April  2021, 14(4): 1329-1343. doi: 10.3934/dcdss.2020368

$ \mathcal{H}_{\infty} $ control for fuzzy markovian jump systems based on sampled-data control method

1. 

Liaocheng University, School of Mathematics Science, Liaocheng 252000, P. R. China

2. 

Shandong University of Science and Technology, College of Mathematics and Systems Science, Qingdao 266590, China

* Corresponding author: Jianwei Xia

Received  August 2019 Revised  January 2020 Published  April 2021 Early access  May 2020

Fund Project: The first author is supported by the National Natural Science Foundation of China under Grants 61573177, 61773191, 61973148

This paper investigates the problems of $ \mathcal{H}_{\infty} $ performance analysis and sampled-data control about fuzzy Markovian jump systems. Firstly, in order to make full use of the information of both intervals $ x(t_{k}) $ to $ x(t) $ and $ x(t) $ to $ x(t_{k+1}) $, we construct the mode-dependent Lyapunov function, which consists of a two-sided closed-loop function. Built on the above Lyapunov function, the stochastically stable conditions with less conservative are given by using linear matrices inequalities (LMIs). Then, a state feedback controller is presented for the studied systems. At last, an example is offered to illustrate the efficiency of our main results.

Citation: Xingyue Liang, Jianwei Xia, Guoliang Chen, Huasheng Zhang, Zhen Wang. $ \mathcal{H}_{\infty} $ control for fuzzy markovian jump systems based on sampled-data control method. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1329-1343. doi: 10.3934/dcdss.2020368
References:
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X. Chang and G. Yang, Nonfragile $H_{\infty}$ filtering of continuous-time fuzzy systems, IEEE Transactions on Signal Processing, 59 (2011), 1528-1538.  doi: 10.1109/TSP.2010.2103068.

[2]

X. Chang, Robust nonfragile $H_{\infty}$ filtering of fuzzy systems with linear fractional parametric uncertainties, IEEE Transactions on Fuzzy Systems, 20 (2012), 1001-1011. 

[3]

G. L. ChenJ. Sun and J. Chen, Mean square exponential stabilization of sampled-data Markovian jump systems, Int J Robust Nonlinear Control, 28 (2018), 5876-5894.  doi: 10.1002/rnc.4351.

[4]

G. ChenJ. Xia and G. Zhuang, Delay-dependent stability and dissipativity analysis of generalized neural networks with Markovian jump parameters and two delay components, J. Frankl. Inst, 353 (2016), 2137-2158.  doi: 10.1016/j.jfranklin.2016.02.020.

[5]

L. S. HuP. Shi and P. M. Frank, Robust sampled-data control for Markovian jump linear systems, Automatica, 42 (2006), 2025-2030.  doi: 10.1016/j.automatica.2006.05.029.

[6]

J. LengH. ZhangD. YanQ. LiuX. Chen and D. Zhang, Digital twin-driven manufacturing cyber-physical system for parallel controlling of smart workshop, Journal of Ambient Intelligence and Humanized Computing, 10 (2019), 1155-1166.  doi: 10.1007/s12652-018-0881-5.

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X. LiX. Yang and T. Huang, Persistence of delayed cooperative models: Impulsive control method, Applied Mathematics and Computation, 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.

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X. LiJ. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Applied Mathematics and Computation, 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.

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X. Li and M. Bohner, An impulsive delay differential inequality and applications, Computers and Mathematics with Applications, 64 (2012), 1875-1881.  doi: 10.1016/j.camwa.2012.03.013.

[10]

X. LiangJ. XiaG. ChenH. Zhang and Z. Wang, Dissipativity-based sampled-data control for fuzzy Markovian jump systems, Applied Mathematics and Computation, 361 (2019), 552-564.  doi: 10.1016/j.amc.2019.05.038.

[11]

F. LiP. ShiC. Lim and L. Wu, Fault detection filtering for nonhomogeneous markovian jump systems via a fuzzy approach, IEEE Transactions on Fuzzy Systems, 26 (2018), 131-141.  doi: 10.1109/TFUZZ.2016.2641022.

