Event-triggered mixed <inline-formula><tex-math id="M1">\begin{document}$ H_\infty $\end{document}</tex-math></inline-formula> and passive control for Markov jump systems with bounded inputs

Liqiang Jin; Yanyan Yin; Kok Lay Teo; Fei Liu

doi:10.3934/jimo.2020024

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doi: 10.3934/jimo.2020024

Event-triggered mixed $ H_\infty $ and passive control for Markov jump systems with bounded inputs

Liqiang Jin ^1, , Yanyan Yin ^2, , Kok Lay Teo ^2,3, and Fei Liu ^1,,

1.	Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Automation, School of Internet of Things Engineering, Jiangnan University, Wuxi, 214122, China
2.	School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth, Western Australia, 6102, Australia
3.	Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin, China

^* Corresponding author: Fei Liu

Received November 2018 Revised August 2019 Published February 2020

Fund Project: The first author is supported in part by the National Natural Science Foundation of China under grant nos. 61773011, 61773183, NSFC 61833007, the Ministry of Education of China under the 111 Project B12018 and Curtin Fellowship

In this brief, the problem of event-triggered mixed $ {H_\infty } $ and passive control for a class of discrete-time stochastic Markov jump systems with bounded inputs is addressed. In order to reduce the frequency of the variation of the controller, an effective triggered scheme, called event-triggered scheme, is proposed, where unlike the traditional triggered scheme, not all sampling states are required to be transmitted to the controller. The event-triggered controller, which is designed using Lyapunov functional analysis approach and slack matrices, can ensure that the resulting system is stochastically stable with a prescribed mixed $ {H_\infty } $ and passive performance index. Sufficient conditions in terms of linear matrix inequalities (LMIs) are derived. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

Keywords: Event-triggered, mixed $ {H_\infty } $ and passive control, Markov jump systems, input constraints.

Mathematics Subject Classification: Primary: 93C55, 93C65; Secondary: 37C75.

Citation: Liqiang Jin, Yanyan Yin, Kok Lay Teo, Fei Liu. Event-triggered mixed $ H_\infty $ and passive control for Markov jump systems with bounded inputs. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020024

References:

