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摘要: 序列推荐(Sequential recommendation, SR)旨在建模用户序列中的动态兴趣, 预测下一个行为. 现有基于知识蒸馏(Knowledge distillation, KD)的多模型集成方法通常将教师模型预测的概率分布作为学生模型样本学习的软标签, 不利于关注低置信度序列样本中的动态兴趣. 为此, 提出一种同伴知识互增强下的序列推荐方法(Sequential recommendation enhanced by peer knowledge, PeerRec), 使多个具有差异的同伴网络按照人类由易到难的认知过程进行两阶段的互相学习. 在第1阶段知识蒸馏的基础上, 第2阶段的刻意训练通过动态最小组策略协调多个同伴从低置信度样本中挖掘出可被加强训练的潜在样本. 然后, 受训的网络利用同伴对潜在样本预测的概率分布调节自身对该样本学习的权重, 从解空间中探索更优的兴趣表示. 3个公开数据集上的实验结果表明, 提出的PeerRec方法相比于最新的基线方法在基于Top-k的指标上不仅获得了更佳的推荐精度, 且具有良好的在线推荐效率.Abstract: Sequential recommendation (SR) aims to model dynamic interests within users＇ interaction sequences and predict the next behaviour. Current knowledge distillation (KD)-based multi-model ensemble methods generally adopt teacher＇s probability distributions as soft labels to guide the sample learning of its students, posing a significant challenge to focus on the samples with less confidence. To this end, we propose a novel sequential recommendation method enhanced by peer knowledge (PeerRec) that makes multiple sequential prediction networks (peer for each other) with distinct knowledge conduct two-stage mutual learning by following the easy to hard human cognition. Based on the knowledge distillation in the first stage, the deliberate practice in the second stage mines the potential samples from the low confidence samples by combining multiple peers through a dynamic minimum group strategy. The training network can further employ the probability distributions from peers to adjust itself learning weight where better representations of dynamic interests can be derived. Extensive experiments on three public recommendation datasets show that the proposed PeerRec not only outperforms the state-of-the-art methods, but also achieves good efficiency in online recommendation.
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信息物理融合系统(Cyber-physical systems, CPS)作为一种新型智能系统应运而生, 它是一类集成计算、网络和物理实体的复杂系统, 将三者进行有机融合与深度协作, 从而达到对大型物理系统与信息系统的实时感知, 动态控制和信息服务等[1], 典型CPS包括工控系统[2]、供水网络[3]等.智能电网从总体上可以视为由信息网和电力网这两个相互依存的网络构成的一个复合网络, 也是一个典型的CPS[4].同步相量测量装置(Phasor measurement units, PMUs)、广域测量系统(Wide-area measurement systems, WAMS)、变电站自动化等技术为智能电网的实现提供了坚实的基础, 但同时也增加了智能电网对信息资源的依赖.一旦信息网出错或崩溃, 电力网一般很难保持正常运行.这也为电力系统安全稳定运行带来了新的问题[5]:首先, 随着PMUs布点的增多, 调度数据网中传送的PMUs数据的比例将会越来越大, PMUs长期不间断且高刷新频率的传送导致海量的状态和控制信息等在通信网络上传送, 可能会产生网络拥塞, 影响数据实时传送; 其次, 与电力的物理系统相比, 信息系统对恶意攻击具有更明显的脆弱性.由于通信网络的开放性, 会导致电力系统面临各种类型网络攻击, 造成失稳甚至毁坏.如2015年12月23日乌克兰电网遭受协同攻击导致近8万用户家庭突发停电事故, 这次事故被认为是第一起由于网络攻击直接导致停电事故的案例.
针对上述问题, 已有一些学者进行了相关的研究[5-10].文献[6]首次将事件触发机制引入多域电力系统负荷频率控制(Load frequency control, LFC)中, 有效减少了数据传输量; 文献[8]提出了一种弹性事件触发机制, 应用于多域电力系统LFC当中, 在考虑网络攻击的情况下保证电力系统的稳定性, 并减少了传输的数据量; 文献[9]提出一种针对电力系统状态估计的错误数据注入攻击防御与检测机制, 从保护和检测两方面入手; 基于保护的防御, 主要是识别和保护关键的传感器, 使系统更能抵御攻击; 基于检测的防御, 设计了基于空间和基于时间的检测方案, 以准确识别数据注入攻击.文献[10]介绍了DoS攻击下电网的LFC方法, 通过将电力系统建模为切换系统, 检测DoS (Denial-of-service, DoS)攻击的存在, 以双域电力系统为例, 分析了系统性能问题.由于电动汽车(Electric vehicles, EVs)具有良好的环境特征, 如温室气体排放量少, 噪声污染低等, 并且可用于提高电力系统的可靠性和灵活性[11].本文将EVs引入智能电网中, 与负荷频率控制相结合, 快速抑制系统扰动所引发的频率变化.在考虑DoS攻击的情况下, 对电力CPS进行稳定性分析, 并对事件触发机制和控制器进行联合设计, 从而达到理想的控制效果.
1. 系统模型
在本文中, 假设存在大量可用的EVs, 即存在足够的电能储备以达到协助传统电力单元实现负荷频率调节的目的, 控制中心通过聚合器对EVs进行集中管理, 聚合器将分散的各EVs的信息和状态收集起来, 发送给控制中心.