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C. Lin, G. Wang, T. Lee and Y. He, LMI Approach to Analysis and Control of Takagi-Sugeno Fuzzy Systems With Time Delay, Lecture Notes in Control and Information Sciences, 351. Springer, Berlin, 2007.

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X. LiangJ. XiaG. ChenH. Zhang and Z. Wang, Dissipativity-based non-fragile sampled-data control for fuzzy Markovian jump systems, Int. J. Fuzzy Syst., 21 (2019), 1709-1723.  doi: 10.1007/s40815-019-00691-1.

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J. H. Park, H. Shen, X. H. Chang and T. H. Lee, Recent Advances in Control and Filtering of Dynamic Systems with Constrained Signals, Cham, Switzerland: Springer, 2019. doi: 10.1007/978-3-319-96202-3.

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H. ShenJ. H. ParkL. Zhang and Z. G. Wu, Robust extended dissipative control for sampled-data Markov jump systems, Int J Control, 87 (2014), 1549-1564.  doi: 10.1080/00207179.2013.878478.

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P. ShiF. LiL. Wu and C. C. Lim, $h_{\infty}$ Neural network-based passive filtering for delayed neutral-type semi-Markovian jump systems, IEEE Trans Neural Netw Learn Syst, 28 (2017), 2101-2114. 

[18]

X. Song, Z. Wang and H. Shen, et al, A unified method to energy-to-peak filter design for networked Markov switched singular systems over a finite-time interval, Journal of the Franklin Institute, 354 (2017), 7899–7916. doi: 10.1016/j.jfranklin.2017.09.018.

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[20]

X. SongS. Song and Bo Li, Adaptive projective synchronization for time-delayed fractional-order neural networks with uncertain parameters and its application in secure communications, Transactions of the Institute of Measurement and Control, 40 (2018), 3078-3087.  doi: 10.1177/0142331217714523.

[21]

S. Song and X. Song, Multi-switching adaptive synchronization of two fractional-order chaotic systems with different structure and different order, International Journal of Control, Automation and Systems, 15 (2017), 1524-1535.  doi: 10.1007/s12555-016-0097-4.

[22]

W. SunJ. XiaG. ZhuangX. Huang and H. Shen, Adaptive fuzzy asymptotically tracking control of full state constrained nonlinear system based on a novel Nussbaum-type function, Journal of the Franklin Institute, 356 (2019), 1810-1827.  doi: 10.1016/j.jfranklin.2018.11.023.

[23]

W. Sun, S. Su, Y. Wu, J. Xia and V. Nguyen, Adaptive fuzzy control with high-order barrier Lyapunov functions for high-order uncertain nonlinear systems with full-state constraints, IEEE Transactions on Cybernetics, 2019, 1–9. doi: 10.1109/TCYB.2018.2890256.

[24]

W. SunS. SuJ. Xia and V. Nguyen, Adaptive fuzzy tracking control of flexible-joint robots with full-state constraints, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49 (2019), 2201-2209.  doi: 10.1109/TSMC.2018.2870642.

[25]

W. Sun, S. Su, G. Dong and W. Bai, Reduced adaptive fuzzy tracking control for high-order stochastic nonstrict feedback nonlinear system with full-state constraints, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019, 1–11. doi: 10.1109/TSMC.2019.2898204.

[26]

W. Sun, S. Su, J. Xia and Y. Wu, Adaptive tracking control of wheeled inverted pendulums with periodic disturbances, IEEE Transactions on Cybernetics, 50 (2020), 1867–1876. doi: 10.1109/TCYB.2018.2884707.

[27]

H. ShenZ. WangX. Huang and J. Wang, Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, J. Frankl.Inst, 351 (2014), 3797-3817.  doi: 10.1016/j.jfranklin.2013.02.031.

[28]

H. ShenY. Z. MenZ. G. Wu and J. H. Park, Nonfragile $\mathcal{H}_{\infty}$ control for fuzzy Markovian jump systems under fast sampling singular perturbation, IEEE Transactions on Fuzzy Systems, 48 (2018), 2058-2069. 

