-
摘要: 大数据面向异构自治的多源海量数据，旨在挖掘数据间复杂且演化的关联.随着数据采集存储和互联网技术的发展，大数据分析和应用已成为各行各业的研发热点.本文从大数据的本质特征开始，评述现有的几种大数据模型，包括5V，5R，4P和HACE定理，同时从知识建模的角度，介绍一种大数据知识工程模型BigKE来生成大知识，并对大知识的前景进行展望.Abstract: Big data deals with heterogeneous and autonomous multi-sources, and aims at mining complex and evolving relationships among data. With the fast development of data collection, data storage and networking technologies, big data analytics has become a hot topic for research and development in various fields. This paper starts with the essential characteristics of big data, reviews existing popular models for big data, including 5V, 5R, 4P and the HACE theorem. Also, from the viewpoint of knowledge modeling, this paper introduces BigKE, a big data knowledge engineering model for big knowldedge, and discusses the challenges and opportunities of big knowledge research and development.
-
Key words:
- Big data /
- knowledge mining /
- heterogeneity /
- fragmented knowledge /
- online learning
-
广义系统[1]的研究是从20世纪70年代[2]开始的, 在近40年的历史中广义系统理论研究得到了迅速发展, 并取得了一系列的丰硕成果[3-5].广义系统理论已成为现代控制理论中一个独立的研究领域, 这些研究成果主要集中在广义定常系统[6]、广义周期时变系统[7-8]和广义时变系统[9-10].随着科学技术的发展和工程技术的需要, 广义时变系统时域控制研究也受到了广泛的关注.
时域稳定性最早是由Kamenkov在文献[11]中提出来的, 至今已有很多可观的成果.文献[12-15]提出了时域稳定和时域有界的定义, 并对一般线性时变系统的时域稳定进行研究.文献[16-18]给出了带脉冲的线性时变系统的时域稳定的定义, 并利用 $L_{2}$ 增益给出时域稳定的判定定理.文献[19-22]探讨了带有干扰的参数不确定性系统时域稳定及时域控制的问题, 文献[23-24]利用微分矩阵不等式研究了一类带有跳变的线性时变系统的时域稳定. Ambrosino等研究了状态依赖时变脉冲动力系统的时域稳定性问题[25], 并在文献[26-29]中对时间依赖和状态依赖脉冲动力系统的时域稳定进行了对比和分析.文献[30-31]利用分段线性化将时变矩阵转化为一组标准的矩阵不等式, 解决了小区间内时变矩阵不等式的求解问题.时变脉冲系统中的研究方法多是时变系统与正常脉冲系统的自然推广, 文中所研究的广义时变脉冲系统不仅是时变脉冲系统的推广, 而且还考虑到广义系统自身的脉冲效应, 因此给研究和实验带来了一定的困难.
本文主要研究状态依赖广义时变脉冲系统的时域稳定问题.在考虑广义系统自身无脉冲效应的前提下, 运用微分矩阵不等式方法给出了广义时变脉冲系统时域稳定的充分条件, 并设计了状态反馈控制器.最后根据分段线性化将小区间内广义时变矩阵不等式转化为广义时不变线性矩阵不等式, 应用Matlab LMI工具箱编程进行求解.
1. 问题描述
$\boldsymbol{R} ^{n}$ 表示 $n$ -维欧几里得空间, $\boldsymbol{R}^{+}$ 是正实数集, $R > 0$ 是对称正定矩阵. ${N}$ 是自然数. $J=[t_0, t_0 +T]$ 表示时间域, 其中, $T \in \boldsymbol{R}^{+}$ .
考虑如下形式的状态依赖广义时变脉冲系统:
$ \begin{align} \left\{ \begin{array}{l} E\dot{{\boldsymbol x}}(t)=A(t){\boldsymbol x}(t), \quad {\boldsymbol x}(t)\notin \mathcal{S}_k\\ {\boldsymbol x}^{+}_{k}(t)=A_{d, k}{\boldsymbol x}(t), \quad {\boldsymbol x}(t)\in \mathcal{S}_k\\ {\boldsymbol x}(t_{0})={\boldsymbol x}_{0}, \quad k=1, 2, \cdot\cdot\cdot, N \end{array} \right. \end{align} $
(1) 其中, ${\boldsymbol x}(t)\in \boldsymbol{R}^{n}$ 为状态向量; $A(\cdot):t\in \boldsymbol{R}^{+}\mapsto \boldsymbol{R}^{n\times n}$ 是连续的函数矩阵, $A_{d, k}\in \boldsymbol{R}^{n\times n}, k=1, 2, \cdot\cdot\cdot, {N} $ 是时不变矩阵; $E$ 为奇异矩阵; ${S}_{k}\subseteq \boldsymbol{R}^{n}, k=1, 2, \cdot\cdot\cdot, {N}$ 是单连通互不相交的跳变集合 $({\boldsymbol x}_{0}\notin \mathcal{S}_{k})$ .根据脉冲时刻定义如下跳变时间集合:
$ \mathcal{T}_{x(\cdot)}=\{t\in \boldsymbol{R}^{+}|{\boldsymbol x}(t)\in \mathcal{S}_{k}, k=1, 2, \cdot\cdot\cdot, {N}\} $
定义1.如果 ${\boldsymbol x}_{0}^{\rm T}E^{\rm T}R{\boldsymbol x}_{0}\leq c_{1}$ 推出 ${\boldsymbol x}^{\rm T}(t)E^{\rm T}\Gamma(t){\boldsymbol x}(t) < c_{1}$ , 对 $\forall t\in J$ , 则称系统 (1) 是对于 $(c_1, J, R, \Gamma(\cdot))$ 时域稳定的.其中 $c_{1}>0, R$ 是对称正定矩阵, $\Gamma(\cdot)$ 是定义在 $J$ 上的函数矩阵, $\Gamma(t_0) < R$ .
定义2.如果存在常数 $s$ 对于任意 $t\in J$ 使得 $\det (sE-A(t)) \neq 0$ , 则广义时变系统 $E\dot{{\boldsymbol x}}(t)=A(t){\boldsymbol x}(t)$ 是一致正则的.
系统 (1) 一致正则与Campbell意义下的解析可解是等价的, 广义时变系统的一致正则性和 $A(t)$ 的连续性保证了其解的存在唯一性, 下面的讨论假定系统是一致正则的.我们将系统作如下分解:
$ \begin{align*} &MEN\!=\!\left[\begin{array}{ccc} I&0\\0&0 \end{array}\right], MA(t)N\!=\!\left[\begin{array}{ccc} A_{11}(t)& A_{12}(t)\\A_{21}(t)& A_{22}(t)\end{array}\right]\\ &N^{-1}{\boldsymbol x}(t)=\left[\begin{array}{ccc}{\boldsymbol x}_{1}(t)& {\boldsymbol x}_{2}(t)\end{array}\right]\end{align*} $
其中, $M, N$ 均为可逆矩阵, 则系统 (1) 等价于如下系统:
$ \begin{align*} \begin{aligned} &\dot{{\boldsymbol x}}_{1}(t)=A_{11}(t){\boldsymbol x}_{1}(t)+A_{12}(t) {\boldsymbol x}_{2}(t)\\ &0 = A_{21}(t) {\boldsymbol x}_{1}(t)+A_{22}(t){\boldsymbol x}_{2}(t) \end{aligned} \end{align*} $
显然, 系统 (1) 对于任意初始条件无脉冲的充要条件是 $A_{22}(t)$ 可逆.