由于EVs的接入, LFC系统中产生了新的时变时延.本文假设所有的同步电机都有再热热涡轮机, 为了便于说明, 对于电力CPS, 我们将每个域的$M$个EVs等效为一个, 控制器的输出按比例分配给EVs和再热电机, 其中表示分配比例.如图 1所示, 不考虑弹性事件触发机制和网络环境下, 包含EVs的电力CPS动态模型可以描述为[12]
$ \begin{align} \begin{cases} \dot{{\pmb x}}(t)=A{\pmb x}(t)+B_{0}{\pmb u}(t)+\\ \quad\qquad\sum\limits_{i=1}^{n}\hat{B}_i{\pmb u} (t-\tau_i(t))+ F{\pmb\Delta P_{d}}(t)\\ {\pmb y}(t)=C{\pmb x}(t) \end{cases} \end{align} $
(1) $ \begin{align*} &{\pmb x}(t)=[{\pmb x_1}^{\rm T}(t) \ {\pmb x_2}^{\rm T}(t) \ \cdots \ {\pmb x_n}^{\rm T}(t) ]^{\rm T}\\ &{\pmb y}(t)=[{\pmb y_1}^{\rm T}(t) \ {\pmb y_2}^{\rm T}(t) \ \cdots \ {\pmb y_n}^{\rm T}(t) ]^{\rm T}\\ &{\pmb u}(t)=[{\pmb u_1}^{\rm T}(t) \ {\pmb u_2}^{\rm T}(t) \ \cdots \ {\pmb u_n}^{\rm T}(t) ]^{\rm T}\\ &{\pmb \Delta P_{d}}(t)=[\Delta P_{d1} \ \Delta P_{d2} \ \cdots \ \Delta P_{dn}]^{\rm T}\\ &{\pmb y_i}(t)=[ACE_i \ \int ACE_i{\rm d}t]^{\rm T}\\ &A=\left[\begin{array}{cccc} A_{11}&A_{12}&\cdots&A_{1n}\\ A_{21}&A_{22}&\cdots&A_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ A_{n1}&A_{n2}&\cdots&A_{nn}\\ \end{array}\right] \end{align*} $
图 1 基于弹性事件触发机制的电力CPS负载频率控制模型Fig. 1 Grid CPS LFC model with a resilient event-triggered$ \begin{align*} &A_{ii}= \left[\begin{array}{ccccccc} -\dfrac{D_i}{M_i}&\dfrac{1}{M_i}&0&0&\dfrac{1}{M_i}&-\dfrac{1}{M_i}&0\\[3mm] 0&-\dfrac{1}{T_{ci}}&\dfrac{1}{T_{ci}}&0&0&0&0\\[3mm] -\dfrac{F_{pi}}{R_iT_{gi}}&0&-\dfrac{1}{T_{ri}}&\dfrac{T_{gi}-F_{pi}T_{ri}} {T_{ri}T_{gi}}&0&0&0\\[3mm] -\dfrac{1}{R_iT_{gi}}&0&0&-\dfrac{1}{T_{gi}}&0&0&0\\[3mm] 0&0&0&0&-\dfrac{1}{T_{EVi}}&0&0\\ 2\pi\sum\limits_{j=1, j\ne i}^{n}T_{ij}&0&0&0&0&0&0\\ \beta_i&0&0&0&0&1&0 \end{array}\right]\\ &{\pmb x_i}(t)=[\Delta f_i \ \Delta P_{gi} \ \Delta P_{mi} \ \Delta X_{gi} \ \Delta P_{EVi} \ \Delta P_{tie-i} \ \int ACE_i{\rm d}t ]^{\rm T} \end{align*} $
$ \begin{align*} &B_0=\left[\begin{array}{cccc} B_{011}&0&\cdots&0\\ 0&B_{022}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&B_{0nn}\\ \end{array}\right]\\ &C={\rm diag}\{C_1, \ C_2, \ \cdots, \ C_n\}\\ &B_{0ii}=\left[0 \ 0 \ \frac{F_{pi}\alpha_{i0}}{T_{gi}} \ \frac{\alpha_{i0}}{T_{gi}} \ 0 \ 0 \ 0\right]^{\rm T}\\ &C_i=\left[\begin{array}{ccccccc} \beta_i&0&0&0&0&1&0\\ 0&0&0&0&0&0&1 \end{array}\right]\\ &F={\rm diag}\{F_1, \ F_2, \ \cdots, \ F_n\}\\ &F_i=\left[-\frac{1}{M_i} \ 0 \ 0 \ 0 \ 0 \ 0 \ 0\right]^{\rm T}\\ &\hat{B}_{i}=\left[\begin{array}{ccccc} 0&\cdots&0&\cdots&0\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ 0&\cdots&B_{iii}&\cdots&0\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ 0&\cdots&0&\cdots&0 \end{array}\right]\\ &B_{iii}=\left[0 \ 0 \ 0 \ 0 \ \frac{K_{EVi}\alpha_{i1}}{T_{EVi}} \ 0 \ 0\right]^{\rm T}\\ &A_{ij}=\left[\begin{array}{cccc} 0&0&\cdots&0\\ 0&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ -2\pi T_{ij}&0&\cdots&0 \end{array}\right] T_{ij}=T_{ji} \end{align*} $
表 1给出了系统中具体的符号意义.每个区域$i$的ACE (Area control error)信号定义为频率偏差与区域之间联络线电力交换之和:
表 1 带EVs电力CPS负载频率控制模型参数($i=1, 2, \cdots, n$)Table 1 Parameters of power CPS LFC model including EV aggregators ($i=1, 2, \cdots, n$)参数符号Parameter notations 物理含义Physical meanings $M_i$ 惯性常数 Inertia constant $D_i$ 负载阻尼常数 Load-damping constant $T_{gi}$ 调速器时间常数 Time constant of governor $T_{ci}$ 涡轮机时间常数 Time constant of turbine $T_{ri}$ 再热时间常数 Time constant of reheat $F_{pi}$ 总涡轮功率分数 Fraction of total turbine power $R_i$ 转速 Speed droop $\beta_i$ 频率偏差系数 Frequency bias factor $K_{EVi}$ 电池系数 Battery coefficient $T_{EVi}$ 电池时间常数 Time constant of battery $T_{ij}$ 联络线同步系数 Synchronizing coefficient of tie-line $\Delta f_i$ 频率偏差 Deviation of frequency $\Delta P_{tie-i}$ 联络线的功率交换 Power transfer of tie-line $ \begin{equation*} ACE_i=\beta_i \Delta f_i+\Delta P_{tie-i} \end{equation*} $
假设传输线路为无损传输, 电力CPS控制各区域之间的联络线功率交换满足$\sum_{i=1}^n \Delta P_{tie-i}=0$.
PI型LFC可以写成
$ \begin{align} {\pmb u}(t)&=-K{\pmb y}(t) \end{align} $
(2) 其中, $K={\rm diag}\{{\pmb K_1}$, ${\pmb K_2}$, $\cdots$, ${\pmb K_n}\}$, ${\pmb K_i}=[K_{Pi}~K_{Ii}]$, $K_{Pi}$, $K_{Ii}$分别为比例增益和积分增益.
注1[13].EVs参与负载频率控制可分为两种模式, SOC (State of charge)可控模式和SOC空闲模式. SOC空闲模式, 即EV从电网中消耗电能或释放电能不考虑EV电池的充电状态, 此时第$m$辆EV的增益$K_{Em}=\bar{K_{e}}$, 其中$\bar{K_{e}}=1$.而SOC可控模式中由于EV使用者的需要, 第$m$辆EV参与负载频率控制时需要考虑SOC, 因此通过SOC计算和当前SOC的值可以获得EV的增益$K_{Em}=\bar{K_e}-\bar{K_e}g_m(t)$, 其中$g_m(t)=(\frac{{\rm SOC}_m-{\rm SOC}_{{\rm low(high)}, m}}{{\rm SOC}_{{\rm max(min)}, m}-{\rm SOC}_{{\rm low(high)}, m}})^{v_m}$, $v_m$表示电池的设计规格, ${\rm SOC_{low(high)}}$为低(高)电池SOC, ${\rm SOC_{max(min)}}$为最大(最小)电池SOC.假设在$t$时刻, 区域$i$中有$Me$辆EVs参与LFC, 其中$Me_1$表示处于SOC空闲模式的EVs, $Me_2=Me-Me_1$表示处于SOC可控模式的EVs, 则区域$i$总的EVs增益$K_{EVi}=\sum_{m=1}^{Me}\frac{K_{Em}}{Me}=\frac{Me_1}{Me}\bar{K_e}+\frac{1}{Me}\bar{K_e}(Me_2-\sum^{Me}_{m=Me_1+1}g_m(t))$.
