软件学报  2020, Vol. 31 Issue (6): 1672-1680 PDF

1. 北京科技大学 自动化学院, 北京 100083;
2. 复杂系统管理与控制国家重点实验室(中国科学院 自动化研究所), 北京 100190

Event-triggering Secure Control of Markov Jump Cyber-Pysical Systems Under Mode- dependent Denial of Service Attacks
MA Chao1,2 , WU Wei2
1. School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China;
2. State Key Laboratory for Management and Control of Complex Systems (Institute of Automation, Chinese Academy of Sciences), Beijing 100190, China
Abstract: This study investigates the secure control problem of Markov jump cyber-physical systems (CPS) under mode-dependent denial of service (DoS) attacks. A novel mode-dependent event-triggering strategy is adopted to reduce the network resource consumptions. In particular, the DoS attacks are supposed to be mode-dependent for more practical applications. The Lyapunov-Krasovskii functional method is utilized to establish the sufficient conditions such that the resulting closed-loop system can be uniformly ultimately bounded under DoS attacks. Furthermore, the desired secure controller can be designed in terms of matrix techniques. Finally, an illustrative example is presented to demonstrate the effectiveness of the theoretical method.
Key words: secure control    event-triggering control    Markov jump cyber-physical system    denial of service attack

(1) 考虑到马尔可夫跳变信息物理系统的跳变特性, 建立了一种新的模态依赖安全控制模型, 从而更好地模拟实际网络攻击的模式;

(2) 提出了一种新颖的模态依赖事件触发控制策略, 用来解决网络攻击下的安全控制问题;

(3) 利用凸优化的方法建立了实现安全控制所需要的充分性条件, 并且给出了相应的事件触发函数与安全控制器的设计过程.

1 预备知识与问题描述 1.1 马尔可夫跳变信息物理系统数学模型

 $\dot x(t) = A(\sigma (t))x(t) + B(\sigma (t))u(t)$ (1)

σ(t)表示连续时间离散状态的马尔可夫过程, 其取值在一个有限的集合I={1, 2, …, N}内.相应的, 其状态转移概率矩阵Π={πij}, ∀i, jI被描述为

 $\Pr (\sigma (t + \Delta t) = j:\sigma (t) = i) = \left\{ {\begin{array}{*{20}{l}} {{\pi _{ij}}\Delta t + o(\Delta t), {\rm{if }} \ \ \ i \ne j} \\ {1 + {\pi _{ii}}\Delta t + o(\Delta t), {\rm{ if }} \ \ \ i = j} \end{array}} \right.$ (2)

1.2 模态依赖事件触发的安全控制器设计

 $\varepsilon \left( {{i_k}h} \right) = \left\{ {\begin{array}{*{20}{l}} {0, {\rm{拒绝服务攻击未发生}}}\\ {1, {\rm{拒绝服务攻击发生}}} \end{array}} \right.,$

 $\Delta _{{t_{k + 1}}h}^{DoS} = t_{{t_{k + 1}}h}^{DoS} - {t_{k + 1}}h, t_{{t_{k + 1}}h}^{DoS} \geqslant {t_{k + 1}}h,$

 ${t_{k + 1}}h = {t_k}h + \min ({i_k}h|\delta {x^T}({t_k}){\mathit{\Phi }_1}(\sigma (t))x({t_k}) - {e^T}({i_k}h){\mathit{\Phi }_2}(\sigma (t))e({i_k}h) + \varepsilon ({i_k}h)\mathit{\Psi }(\Delta _{{t_{k + 1}}h}^{DoS}))$ (3)

●    δ为模态依赖的阈值参数;

●    Φ1(σ(t))与Φ2(σ(t))为模态依赖的常量矩阵;

●    $\mathit{\Psi }(\Delta _{{t_{k + 1}}h}^{DoS}) = {(x({i_k}h) - x({t_{k + 1}}h))^T}\mathit{\Phi }(x({i_k}h) - x({t_{k + 1}}h))$, Φ > 0表示由拒绝服务攻击引起的附加误差函数.

