on Automatic Control, 2014, 59(4): 1008-1014 doi: 10.1109/TAC.2013.2281472 [61] Shu Z, Lam J. An augmented system approach to static output-feedback stabilization with H∞ performance for continuous-time plants. International Journal of Robust & Nonlinear Control, 2009, 19(7): 768-785 [62] >Peaucelle D, Arzelier D. An efficient numerical solution for H2 static output feedback synthesis. In: Proceeding of the 2001 European Control Conference. Porto, Portugal: IEEE, 2001. 3800-3805 [63] Agulhari C M, Oliveira R C L F, Peres P L D. LMI relaxations for reduced-order robust H∞ control of continuous-time uncertain linear systems. IEEE Transactions on Automatic Control, 2012, 57(6): 1532-1537 doi: 10.1109/TAC.2011.2174693 [64] Li X W, Yin S, Gao H J, Kaynak O. Robust static output-feedback control for uncertain linear discrete-time systems via the generalized KYP lemma. In: Proceedings of the 19th IFAC World Congress. Cape Town, South Africa: IFAC, 2014. 7430-7435 [65] Li X W, Gao H J. Robust frequency-domain constrained feedback design via a two-stage heuristic approach. IEEE Transactions on Cybernetics, 2015, 45(10): 2065-2075 doi: 10.1109/TCYB.2014.2364587 [66] Li X W, Gao H J. Generalized Kalman-Yakubovich-Popov lemma for 2-D FM LSS model and its application to finite frequency positive real control. In: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference. Orlando, FL: IEEE, 2011. 6991-6996 [67] Li X W, Gao H J, Wang C H. Generalized Kalman-Yakubovich-Popov lemma for 2-D FM LSS model. IEEE Transactions on Automatic Control, 2012, 57(12): 3090-3103 doi: 10.1109/TAC.2012.2200370 [68] Hao Y Q, Duan Z S, Huang D. Structured controller synthesis with restricted frequency domain specifications. In: Proceedings of the 34th Chinese Control Conference. Hangzhou, China: IEEE, 2015. 2895-2990 [69] Hao Y Q, Duan Z S. Static output-feedback controller synthesis with restricted frequency domain specifications for time-delay systems. IET Control Theory and Applications, 2015, 9(10): 1608-1614 doi: 10.1049/iet-cta.2014.1000 [70] 张晓妮, 杨光红. 混合频小增益动态输出反馈控制综合. 自动化学报, 2008, 34(5): 551-557 doi: 10.3724/SP.J.1004.2008.00551Zhang Xiao-Ni, Yang Guang-Hong. Dynamic output feedback control synthesis with mixed frequency small gain specifications. Acta Automatica Sinica, 2008, 34(5): 551-557 doi: 10.3724/SP.J.1004.2008.00551 [71] 梅平, 邹云. 基于广义KYP引理方法的奇异摄动系统有限频段正实性能分析. 控制与决策, 2010, 25(5): 711-720Mei Ping, Zou Yun. Finite frequency positive realness analysis of singularly perturbed systems based on generalized KYP lemma approach. Control and Decision, 2010, 25(5): 711-720 [72] 董全超. 线性时滞系统鲁棒H∞故障估计与主动容错控制[博士学位论文], 山东大学, 中国, 2010Dong Quan-Chao. Robust H∞ Fault Estimation and Active Fault Tolerant Control for Linear Time-delay Systems [Ph.D. dissertation], Shandong University, China, 2010 [73] Li H, Peng L Y, Ju H H. A finite frequency domain approach to robust and parameter dependent PID controller design for LPV systems. In: Proceedings of the 30th Chinese Control Conference. Yantai, China: IEEE, 2011. 3688-3694 [74] Lim J S, Ryoo J R, Lee Y I. Fixed-order controller design with frequency domain specifications. In: Proceedings of ICROS-SICE International Joint Conference 2009. Fukuoka, Japan: IEEE, 2009. 108-111 [75] Ishizaki T, Kashima K, Imura J, Katoh A, Morita H, Aihara K. Distributed parameter modeling and finite-frequency loop-shaping of electromagnetic molding machine. Control Engineering Practice, 2013, 21(12): 1735-1743 doi: 10.1016/j.conengprac.2013.08.003 [76] Wang H, Yang G H. A finite frequency approach to filter design for uncertain discrete-time systems. International Journal of Adaptive Control & Signal Processing, 2008, 22: 533-550 http://cn.bing.com/academic/profile?id=1974860912&encoded=0&v=paper_preview&mkt=zh-cn [77] hang X N, Yang G H. Delay-dependent filtering for discrete-time systems with finite frequency small gain specifications. In: Proceedings of the 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference. Shanghai, China: IEEE, 2009. 