2. 事件触发通信机制
2.1 传统事件触发机制
早期事件触发机制是所谓的连续事件触发, 需要特殊硬件对状态进行连续监测.此外, 在触发机制设计中, 必须确保任意两个事件触发时刻之间的最小时间间隔严格大于零, 如果最小事件间隔时间为零, 就会出现无限事件发生在有限时间内的奇诺(Zeno)现象[14].为了解决这两个问题, 文献[15]提出一种基于采样数据的离散型事件触发机制:
$ \begin{eqnarray} [{\pmb x}(t_k+jh)-{\pmb x}(t_k)]^{\rm T}\Omega[{\pmb x}(t_k+jh)-{\pmb x}(t_k)]\leq\nonumber\\ \sigma {\pmb x}^{\rm T}(t_k)\Omega {\pmb x}(t_k), j\in {\bf N} \end{eqnarray} $
(3) 当上述条件违背时, 传感器将采样数据传输给控制器.其中$t_k$为事件触发时刻, $\Omega>0$为触发矩阵, $\sigma\in(0, 1)$为触发参数, $h$为采样周期.该触发机制可以保证最小事件间隔$T_{etc}\geq h$, 避免出现Zeno现象, 并且不需要特殊硬件对状态进行连续监测.
2.2 弹性事件触发机制
本文假设DoS攻击的能量是有限的, 即DoS攻击的持续时间是有限的, 当DoS攻击发生时通信中断. DoS攻击的发生将直接导致通信信道上正在传输的数据丢失, 因此并不是所有事件触发时刻的状态都能成功传输到控制器侧.假设DoS攻击发生时, 连续丢包量为$\tau_M$, 那么DoS攻击的持续时间$\tau_{\rm dos}\leq \tau_M T_{etc}$, 为了简单起见, 取$\tau_{\rm dos}\leq \tau_M h$.此时, 传统事件触发机制(3)将不能直接用于判断采样数据传输与否.为了消除DoS攻击所产生的影响, 提出下列弹性事件触发机制:
$ \begin{eqnarray} [{\pmb x}(r_k+jh)-{\pmb x}(r_k)]^{\rm T}\Omega[{\pmb x}(r_k+jh)-{\pmb x}(r_k)]\leq\nonumber\\ \sigma_r {\pmb x}^{\rm T}(r_k)\Omega {\pmb x}(r_k), j\in {\bf N} \end{eqnarray} $
(4) $\sigma_r$为事件触发参数.通常, $\sigma_r < \sigma$, 也就是说弹性事件触发机制将会产生更多的触发状态用以消除DoS攻击对系统所造成的影响.此时, 将存在三个时刻, 采样时刻$kh$, 触发时刻$r_k$和成功传输到控制器侧的时刻$t_k$.令$S_0=\{0, h, 2h, 3h, \cdots, jh\}, ~ j\in {\bf N}$, $S_1=\{0, r_1, r_2, r_3, \cdots, r_k\}, r_k/h\in {\bf N}$, $S_2=\{0, t_1, t_2, t_3, \cdots, t_k\}, t_k/h\in {\bf N}$, 则$S_2\subseteq S_1\subseteq S_0$.当$\sigma_r=\sigma$时, 意味着没有DoS攻击发生.下文将推导出系统所能容忍的DoS攻击最大持续时间.
此时控制器输入${\pmb u}(t)={\pmb u}(t_k), t\in[t_k+d_{\tau_k}, t_{k+1}+d_{\tau_{k+1}})$, $d_{\tau_k}$表示传感器到控制器的传输时延, 最大值为$\bar{d_{\tau}}$, 此时系统(1)可以表示为
$ \begin{eqnarray} \dot{{\pmb x}}(t)=A{\pmb x}(t)+B{\pmb u}(t_k)+F{\pmb\Delta P_{d}}(t) \end{eqnarray} $
(5) 其中, $B=B_{0}+\sum_{i=1}^{n}\hat{B}_i$.利用文献[15]中提出的时滞模型方法对系统进行稳定性分析和控制器设计.
定义
$ \begin{eqnarray*} d(t)= \left\{\begin{aligned} &t-t_{k}, t\in I_{1}\\ &t-t_{k}-lh, t\in I^l_{2}, \quad l=1, 2, \cdots, m_k-1\\ &t-t_{k}-m_kh, t\in I_{3}\\ \end{aligned}\right.\ \end{eqnarray*} $
其中, $I_{1}=[t_{k}+d_{\tau_{k}}, t_{k}+h+\bar{d_\tau}), I^{l}_{2}=[t_{k}+lh+\bar{d_\tau}, t_{k}+lh+h+\bar{d_\tau}), I_{2}=\cup^{m_k-1}_{l=1}I^{l}_{2}$, 和$I_{3}=[t_{k}+m_kh+\bar{d_\tau}, t_{k+1}+d_{\tau_{k+1}})$. $[t_{k}+d_{\tau_{k}}, t_{k+1}+d_{\tau_{k+1}})=I_{1}\cup I_{2}\cup I_{3}$.
因此, 可以得到$0\leq d(t) < \bar{d}$, 其中$\bar{d}=h+\bar{d_\tau}$.对于$\forall t\in[t_{k}+d_{\tau_{k}}, t_{k+1}+d_{\tau_{k+1}})$, 我们定义
$ \begin{eqnarray*} {\pmb e}(t)= \begin{cases} 0, &t\in I_{1}\\ {\pmb x}(t_{k})-{\pmb x}(t_{k}+lh), &t\in I^l_{2}\\ {\pmb x}(t_{k})-{\pmb x}(t_{k}+m_kh), &t\in I_{3}\\ \end{cases} \end{eqnarray*} $
结合$d(t)$和${\pmb e}(t)$定义, 当$t\in[t_{k}+d_{\tau_{k}}, t_{k+1}+d_{\tau_{k+1}})$时, 系统(5)可以变形为
$ \begin{align} \dot{{\pmb x}}(t)=\, &A{\pmb x}(t)-BKC({\pmb e}(t)+\nonumber\\ &{\pmb x}(t-d(t)))+F{\pmb\Delta P_{d}}(t) \end{align} $
(6) 本文的主要目的是在DoS攻击的情况下研究电力系统的稳定性、设计弹性事件触发机制和控制器联合求解方案, 同时系统还满足如下条件:
1) 当负载干扰为零时(即${\pmb\Delta P_d}=0$), 电力系统在DoS攻击下是渐近稳定的;
2) 当系统的初值为零时, 对于任意非零${\pmb\Delta P_d}\in L_2[0, \infty)$, 有$\Vert {\pmb y}(t)\Vert _2 \leq \gamma \Vert {\pmb\Delta P_d}\Vert_2$, 其中$\gamma$是给定的$H_{\infty}$性能.
