New developments in control design techniques of logical control networks

KONG Xiang-shan; WANG Shu-ling; LI Hai-tao; ALSAADI Fuad E.

doi:10.1631/FITEE.1900397

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New developments in control design techniques of logical control networks

Special Issue on Complex Systems and Networks with Their Applications | Updated：2022-05-19

- New developments in control design techniques of logical control networks
- 逻辑控制网络控制设计方法新进展
- Frontiers of Information Technology & Electronic Engineering Vol. 21, Issue 2, Pages: 220-233(2020)
- Affiliations：
  
  School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
  Department of Electrical and Computer Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia
- Author bio：
  
  [ "Xiang-shan KONG, E-mail: kong_xiangshan@126.com" ]
  [ "Shu-ling WANG, E-mail: shulingwang2019@163.com" ]
  [ "Hai-tao LI, E-mail: haitaoli09@gmail.com" ]
  [ "Fuad E. ALSAADI, E-mail: fuad_alsaadi@yahoo.com" ]
- Funds：
- DOI：10.1631/FITEE.1900397
  CLC： O157.5
- Received：08 August 2019，
  
  Revised：2020-;1-;6，
  
  Published：2020-02
- Accepted：
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KONG Xiang-shan, WANG Shu-ling, LI Hai-tao, et al. New developments in control design techniques of logical control networks[J]. Frontiers of Information Technology & Electronic Engineering, 2020, 21(2): 220-233.
DOI：

KONG Xiang-shan, WANG Shu-ling, LI Hai-tao, et al. New developments in control design techniques of logical control networks[J]. Frontiers of Information Technology & Electronic Engineering, 2020, 21(2): 220-233. DOI： 10.1631/FITEE.1900397.

摘要

控制设计是逻辑控制网络研究的一个基本问题。本文对逻辑控制网络控制设计方法的新进展进行综述。首先回顾矩阵半张量积和逻辑控制网络的一些基本结果。接着介绍逻辑控制网络研究中的控制设计方法新进展，包括能达集法、牵制控制法、控制李雅普诺夫函数法、事件触发法和采样控制法。最后通过一个示例说明这些方法的有效性。

Abstract

The control design problem plays a fundamental role in the study of logical control networks (LCNs). This paper presents a detailed survey on new developments in control design techniques of LCNs. First

some preliminary results on the semi-tensor product method and LCNs are reviewed. Then

we move on to some new developments for control design techniques of LCNs

including the reachable set approach

the pinning control technique

the control Lyapunov function approach

the event-triggered control technique

and the sampled-data control technique. Finally

an illustrative example is given to demonstrate the effectiveness of these techniques.

关键词

Keywords

references

KJ Åström , , , B Bernhardsson . . Comparison of periodic and event based sampling for first order stochastic systems . . Proc 14 th IFAC World Congress , , 1999 . . p.5006 - - 5011 . . . .

HW Chen , , , JL Liang . . Output regulation of Boolean control networks with stochastic disturbances . . IET Contr Theory Appl , , 2017 . . 11 ( ( 13 ): ): 2097 - - 2103 . . DOI: 10.1049/iet-cta.2016.1675 http://doi.org/10.1049/iet-cta.2016.1675 . .

HW Chen , , , JL Liang , , , ZD Wang . . Pinning controllability of autonomous Boolean control networks . . Sci China Inform Sci , , 2016 . . 59 ( ( 7 ): ): 070107 DOI: 10.1007/s11432-016-5579-8 http://doi.org/10.1007/s11432-016-5579-8 . .

DZ Cheng , , , HS Qi , , , ZQ Li . . Analysis and Control of Boolean Networks: a Semi-tensor Product Approach . . Springer, London, UK. , , 2011a . . .

DZ Cheng , , , HS Qi , , , ZQ Li , , , 等 . . Stability and stabilization of Boolean networks . . Int J Rob Nonl Contr , , 2011b . . 21 ( ( 2 ): ): 134 - - 156 . . DOI: 10.1002/rnc.1581 http://doi.org/10.1002/rnc.1581 . .

DZ Cheng , , , JE Feng , , , HL Lv . . Solving fuzzy relational equations via semitensor product . . IEEE Trans Fuzzy Syst , , 2012 . . 20 ( ( 2 ): ): 390 - - 396 . . DOI: 10.1109/TFUZZ.2011.2174243 http://doi.org/10.1109/TFUZZ.2011.2174243 . .

DZ Cheng , , , HS Qi , , , ZQ Liu . . From STP to game-based control . . Sci China Inform Sci , , 2018 . . 61 ( ( 1 ): ): 010201 DOI: 10.1007/s11432-017-9265-2 http://doi.org/10.1007/s11432-017-9265-2 . .

XY Ding , , , HT Li . . Optimal control of random evolutionary Boolean games . . Int J Contr, online , , 2019 . . DOI: 10.1080/00207179.2019.1585957 http://doi.org/10.1080/00207179.2019.1585957 . .

XY Ding , , , HT Li , , , QQ Yang , , , 等 . . Stochastic stability and stabilization of $n$-person random evolutionary Boolean games . . Appl Math Comput , , 2017 . . 306 1 - - 12 . . DOI: 10.1016/j.amc.2017.02.020 http://doi.org/10.1016/j.amc.2017.02.020 . .

XY Ding , , , HT Li , , , SL Wang . . Set stability and synchronization of logical networks with probabilistic time delays . . J Franklin Inst , , 2018 . . 355 ( ( 15 ): ): 7735 - - 7748 . . DOI: 10.1016/j.jfranklin.2018.08.009 http://doi.org/10.1016/j.jfranklin.2018.08.009 . .

XY Ding , , , HT Li , , , FE Alsaadi . . Regulation of game result for $n$-person random evolutionary Boolean games . . Asian J Contr, online , , 2019 . . DOI: 10.1002/asjc.2119 http://doi.org/10.1002/asjc.2119 . .

Y Ding , , , YQ Guo , , , YF Xie , , , 等 . . Time-optimal state feedback stabilization of switched Boolean control networks . . Neurocomputing , , 2017 . . 237 265 - - 271 . . DOI: 10.1016/j.neucom.2016.12.044 http://doi.org/10.1016/j.neucom.2016.12.044 . .

