doi: 10.3934/dcdss.2022028
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Robust $ H_\infty $ resilient event-triggered control design for T-S fuzzy systems

1. 

School of Electrical Engineering, Chungbuk National University, Cheongju 28644, South Korea

2. 

Department of Applied Mathematics, Bharathiar University, Coimbatore 641046, India

3. 

Department of Mathematics, Sungkyunkwan University, Suwon, South Korea

* Corresponding author: Oh-Min Kwon and Rathinasamy Sakthivel

Received  November 2021 Revised  December 2021 Early access February 2022

This paper investigates the resilient $ H_\infty $ event-triggered control problem for Takagi-Sugeno fuzzy system with time-varying delay and external disturbance. Contrary to some existing results, the considered event-triggered conditions are verified only at each sampling instant because it is difficult to prove Zeno-freeness for a continuous event-triggered mechanism in the presence of external disturbance. Furthermore, by constructing an appropriate Lyapunov-Krasovskii functional, sufficient conditions are derived in the form of linear matrix inequalities to ensure the asymptotic stability and the $ H_\infty $ performances of closed-loop systems. More precisely, the proposed control design not only improve robust performance but also save the communication resources. Finally, the obtained theoretical results are verified through numerical simulation, which demonstrate the efficiency and advantages of the proposed method.

Citation: Ramalingam Sakthivel, Palanisamy Selvaraj, Yeong-Jae Kim, Dong-Hoon Lee, Oh-Min Kwon, Rathinasamy Sakthivel. Robust $ H_\infty $ resilient event-triggered control design for T-S fuzzy systems. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022028
References:
[1]

X. Cai, S. Zhong, J. Wang and K. Shi, Robust $H_\infty$ control for uncertain delayed T-S fuzzy systems with stochastic packet dropouts, Appl. Math. Comput., 385 (2020), 125432, 21 pp. doi: 10.1016/j.amc.2020.125432.

[2]

Z. DuY. Kao and Ju. H. Park, Tracking control design for interval type-2 fuzzy nonlinear unreliable networked control systems, J. Franklin Inst., 358 (2021), 4159-4177.  doi: 10.1016/j.jfranklin.2021.03.016.

[3]

W. Guan and F. Liu, Non-fragile fuzzy dissipative static output feedback control for Markovian jump systems subject to actuator saturation, Neurocomputing, 193 (2016), 123-132.  doi: 10.1016/j.neucom.2016.02.004.

[4]

A. Hastir, J. J. Winkin and D. Dochain, Exponential stability of nonlinear infinite-dimensional systems: Application to nonisothermal axial dispersion tubular reactors, Automatica J. IFAC, 121 (2020), 109201, 15 pp. doi: 10.1016/j.automatica.2020.109201.

[5]

M. He and J. Li, Resilient guaranteed cost control for uncertain T-S fuzzy systems with time-varying delays and Markov jump parameters, ISA Trans., 88 (2019), 12-22.  doi: 10.1016/j.isatra.2018.11.034.

[6]

S. Kuppusamy and Y. H. Joo, Nonfragile retarded sampled-data switched control of T-S fuzzy systems and its applications, IEEE Trans. Fuzzy Syst., 28 (2020), 2523-2532.  doi: 10.1109/TFUZZ.2019.2940432.

[7]

X. LiD. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Autom. Control, 65 (2020), 4908-4913.  doi: 10.1109/TAC.2020.2964558.

[8]

X. Li and D. Ye, Membership-function-dependent security control for networked T-S fuzzy-model-based systems against DoS attacks, IET Control. Theory Appl., 15 (2021), 360-371.  doi: 10.1049/cth2.12048.

[9]

D. Liu and G.-H. Yang, Event-triggered control for linear systems with actuator saturation and disturbances, IET Control. Theory Appl., 11 (2017), 1351-1359.  doi: 10.1049/iet-cta.2016.1661.

[10]

J. LiuZ.-G. WuD. Yue and Ju H. Park, Stabilization of networked control systems with hybrid-driven mechanism and probabilistic cyber attacks, IEEE Trans. Syst. Man Cybern. Syst., 51 (2021), 943-953.  doi: 10.1109/TSMC.2018.2888633.

[11]

X. Liu and S. Zhong, Stability analysis of delayed switched cascade nonlinear systems with uniform switching signals, Math. Model. Control, 1 (2021), 90-101.  doi: 10.3934/mmc.2021007.