3. 主要结果
引理1[16].令矩阵$R_1, R_2$为正定矩阵, 标量$a\in(0, 1)$, 以及向量${\pmb \omega_1}, {\pmb\omega_2} \in {\bf R}^m$, 那么对于任意矩阵$Y_1, Y_2 \in {\bf R}^{m\times m}$, 下列不等式成立:
$ \begin{align*} &\frac{1}{a}{\pmb\omega_1}^{\rm T}R_1{\pmb\omega_1}+\frac{1}{1-a}{\pmb\omega_2}^{\rm T}R_2{\pmb\omega_2}\geq \nonumber\\ &~~~~~~~{\pmb\omega_1}^{\rm T}[R_1+(1-a)(R_1-Y_1R_2^{-1}Y_1^{\rm T})]{\pmb\omega_1}+ \nonumber \\ &~~~~~~~{\pmb\omega_2}^{\rm T}[R_2+a(R_2-Y_2^{\rm T}R_1^{-1}Y_2)]{\pmb\omega_2}+ \nonumber\\ &~~~~~~~2{\pmb\omega_1}^{\rm T}[aY_1+(1-a)Y_2]{\pmb\omega_2} \end{align*} $
本节对基于事件触发机制的电力CPS进行稳定性分析, 选择Lyapunov-Krasovskii泛函为
$ \begin{align*} V(t)=\, &V_{1}(t)+V_{2}(t)+V_{3}(t)\\ V_{1}(t)=\, &\left[\begin{array}{cc}{\pmb x}^{\rm T}(t)&\int^{t}_{t-\bar{d}} {\pmb x}^{\rm T}(s){\rm d}s\end{array}\right]\times\\& U\left[\begin{array}{c}{\pmb x}(t)\\\int^{t}_{t-\bar{d}}{\pmb x}(s){\rm d}s\end{array}\right]\\ V_{2}(t)=\, &\int_{t-\bar{d}}^{t}\left[\begin{array}{cc}{\pmb x}^{\rm T}(t)&\bar{d}{\pmb x}^{\rm T}(s)\end{array}\right]Q\left[\begin{array}{c}{\pmb x}(t)\\\bar{d}{\pmb x}(s)\end{array}\right]{\rm d}s\\ V_{3}(t)=\, &\bar{d}\int_{t-\bar{d}}^{t}\int_{v}^{t}\dot{{\pmb x}}^{\rm T}(s)R\dot{{\pmb x}}(s){\rm d}s{\rm d}v \end{align*} $
其中, $U$, $Q$对称, $R>0$.应用Jensen不等式,
$ \begin{align*} V_{2}(t)\geq\, &\frac{1}{\bar{d}}\int_{t-\bar{d}}^{t}\left[\begin{array}{c}{\pmb x}(t)\\ \bar{d}{\pmb x}(s)\end{array}\right]^{\rm T}{\rm d}s\times\\ &Q\int_{t-\bar{d}}^{t}\left[\begin{array}{c}{\pmb x}(t)\\ \bar{d}{\pmb x}(s)\end{array}\right]{\rm d}s=\\ &\bar{d}\left[\begin{array}{c}{\pmb x}(t)\\ \int^{t}_{t-\bar{d}}{\pmb x}(s){\rm d}s\end{array}\right]^{\rm T}\times\\&Q\left[\begin{array}{c}{\pmb x}(t)\\ \int^{t}_{t-\bar{d}}{\pmb x}(s){\rm d}s\end{array}\right] \end{align*} $
从而得到:
$ \begin{align*} &V(t)\geq\left[\begin{array}{cc}{\pmb x}^{\rm T}(t)&\int^{t}_{t-\bar{d}}{\pmb x}^{\rm T}(s){\rm d}s\end{array}\right](U+\bar{d}Q)\times\\ &~~\left[\begin{array}{c}{\pmb x}(t)\\\int^{t}_{t-\bar{d}}{\pmb x}(s){\rm d}s\end{array}\right] +\bar{d}\int_{t-\bar{d}}^{t}\int_{v}^{t}\dot{{\pmb x}}^{\rm T}(s)R\dot{{\pmb x}}(s){\rm d}s{\rm d}v\\ \end{align*} $
由上式, 当矩阵$U$和$Q$满足$U+\bar{d}Q>0$, $V(t)>0$, 此时并不需要$U, Q$和$R$都正定.对$V(t)$进行求导,
$ \begin{align*} \dot{V}_{1}(t)=\, &2\left[\begin{array}{cc}{\pmb x}^{\rm T}(t)&\int^{t}_{t-\bar{d}}{\pmb x}^{\rm T}(s){\rm d}s\end{array}\right]U \times\\ &\left[\begin{array}{c}\dot{{\pmb x}}(t)\\{\pmb x}(t)-{\pmb x}(t-\bar{d})\end{array}\right]\\ \dot{V}_{2}(t)=\, &\left[\begin{array}{cc}{\pmb x}^{\rm T}(t)&\bar{d}{\pmb x}^{\rm T}(t)\end{array}\right]Q\left[\begin{array}{c}{\pmb x}(t)\\\bar{d}{\pmb x}(t)\end{array}\right]-\\ &\left[\begin{array}{cc}{\pmb x}^{\rm T}(t)&\bar{d}{\pmb x}^{\rm T}(t-\bar{d})\end{array}\right]Q\left[\begin{array}{c}{\pmb x}(t)\\\bar{d}{\pmb x}(t-\bar{d})\end{array}\right]+\\ &2\int_{t-\bar{d}}^{t}\left[\begin{array}{cc}{\pmb x}^{\rm T}(t)&\bar{d}{\pmb x}^{\rm T}(s)\end{array}\right]Q\left[\begin{array}{c}\dot{{\pmb x}}(t)\\0\end{array}\right]{\rm d}s \end{align*} $
$ \begin{align*} \dot{V}_{3}(t)%=\bar{d}^2\dot{x}^{T}(t)R\dot{x}(t)-\bar{d}\int_{t-\bar{d}}^{t}\dot{x}^T(s)R\dot{x}(s)ds\\ =\, &\bar{d}^2\dot{{\pmb x}}^{\rm T}(t)R\dot{{\pmb x}}(t)-\bar{d}\int_{t-d(t)}^{t}\dot{{\pmb x}}^{\rm T}(s)R\dot{{\pmb x}}(s){\rm d}s-\\ &\bar{d}\int_{t-\bar{d}}^{t-d(t)}\dot{{\pmb x}}^{\rm T}(s)R\dot{{\pmb x}}(s){\rm d}s \end{align*} $
3.1 不考虑DoS攻击情况下的$\pmb{H_{\infty}}$性能分析
为了便于描述, 定义${\pmb\xi_1}:=\text{col}\{{\pmb x}(t)$, ${\pmb x}(t-d(t))$, ${\pmb x}(t-\bar{d})$, $\frac{1}{d(t)}\int_{t-d(t)}^{t}{\pmb x}(s){\rm d}s$, $\frac{1}{\bar{d}-d(t)}\int_{t-\bar{d}}^{t-d(t)}{\pmb x}(s){\rm d}s$, $\frac{1}{d^2(t)}\int_{t-d(t)}^{t}(t-s){\pmb x}(s){\rm d}s$, $\frac{1}{(\bar{d}-d(t))^2}\int_{t-\bar{d}}^{t-d(t)}(t-d(t)-s){\pmb x}(s){\rm d}s$, ${\pmb e}(t)$, ${\pmb\Delta P_d}(t)\}$.