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[5]	H. Dong, Z. Wang, D. W. C. Ho and H. Gao, Robust $ H_{\infty} $ Filtering for Markovian Jump Systems With Randomly Occurring Nonlinearities and Sensor Saturation: The Finite-Horizon Case, IEEE Transactions on Signal Processing, 59 (2011), 3048-3057. doi: 10.1109/TSP.2011.2135854. Google Scholar
[6]	Y. Dong, E. Tian and Q. L. Han, A delay system method for designing event-triggered controllers of networked control systems, IEEE Transactions on Automatic Control, 58 (2013), 475-481. doi: 10.1109/TAC.2012.2206694. Google Scholar
[7]	S. He, Non-fragile passive controller design for nonlinear Markovian jumping systems via observer-based controls, Neurocomputing, 147 (2015), 350-357. doi: 10.1016/j.neucom.2014.06.053. Google Scholar
[8]	W. P. M. H. Heemels, M. C. F. Donkers and A. R. Teel, Periodic event-triggered control for linear systems, IEEE Transactions on Automatic Control, 58 (2013), 847-861. doi: 10.1109/TAC.2012.2220443. Google Scholar
[9]	F. Li, P. Lim, C. C. Shi and L. Wu, Fault detection filtering for nonhomogeneous Markovian jump systems via fuzzy approach, IEEE Transactions on Fuzzy Systems, 26 (2018), 131-141. doi: 10.1109/TFUZZ.2016.2641022. Google Scholar
[10]	H. Li, Z. Chen, L. Wu, H. K. Lam and H Du, Event-triggered fault detection of nonlinear networked systems, IEEE Transactions on Cybernetics, 47 (2017), 1041-1052. doi: 10.1109/TCYB.2016.2536750. Google Scholar
[11]	H. Li, Y. Wang, D. Yao and R. Lu, A sliding mode approach to stabilization of nonlinear Markovian jump singularly perturbed systems, Automatic, 97 (2018), 404-413. doi: 10.1016/j.automatica.2018.03.066. Google Scholar
[12]	H. Liang, L. Zhang, H. R. Karimi and Q. Zhou, Fault estimation for a class of nonlinear sem-Markovian jump systems with partly unknown transition rates and output quantization, International Journal of Robust and Nonlinear Control, 28 (2018), 5962-5980. doi: 10.1002/rnc.4353. Google Scholar
[13]	M. Liu, D. W. C. Ho and P. Shi, Adaptive fault-tolerant compensation control for Markovian jump systems with mismatched external disturbance, Automatic, 58 (2015), 5-14. doi: 10.1016/j.automatica.2015.04.022. Google Scholar
[14]	Y. Liu, Y. Yin, K. L. Teo, S. Wang and F. Liu, Probabilistic control of Markov jump systems by scenario optimization approach, Journal of Industrial and Management Optimization, 15 (2019), 1447-1453. doi: 10.3934/jimo.2018103. Google Scholar
[15]	S. Ma and E. K. Boukas, A singular system approach to robust sliding mode control for uncertain Markov jump systems, Automatica, 45 (2009), 2707-2713. doi: 10.1016/j.automatica.2009.07.027. Google Scholar
[16]	M. Mariton, Jump Linear Systems in Automatic Control, New York: M. Dekker, 1990. Google Scholar
[17]	N. Meskin and K. Khorasani, Fault detection and isolation of discrete-time Markovian jump linear systems with application to a network of multi-agent systems having imperfect communication channels, Automatic, 45 (2009), 2032-2040. doi: 10.1016/j.automatica.2009.04.020. Google Scholar
[18]	C. Peng and T. C. Yang, Event-triggered communication and ${H_\infty }$ control co-design for networked control systems, Automatic, 49 (2013), 1326-1332. doi: 10.1016/j.automatica.2013.01.038. Google Scholar
[19]	P. Shi, Y. Zhang, M. Chadli and R. X. Agarwal, Mixed H-infinity and passive filtering for discrete fuzzy neural networks with stochastic jumps and time delays, IEEE Transactions on Neural Networks and Learning Systems, 27 (2016), 903-909. doi: 10.1109/TNNLS.2015.2425962. Google Scholar