[12]

Y. Liu and Y. Ma, Finite-time non-fragile extended dissipative control for T-S fuzzy system via augmented Lyapunov-Krasovskii functional, ISA Trans., 117 (2021), 1-15.  doi: 10.1016/j.isatra.2021.01.038.

[13]

V. Nithya, R. Sakthivel, R. Sakthivel and F. Alzahrani, Dissipative-based non-fragile filtering for fuzzy networked control systems with switching communication channels, Appl. Math. Comput., 373 (2020), 125011, 16 pp. doi: 10.1016/j.amc.2019.125011.

[14]

H. RenS. Li and C. Lu, Event-triggered adaptive fault-tolerant control for multi-agent systems with unknown disturbances, Discrete Contin. Dyn. Syst.- S, 14 (2021), 1395-1414.  doi: 10.3934/dcdss.2020379.

[15]

R. SakthivelP. SelvarajK. Mathiyalagan and Ju. H. Park, Robust fault-tolerant $H_\infty$ control for offshore steel jacket platforms via sampled-data approach, J. Franklin Inst., 352 (2015), 2259-2279.  doi: 10.1016/j.jfranklin.2015.03.016.

[16]

T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Cybern., SMC-15 (1985), 116-132. 

[17]

X. TanJ. Cao and X. Li, Consensus of leader-following multiagent systems: A distributed event-triggered impulsive control strategy, IEEE Trans. Cybern., 49 (2018), 792-801.  doi: 10.1109/TCYB.2017.2786474.

[18]

S. Theesar and P. Balasubramaniam, Output tracking $H_\infty$ fuzzy control of nonlinear time-delay systems via fuzzy performance evaluator, J. Intell. Fuzzy Syst., 25 (2013), 177-190.  doi: 10.3233/IFS-2012-0624.

[19]

D. Wang, Robust policy learning control of nonlinear plants with case studies for a power system application, IEEE Trans. Ind. Inform., 16 (2020), 1733-1741.  doi: 10.1109/TII.2019.2925632.

[20]

K. WangE. TianJ. LiuL. Wei and D. Yue, Resilient control of networked control systems under deception attacks: A memory-event-triggered communication scheme, Int. J. Robust Nonlinear Control, 30 (2020), 1534-1548.  doi: 10.1002/rnc.4837.

[21]

Y. WuH. R. Karimi and R. Lu, Sampled-data control of network systems in industrial manufacturing, IEEE Trans. Ind. Electron., 65 (2018), 9016-9024.  doi: 10.1109/TIE.2018.2808903.

[22]

J. XiaG. ChenJu. H. ParkH. Shen and G. Zhuang, Dissipativity-based sampled-data control for fuzzy switched Markovian jump systems, IEEE Trans. Fuzzy Syst., 29 (2021), 1325-1339.  doi: 10.1109/TFUZZ.2020.2970856.

[23]

T. XuJ. XiaX. SongZ. Wang and H. Zhang, Sampled-data based dissipativity control of T-S fuzzy Markovian jump systems under actuator saturation with incomplete transition rates, Int. J. Control Autom Syst., 19 (2021), 632-645. 

[24]

H. YanT. WangH. Zhang and H. Shi, Event-triggered $H_\infty$ control for uncertain networked T-S fuzzy systems with time delay, Neurocomputing, 157 (2015), 273-279. 

[25]

F. YangH. ZhangG. Hui and S. Wang, Mode-independent fuzzy fault-tolerant variable sampling stabilization of nonlinear networked systems with both time-varying and random delays, Fuzzy Sets Syst., 207 (2012), 45-63.  doi: 10.1016/j.fss.2012.02.010.

[26]

Y. YangJ. XiaJ. ZhaoX. Li and Z. Wang, Multiobjective nonfragile fuzzy control for nonlinear stochastic financial systems with mixed time delays, Nonlinear Anal. Model. Control, 24 (2019), 696-717.  doi: 10.15388/na.2019.5.2.

[27]

W. ZhangQ.-L. HanY. Tang and Y. Liu, Sampled-data control for a class of linear time-varying systems, Automatica J. IFAC, 103 (2019), 126-134.  doi: 10.1016/j.automatica.2019.01.027.

[28]

X.-M. ZhangQ.-L. Han and B.-L. Zhang, An overview and deep investigation on sampled-data-based event-triggered control and filtering for networked systems, IEEE Trans. Industr. Inform., 13 (2017), 4-16.  doi: 10.1109/TII.2016.2607150.

