Multilevel Peer-to-Peer Co-optimization for Cyber-physical Intelligent Energy Systems
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摘要: 针对能源电力系统的优化管理与控制问题，提出了一种信息物理融合的智慧能源系统（Intelligent energy systems，IES）多级对等协同优化方法.在信息物理融合能源系统（Cyber-physical energy systems，CPES）的基础上，构建了智慧能源系统的局域和广域两级协同优化架构.综合考虑产消者能源实体对等交互过程中的社会福利、供求平衡和需求意愿等因素，基于Stone-Geary函数和双向拍卖机制构建了智慧能源系统能量优化模型，给出了通过收敛判定域引导的全局随机寻优与区域定向寻优策略，有效地提高了算法的局部搜索能力.此外，通过双向拍卖机制的理性定价以及智能合约的辅助服务，有效地实现了用户友好的对等交易模式.仿真实例表明，在社会福利最大化的前提下可获得产消者电力资源最优分配结果，进一步验证了本文方法的有效性和可行性.Abstract: A multilevel peer-to-peer co-optimization method for cyber-physical intelligent energy systems (IES) is proposed to analyze the optimal control and management problem of energy power systems. On the basis of the cyber-physical energy system (CPES), a co-optimization architecture of local and wide-area levels for intelligent energy system is constructed. With the help of Stone-Geary utility function and double auction mechanism, an energy optimization model for intelligent energy system is constructed in consideration of social welfare, supply-demand balance and demand willingness in the peer-to-peer interaction process of prosumers. At the same time, the local search ability of the intelligent optimization algorithm is further improved by the guidance of convergence judgment domain as well as the combination strategy of global random search and directional search. In addition, the user-friendly peer-to-peer trading mode is effectively realized through rational pricing of double auction mechanism and the auxiliary services of smart contract. Simulation results show that the optimal allocation of power resources can be obtained under the premise of maximizing social welfare, which further illustrates the effectiveness and feasibility of this method.
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马尔科夫跳变系统是一类包含连续时间状态变量和离散时间模态变量的混杂系统.在马尔科夫跳变系统中,离散的模态变量是一个在连续时间下具有离散模态的马尔科夫过程,其模态值取自一个有限的集合.马尔科夫跳变系统被广泛应用于那些存在突发故障或环境变化的系统中,包括电力系统、航空航天系统、制造业系统和网络控制系统等[1].近些年,马尔科夫跳变系统逐渐成为了控制理论领域的一个研究热点,主要研究包括稳定性与控制设计[2-15]、故障检测与容错控制[3, 16-20]、滤波及状态和故障估计[3, 7, 9, 16, 20, 21-24].在针对马尔科夫跳变系统估计问题的研究中,文献[9]针对奇异马尔科夫跳变系统设计了观测器.文献[16]基于描述系统的方法,对一类具有延迟和非线性项的马尔科夫跳变系统设计了滑模观测器,给出了系统状态和传感器故障的估计,并将其应用到容错控制中.文献[20]针对一类具有伊藤型随机运动的马尔科夫跳变系统处理了容错控制问题.针对无法在线实时获得系统模态的广义马尔科夫跳变系统,文献[21]研究了部分模态依赖观测器和控制器设计问题.文献[7, 22]考虑了状态估计和滤波问题.文献[23]针对具有非线性扰动的描述马尔科夫跳变系统设计全维和降维观测器来估计系统的状态.文献[24]基于自适应观测器对马尔科夫跳变系统讨论了故障估计问题.在以上的介绍中,文献[7, 21-23]考虑的是马尔科夫跳变系统不具有执行器和传感器的情形,文献[24]只考虑了执行器故障的估计问题,虽然文献[16, 20]基于滑模观测器给出了执行器和传感器故障的同时估计,但是需要事先获知故障以及其导数的上界.由此可见,目前国内外对同时具有执行器和传感器故障的马尔科夫跳变系统进行状态、执行器故障和传感器故障同时估计的研究并不多见.此外,在实际系统中,延迟环节往往是导致系统不稳定的因素之一,状态转移概率也往往是在线估计获得的,具有一些不确定性,因此对于具有延迟环节和状态转移概率不确定性的情形进行相关议题的讨论具有重大意义.
综上所述,本文针对一类具有不确定状态转移概率的延迟马尔科夫跳变系统设计了自适应观测器来同时估计执行器和传感器故障.本文的贡献在于:1) 在状态转移概率不确定的情形下,对一类具有延迟环节和参数不确定性的马尔科夫跳变系统给出了执行器和传感器故障的同时估计;2) 本文假设状态转移概率矩阵是其估计值且具有不确定性,相较于基于精确状态转移概率矩阵的文献[16, 20]更具实用性;3) 本文设计过程中无需事先获知执行器或传感器故障的任何信息,比如,文献[16]要求传感器上界已知等,因此本文具有更小的保守性.
1. 系统模型和问题描述
考虑如下在概率空间$({ \rm {\Omega }},{ {F}},{{P}})$上具有参数不确定性的线性延迟马尔科夫跳变系统
$\left\{ \begin{array}{*{35}{l}} \dot{x}(t)=(A({{r}_{t}})+\Delta A({{r}_{t}}))x(t)~+ \\ ~~~~~~~{{A}_{d}}({{r}_{t}})x(t-\tau )+B({{r}_{t}})u(t)+D({{r}_{t}}){{f}_{a}}(t) \\ y(t)=C({{r}_{t}})x(t)+G({{r}_{t}}){{f}_{s}}(t) \\ x(t)=\phi (t),t\in [\begin{matrix} -\tau &0 \\ \end{matrix}] \\ \end{array} \right.$
(1) 其中,${\rm{\Omega }}$是样本空间,${ {F}}$是样本空间上的 $\sigma$-代数子集,${ {P}}$为概率测度. ${\pmb x}(t) \in {{\bf R}^n}$,${\pmb u}(t)\in {{\bf R}^m}$分别为系统状态和控制输入. ${{\pmb f}_a}(t) \in{{\bf R}^q}$和${{\pmb f}_s}(t) \in {{\bf R}^w}$分别是未知的执行器故障和传感器故障[16, 20].$\left\{ {{r_t}} \right\}$是在有限集${ S} = \left\{ {1,\cdots,s}\right\}$内取值的连续时间离散状态的马尔科夫过程,它具有如下状态转移概率:
${P_r}({r_{t + h}} = j|{r_t} = i) = \left\{ \begin{array}{l}{\pi _{ij}}h + o(h),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}~~~~~ i \ne j\\1 + {\pi _{ii}}h + o(h),{\kern 1pt} {\kern 1pt} {\kern 1pt} i = j\end{array} \right.$
其中$h > 0$,$\mathop {\lim }_{h \to 0} {{o(h)}}/{h} = 0$,${\pi _{ij}}$是从时间$t$处状态$i$到时间$t + h$处状态$j$的状态转移概率,且有${\pi _{ii}} = -\sum\nolimits_{j = 1,i \ne j}^s {{\pi _{ij}}} $,${\pi _{ij}}\ge 0$. 定义$\Pi = \left\{ {{\pi _{ij}}}\right\}$为未知的状态转移概率矩阵,且满足
$\Pi \subseteq \left\{ \hat{\Pi }+\Delta \Pi :\left| \Delta {{\pi }_{ij}} \right|\le {{\kappa }_{ij}},{{\kappa }_{ij}}\ge 0,i\ne j,i,j\in S \right\}$
其中~$\hat \Pi = \left\{ {{{\hat \pi }_{ij}}}\right\}$是已知的常数矩阵,${\hat \pi _{ij}} \ge 0,~i \ne j,$$i,j \in { S}$ 是${\pi _{ij}}$的估计值,$\Delta \Pi = \left\{ {\Delta {\pi _{ij}}} \right\}$表示状态转移速率矩阵中的不确定性,$\Delta {\pi _{ij}}$为速率估计误差,并且在有限集$[{\begin{array}{*{20}{c}}{ - {\kappa _{ij}}}&{{\kappa _{ij}}}\end{array}}]$中取值,对于$i \in { S}$,有${\hat \pi _{ii}}=-\sum\nolimits_{j = 1,i \ne j}^s {{{\hat \pi }_{ij}}}$和$\Delta {\pi _{ii}} = - \sum\nolimits_{j = 1,i \ne j}^s {\Delta {\pi _{ij}}} $. $A({r_t})$,${A_d}({r_t})$,$B({r_t})$,$D({r_t})$,$C({r_t})$和$G({r_t})$是具有适当维数关于${r_t}$的矩阵函数. $\Delta A({r_t})$是表示参数不确定性的未知矩阵,并假设$\Delta A({r_t}) =M({r_t})F({r_t},t)H({r_t})$,其中$M({r_t})$和$H({r_t})$是已知的常矩阵,$F({r_t},t)$是未知的时变矩阵满足${F^{\rm T}}({r_t},t)F({r_t},t)\le I$. $\tau > 0$是已知的延迟时间.函数${\pmb \phi}(t)$是在$[{\begin{array}{*{20}{c}} { - \tau }&0\end{array}}]$上的初始状态,系统初始模态为${r_0}$.假设${G(r_t)}$是列满秩.
注 1.本文假设${G(r_t)}$是列满秩是具有一般性的,许多关于传感器故障估计的文献都用到了此假设\[16, 20, 25].关于参数不确定性$\Delta A({r_t}) =M({r_t})F({r_t},t)H({r_t})$的假设也在针对不确定性系统问题的研究中被频繁应用[26-27].
为了表示方便,我们定义对任意矩阵${ \Psi}$有${ \Psi}({r_t} = i) = {{\Psi}_i}$,${\kern 1pt} i \in {\bf S}$,任意变量${\pmb \chi} (t)$有${\pmb \chi}(t) = {\pmb \chi}$.