应用文献[17]中的Lemma 1,
$ \begin{align} & \bar{d}\int_{t-\bar{d}}^{t-d(t)}{{{{\mathit{\boldsymbol{\dot{x}}}}}^{\text{T}}}}(s)R\mathit{\boldsymbol{\dot{x}}}(s)\text{d}s\ge \frac{1}{a}{{({{\Gamma }_{1}}{{\mathit{\boldsymbol{ }}\!\!\xi\!\!\text{ }}_{1}}(t))}^{\text{T}}}\hat{R}({{\Gamma }_{1}}{{\mathit{\boldsymbol{ }}\!\!\xi\!\!\text{ }}_{1}}(t)) \\ & \bar{d}\int_{t-d(t)}^{t}{{{{\mathit{\boldsymbol{\dot{x}}}}}^{\text{T}}}}(s)R\mathit{\boldsymbol{\dot{x}}}(s)\text{d}s\ge \frac{1}{1-a}{{({{\Gamma }_{2}}{{\mathit{\boldsymbol{ }}\!\!\xi\!\!\text{ }}_{1}}(t))}^{\text{T}}}\hat{R}({{\Gamma }_{2}}{{\mathit{\boldsymbol{ }}\!\!\xi\!\!\text{ }}_{1}}(t)) \\ & {{\Gamma }_{1}}:=\text{col}\{{{e}_{2}}-{{e}_{3}}, {{e}_{2}}+{{e}_{3}}-2{{e}_{5}}, {{e}_{2}}-{{e}_{3}}-6{{e}_{5}}+12{{e}_{7}}\} \\ & {{\Gamma }_{2}}:=\text{col}\{{{e}_{1}}-{{e}_{2}}, {{e}_{1}}+{{e}_{2}}-2{{e}_{4}}, {{e}_{1}}-{{e}_{2}}-6{{e}_{4}}+12{{e}_{6}}\text{ }\!\!\}\!\!\text{ } \\ & \hat{R}=\text{diag}\{R, 3R, 5R\} \\ \end{align} $
其中, $a=\frac{\bar{d}-d(t)}{\bar{d}}$.进一步应用引理1, 可以得到:
$ \begin{align*} &\bar{d}\int_{t-d(t)}^{t}\dot{{\pmb x}}^{\rm T}(s)R\dot{{\pmb x}}(s){\rm d}s+\\ &~~~~~\bar{d}\int_{t-\bar{d}}^{t-d(t)}\dot{{\pmb x}}^{\rm T}(s)R\dot{{\pmb x}}(s){\rm d}s\geq\\ %&~~~~\geq\frac{1}{a}(\Gamma_1\xi_1(t))^T\hat{R}(\Gamma_1\xi_1(t))\\ %&~~~~~+\frac{1}{1-a}(\Gamma_2\xi_1(t))^T\hat{R}(\Gamma_2\xi_1(t))\\ &~~~~~{\pmb\xi_1}^{\rm T}(t)[(2-a)\Gamma_1^{\rm T}\hat{R}\Gamma_1+(1+a)\Gamma_2^{\rm T}\hat{R}\Gamma_2+\\ &~~~~~{\rm Sym}\{\Gamma_1^{\rm T}[aY_1+(1-a)Y_2]\Gamma_2\}-\Pi_2]{\pmb\xi_1}(t)\\ &\Pi_2=a\Gamma_2^{\rm T}Y_2^{\rm T}\hat{R}^{-1}Y_2\Gamma_2+(1-a)\Gamma_1^{\rm T}Y_1\hat{R}^{-1}Y_1^{\rm T}\Gamma_1 \end{align*} $
因此
$ \begin{align*} \dot{V}(t)\leq\,&{\pmb \xi}_1^{\rm T}(t)(\Pi_1+\Pi_2){\pmb \xi_1}(t)\\ \Pi_1:=\, &\ell_1^{\rm T} Q\ell_1-\ell_2^{\rm T} Q\ell_2+\bar{d}^2\ell_0^{\rm T} R\ell_0-\\ &(2-a)\Gamma_1^{\rm T}\hat{R}\Gamma_1-(1+a)\Gamma_2^{\rm T}\hat{R}\Gamma_2+\\ &{\rm Sym}\{\ell_3^{\rm T}U\ell_4+ \bar{d}\ell_4^{\rm T}Q\ell_5-\\ &\Gamma_1^{\rm T}[aY_1+(1-a)Y_2]\Gamma_2\}\\ \ell_0:=\, &Ae_1-BKC(e_2+e_8)+Fe_9\\ \ell_1:=\, &{\rm col}\{e_1, \bar{d}e_1\}, \ell_2={\rm col}\{e_1, \bar{d}e_3\}\\ \ell_3:=\, &{\rm col}\{\ell_0, e_1-e_3\}\\ \ell_4:=\, &{\rm col}\{e_1, d(t)e_4+(\bar{d}-d(t))e_5\}\\ \ell_5:=\, &{\rm col}\{\ell_0, 0\} \end{align*} $
结合传统事件触发机制(3), 对于$t\in[t_k+d_{\tau_k}$, $t_{k+1}+$ $d_{\tau_{k+1}})$, 我们可以得到:
$ \begin{align*} \dot{V}(t)\leq\,&{\pmb\xi_1}^{\rm T}(t)(\hat{\Pi}_1+\Pi_2){\pmb\xi_1}(t)-\\ &{\pmb y}^{\rm T}(t){\pmb y}(t)+\gamma^2{\pmb \Delta P}^{\rm T}_d(t){\pmb \Delta P_d}(t)\\ \hat{\Pi}_1=\, &\Pi_1+\sigma (e_2+e_8)^{\rm T}\Omega (e_2+e_8)-e_8^{\rm T}\Omega e_8+\\ &e_1^{\rm T}C^{\rm T}Ce_1-\gamma^2 e_9^{\rm T}e_9 \end{align*} $
使用Schur补, 可以得到当存在$R>0$, 对称矩阵$U, Q$, 实矩阵$Y_1$, $Y_2$以及标量$\bar{d}$, 满足$U+\bar{d}Q>0$, 并且LMIs (7)和(8)成立时, 系统(6)渐近稳定.
$ \begin{eqnarray} \left[\begin{array}{cc} \hat{\Pi}_1&\Gamma_2^{\rm T}Y_2^{\rm T}\\ *&-\hat{R}\\ \end{array}\right]<0 \end{eqnarray} $
(7) $ \begin{eqnarray} \left[\begin{array}{cc} \hat{\Pi}_1&\Gamma_1^{\rm T}Y_1\\ *&-\hat{R}\\ \end{array}\right]<0 \end{eqnarray} $
(8) 同时, 当LMIs (7)和(8)成立时, 我们可以得到:
$ \begin{align*} \dot{V}(t)\leq -{\pmb y}^{\rm T}(t){\pmb y}(t)+\gamma^2{\pmb \Delta P}^{\rm T}_d(t){\pmb \Delta P_d}(t) \end{align*} $
因为${\pmb x}(t)$在$t$上连续, 所以$\dot{V}(t)$在$t$上也连续.因此对不等式两边同时从$0$到$\infty$对$t$进行积分, 可得:
$ \begin{align*} V(\infty)-V(0)\leq &\int^{\infty}_{0}[-{\pmb y}^{\rm T}(t){\pmb y}(t)+\\ &\gamma^2{\pmb \Delta P}^{\rm T}_d(t){\pmb \Delta P_d}(t)]{\rm d}s \end{align*} $
从而在零初始条件下,
$ \begin{align*} \int^{\infty}_{0}[-{\pmb y}^{\rm T}(t){\pmb y}(t)+\gamma^2{\pmb\Delta P}^{\rm T}_d(t){\pmb\Delta P_d}(t)]{\rm d}s \geq 0 \end{align*} $
即, 对任意非零${\pmb\Delta P_d}(t) \in L_2[0, +\infty)$以及给定的$\gamma$, $\Arrowvert {\pmb y}(t)\Arrowvert_2 \leq \gamma\Arrowvert {\pmb\Delta P_d}\Arrowvert_2 $.在${\pmb\Delta P_d}(t)=0$的情况下, 存在$\varepsilon>0$使得${\pmb x}(t)\ne 0$时, $\dot{V}(t)\leq -\varepsilon \Arrowvert {\pmb x}(t)\Arrowvert_2$.因此, 系统(6)渐近稳定并具$H_{\infty}$范数界.