[29]

Z.-H. ZhangD. LiuC. Deng and Q.-Y. Fan, A dynamic event-triggered resilient control approach to cyber-physical systems under asynchronous DoS attacks, Inf. Sci., 519 (2020), 260-272.  doi: 10.1016/j.ins.2020.01.047.

show all references

References:
[1]

X. Cai, S. Zhong, J. Wang and K. Shi, Robust $H_\infty$ control for uncertain delayed T-S fuzzy systems with stochastic packet dropouts, Appl. Math. Comput., 385 (2020), 125432, 21 pp. doi: 10.1016/j.amc.2020.125432.

[2]

Z. DuY. Kao and Ju. H. Park, Tracking control design for interval type-2 fuzzy nonlinear unreliable networked control systems, J. Franklin Inst., 358 (2021), 4159-4177.  doi: 10.1016/j.jfranklin.2021.03.016.

[3]

W. Guan and F. Liu, Non-fragile fuzzy dissipative static output feedback control for Markovian jump systems subject to actuator saturation, Neurocomputing, 193 (2016), 123-132.  doi: 10.1016/j.neucom.2016.02.004.

[4]

A. Hastir, J. J. Winkin and D. Dochain, Exponential stability of nonlinear infinite-dimensional systems: Application to nonisothermal axial dispersion tubular reactors, Automatica J. IFAC, 121 (2020), 109201, 15 pp. doi: 10.1016/j.automatica.2020.109201.

[5]

M. He and J. Li, Resilient guaranteed cost control for uncertain T-S fuzzy systems with time-varying delays and Markov jump parameters, ISA Trans., 88 (2019), 12-22.  doi: 10.1016/j.isatra.2018.11.034.

[6]

S. Kuppusamy and Y. H. Joo, Nonfragile retarded sampled-data switched control of T-S fuzzy systems and its applications, IEEE Trans. Fuzzy Syst., 28 (2020), 2523-2532.  doi: 10.1109/TFUZZ.2019.2940432.

[7]

X. LiD. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Autom. Control, 65 (2020), 4908-4913.  doi: 10.1109/TAC.2020.2964558.

[8]

X. Li and D. Ye, Membership-function-dependent security control for networked T-S fuzzy-model-based systems against DoS attacks, IET Control. Theory Appl., 15 (2021), 360-371.  doi: 10.1049/cth2.12048.

[9]

D. Liu and G.-H. Yang, Event-triggered control for linear systems with actuator saturation and disturbances, IET Control. Theory Appl., 11 (2017), 1351-1359.  doi: 10.1049/iet-cta.2016.1661.

[10]

J. LiuZ.-G. WuD. Yue and Ju H. Park, Stabilization of networked control systems with hybrid-driven mechanism and probabilistic cyber attacks, IEEE Trans. Syst. Man Cybern. Syst., 51 (2021), 943-953.  doi: 10.1109/TSMC.2018.2888633.

[11]

X. Liu and S. Zhong, Stability analysis of delayed switched cascade nonlinear systems with uniform switching signals, Math. Model. Control, 1 (2021), 90-101.  doi: 10.3934/mmc.2021007.

[12]

Y. Liu and Y. Ma, Finite-time non-fragile extended dissipative control for T-S fuzzy system via augmented Lyapunov-Krasovskii functional, ISA Trans., 117 (2021), 1-15.  doi: 10.1016/j.isatra.2021.01.038.

[13]

V. Nithya, R. Sakthivel, R. Sakthivel and F. Alzahrani, Dissipative-based non-fragile filtering for fuzzy networked control systems with switching communication channels, Appl. Math. Comput., 373 (2020), 125011, 16 pp. doi: 10.1016/j.amc.2019.125011.

[14]

H. RenS. Li and C. Lu, Event-triggered adaptive fault-tolerant control for multi-agent systems with unknown disturbances, Discrete Contin. Dyn. Syst.- S, 14 (2021), 1395-1414.  doi: 10.3934/dcdss.2020379.

[15]

R. SakthivelP. SelvarajK. Mathiyalagan and Ju. H. Park, Robust fault-tolerant $H_\infty$ control for offshore steel jacket platforms via sampled-data approach, J. Franklin Inst., 352 (2015), 2259-2279.  doi: 10.1016/j.jfranklin.2015.03.016.

[16]

T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Cybern., SMC-15 (1985), 116-132. 

[17]

X. TanJ. Cao and X. Li, Consensus of leader-following multiagent systems: A distributed event-triggered impulsive control strategy, IEEE Trans. Cybern., 49 (2018), 792-801.  doi: 10.1109/TCYB.2017.2786474.