定义 1. 定义变量${\pmb \Im} = \{ {\pmb \Im} (t)\}\in {L_2}[\begin{array}{*{20}{c}} 0&\infty\end{array})$,则它的${L_2}$范数为${\left\| {\pmb \Im} \right\|_{_2}} = \sqrt {\int_0^\infty {{\pmb \Im} {{(t)}^{\rm T}}{\pmb \Im} (t){\rm d}t} } $．
定义 2[25]. 对于$\tau > 0$,如果对于${\pmb u}\equiv {\bf 0}$,${\Delta}A_i \equiv {\bf 0}$,${\pmb f}_a \equiv{\bf 0}$,${\pmb f}_s \equiv {\bf 0}$,初始条件$({{\pmb x}_0},{r_0})$和所有定义在$[{\begin{array}{*{20}{c}} { - \tau }&0\end{array}}]$上的有限函数${\pmb \phi}(t)$,有
$\left[{\mathop{\rm E}\nolimits} \int_0^\infty {{{\pmb x}^{\rm T}}(t){\pmb x}}(t){\rm d}t|{{\pmb x}_0},{\pmb \phi}(t),{r_0}\right] \le \infty $
其中${\mathop{\rm E}\nolimits}$表示数学期望,则系统(1) 是随机稳定的.
定义 3[28].对马尔科夫跳变系统形如
\begin{equation}\left\{\begin{array}{l}\dot {\pmb x}(t) = {A_i}{\pmb x}(t) + {B_i}{\pmb u}(t) + {B_{\omega i}}{\pmb \omega} (t)\\{\pmb z}_\omega(t) = {C_i}{\pmb x}(t) + {D_i}{\pmb u}(t) +{D_{\omega i}}{\pmb \omega} (t)\end{array} \right.\end{equation}
(2) 其中~${\pmb \omega}(t)$表示干扰. 如果对于$λ >0$,存在常数$M({{\pmb x}_0},{r_0})$且$M(0,{r_0}) = 0$,满足
$\begin{array}{l}\Big[{\mathop{\rm E}\nolimits} \displaystyle \int_0^\infty {{{\pmb z}_{\omega}^{\rm T}}(t){\pmb z}_{\omega}(t){\rm d}t|{{\pmb x}_0},{r_0}{\Big]^{\frac{1}{2}}}} \le ~~~~~~~~~~~~~~~\\ ~~~~~~~~~~~~~~~~~~~~~~\gamma {[\left\| {{\pmb \omega} (t)} \right\|_2^2 + M({{\pmb x}_0},{r_0})]^{\frac{1}{2}}}\end{array}$
则系统(2) 是随机稳定且具有${H_\infty }$干扰抑制指数$λ$.
引理 1. 对于标量$\sigma > 0$和实矩阵${\Theta }_1$,${\Theta}_2$有
${{\Theta }_1 ^{\rm T}}{\Theta}_2 + {{\Theta}_2 ^{\rm T}}{\Theta }_1 \le {\sigma ^{ - 1}}{{\Theta }_1 ^{\rm T}}{\Theta }_1 + \sigma {{\Theta}_2 ^{\rm T}}{\Theta}_2 $
引理 2. 令矩阵$U$,$V'$和$F'(t)$为任意适当的实数矩阵,其中$U$和$V'$为已知,$F'(t)$为未知且满足${F'^{\rm T}}(t)F'(t) \le I$,对于$\varepsilon> 0$,如下不等式
$UF'(t)V' + {V'^{\rm T}}{F'^{\rm T}}(t){U^{\rm T}}\le \varepsilon U{U^{\rm T}} + {\varepsilon ^{ - 1}}{V'^{\rm T}}V'$
是成立的.
假设 1. ${{\pmb f}_a}$是可微的,且${\dot {\pmb f}_a} \in{L_2}[0\ \infty )$.
注 2.本文所设计方法可以适用于任何有界连续的传感器故障和满足假设1的执行器故障,且设计过程不需要知道有关故障的任何信息,例如故障的上界[16]和故障导数的上界[20]等.在实际应用中,故障往往都是经过一个暂态的变化之后几乎保持不变的,即满足${\dot{\pmb f}_a} \in {L_2}[0\ \infty )$.相比于文献[29]中关于${\dot {\pmb f}_a}$有界的假设,文中对于执行器故障的假设1更具有一般性.
为了能达到传感器故障和状态同时估计的目的,定义一个新的变量$\bar{\pmb x} = \left[{\begin{array}{*{20}{c}}{\pmb x}\\{{{\pmb f}_s}}\end{array}} \right] \in {{\bf R}^{n + w}}$,相应地,记${\bar A_i} = [{{A_i}}\ {{0_{n × w}}}]$,${\bar C_i} =[{{C_i}}\ {{G_i}}]$,$E =[{{I_n}}\ {{0_{n × w}}}]$.于是,系统(1) 可以写为
\begin{equation}\left\{ \begin{array}{l}E\dot {\bar {\pmb x}} = {{\bar A}_i}\bar {\pmb x} + \Delta {A_i}{\pmb x} + {A_{di}}{\pmb x}(t - \tau )+ \\ ~~~~~~~~~~{B_i}{\pmb u} + {D_i}{{\pmb f}_a}\\{\pmb y} = {{\bar C}_i}\bar {\pmb x}\end{array} \right.\end{equation}
(3) 系统(3) 是一个广义描述系统,状态包括原系统的状态和传感器故障.如果能够针对系统(3) 设计一个观测器,就能够同时得到原系统的状态和传感器故障的估计.
2. 主要结论
针对系统(3) ,本节将提出一种能同时估计系统状态、执行器和传感器故障的自适应观测器.
设计如下自适应观测器系统
\begin{equation}\left\{ \begin{array}{l}\dot {\pmb z} = {N_i}{\pmb z} + {L_i}{\pmb y} + {T_i}{B_i}{\pmb u}+ \\ ~~~~~~~~{T_i}{D_i}{{\hat {\pmb f}}_a} + {T_i}{A_{di}}\hat {\pmb x}(t - \tau )\\\hat {\bar {\pmb x}} = {\pmb z} + {Q_i}{\pmb y}\\{{\dot {\hat {\pmb f}}}_a} = {\Phi _i}({\pmb y} - \hat {\pmb y})\end{array} \right.\end{equation}
(4) 其中${\pmb z} \in {{\bf R}^{n + w}}$为观测器中间变量,$\hat {\bar {\pmb x}} = \left[{\begin{array}{*{20}{c}}{\hat {\pmb x}}\\{{{\hat {\pmb f}}_s}}\end{array}} \right]$为$\bar {\pmb x} = \left[{\begin{array}{*{20}{c}}{\pmb x}\\{{{\pmb f}_s}}\end{array}} \right]$的估计,${\hat {\pmb f}_a}$为执行器故障${{\pmb f}_a}$的估计,$\hat {\pmb x}(t - \tau )$为延迟状态${\pmb x}(t - \tau )$的估计. ${N_i}$,${L_i}$,${T_i}$,${Q_i}$和${\Phi _i}$为适当维数的待定矩阵.本文的主要目标就是求取矩阵${N_i}$,${L_i}$,${T_i}$,${Q_i}$和${\Phi _i}$使得系统(4) 可以在${H_\infty }$的意义下估计系统(3) 的状态,同时可以给出执行器故障${{\pmb f}_a}$在${H_\infty }$意义下的估计.
因为${G_i}$是列满秩,我们可以得到$\left[{\begin{array}{*{20}{c}}E\\{{{\bar C}_i}}\end{array}} \right] = \left[{\begin{array}{*{20}{c}}{{I_n}}&{{0_{n × w}}}\\{{C_i}}&{{G_i}}\end{array}} \right]$也是列满秩的.因此存在矩阵${T_i} \in {{\bf R}^{(n + w) × n}}$和${Q_i} \in {{\bf R}^{(n + w) × p}}$满足
\begin{equation}{T_i}E + {Q_i}{\bar C_i} = {I_{n + w}}\end{equation}
(5) 且其中一组特解为
$\begin{align} &{{T}_{i}}={{\left[ \begin{matrix} E \\ {{{\bar{C}}}_{i}} \\ \end{matrix} \right]}^{-1}}\left[ \begin{array}{*{35}{l}} {{I}_{n}} \\ {{0}_{p\times n}} \\ \end{array} \right] \\ &{{Q}_{i}}={{\left[ \begin{matrix} E \\ {{{\bar{C}}}_{i}} \\ \end{matrix} \right]}^{-1}}\left[ \begin{array}{*{35}{l}} {{0}_{p\times n}} \\ {{I}_{p}} \\ \end{array} \right] \\ \end{align}$
(6) 其中"$-1$''为矩阵的Moore-Penrose逆.