3.2 考虑DoS攻击情况下的${\pmb H_\infty}$稳定性分析
本节将在DoS攻击发生时, 基于弹性事件触发机制(4)对系统进行稳定性分析, 找出弹性触发参数$\sigma_r$以及所能容忍的最大DoS攻击持续时间.
定理1.对于给定的标量$\sigma>0$, $\bar{d}>0$, $\gamma>0$, 当存在实矩阵$R>0$, $\Omega>0$, 对称矩阵$Q, U$满足$U+\bar{d}Q>0$, 实矩阵$Y_1$, $Y_2$以及给定的控制器增益矩阵$K$使得LMIs (7)和(8)成立, 那么在弹性事件触发机制(4)下, 系统(1)渐近稳定, 并且$\sigma_r$满足:
$ \begin{equation} \sigma_r \leq (\sqrt[\tau_M+1]{\sqrt{\sigma}+1}-1)^2 \end{equation} $
(9) 同时, 当$\sigma_r$已知的情况下, 由式(10)可以得到系统所能容忍的最大DoS攻击持续时间
$ \begin{equation} \tau_{\rm dos}\leq \tau_{M}h \leq \lfloor[{\rm log}_{1+\sqrt{\sigma_r}}(1+\sqrt{\sigma})]\rfloor h \end{equation} $
(10) 其中, $\lfloor*\rfloor$表示向下取整.
证明.为了符号表示方便, 我们假设在两次成功传输时刻的区间$[t_k, t_{k+1})$存在$\tau_M$个由于DoS攻击所造成的未成功传输但是触发的状态${\pmb x}(r_j)$, 其中$t_k=r_0 < r_1 < r_2 < \cdots < r_{\tau_M} < r_{\tau_M+1}=t_{k+1}$.
因此, 区间$[t_k, t_{k+1})$可以分为多个小区间$[r_j, r_{j+1}), j\in\{0, 1, 2, \cdots, \tau_M\}$.
$ \begin{align*} &\vert {\pmb e}(t)\vert=\vert {\pmb x}(t)-{\pmb x}(t_k)\vert\leq\\ &~~~~~~~~~~~~\sum^{j-1}_{p=0}\vert {\pmb x}(r_{p+1})-{\pmb x}(r_p) \vert+\vert {\pmb x}(t)-{\pmb x}(r_j)\vert \end{align*} $
其中, $t\in[r_j, r_{j+1}), t=r_j+lh, l\in {\bf N}$.应用弹性事件触发机制(4),
$ \begin{align*} &\vert {\pmb x}(t)-{\pmb x}(r_j)\vert \leq \sqrt{\sigma_r}\vert {\pmb x}(r_j)\vert \\ &\vert {\pmb x}(r_{p+1})-{\pmb x}(r_p)\vert \leq \sqrt{\sigma_r}\vert {\pmb x}(r_{p})\vert \end{align*} $
因此
$ %\begin{align*} %&\vert e(t)\vert=\vert x(t)-x(t_k)\vert\\ %&~~~~~~~~~~~~\leq \sum^{j-1}_{p=0}\sqrt{\sigma_r}\vert x(r_p) \vert+\sqrt{\sigma_r}\vert x(r_j)\vert\\ %&~~~~~~~~~~~~=\sum^{j}_{p=0}\sqrt{\sigma_r}\vert x(r_p) \vert %\end{align*} \begin{align*} &\vert {\pmb e}(t)\vert=\vert {\pmb x}(t)-{\pmb x}(t_k)\vert=\sum^{j}_{p=0}\sqrt{\sigma_r}\vert {\pmb x}(r_p) \vert \end{align*} $
又由$\vert {\pmb x}(r_{p+1})-{\pmb x}(r_p)\vert \leq \sqrt{\sigma_r}\vert {\pmb x}(r_{p})\vert$可以得到$\vert {\pmb x}(r_{p+1})\vert \leq (1+\sqrt{\sigma_r})\vert {\pmb x}(r_{p})\vert$, 进一步可得,
$ \begin{align*} &\vert {\pmb x}(r_{p+1}) \vert \leq (1+\sqrt{\sigma_r})^{p+1}\vert {\pmb x}(t_k)\vert, \\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~p=0, 1, \cdots, j-1 \end{align*} $
所以
$ \begin{align} &\vert {\pmb e}(t)\vert=\vert {\pmb x}(t)-{\pmb x}(t_k)\vert\leq\nonumber \\ &~~~~~~~~~~~\sum^{j}_{p=0}\sqrt{\sigma_r}(1+\sqrt{\sigma_r})^{p}\vert {\pmb x}(t_k) \vert \leq\nonumber \\ &~~~~~~~~~~~((1+\sqrt{\sigma_r})^{\tau_M+1}-1)\vert {\pmb x}(t_k)\vert \end{align} $
(11) 由于$t\in[r_j, r_{j+1})$, 数据包没有成功传输, 因此,
$ \begin{align} \vert {\pmb x}(t)-{\pmb x}(t_k)\vert \leq \sqrt{\sigma} \vert {\pmb x}(t_k)\vert \end{align} $
(12) 结合式(11)和(12)即可以得到定理1.
3.3 弹性事件触发机制和负载频率控制器协同设计
本节将在第3.1节和第3.2节稳定性分析的基础上, 对弹性事件触发矩阵和控制器进行联合设计, 从而达到理想的控制效果.
定理2.对于给定的参数$\gamma>0$, $\bar{d}>0$, 在弹性事件触发通信机制(4)情况下, 存在正定矩阵$R$, $\Omega$, 对称矩阵$Q$, $U$满足$U+\bar{d}Q>0$, 以及实矩阵$P$, $Y_1$和$Y_2$使得下列不等式成立, 此时系统(1)渐近稳定, $H_{\infty}$控制器增益$K=K_c(CP)^{+}$:
$ \begin{eqnarray} \left[\begin{array}{cc} \widetilde{\widetilde{\Pi}}_1&\Gamma_2^{\rm T}Y_2^{\rm T}\\ *&-\hat{R}\\ \end{array}\right]<0 \end{eqnarray} $
(13) $ \begin{eqnarray} \left[\begin{array}{cc} \widetilde{\widetilde{\Pi}}_1&\Gamma_1^{\rm T}Y_1\\ *&-\hat{R}\\ \end{array}\right]<0 \end{eqnarray} $
(14) 其中
$ \begin{align*} &\widetilde{\widetilde{\Pi}}_1:=\ell_1^{\rm T} Q\ell_1-\ell_2^{\rm T} Q\ell_2+\bar{d}^2e_8^{\rm T}Re_8-\gamma^2e_{10}^{\rm T}e_{10}-\\ &~~~~~~~(1+a)\Gamma_2^{\rm T}\hat{R}\Gamma_2+{\rm Sym}\{\widetilde{\ell}_3^{\rm T}U\ell_4+\bar{d}\ell_4^{\rm T}Q\widetilde{\ell}_5-\\ &~~~~~~~\Gamma_1^{\rm T}[aY_1+(1-a)Y_2]\Gamma_2-(e_1+e_8)^{\rm T}[Pe_8-\\ &~~~~~~~APe_1+BK_c(e_9+e_2)-Fe_{10}]\}-e_9^{\rm T}\Omega e_9+\\ &~~~~~~~\sigma_r (e_2+e_9)^{\rm T}\Omega (e_2+e_9)+e_1^{\rm T}(CP)^{\rm T}CPe_1-\\ &~~~~~~~(2-a)\Gamma_1^{\rm T}\hat{R}\Gamma_1\\ &\widetilde{\ell}_3={\rm col}\{e_8, e_1-e_3\}, \widetilde{\ell}_5={\rm col}\{e_8, 0\} \end{align*} $
证明.选择非奇异实矩阵$P$[18], 令${\pmb x}(t)=P{\pmb z}(t)$.则${\pmb u}(t)=-KC{\pmb x}(t)=-KCP{\pmb z}(t)=-K_c{\pmb z}(t)$, $K_c=KCP$, 当$t\in [t_k+d_{\tau_k}, t_{k+1}+d_{\tau_{k+1}})$系统(1)变形为
$ \begin{align} \dot{{\pmb z}}(t)=\, &P^{-1}AP{\pmb z}(t)-P^{-1}BK_c{\pmb z}(t_k)+\nonumber\\ &P^{-1}F{\pmb\Delta P_d}(t) \end{align} $
(15) 与第2节中相似, 对$d(t)$和${\pmb e_z}(t)$进行定义, 其中$d(t)$的定义与第3节中相同.
