Linear programming-based stochastic stabilization of hidden semi-Markov jump positive systems

Xuan Jia; Junfeng Zhang; Tarek Raïssi; Xuan Jia; Junfeng Zhang; Tarek Raïssi

doi:10.3934/math.20241289

AIMS Mathematics

2024, Volume 9, Issue 10: 26483-26498. doi: 10.3934/math.20241289

Previous Article Next Article

Research article Special Issues

Linear programming-based stochastic stabilization of hidden semi-Markov jump positive systems

1.
School of Information and Communication Engineering, Hainan University, Haikou 570228, China
2.
Cedric-Lab, Conservatoire National des Arts et Metiers, Paris 75141, France

Received: 11 July 2024 Revised: 15 August 2024 Accepted: 27 August 2024 Published: 12 September 2024
MSC : 60K15, 93C10, 93C28, 93E15

This paper focuses on the stochastic stabilization of hidden semi-Markov jump nonlinear positive systems. First, a notion of stochastic stability is introduced for this class of systems. A criterion is addressed to ensure the stochastic stability using a stochastic copositive Lyapunov function. Then, an observed mode is proposed to estimate the emitted value of the hidden semi-Markov process and the mode-dependent controller is designed using an improved matrix decomposition approach. Some auxiliary variables are added to decouple the coupling terms in hidden semi-Markov jump nonlinear positive systems into a tractable condition. All conditions are described in terms of linear programming. Moreover, the proposed design is developed for systems with partially known emission probabilities. Two examples are provided to show the validity of the obtained results.
- hidden semi-Markov jump positive systems,
- stochastic stabilization,
- linear programming
Citation: Xuan Jia, Junfeng Zhang, Tarek Raïssi. Linear programming-based stochastic stabilization of hidden semi-Markov jump positive systems[J]. AIMS Mathematics, 2024, 9(10): 26483-26498. doi: 10.3934/math.20241289

Related Papers:

Abstract

This paper focuses on the stochastic stabilization of hidden semi-Markov jump nonlinear positive systems. First, a notion of stochastic stability is introduced for this class of systems. A criterion is addressed to ensure the stochastic stability using a stochastic copositive Lyapunov function. Then, an observed mode is proposed to estimate the emitted value of the hidden semi-Markov process and the mode-dependent controller is designed using an improved matrix decomposition approach. Some auxiliary variables are added to decouple the coupling terms in hidden semi-Markov jump nonlinear positive systems into a tractable condition. All conditions are described in terms of linear programming. Moreover, the proposed design is developed for systems with partially known emission probabilities. Two examples are provided to show the validity of the obtained results.

