Skip to main content
Springer Nature Link
Log in

Event-triggered Control of Positive Switched Systems with Actuator Saturation and Time-delay

  • Research Article
  • Published:
International Journal of Automation and Computing Aims and scope Submit manuscript

Abstract

This paper investigates the event-triggered control of positive switched systems with randomly occurring actuator saturation and time-delay, where the actuator saturation and time-delay obey different Bernoulli distributions. First, an event-triggering condition is constructed based on a 1-norm inequality. Under the presented event-triggering scheme, an interval estimation method is utilized to deal with the error term of the systems. Using a co-positive Lyapunov functional, the event-triggered controller and the cone attraction domain gain matrices are designed via matrix decomposition techniques. The positivity and stability of the resulting closed-loop systems are reached by guaranteeing the positivity of the lower bound of the systems and the stability of the upper bound of the systems, respectively. The proposed approach is developed for interval and polytopic uncertain systems, respectively. Finally, two examples are provided to illustrate the effectiveness of the theoretical findings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. L. Farina, S. Rinaldi. Positive Linear Systems: Theory and Applications, New York: John Wiley & Sons, 2000. DOI: https://doi.org/10.1002/9781118033029.

    MATH  Google Scholar 

  2. J. Lam, Y. Chen, X. W. Liu, X. D. Zhao, J. F. Zhang. Positive Systems Theory and Applications (POSTA 2018), Cham: Springer, 2019. DOI: https://doi.org/10.1007/978-3-030-04327-8.

    Google Scholar 

  3. S. Y. Xiao, X. H. Ge, Q. L. Han, Y. J. Zhang. Distributed resilient estimator design for positive systems under topological attacks. IEEE Transactions on Cybernetics, published online. DOI: https://doi.org/10.1109/TCYB.2020.2981646.

  4. R. Shorten, F. Wirth, D. Leith. A positive systems model of TCP-like congestion control: Asymptotic results. IEEE/ACM Transactions on Networking, vol. 14, no. 3, pp. 616–629, 2006. DOI: https://doi.org/10.1109/TNET.2006.876178.

    Google Scholar 

  5. J. F. Zhang, L. W. Zhang, T. Raïssi. A linear framework on the distributed model predictive control of positive systems. Systems & Control Letters, vol. 138, Article number 104665, 2020. DOI: https://doi.org/10.1016/j.sysconle.2020.104665.

  6. C. C. Sun. Stabilization for a class of discrete-time switched large-scale systems with parameter uncertainties. International Journal of Automation and Computing, vol. 16, no. 4, pp. 543–552, 2019. DOI: https://doi.org/10.1007/s11633-016-0966-6.

    Google Scholar 

  7. H. L. Ren, G. D. Zong, H. R. Karimi. Asynchronous finite-time filtering of networked switched systems and its application: An event-driven method. IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 66, no. 1, pp. 391–402, 2019. DOI: https://doi.org/10.1109/TCSI.2018.2857771.

    MathSciNet  Google Scholar 

  8. Q. Zhou, S. Y. Zhao, H. Y. Li, R. Q. Lu, C. W. Wu. Adaptive neural network tracking control for robotic manipulators with dead zone. IEEE Transactions on Neural Networks and Learning Systems, vol. 30, no. 12, pp. 3611–3620, 2019. DOI: https://doi.org/10.1109/TNNLS.2018.2869375.

    MathSciNet  Google Scholar 

  9. B. D. O. Anderson, M. B. Ye. Recent advances in the modelling and analysis of opinion dynamics on influence networks. International Journal of Automation and Computing, vol. 16, no. 2, pp. 129–149, 2019. DOI: https://doi.org/10.1007/s11633-019-1169-8.

    Google Scholar 

  10. X. D. Zhao, L. X. Zhang, P. Shi, M. Liu. Stability of switched positive linear systems with average dwell time switching. Automatica, vol. 48, no. 6, pp. 1132–1137, 2012. DOI: https://doi.org/10.1016/j.automatica.2012.03.008.

    MathSciNet  MATH  Google Scholar 

  11. X. W. Liu, C. Y. Dang. Stability analysis of positive switched linear systems with delays. IEEE Transactions on Automatic Control, vol. 56, no. 7, pp. 1684–1690, 2011. DOI: https://doi.org/10.1109/TAC.2011.2122710.

    MathSciNet  MATH  Google Scholar 

  12. E. Fornasini, M. E. Valcher. Linear copositive Lyapunov functions for continuous-time positive switched systems. IEEE Transactions on Automatic Control, vol. 55, no. 8, pp. 1933–1937, 2010. DOI: https://doi.org/10.1109/TAC.2010.2049918.