定义观测误差${\pmb { e}} = \bar {\pmb x} - \hat {\bar {\pmb x}}$,由系统(4) 中的第二式和式(5) 可得
${\pmb { e}} = \bar {\pmb x} -{\pmb z} - {Q_i}{\bar C_i}\bar {\pmb x} = ({I_{n + w}} -{Q_i}{\bar C_i})\bar {\pmb x} - {\pmb z} = {T_i}E\bar {\pmb x} -{\pmb z}$
因此误差的动态方程为
\begin{equation}\begin{array}{l}\dot {\pmb { e}} = {T_i}E\dot {\bar {\pmb x}} - \dot {\pmb z} =\\ ~~~~~~~~{T_i}{{\bar A}_i}\bar {\pmb x} + {T_i}\Delta {A_i}{\pmb x} + {T_i}{A_{di}}{\pmb x}(t - \tau ) + {T_i}{B_i}{\pmb u}+ \\ ~~~~~~~~{T_i}{D_i}{{\pmb f}_a} - {N_i}{\pmb z} - {L_i}{\pmb y}- \\ ~~~~~~~~{T_i}{B_i}{\pmb u} - {T_i}{D_i}{{\hat {\pmb f}}_a} - {T_i}{A_{di}}\hat {\pmb x}(t - \tau ) =\\ ~~~~~~~~ {T_i}{{\bar A}_i}\bar {\pmb x} + {T_i}\Delta {A_i}{\pmb x} + {T_i}{A_{di}}{\pmb x}(t - \tau )+ \\ ~~~~~~~~ {T_i}{B_i}{\pmb u} + {T_i}{D_i}{{\pmb f}_a} - {N_i}{\pmb z} - {L_i}{\pmb y}- \\ ~~~~~~~~{T_i}{B_i}{\pmb u} - {T_i}{D_i}{{\hat {\pmb f}}_a} - {T_i}{A_{di}}\hat {\pmb x}(t - \tau )+ \\ ~~~~~~~~ {N_i}{T_i}E\bar {\pmb x} - {N_i}{T_i}E\bar {\pmb x} =\\ ~~~~~~~~{N_i}{\pmb { e}} + ({T_i}{{\bar A}_i} - {L_i}{{\bar C}_i} - {N_i}{T_i}E)\bar {\pmb x}+ \\ ~~~~~~~~{T_i}{D_i}{{\tilde {\pmb f}}_a} + {T_i}{A_{di}}\tilde {\pmb x}(t - \tau ) + {T_i}\Delta {A_i}{\pmb x}\end{array}\end{equation}
(7) 其中${\tilde {\pmb f}_a} = {{\pmb f}_a} - {\hat {\pmb f}_a}$,$\tilde {\pmb x}(t - \tau ) = {\pmb x}(t - \tau ) - \hat {\pmb x}(t - \tau )$. 如果待定矩阵${N_i} \in {{\bf R}^{(n + w) × (n + w)}}$ 和${L_i} \in {{\bf R}^{(n + w) × p}}$满足
\begin{equation}{T_i}{\bar A_i} - {L_i}{\bar C_i} - {N_i}{T_i}E = 0\end{equation}
(8) 则式(7) 可以化简为
\begin{equation}\dot {\pmb { e}} = {N_i}{\pmb { e}} + {T_i}{D_i}{\tilde {\pmb f}_a} + {T_i}{A_{di}}\tilde {\pmb x}(t - \tau ) + {T_i}\Delta {A_i}{\pmb x}\end{equation}
(9) 不难发现满足式(8) 的一组解为
\begin{equation}{N_i} = {T_i}{\bar A_i} - {K_i}{\bar C_i}\end{equation}
(10) 和
\begin{equation}{L_i} = {K_i} + {N_i}{Q_i}\end{equation}
(11) 其中${K_i}$为具有适当维数的任意矩阵.
由系统(4) 中第三式和假设1可以得到
\begin{equation}{\dot {\tilde{\pmb f}}_a}={\dot {\pmb f}_a} - {\Phi _i}{\bar C_i}{\pmb {e}}\end{equation}
(12) 为了能够方便地求取矩阵${K_i}$和${\Phi_i}$,定义一个新的变量${\pmb \zeta} = \left[{\begin{array}{*{20}{c}}{\pmb { e}}\\{{{\tilde {\pmb f}}_a}}\end{array}} \right] \in {{\bf R}^{n + w + q}}$,则误差方程(9) 和(12) 可以合并为
\begin{equation}\dot {\pmb \zeta} = ({\hat A_i} - {\hat L_i}{\hat C_i}){\pmb \zeta} + {\hat D_i}{\pmb v} + {\hat A_{di}}{\pmb \zeta}(t - \tau ) + \Delta {\hat A_i}{\pmb x}\end{equation}
(13) 其中${\hat A_i} = \left[{\begin{array}{*{20}{c}}{{T_i}{{\bar A}_i}}&{{T_i}{D_i}}\\{{0_{q × (n + w)}}}&{{0_{q × q}}}\end{array}} \right]$,${\hat L_i} = \left[{\begin{array}{*{20}{c}}{{K_i}}\\{{\Phi _i}}\end{array}} \right]$,${\hat C_i} = \left[{\begin{array}{*{20}{c}}{{{\bar C}_i}}&{{0_{p × q}}}\end{array}} \right]$,${\hat D_i} = \left[{\begin{array}{*{20}{c}}{{0_{(n + w) × q}}}\\{{I_q}}\end{array}} \right]$,${\hat A_{di}} = \left[{\begin{array}{*{20}{c}}{{T_i}{{\bar A}_{di}}}&{{0_{(n + w) × q}}}\\{{0_{q × (n + w)}}}&{{0_{q × q}}}\end{array}} \right]$,$\Delta {\hat A_i} = {\hat M_i}{\hat F_i}(t){\hat N_i}$,${\bar A_{di}} = \left[{\begin{array}{*{20}{c}}{{A_{di}}}&{{0_{n × w}}}\end{array}} \right]$,${\hat M_i} = \left[{\begin{array}{*{20}{c}}{{T_i}{M_i}}&{{0_{(n + w) × 1}}}\\{{0_{q × 1}}}&{{0_{q × 1}}}\end{array}} \right]$,${\hat F_i}(t) ={ \rm diag}\left\{{F_i}(t),{F_i}(t)\right\}$,${\hat N_i} = \left[{\begin{array}{*{20}{c}}{{N_i}}\\{{0_{1 × n}}}\end{array}} \right]$和~${\pmb v} = {\dot {\pmb f}_a}$.由${F_i}(t)$的性质可得${\hat F_i}(t)$同样满足$\hat F_i^{\rm T}(t){\hat F_i}(t) \le I$.
由式(10) 、(11) 和(13) 可以看出,只要得到矩阵${\hat L_i}$使得式(13) 是鲁棒随机稳定的,观测器(4) 即可以实现.矩阵${K_i}$和${\Phi _i}$可由下式获得
\begin{equation}\left\{ \begin{array}{l}{K_i} = [{\begin{array}{*{20}{c}} {{I_{n + w}}}&{{0_{(n + w)× q}}}\end{array}}]{{\hat L}_i}\\{\Phi _i} = [{\begin{array}{*{20}{c}} {{0_{q × (n +w)}}}&{{I_q}}\end{array}}]{{\hat L}_i}\end{array} \right.\end{equation}
(14) 下面的定理给出了本文的主要结论.该定理不仅给出了式(13) 的鲁棒稳定性的证明,还给出了矩阵${\hat L_i}$的求取方法.
定理 1. 如果对于${λ _{ij}} > 0$,${\mu _{ij}} >0$,${\varepsilon _{1i}} > 0$ 和${\varepsilon _{2i}} > 0$,$i,j\in { S}$,存在对称正定矩阵 ${P_i} \in {{\bf R}^{(n + w + q)× (n + w + q)}}$,${R_i} \in {{\bf R}^{n× n}}$,$X \in{{\bf R}^{(n + w + q)× (n + w + q)}}$,$Z \in {\bf R}^{n× n}$,矩阵${Y_i} \in {{\bf R}^{(n + w + q) × p}}$和$\gamma> 0$ 使得下述凸优化问题(15) 有解(其中,$ * $表示矩阵的对称部分,这里对对称矩阵${\wp }$,${\wp } ＜0$表示对称矩阵${\wp }$ 是负定的),则误差系统(13) 是鲁棒随机稳定的,且具有${H_\infty}$干扰抑制水平$\gamma $.其中
$$\begin{array}{l}{\Gamma _{1i}} = {P_i}{{\hat A}_i} - {Y_i}{{\hat C}_i} + \hat A_i^{\rm T}{P_i} - {({Y_i}{{\hat C}_i})^{\rm T}} + {I_{n + w + q}}+ \\ ~~~~~~~~~~\sum\limits_{j = 1,j \ne i}^s {\frac{{\kappa _{ij}^2}}{4}{λ _{ij}}{I_{n + w + q}} + } \sum\limits_{j = 1}^s {{{\hat \pi }_{ij}}{P_j} + X}\end{array}$$ $$\begin{array}{l}{\Gamma _{2i}} = {R_i}{A_i} + A_i^{\rm T}{R_i} +Z+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\~~~~~~~~~~ \sum\limits_{P_i \hat L_i= Y_i}^s {{{\hat \pi }_{ij}}{R_j} + \sum\limits_{j = 1,j \ne i}^s {\frac{{\kappa _{ij}^2}}{4}{\mu _{ij}}{I_n}} }~~~~~~\end{array}$$ $$\begin{array}{l}{{\bar P}_i} = [\begin{array}{*{20}{c}} {{P_1} - {P_i}}& \cdots&{{P_{i-1}} - {P_i}}\end{array}\\~~~~{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}\begin{array}{*{20}{c}} { {P_{i + 1}-{P_i} }}& \cdots &{{P_s} -{P_i}}]\end{array}\end{array}$$ $$\begin{array}{l}{{\bar R}_i} = [\begin{array}{*{20}{c}} {{R_1} - {R_i}}& \cdots&{{R_{i-1}} - {R_i}}\end{array}\\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}\begin{array}{*{20}{c}} {{R_{i+1}} - {R_i}}& \cdots &{{R_s} -{R_i}}]\end{array}\end{array} $$ $$\begin{array}{l}{λ _{1i}} ={\rm diag}\big\{-{λ _{i1}}{I_{n + w + q}},\cdots ,- {λ _{i(i - 1) }}{I_{n + w + q}} ,\\ {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {λ _{i(i + 1) }}{I_{n + w +q}},\cdots ,- {λ _{is}}{I_{n + w + q}}\big\}\end{array} $$ $$\begin{array}{l}{λ _{2i}} = {\rm diag}\big\{-{\mu _{i1}}{I_n},\cdots,- {\mu_{i(i - 1) }}{I_n},\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} ~~~~~~~~~~~~~~~~~{\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {\mu_{i(i + 1) }}{I_n},\cdots ,- {\mu_{is}}{I_n}\big\}~~~~~~~~~~~~~~~~~~~\end{array} $$