[29]

H. ShenF. LiH. YanH. Karimi and H. Lam, Finite-time event-triggered $\mathcal{H}_{\infty}$ control for T-S fuzzy Markov jump systems, IEEE Transactions on Fuzzy Systems, 26 (2018), 3122-3135. 

[30]

J. WangH. WuL. Guo and Y. Luo, Robust $H_{\infty}$ fuzzy control for uncertain nonlinear Markovian jump systems with time-varying delay, Fuzzy Sets and Systems, 212 (2013), 41-61.  doi: 10.1016/j.fss.2012.07.010.

[31]

Z. G. WuP. ShiH. Su and J. Chu, Asynchronous $l_{2}-l_{\infty}$ filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities, Automatica, 50 (2014), 180-186.  doi: 10.1016/j.automatica.2013.09.041.

[32]

Z. WuP. ShiH. Su and R. Lu, Dissipativity-based sampled-data fuzzy control design and its application to truck-trailer system, IEEE Transactions on Fuzzy Systems, 23 (2015), 1669-1679.  doi: 10.1109/TFUZZ.2014.2374192.

[33]

J. XiaG. Chen and W. Sun, Extended dissipative analysis of generalized Markovian switching neural networks with two delay components, Neurocomputing, 260 (2017), 275-283.  doi: 10.1016/j.neucom.2017.05.005.

[34]

J. Xia, J. Zhang, J. Feng, Z. Wang and G. Zhuang, Command filter-based adaptive fuzzy control for nonlinear systems with unknown control directions, IEEE Transactions on Systems, Man and Cybernetics: Systems, In Press.

[35]

J. XiaJ. ZhangW. SunB. Y. Zhang and Z. Wang, Finite-time adaptive fuzzy control for nonlinear systems with full state constraints, IEEE Transactions on Systems, Man and Cybernetics: Systems, 49 (2019), 1541-1548.  doi: 10.1109/TSMC.2018.2854770.

[36]

S. Y. XuJ. Lam and X. R. Mao, Delay-dependent $H_{\infty}$ control and filtering for uncertain markovian jump systems with time-varying delays, IEEE Transactions on Circuits and Systems, 54 (2007), 2070-2077.  doi: 10.1109/TCSI.2007.904640.

[37]

D. YangX. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Analysis: Hybrid Systems, 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.

[38]

X. YangX. LiQ. Xi and P. Duan, Review of stability and stabilization for impulsive delayed systems, Mathematical Biosciences and Engineering, 15 (2018), 1495-1515.  doi: 10.3934/mbe.2018069.

[39]

H. ZengK. TeoY. HeH. Xu and W. Wang, Sampled-data synchronization control for chaotic neural networks subject to actuator saturation, Neurocomputing, 260 (2017), 25-31.  doi: 10.1016/j.neucom.2017.02.063.

[40]

H. ZengY. HeM. Wu and J. She, Free-matrix-based integral inequality for stability analysis of systems with time-varying delay, IEEE Trans. Automat. Contr, 60 (2015), 2768-2772.  doi: 10.1109/TAC.2015.2404271.

[41]

H. ZengK. Teo and Y. He, A new looped-functional for stability analysis of sampled-data systems, Automatica, 82 (2017), 328-331.  doi: 10.1016/j.automatica.2017.04.051.

[42]

H. ZengK. TeoY. He and W. Wang, Sampled-data-based dissipative control of T-S fuzzy systems, Applied Mathematical Modelling, 65 (2019), 415-427.  doi: 10.1016/j.apm.2018.08.012.

[43]

G. ZhuangJ. XiaW. SunQ. MaZ. Wang and Y. Wang, Normalization and stabilization of neutral descriptor hybrid systems based on P-D feedback control, Journal of the Franklin Institute, 357 (2020), 1070-1089.  doi: 10.1016/j.jfranklin.2019.10.020.

[44]

B. ZhangW. X. Zheng and S. Xu, $h_{\infty}$ Filtering of Markovian jump delay systems based on a new performance index, IEEE Trans Circuits Syst I Reg Pap, 60 (2013), 1250-1263.  doi: 10.1109/TCSI.2013.2246213.