References

[1]	L. Farina, S. Rinaldi, Positive linear systems: theory and applications, Hoboken: John Wiley & Sons, 2011.
[2]	J. Shen, J. Lam, $L_1$-gain analysis for positive systems with distributed delays, Automatica, 50 (2014), 175–179. https://doi.org/10.1016/j.automatica.2013.09.037 doi: 10.1016/j.automatica.2013.09.037
[3]	R. Shorten, F. Wirth, D. Leith, A positive systems model of TCP-like congestion control: Asymptotic results, IEEE/ACM Trans. Netw., 14 (2006), 616–629. https://doi.org/10.1109/TNET.2006.876178 doi: 10.1109/TNET.2006.876178
[4]	M. A. Rami, J. Shamma, Hybrid positive systems subject to Markovian switching, IFAC Proceed, 42 (2009), 138–143. https://doi.org/10.3182/20090916-3-ES-3003.00025 doi: 10.3182/20090916-3-ES-3003.00025
[5]	P. Bolzern, P. Colaneri, G. Nicolao, Stochastic stability of positive Markov jump linear systems, Automatica, 50 (2014), 1181–1187. https://doi.org/10.1016/j.automatica.2014.02.016 doi: 10.1016/j.automatica.2014.02.016
[6]	J. Cavalcanti, H. Balakrishnan, Sign-stability of positive Markov jump linear systems, Automatica, 111 (2020), 108638. https://doi.org/10.1016/j.automatica.2019.108638 doi: 10.1016/j.automatica.2019.108638
[7]	J. Zhang, B. Du, S. Zhang, S. Ding, A double sensitive fault detection filter for positive Markovian jump systems with a hybrid event-triggered mechanism, IEEE/CAA J. Automat. Sin., 2024. https://doi.org/10.1109/JAS.2024.124677 doi: 10.1109/JAS.2024.124677
[8]	M. Ogura, F. Martin, Stability analysis of positive semi-Markovian jump linear systems with state resets, SIAM J. Control Optim., 52 (2011) 1809–1831. https://doi.org/10.1137/130925177 doi: 10.1137/130925177
[9]	J. Zhang, G. Zheng, Y. Feng, Y. Chen, Event-triggered state-feedback and dynamic output-feedback control of PMJSs with intermittent faults, IEEE Trans. Automat. Control, 68 (2023), 1039–1046. https://doi.org/10.1109/TAC.2022.3146709 doi: 10.1109/TAC.2022.3146709
[10]	Q. Li, J. Zhang, P. Lin, Adaptive event-triggered fault detection for nonlinear positive semi-Markov jump systems, Int. J. Adapt. Control Signal Process., 36 (2022), 2677–2700. https://doi.org/10.1002/acs.3483 doi: 10.1002/acs.3483
[11]	L. Li, W. Qi, X. Chen, X. Gao, Y. Wei, Stability analysis and control synthesis for positive semi-Markov jump systems with time-varying delay, Appl. Math. Comput., 332 (2018), 363–375. https://doi.org/10.1016/j.amc.2018.02.055 doi: 10.1016/j.amc.2018.02.055
[12]	M. A. Rami, Solvability of static output-feedback stabilization for LTI positive systems, Syst. Control Lett., 60 (2011), 704–708. https://doi.org/10.1016/j.sysconle.2011.05.007 doi: 10.1016/j.sysconle.2011.05.007
[13]	O. Mason, R. Shorten, On linear copositive Lyapunov functions and the stability of switched positive linear systems, IEEE Trans. Automat. Control, 52 (2007), 1346–1349. https://doi.org/0.1109/TAC.2007.900857
[14]	O. L. do Valle Costa, M. D. Fragoso, M. G. Todorov, A detector-based approach for the $H_2$ control of Markov jump linear systems with partial information, IEEE Trans. Automat. Control, 60 (2015), 1219–1234. https://doi.org/10.1109/TAC.2014.2366253 doi: 10.1109/TAC.2014.2366253
[15]	F. Stadtmann, O. L. do Valle Costa, $H_2$-control of continuous-time hidden Markov jump linear systems, IEEE Trans. Automat. Control, 62 (2017), 4031-–4037. https://doi.org/10.1109/TAC.2016.2616303 doi: 10.1109/TAC.2016.2616303
[16]	F. Li, C. Du, C. Yang, L. Wu, W. Gui, Finite-time asynchronous sliding mode control for Markovian jump systems, Automatica, 109 (2019), 108503. https://doi.org/10.1016/j.automatica.2019.108503 doi: 10.1016/j.automatica.2019.108503