HB Fan , , , JE Feng , , , M Meng , , , 等 . . General decomposition of fuzzy relations: semi-tensor product approach . . Fuzzy Set Syst, online , , 2018 . . DOI: 10.1016/j.fss.2018.12.012 http://doi.org/10.1016/j.fss.2018.12.012 . .

E Fornasini , , , MM Valcher . . On the periodic trajectories of Boolean control networks . . Automatica , , 2013 . . 49 ( ( 5 ): ): 1506 - - 1509 . . DOI: 10.1016/j.automatica.2013.02.027 http://doi.org/10.1016/j.automatica.2013.02.027 . .

E Fornasini , , , ME Valcher . . Recent developments in Boolean networks control . . J Contr Dec , , 2016 . . 3 ( ( 1 ): ): 1 - - 18 . . DOI: 10.1080/23307706.2015.1098577 http://doi.org/10.1080/23307706.2015.1098577 . .

R Goebel , , , C Prieur , , , AR Teel . . Smooth patchy control Lyapunov functions . . Automatica , , 2009 . . 45 ( ( 3 ): ): 675 - - 683 . . DOI: 10.1016/j.automatica.2008.10.023 http://doi.org/10.1016/j.automatica.2008.10.023 . .

YQ Guo , , , P Wang , , , WH Gui , , , 等 . . Set stability and set stabilization of Boolean control networks based on invariant subsets . . Automatica , , 2015 . . 61 106 - - 112 . . DOI: 10.1016/j.automatica.2015.08.006 http://doi.org/10.1016/j.automatica.2015.08.006 . .

I Karafyllis , , , ZP Jiang . . Global stabilization of nonlinear systems based on vector control Lyapunov functions . . IEEE Trans Autom Contr , , 2013 . . 58 ( ( 10 ): ): 2550 - - 2562 . . DOI: 10.1109/TAC.2013.2264855 http://doi.org/10.1109/TAC.2013.2264855 . .

TH Lee , , , JH Park . . Improved criteria for sampled-data synchronization of chaotic Lur'e systems using two new approaches . . Nonl Anal Hybr Syst , , 2017 . . 24 132 - - 145 . . DOI: 10.1016/j.nahs.2016.11.006 http://doi.org/10.1016/j.nahs.2016.11.006 . .

TH Lee , , , JH Park , , , OM Kwon , , , 等 . . Stochastic sampled-data control for state estimation of time-varying delayed neural networks . . Neur Netw , , 2013 . . 46 99 - - 108 . . DOI: 10.1016/j.neunet.2013.05.001 http://doi.org/10.1016/j.neunet.2013.05.001 . .

BW Li , , , Y Liu , , , KI Kou , , , 等 . . Event-triggered control for the disturbance decoupling problem of Boolean control networks . . IEEE Trans Cybern , , 2018 . . 48 ( ( 9 ): ): 2764 - - 2769 . . DOI: 10.1109/TCYB.2017.2746102 http://doi.org/10.1109/TCYB.2017.2746102 . .

BW Li , , , JQ Lu , , , J Zhong , , , 等 . . Fast-time stability of temporal Boolean networks . . IEEE Trans Neur Netw Learn Syst , , 2019a . . 30 ( ( 8 ): ): 2285 - - 2294 . . DOI: 10.1109/TNNLS.2018.2881459 http://doi.org/10.1109/TNNLS.2018.2881459 . .

BW Li , , , JQ Lu , , , Y Liu , , , 等 . . The outputs robustness of Boolean control networks via pinning control . . IEEE Trans Contr Netw Syst, online , , 2019b . . DOI: 10.1109/TCNS.2019.2913543 http://doi.org/10.1109/TCNS.2019.2913543 . .

BW Li , , , JG Lou , , , Y Liu , , , 等 . . Robust invariant set analysis of Boolean networks . . Complexity , , 2019c . . 2019 2731395 DOI: 10.1155/2019/2731395 http://doi.org/10.1155/2019/2731395 . .

FF Li . . Feedback control design for the complete synchronisation of two coupled Boolean networks . . Int J Syst Sci , , 2016a . . 47 ( ( 12 ): ): 2996 - - 3003 . . DOI: 10.1080/00207721.2015.1053834[fanxiexiantihuan http://doi.org/10.1080/00207721.2015.1053834[fanxiexiantihuan . .

FF Li . . Pinning control design for the stabilization of Boolean networks . . IEEE Trans Neur Netw Learn Syst , , 2016b . . 27 ( ( 7 ): ): 1585 - - 1590 . . . .

FF Li . . Pinning control design for the synchronization of two coupled Boolean networks . . IEEE Trans Circ Syst , , 2016c . . 63 ( ( 3 ): ): 309 - - 313 . . DOI: 10.1109/TCSII.2015.2482658 http://doi.org/10.1109/TCSII.2015.2482658 . .

FF Li . . Stability of Boolean networks with delays using pinning control . . IEEE Trans Contr Netw Syst , , 2018 . . 5 ( ( 1 ): ): 179 - - 185 . . DOI: 10.1109/TCNS.2016.2585861 http://doi.org/10.1109/TCNS.2016.2585861 . .

FF Li , , , JT Sun . . Stability and stabilization of Boolean networks with impulsive effects . . Syst Contr Lett , , 2012 . . 61 ( ( 1 ): ): 1 - - 5 . . DOI: 10.1016/j.sysconle.2011.09.019 http://doi.org/10.1016/j.sysconle.2011.09.019 . .

FF Li , , , Y Tang . . Set stabilization for switched Boolean control networks . . Automatica , , 2017 . . 78 223 - - 230 . . DOI: 10.1016/j.automatica.2016.12.007 http://doi.org/10.1016/j.automatica.2016.12.007 . .

FF Li , , , LH Xie . . Set stabilization of probabilistic Boolean networks using pinning control . . IEEE Trans Neur Netw Learn Syst , , 2019 . . 30 ( ( 8 ): ): 2555 - - 2561 . . . .

FF Li , , , HC Yan , , , HR Karimi . . Single-input pinning controller design for reachability of Boolean networks . . IEEE Trans Neur Netw Learn Syst , , 2018 . . 29 ( ( 7 ): ): 3264 - - 3269 . . DOI: 10.1109/TNNLS.2017.2705109 http://doi.org/10.1109/TNNLS.2017.2705109 . .