$ \begin{eqnarray*} {\pmb e_z}(t)= \begin{cases} 0, &t\in I_{1}\\ {\pmb z}(t_{k})-{\pmb z}(t_{k}+lh), &t\in I^l_{2}\\ {\pmb z}(t_{k})-{\pmb z}(t_{k}+m_kh), &t\in I_{3}\\ \end{cases} \end{eqnarray*} $
则系统(1)可以进一步变形为
$ \begin{align} &\dot{{\pmb z}}(t)=P^{-1}AP{\pmb z}(t)-P^{-1}BK_c({\pmb e_z}(t)+\nonumber\\ &~~~~~~~~~~~~~~~~~~~~~{\pmb z}(t-d(t)))+P^{-1}F{\pmb\Delta P_d}(t) \end{align} $
(16) 本节基于状态${\pmb z}$选择与之前相同的Lyapunov-Krasovskii泛函以及相同的证明方法, 不同的是在第2.1节中控制器增益$K$是预先给出的, 而定理2中可以同时求出控制器增益$K$以及事件触发矩阵.定义${\pmb\xi_2}:=\text{col}\{{\pmb z}(t)$, ${\pmb z}(t-d(t))$, ${\pmb z}(t-\bar{d})$, $\frac{1}{d(t)}\int_{t-d(t)}^{t}{\pmb z}(s){\rm d}s$, $\frac{1}{\bar{d}-d(t)}\int_{t-\bar{d}}^{t-d(t)}{\pmb z}(s){\rm d}s$, $\frac{1}{d^2(t)}\int_{t-d(t)}^{t}(t-s){\pmb z}(s){\rm d}s$, $\frac{1}{(\bar{d}-d(t))^2}\int_{t-\bar{d}}^{t-d(t)}(t-d(t)-s){\pmb z}(s){\rm d}s$, $\dot{{\pmb z}}(t)$, ${\pmb e_z}(t)$, ${\pmb\Delta P_d}(t)\}$.
从而, 可得到
$ \begin{align*} \dot{V}(t)\leq {\pmb\xi_2}^{\rm T}(\widetilde{\Pi}_1+\Pi_2){\pmb\xi_2} \end{align*} $
$ \begin{align*} &\widetilde{\Pi}_1:=\ell_1^{\rm T} Q\ell_1-\ell_2^{\rm T} Q\ell_2+\bar{d}^2e_8^{\rm T}Re_8-(2-a)\Gamma_1^{\rm T}\hat{R}\Gamma_1-\\ &~~~~~~~(1+a)\Gamma_2^{\rm T}\hat{R}\Gamma_2+{\rm Sym}\{\widetilde{\ell}_3^{\rm T}U\ell_4+\bar{d}\ell_4^{\rm T}Q\widetilde{\ell}_5-\\ &~~~~~~~\Gamma_1^{\rm T}[aY_1+(1-a)Y_2]\Gamma_2\}-e_9^{\rm T}\Omega e_9-\gamma^2 e_{10}^{\rm T}e_{10}+\\ &~~~~~~~\sigma_r (e_2+e_9)^{\rm T}\Omega (e_2+e_9)+e_1^{\rm T}(CP)^{\rm T}CPe_1 \end{align*} $
此外
$ \begin{align*} &-2[{\pmb z}(t)+\dot{{\pmb z}}(t)]^{\rm T}[P\dot{{\pmb z}}(t)-AP{\pmb z}(t)+BK_c\times\\ &~~~~~~~~~~~~({\pmb z}(t-d(t))+{\pmb e_z}(t))-F{\pmb\Delta P_d}(t)]=0 \end{align*} $
因此可以得到: $\dot{V}(t)\leq {\pmb\xi_2}^{\rm T}(\widetilde{\widetilde{\Pi}}_1+\Pi_2){\pmb\xi_2}$, 使用Schur补即可得到式(13)和(14).
4. 仿真案例
本节我们将第3节中的相关结论应用于三域LFC控制系统中, 验证所提出的弹性事件触发机制的有效性, 联合求出控制器增益$K$和事件触发矩阵.弹性事件触发通信机制与三域电力系统LFC的统一框架如图 1所示, 其中的相关参数[19]见表 2.
表 2 带EVs三域LFC模型参数($i=1, 2, 3$)Table 2 Parameters of three-area LFC model including EV aggregators ($i=1, 2, 3$)参数 取值 $M_i$ 10 $D_i$ 1.0 $T_{gi}$ 0.1 $T_{ci}$ 0.3 $T_{ri}$ 10 $F_{pi}$ 0.05 $R_i$ 0.05 $\beta_i$ 21 $K_{EVi}$ 1 $T_{EVi}$ 1 $T_{ij}$ 0.026 令$h=0.01$, 控制器增益$K_{Pi}=0.2$, $K_{Ii}=0.2$, $i=1, 2, 3$, 分配比例$\alpha_0=0.9$, $\alpha_1=0.1$.如表 3所示, 给定$\sigma$和$\sigma_r$, 基于定理1我们可以得到DoS攻击最大持续时间$\tau_{\rm dos}$的值.从中可以看出弹性事件触发机制可以容许DoS攻击所造成的数据包丢失, 当攻击持续时间小于$\tau_{\rm dos}$时, 系统保持稳定.此外, 当给定$\sigma$和$\tau_M$时, 也可以求出$\sigma_r$的值.从表 3可以看出, $\sigma_r$越大, $\tau_{\rm dos}$越小, 这是因为$\sigma_r$越大, 信道上传输的数据量越小, 从而在保证系统稳定的情况下, 所能允许的DoS攻击造成的数据丢失量越小, 因此系统所能承受的最大攻击持续时间越小.当$\sigma=\sigma_r=0.01$时, 系统触发次数为389次, 而根据文献[7$-$8]所提方法可以得到触发次数分别为398次和457次, 因此可以看出本文所提方法可以得到更少的触发次数, 有利于节约通信资源.