[20]	Z. Wang, Y. Liu and X. Liu, Exponential stabilization of a class of stochastic system with Markovian jump parameters and mode-dependent mixed time-delays, IEEE Transactions on Automatic Control, 55 (2010), 1656-1662. doi: 10.1109/TAC.2010.2046114. Google Scholar
[21]	L. Wu, P. Shi, H. Gao and C. Wang, ${H_\infty }$ filtering for 2D Markovian jump systems, Automatic, 44 (2008), 1849-1858. doi: 10.1016/j.automatica.2007.10.027. Google Scholar
[22]	L. Wu, X. Yao and W. X. Zheng, Generalized ${H_2 }$ fault detection for two-dimensional Markovian jump systems, Automatic, 48 (2012), 1741-1750. doi: 10.1016/j.automatica.2012.05.024. Google Scholar
[23]	Z. G. Wu, P. Shi, Z. Shu, H. Su and R. Lu, Passivity-based asynchronous control for Markov jump systems, IEEE Transactions on Automatic Control, 62 (2017), 2020-2025. doi: 10.1109/TAC.2016.2593742. Google Scholar
[24]	S. Xu, T. Chen and J. Lam, Robust ${H_\infty }$ filtering for uncertain Markovian jump systems with mode-dependent time delays, IEEE Transactions on Automatic Control, 48 (2003), 900-907. doi: 10.1109/TAC.2003.811277. Google Scholar
[25]	Y. Yin and Z. Lin, Constrained control of uncertain nonhomogeneous Markovian jump systems, International Journal of Robust and Nonlinear Control, 27 (2017), 3937-3950. Google Scholar
[26]	Y. Yin, Y. Liu, K. L. Teo and S. Wang, Event-triggered probabilistic robust control of linear systems with input constrains: By scenario optimization approach, International Journal of Robust and Nonlinear Control, 28 (2018), 144-153. doi: 10.1002/rnc.3858. Google Scholar
[27]	S. Yu, T. Qu, F. Xu, H. Chen and Y. Hu, Stability of finite horizon model predictive control with incremental input constraints, Automatic, 79 (2017), 265-272. doi: 10.1016/j.automatica.2017.01.040. Google Scholar
[28]	H. Zhang, Y. Shi and J. Wang, On energy-to-peak filtering for nonuniformly sampled nonlinear systems: A Markovian jump system approach, IEEE Transactions on Fuzzy Systems, 22 (2014), 212-222. doi: 10.1109/TFUZZ.2013.2250291. Google Scholar
[29]	X. Zhao, X. Zheng, D. Yao and L. Wu, Adaptive tracking control for a class of uncertain switched nonlinear systems, Automatica, 52 (2015), 185-191. doi: 10.1016/j.automatica.2014.11.019. Google Scholar
[30]	H. Zhang, G. Zhang and J. Wang, $H_\infty$ observer design for LPV systems with uncertain measurements on scheduling variables: application to an electric ground vehicle, IEEE/ASME Transactions on Mechatronics, 21 (2016), 1659-1670. doi: 10.1109/TMECH.2016.2522759. Google Scholar
[31]	H. Zhang and J. Wang, Active steering actuator fault detection for an automatically-steered electric ground vehicle, IEEE Transactions on Vehicular Technology, 66 (2016), 3685-3702. doi: 10.1109/TVT.2016.2604759. Google Scholar
[32]	M. Zhang, P. Shi, L. Ma, J. Cai and H. Su, Network-based fuzzy control for nonlinear Markov jump systems subject to quantization and dropout compensation, Fuzzy Sets and Systems, 371 (2019), 96-109. doi: 10.1016/j.fss.2018.09.007. Google Scholar
[33]	L. Zhang, ${H_\infty }$ estimation for discrete-time piecewise homogeneous Markov jump linear systems, Automatic, 45 (2009), 2570-2576. doi: 10.1016/j.automatica.2009.07.004. Google Scholar
[34]	X. M. Zhang and Q. L. Han, Event-based ${H_\infty }$ filtering for sampled-data systems, Automatica, 51 (2015), 55-69. doi: 10.1016/j.automatica.2014.10.092. Google Scholar
[35]	Y. Zhang, Y. He, M. Wu and J. Zhang, Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices, Automatica, 47 (2011), 79-84. doi: 10.1016/j.automatica.2010.09.009. Google Scholar