\begin{equation} \left[{\begin{array}{*{20}{c}}{{\Gamma _{1i}}}&{{P_i}{{\hat A}_{di}}}&0&0&0&0&{{P_i}{{\hat D}_i}}&{{P_i}{{\hat M}_i}}&{{{\bar P}_i}}&0&0&0&0\\ * &{ - X}&0&0&0&0&0&0&0&0&0&0&0\\ *&* &{{\Gamma _{2i}}}&{{R_i}{A_d}_i}&{{R_i}{B_i}}&{{R_i}{D_i}}&0&0&0&{\hat N_i^{\rm T}}&{{R_i}{M_i}}&{N_i^{\rm T}}&{{{\bar R}_i}}\\ *&*&* &{ - Z}&0&0&0&0&0&0&0&0&0\\ *&*&*&* &{ - {\gamma ^2}{I_m}}&0&0&0&0&0&0&0&0\\ *&*&*&*&* &{ - {\gamma ^2}{I_q}}&0&0&0&0&0&0&0\\ *&*&*&*&*&* &{ - {\gamma ^2}{I_q}}&0&0&0&0&0&0\\ *&*&*&*&*&*&* &{ - \varepsilon _{1i}^{ - 1}{I_2}}&0&0&0&0&0\\ *&*&*&*&*&*&*&* &{ {λ _{1i}}}&0&0&0&0\\ *&*&*&*&*&*&*&*&* &{ - {\varepsilon _{1i}}{I_2}}&0&0&0\\ *&*&*&*&*&*&*&*&*&* &{ - \varepsilon _{2i}^{ - 1}{I_1}}&0&0\\ *&*&*&*&*&*&*&*&*&*&* &{ - {\varepsilon _{2i}}{I_1}}&0\\ *&*&*&*&*&*&*&*&*&*&*&* &{ {λ _{2i}}}\end{array}} \right]＜ 0\end{equation}
(15) 证明. 选取Lyapunov-Krasovskii函数
$\begin{array}{l}V({\pmb \zeta},{\pmb x},i) = V({\pmb \zeta},i) + {\displaystyle\int}_{t - \tau }^t {{{\pmb \zeta} ^{\rm T}}(} \theta)X{\pmb \zeta} (\theta ){\rm d}\theta~ +\$2mm] ~~~~~~~~~~~~~~~~~~~~~V({\pmb x},i) + \displaystyle\int_{t - \tau }^t {{{\pmb x}^{\rm T}}(} \theta )Z{\pmb x}(\theta ){\rm d}\theta\end{array}$
其中$V({\pmb \zeta},i) = {{\pmb \zeta}^{\rm T}}{P_i}{\pmb \zeta}$,$V({\pmb x},i) = {{\pmb x}^{\rm T}}{R_i}{\pmb x}$.定义对于具有马尔科夫过程的Lyapunov 函数的弱微分算子为
$\ell V({\pmb \zeta} ,i) = V({\pmb \zeta},{r_t}) \cdot \frac{{{\rm d}{\pmb \zeta} }}{{\rm d}{t}}{|_{{r_t} = i}} + \sum\limits_{j \in{ S}} {{\pi _{ij}}V({\pmb \zeta},j)} $ $\ell V({\pmb x},i) =V({\pmb x},{r_t}) \cdot \frac{{{\rm d}{\pmb x}}}{{{\rm d}t}}{|_{{r_t} = i}} + \sum\limits_{j \in { S}} {{\pi_{ij}}V({\pmb x},j)} $
因此对于任意 $i \in { S}$有
\begin{equation*}\begin{array}{l} \ell V({\pmb\zeta},{\pmb x},i) = {{\pmb \zeta}^{\rm T}}({P_i}({{\hat A}_i} -{{\hat L}_i}{{\hat C}_i})+ \\~~~~~{({{\hat A}_i} -{{\hat L}_i}{{\hat C}_i})^{\rm T}}{P_i}){\pmb \zeta} + 2{{\pmb \zeta} ^{\rm T}}{P_i}{{\hat D}_i}{\pmb v}+ \\ ~~~~~2{{\pmb \zeta} ^{\rm T}}{P_i}\Delta {{\hat A}_i}{\pmb x} + 2{{\pmb \zeta} ^{\rm T}}{P_i}{{\hat A}_{di}}{\pmb \zeta} (t - \tau )+ \\ ~~~~~{{\pmb \zeta} ^{\rm T}}X{\pmb \zeta} - {{\pmb \zeta} ^{\rm T}}(t - \tau )X{\pmb \zeta} (t - \tau )+ \\ ~~~~~{{\pmb x}^{\rm T}}({R_i}{A_i} + A_i^{\rm T}{R_i}){\pmb x} + {{\pmb x}^{\rm T}}Z{\pmb x}+ \\ ~~~~~2{{\pmb x}^{\rm T}}{R_i}{B_i}{\pmb u} + 2{{\pmb x}^{\rm T}}{R_i}{D_i}{{\pmb f}_a}+ \\ ~~~~~2{{\pmb x}^{\rm T}}{R_i}{A_d}_i{\pmb x}(t - \tau ) + 2{{\pmb x}^{\rm T}}{R_i}\Delta {A_i}{\pmb x}- \\~~~~~{{\pmb x}^{\rm T}}(t - \tau )Z{\pmb x}(t - \tau )+ \\~~~~~ {{\pmb \zeta} ^{\rm T}}\sum\limits_{j = 1}^s {{\pi_{ij}}{P_j}{\pmb \zeta} } + {{\pmb x}^{\rm T}}\sum\limits_{j =1}^s {{\pi _{ij}}{R_j}{\pmb x}}\end{array}\end{equation*}
(16) 对于上式中的$2{{\pmb \zeta} ^{\rm T}}{P_i}\Delta {\hat A_i}{\pmb x}$和$2{{\pmb x}^{\rm T}}{R_i}\Delta {A_i}{\pmb x}$ 应用引理2可得,对于${\varepsilon _{1i}} > 0 $和${\varepsilon _{2i}} > 0$,$i\in { S}$有
\begin{equation*}2{{\pmb \zeta} ^{\rm T}}{P_i}\Delta {\hat A_i}{\pmb x} \le {\varepsilon _{1i}}{{\pmb \zeta} ^{\rm T}}{P_i}{\hat M_i}\hat M_i^{\rm T}{P_i}{\pmb \zeta} + \varepsilon_{1i}^{ - 1}{{\pmb x}^{\rm T}}\hat N_i^{\rm T} {\hat N_i}{ {\pmb x}}\end{equation*}
(17) 和
\begin{equation*}2{{\pmb x}^{\rm T}}{R_i}\Delta {A_i}{\pmb x} \le {\varepsilon _{2i}}{{\pmb x}^{\rm T}}{R_i}{M_i}M_i^{\rm T}{R_i}{\pmb x} + \varepsilon _{2i}^{ - 1}{{\pmb x}^{\rm T}}N_i^{\rm T}{N_i}{\pmb x}\end{equation*}
(18) 此外,基于状态转移概率矩阵的特性我们可以分析得到
$\begin{array}{l}\sum\limits_{j = 1}^s {{\pi _{ij}}{P_j}} = \sum\limits_{j = 1}^s {{{\hat \pi }_{ij}}{P_j}} + \sum\limits_{j = 1}^s {\Delta {\pi _{ij}}{P_j}} =\\ ~~~~~~~~~~\sum\limits_{j = 1}^s {{{\hat \pi }_{ij}}{P_j}} + \sum\limits_{j = 1,j \ne i}^s {(\frac{{\Delta {\pi _{ij}}}}{2}({P_j} - {P_i})}+ \\ ~~~~~~~~~~\dfrac{{\Delta {\pi _{ij}}}}{2}({P_j} - {P_i}))\end{array}$
对上式第二行最后一项应用引理1,可以得到对于$i,j \in { S},
$\begin{align} &\sum\limits_{j=1}^{s}{{{\pi }_{ij}}{{P}_{j}}}\le \sum\limits_{j=1}^{s}{{{{\hat{\pi }}}_{ij}}{{P}_{j}}}+ \\ &\sum\limits_{j=1,j\ne i}^{s}{(\frac{\kappa _{ij}^{2}}{4}{{\lambda }_{ij}}{{I}_{n+w+q}}+\lambda 955;_{ij}^{-1}{{({{P}_{j}}-{{P}_{i}})}^{2}})} \\ \end{align}$
(19) 其中${\kappa _{ij}}$为$\Delta {\pi _{ij}}$的上界在第一部分已经被定义,${λ _{ij}} > 0$为任意标量.同理对于$i,j \in { S}$,${\mu _{ij}} > 0$有
\begin{equation*}\begin{array}{l}\sum\limits_{j = 1}^s {{\pi _{ij}}{R_j}} \le \sum\limits_{j = 1}^s {{{\hat \pi }_{ij}}{R_j}} + \\ ~~~~~~~~~~\sum\limits_{j = 1,j \ne i}^s {(\dfrac{{\kappa _{ij}^2}}{4}{\mu _{ij}}{I_n},+,} \mu _{ij}^{ - 1}{({R_j} - {R_i})^2})\end{array}\end{equation*}
(20) 将式(17) ~(20) 代入式(16) 可得