4420-4425 [78] Gao H J, Li X W, Yu X H. Finite frequency approaches to H∞ filtering for continuous-time state-delayed systems. In: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference. Orlando, FL: IEEE, 2011. 2583-2588 [79] Gao H J, Li X W. H∞ filtering for discrete-time state-delayed systems with finite frequency specifications. IEEE Transactions on Automatic Control, 2011, 56(12): 2935-2941 doi: 10.1109/TAC.2011.2159909 [80] Gao H J, Li X W. Robust Filtering for Uncertain Systems. Switzerland: Springer International Publishing, 2014. [81] 李贤伟. 时滞系统的有限频域H∞滤波分析与综合[硕士学位论文], 哈尔滨工业大学, 中国, 2011Li Xian-Wei. Finite Frequency H∞ Filtering Analysis and Synthesis of Time-Delay Systems [Master dissertation], Harbin Institute of Technology, China, 2011 [82] Li X W, Gao H J. Robust finite frequency H∞ filtering for uncertain 2-D Roesser systems. Automatica, 2012, 48(6): 1163-1170 doi: 10.1016/j.automatica.2012.03.012 [83] Li X W, Gao H J, Karimi H R. Robust H∞ filtering for 2-D FM systems: a finite frequency approach. In: Proceedings of the 51st IEEE Annual Conference on Decision and Control. Maui, Hawaii, USA: IEEE, 2012. 3526-3530 [84] Li X W, Gao H J. Robust finite frequency H∞ filtering for uncertain 2-D systems: the FM model case. Automatica, 2013, 49(8): 2446-2452 doi: 10.1016/j.automatica.2013.04.014 [85] Li X W, Gao H J. Reduced-order generalized H∞ filtering for linear discrete-time systems with application to channel equalization. IEEE Transactions on Signal Processing, 2014, 62(13): 3393-3402 doi: 10.1109/TSP.2014.2324996 [86] 丁大伟. 线性切换系统的若干问题研究 [博士学位论文], 东北大学, 中国, 2010Ding Da-Wei. Study on Some Problems of Switched Linear Systems [Ph.D. dissertation], Northeastern University, China, 2010 [87] Ding D W, Yang G H. Fuzzy filter design for nonlinear systems in finite-frequency domain. IEEE Transactions on Fuzzy Systems, 2010, 18(5): 935-945 doi: 10.1109/TFUZZ.2010.2058807 [88] Wang H, Ju H H. Reliable H∞ filtering for LPV systems with sensor faults in finite frequency domain. International Journal of Systems Science, 2013, 44(12): 2310-2320 doi: 10.1080/00207721.2012.702240 [89] Grigoriadis K M. Optimal H∞ model reduction via linear matrix inequalities: continuous- and discrete-time cases. Systems & Control Letters, 1995, 26(5): 321-333 [90] Du X, Yang G H. H∞ model reduction of linear continuous-time systems over finite-frequency interval. IET Control Theory and Applications, 2010, 4(3): 499-508 doi: 10.1049/iet-cta.2008.0537 [91] 杜鑫. 基于LMI技术的线性系统模型降阶与静态输出反馈控制器设计[博士学位论文], 东北大学, 中国, 2009Du Xin. LMI-based Approaches to Model Reduction and Static Output Feedback Controller Design for Linear Systems [Ph.D. dissertation], Northeastern University, China, 2009 [92] Du X, Liu F W, Zhu X L, Chao W. Finite frequency approaches to H∞ model reduction for continuous-time state delayed systems. In: Proceedings of the 24th Chinese Control and Decision Conference. Taiyuan, China: IEEE, 2012. 470-475 [93] Du X, Liu F W, Zhu X L, Zheng M. H∞ model reduction of discrete-time linear state delayed systems over finite-frequency ranges. In: Proceedings of the 24th Chinese Control and Decision Conference. Taiyuan, China: IEEE, 2012. 954-959 [94] Li X W, Gao H J. A frequency-specific enhanced approach to transfer function approximation. In: Proceedings of the 23rd IEEE International Symposium on Industrial Electronics. Istanbul, Turkey: IEEE, 2014. 