H Li , , , Y Li , , , S Wang . . Recent development on analysis and control of finite-value dynamic systems . . J Shandong Norm Univ Nat Sci, in press (in Chinese) , , 2019 . . .

HT Li , , , XY Ding . . A control Lyapunov function approach to feedback stabilization of logical control networks . . SIAM J Contr Optim , , 2019 . . 57 ( ( 2 ): ): 810 - - 831 . . . .

HT Li , , , YZ Wang . . Output feedback stabilization control design for Boolean control networks . . Automatica , , 2013 . . 49 ( ( 12 ): ): 3641 - - 3645 . . DOI: 10.1016/j.automatica.2013.09.023 http://doi.org/10.1016/j.automatica.2013.09.023 . .

HT Li , , , YZ Wang . . Minimum-time state feedback stabilization of constrained Boolean control networks . . Asian J Contr , , 2016a . . 18 ( ( 5 ): ): 1688 - - 1697 . . DOI: 10.1002/asjc.1234 http://doi.org/10.1002/asjc.1234 . .

HT Li , , , YZ Wang . . Output tracking of switched Boolean networks under open-loop/closed-loop switching signals . . Nonl Anal Hybr Syst , , 2016b . . 22 137 - - 146 . . DOI: 10.1016/j.nahs.2016.04.001 http://doi.org/10.1016/j.nahs.2016.04.001 . .

HT Li , , , YZ Wang . . Robust stability and stabilisation of Boolean networks with disturbance inputs . . Int J Syst Sci , , 2016c . . 48 ( ( 4 ): ): 750 - - 756 . . DOI: 10.1080/00207721.2016.1212433 http://doi.org/10.1080/00207721.2016.1212433 . .

HT Li , , , YZ Wang . . Further results on feedback stabilization control design of Boolean control networks . . Automatica , , 2017a . . 83 303 - - 308 . . DOI: 10.1016/j.automatica.2017.06.043 http://doi.org/10.1016/j.automatica.2017.06.043 . .

HT Li , , , YZ Wang . . Lyapunov-based stability and construction of Lyapunov functions for Boolean networks . . SIAM J Contr Optim , , 2017b . . 55 ( ( 6 ): ): 3437 - - 3457 . . DOI: 10.1137/16M1092581 http://doi.org/10.1137/16M1092581 . .

HT Li , , , XF Xiao , , , XY Lei , , , 等 . . Second-order consensus seeking in multi-agent systems with nonlinear dynamics over random switching directed networks . . IEEE Trans Circ Syst , , 2013a . . 60 ( ( 6 ): ): 1595 - - 1607 . . DOI: 10.1109/TCSI.2012.2226507 http://doi.org/10.1109/TCSI.2012.2226507 . .

HT Li , , , YZ Wang , , , ZB Liu . . Simultaneous stabilization for a set of Boolean control networks . . Syst Contr Lett , , 2013b . . 62 ( ( 12 ): ): 1168 - - 1174 . . DOI: 10.1016/j.sysconle.2013.09.008 http://doi.org/10.1016/j.sysconle.2013.09.008 . .

HT Li , , , YZ Wang , , , ZB Liu . . Stability analysis for switched Boolean networks under arbitrary switching signals . . IEEE Trans Autom Contr , , 2014 . . 59 ( ( 7 ): ): 1978 - - 1982 . . DOI: 10.1109/TAC.2014.2298731 http://doi.org/10.1109/TAC.2014.2298731 . .

HT Li , , , YZ Wang , , , LH Xie . . Output tracking control of Boolean control networks via state feedback: constant reference signal case . . Automatica , , 2015 . . 59 54 - - 59 . . DOI: 10.1016/j.automatica.2015.06.004 http://doi.org/10.1016/j.automatica.2015.06.004 . .

HT Li , , , YZ Wang , , , PL Guo . . State feedback based output tracking control of probabilistic Boolean networks . . Inform Sci , , 2016 . . 349-350 1 - - 11 . . DOI: 10.1016/j.ins.2016.02.035 http://doi.org/10.1016/j.ins.2016.02.035 . .

HT Li , , , YZ Wang , , , PL Guo . . Output reachability analysis and output regulation control design of Boolean control networks . . Sci China Inform Sci , , 2017a . . 60 ( ( 2 ): ): 022202 DOI: 10.1007/s11432-015-0611-4 http://doi.org/10.1007/s11432-015-0611-4 . .

HT Li , , , LH Xie , , , YZ Wang . . Output regulation of Boolean control networks . . IEEE Trans Autom Contr , , 2017b . . 62 ( ( 6 ): ): 2993 - - 2998 . . . .

HT Li , , , PP Song , , , QQ Yang . . Pinning control design for robust output tracking of $k$-valued logical networks . . J Franklin Inst , , 2017c . . 354 ( ( 7 ): ): 3039 - - 3053 . . DOI: 10.1016/j.jfranklin.2017.02.009 http://doi.org/10.1016/j.jfranklin.2017.02.009 . .

HT Li , , , XY Ding , , , A Alsaedi , , , 等 . . Stochastic set stabilisation of $n$-person random evolutionary Boolean games and its applications . . IET Contr Theory Appl , , 2017d . . 11 ( ( 13 ): ): 2152 - - 2160 . . . .

HT Li , , , GD Zhao , , , M Meng , , , 等 . . A survey on applications of semi-tensor product method in engineering . . Sci China Inform Sci , , 2018a . . 61 ( ( 1 ): ): 010202 DOI: 10.1007/s11432-017-9238-1 http://doi.org/10.1007/s11432-017-9238-1 . .

HT Li , , , XY Ding , , , QQ Yang , , , 等 . . Algebraic formulation and Nash equilibrium of competitive diffusion games . . Dynam Game Appl , , 2018b . . 8 ( ( 2 ): ): 423 - - 433 . . DOI: 10.1007/s13235-017-0228-4 http://doi.org/10.1007/s13235-017-0228-4 . .

HT Li , , , YT Zheng , , , FE Alsaadi . . Algebraic formulation and topological structure of Boolean networks with state-dependent delay . . J Comput Appl Math , , 2019a . . 350 87 - - 97 . . DOI: 10.1016/j.cam.2018.10.003 http://doi.org/10.1016/j.cam.2018.10.003 . .