 $u(t) = K(\sigma (t))x\left( {{t_k}h} \right), {t_k}h \le t < {t_{k + 1}}h$ (4)

 $\dot x(t) = A(\sigma (t))x(t) + B(\sigma (t))K(\sigma (t))x({t_k}h), {t_k}h \leqslant t < {t_{k + 1}}h$ (5)

 $\dot x(t) = {A_i}x(t) + {B_i}{K_i}x({t_k}h), {t_k}h \leqslant t < {t_{k + 1}}h$ (6)

1.3 控制目标

2 控制算法设计

 ${J_{DoS}} = \frac{{{\mathit{\varepsilon}} ({i_k}h){\mathit{\Psi}} ({\mathit{\Delta}} _{{t_{k + 1}}h}^{DoS})}}{{\kappa {{\mathit{\lambda}} _{\min }}({P_i})}},$

 $\begin{gathered} {{\mathit{\Xi}} _i} = \left[ {\begin{array}{*{20}{c}} {{{\mathit{\Xi}} _{1i}}}&{{{\mathit{\Xi}} _{2i}}} \\ *&{{{\mathit{\Xi}} _{3i}}} \end{array}} \right], \\ {{\mathit{\Xi}} _{1i}} = 2{P_i}{A_i} + 2{P_i}{B_i}{K_i} + \delta {{\mathit{\Phi}} _{1i}} + \sum\limits_{i = 1}^N {{\pi _{ij}}{P_j}} , \\ {{\mathit{\Xi}} _{2i}} = [\begin{array}{*{20}{c}} { - {P_i}{B_i}{K_i} - \delta {{\mathit{\Phi}} _{1i}}}&{ - {P_i}{B_i}{K_i} - \delta {{\mathit{\Phi}} _{1i}}}&{hA_i^TR + hK_i^TB_i^TR} \end{array}], \\ {{\mathit{\Xi}} _{3i}} = \left[ {\begin{array}{*{20}{c}} { - R + \delta {{\mathit{\Phi}} _{1i}}}&{\delta {{\mathit{\Phi}} _{1i}}}&{ - hK_i^TB_i^TR} \\ *&{ - {{\mathit{\Phi}} _{2i}} + \delta {{\mathit{\Phi}} _{1i}}}&{ - hK_i^TB_i^TR} \\ *&*&{ - R} \end{array}} \right]. \\ \end{gathered}$

 $\dot x(t) = {A_i}x(t) + {B_i}{K_i}(x(t - d(t)) - e({i_k}h)), {t_k}h \leqslant t < {t_{k + 1}}h$ (7)

 $V(t) = \sum\limits_{k = 1}^2 {{V_k}(t)} ,$

 ${V_1}(t) = {x^T}(t){P_i}x(t), \\ {V_2}(t) = h\int_{ - h}^0 {\int_{t + \theta }^t {{{\dot x}^T}(s)R\dot x(s){\rm{d}}s{\rm{d}}\theta } } .$

 $\mathcal{L}V(t) = \mathop {\lim }\limits_{{\mathit{\Delta}} t \to \infty } \frac{1}{{{\mathit{\Delta}} t}}{\mathit{\mathbb{E}}}\{ V(t + {\mathit{\Delta}} t, i)|t - V(t)\} .$

 $\begin{gathered} \mathcal{L}{V_1}(t) = {{\dot x}^T}(t){P_i}x(t) + {x^T}(t){P_i}\dot x(t) + \sum\limits_{i = 1}^N {{\pi _{ij}}} {x^T}(t){P_j}x(t) \\ {\rm{ }} = 2{x^T}(t){P_i}\dot x(t) + \sum\limits_{i = 1}^N {{\pi _{ij}}} {x^T}(t){P_j}x(t) \\ {\rm{ }} = 2{x^T}(t){P_i}{A_i}x(t) + 2{x^T}(t){P_i}{B_i}{K_i}x(t) - 2{x^T}(t){P_i}{B_i}{K_i}\int_{t - d(t)}^t {{{\dot x}^T}(s){\rm{d}}s} \\ {\rm{ }} - 2{x^T}(t){P_i}{B_i}{K_i}e({i_k}h)) + \sum\limits_{i = 1}^N {{\pi _{ij}}} {x^T}(t){P_j}x(t), \\ \mathcal{L}{V_2}(t) = {h^2}{{\dot x}^T}(t)R\dot x(t) - h\int_{t - h}^t {{{\dot x}^T}(s)R\dot x(s){\rm{d}}s} \\ {\rm{ }} \leqslant {h^2}{{\dot x}^T}(t)R\dot x(t) - h\int_{t - h}^t {{{\dot x}^T}(s){\rm{d}}sR\int_{t - h}^t {\dot x(s){\rm{d}}s} } \\ {\rm{ }} \leqslant {h^2}{{\dot x}^T}(t)R\dot x(t) - h\int_{t - d(t)}^t {{{\dot x}^T}(s){\rm{d}}sR\int_{t - d(t)}^t {\dot x(s){\rm{d}}s} } . \\ \end{gathered}$