另外考虑一类带有外部干扰的状态依赖广义时变脉冲系统
$ \begin{align} \left\{ \begin{array}{l} E\dot{{\boldsymbol x}}(t)=A(t){\boldsymbol x}(t)+G(t){\boldsymbol\omega}(t), \quad {\boldsymbol x}(t)\notin \mathcal{S}_k\\ {\boldsymbol x}^{+}_{k}(t)=A_{d, k}{\boldsymbol x}(t), \quad {\boldsymbol x}(t)\in \mathcal{S}_k\\ {\boldsymbol x}(t_{0})={\boldsymbol x}_{0}, \quad k=1, 2, \cdot\cdot\cdot, {N} \end{array} \right. \end{align} $
(2) 外部干扰 ${\boldsymbol \omega}(t)$ 满足:
$ \begin{align} \begin{array}{l} \int_{t_0}^{t_0 +T}{\boldsymbol \omega}^{\rm T}(s){\boldsymbol \omega}(s){\rm d}s\leq d, \quad d\geq 0 \end{array} \end{align} $
(3) 定义3.如果 ${\boldsymbol x}_{0}^{\rm T}E^{\rm T}R{\boldsymbol x}_{0}\leq c_{1}$ 推出 ${{\boldsymbol{x}}^{\text{T}}}(t){{E}^{\text{T}}}\Gamma (t)\boldsymbol{x}(t) < {{c}_{2}}$ , 对 $\forall t\in J$ , 则称系统 (2) 是对于 $(c_1, c_2, {\boldsymbol \omega(t)}, J, R, \Gamma(\cdot))$ 时域稳定的.其中 $0 < c_{1} < c_{2}, R$ 是对称正定矩阵, $\Gamma(\cdot)$ 是定义在 $J$ 上的函数矩阵, $\Gamma(t_0) < R$ , 外部干扰 ${\boldsymbol \omega}(t)$ 满足式 (3).
2. 时域稳定性分析
本节分别对系统 (1) 和 (2) 给出时域稳定的充分条件.
定理1.对于系统 (1), 如果存在分段连续可微对称非奇异函数矩阵 $P(\cdot)$ 在 $J$ 上满足下列一组矩阵不等式, $\forall t\in J$ .
$ E^{\rm T}P(t)=P^{\rm T}(t)E\geq 0 $
(4a) $ A^{\rm T}(t)P(t)+P^{\rm T}(t)A(t)+E^{\rm T}\dot{P}(t) < 0 $
(4b) $ \begin{array}{l} A_{d, k}^{\rm T}E^{\rm T}P(t)A_{d, k}-E^{\rm T}P(t) < 0\\ \qquad {\boldsymbol x}(t)\in \mathcal{S}_k, \qquad k=1, 2, \cdot\cdot\cdot, {N} \end{array} $
(4c) $ E^{\rm T}\Gamma(t)\leq E^{\rm T}P(t)\leq E^{\rm T}P(t_0) < E^{\rm T}R $
(4d) 则称系统 (1) 对于 $(c_1, J, R, \Gamma(\cdot))$ 时域稳定.
证明. 由定理条件 (4a) 和下式
$ \begin{align*} M^{\rm -T}P(t)N=\left[\begin{array}{cc}P_{1}(t)&P_{2}(t)\\ P_{3}(t)&P_{4}(t)\end{array}\right] \end{align*} $
有 $P_2(t)=0$ 和 $P_1(t)$ 对称, 将条件 (4b) 进行分解可得:
$ \begin{align*} \begin{aligned} &N^{\rm T}A^{\rm T}(t)M^{\rm T}M^{\rm -T}P(t)N +\\ &\quad N^{\rm T}P^{\rm T}(t)M^{-1}MA(t)N+\\ &\quad N^{\rm T}E^{\rm T}M^{\rm T}M^{\rm -T}\dot{P}(t)N =\\ &\quad \left[\begin{array}{cc}A_{11}^{\rm T}(t)&A_{21}^{\rm T}(t)\\A_{12}^{\rm T}(t)&A_{22}^{\rm T}(t)\end{array}\right]\left[\begin{array}{cc}P_{1}(t)&P_{2}(t)\\P_{3}(t)&P_{4}(t)\end{array}\right]+\\ &\quad \left[\begin{array}{cc}P_{1}^{\rm T}(t)&P_{3}^{\rm T}(t)\\P_{2}^{\rm T}(t)&P_{4}^{\rm T}(t)\end{array}\right]\left[\begin{array}{cc}A_{11}(t)&A_{12}(t)\\A_{21}(t)&A_{22}(t)\end{array}\right]+\\ &\quad \left[\begin{array}{cc}I&0\\0&0\end{array}\right]\left[\begin{array}{cc}\dot{P}_{1}(t)&\dot{P}_{2}(t)\\ \dot{P}_{3}(t)&\dot{P}_{4}(t)\end{array}\right]=\\ &\quad \left[\begin{array}{cc}\ast&\ast\\\ast&A_{22}^{\rm T}(t)P_{4}(t)+P_{4}^{\rm T}(t)A_{22}(t)\end{array}\right] < 0 \end{aligned} \end{align*} $
由上式显然有 $A_{22}^{\rm T}(t)P_{4}(t)+P_{4}^{\rm T}(t)A_{22}(t) < 0$ .因此, $A_{22}(t)$ 可逆, 系统对任意初始状态无脉冲.
构造下列广义Lyapunov函数
$ \begin{align*} V(t, {\boldsymbol x})={\boldsymbol x}^{\rm T}(t)E^{\rm T}P(t){\boldsymbol x}(t) \end{align*} $
当 $t\notin \mathcal{T}_{x(\cdot)}$ , 即系统状态向量 ${\boldsymbol x}(t)$ 没有达到跳变集合.则对 $V(t, {\boldsymbol x})$ 求导:
$ \begin{align*} &\dot V(t, {\boldsymbol x})={\boldsymbol x}^{\rm T}(t)[A^{\rm T}(t)P(t)+P^{\rm T}(t)A(t)+\\ &\qquad E^{\rm T}\dot{P}(t)]{\boldsymbol x}(t) \end{align*} $
由条件 (4b) 可知 $\dot V(t, {\boldsymbol x})< 0$ .
然而当 $t\in\mathcal{T}_{x(\cdot)}$ , 即系统依赖状态发生跳变, 由条件 (4c) 可得:
$ \begin{align*} & V (t, {\boldsymbol x}^+_k)-V (t, {\boldsymbol x})={\boldsymbol x}^{\rm T}(t)\times\\ & \qquad [A_{d, k}^{\rm T}E^{\rm T}P (t) A_{d, k}-E^{\rm T}P (t)]{\boldsymbol x}(t) < 0 \end{align*} $
则可以推出 $V(t, {\boldsymbol x})$ 在 $J$ 上是严格递减的.
对于系统 (1) 给定初始条件 $t_0$ 使得 ${\boldsymbol x}_{0}^{\rm T}E^{\rm T}R{\boldsymbol x}_{0}\leq c_{1}$ , 对所有的 $t\in J$ 都有:
$ \begin{align*} \begin{aligned} {\boldsymbol x}^{\rm T}(t)E^{\rm T}\Gamma(t){\boldsymbol x}(t)&\leq {\boldsymbol x}^{\rm T}(t)E^{\rm T}P(t){\boldsymbol x}(t)\leq\\ & {\boldsymbol x}_{0}^{\rm T}E^{\rm T}P(t_0){\boldsymbol x}_{0} < \\ &{\boldsymbol x}_{0}^{\rm T}E^{\rm T}RE{\boldsymbol x}_{0} < c_1 \end{aligned} \end{align*} $
因此系统 (1) 对于 $(c_1, J, R, \Gamma(\cdot))$ 时域稳定.
定理2.对于系统 (2), 如果存在分段连续可微对称非奇异函数矩阵 $P(\cdot)$ 在 $J$ 上满足下列不等式, $\forall t\in J$ .