[45]

J. ZhangJ. XiaW. SunG. Zhuang and Z. Wang, Finite-time tracking control for stochastic nonlinear systems with full state constraints, Applied Mathematics and Computation, 338 (2018), 207-220.  doi: 10.1016/j.amc.2018.05.040.

[46]

J. ZhangX. Liang and J. Xia, Adaptive tracking control for stochastic nonlinear systems with full state constraints, Journal of Liaocheng University (Natural Science Edition), 32 (2019), 8-13. 

[47]

G. ZhuangS. XuJ. XiaQ. Ma and Z. Zhang, Non-fragile delay feedback control for neutral stochastic Markovian jump systems with time-varying delays, Applied Mathematics and Computation, 355 (2019), 21-32.  doi: 10.1016/j.amc.2019.02.057.

[48]

G. ZhuangQ. MaB. ZhangS. Xu and J. Xia, Admissibility and stabilization of stochastic singular Markovian jump systems with time delays, Systems and Control Letters, 114 (2018), 1-10.  doi: 10.1016/j.sysconle.2018.02.004.

[49]

G. ZhuangS. XuB. ZhangH. Xu and Y. Chu, Robust $H_{\infty}$ deconvolution filtering for uncertain singular Markovian jump systems with time-varying delays, International Journal of Robust and Nonlinear Control, 26 (2016), 2564-2585.  doi: 10.1002/rnc.3461.

[50]

G. ZhuangS. XuJ. XiaQ. Ma and Z. Zhang, Non-fragile delay feedback control for neutra stochastic Markovian jump systems with time-varying delays, Applied Mathematics and Computation, 355 (2019), 21-32.  doi: 10.1016/j.amc.2019.02.057.

[51]

G. ZhuangJ. XiaJ. FengW. Sun and B. Zhang, Admissibilization for implicit jump systems with mixed retarded delays based on reciprocally convex integral inequality and Barbalat's lemma, IEEE Trans. Syst., Man, Cybern., 16 (2020), 1-11.  doi: 10.1109/TSMC.2020.2964057.

show all references

References:
[1]

X. Chang and G. Yang, Nonfragile $H_{\infty}$ filtering of continuous-time fuzzy systems, IEEE Transactions on Signal Processing, 59 (2011), 1528-1538.  doi: 10.1109/TSP.2010.2103068.

[2]

X. Chang, Robust nonfragile $H_{\infty}$ filtering of fuzzy systems with linear fractional parametric uncertainties, IEEE Transactions on Fuzzy Systems, 20 (2012), 1001-1011. 

[3]

G. L. ChenJ. Sun and J. Chen, Mean square exponential stabilization of sampled-data Markovian jump systems, Int J Robust Nonlinear Control, 28 (2018), 5876-5894.  doi: 10.1002/rnc.4351.

[4]

G. ChenJ. Xia and G. Zhuang, Delay-dependent stability and dissipativity analysis of generalized neural networks with Markovian jump parameters and two delay components, J. Frankl. Inst, 353 (2016), 2137-2158.  doi: 10.1016/j.jfranklin.2016.02.020.

[5]

L. S. HuP. Shi and P. M. Frank, Robust sampled-data control for Markovian jump linear systems, Automatica, 42 (2006), 2025-2030.  doi: 10.1016/j.automatica.2006.05.029.

[6]

J. LengH. ZhangD. YanQ. LiuX. Chen and D. Zhang, Digital twin-driven manufacturing cyber-physical system for parallel controlling of smart workshop, Journal of Ambient Intelligence and Humanized Computing, 10 (2019), 1155-1166.  doi: 10.1007/s12652-018-0881-5.

[7]

X. LiX. Yang and T. Huang, Persistence of delayed cooperative models: Impulsive control method, Applied Mathematics and Computation, 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.

[8]

X. LiJ. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Applied Mathematics and Computation, 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.

[9]

X. Li and M. Bohner, An impulsive delay differential inequality and applications, Computers and Mathematics with Applications, 64 (2012), 1875-1881.  doi: 10.1016/j.camwa.2012.03.013.