 ${e^T}({i_k}h){\mathit{\Phi }_{2i}}(\sigma (t))e({i_k}h) \le \delta (\sigma (t)){x^T}({t_k}){\mathit{\Phi }_{1i}}(\sigma (t))x({t_k}) + \varepsilon ({i_k}h)\mathit{\Psi }(\Delta _{{t_{k + 1}}h}^{DoS}).$

 ${\cal L}V(t) \le {\eta ^T}(t)\mathit{\Xi }\eta (t) + \varepsilon ({i_k}h)\mathit{\Psi }(\Delta _{{t_{k + 1}}h}^{DoS}),\$

 $\mathcal{L}V(t) \leqslant - \kappa V(t) + \varepsilon ({i_k}h)\mathit{\Phi } (\Delta _{{t_{k + 1}}h}^{DoS}).$

 ${x^T}(t){P_i}x(t) \leqslant V(t) \leqslant V(0) + \frac{{\varepsilon ({i_k}h)\mathit{\Phi } (\Delta _{{t_{k + 1}}h}^{DoS})}}{\kappa }$ .

 $||x(t)|| \leqslant \sqrt {\frac{{V(0) + \frac{{\varepsilon ({i_k}h)\mathit{\Phi } (\Delta _{{t_{k + 1}}h}^{DoS})}}{\kappa }}}{{{\lambda _{\min }}({P_i})}}}$ .

 ${J_{DoS}} = \frac{{\varepsilon ({i_k}h)\mathit{\Phi } (\Delta _{{t_{k + 1}}h}^{DoS})}}{{\kappa {\lambda _{\min }}({P_i})}}$ .

 ${J_{DoS}} = \frac{{\varepsilon ({i_k}h)\mathit{\Phi } (\Delta _{{t_{k + 1}}h}^{DoS})}}{{\kappa {\lambda _{\min }}({P_i})}},$

 $\begin{gathered} {{\bar {\mathit{\Xi}} }_i} = \left[ {\begin{array}{*{20}{c}} {{{\bar {\mathit{\Xi}} }_{1i}}}&{{{\bar {\mathit{\Xi}} }_{2i}}} \\ *&{{{\bar {\mathit{\Xi}} }_{3i}}} \end{array}} \right], \\ {{\bar {\mathit{\Xi}} }_{1i}} = \left[ {\begin{array}{*{20}{c}} {2{A_i}{{\bar P}_i} + 2{B_i}{S_i} + \delta {{\bar {\mathit{\Phi}} }_{1i}} + {\pi _{ii}}{{\bar P}_i}}&{ - {B_i}{S_i} - \delta {{\bar {\mathit{\Phi}} }_{1i}}}&{ - {B_i}{S_i} - \delta {{\bar {\mathit{\Phi}} }_{1i}}} \\ *&{ - \bar R + \delta {{\bar {\mathit{\Phi}} }_{1i}}}&{\delta {{\bar {\mathit{\Phi}} }_{1i}}} \\ *&*&{ - {{\bar {\mathit{\Phi}} }_{2i}} + \delta {{\bar {\mathit{\Phi}} }_{1i}}} \end{array}} \right], \\ {{\bar {\mathit{\Xi}} }_{2i}} = \left[ {\begin{array}{*{20}{c}} {h{{\bar P}_i}A_i^T + hS_i^TB_i^T}&{\sqrt {{\pi _{i1}}} {{\bar P}_i}}& \ldots &{\sqrt {{\pi _{iN}}} {{\bar P}_i}} \\ { - hS_i^TB_i^T}&0& \ldots &0 \\ { - hS_i^TB_i^T}& \vdots & \ddots & \vdots \end{array}} \right], \\ {{\bar {\mathit{\Xi}} }_{3i}} = \left[ {\begin{array}{*{20}{c}} {\bar R - 2{{\bar P}_i}}&0& \ddots &0 \\ *&{ - {{\bar P}_1}}& \ddots &0 \\ *&*& \ddots &0 \\ *&*&*&{ - {{\bar P}_N}} \end{array}} \right]. \\ \end{gathered}$

 ${K_i} = {S_i}\bar P_i^{ - 1}$ .