$ E^{\rm T}P(t)=P^{\rm T}(t)E\geq 0 $
(5a) $ \begin{bmatrix} \Pi(t)&P^{\rm T}(t)G(t)\\ G^{\rm T}(t)P(t)&-I \end{bmatrix}< 0 $
(5b) $ \begin{array}{l} A_{d, k}^{\rm T}E^{\rm T}P(t)A_{d, k}-E^{\rm T}P(t)< 0\\ \qquad {\boldsymbol x}(t)\in \mathcal{S}_k, \qquad k=1, 2, \cdot\cdot\cdot, {N} \end{array} $
(5c) $ E^{\rm T}\Gamma(t)\leq E^{\rm T}P(t)\leq E^{\rm T}P(t_0) < E^{\rm T}R $
(5d) 则称系统 (2) 是对于 $(c_1, c_2, {\boldsymbol \omega}(t), J, R, \Gamma(\cdot))$ 时域稳定的.其中
$ \begin{align*} \Pi(t)=A^{\rm T}(t)P(t)+P^{\rm T}(t)A(t)+E^{\rm T}\dot{P}(t) \end{align*} $
证明.由条件 (5b) 可知 $\Pi(t) < 0$ , 同理可证系统 (2) 对任意初始状态无脉冲.
考虑下列广义Lyapunov函数
$ \begin{align*} V(t, {\boldsymbol x})={\boldsymbol x}^{\rm T}(t)E^{\rm T}P(t){\boldsymbol x}(t) \end{align*} $
当 $t\notin \mathcal{T}_{x(\cdot)}$ , 即系统状态向量 ${\boldsymbol x}(t)$ 没有达到跳变集合.则对 $V(t, {\boldsymbol x})$ 求导:
$ \begin{align*} \begin{array}{l} \dot V(t, {\boldsymbol x})={\boldsymbol x}^{\rm T}(t)\Pi(t){\boldsymbol x}(t)+{\boldsymbol \omega}^{\rm T}(t)G^{\rm T}(t)P(t)\times\\ {\boldsymbol x}(t)+{\boldsymbol x}^{\rm T}(t)P^{\rm T}(t)G(t){\boldsymbol \omega}(t) \end{array} \end{align*} $
构造下列向量
$ \begin{align*} {\boldsymbol z}(t)=\left[\begin{array}{c}{\boldsymbol x}(t)\\ {\boldsymbol \omega}(t) \end{array}\right] \end{align*} $
由式 (5b) 得:
$ \begin{array}{lll} {\boldsymbol z}^{\rm T}(t) \left[\begin{array}{cc}\Pi(t)&P^{\rm T}(t)G(t)\\ G^{\rm T}(t)P(t)&-I\end{array}\right]{\boldsymbol z}(t)=\\ \quad \dot V(t, {\boldsymbol x})-{\boldsymbol \omega}^{\rm T}(t){\boldsymbol \omega}(t) < 0 \end{array} $
显然有:
$ \begin{align*} \dot V(t, {\boldsymbol x}) < {\boldsymbol \omega}^{\rm T}(t){\boldsymbol \omega}(t) \end{align*} $
对上式在 $[t_0, t]$ 上求积分得:
$ \begin{align*} \int_{t_0}^{t}\dot V(s, {\boldsymbol x}){\rm d}s < \int_{t_0}^{t}{\boldsymbol \omega}^{\rm T}(s){\boldsymbol \omega}(s){\rm d}s \end{align*} $
则
$ \begin{align*} \begin{aligned} V(t, {\boldsymbol x})& < V(t_0, {\boldsymbol x})+\int_{t_0}^{t}{\boldsymbol \omega}^{\rm T}(s){\boldsymbol \omega}(s){\rm d}s < \\ &{\boldsymbol x}_{0}^{\rm T}E^{\rm T}P(t_0){\boldsymbol x}_{0}+d < \\ &{\boldsymbol x}_{0}^{\rm T}E^{\rm T}R{\boldsymbol x}_{0}+d < \\ &c_1+d \end{aligned} \end{align*} $
当 $t\in\mathcal{T}_{x(\cdot)}$ , 即系统状态发生跳变
$ \begin{align*} \begin{aligned} V(t, {\boldsymbol x}^+)&= {\boldsymbol x}^{\rm T}(t)A_{d, k}^{\rm T}E^{\rm T}P(t)A_{d, k}{\boldsymbol x}(t)< \\ &{\boldsymbol x}^{\rm T}(t)E^{\rm T}P(t){\boldsymbol x}(t) < V(t, {\boldsymbol x}) < \\ &c_1+d \end{aligned} \end{align*} $
则 $\exists c_2>c_1$ 使得 $V(t, {\boldsymbol x})$ 在 $J$ 上有 $V(t, {\boldsymbol x}) < c_2$ .
对系统 (2) 给定初始条件 $t_0$ 使得 ${\boldsymbol x}_{0}^{\rm T}E^{\rm T}R{\boldsymbol x}_{0}\leq c_{1}$ , 对所有的 $t\in J$ 都有:
$ \begin{align*} {\boldsymbol x}^{\rm T}(t)E^{\rm T}\Gamma (t){\boldsymbol x}(t)\leq {\boldsymbol x}^{\rm T}(t)E^{\rm T}P(t){\boldsymbol x}(t) < c_2\end{align*} $
则称系统 (2) 对于 $(c_1, c_2, {\boldsymbol\omega}(t), J, R, \Gamma(\cdot))$ 时域稳定.
接下来应用S-procedure理论[30]进一步研究系统 (1) 和系统 (2) 的时域稳定.
引理1[25].存在一个连续闭集 $\mathcal{S}\subseteq \boldsymbol{R}^n$ 、对称矩阵 $Q_0\in \boldsymbol{R}^{n\times n}$ 、 $Q_i\in \boldsymbol{R}^{n\times n}$ 满足:
$ {\boldsymbol x}^{\rm T}(t)Q_{0}{\boldsymbol x}(t) < 0, \quad {\boldsymbol x}(t)\in \mathcal{S} $
(6a) $ {\boldsymbol x}^{\rm T}(t)Q_{i}{\boldsymbol x}(t) < 0, \quad {\boldsymbol x}(t)\in \mathcal{S}, \quad i=1, \cdot\cdot\cdot, p $
(6b) 则一定存在非负标量 $c_i, i=1, \cdot\cdot\cdot, p$ , 使得:
$ \begin{align} Q_0-\sum_{i=1}^{p}c_iQ_i < 0 \end{align} $
(6c) 显然条件 (6c) 可以推出条件 (6a). S-procedure就是通过判断条件 (6c) 的可行性来验证条件 (6a) 是否成立的.一般来说, 条件 (6c) 比条件 (6a) 更容易检验.所以, 通过应用S-procedure理论可以找到检验 (6a) 成立的更有效的方法.
由引理1的结论, 考虑条件 (4c), 当给定 $k$ 时, 令
$ \begin{align*} \begin{array}{l} Q_0=A_{d, k}^{\rm T}E^{\rm T}P(t)A_{d, k}-E^{\rm T}P(t), \quad \mathcal{S}=\mathcal{S}_k \end{array} \end{align*} $
则条件 (4c) 等价于如下不等式:
$ \begin{align*} \begin{array}{l} A_{d, k}^{\rm T}E^{\rm T}P(t)A_{d, k}- E^{\rm T}P(t)-\sum\limits_{i=1}^{p}c_{i, k}(t)Q_{i, k} < 0 \end{array} \end{align*} $
于是可以推出如下定理.
定理3.如果存在对称矩阵集合 $Q_{i, k}, i=1, \cdot\cdot\cdot, p_k, k=1, \cdot\cdot\cdot, {N}$ 满足引理1条件, 并且存在一个连续可微对称非奇异函数矩阵 $P(\cdot)$ 和非负标量 $c_{i, k}(\cdot)\geq 0 $ 有:
$ E^{\rm T}P(t)=P^{\rm T}(t)E\geq 0 $
(7a) $ A^{\rm T}(t)P(t)+P^{\rm T}(t)A(t)+E^{\rm T}\dot{P}(t) < 0 $
(7b) $ \begin{array}{l} A_{d, k}^{\rm T}E^{\rm T}P(t)A_{d, k}- E^{\rm T}P(t)-\sum\limits_{i=1}^{p}c_{i, k}(t)Q_{i, k}<0\\ \qquad {\boldsymbol x}(t)\in \mathcal{S}_k, \quad k=1, 2, \cdot\cdot\cdot, {N} \end{array} $
(7c) $ E^{\rm T}\Gamma(t)\leq E^{\rm T}P(t)\leq E^{\rm T}P(t_0)< E^{\rm T}R $
(7d) 成立, 则系统 (1) 对于 $(c_1, J, R, \Gamma(\cdot))$ 时域稳定.其中 $J=[t_0\quad t_0 +T]$ .