HT Li , , , XJ Xu , , , XY Ding . . Finite-time stability analysis of stochastic switched Boolean networks with impulsive effect . . Appl Math Comput , , 2019b . . 347 557 - - 565 . . DOI: 10.1016/j.amc.2018.11.018 http://doi.org/10.1016/j.amc.2018.11.018 . .

JN Li , , , H Modares , , , TY Chai , , , 等 . . Off-policy reinforcement learning for synchronization in multiagent graphical games . . IEEE Trans Neur Netw Learn Syst , , 2017 . . 28 ( ( 10 ): ): 2434 - - 2445 . . DOI: 10.1109/TNNLS.2016.2609500 http://doi.org/10.1109/TNNLS.2016.2609500 . .

R Li , , , M Yang , , , TG Chu . . State feedback stabilization for Boolean control networks . . IEEE Trans Autom Contr , , 2013 . . 58 ( ( 7 ): ): 1853 - - 1857 . . DOI: 10.1109/TAC.2013.2238092 http://doi.org/10.1109/TAC.2013.2238092 . .

R Li , , , M Yang , , , TG Chu . . State feedback stabilization for probabilistic Boolean networks . . Automatica , , 2014 . . 50 ( ( 4 ): ): 1272 - - 1278 . . DOI: 10.1016/j.automatica.2014.02.034 http://doi.org/10.1016/j.automatica.2014.02.034 . .

XD Li , , , JH Shen , , , H Akca , , , 等 . . LMI-based stability for singularly perturbed nonlinear impulsive differential systems with delays of small parameter . . Appl Math Comput , , 2015 . . 250 798 - - 804 . . DOI: 10.1016/j.amc.2014.10.113 http://doi.org/10.1016/j.amc.2014.10.113 . .

XD Li , , , HT Li , , , GD Zhao . . Function perturbation impact on feedback stabilization of Boolean control networks . . IEEE Trans Neur Netw Learn Syst , , 2019a . . 30 ( ( 8 ): ): 2548 - - 2554 . . DOI: 10.1109/TNNLS.2018.2881168 http://doi.org/10.1109/TNNLS.2018.2881168 . .

XD Li , , , XY Yang , , , TW Huang . . Persistence of delayed cooperative models: impulsive control method . . Appl Math Comput , , 2019b . . 342 130 - - 146 . . DOI: 10.1016/j.amc.2018.09.003 http://doi.org/10.1016/j.amc.2018.09.003 . .

XH Li , , , JQ Lu , , , XY Chen , , , 等 . . Robust output tracking of delayed Boolean networks under pinning control . . IEEE Trans Circ Syst , , 2018 . . 65 ( ( 9 ): ): 1249 - - 1253 . . DOI: 10.1109/TCSII.2018.2827931 http://doi.org/10.1109/TCSII.2018.2827931 . .

YL Li , , , HT Li , , , WW Sun . . Event-triggered control for robust set stabilization of logical control networks . . Automatica , , 2018a . . 95 556 - - 560 . . DOI: 10.1016/j.automatica.2018.06.030 http://doi.org/10.1016/j.automatica.2018.06.030 . .

YL Li , , , HT Li , , , XJ Xu , , , 等 . . Semi-tensor product approach to minimal-agent consensus control of networked evolutionary games . . IET Contr Theory Appl , , 2018b . . 12 ( ( 16 ): ): 2269 - - 2275 . . DOI: 10.1049/iet-cta.2018.5230 http://doi.org/10.1049/iet-cta.2018.5230 . .

YL Li , , , HT Li , , , PY Duan . . Synchronization of switched logical control networks via event-triggered control . . J Franklin Inst , , 2018c . . 355 ( ( 12 ): ): 5203 - - 5216 . . DOI: 10.1016/j.jfranklin.2018.04.028 http://doi.org/10.1016/j.jfranklin.2018.04.028 . .

YL Li , , , HT Li , , , SL Wang . . Constrained sampled-data reachability and stabilization of logical control networks . . IEEE Trans Circ Syst , , 2019 . . 66 ( ( 12 ): ): 2002 - - 2006 . . DOI: 10.1109/TCSII.2019.2892357 http://doi.org/10.1109/TCSII.2019.2892357 . .

YY Li , , , BW Li , , , Y Liu , , , 等 . . Set stability and stabilization of switched Boolean networks with state-based switching . . IEEE Access , , 2018 . . 6 35624 - - 35630 . . DOI: 10.1109/ACCESS.2018.2851391 http://doi.org/10.1109/ACCESS.2018.2851391 . .

S Liang , , , GD Zhao , , , HT Li , , , 等 . . Structural stability analysis of gene regulatory networks modeled by Boolean networks . . Math Meth Appl Sci , , 2019 . . 42 ( ( 7 ): ): 2221 - - 2230 . . DOI: 10.1002/mma.5488 http://doi.org/10.1002/mma.5488 . .

L Lin , , , SY Zhu , , , Y Liu , , , 等 . . Output regulation of Boolean control networks with nonuniform sampled-data control . . IEEE Access , , 2019 . . 7 50691 - - 50696 . . DOI: 10.1109/ACCESS.2019.2910593 http://doi.org/10.1109/ACCESS.2019.2910593 . .

JY Liu , , , Y Liu , , , YQ Guo , , , 等 . . Sampled-data state-feedback stabilization of probabilistic Boolean control networks: a control Lyapunov function approach . . IEEE Trans Cybern, online , , 2019 . . DOI: 10.1109/TCYB.2019.2932914 http://doi.org/10.1109/TCYB.2019.2932914 . .

RJ Liu , , , JQ Lu , , , JG Lou , , , 等 . . Set stabilization of Boolean networks under pinning control strategy . . Neurocomputing , , 2017 . . 260 142 - - 148 . . DOI: 10.1016/j.neucom.2017.04.005 http://doi.org/10.1016/j.neucom.2017.04.005 . .

RJ Liu , , , JQ Lu , , , Y Liu , , , 等 . . Delayed feedback control for stabilization of Boolean control networks with state delay . . IEEE Trans Neur Netw Learn Syst , , 2018 . . 29 ( ( 7 ): ): 3283 - - 3288 . . DOI: 10.1109/TNNLS.2017.2659386 http://doi.org/10.1109/TNNLS.2017.2659386 . .