[10]

X. LiangJ. XiaG. ChenH. Zhang and Z. Wang, Dissipativity-based sampled-data control for fuzzy Markovian jump systems, Applied Mathematics and Computation, 361 (2019), 552-564.  doi: 10.1016/j.amc.2019.05.038.

[11]

F. LiP. ShiC. Lim and L. Wu, Fault detection filtering for nonhomogeneous markovian jump systems via a fuzzy approach, IEEE Transactions on Fuzzy Systems, 26 (2018), 131-141.  doi: 10.1109/TFUZZ.2016.2641022.

[12]

C. Lin, G. Wang, T. Lee and Y. He, LMI Approach to Analysis and Control of Takagi-Sugeno Fuzzy Systems With Time Delay, Lecture Notes in Control and Information Sciences, 351. Springer, Berlin, 2007.

[13]

X. LiangJ. XiaG. ChenH. Zhang and Z. Wang, Dissipativity-based non-fragile sampled-data control for fuzzy Markovian jump systems, Int. J. Fuzzy Syst., 21 (2019), 1709-1723.  doi: 10.1007/s40815-019-00691-1.

[14]

J. H. Park, H. Shen, X. H. Chang and T. H. Lee, Recent Advances in Control and Filtering of Dynamic Systems with Constrained Signals, Cham, Switzerland: Springer, 2019. doi: 10.1007/978-3-319-96202-3.

[15]

J. H. Park, T. H. Lee, Y. Liu and J. Chen, Dynamic Systems with Time Delays: Stability and Control, Singapore, Springer-Nature, 2019. doi: 10.1007/978-981-13-9254-2.

[16]

H. ShenJ. H. ParkL. Zhang and Z. G. Wu, Robust extended dissipative control for sampled-data Markov jump systems, Int J Control, 87 (2014), 1549-1564.  doi: 10.1080/00207179.2013.878478.

[17]

P. ShiF. LiL. Wu and C. C. Lim, $h_{\infty}$ Neural network-based passive filtering for delayed neutral-type semi-Markovian jump systems, IEEE Trans Neural Netw Learn Syst, 28 (2017), 2101-2114. 

[18]

X. Song, Z. Wang and H. Shen, et al, A unified method to energy-to-peak filter design for networked Markov switched singular systems over a finite-time interval, Journal of the Franklin Institute, 354 (2017), 7899–7916. doi: 10.1016/j.jfranklin.2017.09.018.

[19]

X. Song, M. Wang and S. Song, et al, Reliable state estimation for Markovian jump reactiondiffusion neural networks with sensor saturation and asynchronous failure, IEEE Access, 6 (2018), 50066–50076. doi: 10.1109/ACCESS.2018.2868060.

[20]

X. SongS. Song and Bo Li, Adaptive projective synchronization for time-delayed fractional-order neural networks with uncertain parameters and its application in secure communications, Transactions of the Institute of Measurement and Control, 40 (2018), 3078-3087.  doi: 10.1177/0142331217714523.

[21]

S. Song and X. Song, Multi-switching adaptive synchronization of two fractional-order chaotic systems with different structure and different order, International Journal of Control, Automation and Systems, 15 (2017), 1524-1535.  doi: 10.1007/s12555-016-0097-4.

[22]

W. SunJ. XiaG. ZhuangX. Huang and H. Shen, Adaptive fuzzy asymptotically tracking control of full state constrained nonlinear system based on a novel Nussbaum-type function, Journal of the Franklin Institute, 356 (2019), 1810-1827.  doi: 10.1016/j.jfranklin.2018.11.023.

[23]

W. Sun, S. Su, Y. Wu, J. Xia and V. Nguyen, Adaptive fuzzy control with high-order barrier Lyapunov functions for high-order uncertain nonlinear systems with full-state constraints, IEEE Transactions on Cybernetics, 2019, 1–9. doi: 10.1109/TCYB.2018.2890256.