注1.由引理1知定理3的条件有利于这类问题的求解.在定理1中, 跳变时需要解无穷多个矩阵不等式, 而在定理3中, 跳变时只需解在指定集合上的有限个矩阵不等式.同理, 由定理2可以推出如下定理.
定理4.如果存在对称矩阵集合 $Q_{i, k}, i=1, \cdot\cdot\cdot, p_k, k=1, \cdot\cdot\cdot, {N}$ 满足引理1条件, 并且存在一个连续可微对称非奇异函数矩阵 $P(\cdot)$ 和非负标量 $c_{i, k}(\cdot)\geq 0$ 有:
$ E^{\rm T}P(t)=P^{\rm T}(t)E\geq 0 $
(8a) $ \begin{bmatrix} \Pi(t)&P^{\rm T}(t)G(t)\\ G^{\rm T}(t)P(t)&-I \end{bmatrix} < 0 $
(8b) $ \begin{array}{l} A_{d, k}^{\rm T}E^{\rm T}P(t)A_{d, k}-E^{\rm T}P(t)-\sum\limits_{i=1}^{p}c_{i, k}(t)Q_{i, k} < 0\\ \qquad {\boldsymbol x}(t)\in \mathcal{S}_k, \quad k=1, 2, \cdot\cdot\cdot, {N} \end{array} $
(8c) $ E^{\rm T}\Gamma(t)\leq E^{\rm T}P(t)\leq E^{\rm T}P(t_0) < E^{\rm T}R $
(8d) 成立, 则称系统 (2) 对于 $(c_1, c_2, {\boldsymbol \omega}(t), J, R, \Gamma(\cdot))$ 时域稳定.其中 $\Pi(t)$ 与定理2相同.
3. 时域控制器设计
考虑如下状态依赖广义时变脉冲系统
$ \begin{align} \left\{ \begin{array}{l} E\dot{{\boldsymbol x}}(t)=A(t){\boldsymbol x}(t)+G(t){\boldsymbol \omega}(t)+\\ \qquad B(t)u(t), \quad {\boldsymbol x}(t)\notin \mathcal{S}_k\\ {\boldsymbol x}^{+}_{k}(t)=A_{d, k}{\boldsymbol x}(t), \quad {\boldsymbol x}(t)\in \mathcal{S}_k\\ {\boldsymbol x}(t_{0})={\boldsymbol x}_{0}\quad k=1, 2, \cdot\cdot\cdot, {N} \end{array} \right. \end{align} $
(9) 对上述系统找到一个状态反馈控制律
$ \begin{align*} {\boldsymbol u}(t)=K(t){\boldsymbol x}(t) \end{align*} $
使得闭环系统
$ \begin{align} \left\{ \begin{array}{l} E\dot{{\boldsymbol x}}(t)=A_c(t){\boldsymbol x}(t)+G(t){\boldsymbol \omega}(t), \quad {\boldsymbol x}(t)\notin \mathcal{S}_k\\ {\boldsymbol x}^{+}_{k}(t)=A_{d, k}{\boldsymbol x}(t), \quad {\boldsymbol x}(t)\in \mathcal{S}_k\\ {\boldsymbol x}(t_{0})={\boldsymbol x}_{0}, \quad k=1, 2, \cdot\cdot\cdot, {N} \end{array} \right. \end{align} $
(10) 是时域稳定, 其中 $A_c(t)=A(t)+B(t)K(t)$ .
定理5.对于系统 (10), 如果存在分段连续可微对称非奇异的函数矩阵 $\bar{P}(\cdot)$ 、函数矩阵 $L_{k}(\cdot)$ 和非负标量 $C_{i, k}\leq 0$ 满足下列不等式组, $\forall t\in J$ .
$ \begin{bmatrix} \Pi_{1}(t)&G(t)\\ G^{\rm T}(t)&-I \end{bmatrix} < 0 $
(11a) $ \begin{array}{l} A_{d, k}^{\rm T}E^{\rm T}\bar{P}^{-1}(t)A_{d, k}-E^{\rm T}\bar{P}^{-1}(t)-\\ \qquad \quad \sum\limits_{i=1}^{p}c_{i, k}(t)Q_{i, k} < 0\\ \qquad {\boldsymbol x}(t)\in S_k, \quad k=1, 2, \cdot\cdot\cdot, {N} \end{array} $
(11b) $ E^{\rm T}\Gamma(t)\leq E^{\rm T}\bar{P}^{-1}(t)\leq E^{\rm T}\bar{P}^{-1}(t_0) < E^{\rm T}R $
(11c) 则称系统 (10) 对于 $(c_1, c_2, {\boldsymbol\omega}(t), J, R, \Gamma(\cdot))$ 时域稳定, 且状态反馈控制律为
$ \begin{align} K_k(t)=L_k(t)\bar{P}^{-1}(t) \end{align} $
(12) 其中, $\Pi_1(t)=-E\dot{\bar{P}}(t)+\bar{P}^{\rm T}(t)A^{\rm T}(t)+A(t)\bar{P}(t)+L^{\rm T}_{k}(t)B^{\rm T}(t)+B(t)L_{k}(t)$ .
证明.对于 $t\in J_k$ , 将状态反馈控制律 $K_{k}(t)=L_{k}(t)\bar{P}^{-1}(t)$ 带入式 (9), 可以得到闭环系统 (10), 其中, $A_{c}(t)=A(t)+B(t)L_{i}(t)\bar{P}(t)^{-1}$ .显然 (11a) 等价于下列不等式:
$ \begin{align} \left[\begin{array}{cc}\Pi_{2}(t)&G(t)\\ G^{\rm T}(t)&-I\end{array}\right] < 0 \end{align} $
(13) 其中
$ \begin{align*} \Pi_{2}(t)=-E\dot{\bar{P}}(t)+\bar{P}^{\rm T}(t)A_{c}^{\rm T}(t)+A_{c}(t)\bar{P}(t) \end{align*} $
令 $P(t)=\bar{P}^{-1}(t)$ , 将式 (13) 分别左乘 ${\rm diag}\{\bar{P}^{\rm -T}(t), I\}$ , 右乘 ${\rm diag} \{\bar{P}^{-1}(t), I\}$ , 得到下列不等式:
$ \begin{align*} \left[\begin{array}{cc}\Pi_{3}(t)&\bar{P}^{\rm-T}(t)G(t)\\ G^{\rm T}(t)\bar{P}^{-1}(t)&-I\end{array}\right] < 0 \end{align*} $
其中, $ \Pi_{3}(t)=-\bar{P}^{\rm -T}(t)E\dot{\bar{P}}(t)\bar{P}^{-1}(t)+ A_{c}^{\rm T}(t) \bar{P}^{-1}(t)+\bar{P}^{\rm -T}(t)A_{c}(t).$
因为 $P(t)=\bar{P}^{-1}(t)$ , 故
$ \begin{align*} I=\bar{P}(t)P(t), \quad 0=\dot{\bar{P}}(t)P(t)+\bar{P}(t)\dot{P}(t) \end{align*} $
又因为 $E^{\rm T}P(t)=P^{\rm T}(t)E\geq0$ , 我们很容易可以得出:
$ \begin{align*} -P^{\rm T}(t)E\dot{\bar{P}}(t)P(t)=-E^{\rm T}P(t)\dot{\bar{P}}(t)P(t)=E^{\rm T}\dot{P}(t)\end{align*} $
由此可得:
$ \begin{align} \left[\begin{array}{cc}\Pi_{4}(t)&P^{\rm T}(t)G(t)\\ G^{\rm T}(t)P(t)&-I\end{array}\right] < 0 \end{align} $
(14) 其中
$ \begin{align*} \Pi_{4}(t)=E^{\rm T}\dot{P}(t)+A_c^{\rm T}(t)P(t)+P(t)A_c(t) \end{align*} $
另一方面, 因为 $P(t)=\bar{P}^{-1}(t)$ , 条件 (11b) 和 (11c) 等价于式 (7c) 和 (7d).综上所述, 闭环系统 (10) 是对于 $(c_1, c_2, {\boldsymbol\omega}(t), J, R, \Gamma(\cdot))$ 时域稳定的.