3 仿真验证

 $\frac{{d{I_L}(t)}}{{dt}} = \frac{{u - {u_c}(t) - {I_L}(t)}}{{{L_i}}}, \frac{{d{u_c}(t)}}{{dt}} = \frac{{{I_L}(t)}}{{{C_i}}}.$

x(t)=[uc(t), IL(t)]T, 则有:

 ${A_i} = \left[ {\begin{array}{*{20}{c}} 0&{\frac{1}{{{C_i}}}} \\ { - \frac{1}{{{L_i}}}}&{ - \frac{R}{{{L_i}}}} \end{array}} \right], {B_i} = \left[ {\begin{array}{*{20}{c}} 0 \\ {\frac{1}{{{L_i}}}} \end{array}} \right].$
 Fig. 1 RLC circuit 图 1 RLC电路

 ${A_1} = \left[ {\begin{array}{*{20}{c}} 0&{\frac{1}{{0.5}}} \\ { - \frac{1}{4}}&{ - \frac{{0.01}}{4}} \end{array}} \right], {A_2} = \left[ {\begin{array}{*{20}{c}} 0&{\frac{1}{{0.8}}} \\ { - \frac{1}{8}}&{ - \frac{{0.01}}{8}} \end{array}} \right], {B_1} = \left[ {\begin{array}{*{20}{c}} 0 \\ {\frac{1}{4}} \end{array}} \right], {B_2} = \left[ {\begin{array}{*{20}{c}} 0 \\ {\frac{1}{8}} \end{array}} \right], {\mathit{\Pi }} = \left[ {\begin{array}{*{20}{c}} { - 0.5}&{0.5} \\ {0.3}&{ - 0.3} \end{array}} \right].$

 ${\kappa _1} = [ - 0.8125 - 0.9496], {\kappa _2} = [ - 1.4783 - 1.2428]$

 Fig. 2 Release intervals of the event-triggered control 图 2 事件触发释放间隔

 Fig. 3 State response of the closed-loop Markov jump cyber-pysical system 图 3 马尔可夫跳变信息物理系统闭环状态轨迹

 Fig. 4 Performance response of the closed-loop Markov jump cyber-pysical system 图 4 马尔可夫跳变信息物理系统性能轨迹