RJ Liu , , , JQ Lu , , , WX Zheng , , , 等 . . Output feedback control for set stabilization of Boolean control networks . . IEEE Trans Neur Netw Learn Syst, online , , 2019 . . DOI: 10.1109/TNNLS.2019.2928028 http://doi.org/10.1109/TNNLS.2019.2928028 . .

S Liu , , , T Li , , , LH Xie , , , 等 . . Continuous-time and sampled-data-based average consensus with logarithmic quantizers . . Automatica , , 2013 . . 49 ( ( 11 ): ): 3329 - - 3336 . . DOI: 10.1016/j.automatica.2013.07.016 http://doi.org/10.1016/j.automatica.2013.07.016 . .

Y Liu , , , JQ Lu , , , B Wu . . Some necessary and sufficient conditions for the output controllability of temporal Boolean control networks . . ESAIM Contr Optim Calc Var , , 2014 . . 20 ( ( 1 ): ): 158 - - 173 . . DOI: 10.1051/cocv/2013059 http://doi.org/10.1051/cocv/2013059 . .

Y Liu , , , HW Chen , , , JQ Lu , , , 等 . . Controllability of probabilistic Boolean control networks based on transition probability matrices . . Automatica , , 2015 . . 52 340 - - 345 . . DOI: 10.1016/j.automatica.2014.12.018 http://doi.org/10.1016/j.automatica.2014.12.018 . .

Y Liu , , , BW Li , , , JG Lou . . Disturbance decoupling of singular Boolean control networks . . IEEE/ACM Trans Comput Biol Bioinform , , 2016a . . 13 ( ( 6 ): ): 1194 - - 1200 . . DOI: 10.1109/TCBB.2015.2509969 http://doi.org/10.1109/TCBB.2015.2509969 . .

Y Liu , , , LJ Sun , , , JQ Lu , , , 等 . . Feedback controller design for the synchronization of Boolean control networks . . IEEE Trans Neur Netw Learn Syst , , 2016b . . 27 ( ( 9 ): ): 1991 - - 1996 . . DOI: 10.1109/TNNLS.2015.2461012 http://doi.org/10.1109/TNNLS.2015.2461012 . .

Y Liu , , , JD Cao , , , LJ Sun , , , 等 . . Sampled-data state feedback stabilization of Boolean control networks . . Neur Comput , , 2016c . . 28 ( ( 4 ): ): 778 - - 799 . . DOI: 10.1162/NECO_a_00819 http://doi.org/10.1162/NECO_a_00819 . .

Y Liu , , , BW Li , , , HW Chen , , , 等 . . Function perturbations on singular Boolean networks . . Automatica , , 2017a . . 84 36 - - 42 . . DOI: 10.1016/j.automatica.2017.06.035 http://doi.org/10.1016/j.automatica.2017.06.035 . .

Y Liu , , , BW Li , , , JQ Lu , , , 等 . . Pinning control for the disturbance decoupling problem of Boolean networks . . IEEE Trans Autom Contr , , 2017b . . 62 ( ( 12 ): ): 6595 - - 6601 . . DOI: 10.1109/TAC.2017.2715181 http://doi.org/10.1109/TAC.2017.2715181 . .

Y Liu , , , JD Cao , , , BW Li , , , 等 . . Normalization and solvability of dynamic-algebraic Boolean networks . . IEEE Trans Neur Netw Learn Syst , , 2018 . . 29 ( ( 7 ): ): 3301 - - 3306 . . . .

Y Liu , , , LY Tong , , , JG Lou , , , 等 . . Sampled-data control for the synchronization of Boolean control networks . . IEEE Trans Cybern , , 2019a . . 49 ( ( 2 ): ): 726 - - 732 . . . .

Y Liu , , , LQ Wang , , , JQ Lu , , , 等 . . Sampled-data stabilization of probabilistic Boolean control networks . . Syst Contr Lett , , 2019b . . 124 106 - - 111 . . DOI: 10.1016/j.sysconle.2018.12.012 http://doi.org/10.1016/j.sysconle.2018.12.012 . .

YS Liu , , , YT Zheng , , , HT Li , , , 等 . . Control design for output tracking of delayed Boolean control networks . . J Comput Appl Math , , 2018 . . 327 188 - - 195 . . DOI: 10.1016/j.cam.2017.06.016 http://doi.org/10.1016/j.cam.2017.06.016 . .

ZB Liu , , , Y Wang , , , H Li . . New approach to derivative calculation of multi-valued logical functions with application to fault detection . . IET Contr Theory Appl , , 2014 . . 8 ( ( 8 ): ): 554 - - 560 . . DOI: 10.1049/iet-cta.2013.0104 http://doi.org/10.1049/iet-cta.2013.0104 . .

JQ Lu , , , J Zhong , , , DWC Ho , , , 等 . . On controllability of delayed Boolean control networks . . SIAM J Contr Optim , , 2016a . . 54 ( ( 2 ): ): 475 - - 494 . . DOI: 10.1137/140991820 http://doi.org/10.1137/140991820 . .

JQ Lu , , , J Zhong , , , C Huang , , , 等 . . On pinning controllability of Boolean control networks . . IEEE Trans Autom Contr , , 2016b . . 61 ( ( 6 ): ): 1658 - - 1663 . . DOI: 10.1109/TAC.2015.2478123 http://doi.org/10.1109/TAC.2015.2478123 . .

JQ Lu , , , HT Li , , , Y Liu , , , 等 . . Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems . . IET Contr Theory Appl , , 2017 . . 11 ( ( 13 ): ): 2040 - - 2047 . . DOI: 10.1049/iet-cta.2016.1659 http://doi.org/10.1049/iet-cta.2016.1659 . .

JQ Lu , , , ML Li , , , Y Liu , , , 等 . . Nonsingularity of Grain-like cascade FSRs via semi-tensor product . . Sci China Inform Sci , , 2018a . . 61 ( ( 1 ): ): 010204 DOI: 10.1007/s11432-017-9269-6 http://doi.org/10.1007/s11432-017-9269-6 . .