为了将小区间内的时变矩阵不等式转化为标准的矩阵不等式组.可以将小区间内的时变矩阵不等式做如下处理.假设 $P(t)$ (或 $\bar{P}(t)$ ) 是分段线性的, $P(t)$ (或 $\bar{P}(t)$ ) 在 $\mathcal{T}_{x(\cdot)}$ 处发生跳变 (以下只给出 $P(t)$ (或 $\bar{P}(t)$ ) 的形式, $L(t)$ 、 $\dot{P}(t)$ 形式与 $P(t)$ 类似).
$ \begin{align*} \label{eq} \left\{ \begin{aligned} & P(0)({\rm or}\ (\bar{P}(0)))=\Pi_{1}^{0}\\ & P(t)({\rm or}\ (\bar{P}(t)))=\Pi_{k}^{0}+\Pi_{k}^{s}(t-(k-1)T_{s}), \\ & \qquad \qquad k\in N : kx \bar{k}, \quad t\in[(k-1)T_{s}, kT_{s}]\\ & P(t)({\rm or}\ (\bar{P}(t)))=\Pi_{\bar{k}+1}^{0}+\Pi_{\bar{k}+1}^{0}(t-\bar{k}T_{s}), t\in[\bar{k}T_{s}, T] \end{aligned} \right. \end{align*} $
因此上述条件可以转化为一组标准的矩阵不等式求解问题.由于 $E^{\rm T}P(t)=P^{\rm T}(t)E\geq0$ ( $E^{\rm T}\Pi_{k}^{0}=\Pi_{k}^{0\rm T}E\geq0$ $E^{\rm T}\Pi_{k}^{s}=\Pi_{k}^{s{\rm T}}E\geq0$ ) 是一组非严格的矩阵不等式, 这样对于求解会造成一定的麻烦.为了将非严格的矩阵不等式组转化为严格的矩阵不等式组, 我们介绍如下引理.
引理2[31].如果 $X\in \boldsymbol{R}^{n\times n}$ 是对称矩阵且满足 $E_{L}^{\rm T}XE_{L}>0$ , $T\in {\bf R }^{(n-r)\times(n-r)}$ 是非奇异矩阵.则 $XE+M^{\rm T}TS^{\rm T}$ 也是非奇异的且它的逆可以表示为
$ \begin{align*} (XE+M^{\rm T}TS^{\rm T})^{-1}=XE^{\rm T}+STM \end{align*} $
其中, $X$ 是对称矩阵, $T$ 是非奇异矩阵.
$ \begin{align*} E_{R}^{\rm T}XE_{R}=(E_{L}^{\rm T}XE_{L})^{-1}, \ T=(S^{\rm T}S)^{-1}T^{-1}(MM^{\rm T})^{-1}\end{align*} $
$M$ 和 $S$ 是行满秩矩阵满足 $ME=0$ , $ES=0$ ; $E$ 可以分解为 $E=E_{L}E_{R}^{\rm T}$ , 其中 $E_{L}\in \boldsymbol{R}^{n\times r}$ , $E_{R}\in {\bf R }^{n\times r}$ 是列满秩的.
令 $\Pi_{k}^{0}=X_{k}^{0}E+M^{\rm T}T_{k}^{0}S^{\rm T}$ , $\Pi_{k}^{s}=X_{k}^{s}E+M^{\rm T}T_{k}^{s}S^{\rm T}$ .根据引理2可以得到 $(X_{k}^{0}E+M^{\rm T}T_{k}^{0}S^{\rm T})^{-1}=X_{k}^{0}E^{\rm T}+ST_{k}^{0}M$ 和 $(X_{k}^{s}E+M^{\rm T}T_{k}^{s}S^{\rm T})^{-1}=X_{k}^{s}E^{\rm T}+ST_{k}^{0}M$ .这样 $E^{\rm T}P(t)=P^{\rm T}(t)E\geq0$ 就得到了满足.因此, 非严格的矩阵不等式组转化为了严格的矩阵不等式组.利用Matlab LMI工具箱, 就可以对 $X_{k}^{s}, T_{k}^{0}, T_{k}^{s}, X_{k}^{0}$ (或 $X_{k}^{0}, X_{k}^{s}, T_{k}^{0}, T_{k}^{s}), \Lambda_{k}^{0}, \Lambda_{k}^{s}$ 进行求解, 从而得到 $P(t)$ (或 $(\bar{P}(t))$ 和 $L(t)$ .
4. 数值算例
例1.考虑广义时变脉冲系统 (9),
$ \begin{align*} &E=\left[\begin{array}{ccc} 1&0\\0&0 \end{array}\right], \quad A=\left[\begin{array}{ccc} 1&-1.5\\0.5&t \end{array}\right]\\ &A_{d, 1}=\left[\begin{array}{ccc} 0.8&0\\-1&0.5 \end{array}\right], \quad G=\left[\begin{array}{ccc} 1\\1 \end{array}\right] B=\left[\begin{array}{ccc} 0\\1 \end{array}\right] \end{align*} $
跳变集合
$ \begin{align*} \mathcal{S}_1= \left(\begin{array}{*{20}c}\left(\begin{array}{*{20}c}0.5\\ 0.2\end{array}\right), &\left(\begin{array}{*{20}c}0.4\\ 0.4 \end{array}\right)\end{array}\right) \end{align*} $
其中, 选取 ${\boldsymbol\omega}(t)=1$ , $J=[0 {\rm s}\quad 5 {\rm s}]$ , $R=\left[{array}{ccc} 2&0\\0&2 {array}\right]$ , $c_1=3$ , $c_2=4$ .由引理1可选取
$ \begin{align*} Q=\left[\begin{array}{ccc} 0.4000&-0.7000\\-0.7000&1.0000 \end{array}\right]\end{align*} $
显然
$ \begin{align*} E_{R}=E_{L}=\left[\begin{array}{ccc} 1\\0 \end{array}\right] \end{align*} $
我们选 $M^{\rm T}=S=[0 \quad 1]^{\rm T}$ .则存在 $\Gamma(t)=\left[{array}{ccc} 1&0\\0&1{array}\right]$ , 根据Matlab LMI工具箱求解矩阵不等式组, 可得正数 $c_1(\cdot)$ 及矩阵 $P(\cdot), L_1(\cdot)$ , 使得式 (11) 成立, 则闭环系统 (10) 是时域稳定的, 且状态反馈控制律 $K(t)=[K_1(t)\quad K_2(t)]$ 见图 1.
5. 结论
本文针对状态依赖广义时变脉冲系统时域稳定问题进行研究, 给出了广义时变脉冲系统时域稳定充分条件及状态反馈控制器的设计.并且对上述充分条件提出DLMIs优化计算的方法, 使得这类问题在数值计算上易于处理.利用分段线性化将小区间内广义时变矩阵不等式转化为广义时不变线性矩阵不等式来求解.最后给出数值算例来验证结论的有效性.