show all references

References:

[1]	P. Bolzern, P. Colaneri and G. De Nicolao, Stochastic stability of positive Markov jump linear systems, Automatica, 50 (2014), 1181-1187. doi: 10.1016/j.automatica.2014.02.016. Google Scholar
[2]	X. H. Chang and G. H. Yang, New results on output feedback $H_\infty$ control for linear discrete-time systems, IEEE Transactions on Automatic Control, 59 (2014), 1355-1359. doi: 10.1109/TAC.2013.2289706. Google Scholar
[3]	J. Cheng, J. H. Park, L. Zhang and Y. Zhu, An asynchronous operation approach to event-triggered control for fuzzy Markovian jump systems with general switching policies, IEEE Transactions on Fuzzy Systems, 26 (2018), 6-18. doi: 10.1109/TFUZZ.2016.2633325. Google Scholar
[4]	D. V. Dimarogonas, E. Frazzoli and K. H. Johansson, Distributed event-triggered control for multi-agent systems, IEEE Transactions on Automatic Control, 57 (2012), 1291-1297. doi: 10.1109/TAC.2011.2174666. Google Scholar
[5]	H. Dong, Z. Wang, D. W. C. Ho and H. Gao, Robust $ H_{\infty} $ Filtering for Markovian Jump Systems With Randomly Occurring Nonlinearities and Sensor Saturation: The Finite-Horizon Case, IEEE Transactions on Signal Processing, 59 (2011), 3048-3057. doi: 10.1109/TSP.2011.2135854. Google Scholar
[6]	Y. Dong, E. Tian and Q. L. Han, A delay system method for designing event-triggered controllers of networked control systems, IEEE Transactions on Automatic Control, 58 (2013), 475-481. doi: 10.1109/TAC.2012.2206694. Google Scholar
[7]	S. He, Non-fragile passive controller design for nonlinear Markovian jumping systems via observer-based controls, Neurocomputing, 147 (2015), 350-357. doi: 10.1016/j.neucom.2014.06.053. Google Scholar
[8]	W. P. M. H. Heemels, M. C. F. Donkers and A. R. Teel, Periodic event-triggered control for linear systems, IEEE Transactions on Automatic Control, 58 (2013), 847-861. doi: 10.1109/TAC.2012.2220443. Google Scholar
[9]	F. Li, P. Lim, C. C. Shi and L. Wu, Fault detection filtering for nonhomogeneous Markovian jump systems via fuzzy approach, IEEE Transactions on Fuzzy Systems, 26 (2018), 131-141. doi: 10.1109/TFUZZ.2016.2641022. Google Scholar
[10]	H. Li, Z. Chen, L. Wu, H. K. Lam and H Du, Event-triggered fault detection of nonlinear networked systems, IEEE Transactions on Cybernetics, 47 (2017), 1041-1052. doi: 10.1109/TCYB.2016.2536750. Google Scholar
[11]	H. Li, Y. Wang, D. Yao and R. Lu, A sliding mode approach to stabilization of nonlinear Markovian jump singularly perturbed systems, Automatic, 97 (2018), 404-413. doi: 10.1016/j.automatica.2018.03.066. Google Scholar
[12]	H. Liang, L. Zhang, H. R. Karimi and Q. Zhou, Fault estimation for a class of nonlinear sem-Markovian jump systems with partly unknown transition rates and output quantization, International Journal of Robust and Nonlinear Control, 28 (2018), 5962-5980. doi: 10.1002/rnc.4353. Google Scholar
[13]	M. Liu, D. W. C. Ho and P. Shi, Adaptive fault-tolerant compensation control for Markovian jump systems with mismatched external disturbance, Automatic, 58 (2015), 5-14. doi: 10.1016/j.automatica.2015.04.022. Google Scholar
[14]	Y. Liu, Y. Yin, K. L. Teo, S. Wang and F. Liu, Probabilistic control of Markov jump systems by scenario optimization approach, Journal of Industrial and Management Optimization, 15 (2019), 1447-1453. doi: 10.3934/jimo.2018103. Google Scholar
[15]	S. Ma and E. K. Boukas, A singular system approach to robust sliding mode control for uncertain Markov jump systems, Automatica, 45 (2009), 2707-2713. doi: 10.1016/j.automatica.2009.07.027. Google Scholar
[16]	M. Mariton, Jump Linear Systems in Automatic Control, New York: M. Dekker, 1990. Google Scholar
[17]	N. Meskin and K. Khorasani, Fault detection and isolation of discrete-time Markovian jump linear systems with application to a network of multi-agent systems having imperfect communication channels, Automatic, 45 (2009), 2032-2040. doi: 10.1016/j.automatica.2009.04.020. Google Scholar
[18]	C. Peng and T. C. Yang, Event-triggered communication and ${H_\infty }$ control co-design for networked control systems, Automatic, 49 (2013), 1326-1332. doi: 10.1016/j.automatica.2013.01.038. Google Scholar
[19]	P. Shi, Y. Zhang, M. Chadli and R. X. Agarwal, Mixed H-infinity and passive filtering for discrete fuzzy neural networks with stochastic jumps and time delays, IEEE Transactions on Neural Networks and Learning Systems, 27 (2016), 903-909. doi: 10.1109/TNNLS.2015.2425962. Google Scholar