[17]	Z. G. Wu, P. Shi, Z. Shu, H. Su, R. Lu, Passivity-based asynchronous control for Markov jump systems, IEEE Trans. Automat. Control, 62 (2017), 2020–2025. https://doi.org/10.1109/TAC.2016.2593742 doi: 10.1109/TAC.2016.2593742
[18]	A. M. Oliveira, O. L. V. Costa, Mixed control of hidden Markov jump systems, Int. J. Robust Nonlinear Control, 28 (2018), 1261–1280. https://doi.org/10.1002/rnc.3952 doi: 10.1002/rnc.3952
[19]	S. C. Huo, F. Li, Hidden Markov model-based control for networked fuzzy Markov jump systems against randomly occurring multichannel attacks, Int. J. Robust Nonlinear Control, 31 (2021), 1657–1673. https://doi.org/10.1002/rnc.5363 doi: 10.1002/rnc.5363
[20]	F. Li, S. Xu, H. Shen, Z. Zhang, Extended dissipativity-based control for hidden Markov jump singularly perturbed systems subject to general probabilities, IEEE Trans. Syst. Man Cybern.: Syst., 51 (2021), 5752–5761. https://doi.org/10.1109/TSMC.2019.2957659 doi: 10.1109/TSMC.2019.2957659
[21]	Y. Tian, H. Yan, H. Zhang, J. Cheng, H. Shen, Asynchronous output feedback control of hidden semi-Markov jump systems with random mode-dependent delays, IEEE Trans. Automat. Control, 67 (2022), 4107–4114. https://doi.org/10.1109/TAC.2021.3110006 doi: 10.1109/TAC.2021.3110006
[22]	L. Zhang, B. Cai, T. Tan, Y. Shi, Stabilization of non-homogeneous hidden semi-Markov jump systems with limited sojourn-time information, Automatica, 117 (2020), 108963. https://doi.org/10.1016/j.automatica.2020.108963 doi: 10.1016/j.automatica.2020.108963
[23]	B. Wang, Q. Zhu, The observed mode dependent controller design problem for a class of continuous-time hidden semi-Markov jump systems, Int. J. Robust Nonlinear Control, 33 (2023), 8421–8432. https://doi.org/10.1002/rnc.6827 doi: 10.1002/rnc.6827
[24]	L. X. Zhang, B. Cai, Y. Shi, Stabilization of hidden semi-Markov jump systems: emission probability approach, Automatica, 101 (2019), 87–95. https://doi.org/10.1016/j.automatica.2018.11.027 doi: 10.1016/j.automatica.2018.11.027
[25]	S. Z. Yu, Hidden semi-Markov models, Artific. Intell., 174 (2010), 215–243. https://doi.org/10.1016/j.artint.2009.11.011 doi: 10.1016/j.artint.2009.11.011
[26]	B. Cai, L. Zhang, Y. Shi, Observed-mode-dependent state estimation of hidden semi-Markov jump linear systems, IEEE Trans. Automat. Control, 65 (2020), 442–449. https://doi.org/10.1109/TAC.2019.2919114 doi: 10.1109/TAC.2019.2919114
[27]	F. Li, W. X. Zheng, S. Xu, Stabilization of discrete-time hidden semi-Markov jump singularly perturbed systems with partially known emission probabilities, IEEE Trans. Automat. Control, 67 (2022), 4234–4240. https://doi.org/10.1109/TAC.2021.3113471 doi: 10.1109/TAC.2021.3113471
[28]	B. Wang, Q. Zhu, S. Li, Stabilization of discrete-time hidden semi-Markov jump linear systems with partly unknown emission probability matrix, IEEE Trans. Automat. Control, 69 (2023), 1952–1959. https://doi.org/10.1109/TAC.2023.3272190 doi: 10.1109/TAC.2023.3272190
[29]	G. Zong, W. Qi, H. R. Karimi, $L_1$ control of positive semi-Markov jump systems with state delay, IEEE Trans. Syst. Man Cybern.: Syst., 51 (2021), 7569–7578. https://doi.org/10.1109/TSMC.2020.2980034 doi: 10.1109/TSMC.2020.2980034
[30]	Y. Tian, H. Yan, H. Zhang, M. Wang, J. Yi, Time-varying gain controller synthesis of piecewise homogeneous semi-Markov jump linear systems, Automatica, 146 (2022), 110594. https://doi.org/10.1016/j.automatica.2022.110594 doi: 10.1016/j.automatica.2022.110594
[31]	W. Qi, Y. Yi, P. Shi, G. Zong, Z. G. Wu, Adaptive event-driven control for continuous positive stochastic jump models with cyber attacks, IEEE Trans. Circuits Syst. II Express Briefs, 71 (2024), 301–305. https://doi.org/10.1109/TCSII.2023.3300727 doi: 10.1109/TCSII.2023.3300727

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)