-
[1] Beyer M A, Laney D. The importance of "Big Data":a definition[Online], available:https://www.gartner.com/doc/2057415, February 17, 2016 [2] Marr B. Big data:the 5 Vs everyone must know[Online], http://www.linkedin.com/pulse/20140306073407-64875646-big-data-the-5-vs-everyone-must-know, January 21, 2016 [3] Mervis J. Agencies rally to tackle big data. Science, 2013, 336(6077):22-22 http://cn.bing.com/academic/profile?id=1978954664&encoded=0&v=paper_preview&mkt=zh-cn [4] 王飞跃.软件定义的系统与知识自动化:从牛顿到默顿的平行升华.自动化学报, 2015, 42(1):1-8 http://www.aas.net.cn/CN/abstract/abstract18578.shtmlWang Fei-Yue. Software-deined systems and knowledge automation:a parallel paradigm shift from Newton to Merton. Acta Automatica Sinica, 2015, 42(1):1-8 http://www.aas.net.cn/CN/abstract/abstract18578.shtml [5] Fish A N. Knowledge Automation:How to Implement Decision Management in Business Processes. USA:Wiley, 2012. [6] Fernández A, Del Río S, López V, Bawakid A, Del Jesus M J, Benítez J M, Herrera F. Big data with cloud computing:an insight on the computing environment, MapReduce, and programming frameworks. Wiley Interdisciplinary Reviews:Data Mining and Knowledge Discovery, 2014, 4(5):380-409 doi: 10.1002/widm.2014.4.issue-5 [7] Kent S M. Sloan digital sky survey. Science with Astronomical Near-Infrared Sky Surveys. France:Springer, 1994. 27-30 [8] Labrinidis A, Jagadish H V. Challenges and opportunities with big data. Proceedings of the VLDB Endowment, 2012, 5(12):2032-2033 doi: 10.14778/2367502 [9] Knoll A, Meinkoehn J. Data fusion using large multi-agent networks:an analysis of network structure and performance. In:Proceedings of the 1994 IEEE International Conference on MFI'94, Multisensor Fusion and Integration for Intelligent Systems (MFI). Las Vegas, NV:IEEE, 1994. 113-120 [10] Nature Editorial. Community cleverness required. Nature, 2008, 455(7209):1-1 doi: 10.1038/455001a [11] Che D R, Safran M, Peng Z Y. From big data to big data mining:challenges, issues, and opportunities. In:Proceedings of the 18th International Conference on Database Systems for Advanced Applications. Wuhan, China:Springer, 2013. 1-15 [12] Stidston M. Business leaders need R's not V's:the 5 R's of big data[Online], available:https://www.mapr.com/blog/business-leaders-need-r%E2%80%99s-not-v%E2%80%99s-5-r%E2%80%99s-big-data#.U2qmcq1dWIU, December 21, 2015 [13] 王济, 王琦.中医体质研究与4P医学的实施.中国中西医结合杂志, 2012, 32(5):693-695 http://www.cnki.com.cn/Article/CJFDTOTAL-ZZXJ201205035.htmWang Ji, Wang Qi. Chinese constitution research and the practice of 4P medical model. Chinese Journal of Integrated Traditional and Western Medicine, 2012, 32(5):693-695 http://www.cnki.com.cn/Article/CJFDTOTAL-ZZXJ201205035.htm [14] Auffray C, Charron D, Hood L. Predictive, preventive, personalized and participatory medicine:back to the future. Genome Medicine, 2010, 2(8):57-57 doi: 10.1186/gm178 [15] Wu X D, Zhu X Q, Wu G Q, Ding W. Data mining with big data. IEEE Transactions on Knowledge and Data Engineering, 2014, 26(1):97-107 doi: 10.1109/TKDE.2013.109 [16] Wikipedia. Big data[Online], available:https://en.wikipe-dia.org/wiki/Big_data#Definition, December 12, 2015 [17] IDC权威定义大数据概念:满足4V标准[Online], available:http://www.d1net.com/bigdata/news/237143.html, December 12, 2015 [18] Tien J M. Big data:unleashing information. Journal of Systems Science and Systems Engineering, 2013, 22(2):127-151 doi: 10.1007/s11518-013-5219-4 [19] 王元卓, 靳小龙, 程学旗.网络大数据:现状与展望.计算机学报, 2013, 36(6):1125-1138 http://www.cnki.com.cn/Article/CJFDTOTAL-JSJX201306001.htmWang Yuan-Zhuo, Jin Xiao-Long, Cheng Xue-Qi. Network big data:present and future. Chinese Journal of Computers, 2013, 36(6):1125-1138 http://www.cnki.com.cn/Article/CJFDTOTAL-JSJX201306001.htm [20] 王卫卫, 李小平, 冯象初, 王斯琪.稀疏子空间聚类综述.自动化学报, 2015, 41(8):1373-1384Wang Wei-Wei, Li Xiao-Ping, Feng Xiang-Chu, Wang Si-Qi. A survey on sparse subspace clustering. Acta Automatica Sinica, 2015, 41(8):1373-1384 [21] Armbrust M, Fox A, Griffith R, Joseph A D, Katz R H, Konwinski A, Lee G, Patterson D A, Rabkin A, Stoica I, Zaharia M. Above the Clouds:A Berkeley View of Cloud Computing, Technical Report UCB/EECS-2009-28, EECS Department, University of California, Berkeley, 2009 http://cn.bing.com/academic/profile?id=2131629857&encoded=0&v=paper_preview&mkt=zh-cn [22] Blaabjerg F, Teodorescu R, Liserre M, Timbus A V. Overview of control and grid synchronization for distributed power generation systems. IEEE Transactions on Industrial Electronics, 2006, 53(5):1398-1409 doi: 10.1109/TIE.2006.881997 [23] Leskovec J, Huttenlocher D, Kleinberg J. Signed networks in social media. In:Proceedings of the 2010 SIGCHI Conference on Human Factors in Computing Systems. New York:ACM, 2010. 1361-1370 [24] Zikopoulos P, Eaton C. Understanding Big Data:Analytics for Enterprise Class Hadoop and Streaming Data. USA:McGraw-Hill Osborne Media, 2011. [25] The four V's of big data[Online], available:http://www.ib-mbigdatahub.com/sites/default/files/infographic_file/4-Vs-of-big-data.jpg, January 21, 2016 [26] Lazer D, Kennedy R, King G, Vespignan A. The parable of google flu:traps in big data analysis. Science, 2014, 343(6176):1203-1205 doi: 10.1126/science.1248506 [27] IBM. What is big data?[Online], available:http://www-01.ibm.com/software/data/bigdata/what-is-big-data.html, December 2, 2015 [28] Barwick H. The "four Vs" of big data. Implementing information infrastructure symposium[Online], available:http://www.computerworld.com.au/article/396198/iiis_four_vs_big_data, December 2, 2015 [29] 数据并非越大越好:谷歌流感趋势错在哪儿了?[Online], available:http://www.guokr.com/article/438117/, December 2, 2015 [30] Ghemawat S, Gobioff H, Leung S T. The Google file system. In:Proceedings of the 19th ACM Symposium on Operating Systems Principles. New York:ACM, 2003. 29-43 [31] Dean J, Ghemawat S. MapReduce:simplified data processing on large clusters. In:Proceedings of the 6th Symposium on Operating Systems Design and Implementation. Berkeley, CA, USA:USENIX Association, 2004. 137-149 [32] Big data solution offering[Online], available:http://mike2.openmethodology.org/wike/Big_Data_Solution_Offering, November 28, 2015 [33] White T. Hadoop:The Definitive Guide (2nd Edition). USA:Yahoo Press, 2010. 