JQ Lu , , , LJ Sun , , , Y Liu , , , 等 . . Stabilization of Boolean control networks under aperiodic sampled-data control . . SIAM J Contr Optim , , 2018b . . 56 ( ( 6 ): ): 4385 - - 4404 . . DOI: 10.1137/18M1169308 http://doi.org/10.1137/18M1169308 . .

JQ Lu , , , ML Li , , , TW Huang , , , 等 . . The transformation between the Galois NLFSRs and the Fibonacci NLFSRs via semi-tensor product of matrices . . Automatica , , 2018c . . 96 393 - - 397 . . DOI: 10.1016/j.automatica.2018.07.011 http://doi.org/10.1016/j.automatica.2018.07.011 . .

YY Lu , , , W Zhang . . A piecewise smooth control-Lyapunov function framework for switching stabilization . . Automatica , , 2017 . . 76 258 - - 265 . . DOI: 10.1016/j.automatica.2016.09.029 http://doi.org/10.1016/j.automatica.2016.09.029 . .

M Lukk , , , M Kapushesky , , , J Nikkilä , , , 等 . . A global map of human gene expression . . Nat Biotechnol , , 2010 . . 28 ( ( 4 ): ): 322 - - 324 . . DOI: 10.1038/nbt0410-322 http://doi.org/10.1038/nbt0410-322 . .

Y Mao , , , LQ Wang , , , Y Liu , , , 等 . . Stabilization of evolutionary networked games with length-$r$ information . . Appl Math Comput , , 2018 . . 337 442 - - 451 . . DOI: 10.1016/j.amc.2018.05.027 http://doi.org/10.1016/j.amc.2018.05.027 . .

C Moraga , , , E Trillas , , , S Guadarrama . . Multiple-valued logic and artificial intelligence fundamentals of fuzzy control revisited . . Artif Intell Rev , , 2003 . . 20 ( ( 3-4 ): ): 169 - - 197 . . . .

NK Mu , , , XF Liao , , , TW Huang . . Leader-following consensus in second-order multiagent systems via event-triggered control with nonperiodic sampled data . . IEEE Trans Circ Syst , , 2015 . . 62 ( ( 10 ): ): 1007 - - 1011 . . DOI: 10.1023/10.1109/TCSII.2015.2458036 http://doi.org/10.1023/10.1109/TCSII.2015.2458036 . .

FJ Mller , , , A Schuppert . . Few inputs can reprogram biological networks . . Nature , , 2011 . . 478 ( ( 7369 ): ): E4 DOI: 10.1038/nature10543 http://doi.org/10.1038/nature10543 . .

FJ Mller , , , BM Schuldt , , , R Williams , , , 等 . . A bioinformatic assay for pluripotency in human cells . . Nat Meth , , 2011 . . 8 ( ( 4 ): ): 315 - - 317 . . DOI: 10.1038/nmeth.1580 http://doi.org/10.1038/nmeth.1580 . .

R Postoyan , , , P Tabuada , , , D Nešić , , , 等 . . A framework for the event-triggered stabilization of nonlinear systems . . IEEE Trans Autom Contr , , 2015 . . 60 ( ( 4 ): ): 982 - - 996 . . DOI: 10.1109/TAC.2014.2363603 http://doi.org/10.1109/TAC.2014.2363603 . .

DP Rosin , , , D Rontani , , , DJ Gauthier , , , 等 . . Control of synchronization patterns in neural-like Boolean networks . . Phys Rev Lett , , 2013 . . 110 ( ( 5 ): ): 104102 DOI: 10.1103/PhysRevLett.110.104102 http://doi.org/10.1103/PhysRevLett.110.104102 . .

RG Sanfelice . . On the existence of control Lyapunov functions and state-feedback laws for hybrid systems . . IEEE Trans Autom Contr , , 2013 . . 58 ( ( 12 ): ): 3242 - - 3248 . . DOI: 10.1109/TAC.2013.2264851 http://doi.org/10.1109/TAC.2013.2264851 . .

LJ Sun , , , JQ Lu , , , WK Ching . . Switching-based stabilization of aperiodic sampled-data Boolean control networks with all subsystems unstable . . Front Inform Technol Electron Eng , , 2020 . . 21 ( ( 2 ): ): 260 - - 267 . . DOI: 10.1631/FITEE.1900312 http://doi.org/10.1631/FITEE.1900312 . .

LY Tong , , , Y Liu , , , FE Alsaadi , , , 等 . . Robust sampled-data control invariance for Boolean control networks . . J Franklin Inst , , 2017 . . 354 ( ( 15 ): ): 7077 - - 7087 . . DOI: 10.1016/j.jfranklin.2017.08.017 http://doi.org/10.1016/j.jfranklin.2017.08.017 . .

LY Tong , , , Y Liu , , , YY Li , , , 等 . . Robust control invariance of probabilistic Boolean control networks via event-triggered control . . IEEE Access , , 2018a . . 6 37767 - - 37774 . . DOI: 10.1109/ACCESS.2018.2828128 http://doi.org/10.1109/ACCESS.2018.2828128 . .

LY Tong , , , Y Liu , , , JG Lou , , , 等 . . Static output feedback set stabilization for context-sensitive probabilistic Boolean control networks . . Appl Math Comput , , 2018b . . 332 263 - - 275 . . DOI: 10.1016/j.amc.2018.03.043 http://doi.org/10.1016/j.amc.2018.03.043 . .

LY Tong , , , JL Liang , , , HW Chen . . State feedback controller design for anti-synchronization of Boolean control networks: an event-based idea . . Asian J Contr, online , , 2019 . . DOI: 10.1002/asjc.2105 http://doi.org/10.1002/asjc.2105 . .

U Vaidya , , , PG Mehta , , , UV Shanbhag . . Nonlinear stabilization via control Lyapunov measure . . IEEE Trans Autom Contr , , 2010 . . 55 ( ( 6 ): ): 1314 - - 1328 . . DOI: 10.1109/TAC.2010.2042226 http://doi.org/10.1109/TAC.2010.2042226 . .

A Veliz-Cuba , , , B Stigler . . Boolean models can explain bistability in the lac operon . . J Comput Biol , , 2011 . . 18 ( ( 6 ): ): 783 - - 794 . . DOI: 10.1089/cmb.2011.0031 http://doi.org/10.1089/cmb.2011.0031 . .

