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 自动化学报  2019, Vol. 45 Issue (1): 174-184 PDF

1. 重庆科技学院数理与大数据学院 重庆 401331 中国;
2. 重庆大学信息物理社会可信服务计算教育部重点实验室 重庆 400044 中国;
3. 重庆大学自动化学院 重庆 400044 中国;
4. 南洋理工大学电气工程与电子学院 新加坡 639798 新加坡

Distributed Secure State Estimation and Control for CPSs Under Sensor Attacks -A Finite Time Approach
AO Wei1, SONG Yong-Duan2,3, WEN Chang-Yun4
1. College of Mathematics and Science, the Chongqing University of Science and Technology, Chongqing 401331, China;
2. Key Laboratory of Dependable Service Computing in Cyber Physical Society, Ministry of Education, Chongqing 400044, China;
3. School of the Automation, Chongqing University, Chongqing 400044, China;
4. School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore
Manuscript received : May 31, 2018, accepted: August 14, 2018.
Foundation Item: Supported by National Natural Science Foundation of China (61773081, 61860206008), and the Technology Transformation Program of Chongqing Higher Education University (KJZH17102)
Corresponding author. SONG Yong-Duan   He is the standing director of Automation Association of China and the chair of the committee on Reliable Control Systems under Automation Association of China. He is also the founding chair of the Chongqing Chapter of IEEE Computational Intelligence Society (CIS). He is serving as an Associate Editor for 6 international scientific journals including IEEE Translation on Automatic Control. He received his Ph. D. degree in electrical and computer engineering from Tennessee Technological University, Cookeville, TN, USA in 1992. His research interest covers intelligent systems, guidance navigation and control, bioinspired adaptive control, system cooperation and reliability. Corresponding author of this paper.
Abstract: This paper investigates the problem of distributed secure state estimation and control for nonlinearly coupled interconnected cyber physical systems (CPS) under sensor attacks. Distributed schemes consisting of pre-selectors and observers are presented to solve the secure state estimation problem. Then, with the obtained state estimation and following the backstepping design procedure, distributed secure control algorithms are derived. Theoretical analysis shows that, with the proposed distributed secure observers and controllers, not only the state estimation of the CPS under attacks is obtained in a given finite time, but also the state tracking is ensured in a finite time. Finally, the developed algorithms are applied to an islanded micro-grid system as an illustration, verifying the effectiveness of the proposed method.
Key words: Cyber physical system (CPS) under attacks     distributed secure state estimation     distributed secure control     a finite time approach

 图 1 受到传感器攻击的信息物理系统与有限时间状态安全估计与控制框图 Fig. 1 The diagram of the CPS under sensor attacks and the finite time secure state estimation and control

1) 针对一类含有非线性耦合的信息物理系统, 提出一种由安全测量预选器和有限时间观测器组成的分布式有限时间状态安全估计策略.当满足一定条件时, 该策略可确保系统状态在预设有限时间之内被准确估计出来;

2) 利用获得的安全状态估计, 采用反步设计方法建立分布式有限时间安全控制律;

3) 基于李亚普洛夫稳定性理论, 严格论证该控制律可以保证系统在有限时间内跟踪期望信号, 且各系统状态, 观测器信号以及控制信号有界.该方法的有效性也通过仿真实验得到验证.

1 系统模型

 $$$\label{eqSubCPSModel} %{\Sigma _{a, i}}: \begin{cases} {{\dot x}_{i, 1}}(t) = {a_{i, 1}}{x_{i, 1}}(t) + {x_{i, 2}}(t)\\ \qquad\qquad\qquad \vdots \\ {{\dot x}_{i, n - 1}}(t) = {a_{i, n - 1}}{x_{i, 1}}(t) + {x_{i, n}}(t)\\ {{\dot x}_{i, n}}(t) = {a_{i, n}}{x_{i, 1}}(t) + {b_i}{u_i}(t) + {f_{i}}( \bar{\mathit{\boldsymbol{x}}}, t) \\ {{\mathit{\boldsymbol{y}}}_i}(t) = {C_i}{{\mathit{\boldsymbol{x}}}_i}(t) + {{\mathit{\boldsymbol{\eta}}} _i}(t) \end{cases}$$$ (1)

2 问题描述

1) 分布式有限时间安全状态估计问题:即针对如式(1)所示受攻击信息物理系统, 提出其分布式状态安全估计可解的充分条件, 并设计安全估计器, 在有限时间内获得系统的真实状态.

2) 分布式有限时间安全问题:即针对如式(1)所示受攻击信息物理系统, 在安全状态估计实现的基础上, 设计分布式安全控制器, 使系统能够在有限时间内跟踪任意给定信号.