$\begin{array}{l} \ell V({\pmb \zeta},x,i) \le {{\pmb \zeta} ^{\rm T}}({P_i}({{\hat A}_i} -{{\hat L}_i}{{\hat C}_i}) ~+\\~~~~~~ {({{\hat A}_i} - {{\hat L}_i}{{\hat C}_i})^{\rm T}}{P_i}){\pmb \zeta}+ 2{{\pmb \zeta}^{\rm T}}{P_i}{{\hat D}_i}{\pmb v }~+\\~~~~~~{\varepsilon _{1i}}{{\pmb \zeta}^{\rm T}}{P_i}{{\hat M}_i}\hat M_i^{\rm T}{P_i}{\pmb \zeta} +\varepsilon _{1i}^{ - 1}{{\pmb x}^{\rm T}}\hat N_i^{\rm T}{{\hat N}_i}{\pmb x} ~+\\~~~~~~2{{{\pmb \zeta}}^{\rm T}}{P_i}{{\hat A}_{di}}{{\pmb\zeta}} (t -\tau ) + {{ {\pmb \zeta}} ^{\rm T}}X{\pmb \zeta} ~- \\~~~~~~{{ {\pmb\zeta}} ^{\rm T}}(t - \tau )X{\pmb \zeta} (t - \tau)+ {{\pmb x}^{\rm T}}Z{\pmb x} ~+\\ ~~~~~~ {{\pmb x}^{\rm T}}({R_i}{A_i} + A_i^{\rm T}{R_i}){\pmb x}+ 2{{\pmb x}^{\rm T}}{R_i}{B_i}{\pmb u}~+ \\ ~~~~~~2{{\pmb x}^{\rm T}}{R_i}{D_i}{{\pmb f}_a} + {\varepsilon _{2i}}{{\pmb x}^{\rm T}}{R_i}{M_i}M_i^{\rm T}{R_i}{\pmb x} ~+ \\~~~~~~ \varepsilon _{2i}^{ - 1}{{\pmb x}^{\rm T}}N_i^{\rm T}{N_i}{\pmb x}+ 2{{\pmb x}^{\rm T}}{R_i}{A_d}_i{\pmb x}(t - \tau ) ~- \\~~~~~~{{\pmb x}^{\rm T}}(t - \tau )Z{\pmb x}(t - \tau )+ {{\pmb \zeta} ^{\rm T}}\sum\limits_{j = 1}^s {{{\hat \pi }_{ij}}{P_j}} {\pmb \zeta} ~+\\~~~~~~{{\pmb \zeta} ^{\rm T}}\sum\limits_{j = 1,j \ne i}^s {(\displaystyle\frac{{\kappa _{ij}^2}}{4}{λ _{ij}}{I_{n+w+q}}} + λ _{ij}^{ - 1}{({P_j} - {P_i})^2}){\pmb \zeta} ~+\\~~~~~~{{\pmb x}^{\rm T}}\sum\limits_{j = 1}^s {{{\hat \pi }_{ij}}{R_j}} {\pmb x}~+ \\~~~~~~ {{\pmb x}^{\rm T}}\sum\limits_{j = 1,j \ne i}^s {(\dfrac{{\kappa _{ij}^2}}{4}{\mu _{ij}}{I_n} + } \mu _{ij}^{ - 1}{({R_j} - {R_i})^2}){\pmb x}\end{array}$
令$W = \ell V({\pmb \zeta} ,{\pmb x},i) + {{\pmb \zeta} ^{\rm T}}{\pmb \zeta} - {\gamma ^2}{{\pmb \varpi} ^{\rm T}}{\pmb \varpi} $,其中${\pmb \varpi} = \left[{\begin{array}{*{20}{c}}{\pmb u}\\{{{\pmb f}_a}}\\{\pmb v}\end{array}} \right]$,则有
$\begin{array}{l}W \le {{\pmb \zeta} ^{\rm T}}({P_i}({{\hat A}_i} - {{\hat L}_i}{{\hat C}_i}) + {({{\hat A}_i} - {{\hat L}_i}{{\hat C}_i})^{\rm T}}{P_i}){\pmb \zeta}+\\ ~~~~~~~~2{{\pmb \zeta} ^{\rm T}}{P_i}{{\hat D}_i}{\pmb v} + {\varepsilon _{1i}}{{\pmb \zeta} ^{\rm T}}{P_i}{{\hat M}_i}\hat M_i^{\rm T}{P_i}{\pmb \zeta}+ \\ ~~~~~~~~ \varepsilon _{1i}^{ - 1}{{\pmb x}^{\rm T}}\hat N_i^{\rm T}{{\hat N}_i}{\pmb x} + 2{{\pmb \zeta} ^{\rm T}}{P_i}{{\hat A}_{di}}{\pmb \zeta} (t - \tau )+ \\ ~~~~~~~~ {{\pmb \zeta} ^{\rm T}}X{\pmb \zeta} - {{\pmb \zeta} ^{\rm T}}(t - \tau )X{\pmb \zeta} (t - \tau )+ \\ ~~~~~~~~{{\pmb x}^{\rm T}}({R_i}{A_i} + A_i^{\rm T}{R_i}){\pmb x} + {{\pmb x}^{\rm T}}Z{\pmb x} + 2{{\pmb x}^{\rm T}}{R_i}{B_i}{\pmb u}+ \\\end{array}$$\\ \begin{array}{l} ~~~~~~~~ {\varepsilon _{2i}}{{\pmb x}^{\rm T}}{R_i}{M_i}M_i^{\rm T}{R_i}{\pmb x} + \varepsilon _{2i}^{ - 1}{{\pmb x}^{\rm T}}N_i^{\rm T}{N_i}{\pmb x}+ \\~~~~~~~~2{{\pmb x}^{\rm T}}{R_i}{D_i}{{\pmb f}_a} + 2{{\pmb x}^{\rm T}}{R_i}{A_d}_i{\pmb x}(t - \tau )- \\ ~~~~~~~~{{\pmb x}^{\rm T}}(t - \tau )Z{\pmb x}(t - \tau ) + {{\pmb \zeta} ^{\rm T}}\sum\limits_{j = 1}^s {{{\hat \pi }_{ij}}{P_j}} {\pmb \zeta}+ \\ ~~~~~~~~{{\pmb \zeta} ^{\rm T}}\sum\limits_{j = 1,j \ne i}^s {(\dfrac{{\kappa _{ij}^2}}{4}{λ _{ij}}{I_{n + w + q}} + } λ _{ij}^{ - 1}{({P_j} - {P_i})^2}){\pmb \zeta}+ \\ ~~~~~~~~{{\pmb x}^{\rm T}}\sum\limits_{j = 1}^s {{{\hat \pi }_{ij}}{R_j}} {\pmb x} + {{\pmb x}^{\rm T}}\sum\limits_{j = 1,j \ne i}^s {(\dfrac{{\kappa _{ij}^2}}{4}{\mu _{ij}}{I_n}}+ \\ ~~~~~~~~\mu _{ij}^{ - 1}{({R_j} - {R_i})^2}){\pmb x} + {{\pmb \zeta} ^{\rm T}}{\pmb \zeta} - \gamma {{\pmb \varpi} ^{\rm T}}{\pmb \varpi} = {{\pmb \eta} ^{\rm T}}{\Omega _i}{\pmb \eta}\end{array}$
其中
$\begin{align} & {{\Omega }_{i}}=\left[ \begin{matrix} {{r}_{1i}} & {{P}_{i}}{{{\hat{A}}}_{di}} & 0 & 0 & 0 & 0 & {{P}_{i}}{{{\hat{D}}}_{i}} \\ * & -X & 0 & 0 & 0 & 0 & 0 \\ * & * & {{r}_{2i}} & {{R}_{i}}{{A}_{d}}_{i} & {{R}_{i}}{{B}_{i}} & {{R}_{i}}{{D}_{i}} & 0 \\ * & * & * & -Z & 0 & 0 & 0 \\ * & * & * & * & -{{\gamma }^{2}}{{I}_{m}} & 0 & 0 \\ * & * & * & * & * & -{{\gamma }^{2}}{{I}_{q}} & 0 \\ * & * & * & * & * & * & -{{\gamma }^{2}}{{I}_{q}} \\ \end{matrix} \right] \\ & {{r}_{1i}}={{P}_{i}}({{{\hat{A}}}_{i}}-{{{\hat{L}}}_{i}}{{{\hat{C}}}_{i}})+{{({{{\hat{A}}}_{i}}-{{{\hat{L}}}_{i}}{{{\hat{C}}}_{i}})}^{\text{T}}}{{P}_{i}}+{{I}_{n+w+q}}+ \\ & {{\varepsilon }_{1i}}{{P}_{i}}{{{\hat{M}}}_{i}}\hat{M}_{i}^{\text{T}}{{P}_{i}}+X+\sum\limits_{j=1}^{s}{{{{\hat{\pi }}}_{ij}}{{P}_{j}}}+ \\ & \sum\limits_{j=1,j\ne i}^{s}{(\frac{\kappa _{ij}^{2}}{4}{{\lambda }_{ij}}{{I}_{n+w+q}}+}\lambda _{ij}^{-1}{{({{P}_{j}}-{{P}_{i}})}^{2}}) \\ & {{r}_{2i}}={{R}_{i}}{{A}_{i}}+A_{i}^{\text{T}}{{R}_{i}}+Z+\sum\limits_{j=1}^{s}{{{{\hat{\pi }}}_{ij}}{{R}_{j}}}+ \\ & \sum\limits_{j=1,j\ne i}^{s}{(\frac{\kappa _{ij}^{2}}{4}{{\mu }_{ij}}{{I}_{n}}+}\mu _{ij}^{-1}{{({{R}_{j}}-{{R}_{i}})}^{2}})+ \\ & {{\varepsilon }_{2i}}{{R}_{i}}{{M}_{i}}M_{i}^{\text{T}}{{R}_{i}}+\varepsilon _{1i}^{-1}\hat{N}_{i}^{\text{T}}{{{\hat{N}}}_{i}}+\varepsilon _{2i}^{-1}N_{i}^{\text{T}}{{N}_{i}} \\ & \eta ={{\left[ \begin{matrix} \zeta & \zeta (t-\tau ) & x & x(t-\tau ) & \varpi \\ \end{matrix} \right]}^{\text{T}}} \\ \end{align}$
注意到${P_i}{\hat L_i} = {Y_i}$,且对式(15) 计算舒尔补可得$W＜0$,即$\ell V + {{\pmb \zeta} ^{\rm T}}{\pmb \zeta} - \gamma {{\pmb\varpi} ^{\rm T}}{\pmb \varpi}＜0$.由Dynkin$'$s公式,有
$\begin{array}{l}{\rm E}\left\{ {V({\pmb \zeta},{\pmb x},i)} \right\} - {\rm E}\left\{ {V({{\pmb \zeta} _0},{{\pmb x}_0},{r_0})} \right\}+ \\ {\rm E}\displaystyle\int_0^\infty {{{\pmb \zeta}^{\rm T}}(\theta ){\pmb \zeta} (\theta ){\rm d}\theta -{\rm E}} \displaystyle\int_0^\infty {{\gamma ^2}{{\pmb \varpi}^{\rm T}}(\theta ){\pmb \varpi} (\theta ){\rm d}\theta } ＜0\end{array}$
其中${\pmb \zeta}_0,{\pmb x}_0,r_0$分别为相应量的初始值.因此我们可以得到
$\begin{array}{l}{\rm E}\displaystyle\int_0^\infty {{{\pmb \zeta} ^{\rm T}}(\theta ){\pmb \zeta} (\theta ){\rm d}\theta - {\rm E}}\displaystyle\int_0^\infty{{\gamma ^2}{{\pmb \varpi} ^{\rm T}}(\theta ){\pmb \varpi} (\theta ){\rm d}\theta }＜ \\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\mathop{\rm E}\nolimits} V({{\pmb \zeta} _0},{{\pmb x} _0},{r_0})\end{array}$
由定义1即
$\begin{align} &{{[\text{ }\int\limits_{0}^{\infty }{{{\zeta }^{\text{T}}}(\theta )}\zeta (\theta )\text{d}\theta ]}^{\frac{1}{2}}}\le [{{\gamma }^{2}}\left\| \varpi (\theta ) \right\|_{2}^{2}+~ \\ &V({{\zeta }_{0}},{{x}_{0}},{{r}_{0}}){{]}^{\frac{1}{2}}} \\ \end{align}$
(21) 因此结合定义2和3,系统(13) 鲁棒随机稳定且具有干扰抑制水平$\gamma$.