18-22 [95] Li X W, Gao H J. Min-max approximation of transfer functions with application to filter design. IEEE Transactions on Signal Processing, 2015, 63(1): 31-40 doi: 10.1109/TSP.2014.2364787 [96] Li X W, Yin S, Gao H J. Passivity-preserving model reduction with finite frequency H∞ approximation performance. Automatica, 2014, 50(9): 2294-2303 doi: 10.1016/j.automatica.2014.07.001 [97] Li X W, Yu C B, Gao H J. Frequency-limited H∞ model reduction for positive systems. IEEE Transactions on Automatic Control, 2015, 60(4): 1093-1098 doi: 10.1109/TAC.2014.2352751 [98] Yang R, Xie L H, Zhang C S. Generalized two-dimensional Kalman-Yakubovich-Popov lemma for discrete Roesser model. IEEE Transactions on Circuits and Systems—I: Regular Papers, 2008, 55(10): 3223-3233 doi: 10.1109/TCSI.2008.923284 [99] Li X W, Lam J, Cheung K C. Generalized H∞ model reduction for stable two-dimensional discrete systems. Multidimensional Systems and Signal Processing, 2016, 27(2): 359-382 doi: 10.1007/s11045-014-0306-3 [100] Shen J, Lam J. Improved results on H∞ model reduction for continuous-time linear systems over finite frequency ranges. Automatica, 2015, 5: 79-84 http://cn.bing.com/academic/profile?id=2089410483&encoded=0&v=paper_preview&mkt=zh-cn [101] Ding D W, Du X, Li X. Finite-frequency model reduction of two-dimensional digital filters. IEEE Transactions on Automatic Control, 2015, 60(6): 1624-1629 doi: 10.1109/TAC.2014.2359305 [102] Wang H, Yang G H. A finite frequency domain approach to fault detection observer design for linear continuous-time systems. Asian Journal of Control, 2008, 10(5): 559-568 doi: 10.1002/asjc.v10:5 [103] Wang H, Yang G H. A finite frequency domain approach to fault detection for linear discrete-time systems. International Journal of Control, 2008, 81(7): 1162-1171 doi: 10.1080/00207170701691513 [104] 王恒, 居鹤华, 杨光红. 线性多胞型不确定连续系统故障检测滤波器设计. 自动化学报, 2010, 36(5): 742-750 http://www.aas.net.cn/CN/abstract/abstract13721.shtmlWang Heng, Ju He-Hua, Yang Guang-Hong. Fault detection filter design for linear polytopic uncertain continuous-time systems. Acta Automatica Sinica, 2010, 36(5): 742-750 http://www.aas.net.cn/CN/abstract/abstract13721.shtml [105] Wang H, Yang G H. Simultaneous fault detection and control for uncertain linear discrete-time systems. IET Control Theory and Applications, 2009, 3(5): 583-594 doi: 10.1049/iet-cta.2007.0463 [106] Yang G H, Wang H, Xie L H. Fault detection for output feedback control systems with actuator stuck faults: a steady-state-based approach. International Journal of Robust & Nonlinear Control, 2010, 20(15): 1739-1757 http://cn.bing.com/academic/profile?id=2022087177&encoded=0&v=paper_preview&mkt=zh-cn [107] Yang H J, Xia Y Q, Liu B. Fault detection for T-S fuzzy discrete systems in finite-frequency domain. IEEE Transactions on Systems, Man, and Cybernetics--Part B: Cybernetics, 2011, 41(4): 911-920 doi: 10.1109/TSMCB.2010.2099653 [108] Yang H J, Xia Y Q, Zhang J H. Generalised finite-frequency KYP lemma in delta domain and applications to fault detection. International Journal of Control, 2011, 84(3): 511-525 doi: 10.1080/00207179.2011.561501 [109] Zhang Z, Jaimoukha I M. Optimal state space solution to the fault detection problem at single frequency. In: Proceedings of the 18th IFAC World Congress. Milano, Italy: IFAC, 2011. 7619-7624 [110] Long Y, Yang G H. Fault detection and isolation for networked control systems with finite frequency specifications. International Journal of Robust & Nonlinear Control, 2014, 24(3): 495-514 http://cn.bing.com/academic/profile?id=1917424513&encoded=0&v=paper_preview&mkt=zh-cn [111] Long Y, Yang G H. Fault detection in finite frequency domain for networked control systems with missing measurements. Journal of the Franklin Institute, 2013, 350(9): 2605-2626 doi: 10.1016/j.jfranklin.2013.01.015 [112] Zhang K, Jiang B, Shi P, Xu J F. Multi-constrained fault estimation observer design with finite frequency specifications for continuous-time systems. International Journal of Control, 2014, 87(8): 1635-1645 doi: 10.1080/00207179.2014.880950 [113] Zhang K, Jiang B, Shi P, Xu J. Fault estimation observer design for discrete-time systems in finite-frequency domain. International Journal of Robust & Nonlinear