May  2021, 17(3): 1343-1355. doi: 10.3934/jimo.2020024

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

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  May 2021 Early access  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.

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 and Management Optimization, 2021, 17 (3) : 1343-1355. doi: 10.3934/jimo.2020024
References:
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P. BolzernP. 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.

[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.

[3]

J. ChengJ. H. ParkL. 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.

[4]

D. V. DimarogonasE. 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.

[5]

H. DongZ. WangD. 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.

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Y. DongE. 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.

[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.

[8]

W. P. M. H. HeemelsM. 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.

[9]

F. LiP. LimC. 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.

[10]

H. LiZ. ChenL. WuH. 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.

[11]

H. LiY. WangD. 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.

[12]

H. LiangL. ZhangH. 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.

[13]

M. LiuD. 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.

[14]

Y. LiuY. YinK. L. TeoS. 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.

[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.

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M. Mariton, Jump Linear Systems in Automatic Control, New York: M. Dekker, 1990.

[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.

[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.

[19]

P. ShiY. ZhangM. 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.

[20]

Z. WangY. 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.

[21]

L. WuP. ShiH. 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.

[22]

L. WuX. 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.

[23]

Z. G. WuP. ShiZ. ShuH. 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.

[24]

S. XuT. 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.

[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. 

[26]

Y. YinY. LiuK. 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.

[27]

S. YuT. QuF. XuH. 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.

[28]

H. ZhangY. 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.

[29]

X. ZhaoX. ZhengD. 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.

[30]

H. ZhangG. 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.

[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.

[32]

M. ZhangP. ShiL. MaJ. 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.

[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.

[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.

[35]

Y. ZhangY. HeM. 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.

show all references

References:
[1]

P. BolzernP. 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.

[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.

[3]

J. ChengJ. H. ParkL. 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.

[4]

D. V. DimarogonasE. 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.

[5]

H. DongZ. WangD. 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.

[6]

Y. DongE. 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.

[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.

[8]

W. P. M. H. HeemelsM. 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.

[9]

F. LiP. LimC. 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.

[10]

H. LiZ. ChenL. WuH. 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.

[11]

H. LiY. WangD. 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.

[12]

H. LiangL. ZhangH. 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.

[13]

M. LiuD. 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.

[14]

Y. LiuY. YinK. L. TeoS. 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.

[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.

[16]

M. Mariton, Jump Linear Systems in Automatic Control, New York: M. Dekker, 1990.

[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.

[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.

[19]

P. ShiY. ZhangM. 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.

[20]

Z. WangY. 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.

[21]

L. WuP. ShiH. 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.

[22]

L. WuX. 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.

[23]

Z. G. WuP. ShiZ. ShuH. 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.

[24]

S. XuT. 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.

[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. 

[26]

Y. YinY. LiuK. 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.

[27]

S. YuT. QuF. XuH. 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.

[28]

H. ZhangY. 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.

[29]

X. ZhaoX. ZhengD. 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.

[30]

H. ZhangG. 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.

[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.

[32]

M. ZhangP. ShiL. MaJ. 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.

[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.

[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.

[35]

Y. ZhangY. HeM. 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.

Figure 1.  System mode trajectory
Figure 2.  System states curve
Figure 3.  Triggered instants diagram
Figure 4.  Control curve
Figure 5.  System mode trajectory
Figure 6.  System states curve
Figure 7.  Triggered instants diagram
Figure 8.  Control curve
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