[20]	Z. Wang, Y. Liu and X. Liu, Exponential stabilization of a class of stochastic system with Markovian jump parameters and mode-dependent mixed time-delays, IEEE Transactions on Automatic Control, 55 (2010), 1656-1662. doi: 10.1109/TAC.2010.2046114. Google Scholar
[21]	L. Wu, P. Shi, H. Gao and C. Wang, ${H_\infty }$ filtering for 2D Markovian jump systems, Automatic, 44 (2008), 1849-1858. doi: 10.1016/j.automatica.2007.10.027. Google Scholar
[22]	L. Wu, X. Yao and W. X. Zheng, Generalized ${H_2 }$ fault detection for two-dimensional Markovian jump systems, Automatic, 48 (2012), 1741-1750. doi: 10.1016/j.automatica.2012.05.024. Google Scholar
[23]	Z. G. Wu, P. Shi, Z. Shu, H. Su and R. Lu, Passivity-based asynchronous control for Markov jump systems, IEEE Transactions on Automatic Control, 62 (2017), 2020-2025. doi: 10.1109/TAC.2016.2593742. Google Scholar
[24]	S. Xu, T. Chen and J. Lam, Robust ${H_\infty }$ filtering for uncertain Markovian jump systems with mode-dependent time delays, IEEE Transactions on Automatic Control, 48 (2003), 900-907. doi: 10.1109/TAC.2003.811277. Google Scholar
[25]	Y. Yin and Z. Lin, Constrained control of uncertain nonhomogeneous Markovian jump systems, International Journal of Robust and Nonlinear Control, 27 (2017), 3937-3950. Google Scholar
[26]	Y. Yin, Y. Liu, K. L. Teo and S. Wang, Event-triggered probabilistic robust control of linear systems with input constrains: By scenario optimization approach, International Journal of Robust and Nonlinear Control, 28 (2018), 144-153. doi: 10.1002/rnc.3858. Google Scholar
[27]	S. Yu, T. Qu, F. Xu, H. Chen and Y. Hu, Stability of finite horizon model predictive control with incremental input constraints, Automatic, 79 (2017), 265-272. doi: 10.1016/j.automatica.2017.01.040. Google Scholar
[28]	H. Zhang, Y. Shi and J. Wang, On energy-to-peak filtering for nonuniformly sampled nonlinear systems: A Markovian jump system approach, IEEE Transactions on Fuzzy Systems, 22 (2014), 212-222. doi: 10.1109/TFUZZ.2013.2250291. Google Scholar
[29]	X. Zhao, X. Zheng, D. Yao and L. Wu, Adaptive tracking control for a class of uncertain switched nonlinear systems, Automatica, 52 (2015), 185-191. doi: 10.1016/j.automatica.2014.11.019. Google Scholar
[30]	H. Zhang, G. Zhang and J. Wang, $H_\infty$ observer design for LPV systems with uncertain measurements on scheduling variables: application to an electric ground vehicle, IEEE/ASME Transactions on Mechatronics, 21 (2016), 1659-1670. doi: 10.1109/TMECH.2016.2522759. Google Scholar
[31]	H. Zhang and J. Wang, Active steering actuator fault detection for an automatically-steered electric ground vehicle, IEEE Transactions on Vehicular Technology, 66 (2016), 3685-3702. doi: 10.1109/TVT.2016.2604759. Google Scholar
[32]	M. Zhang, P. Shi, L. Ma, J. Cai and H. Su, Network-based fuzzy control for nonlinear Markov jump systems subject to quantization and dropout compensation, Fuzzy Sets and Systems, 371 (2019), 96-109. doi: 10.1016/j.fss.2018.09.007. Google Scholar
[33]	L. Zhang, ${H_\infty }$ estimation for discrete-time piecewise homogeneous Markov jump linear systems, Automatic, 45 (2009), 2570-2576. doi: 10.1016/j.automatica.2009.07.004. Google Scholar
[34]	X. M. Zhang and Q. L. Han, Event-based ${H_\infty }$ filtering for sampled-data systems, Automatica, 51 (2015), 55-69. doi: 10.1016/j.automatica.2014.10.092. Google Scholar
[35]	Y. Zhang, Y. He, M. Wu and J. Zhang, Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices, Automatica, 47 (2011), 79-84. doi: 10.1016/j.automatica.2010.09.009. Google Scholar

Figure 1. System mode trajectory

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Figure 2. System states curve

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Figure 3. Triggered instants diagram

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Figure 4. Control curve

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Figure 5. System mode trajectory

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Figure 6. System states curve

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Figure 7. Triggered instants diagram

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Figure 8. Control curve

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Event-triggered mixed $ H_\infty $ and passive control for Markov jump systems with bounded inputs

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