Control, 2015, 25(9): 1379-1398 http://cn.bing.com/academic/profile?id=1486867056&encoded=0&v=paper_preview&mkt=zh-cn [114] Zhang K, Jiang B, Shi P, Xu J F. Analysis and design of robust H∞ fault estimation observer with finite-frequency specifications for discrete-time fuzzy systems. IEEE Transactions on Cybernetics, 2015, 45(7): 1225-1235 doi: 10.1109/TCYB.2014.2347697 [115] Zhou T. Generalized positiveness of spatially interconnected systems over quadratically constrained frequency domains. Systems & Control Letters, 2012, 61(12): 1187-1193 http://cn.bing.com/academic/profile?id=2031127339&encoded=0&v=paper_preview&mkt=zh-cn [116] Sun W C, Zhao Y, Li J F, Zhang L X, Gao H J. Active suspension control with frequency band constraints and actuator input delay. IEEE Transactions on Industrial Electronics, 2012, 59(1): 530-537 doi: 10.1109/TIE.2011.2134057 [117] Xiong J L, Petersen I R, Lanzon A. Finite frequency negative imaginary systems. IEEE Transactions on Automatic Control, 2012, 57(11): 2917-2922 doi: 10.1109/TAC.2012.2193705 [118] Hoang H G, Tuan H D, Apkarian P. A Lyapunov variable-free KYP lemma for SISO continuous systems. IEEE Transactions on Automatic Control, 2008, 53(11): 2669-2673 doi: 10.1109/TAC.2008.2007156 [119] Hoang H G, Tuan H D, Nguyen T Q. Frequency-selective KYP lemma, IIR filter, and filter bank design. IEEE Transactions on Signal Processing, 2009, 57(3): 956-965 doi: 10.1109/TSP.2008.2009012 [120] Pipeleers G, Vandenberghe L. Generalized KYP lemma with real data. IEEE Transactions on Automatic Control, 2011, 56(12): 2942-2946 doi: 10.1109/TAC.2011.2161945 [121] Pipeleers G, Iwasaki T, Hara S. Generalizing the KYP lemma to multiple frequency intervals. SIAM Journal on Control and Optimization, 2014, 52(6): 3618-3638 doi: 10.1137/130938451 [122] Graham M R, de Oliveira M C, de Callafon R A. An alternative Kalman-Yakubovich-Popov lemma and some extensions. Automatica, 2009, 45(6): 1489-1496 doi: 10.1016/j.automatica.2009.02.006 [123] Graham M R, de Oliveira M C. Linear matrix inequality tests for frequency domain inequalities with affine multipliers. Automatica, 2010, 46(5): 897-901 doi: 10.1016/j.automatica.2010.02.009 [124] Tanaka T, Langbort C. Symmetric formulation of the S-procedure, Kalman-Yakubovich-Popov lemma and their exact losslessness conditions. IEEE Transactions on Automatic Control, 2013, 58(6): 1486-1496 doi: 10.1109/TAC.2013.2237832 [125] Iwasaki T, Hara S, Fradkov A L. Time domain interpretations of frequency domain inequalities on (semi) finite ranges. Systems & Control Letters, 2005, 54(7): 681-691 http://cn.bing.com/academic/profile?id=2072400962&encoded=0&v=paper_preview&mkt=zh-cn [126] Kaizuka Y, Kojima C, Hara S. Time domain characterization of finite frequency properties via behavioral approach. In: Proceedings of the SICE Annual Conference 2008. Tokyo, Japan: IEEE, 2008. 2364-2369 [127] Kojima C, Hara S. An achievability condition for n-dimensional behaviors with a finite frequency specification: dissipation inequalities approach. In: Proceedings of the 49th IEEE Conference on Decision and Control. Atlanta, GA: IEEE, 2010. 7730-7735 [128] Kojima C, Hara S. Controller synthesis for multiple finite frequency specifications: dissipation inequalities approach. In: Proceedings of the SICE Annual Conference 2010. Taipei, China: IEEE, 2010. 173-178 [129] Sun W C, Li J F, Zhao Y, Gao H J. Vibration control for active seat suspension systems via dynamic output feedback with limited frequency characteristic. Mechatronics, 2011, 21(1): 250-260 doi: 10.1016/j.mechatronics.2010.11.001 [130] Li X W, Gao H J. Load mitigation for a floating wind turbine via generalized H∞ structural control. IEEE Transactions on Industrial Electronics, 2016, 63(1): 332-342 doi: 10.1109/TIE.2015.2465894 [131] Nagahara M, Yamamoto Y. Frequency domain min-max optimization of noise-shaping delta-sigma modulators. IEEE Transactions on Signal Processing, 2012, 60(6): 2828-2839 doi: 10.1109/TSP.2012.2188522 [132] Li X W, Gao H J, Yu C B. An iterative LMI approach to IIR noise transfer function optimization for delta-sigma modulators. In: Proceedings of the 3rd Australian Control Conference. Fremantle, WA: IEEE, 2013. 