[18]

S. Theesar and P. Balasubramaniam, Output tracking $H_\infty$ fuzzy control of nonlinear time-delay systems via fuzzy performance evaluator, J. Intell. Fuzzy Syst., 25 (2013), 177-190.  doi: 10.3233/IFS-2012-0624.

[19]

D. Wang, Robust policy learning control of nonlinear plants with case studies for a power system application, IEEE Trans. Ind. Inform., 16 (2020), 1733-1741.  doi: 10.1109/TII.2019.2925632.

[20]

K. WangE. TianJ. LiuL. Wei and D. Yue, Resilient control of networked control systems under deception attacks: A memory-event-triggered communication scheme, Int. J. Robust Nonlinear Control, 30 (2020), 1534-1548.  doi: 10.1002/rnc.4837.

[21]

Y. WuH. R. Karimi and R. Lu, Sampled-data control of network systems in industrial manufacturing, IEEE Trans. Ind. Electron., 65 (2018), 9016-9024.  doi: 10.1109/TIE.2018.2808903.

[22]

J. XiaG. ChenJu. H. ParkH. Shen and G. Zhuang, Dissipativity-based sampled-data control for fuzzy switched Markovian jump systems, IEEE Trans. Fuzzy Syst., 29 (2021), 1325-1339.  doi: 10.1109/TFUZZ.2020.2970856.

[23]

T. XuJ. XiaX. SongZ. Wang and H. Zhang, Sampled-data based dissipativity control of T-S fuzzy Markovian jump systems under actuator saturation with incomplete transition rates, Int. J. Control Autom Syst., 19 (2021), 632-645. 

[24]

H. YanT. WangH. Zhang and H. Shi, Event-triggered $H_\infty$ control for uncertain networked T-S fuzzy systems with time delay, Neurocomputing, 157 (2015), 273-279. 

[25]

F. YangH. ZhangG. Hui and S. Wang, Mode-independent fuzzy fault-tolerant variable sampling stabilization of nonlinear networked systems with both time-varying and random delays, Fuzzy Sets Syst., 207 (2012), 45-63.  doi: 10.1016/j.fss.2012.02.010.

[26]

Y. YangJ. XiaJ. ZhaoX. Li and Z. Wang, Multiobjective nonfragile fuzzy control for nonlinear stochastic financial systems with mixed time delays, Nonlinear Anal. Model. Control, 24 (2019), 696-717.  doi: 10.15388/na.2019.5.2.

[27]

W. ZhangQ.-L. HanY. Tang and Y. Liu, Sampled-data control for a class of linear time-varying systems, Automatica J. IFAC, 103 (2019), 126-134.  doi: 10.1016/j.automatica.2019.01.027.

[28]

X.-M. ZhangQ.-L. Han and B.-L. Zhang, An overview and deep investigation on sampled-data-based event-triggered control and filtering for networked systems, IEEE Trans. Industr. Inform., 13 (2017), 4-16.  doi: 10.1109/TII.2016.2607150.

[29]

Z.-H. ZhangD. LiuC. Deng and Q.-Y. Fan, A dynamic event-triggered resilient control approach to cyber-physical systems under asynchronous DoS attacks, Inf. Sci., 519 (2020), 260-272.  doi: 10.1016/j.ins.2020.01.047.

Figure 1.  State responses curve of T-S fuzzy system (1) with control
Figure 2.  State responses curve of T-S fuzzy system (1) without control
Figure 3.  Control Curve
Figure 4.  The event-triggered instants and intervals
Figure 5.  State responses curve of T-S fuzzy system (1) with control
Figure 6.  State responses curve of T-S fuzzy system (1) without control
Figure 7.  Control responses
Figure 8.  The event-triggered instants and intervals
Table 1.  Calculated upper bound of $ \bar{k} $ for different values of $ \delta $
$ \delta $ 0.05 0.08 0.10 0.15 0.18
$ \bar{k} $ 0.3410 0.3312 0.3201 0.3050 0.3016
$ \delta $ 0.05 0.08 0.10 0.15 0.18
$ \bar{k} $ 0.3410 0.3312 0.3201 0.3050 0.3016
Table 2.  Calculated minimum value of $ \gamma $ for various values of $ \bar{k} $
$ \bar{k} $ 0.1 0.13 0.15 0.25 0.30
$ \gamma $ 0.2236 0.2491 0.2743 0.6646 1.6538
$ \bar{k} $ 0.1 0.13 0.15 0.25 0.30
$ \gamma $ 0.2236 0.2491 0.2743 0.6646 1.6538
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