1-4 [34] Gupta P, Kumar P, Gopal G. Sentiment analysis on Hadoop with Hadoop streaming. International Journal of Computer Applications, 2015, 121(11):4-8 doi: 10.5120/21582-4651 [35] Liao S H. Expert system methodologies and applications——a decade review from 1995 to 2004. Expert Systems with Applications, 2005, 28(1):93-103 doi: 10.1016/j.eswa.2004.08.003 [36] 吴信东, 叶明全, 胡东辉, 吴共庆, 胡学钢, 王浩.普适医疗信息管理与服务的关键技术与挑战.计算机学报, 2012, 35(5):827-845 doi: 10.3724/SP.J.1016.2012.00827Wu Xin-Dong, Ye Ming-Quan, Hu Dong-Hui, Wu Gong-Qing, Hu Xue-Gang, Wang Hao. Pervasive medical information management and services:key techniques and challenges. Chinese Journal of Computers, 2012, 35(5):827-845 doi: 10.3724/SP.J.1016.2012.00827 [37] Auffray C, Chen Z, Hood L. Systems medicine:the future of medical genomics and healthcare. Genome Medicine, 2009, 1(1):2-2 doi: 10.1186/gm2 [38] 罗旭, 陈博, 罗莉娅, 张宏雁, 吴昊, 李景波. 4P医学理念下医院健康管理体系重构思考.中国医院, 2014, 18(7):61-63 http://www.cnki.com.cn/Article/CJFDTOTAL-ZGYU201407026.htmLuo Xu, Chen Bo, Luo Li-Ya, Zhang Hong-Yan, Wu Hao, Li Jing-Bo. Discussion on reconstructing hospital healthcare management under 4P medical conception. Chinese Hospitals, 2014, 18(7):61-63 http://www.cnki.com.cn/Article/CJFDTOTAL-ZGYU201407026.htm [39] Wu X D, Chen H H, Wu G Q, Liu J, Zheng Q H, He X F, Zhou A Y, Zhao Z Q, Wei B F, Li Y, Zhang Q P, Zhang S C, Lu R Q, Zheng N N. Knowledge engineering with big data. IEEE Intelligent Systems, 2015, 30(5):46-55 doi: 10.1109/MIS.2015.56 [40] Klasnja P, Pratt W. Healthcare in the pocket:mapping the space of mobile-phone health interventions. Journal of Biomedical Informatics, 2012, 45(1):184-198 doi: 10.1016/j.jbi.2011.08.017 [41] Vassis D, Belsis P, Skourlas C, Pantziou G. Providing advanced remote medical treatment services through pervasive environments. Personal and Ubiquitous Computing, 2010, 14(6):563-573 doi: 10.1007/s00779-009-0273-0 [42] 合肥工业大学吴信东:大数据Processing Framework多层架构[Online], available:http://www.csdn.net/article/2012-07-27/2825305, December 7, 2015 [43] Petersen W P, Arbenz P. Introduction to Parallel Computing. Oxford:Oxford University Press, 2004. [44] Corbett J C, Dean J, Epstein M, Fikes A, Frost C, Furman J J, Ghemawat S, Gubarev A, Heiser C, Hochschild P, Hsieh W, Kanthak S, Kogan E, Li H Y, Lloyd A, Melnik S, Mwaura D, Nagle D, Quinlan S, Rao R, Rolig L, Saito Y, Szymaniak M, Taylor C, Wang R, Woodford D. Spanner:Google's globally-distributed database. ACM Transactions on Computer Systems, 2012, 31(3):Article No.8 http://cn.bing.com/academic/profile?id=2112564224&encoded=0&v=paper_preview&mkt=zh-cn [45] Chang F, Dean J, Ghemawat S, Hsieh W C, Wallach D A, Burrows M, Chandra T, Fikes A, Gruber R E. BigTable:a distributed storage system for structured data. ACM Transactions on Computer Systems, 2008, 26(2):Article No.4 http://cn.bing.com/academic/profile?id=1981420413&encoded=0&v=paper_preview&mkt=zh-cn [46] Peel M, Rowley J. Information sharing practice in multi-agency working. ASLIB Proceedings, 2010, 62(1):11-28 doi: 10.1108/00012531011015172 [47] Wang M D, Li B, Zhao Y X, Pu G G. Formalizing Google file system. In:Proceedings of the 20th IEEE Pacific Rim International Symposium on Dependable Computing (PRDC). Singapore:IEEE, 2014. 190-191 [48] Cormode G, Srivastava D. Anonymized data:generation, models, usage. In:Proceedings of the 2009 ACM SIGMOD International Conference on Management of Data. Providence, RI:ACM, 2009. 1015-1018 [49] Sweeney L. k-anonymity:a model for protecting privacy. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2002, 10(5):557-570 doi: 10.1142/S0218488502001648 [50] Kopanas I, Avouris N M, Daskalaki S. The role of domain knowledge in a large scale data mining project. Methods and Applications of Artificial Intelligence. Thessaloniki, Greece:Springer, 2002. 288-299 [51] Salton G M, Wong A, Yang C S. A vector space model for automatic indexing. Communications of the ACM, 1975, 18(11):613-620 doi: 10.1145/361219.361220 [52] Deerwester S C, Dumais S T, Furnas G W, Landauer T K, Harshman R. Indexing by latent semantic analysis. Journal of the American Society for Information Science, 1990, 41(6):391-407 doi: 10.1002/(ISSN)1097-4571 [53] Freedman E G, Shah P. Toward a model of knowledge-based graph comprehension. Diagrammatic Representation and Inference. Callaway Gardens, GA, USA:Springer, 2002. 18-30 [54] Aral S, Walker D. Identifying influential and susceptible members of social networks. Science, 2012, 337(6092):337-341 doi: 10.1126/science.1215842 [55] Centola D. The spread of behavior in an online social network experiment. Science, 2010, 329(5996):1194-1197 doi: 10.1126/science.1185231 [56] Strassel S, Adams D, Goldberg H, Herr J, Keesing R, Oblinger D, Simpson H, Schrag R, Wright J. The DARPA machine reading program——encouraging linguistic and reasoning research with a series of reading tasks. In:Proceedings of the 7th International Conference on Language Resources and Evaluation. Valletta, Malta:European Language Resources Association, 2010. 986-993 [57] Studer R, Benjamins V R, Fensel D. Knowledge engineering:principles and methods. Data and Knowledge Engineering, 1998, 25(1-2):161-197 doi: 10.1016/S0169-023X(97)00056-6 [58] 潘云鹤, 王金龙, 徐从富.数据流频繁模式挖掘研究进展.自动化学报, 2006, 32(4):594-602 http://www.aas.net.cn/CN/abstract/abstract14320.shtmlPan Yun-He, Wang Jin-Long, Xu Cong-Fu. State-of-the-art on frequent pattern mining in data streams. Acta Automatica Sinica, 2006, 32(4):594-602 http://www.aas.net.cn/CN/abstract/abstract14320.shtml [59] 王珊, 王会举, 覃雄派, 周烜.架构大数据:挑战、现状与展望.计算机学报, 2011, 34(10):1741-1752 doi: 10.3724/SP.J.1016.2011.01741Wang Shan, Wang Hui-Ju, Qin Xiong-Pai, Zhou Xuan. Architecting big data:challenges, studies and forecasts. Chinese Journal of Computers, 2011, 34(10):1741-1752 doi: 10.3724/SP.J.1016.2011.01741 [60] Guha S, Mishra N, Motwani R, O'Callaghan L. Clustering data streams. In:Proceedings of the 41st Annual Symposium on Foundations of Computer Science. Redono Beach, USA:IEEE, 2000. 359-366 [61] 朱群, 张玉红, 胡学钢, 李培培.一种基于双层窗口的概念漂移数据流分类算法.自动化学报, 2011, 37(9):1077-1084 http://www.aas.net.cn/CN/abstract/abstract17531.shtmlZhu Qun, Zhang Yu-Hong, Hu Xue-Gang, Li Pei-Pei. A double-window-based classification algorithm for concept drifting data streams. Acta Automatica Sinica, 2011, 37(9):1077-1084 http://www.aas.net.cn/CN/abstract/abstract17531.shtml [62] 张昕, 李晓光, 王大玲, 于戈.数据流中一种快速启发式频繁模式挖掘方法.