注 3.本文引入了Lyapunov-Krasovskii函数来处理带有时滞项的稳定性证明,利用Lyapunov-Krasovskii函数中的积分项可以把系统的时滞项连同状态引入到线性矩阵不等式中,从而利用Lyapunov稳定性理论可以使得整个系统的状态满足定义2的要求.
由定理1可以得出,系统(4) 是系统(3) 的鲁棒观测器,并且可以估计出系统状态、执行器和传感器故障.具体的算法如下:
1) 由式(5) 和(6) 计算得出矩阵${T_i}$和${Q_i}$;
2) 求解凸优化问题(15) ,若有解,则可以得到${\hat L_i} = P_i^{ -1}{Y_i}$,并通过式(14) 计算得到矩阵${K_i}$和${\Phi _i}$;
3) 将${K_i}$代入式(10) 和(11) 即可得到矩阵${N_i}$和${L_i}$.
至此观测器的系数矩阵均求取得到,观测器(4) 可以实现.系统的状态和传感器故障可由$\hat{\pmb x} =[{\begin{array}{*{20}{c}} {{I_n}}&{{0_{n × w}}}\end{array}}]\hat {\bar {\pmb x} }$和${\hat {\pmb f} _s} =[{\begin{array}{*{20}{c}}{{0_{w × n}}}&{{I_w}}\end{array}}]\hat {\bar {\pmb x} }$得到. ${\hat {\pmb f} _a}$可由系统(4) 中的第三式在线调节得到.
注 4[13].系统(4) 中的观测器是依赖于马尔科夫跳变系统的模态的,因此当系统跳变到某一个模态时,观测器相应地切换到这个模态.此外,观测器又依赖于转移概率${\pi_{ij}}$,使得其能够处理跳变所带来的影响.因此,观测器(4) 可以保证在系统跳变的情形下始终能估计出系统状态、执行器和传感器故障.
注 5 .本文所设计的方法中传感器估计的思路与文献[16]相类似,都是利用了广义描述系统的思想,但是设计观测器的技术手段是不同的.文献[16]将传感器故障引入到描述系统中,然后针对该描述系统设计了滑模观测器,利用滑模控制律抑制了传感器故障,然后对系统状态(包含传感器故障)做出了估计.而本文并没有将传感器故障引入描述系统中,针对描述系统设计了自适应观测器,该观测器不仅可以估计系统状态和传感器故障,还可以在线自动调节出执行器故障.相较于文献[16],本文存在以下优点: 1) 本文考虑的是一类具有参数不确定且状态转移概率不确定的延迟马尔科夫跳变系统,而文献[16]假定状态转移概率精确可得,这具有一定限制性;2) 本文同时估计了状态、执行器和传感器故障,文献[16]并没有涉及到执行器故障的估计;3) 由于文献[16]通过设计滑模观测器估计系统状态和抑制传感器故障,因此需要提前获知传感器故障的上界,而本文设计无需知道其上界.
3. 仿真分析
3.1 数值例子
为验证本文所提出方法的有效性,考虑一个形如式(1) 的具有两个模态的数值延迟马尔科夫跳变系统,相关参数如下:
$${A_1}= \left[{\begin{array}{*{20}{c}}{ - 5}&0&1\\0&{ - 7.5}&0\\2&0&{ - 5}\end{array}} \right],{A_2} = \left[{\begin{array}{*{20}{c}}{ - 6}&0&{1.1}\\0&{ - 8}&0\\0&0&{ - 5}\end{array}} \right] $$ $${A_{d1}} = \left[{\begin{array}{*{20}{c}}{0.2}&0&{0.1}\\{0.1}&0&0\\0&{0.1}&0\end{array}} \right],{B_1} = \left[{\begin{array}{*{20}{c}}1\\0\\1\end{array}} \right]$$ $${A_{d2}} = \left[{\begin{array}{*{20}{c}}{0.1}&0&{0.05}\\{0.05}&0&0\\0&{0.05}&0\end{array}} \right],{B_2} = \left[{\begin{array}{*{20}{c}}{0.5}\\0\\{0.5}\end{array}} \right]$$ $${D_1} = \left[{\begin{array}{*{20}{c}}{0.2}\\{0.1}\\{0.1}\end{array}} \right],{D_2} = \left[{\begin{array}{*{20}{c}}{0.3}\\{0.05}\\{0.1}\end{array}} \right]$$ $${C_1} = {C_2} = \left[{\begin{array}{*{20}{c}}1&1&0\\0&1&0\end{array}} \right]$$$${G_1} = {G_2} = \left[{\begin{array}{*{20}{c}}{0.1}\\{ - 0.3}\end{array}} \right],{M_1} = {M_2} = \left[{\begin{array}{*{20}{c}}{0.1}\\{0.2}\\{0.1}\end{array}} \right]$$ $${N_1} = {N_2} = \left[{\begin{array}{*{20}{c}}{0.1}&{0.2}&{0.2}\end{array}} \right]$$$${F_1}(t) = {F_2}(t) = \sin (t)$$
估计的状态转移概率矩阵为$\hat \Pi {\rm{ ~ = }}\left[{\begin{array}{*{20}{c}}{{\rm{ - }}0.4}&{0.4}\\{0.3}&{{\rm{ - }}0.3}\end{array}} \right]$,${\kappa _{12}} = {\kappa _{21}} = 1$和${λ _{12}} = {λ _{21}} = {\varepsilon _{11}} = {\varepsilon _{12}} = {\varepsilon _{21}} = {\varepsilon _{22}} = {\mu _{12}} = {\mu _{21}} = 1$,且延迟时间为3s. 执行器故障设定为${{ f}_a} = \sin (5t) + { {\rm e}^{ - 2t}} + 2\cos (t)$,传感器故障设定为${{ f}_s} = \sin (t) + 2\cos (5t)$. 本文假设马尔科夫系统有2 个模态${ S} = \left\{ {1,2} \right\}$.
在仿真中分别设置初始状态${{\pmb x}_0} = {[{\begin{array}{*{20}{c}} 3&{ - 2}&2\end{array}}]^{\rm T}}$,${{\pmb z}_0} = {[{\begin{array}{*{20}{c}}0&0&2\end{array}}]^{\rm T}}$,${r_0} = 1$和${\pmb \phi} (t) = {[{\begin{array}{*{20}{c}}1&0&0\end{array}}]^{\rm T}}$,$t \in [{\begin{array}{*{20}{c}}{ - 3}&0\end{array}}]$.系统状态估计如图 1~3所示.图 4为执行器故障估计效果,图 5为传感器故障估计效果.图 6为马尔科夫跳变系统的切换信号.由图 1~5可以看出本文方法对状态、执行器和传感器故障有很好的估计效果,仿真结果证明了方法的可行性.
3.2 实际例子
为了进一步验证本文所设计方法,接下来针对一个实际例子进行仿真,以此验证设计方法的实用性.考虑一个线性化的F-404飞行器引擎模型,其中矩阵$A$为
$A(t)=\left[ \begin{matrix} -1.46&0&2.428 \\ 0.1643+0.5\beta (t)&-0.4+\beta (t)&-0.3788 \\ 0.3107&0&-2.23 \\ \end{matrix} \right]$
$\beta (t)$是一个不确定的模型参数.假设$\beta (t)$满足一个$N = 2$的Markov过程:
$\beta (t) = \left\{ \begin{array}{l} - 1,\; \; \; \; r(t) = 1\\ - 2,\; \; \; \; r(t) = 2\end{array} \right.$
其他矩阵设置如下:
$${B_1} = \left[{\begin{array}{*{20}{c}}0\\1\\{0.3}\end{array}} \right],{B_2} = \left[{\begin{array}{*{20}{c}}{ - 1}\\{0.2}\\{ - 2}\end{array}} \right],{D_1} = \left[{\begin{array}{*{20}{c}}0\\{ - 0.1}\\0\end{array}} \right]$$ $${D_2} = \left[{\begin{array}{*{20}{c}}{ - 0.1}\\0\\{ - 0.3}\end{array}} \right],{C_1} = {C_2} = \left[{\begin{array}{*{20}{c}}1&0&0\\1&0&1\end{array}} \right]$$ $${G_1} = {G_2} = \left[{\begin{array}{*{20}{c}}{ - 1}\\1\end{array}} \right]$$ $${A_{d1}} = \left[{\begin{array}{*{20}{c}}{0.1}&0&{0.1}\\{0.1}&0&0\\0&{0.1}&{0.2}\end{array}} \right] $$ $${A_{d2}} = \left[{\begin{array}{*{20}{c}}{0.1}&0&{0.05}\\{0.03}&0&0\\0&{0.05}&{0.1}\end{array}} \right]$$ $${M_1} = {M_2} = \left[{\begin{array}{*{20}{c}}{0.1}\\0\\{0.3}\end{array}} \right],{F_1}(t) = {F_2}(t) = \sin (t)$$ $${N_1} = {N_2} = \left[{\begin{array}{*{20}{c}}{0.1}&{0.3}&{0.1}\end{array}} \right]$$
估计的转移概率矩阵为$\hat \Pi = \left[{\begin{array}{*{20}{c}}{ - 3}&3\\4&{ - 4}\end{array}} \right]$,其他参数选取如同实例1.