A Vinodkumar , , , T Senthilkumar , , , XD Li . . Robust exponential stability results for uncertain infinite delay differential systems with random impulsive moments . . Adv Differ Equat , , 2018 . . 2018 39 DOI: 10.1186/s13662-018-1488-z http://doi.org/10.1186/s13662-018-1488-z . .

B Wang , , , JE Feng . . On detectability of probabilistic Boolean networks . . Inform Sci , , 2019 . . 483 383 - - 395 . . DOI: 10.1016/j.ins.2019.01.055 http://doi.org/10.1016/j.ins.2019.01.055 . .

JY Wang , , , JW Feng , , , C Xu , , , 等 . . The synchronization of instantaneously coupled harmonic oscillators using sampled data with measurement noise . . Automatica , , 2016 . . 66 155 - - 162 . . DOI: 10.1016/j.automatica.2016.01.012 http://doi.org/10.1016/j.automatica.2016.01.012 . .

LQ Wang , , , Y Liu , , , ZG Wu , , , 等 . . Stabilization and finite-time stabilization of probabilistic Boolean control networks . . IEEE Trans Syst Man Cybern Syst, online , , 2019 . . DOI: 10.1109/TSMC.2019.2898880 http://doi.org/10.1109/TSMC.2019.2898880 . .

SL Wang , , , HT Li . . Column stacking approach to resolution of systems of fuzzy relational inequalities . . J Franklin Inst , , 2019 . . 356 ( ( 6 ): ): 3314 - - 3332 . . DOI: 10.1016/j.jfranklin.2019.02.007 http://doi.org/10.1016/j.jfranklin.2019.02.007 . .

YH Wang , , , DZ Cheng . . Stability and stabilization of a class of finite evolutionary games . . J Franklin Inst , , 2017 . . 354 ( ( 3 ): ): 1603 - - 1617 . . DOI: 10.1016/j.jfranklin.2016.12.007 http://doi.org/10.1016/j.jfranklin.2016.12.007 . .

YQ Wu , , , XY Meng , , , LH Xie , , , 等 . . An input-based triggering approach to leader-following problems . . Automatica , , 2017 . . 75 221 - - 228 . . DOI: 10.1016/j.automatica.2016.09.040 http://doi.org/10.1016/j.automatica.2016.09.040 . .

ZG Wu , , , P Shi , , , HY Su , , , 等 . . Local synchronization of chaotic neural networks with sampled-data and saturating actuators . . IEEE Trans Cybern , , 2014 . . 44 ( ( 12 ): ): 2635 - - 2645 . . DOI: 10.1109/TCYB.2014.2312004 http://doi.org/10.1109/TCYB.2014.2312004 . .

XJ Xu , , , HT Li , , , YL Li , , , 等 . . Output tracking control of Boolean control networks with impulsive effects . . Math Meth Appl Sci , , 2018a . . 41 ( ( 4 ): ): 1554 - - 1564 . . DOI: 10.1002/mma.4685 http://doi.org/10.1002/mma.4685 . .

XJ Xu , , , YS Liu , , , HT Li , , , 等 . . Robust set stabilization of Boolean control networks with impulsive effects . . Nonl Anal Model Contr , , 2018b . . 23 ( ( 4 ): ): 553 - - 567 . . DOI: 10.15388/NA.2018.4.6 http://doi.org/10.15388/NA.2018.4.6 . .

XJ Xu , , , YS Liu , , , HT Li , , , 等 . . Synchronization of switched Boolean networks with impulsive effects . . Int J Biomath , , 2018c . . 11 ( ( 6 ): ): 1850080 . .

JJ Yang , , , JQ Lu , , , LL Li , , , 等 . . Event-triggered control for the synchronization of Boolean control networks . . Nonl Dynam , , 2019 . . 96 ( ( 2 ): ): 1335 - - 1344 . . DOI: 10.1007/s11071-019-04857-2 http://doi.org/10.1007/s11071-019-04857-2 . .

JJ Yang , , , JQ Lu , , , JG Lou , , , 等 . . Synchronization of drive-response Boolean control networks with impulsive disturbances . . Appl Math Comput , , 2020 . . 364 124679 DOI: 10.1016/j.amc.2019.124679 http://doi.org/10.1016/j.amc.2019.124679 . .

QQ Yang , , , HT Li , , , YS Liu . . Pinning control design for feedback stabilization of constrained Boolean control networks . . Adv Differ Equat , , 2016 . . 2016 182 DOI: 10.1186/s13662-016-0909-0 http://doi.org/10.1186/s13662-016-0909-0 . .

QQ Yang , , , HT Li , , , PP Song , , , 等 . . Global convergence of serial Boolean networks based on algebraic representation . . J Differ Equat Appl , , 2017 . . 23 ( ( 3 ): ): 633 - - 647 . . DOI: 10.1080/10236198.2016.1270275 http://doi.org/10.1080/10236198.2016.1270275 . .

YY Yu , , , JE Feng , , , B Wang , , , 等 . . Sampled-data controllability and stabilizability of Boolean control networks: nonuniform sampling . . J Franklin Inst , , 2018 . . 335 ( ( 12 ): ): 5324 - - 5335 . . DOI: 10.1016/j.jfranklin.2018.05.036 http://doi.org/10.1016/j.jfranklin.2018.05.036 . .

YY Yu , , , M Meng , , , JE Feng , , , 等 . . An adjoint network approach to design stabilizable switching signals of switched Boolean networks . . Appl Math Comput , , 2019a . . 357 12 - - 22 . . DOI: 10.1016/j.amc.2019.02.067 http://doi.org/10.1016/j.amc.2019.02.067 . .

YY Yu , , , JE Feng , , , JF Pan , , , 等 . . Block decoupling of Boolean control networks . . IEEE Trans Autom Contr , , 2019b . . 64 ( ( 8 ): ): 3129 - - 3140 . . DOI: 10.1109/TAC.2018.2880411 http://doi.org/10.1109/TAC.2018.2880411 . .

YY Yu , , , B Wang , , , JE Feng . . Input observability of Boolean control networks . . Neurocomputing , , 2019c . . 333 22 - - 28 . . DOI: 10.1016/j.neucom.2018.12.014 http://doi.org/10.1016/j.neucom.2018.12.014 . .