3 分布式有限时间状态安全估计

3.1 安全测量预选器设计

 ${$$\label{eqCont} {s_i} \le \left\{ {\begin{array}{*{20}{c}} {\dfrac{1}{2}({p_i} - 1)}, &{{p_i}~\text{为奇数}}\\[3mm] {\dfrac{1}{2}{p_i} - 1}, &{{p_i}~\text{为偶数}} \end{array}} \right.$$}$ (2)

 $$$\label{eqMedOper} Med\left[ {{{\mathit{\boldsymbol{y}}}_i}(t)} \right] = \begin{cases} {{v_{i, in}}, }&{{p_i \text{为奇数}}}\\ { \dfrac{1}{2}({v_{i, in}} + {v_{i, in + 1}}), }&{{p_i \text{为偶数}}} \end{cases}$$$ (3)

 $$$\label{eqPreSel} {z_{p, i}}(t) = Med\left[ {{{y}_i}(t)} \right]$$$ (4)

 $$$\label{eqSecPreVal} {x_{i, 1}}(t) = {z_{p, i}}(t)$$$ (5)

3.2 分布式状态安全估计器设计

 $$$\label{eqDisSecObsv} \left\{ {\begin{array}{*{20}{l}} {{\dot {\hat {\bar {\mathit{\boldsymbol{\zeta}}}}} }_i}(t) = {{\bar {F}}_i}{{\hat {\bar {\mathit{\boldsymbol{\zeta}}}} }_i}(t) + {{\bar {H}}_i} {\bar {\mathit{\boldsymbol{f}}}_i(t)} + {{\bar {L}}_i}{z_{p, i}}(t) + \\ \qquad ~\quad{{\bar {H}}_i}{{B}_i}{u_i}(t)\\ {{{\hat {\mathit{\boldsymbol{x}}}}_i}(t) = {{\bar {M}}_i}[{{\hat {\bar {\mathit{\boldsymbol{\zeta}}}}}_i}(t) - {e^{{{\bar {F}}_i}{T_{e, i}}}}{{\hat {\bar {\mathit{\boldsymbol{\zeta}}}}}_i}(t - {T_{e, i}})]} \end{array}} \right.$$$ (6)

 $$$\label{eqTeCond} \det \left[ {\begin{array}{*{20}{c}} {{{\bar {H}}_i}}&{\bar {F}_i^{{T_{e, i}}}{{\bar {H}}_i}} \end{array}} \right] \ne 0$$$ (7)

 $$$\label{eqMatrMCond} {\bar {M}_i} = \left[ {\begin{array}{*{20}{c}} {{{I}_{n \times n}}}&{{{\mathit{\boldsymbol{0}}}_{n \times n}}} \end{array}} \right]{\left[ {\begin{array}{*{20}{c}} {{{\bar {H}}_i}}&{\bar {F}_i^{{T_{e}}}{{\bar {H}}_i}} \end{array}} \right]^{ - 1}}$$$ (8)

 $$$\label{eqAlphi1} {\alpha _{i, 1}}(t) = - {\beta _{i, 1}}\xi _{i, 1}^{{r_2}}(t) + {\dot x_{i, d}}(t)$$$ (17)

 $$$\label{eqDisSecCtrl} {u_i}(t) = \left\{ {\begin{array}{*{20}{l}} 0, &{t \le {T_e}}\\ {\dfrac{1}{{{b_i}}}{\alpha _{i, n}}(t) }, %= - \frac{1}{{{b_i}}}{\beta _{i, n}}\xi_{i, n}^{{r_{n + 1}}}(t) - \dfrac{1}{{{b_i}}}[{f_i}({{\bar x}_i}, t) +\sum\limits_{j = 1}^n {{\vartheta _{i, n, j}}{x_{i, j}}(t)} + x_{i, d}^{(n)}(t)] &t> T_e \end{array}} \right.$$$ (29)

4.2 稳定性分析

$t \ge {T_e}$时, 构造Lyapunov函数${V_{i, n}}(t) = {V_{i, n{\mathit{\boldsymbol{ - 1}}}}}(t){\mathit{\boldsymbol{ + }}}\int_{{\alpha _{i, n - 1}}}^{{e_{i, n}}} {{{[{s^{\frac{1}{{{r_n}}}}} - {{({\alpha _{i, n - 1}})}^{\frac{1}{{{r_n}}}}}]}^{2 - {r_n} + \tau }}{{\rm d}s}}$, 对其求导, 并考虑系统模型(1)和控制律(29), 可得:

 \label{eqDerVin} \begin{aligned} {{\dot V}_{i, n}}(t) &= {{\dot V}_{i, n - 1}}(t) + {[{({e_{i, n}})^{\frac{1}{{{r_n}}}}} - {({\alpha _{i, n - 1}})^{\frac{1}{{{r_n}}}}}]^{2 - {r_n} + \tau }} \cdot \\ &({b_i}{u_i} + \sum\limits_{j = 1}^n {{\vartheta _{i, n, j}}{x_{i, j}}} + {f_{i}} -x_{i, d}^{(n)}) \le\\ & - {\beta _{i, n}}\xi _{i, n}^2 + \xi _{i, n -1}^{2 - {r_n}}({e_{i, n}} - {\alpha _{i, n - 1}}) - \\ & 2(\xi _{i, 1}^2 + \cdots + \xi _{i, n - 1}^2) \end{aligned} (31)

 \label{eqErinCond} \begin{aligned} \xi _{i, n - 1}^{2 - {r_n}}({e_{i, n}} - {\alpha _{i, n - 1}}) &\le {2^{1 - {r_{n - 1}} - \tau }}{\left| {{\xi _{i, n - 1}}} \right|^{2 - {r_{n - 1}} - \tau }} \cdot \\ & {\left| {{\xi _{i, n}}} \right|^{{r_n}}} \le \frac{1}{2}\xi _{i, n - 1}^2 + {c_{i, n}}\xi _{i, n}^2 \end{aligned} (32)

 \label{eqDerVinb} \begin{aligned} {\dot V_{i, n}}(t) &\le - 2(\xi _{i, 1}^2 + \cdots + \xi _{i, n - 1}^2) + \frac{1}{2}\xi _{i, n - 1}^2 + \\ &{c_{i, n}}\xi _{i, n}^2- {\beta _{i, n}}\xi _{i, n}^2 \le - (\xi _{i, 1}^2 + \cdots + \xi _{i, n}^2) \end{aligned} (33)

 \label{eqErijCond} \begin{aligned} \left| {{e_{i, j}} - {\alpha _{i, j - 1}}} \right| %&\le {2^{2 - {r_j}}}{\left|{{{({e_{i, j}})}^{\frac{1}{{{r_j}}}}} - {{({\alpha _{i, j -1}})}^{\frac{1}{{{r_j}}}}}} \right|^{{r_j}}} \\ & \le {2^{2 - {r_j}}}{\left| {{\xi _{i, j}}} \right|^{{r_j}}} \end{aligned} (34)

 \begin{align}\label{eqIntErijCond} &\int_{{\alpha _{i, j - 1}}}^{{e_{i, j}}} {{{[{s^{\frac{1}{{{r_j}}}}} - {{({\alpha _{i, j - 1}})}^{\frac{1}{{{r_j}}}}}]}^{2 - {r_j} + \tau }}{{\rm d}s}} \le\nonumber\\ &\qquad \left| {{e_{i, j}} - {\alpha _{i, j - 1}}} \right|{\left| {{{({e_{i, j}})}^{\frac{1}{{{r_j}}}}} - {{({\alpha _{i, j - 1}})}^{\frac{1}{{{r_j}}}}}} \right|^{2 - {r_j} + \tau }}\le\nonumber\\ &\qquad {2^{2 - {r_j}}}{\left| {{\xi _{i, j}}} \right|^{{r_j}}}{\left| {{\xi _{i, j}}} \right|^{2 - {r_j} + \tau }} = {2^{2 - {r_j}}}{\left| {{\xi _{i, j}}} \right|^{2 + \tau }} \end{align} (35)

 $$$\label{eqVinCond} {V_{i, n}}(t) \le \frac{1}{{{\gamma _i}}}(\xi _{i, 1}^{2 + \tau } + \cdots + \xi _{i, n}^{2 + \tau })$$$ (36)

 $$$\label{eqDerVn} {\dot V_n}(t) \le - \sum\limits_{i = 1}^N {(\xi _{i, 1}^2 + \cdots + \xi _{i, n}^2)}$$$ (37)

 \label{eqDerVnb} \begin{aligned} {\dot V_n}(t) + \lambda {[{V_n}(t)]^{\frac{2}{{2 + \tau }}}} &\le - \sum\limits_{i = 1}^N {(\xi _{i, 1}^2 + \cdots + \xi _{i, n}^2)} +\\ & \lambda\frac{1}{{{\gamma ^{\frac{2}{{2 + \tau }}}}}}\sum\limits_{i = 1}^N {(\xi _{i, 1}^2 + \cdots + \xi _{i, n}^2)} \le\\ & - \frac{1}{2}\sum\limits_{i = 1}^N {(\xi _{i, 1}^2 + \cdots + \xi _{i, n}^2)} \end{aligned} (38)

 \begin{align}\label{eqDerThet} {\dot \Theta _i}({{{\pmb e}}_i}, t) =&{e_{i, 1}}{e_{i, 2}} + \cdots +{e_{i, n}} \cdot[{b_i}{u_i} +\nonumber\\ &\sum\limits_{j = 1}^n {{\vartheta _{i, n, j}}{x_{i, j}}} + {f_{i}}- x_{i, d}^{(n)}] \end{align} (39)