从参数中不难发现系统满足rank$({D_1}) \ne {\rm rank}({C_1}{D_1}) =0$,这与基于滑模观测器利用等价输出注入信号重构故障方法[30]的匹配条件是矛盾的.因此,传统的基于滑模观测器的方法不能用于该系统.此外,由于本文所讨论的系统是随机系统,因此系统的输出也是随机的,这就意味着基于代数重构的故障估计方法[31]也是不可行的,因为该方法中涉及到输出的微分信息.相比于文献[16]中的设计方法要求传感器故障是有界的且上界已知,以及其一阶微分也是有界的且上界已知[20],本文的设计方法中对这两种故障仅作了如下要求${\dot {\pmb f}_a} \in{L_2}[0\ \infty)$,这就使得本文的设计方法在实际应用中具有更广泛的应用范围.
为了验证本文所设计方法的优越性,选取执行器故障和传感器故障分别为${{f}_a} = 0.3\sin (t) + 0.5\cos (3t)$和${{ f}_s} = \sin (2t)$.
在仿真中分别设置初始状态${{\pmb x}_0} = {[{\begin{array}{*{20}{c}} 1&1&1\end{array}}]^{\rm T}}$,${{\pmb z}_0} = {[{\begin{array}{*{20}{c}}1&2&{0.3}&0\end{array}}]^{\rm T}}$,${r_0} = 1$和${\pmb \phi} (t) = {[{\begin{array}{*{20}{c}}1&0&0\end{array}}]^{\rm T}}$,$t \in [{\begin{array}{*{20}{c}}{ - 3}&0\end{array}}]$.系统状态估计如图 7~9所示.图 10和图 11分别为执行器和传感器故障估计效果.图 12为马尔科夫切换信号.由图 7~11可以看出本文方法对状态、执行器和传感器故障有很好的估计效果,仿真结果也证明了正如注5所示,该方法相较于文献[16, 20]具有优越性.
图 10 F-404模型执行器故障估计曲线Fig. 10 The curve of the estimation of actuator fault of Model F-404
图 11 F-404模型传感器故障估计曲线Fig. 11 The curve of the estimation of sensor fault of Model F-4044. 结论
本文针对具有参数不确定和延迟环节的马尔科夫跳变系统,在状态转移概率矩阵不确定的情形下,讨论了执行器和传感器故障同时估计的方法.首先构造一个广义描述系统,接着针对该系统设计自适应状态观测器使得执行器和传感器故障可以同时估计出.该方法的充分条件由线性矩阵不等式给出.仿真分析证明了该方法的可行性.
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图 1 IES系统多级对等协同优化架构
Fig. 1 The architecture of multilevel peer-to-peer co-optimization in IES
图 6 联盟1产消者3实时电力供求平衡
Fig. 6 Real-time supply and demand balance of 3rd prosumer in the first union
图 7 联盟2产消者10实时电力供求平衡
Fig. 7 Real-time supply and demand balance of 10th prosumer in the second union
图 8 联盟3产消者6实时电力供求平衡
Fig. 8 Real-time supply and demand balance of 6th prosumer in the third union
图 9 联盟4产消者14实时电力供求平衡
Fig. 9 Real-time supply and demand balance of 14th prosumer in the fourth union
表 1 优化模型相关参数
Table 1 The values of related parameters
参数 取值 $\eta$ 0.9 $l_{1}$ 0.01 $l_{2}$ 0.015 $\rho$ 0.8 $e_{1}$ 0.03 $e_{2}$ 0.045 
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表 2 不同控制参数下的迭代次数和计算时间
Table 2 The number of iterations and calculation time with different control parameters
$\varepsilon$ 迭代次数(次) 总计算时间(s) 0.8 20 483.46 0.6 15 369.90 0.4 11 281.34 0.2 8 162.64
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表 3 联盟1产消者3某时段多级对等电力交易结果
Table 3 Multilevel peer-to-peer power trading results of the third prosumer node in the first union at a certain time
时段 偏差电量(kWh) 买方报价(元/kWh) 买方密封交易信息摘要 卖方 购买电量(kWh) 卖方收益(元) 买方支付(元) 3:15 0.457 0.348 eb45cc1d1c684ae290d373e18b260d95 联盟1产消者2 0.097 0.034 0.158 - - - a378420878294a04965be0a999054616 联盟1产消者11 0.082 0.029 - - - - d9f7bf9a2b054722913725c4335a802f 联盟2产消者5 0.144 0.050 - - - - e00a77da2c944828b304543a749562d3 联盟3产消者3 0.130 0.045 - 3:20 0.441 0.327 77298613587a49b5ae3d79a66b20b3f7 联盟1产消者4 0.002 0.001 0.144 - - - 0928c7e54bb04ab9a9405046982b1ece 联盟2产消者8 0.128 0.042 - - - - 836cdb97cb5c4eb5a10cde523126ce51 联盟3产消者13 0.201 0.065 - - - - 1b9ce86e11a0493a928491ae96ff1531 联盟4产消者7 0.110 0.036 - 3:25 0.385 0.296 700d192ebcc94008bf2834f13ebfe180 联盟1产消者6 0.084 0.025 0.114 - - - 1e25a82f6df84e13a0585f717800f703 联盟1产消者15 0.124 0.037 - - - - 5ceb8c3095014a7b90f43b8e977c8fd0 联盟3产消者9 0.092 0.027 - - - - 23d398b02ba14c84a585faf63aa05f39 联盟4产消者1 0.085 0.025 - 3:30 0.426 0.324 8bd98c74da514150914b09951289d1dd 联盟1产消者5 0.039 0.013 0.138 - - - 2c39321d56564099af552d0f605f7ece 联盟2产消者10 0.123 0.040 - - - - 84e0e44dd96e4957ae25c95c1668e7d2 联盟3产消者6 0.046 0.015 - - - - 75d10e745eb44602a352f65b3b5e084a 联盟4产消者6 0.218 0.070 - 3:35 0.377 0.291 e7318fc4dda44955ac1083961c703939 联盟1产消者12 0.090 0.026 0.110 - - - d2a7c521239b4afa9f078a7402c4b246 联盟1产消者7 0.059 0.017 - - - - 2a2cbf7036854a7bafcb34fbd868bea1 联盟1产消者4 0.081 0.024 - - - - b84c13d1b952407a9c1589d35d2d5ee4 联盟4产消者1 0.147 0.043 -
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[1] Yu X H, Xue Y S.Smart grids:a cyber-physical systems perspective.Proceedings of the IEEE, 2016, 104(5):10581070 doi: 10.1109/JPROC.2015.2503119 [2] Xue Y S, Yu X H.Beyond smart grid-cyber-physical-social system in energy future (point of view).Proceedings of the IEEE, 2017, 105(12):2290-2292 doi: 10.1109/JPROC.2017.2768698 [3] 管晓宏, 赵千川, 贾庆山, 吴江, 刘烃.信息物理融合能源系统.北京:科学出版社, 2016.Guan Xiao-Hong, Zhao Qian-Chuan, Jia Qing-Shan, Wu Jiang, Liu Ting.Cyber-Physical Energy Systems.Beijing:Science Press, 2016. [4] 杨新法, 苏剑, 吕志鹏, 刘海涛, 李蕊.微电网技术综述.中国电机工程学报, 2014, 34(1):57-70 http://d.old.wanfangdata.com.cn/Periodical/zgdjgcxb201401008Yang Xin-Fa, Su Jian, Lv Zhi-Peng, Liu Hai-Tao, Li Rui.Overview on micro-grid technology.Proceedings of the CSEE, 2014, 34(1):57-70 http://d.old.wanfangdata.com.cn/Periodical/zgdjgcxb201401008 [5] Wu Y, Tan X Q, Qian L P, Tsang D H K, Song W Z, Yu L.Optimal pricing and energy scheduling for hybrid energy trading market in future smart grid.IEEE Transactions on Industrial Informatics, 2015, 11(6):1585-1596 doi: 10.1109/TII.2015.2426052 [6] Farraj A, Hammad E, Deepa K.A cyber-physical control framework for transient stability in smart grid.IEEE Transactions on Smart Grid, 2018, 9(2):1205-1215 doi: 10.1109/TSG.2016.2581588 [7] 王毅, 张宁, 康重庆.能源互联网中能量枢纽的优化规划与运行研究综述及展望.中国电机工程学报, 2015, 35(22):5669-5681 http://d.old.wanfangdata.com.cn/Periodical/zgdjgcxb201522001Wang Yi, Zhang Ning, Kang Chong-Qing.Review and prospect of optimal planning and operation of energy hub in energy internet.Proceedings of the CSEE, 2015, 35(22):5669-5681 http://d.old.wanfangdata.com.cn/Periodical/zgdjgcxb201522001 [8] Zhang H G, Li Y S, Gao D W, Zhou G J.Distributed optimal energy management for energy internet.IEEE Transactions on Industrial Informatics, 2017, 13(6):3081-3097 doi: 10.1109/TII.2017.2714199 [9] 孙秋野, 滕菲, 张化光.能源互联网及其关键控制问题.自动化学报, 2017, 43(2):176-194 http://www.aas.net.cn/CN/abstract/abstract18999.shtmlSun Qiu-Ye, Teng Fei, Zhang Hua-Guang.Energy internet and its key control issues.Acta Automatica Sinica, 2017, 43(2):176-194 http://www.aas.net.cn/CN/abstract/abstract18999.shtml [10] Zhang J J, Gao D W, Zhang Y C, Wang X, Zhao X Y, Duan D L, et al.Social energy:mining energy from the society.IEEE/CAA Journal of Automatica Sinica, 2017, 4(3):466482 doi: 10.1109/JAS.2017.7510547 [11] 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 [12] Vrba P, Mařík V, Siano P, Leitão P, Zhabelova G, Vyatkin V, et al.A review of agent and service-oriented concepts applied to intelligent energy systems.IEEE Transactions on Industrial Informatics, 2014, 10(3):1890-1903 doi: 10.1109/TII.2014.2326411 [13] Palensky P, Widl E, Stifter M, Elsheikh A.Modeling intelligent energy systems:co-simulation platform for validating flexible-demand EV charging management.IEEE Transactions on Smart Grid, 2013, 4(4):1939-1947 doi: 10.1109/TSG.2013.2258050 [14] 程乐峰, 余涛, 张孝顺, 殷林飞, 瞿凯平.信息-物理-社会融合的智慧能源调度机器人及其知识自动化:框架、技术与挑战.