LQ Zhang , , , JE Feng , , , XH Feng , , , 等 . . Further results on disturbance decoupling of mix-valued logical networks . . IEEE Trans Autom Contr , , 2014 . . 59 ( ( 6 ): ): 1630 - - 1634 . . DOI: 10.1109/TAC.2013.2292733 http://doi.org/10.1109/TAC.2013.2292733 . .

Y Zhao , , , BK Ghosh , , , DZ Cheng . . Control of large-scale Boolean networks via network aggregation . . IEEE Trans Neur Netw Learn Syst , , 2016 . . 27 ( ( 7 ): ): 1527 - - 1536 . . DOI: 10.1109/TNNLS.2015.2442593 http://doi.org/10.1109/TNNLS.2015.2442593 . .

YT Zheng , , , HT Li , , , XY Ding , , , 等 . . Stabilization and set stabilization of delayed Boolean control networks based on trajectory stabilization . . J Franklin Inst , , 2017 . . 354 ( ( 17 ): ): 7812 - - 7827 . . . .

J Zhong , , , JQ Lu , , , Y Liu , , , 等 . . Synchronization in an array of output-coupled Boolean networks with time delay . . IEEE Tran Neur Netw Learn Syst , , 2014 . . 25 ( ( 12 ): ): 2288 - - 2294 . . DOI: 10.1109/TNNLS.2014.2305722 http://doi.org/10.1109/TNNLS.2014.2305722 . .

J Zhong , , , JQ Lu , , , TW Huang , , , 等 . . Controllability and synchronization analysis of identical-hierarchy mixed-valued logical control networks . . IEEE Trans Cybern , , 2017a . . 47 ( ( 11 ): ): 3482 - - 3493 . . DOI: 10.1109/TCYB.2016.2560240 http://doi.org/10.1109/TCYB.2016.2560240 . .

J Zhong , , , DWC Ho , , , JQ Lu , , , 等 . . Switching-signal-triggered pinning control for output tracking of switched Boolean networks . . IET Contr Theory Appl , , 2017b . . 11 ( ( 13 ): ): 2089 - - 2096 . . DOI: 10.1049/iet-cta.2017.0260 http://doi.org/10.1049/iet-cta.2017.0260 . .

J Zhong , , , DWCH Ho , , , JQ Lu , , , 等 . . Pinning controllers for activation output tracking of Boolean network under one-bit perturbation . . IEEE Trans Cybern , , 2019a . . 49 ( ( 9 ): ): 3398 - - 3408 . . DOI: 10.1109/TCYB.2018.2842819 http://doi.org/10.1109/TCYB.2018.2842819 . .

J Zhong , , , Y Liu , , , KI Kou , , , 等 . . On the ensemble controllability of Boolean control networks using STP method . . Appl Math Comput , , 2019b . . 358 51 - - 62 . . DOI: 10.1016/j.amc.2019.03.059 http://doi.org/10.1016/j.amc.2019.03.059 . .

J Zhong , , , BW Li , , , Y Liu , , , 等 . . Output feedback stabilizer design of Boolean networks based on network structure . . Front Inform Technol Electron Eng , , 2020 . . 21 ( ( 2 ): ): 247 - - 259 . . DOI: 10.1631/FITEE.1900229 http://doi.org/10.1631/FITEE.1900229 . .

B Zhu , , , XH Xia , , , Z Wu . . Evolutionary game theoretic demand-side management and control for a class of networked smart grid . . Automatica , , 2016 . . 70 94 - - 100 . . DOI: 10.1016/j.automatica.2016.03.027 http://doi.org/10.1016/j.automatica.2016.03.027 . .

QX Zhu , , , W Lin . . Stabilizing Boolean networks by optimal event-triggered feedback control . . Syst Contr Lett , , 2019 . . 126 40 - - 47 . . DOI: 10.1016/j.sysconle.2019.03.002 http://doi.org/10.1016/j.sysconle.2019.03.002 . .

QX Zhu , , , Y Liu , , , JQ Lu , , , 等 . . Controllability and observability of Boolean control networks via sampled-data control . . IEEE Trans Contr Netw Syst, online , , 2018a . . DOI: 10.1109/TCNS.2018.2885273 http://doi.org/10.1109/TCNS.2018.2885273 . .

QX Zhu , , , Y Liu , , , JQ Lu , , , 等 . . Observability of Boolean control networks . . Sci China Inform Sci , , 2018b . . 61 ( ( 9 ): ): 092201 DOI: 10.1007/s11432-017-9135-4 http://doi.org/10.1007/s11432-017-9135-4 . .

QX Zhu , , , Y Liu , , , JQ Lu , , , 等 . . On the optimal control of Boolean control networks . . SIAM J Contr Optim , , 2018c . . 56 ( ( 2 ): ): 1321 - - 1341 . . DOI: 10.1137/16M1070281 http://doi.org/10.1137/16M1070281 . .

QX Zhu , , , Y Liu , , , JQ Lu , , , 等 . . Further results on the controllability of Boolean control networks . . IEEE Trans Autom Contr , , 2019 . . 64 ( ( 1 ): ): 440 - - 442 . . DOI: 10.1109/TAC.2018.2830642 http://doi.org/10.1109/TAC.2018.2830642 . .

SY Zhu , , , JG Lou , , , Y Liu , , , 等 . . Event-triggered control for the stabilization of probabilistic Boolean control networks . . Complexity , , 2018a . . 2018 9259348 DOI: 10.1155/2018/9259348 http://doi.org/10.1155/2018/9259348 . .

SY Zhu , , , Y Liu , , , JG Lou , , , 等 . . Sampled-data state feedback control for the set stabilization of Boolean control networks . . IEEE Trans Syst Man Cybern Syst, online , , 2018b . . DOI: 10.1109/TSMC.2018.2852703 http://doi.org/10.1109/TSMC.2018.2852703 . .

SY Zhu , , , JQ Lu , , , DWC Ho . . Topological sorting for finite-time stability of probabilistic logical networks . . IEEE Trans Circ Syst, online , , 2019 . . DOI: 10.1109/TCSII.2019.2919018 http://doi.org/10.1109/TCSII.2019.2919018 . .

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Controllability of Boolean control networks with multiple time delays in both states and controls

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