    MathSciNet  MATH  Google Scholar 

  13. F. Blanchini, P. Colaneri, M. E. Valcher. Co-positive Lyapunov functions for the stabililation of positive switched systems. IEEE Transactions on Automatic Control, vol. 57, no. 12, pp. 3038–3050, 2012. DOI: https://doi.org/10.1109/TAC.2012.2199169

    MathSciNet  MATH  Google Scholar 

  14. O Mason, R Shorten. On linear copositive Lyapunov functions and the stability of switched positive linear systems IEEE Transactions on Automatic Control, vol 52, no 7, pp 1346–1349, 2007 DOI: https://doi.org/10.1109/TAC.2007.900857

    MathSciNet  MATH  Google Scholar 

  15. A Zappavigna, P Colaneri, J C Geromel, R Shorten. Dwell time analysis for continuous-time switched linear positive systems In Proceedings of American Control Conference, IEEE, Baltimore, USA, pp. 6256–6261, 2010. DOI: https://doi.org/10.1109/ACC.2010.5531524

    Google Scholar 

  16. P Colaneri, R H Middleton, Z Y. Chen, D Caporale, F Blanchini. Convexity of the cost functional in an optimal control problem for a class of positive switched systems Automatica, vol. 50, no. 4, pp. 1227–1234, 2014. DOI: https://doi.org/10.1016/j.automatica.2014.02.025.

    MathSciNet  MATH  Google Scholar 

  17. E. Fornasini, M. E. Valcher. Stability and stabilizability of special classes of discrete-time positive switched systems. In Proceedings of American Control Conference, IEEE, San Francisco, USA, pp. 2619–2624, 2011. DOI: https://doi.org/10.1109/ACC.2011.5990765.

    Google Scholar 

  18. R. Dorf, M. Farren, C. Phillips. Adaptive sampling frequency for sampled-data control systems. IRE Transactions on Automatic Control, vol. 7, no. 1, pp. 38–47, 1962. DOI: https://doi.org/10.1109/TAC.1962.1105415.

    Google Scholar 

  19. J. Lunze, D. Lehmann. A state-feedback approach to event-based control. Automatica, vol. 46, no. 1, pp. 211–215, 2010. DOI: https://doi.org/10.1016/j.automatica.2009.10.035.

    MathSciNet  MATH  Google Scholar 

  20. R. Postoyan, P. Tabuada, D. Nešić, A. Anta. A framework for the event-triggered stabilization of nonlinear systems. IEEE Transactions on Automatic Control, vol. 60, no. 4, pp. 982–996, 2015. DOI: https://doi.org/10.1109/TAC.2014.2363603.

    MathSciNet  MATH  Google Scholar 

  21. L. T. Xing, C. Y. Wen, Z. T. Liu, H. Y. Su, J. P. Cai. Event-triggered adaptive control for a class of uncertain nonlinear systems. IEEE Transactions on Automatic Control, vol. 62, no. 4, pp. 2071–2076, 2017. DOI: https://doi.org/10.1109/TAC.2016.2594204.

    MathSciNet  MATH  Google Scholar 

  22. W. P. M. H. Heemels, M. C. F. Donkers, A. R. Teel. Periodic event-triggered control for linear systems. IEEE Transactions on Automatic Control, vol. 58, no. 4, pp. 847–861, 2013. DOI: https://doi.org/10.1109/TAC.2012.2220443.

    MathSciNet  MATH  Google Scholar 

  23. T. F. Li, J. Fu. Event-triggered control of switched linear systems. Journal of the Franklin Institute, vol. 354, no. 15, pp. 6451–6462, 2017. DOI: https://doi.org/10.1016/j.jfranklin.2017.05.018.

    MathSciNet  MATH  Google Scholar 

  24. Y. W. Qi, Z. Cao, X. L. Li. Decentralized event-triggered H∞ control for switched systems with network communication delay. Journal of the Franklin Institute, vol. 356, no. 3, pp. 1424–1445, 2019. DOI: https://doi.org/10.1016/j.jfranklin.2018.12.008.

    MathSciNet  MATH  Google Scholar 

  25. Y. Y. Yin, Z. L. Lin, Y. Q. Liu, K. L. Teo. Event-triggered constrained control of positive systems with input saturation. International Journal of Robust and Nonlinear Control, vol. 28, no. 11, pp. 3532–3542, 2018. DOI: https://doi.org/10.1002/rnc.4097.