[24]

W. SunS. SuJ. Xia and V. Nguyen, Adaptive fuzzy tracking control of flexible-joint robots with full-state constraints, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49 (2019), 2201-2209.  doi: 10.1109/TSMC.2018.2870642.

[25]

W. Sun, S. Su, G. Dong and W. Bai, Reduced adaptive fuzzy tracking control for high-order stochastic nonstrict feedback nonlinear system with full-state constraints, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019, 1–11. doi: 10.1109/TSMC.2019.2898204.

[26]

W. Sun, S. Su, J. Xia and Y. Wu, Adaptive tracking control of wheeled inverted pendulums with periodic disturbances, IEEE Transactions on Cybernetics, 50 (2020), 1867–1876. doi: 10.1109/TCYB.2018.2884707.

[27]

H. ShenZ. WangX. Huang and J. Wang, Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, J. Frankl.Inst, 351 (2014), 3797-3817.  doi: 10.1016/j.jfranklin.2013.02.031.

[28]

H. ShenY. Z. MenZ. G. Wu and J. H. Park, Nonfragile $\mathcal{H}_{\infty}$ control for fuzzy Markovian jump systems under fast sampling singular perturbation, IEEE Transactions on Fuzzy Systems, 48 (2018), 2058-2069. 

[29]

H. ShenF. LiH. YanH. Karimi and H. Lam, Finite-time event-triggered $\mathcal{H}_{\infty}$ control for T-S fuzzy Markov jump systems, IEEE Transactions on Fuzzy Systems, 26 (2018), 3122-3135. 

[30]

J. WangH. WuL. Guo and Y. Luo, Robust $H_{\infty}$ fuzzy control for uncertain nonlinear Markovian jump systems with time-varying delay, Fuzzy Sets and Systems, 212 (2013), 41-61.  doi: 10.1016/j.fss.2012.07.010.

[31]

Z. G. WuP. ShiH. Su and J. Chu, Asynchronous $l_{2}-l_{\infty}$ filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities, Automatica, 50 (2014), 180-186.  doi: 10.1016/j.automatica.2013.09.041.

[32]

Z. WuP. ShiH. Su and R. Lu, Dissipativity-based sampled-data fuzzy control design and its application to truck-trailer system, IEEE Transactions on Fuzzy Systems, 23 (2015), 1669-1679.  doi: 10.1109/TFUZZ.2014.2374192.

[33]

J. XiaG. Chen and W. Sun, Extended dissipative analysis of generalized Markovian switching neural networks with two delay components, Neurocomputing, 260 (2017), 275-283.  doi: 10.1016/j.neucom.2017.05.005.

[34]

J. Xia, J. Zhang, J. Feng, Z. Wang and G. Zhuang, Command filter-based adaptive fuzzy control for nonlinear systems with unknown control directions, IEEE Transactions on Systems, Man and Cybernetics: Systems, In Press.

[35]

J. XiaJ. ZhangW. SunB. Y. Zhang and Z. Wang, Finite-time adaptive fuzzy control for nonlinear systems with full state constraints, IEEE Transactions on Systems, Man and Cybernetics: Systems, 49 (2019), 1541-1548.  doi: 10.1109/TSMC.2018.2854770.

[36]

S. Y. XuJ. Lam and X. R. Mao, Delay-dependent $H_{\infty}$ control and filtering for uncertain markovian jump systems with time-varying delays, IEEE Transactions on Circuits and Systems, 54 (2007), 2070-2077.  doi: 10.1109/TCSI.2007.904640.

[37]

D. YangX. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Analysis: Hybrid Systems, 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.

[38]

X. YangX. LiQ. Xi and P. Duan, Review of stability and stabilization for impulsive delayed systems, Mathematical Biosciences and Engineering, 15 (2018), 1495-1515.  doi: 10.3934/mbe.2018069.

[39]

H. ZengK. TeoY. HeH. Xu and W. Wang, Sampled-data synchronization control for chaotic neural networks subject to actuator saturation, Neurocomputing, 260 (2017), 25-31.  doi: 10.1016/j.neucom.2017.02.063.