4 结论

 [1] Pasqualetti F, Dörfler F, Bullo F. Attack detection and identification in cyber-physical systems. IEEE Trans. on Automatic Control, 2013, 58(11): 2715-2729. [2] Humayed A, Lin J, Li F, et al. Cyber-Physical systems security—A survey. IEEE Internet of Things Journal, 2017, 4(6): 1802-1831. [doi:10.1109/JIOT.2017.2703172] [3] Chen TM, Sanchez-Aarnoutse JC, Buford J. Petri net modeling of cyber-physical attacks on smart grid. IEEE Trans. on Smart Grid, 2011, 2(4): 741-749. [4] Jia D, Lu K, Wang J, et al. A survey on platoon-based vehicular cyber-physical systems. IEEE Communications Surveys & Tutorials, 2016, 18(1): 263-284. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=48d725371af5bcb6f98c760045f5e409 [5] Wang L, Törngren M, Onori M. Current status and advancement of cyber-physical systems in manufacturing. Journal of Manufacturing Systems, 2015, 37: 517-527. [doi:10.1016/j.jmsy.2015.04.008] [6] Fawzi H, Tabuada P, Diggavi S. Secure estimation and control for cyber-physical systems under adversarial attacks. IEEE Trans. on Automatic Control, 2014, 59(6): 1454-1467. [7] Li Y, Shi L, Cheng P, et al. Jamming attacks on remote state estimation in cyber-physical systems: A game-theoretic approach. IEEE Trans. on Automatic Control, 2015, 60(10): 2831-2836. [8] Cao R, Cheng L. Secure control of Euler-Lagrange systems under denial-of-service attacks. Aerospace Control and Application, 2018, 44(5): 76-80(in Chinese with English abstract). [doi:10.3969/j.issn.1674-1579.2018.05.011] [9] Chen B, Ho DW C, Zhang WA, et al. Distributed dimensionality reduction fusion estimation for cyber-physical systems under DoS attacks. IEEE Trans. on Systems, Man, and Cybernetics: Systems, 2019, 49(2): 455-468. [10] Chen B, Ho DWC, Hu G, et al. Secure fusion estimation for bandwidth constrained cyber-physical systems under replay attacks. IEEE Trans. on Cybernetics, 2018, 48(6): 1862-1876. [11] Miao F, Zhu Q, Pajic M, et al. Coding schemes for securing cyber-physical systems against stealthy data injection attacks. IEEE Trans. on Control of Network Systems, 2017, 4(1): 106-117. [12] Han S, Xie M, Chen HH, et al. Intrusion detection in cyber-physical systems: Techniques and challenges. IEEE Systems Journal, 2014, 8(4): 1049-1059. https://ieeexplore.ieee.org/document/6942184 [13] Ding D, Han QL, Xiang Y, et al. A survey on security control and attack detection for industrial cyber-physical systems. Neurocomputing, 2018, 275: 1674-1683. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=b5168ebdddb31c76074a5f35e06f0626 [14] Wells LJ, Camelio JA, Williams CB, et al. Cyber-physical security challenges in manufacturing systems. Manufacturing Letters, 2014, 2(2): 74-77. https://www.sciencedirect.com/science/article/pii/S0007850616301974 [15] Lu AY, Yang GH. Event-Triggered secure observer-based control for cyber-physical systems under adversarial attacks. Information Sciences, 2017, 420: 96-109. [doi:10.1016/j.ins.2017.08.057] [16] Wang D, Wang Z, Shen B, et al. Recent advances on filtering and control for cyber-physical systems under security and resource constraints. Journal of the Franklin Institute, 2016, 353(11): 2451-2466. [doi:10.1016/j.jfranklin.2016.04.011] [17] Shoukry Y, Tabuada P. Event-Triggered state observers for sparse sensor noise/attacks. IEEE Trans. on Automatic Control, 2016, 61(8): 2079-2091. [18] Shi P, Li F. A survey on Markovian jump systems: Modeling and design. Int'l Journal of Control, Automation and Systems, 2015, 13(1): 1-16. http://link.springer.com/article/10.1007/s12555-014-0576-4 [19] De Farias DP, Geromel JC, Do Val JBR, et al. Output feedback control of Markov jump linear systems in continuous-time. IEEE Trans. on Automatic Control, 2000, 45(5): 944-949. [20] Shen H, Zhu Y, Zhang L, et al. Extended dissipative state estimation for Markov jump neural networks with unreliable links. IEEE Trans. on Neural Networks and Learning Systems, 2017, 28(2): 346-358. [21] Yang X, Cao J, Lu J. Synchronization of randomly coupled neural networks with Markovian jumping and time-delay. IEEE Trans. on Circuits and Systems Ⅰ: Regular Papers, 2013, 60(2): 363-376. [22] Chen B, Niu Y, Zou Y. Security control for Markov jump system with adversarial attacks and unknown transition rates via adaptive sliding mode technique. Journal of the Franklin Institute, 2019, 356(6): 3333-3352. [doi:10.1016/j.jfranklin.2019.01.045] [23] Yang, K, Wang, R, Jiang, Y, et al. Enhanced resilient sensor attack detection using fusion interval and measurement history. In: Proc. of the Int'l Conf. on Hardware/Software Codesign and System Synthesis. IEEE, 2018. 1-3. [24] Yang K, Wang R, Jiang Y, et al. Sensor attack detection using history based pairwise inconsistency. Future Generation Computer Systems, 2018, 86: 392-402. [doi:10.1016/j.future.2018.03.050] [25] Fridman E, Seuret A, Richard JP. Robust sampled-data stabilization of linear systems: an input delay approach. Automatica, 2004, 40(8): 1441-1446. [doi:10.1016/j.automatica.2004.03.003] [8] 曹然, 程龙. 拒绝服务攻击下的Euler-Lagrange系统的安全控制. 空间控制技术与应用, 2018, 44(5): 76-80. [doi:10.3969/j.issn.1674-1579.2018.05.011]