    MathSciNet  MATH  Google Scholar 

  26. L. Y. Liu, J. F. Zhang, Y. Shao, X. J. Deng. Event-triggered control of positive switched systems based on linear programming. IET Control Theory & Applications, vol. 14, no. 1, pp. 145–155, 2020. DOI: https://doi.org/10.1049/iet-cta.2019.0606.

    Google Scholar 

  27. H. J. Wang, P. Shi, C. C. Lim, Q. Q. Xue. Event-triggered control for networked Markovian jump systems. International Journal of Robust and Nonlinear Control, vol. 25, no. 17, pp. 3422–3438, 2015. DOI: https://doi.org/10.1002/rnc.3273.

    MathSciNet  MATH  Google Scholar 

  28. A. Selivanov, E. Fridman. Event-triggered H∞ control: A switching approach. IEEE Transactions on Automatic Control, vol. 61, no. 10, pp. 3221–3226, 2016. DOI: https://doi.org/10.1109/TAC.2015.2508286.

    MathSciNet  Google Scholar 

  29. H. Ma, H. Y. Li, R. Q. Lu, T. W. Huang. Adaptive event-triggered control for a class of nonlinear systems with periodic disturbances. Science China Information Sciences, vol. 63, no. 5, Article number 150212, 2020. DOI: https://doi.org/10.1007/s11432-019-2680-1.

    Google Scholar 

  30. T. S. Hu, Z. L. Lin, B. M. Chen. Analysis and design for discrete-time linear systems subject to actuator saturation. Systems & Control Letters, vol. 45, no. 2, pp. 97–112, 2002. DOI: https://doi.org/10.1016/S0167-6911(01)00168-2.

    MathSciNet  MATH  Google Scholar 

  31. Z. D. Wang, B. Shen, X. S. Liu. H∞ filtering with randomly occurring sensor saturations and missing measurements. Automatica, vol. 48, no. 3, pp. 556–562, 2012. DOI: https://doi.org/10.1016/j.automatica.2012.01.008.

    MathSciNet  MATH  Google Scholar 

  32. J. Wang, J. Zhao. Stabilisation of switched positive systems with actuator saturation. IET Control Theory & Applications, vol. 10, no. 6, pp. 717–723, 2016. DOI: https://doi.org/10.1049/iet-cta.2015.0064.

    MathSciNet  Google Scholar 

  33. J. F. Zhang, T. Raiïssi. Saturation control of switched nonlinear systems. Nonlinear Analysis: Hybrid Systems, vol. 32, pp. 320–336, 2019. DOI: https://doi.org/10.1016/j.nahs.2019.01.005.

    MathSciNet  Google Scholar 

  34. G. W. Zhang, P. Yang, J. Wang, J. J. Sun, Y. Zhang. Integrated observer-based fixed-time control with backstepping method for exoskeleton robot. International Journal of Automation and Computing, vol. 17, no. 1, pp. 71–82, 2020. DOI: https://doi.org/10.1007/s11633-019-1201-z.

    Google Scholar 

  35. X. M. Sun, W. Wang, G. P. Liu, J. Zhao. Stability analysis for linear switched systems with time-varying delay. IEEE Transactions on Systems, Man, and Cybernetics — Part B: Cybernetics, vol. 38, no. 2, pp. 528–533, 2008. DOI: https://doi.org/10.1109/TSMCB.2007.912078.

    Google Scholar 

  36. Z. G. Feng, H. Y. Zhang, H. P. Du, Z. Y. Jiang. Admissibilisation of singular interval type-2 Takagi-sugeno fuzzy systems with time delay. IET Control Theory & Applications, vol. 14, no. 8, pp. 1022–1032, 2020. DOI: https://doi.org/10.1049/ietcta.2019.0791.

    Google Scholar 

  37. D. W. Zhang, Q. L. Han, X. M. Zhang. Network-based modeling and proportional-integral control for direct-drive-wheel systems in wireless network environments. IEEE Transactions on Cybernetics, vol. 50, no. 6, pp. 2462–2474, 2020. DOI: https://doi.org/10.1109/TCYB.2019.2924450.

    MathSciNet  Google Scholar 

  38. E. Fridman, U. Shaked. An improved stabilization method for linear time-delay systems. IEEE Transactions on Automatic Control, vol. 47, no. 11, pp. 1931–1937, 2002. DOI: https://doi.org/10.1109/TAC.2002.804462.

    MathSciNet  MATH  Google Scholar 

  39. D. Yue, E. G. Tian, Z. D. Wang, J. Lam. Stabilization of systems with probabilistic interval input delays and its applications to networked control systems. IEEE Transactions on Systems, Man, and Cybernetics — Part A: Systems and Humans, vol. 39, no. 4, pp. 939–945, 2009. DOI: https://doi.org/10.1109/TSMCA.2009.2019875.