67-72 [133] Li X W, Yu C B, Gao H J. Design of delta-sigma modulators via generalized Kalman-Yakubovich-Popov lemma. Automatica, 2014, 50(10): 2700-2708 doi: 10.1016/j.automatica.2014.09.002 [134] Li X W, Gao H J, Gu K Q. Delay-independent stability analysis of linear time-delay systems based on frequency discretization. Automatica, 2016, 70: 288-294 doi: 10.1016/j.automatica.2015.12.031 [135] Li X W, Lam J, Gao H J, Gu Y. A frequency-partitioning approach to stability analysis of two-dimensional discrete systems. Multidimensional Systems and Signal Processing, 2015, 26(1): 67-93 doi: 10.1007/s11045-013-0237-4 [136] ÅstrÖm K J, Murray R M. Feedback Systems: An Introduction for Scientists and Engineers. Princeton: Princeton University Press, 2008. [137] Wu F, Jaramillo J J. Computationally efficient algorithm for frequency-weighted optimal H∞ model reduction. Asian Journal of Control, 2003, 5(3): 341-349 http://cn.bing.com/academic/profile?id=2138201448&encoded=0&v=paper_preview&mkt=zh-cn [138] De Oliveira M C, Skelton R E. Stability tests for constrained linear systems. Perspectives in Robust Control. London: Springer-Verlag, 2001. 241-257 http://cn.bing.com/academic/profile?id=1927395284&encoded=0&v=paper_preview&mkt=zh-cn [139] Chao H H, Vandenberghe L. Extensions of semidefinite programming methods for atomic decomposition. In: Proceedings of the 41st IEEE International Conference on Acoustics, Speech and Signal Processing. Shanghai, China: IEEE, 2016. 4757-4761 期刊类型引用(7)
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![](data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' width='210' height='179'><foreignObject width='2000' height='100%'><div xmlns='http://www.w3.org/1999/xhtml' style='font-size:16px;font-family:Helvetica'><table>
<thead>
<tr><td valign="middle" class="table_top_border">$\Lambda $</td><td valign="middle" class="table_top_border">$\Phi $</td><td valign="middle" class="table_top_border">$\Omega $</td><td valign="middle" class="table_top_border">$\Psi $</td></tr>
</thead>
<tbody>
<tr><td valign="middle" class="table_top_border2" rowspan="3">连续系统</td><td valign="middle" class="table_top_border2" rowspan="3">$\left[ \begin{matrix}
0 & 1 \\
1 & 0 \\
\end{matrix} \right]$</td><td valign="middle" class="table_top_border2">低频</td><td valign="middle" class="table_top_border2">$\left[ \begin{matrix}
-1 & 0 \\
0 & \omega _{l}^{2} \\
\end{matrix} \right]$</td></tr>
<tr><td valign="middle">中频</td><td valign="middle">$\left[ \begin{matrix}
-1 & \text{j}{{\omega }_{c}} \\
-\text{j}{{\omega }_{c}} & -{{\omega }_{1}}{{\omega }_{2}} \\
\end{matrix} \right]$</td></tr>
<tr><td valign="middle">高频</td><td valign="middle">$\left[ \begin{matrix}
1 & 0 \\
0 & -\omega _{h}^{2} \\
\end{matrix} \right]$</td></tr>
<tr><td valign="middle" rowspan="3" class="table_bottom_border">离散系统</td><td valign="middle" rowspan="3" class="table_bottom_border">$\left[ \begin{matrix}
1 & 0 \\
0 & -1 \\
\end{matrix} \right]$</td><td valign="middle">低频</td><td valign="middle">$\left[ \begin{matrix}
0 & 1 \\
1 & -2\cos {{\omega }_{r}} \\
\end{matrix} \right]$</td></tr>
<tr><td valign="middle">中频</td><td valign="middle">$\left[ \begin{matrix}
0 & {{\text{e}}^{\text{j}\omega c}} \\
{{\text{e}}^{-\text{j}\omega \text{c}}} & -2\cos {{\omega }_{r}} \\
\end{matrix} \right]$</td></tr>
<tr><td valign="middle" class="table_bottom_border">高频</td><td valign="middle" class="table_bottom_border">$\left[ \begin{matrix}
0 & -1 \\
-1 & 2\cos {{\omega }_{h}} \\
\end{matrix} \right]$</td></tr>
</tbody>
</table></div></foreignObject></svg>)