[40]

H. ZengY. HeM. Wu and J. She, Free-matrix-based integral inequality for stability analysis of systems with time-varying delay, IEEE Trans. Automat. Contr, 60 (2015), 2768-2772.  doi: 10.1109/TAC.2015.2404271.

[41]

H. ZengK. Teo and Y. He, A new looped-functional for stability analysis of sampled-data systems, Automatica, 82 (2017), 328-331.  doi: 10.1016/j.automatica.2017.04.051.

[42]

H. ZengK. TeoY. He and W. Wang, Sampled-data-based dissipative control of T-S fuzzy systems, Applied Mathematical Modelling, 65 (2019), 415-427.  doi: 10.1016/j.apm.2018.08.012.

[43]

G. ZhuangJ. XiaW. SunQ. MaZ. Wang and Y. Wang, Normalization and stabilization of neutral descriptor hybrid systems based on P-D feedback control, Journal of the Franklin Institute, 357 (2020), 1070-1089.  doi: 10.1016/j.jfranklin.2019.10.020.

[44]

B. ZhangW. X. Zheng and S. Xu, $h_{\infty}$ Filtering of Markovian jump delay systems based on a new performance index, IEEE Trans Circuits Syst I Reg Pap, 60 (2013), 1250-1263.  doi: 10.1109/TCSI.2013.2246213.

[45]

J. ZhangJ. XiaW. SunG. Zhuang and Z. Wang, Finite-time tracking control for stochastic nonlinear systems with full state constraints, Applied Mathematics and Computation, 338 (2018), 207-220.  doi: 10.1016/j.amc.2018.05.040.

[46]

J. ZhangX. Liang and J. Xia, Adaptive tracking control for stochastic nonlinear systems with full state constraints, Journal of Liaocheng University (Natural Science Edition), 32 (2019), 8-13. 

[47]

G. ZhuangS. XuJ. XiaQ. Ma and Z. Zhang, Non-fragile delay feedback control for neutral stochastic Markovian jump systems with time-varying delays, Applied Mathematics and Computation, 355 (2019), 21-32.  doi: 10.1016/j.amc.2019.02.057.

[48]

G. ZhuangQ. MaB. ZhangS. Xu and J. Xia, Admissibility and stabilization of stochastic singular Markovian jump systems with time delays, Systems and Control Letters, 114 (2018), 1-10.  doi: 10.1016/j.sysconle.2018.02.004.

[49]

G. ZhuangS. XuB. ZhangH. Xu and Y. Chu, Robust $H_{\infty}$ deconvolution filtering for uncertain singular Markovian jump systems with time-varying delays, International Journal of Robust and Nonlinear Control, 26 (2016), 2564-2585.  doi: 10.1002/rnc.3461.

[50]

G. ZhuangS. XuJ. XiaQ. Ma and Z. Zhang, Non-fragile delay feedback control for neutra stochastic Markovian jump systems with time-varying delays, Applied Mathematics and Computation, 355 (2019), 21-32.  doi: 10.1016/j.amc.2019.02.057.

[51]

G. ZhuangJ. XiaJ. FengW. Sun and B. Zhang, Admissibilization for implicit jump systems with mixed retarded delays based on reciprocally convex integral inequality and Barbalat's lemma, IEEE Trans. Syst., Man, Cybern., 16 (2020), 1-11.  doi: 10.1109/TSMC.2020.2964057.

Table 1.  $ \gamma_{max} $ for $ h_{min} = 0 $ and different $ h_{max} $
$ h_{max} $ 0.05 0.15 0.25 0.35
$ \gamma $ $ 1.7320 $ 1.7678 1.8246 1.9285
$ h_{max} $ 0.05 0.15 0.25 0.35
$ \gamma $ $ 1.7320 $ 1.7678 1.8246 1.9285
Table 2.  $ \gamma_{max} $ for $ h_{max} = h_{min} $
$ h $ 0.05 0.15 0.25 0.35
$ \gamma $ $ 1.7299 $ 1.7576 1.7982 1.8659
$ h $ 0.05 0.15 0.25 0.35
$ \gamma $ $ 1.7299 $ 1.7576 1.7982 1.8659
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