$\varepsilon_i (t)= \sum\limits_{j = 1}^n {{\vartheta _{i, n, j}}{x_{i, j}}}(t) + {f_{i}}(\bar x, t)+{b_i}{u_i}(t) -x_{i, d}^{(n)}(t)$, 根据控制律定义式(29)以及式(28), 可得:

 \label{eqAlphinCond} \begin{aligned} %{b_i}{u_i}(t) + \sum\limits_{j = 1}^n {{\vartheta _{i, n, j}}{x_{i, j}}(t)} +{f_i}({{\bar x}_i}, t){\mathit{\boldsymbol{ - }}}x_{i, d}^{(n)}(t) \varepsilon_i (t)%&\le {\beta _{i, n}}{\left| {{\xi_{i, n}}(t)} \right|^{{r_{n + 1}}}} \\ &\le {\beta _{i, n}}{\left| {{e_{i, n}}} \right|^{\frac{{{r_{n + 1}}}}{{{r_n}}}}} + {\beta _{i, n}}{\left| {{\alpha _{i, n - 1}}} \right|^{\frac{{{r_{n + 1}}}}{{{r_n}}}}} \le {\beta _{i, n}} \cdot \\ & {\left| {{e_{i, n}}} \right|^{\frac{{{r_{n + 1}}}}{{{r_n}}}}} +{\beta _{i, n}}\beta _{i, n{{ - 1}}}^{\frac{{{r_{n + 1}}}}{{{r_n}}}}{\left| {{e_{i, n{{ - 1}}}}} \right|^{\frac{{{r_{n + 1}}}}{{{r_{n{{ - 1}}}}}}}}+\cdots {{ + }}\\ &{\beta _{i, n}}\beta _{i, n{{ - 1}}}^{\frac{{{r_{n + 1}}}}{{{r_n}}}} \cdots \beta _{i, {{1}}}^{\frac{{{r_{n + 1}}}}{{{r_{{2}}}}}}{\left| {{e_{i, {{1}}}}} \right|^{\frac{{{r_{n + 1}}}}{{{r_{{1}}}}}}} \end{aligned} (40)

 $$$\label{eqSumThetCond} \Theta ({\pmb e}, t) \le \left[\Theta ({\pmb e}(0) + \frac{L}{K}\right]{{\rm e}^{Kt}} - \frac{L}{K}$$$ (47)

5 仿真示例

 $$$\label{eqMicGridDyn} \left\{ {\begin{array}{*{20}{l}} {{{\dot \delta }_i}(t) = {\omega _i}(t)}\\ {{\tau _{Pi}}{{\dot \omega }_i}(t) + {\omega _i}(t) + {k_{Pi}}({P_i}(t) - P_i^d(t)) + {u_i}(t) = 0} \end{array}} \right.$$$ (48)
 图 2 受到传感器攻击的微电网系统框图 Fig. 2 The diagram of the micro-grid system under sensor attacks

 \begin{align}\label{eqRealPow} {P_i}(t) = &{P_{1i}}V_i^2(t) + {P_{2i}}{V_i}(t) + {P_{3i}} + \nonumber\\ &\sum\limits_{j=1}^N {{V_i}(t){V_j}(t)\left| {{B_{ij}}} \right|\sin ({\delta _i} - {\delta _j})} \end{align} (49)

 $$$\label{eqMicGridDyn2} \left\{ \begin{array}{l} {{\dot {\mathit{\boldsymbol{x}}}}_i}(t) = {{A}_i}{{\mathit{\boldsymbol{x}}}_i}(t) + {{B}_i}{u_i}(t) + {{B}_i} {f_{i}}(\bar{\pmb x}, t)\\ {{\mathit{\boldsymbol{y}}}_i}(t) = {{C}_i}{{\mathit{\boldsymbol{x}}}_i}(t) + {{\mathit{\boldsymbol{\eta}} }_i}(t) \end{array} \right.$$$ (50)

 图 3 发电机1受控状态及其估计值 Fig. 3 The angle of the 1st generator and its estimation
 图 4 发电机2受控状态及其估计值 Fig. 4 The angle of the 2nd generator and its estimation
 图 5 发电机3受控状态及其估计值 Fig. 5 The angle of the 3rd generator and its estimation
 图 6 发电机4受控状态及其估计值 Fig. 6 The angle of the 4th generator and its estimation
 图 7 设计的安全控制输入 Fig. 7 The designed secure control law
6 结束语

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