中国电机工程学报, 2018, 38(1):25-40 http://www.cnki.com.cn/Article/CJFDTotal-ZGDC201801003.htmCheng Le-Feng, Yu Tao, Zhang Xiao-Shun, Yin Lin-Fei, Qu Kai-Ping.Cyber-physical-social systems based smart energy robotic dispatcher and its knowledge automation:framework, techniques and challenges.Proceedings of the CSEE, 2018, 38(1):25-40 http://www.cnki.com.cn/Article/CJFDTotal-ZGDC201801003.htm [15] Roche R, Blunier B, Miraoui A, Hilaire V, Koukam A.Multi-agent systems for grid energy management:a short review.In:Proceedings of the 36th Annual Conference on IEEE Industrial Electronics Society.Glendale, USA:IEEE, 2010.3341-3346 [16] Duan R, Deconinck G.Future electricity market interoperability of a multi-agent model of the smart grid.In:Proceedings of the 2010 International Conference on Networking, Sensing and Control (ICNSC).Chicago, USA:IEEE, 2010.625-630 [17] Vergados D J, Mamounakis I, Makris P, Varvarigos E.Prosumer clustering into virtual microgrids for cost reduction in renewable energy trading markets.Sustainable Energy, Grids and Networks, 2016, 7:90-103 doi: 10.1016/j.segan.2016.06.002 [18] Marzband M, Javadi M, Domínguez-García J L, Moghaddam M M.Non-cooperative game theory based energy management systems for energy district in the retail market considering DER uncertainties.IET Generation, Transmission and Distribution, 2016, 10(12):2999-3009 doi: 10.1049/iet-gtd.2016.0024 [19] Di Battista G, Di Donato V, Patrignani M, Pizzonia M, Roselli V, Tamassia R.Bitconeview:visualization of flows in the Bitcoin transaction graph.In:Proceedings of the 2015 IEEE Symposium on Visualization for Cyber Security.Chicago, USA:IEEE, 2015.1-8 [20] Nguyen Q K.Blockchain-a financial technology for future sustainable development.In:Proceedings of the 3rd International Conference on Green Technology and Sustainable Development.Taiwan, China:IEEE, 2016.51-54 [21] 张宁, 王毅, 康重庆, 程将南, 贺大玮.能源互联网中的区块链技术:研究框架与典型应用初探.中国电机工程学报, 2016, 36(15):4011-4022 http://d.old.wanfangdata.com.cn/Periodical/shjn201803002Zhang Ning, Wang Yi, Kang Chong-Qing, Cheng Jiang-Nan, He Da-Wei.Blockchain technique in the energy internet:preliminary research framework and typical applications.Proceedings of the CSEE, 2016, 36(15):4011-4022 http://d.old.wanfangdata.com.cn/Periodical/shjn201803002 [22] 平健, 陈思捷, 张宁, 严正, 姚良忠.基于智能合约的配电网去中心化交易机制.中国电机工程学报, 2017, 37(13):3682-3690 http://www.cnki.com.cn/Article/CJFDTOTAL-ZGDC201713004.htmPing Jian, Chen Si-Jie, Zhang Ning, Yan Zheng, Yao Liang-Zhong.Decentralized transactive mechanism in distribution network based on smart contract.Proceedings of the CSEE, 2017, 37(13):3682-3690 http://www.cnki.com.cn/Article/CJFDTOTAL-ZGDC201713004.htm [23] 孙秋野, 赵美伊, 陈月, 马大中.能源互联网多能源系统最优功率流.中国电机工程学报, 2017, 37(6):1590-1598 http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=QKC20172017042100215298Sun Qiu-Ye, Zhao Mei-Yi, Chen Yue, Ma Da-Zhong.Optimal energy flow of multiple energy systems in energy internet.Proceedings of the CSEE, 2017, 37(6):1590-1598 http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=QKC20172017042100215298 [24] Sun Q Y, Han R K, Zhang H G, Zhou J G, Guerrero J M.A multiagent-based consensus algorithm for distributed coordinated control of distributed generators in the energy internet.IEEE Transactions on Smart Grid, 2015, 6(6):30063019 doi: 10.1109/TSG.2015.2412779 [25] Di Giorgio A, Liberati F.Near real time load shifting control for residential electricity prosumers under designed and market indexed pricing models.Applied Energy, 2014, 128:119-132 doi: 10.1016/j.apenergy.2014.04.032 [26] Zhang C H, Wu J Z, Zhou Y, Cheng M, Long C.Peer-to-Peer energy trading in a microgrid.Applied Energy, 2018, 220:1-12 doi: 10.1016/j.apenergy.2018.03.010 [27] 王永福, 张伯明, 孙宏斌, 夏清, 黄永皓, 孟远景, 等.基于内点法的实时平衡交易算法.电网技术, 2004, 28(1):48-50, 55 doi: 10.3321/j.issn:1000-3673.2004.01.012Wang Yong-Fu, Zhang Bo-Ming, Sun Hong-Bin, Xia Qing, Huang Yong-Hao, Meng Yuan-Jing, et al.A new method for balancing trade based on interior-point algorithm.Power System Technology, 2004, 28(1):48-50, 55 doi: 10.3321/j.issn:1000-3673.2004.01.012 [28] 王魁, 张步涵, 周杨, 李俊芳.基于混沌量子粒子群算法的含风电场电力系统实时调度.电力系统自动化, 2011, 35(22):141-146 http://d.old.wanfangdata.com.cn/Periodical/dlxtzdh201122028Wang Kui, Zhang Bu-Han, Zhou Yang, Li Jun-Fang.Real-time dispatch in wind power integrated system based on choas quantum-behaved particle swarm optimization.Automation of Electric Power Systems, 2011, 35(22):141-146 http://d.old.wanfangdata.com.cn/Periodical/dlxtzdh201122028 [29] Chen J Y, Wang J, Chen Q, Wu D.Optimal dispatch of medium-voltage microgrid using an adaptive PSO algorithm.In:Proceedings of the 7th International Conference on Intelligent Human-Machine Systems and Cybernetics.Hangzhou, China:IEEE, 2015.324-329 [30] Li Y L, Shao W, You L, Wang B Z.An improved PSO algorithm and its application to UWB antenna design.IEEE Antennas and Wireless Propagation Letters, 2013, 12:12361239 doi: 10.1109/LAWP.2013.2283375 [31] 王飞跃, 王晓, 袁勇, 王涛, 林懿伦.社会计算与计算社会:智慧社会的基础与必然.科学通报, 2015, 60(5-6):460-469 http://d.old.wanfangdata.com.cn/Periodical/fzxtyfzxkx200404002Wang Fei-Yue, Wang Xiao, Yuan Yong, Wang Tao, Lin Yi-Lun.Social computing and computational societies:the foundation and consequence of smart societies.Chinese Science Bulletin, 2015, 60(5-6):460-469 http://d.old.wanfangdata.com.cn/Periodical/fzxtyfzxkx200404002 [32] 孙秋野, 王冰玉, 黄博南, 马大中.狭义能源互联网优化控制框架及实现.中国电机工程学报, 2015, 35(18):4571-4580 http://d.old.wanfangdata.com.cn/Periodical/zgdjgcxb201518002Sun Qiu-Ye, Wang Bing-Yu, Huang Bo-Nan, Ma Da-Zhong.The optimization control and implementation for the special energy internet.Proceedings of the CSEE, 2015, 35(18):4571-4580 http://d.old.wanfangdata.com.cn/Periodical/zgdjgcxb201518002 [33] 孙秋野.能源互联网.北京:科学出版社, 2015.Sun Qiu-Ye.The Energy Internet.Beijing:Science Press, 2015. [34] 王飞跃.平行控制:数据驱动的计算控制方法.自动化学报, 2013, 39(4):293-302 http://www.aas.net.cn/CN/abstract/abstract17915.shtmlWang Fei-Yue.Parallel control:a method for data-driven and computational control.Acta Automatica Sinica, 2013, 39(4):293-302 http://www.aas.net.cn/CN/abstract/abstract17915.shtml [35] 张俊, 高文忠, 张应晨, 郑心湖, 杨柳青, 郝君, 等.运行于区块链上的智能分布式电力能源系统:需求、概念、方法以及展望.自动化学报, 2017, 43(9):1544-1554 http://www.aas.net.cn/CN/abstract/abstract19130.shtmlZhang Jun, Gao Wen-Zhong, Zhang Ying-Chen, Zheng Xin-Hu, Yang Liu-Qing, Hao Jun, et al.Blockchain based intelligent distributed electrical energy systems:needs, concepts, approaches and vision.Acta Automatica Sinica, 2017, 43(9):1544-1554 http://www.aas.net.cn/CN/abstract/abstract19130.shtml [36] Cheng S, Zeng B, Huang Y Z.Research on application model of blockchain technology in distributed electricity market.In:Proceedings of the 2nd International Conference on New Energy and Future Energy System (NEFES).Kunming, China:IOP Publishing Ltd, 2017.1-11 [37] Geary R C.A note on "a constant-utility index of the cost of living".The Review of Economic Studies, 1950, 18(1):65-66 doi: 10.2307/2296107 [38] 项顶, 宋永华, 胡泽春, 徐智威.电动汽车参与V2G的最优峰谷电价研究.中国电机工程学报, 2013, 33(31):15-25 http://d.old.wanfangdata.com.cn/Periodical/zgdjgcxb201331030Xiang Ding, Song Yong-Hua, Hu Ze-Chun, Xu Zhi-Wei.Research on optimal time of use price for electric vehicle participating V2G.Proceedings of the CSEE, 2013, 33(31):1525 http://d.old.wanfangdata.com.cn/Periodical/zgdjgcxb201331030 [39] Friedman D, Rust J.The Double Auction Market:Institutions, Theories and Evidence.Boulder, CO, USA:Westview Press, 1993. [40] Iosifidis G, Koutsopoulos I.Double auction mechanisms for resource allocation in autonomous networks.IEEE Journal on Selected Areas in Communications, 2010, 28(1):95-102 doi: 10.1109/JSAC.2010.100110 期刊类型引用(7)
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