表 3 给定不同的$\sigma$和$\sigma_r$, 最大连续丢包量$\tau_M$和攻击持续时间$\tau_{{\rm dos}}$的值Table 3 $\tau_M$ and $\tau_{{\rm dos}}$ for different $\sigma$ and $\sigma_r$$\sigma$ 0.1 0.1 0.3 0.3 0.5 0.5 0.5 $\sigma_r$ 0.01 0.03 0.01 0.03 0.01 0.03 0.05 $\tau_M$ 2 1 4 2 5 3 2 $\tau_{{\rm dos}}$ 0.02 0.01 0.04 0.02 0.05 0.03 0.02 基于定理2, 给定$\sigma_r=0.01$时, 使用Matlab/LMI工具箱, 可以同时得到控制器增益和弹性事件触发矩阵如下:
$ \begin{align*} &K=\left[\begin{array}{cccccc} 2.38&-0.06&0&0&0&0\\ 0&0&2.38&-0.06&0&0\\ 0&0&0&0&2.38&-0.06\\ \end{array}\right]\\ &\Omega={\rm diag}\{\Omega_1, \Omega_2, \Omega_3\} \end{align*} $
此时, 系统的响应曲线如图 2所示, 由图中可以看出系统处于稳定状态.
$ \begin{align*} &\Omega_i=\left[\begin{array}{ccccccc} 3.64&-172.49&-11.01&85.51&20.83&-35.07&-2.18\\ -172.49&34\, 113.53&-477.46&-13\, 091.96&-15\, 595.51&-2\, 455.89&133.46\\ -11.01&-477.46&2\, 559.74&40\, 074.27&-171.10&339.19&-5.25\\ 85.51&-13\, 091.96&40\, 074.27&992\, 118.76&-1\, 436&-215.22&-11.12\\ 20.83&-15\, 595.51&-171.10&-1\, 436&8\, 599&2\, 037.21&-25.42\\ -35.07&-2\, 455.89&339.19&-215.22&2\, 037.21&1\, 125.41&-2.55\\ -2.18&133.46&-5.25&-11.12&-25.42&-2.55&7.96\\ \end{array}\right], \quad i=1, 2, 3 \end{align*} $
5. 结论
本文将电动汽车引入电力CPS中, 提出了一种弹性事件触发机制, 能够容忍DoS攻击所造成的数据包丢失, 并给出了系统所能承受的最大DoS攻击持续时间.构建新型Lyapunov-Krasovskii泛函, 对系统进行稳定性分析, 并联合求出弹性控制器增益和事件触发矩阵.所得到的时滞依赖稳定性条件中, 矩阵$U$, $Q$只需要对称即可, 放松了对其正定性的要求.由于所提出的弹性事件触发通信机制, 在保证电力CPS稳定的情况下, LFC控制器输入只在需要的时候进行更新, 并可以消除DoS攻击对系统稳定性所造成的不利影响.最后, 通过三域电力系统仿真, 验证了所提出方法的有效性.
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图 1 用户动态兴趣在潜在空间中的表示与推断
Fig. 1 The representation and inference of dynamic interests in latent representation spaces
图 4 PeerRec变体在HR@1指标上的对比
Fig. 4 The comparison between the variants of PeerRec in terms of HR@1
图 6 Batch大小设置为256时, 模型的迭代速率
Fig. 6 The running speed of different models with batch size 256
表 1 实验集数据统计表
Table 1 Statistics of dataset
ML-1m LastFM Toys 用户数量 6 040 1 090 19 412 行为类别数量 3 416 3 646 11 924 最长序列的行为数量 2 275 897 548 最短序列的行为数量 16 3 3 序列的平均行为数量 163.50 46.21 6.63 序列行为数量的方差 192.53 77.69 8.50 
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表 2 与基线模型在精度指标上的对比
Table 2 The comparison with baselines in terms of accuracy based metrics
数据集 模型 HR@1 HR@5 HR@10 NDCG@5 NDCG@10 MRR ML-1m POP 0.0407 0.1603 0.2775 0.1008 0.1383 0.1233 BERT4Rec[17] 0.3695 0.6851 0.7823 0.5375 0.5690 0.5108 S3-Rec[1] 0.2897 0.6575 0.7911 0.4557 0.5266 0.4535 HyperRec[9] 0.3180 0.6631 0.7738 0.5014 0.5375 0.4731 R-CE[11] 0.3988 0.6478 0.7404 0.5327 0.5627 0.5179 STOSA[14] 0.3222 0.6546 0.7844 0.4967 0.5389 0.4716 PeerRec (同伴1) 0.4250 0.7197 0.8141 0.5843 0.6150 0.5600 PeerRec (同伴2) 0.4252 0.7225 0.8141 0.5860 0.6157 0.5610 LastFM POP 0.0202 0.0908 0.1780 0.0544 0.0825 0.0771 BERT4Rec[17] 0.1091 0.3294 0.4614 0.2227 0.2648 0.2266 S3-Rec[1] 0.1156 0.2844 0.4229 0.2003 0.2452 0.2148 HyperRec[9] 0.1146 0.3147 0.4688 0.2150 0.2646 0.2241 R-CE[11] 0.0651 0.1835 0.2862 0.1243 0.1570 0.1397 STOSA[14] 0.0752 0.2165 0.3412 0.1458 0.1860 0.1556 PeerRec (同伴1) 0.1294 0.3495 0.4789 0.2339 0.2755 0.2341 PeerRec (同伴2) 0.1248 0.3358 0.4835 0.2318 0.2796 0.2378 Toys POP 0.0260 0.1046 0.1848 0.0652 0.0909 0.0861 BERT4Rec[17] 0.1390 0.3379 0.4596 0.2409 0.2802 0.2444 S3-Rec[1] 0.0990 0.3023 0.4393 0.2021 0.2463 0.2081 HyperRec[9] 0.1147 0.2875 0.3909 0.2031 0.2365 0.2087 R-CE[11] 0.1130 0.3189 0.4529 0.2179 0.2611 0.2233 STOSA[14] 0.1838 0.3587 0.4550 0.2749 0.3059 0.2732 PeerRec (同伴 1) 0.1794 0.3703 0.4785 0.2785 0.3134 0.2810 PeerRec (同伴 2) 0.1782 0.3706 0.4778 0.2781 0.3127 0.2803
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表 3 知识蒸馏与刻意训练对比
Table 3 The comparison between knowledge distillation and deliberate practice
数据集 HR@1 NDCG@5 MRR 知识蒸馏[39] ML-1m 0.3952 0.5656 0.5386 LastFM 0.1119 0.2301 0.2314 Toys 0.1693 0.2761 0.2767 刻意训练 PeerRec ML-1m 0.4251 0.5852 0.5605 LastFM 0.1271 0.2329 0.2360 Toys 0.1788 0.2783 0.2807
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表 4 PeerRec模型采用不同初始化的性能对比
Table 4 The performance comparison between different initializations of our PeerRec
数据集 初始化方式 HR@1 NDCG@5 MRR ML-1m TND 0.4251 0.5852 0.5605 Xavier 0.4263 0.5852 0.5600 Kaiming 0.4278 0.5911 0.5652 LastFM TND 0.1271 0.2329 0.2360 Xavier 0.1294 0.2397 0.2424 Kaiming 0.1247 0.2257 0.2342 Toys TND 0.1788 0.2783 0.2807 Xavier 0.1775 0.2794 0.2811 Kaiming 0.1806 0.2776 0.2804
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