    Google Scholar 

  40. M. Xiang, Z. R. Xiang. Stability, L1-gain and control synthesis for positive switched systems with time-varying delay. Nonlinear Analysis: Hybrid Systems, vol. 9, pp. 9–17, 2013. DOI: https://doi.org/10.1016/j.nahs.2013.01.001.

    MathSciNet  MATH  Google Scholar 

  41. T. T. Liu, B. W. Wu, L. L. Liu, Y. E. Wang. Asynchronously finite-time control of discrete impulsive switched positive time-delay systems. Journal of the Franklin Institute, vol. 352, no. 10, pp. 4503–4514, 2015. DOI: https://doi.org/10.1016/j.jfranklin.2015.06.015.

    MathSciNet  MATH  Google Scholar 

  42. J. F. Zhang, X. D. Zhao, F. B. Zhu, Z. Z. Han. L1/ℓ1-gain analysis and synthesis of Markovian jump positive systems with time delay. ISA Transactions, vol. 63, pp. 93–102, 2016. DOI: https://doi.org/10.1016/j.isatra.2016.03.015.

    Google Scholar 

  43. W. Elloumi, A. Benzaouia, M. Chaabane. Delay-dependent stabilization conditions of controlled positive continuous-time systems. International Journal of Automation and Computing, vol. 11, no. 6, pp. 653–660, 2014. DOI: https://doi.org/10.1007/s11633-014-0816-3.

    Google Scholar 

  44. J. P. Hespanha, A. S. Morse. Stability of switched systems with average dwell-time. In Proceedings of the 38th IEEE Conference on Decision and Control, IEEE, Phoenix, USA, pp. 2655–2660, 1999. DOI: https://doi.org/10.1109/CDC.1999.831330.

  45. X. M. Zhang, Q. L. Han, X. H. Ge, D. R. Ding, L. Ding, D. Yue, C. Peng. Networked control systems: A survey of trends and techniques. IEEE/CAA Journal of Automatica Sinica, vol. 7, no. 1, pp. 1–17, 2020. DOI: https://doi.org/10.1109/JAS.2019.1911651.

    MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Nos. 62073111 and 61751304), Fundamental Research Funds for the Provincial Universities of Zhejiang (No. GK209907299001-007), Natural Science Foundation of Zhejiang Province, China (Nos. LY20F030008 and LY20F030011), and Foundation of Zhejiang Provincial Department of Education (No. Y201942017).

Author information

Authors and Affiliations

  1. School of Automation, Hangzhou Dianzi University, Hangzhou, 310018, China

    Jun-Feng Zhang, Lai-You Liu, Shi-Zhou Fu & Shuo Li

  2. Inria, University of Lille, Lille, 59000, France

    Jun-Feng Zhang

Authors
  1. Jun-Feng Zhang
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Lai-You Liu
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Shi-Zhou Fu
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Shuo Li
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Jun-Feng Zhang.

Additional information

Jun-Feng Zhang received the Ph. D. degree in Shanghai Jiao Tong University, China in 2014. From December 2014, he worked in School of Automation, Hangzhou Dianzi University, China. From August 2019 to August 2020, he visited Inria, University of Lille, France. He is a member of IEEE and CAA. He was the co-chair of Program Committee in the 6th International Conference on Positive Systems. He has published more than 50 journal and conference papers in the field of positive systems.

His research interests include positive systems, switched systems, and model predictive control.

Lai-You Liu received the B. Sc. degree in Zhengzhou University of Aeronautics, China in 2017. He is a master student in Hangzhou Dianzi University, China.

His research interests include positive systems and hybrid systems. E-mail: laiyouliu@126.com

Shi-Zhou Fu received the B. Sc. degree in Hangzhou Dianzi University, China in 2010. He received the Ph. D. degree in Hong Kong University, China in 2015. He was appointed as a lecturer at Hangzhou Dianzi University, China in 2016.

His research interests include fuzzy control, quantum control and robust control. E-mail: fushizhou@hdu.edu.cn

Shuo Li received the Ph. D. degree in control science and engineering from Nanjing University of Science and Technology, China in 2017. She was appointed as a lecturer at Hangzhou Dianzi University, China in 2017.

Her research interests include positive systems, switched systems, and fuzzy systems.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, JF., Liu, LY., Fu, SZ. et al. Event-triggered Control of Positive Switched Systems with Actuator Saturation and Time-delay. Int. J. Autom. Comput. 18, 141–154 (2021). https://doi.org/10.1007/s11633-020-1245-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11633-020-1245-0

Keywords