软件学报, 2005, 16(12):2099-2105 doi: 10.1360/jos162099Zhang Xin, Li Xiao-Guang, Wang Da-Ling, Yu Ge. A high-speed heuristic algorithm for mining frequent patterns in data stream. Journal of Software, 2005, 16(12):2099-2105 doi: 10.1360/jos162099 [63] Wu X D, Yu K, Ding W, Wang H, Zhu X Q. Online feature selection with streaming features. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2013, 35(5):1178-1192 doi: 10.1109/TPAMI.2012.197 [64] Zhang Q, Zhang P, Long G D, Ding W, Zhang C Q, Wu X D. Towards mining trapezoidal data streams. In:Proceedings of the 2015 IEEE International Conference on Data Mining (ICDM'15). Atlantic City, NJ, USA:IEEE, 2015. 1111-1116 [65] Wu X D, Yu K, Wang H, Ding W. Online streaming feature selection. In:Proceedings of the 27th International Conference on Machine Learning. Haifa, Israel, 2010. 1159-1166 [66] Kivinen J, Smola A J, Williamson R C. Online learning with kernels. IEEE Transactions on Signal Processing, 2004, 52(8):2165-2176 doi: 10.1109/TSP.2004.830991 [67] Kimeldorf G, Wahba G. Some results on Tchebycheffian spline functions. Journal of Mathematical Analysis and Applications, 1971, 33(1):82-95 doi: 10.1016/0022-247X(71)90184-3 [68] Zhou Z H, Chawla N V, Jin Y C, Williams G J. Big data opportunities and challenges:discussions from data analytics perspectives[Discussion forum]. IEEE Computational Intelligence Magazine, 2014, 9(4):62-74 doi: 10.1109/MCI.2014.2350953 [69] Vijayakumar S, D'Souza A, Schaal S. Incremental online learning in high dimensions. Neural Computation, 2005, 17(12):2602-2634 doi: 10.1162/089976605774320557 [70] Hunter A, Summerton R. Fusion rules for context-dependent aggregation of structured news reports. Journal of Applied Non-Classical Logics, 2004, 14(3):329-366 doi: 10.3166/jancl.14.329-366 [71] Žliobaitė I. Learning under concept drift:an overview. Computer Science——Artificial Intelligence[Online], available:http://arxiv.org/abs/1010.4784, May 31, 2015 [72] 李建中, 刘显敏.大数据的一个重要方面:数据可用性.计算机研究与发展, 2013, 50(6):1147-1162 http://www.cnki.com.cn/Article/CJFDTOTAL-JFYZ201306006.htmLi Jian-Zhong, Liu Xian-Min. An important aspect of big data:data usability. Journal of Computer Research and Development, 2013, 50(6):1147-1162 http://www.cnki.com.cn/Article/CJFDTOTAL-JFYZ201306006.htm [73] Samarati P, Sweeney L. Protecting privacy when disclosing information:k-anonymity and its enforcement through generalization and suppression. In:Proceedings of the 1998 IEEE Symposium on Research in Security and Privacy. Palo Alto, CA:IEEE, 1998. 1-19 [74] 王超, 杨静, 张健沛.基于轨迹特征及动态邻近性的轨迹匿名方法研究.自动化学报, 2015, 41(2):330-341 http://www.aas.net.cn/CN/abstract/abstract18612.shtmlWang Chao, Yang Jing, Zhang Jian-Pei. Research on trajectory privacy preserving method based on trajectory characteristics and dynamic proximity. Acta Automatica Sinica, 2015, 41(2):330-341 http://www.aas.net.cn/CN/abstract/abstract18612.shtml [75] Wu X D, Zhu X Q. Mining with noise knowledge:error-aware data mining. IEEE Transactions on Systems, Man, and Cybernetics——Part A:Systems and Humans, 2008, 38(4):917-932 doi: 10.1109/TSMCA.2008.923034 [76] He H B, Garcia E A. Learning from imbalanced data. IEEE Transactions on Knowledge and Data Engineering, 2009, 21(9):1263-1284 doi: 10.1109/TKDE.2008.239 [77] 王飞跃.迈向知识自动化[Online], available:http://www.cas.cn/xw/zjsd/201401/t20140103_4009925.shtml, June 1, 2016 [78] 邓建玲, 王飞跃, 陈耀斌, 赵向阳.从工业4.0到能源5.0:智能能源系统的概念、内涵及体系框架.自动化学报, 2015, 41(12):2003-2016 http://www.aas.net.cn/CN/abstract/abstract18774.shtmlDeng Jian-Ling, Wang Fei-Yue, Chen Yao-Bin, Zhao Xiang-Yang. From industries 4.0 to energy 5.0:concept and framework of intelligent energy systems. Acta Automatica Sinica, 2015, 41(12):2003-2016 http://www.aas.net.cn/CN/abstract/abstract18774.shtml [79] Twitter Blog. Dispatch from the Denver debate[Online], available:http://blog.twitter.com/2012/100dispatch-reom-denver-debate.html, October 1, 2012 [80] Chun D X, Jun C J, Zhong C Y, Chao T M, Cong P. Data engineering in information system construction. In:Proceedings of the 2012 IEEE Symposium on Robotics and Applications (ISRA). Kuala Lumpur:IEEE, 2012. 135-137 [81] Aggarwal C C. . US:Springer, 2007. [82] Silva J A, Faria E R, Barros R C, Hruschka E R, de Carvalho A C P L F, Gama J. Data stream clustering:a survey. ACM Computing Surveys, 2013, 46(1):Article No.13 http://cn.bing.com/academic/profile?id=2088340225&encoded=0&v=paper_preview&mkt=zh-cn [83] Patil P D, Kulkarni P. Adaptive supervised learning model for training set selection under concept drift data streams. In:Proceedings of the 2013 International Conference on Cloud and Ubiquitous Computing and Emerging Technologies. Pune:IEEE, 2013. 36-41 [84] Hakkani-Tür D, Heck L, Tur G. Using a knowledge graph and query click logs for unsupervised learning of relation detection. In:Proceedings of the 2013 IEEE International Conference on Acoustics, Speech, and Signal Processing. Vancouver, BC:IEEE, 2013. 8327-8331 [85] Dantas J R V, Farias P P M. Conceptual navigation in knowledge management environments using NavCon. Information Processing and Management, 2010, 46(4):413-425 doi: 10.1016/j.ipm.2009.08.007 [86] Xu C J, Li A P, Liu X M. Knowledge fusion and evaluation system with fusion-knowledge measure. In:Proceedings of the 2nd International Symposium on Computational Intelligence and Design. Changsha, China:IEEE, 2009. 127-131 [87] Shahabi C, Zarkesh A M, Adibi J, Shah V. Knowledge discovery from users web-page navigation. In:Proceedings of the 7th International Workshop on Research Issues in Data Engineering. Birmingham:IEEE, 1997. 20-29 [88] Baldauf M, Dustdar S, Rosenberg F. A survey on context-aware systems. International Journal of Ad Hoc and Ubiquitous Computing, 2007, 2(4):263-277 doi: 10.1504/IJAHUC.2007.014070 [89] Herlocker J L, Konstan J A, Terveen L G, Riedl J T. Evaluating collaborative filtering recommender systems. ACM Transactions on Information Systems, 2004, 22(1):5-53 doi: 10.1145/963770 [90] 岳元龙, 左信, 罗雄麟.提高测量可靠性的多传感器数据融合有偏估计方法.自动化学报, 2014, 40(9):1843-1852 http://www.aas.net.cn/CN/abstract/abstract18453.shtmlYue Yuan-Long, Zuo Xin, Luo Xiong-Lin. Improving measurement reliability with biased estimation for multi-sensor data fusion. Acta Automatica Sinica, 2014, 40(9):1843-1852 http://www.aas.net.cn/CN/abstract/abstract18453.shtml [91] Xu C, Zhang Y Q, Li R Z. On the feasibility of distributed kernel regression for big data. Statistics[Online], available:http://arxiv.org/abs/1505.00869, May 31, 2016 -
下载:
下载:
计量
- 文章访问数: 3840
- HTML全文浏览量: 613
- PDF下载量: 2947
- 被引次数: 0
