Skip to main content
SpringerLink
Log in

Stability and stabilization of a class of switched stochastic systems with saturation control

  • Research Paper
  • Published:
Science China Information Sciences Aims and scope Submit manuscript

Abstract

A switching system always comprises of several subsystems and a rule supervising the switching between the subsystems. A major problem that is often inherent to all dynamical systems is actuator saturation. Saturation is a nonlinear property that nonlinearly maps small input signals to the output, which may affect the system properties and even destroy them. In this study, stability and stabilization of a class of switched stochastic systems with saturation control was investigated. First, the variation parameter method was used to present the integral form of switched stochastic systems. Second, to guarantee that the zero solution is globally exponentially stable in mean square, two sufficient conditions were obtained using direct computation with Gronwall inequality and indirect method with matrix theory, respectively. Further, another simple sufficient condition was obtained for the stability of the systems using the row norm, column norm, and Frobenius norm. Finally, two examples were used to illustrate the preciseness and effectiveness of the results. Moreover, various control designs were observed to stabilize the systems, which differ from the technique of linear matrix inequalities.

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

Access this article

Log in via an institution

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

References

  1. Liberzon D. Switching in Systems and Control. Boston: Birkhäuser, 2003

    Book  MATH  Google Scholar 

  2. Sun Z, Ge S S. Switched Linear Systems: Control and Design. Berlin: Springer, 2005

    Book  MATH  Google Scholar 

  3. Sun Y G, Wang L. On stability of a class of switched nonlinear systems. Automatica, 2013, 49: 305–307

    Article  MATH  Google Scholar 

  4. Zhang J F, Han Z Z, Zhu F B, et al. Absolute exponential stability and stabilization of switched nonlinear systems. Syst Control Lett, 2014, 66: 51–57

    Article  MATH  Google Scholar 

  5. Lu X D, Zhang X F. Stability analysis of switched systems on time scales with all modes unstable. Nonlinear Anal-Hybrid Syst, 2019, 33: 371–379

    Article  MATH  Google Scholar 

  6. Aleksandrov A Y, Chen Y, Platonov A V, et al. Stability analysis for a class of switched nonlinear systems. Automatica, 2011, 47: 2286–2291

    Article  MATH  Google Scholar 

  7. LaSalle J P. The Stability of Dynamical System. Philadelphia: SIAM, 1976

    Book  MATH  Google Scholar 

  8. Aleksandrov A, Chen Y, Platonov A, et al. Stability analysis and uniform ultimate boundedness control synthesis for a class of nonlinear switched difference systems. J Differ Equ Appl, 2012, 18: 1545–1561

    Article  MATH  Google Scholar 

  9. Zong G D, Wang R H, Zheng W X, et al. Finite-time H∞ control for discrete-time switched nonlinear systems with time delay. Int J Robust Nonlinear Control, 2015, 25: 914–936

    Article  MATH  Google Scholar 

  10. Yang D, Zong G D, Karimi H R. H∞ refined antidisturbance control of switched LPV systems with application to aero-engine. IEEE Trans Ind Electron, 2020, 67: 3180–3190

    Article  Google Scholar 

  11. Hou L L, Ren H L, Zong G D. Finite-time stability of interconnected impulsive switched systems. IET Control Theory A, 2016, 46: 648–654

    Google Scholar 

  12. Karatzas I, Shreve S E. Brownian Motion and Stochastic Calculus. 2nd ed. Berlin: Springer, 1991

    MATH  Google Scholar 

  13. Mao X R. Stochastic Differential Equations and Applications. Chichester: Horwood Publ, 1997

    MATH  Google Scholar 

  14. Guo Y X, Xu C, Wu J. Stability analysis of neutral stochastic delay differential equations by a generalisation of Banach’s contraction principle. Int J Control, 2017, 90: 1555–1560

    Article  MATH  Google Scholar 

  15. Guo Y X, Xu C. Controllability of stochastic delay systems with impulse in a separable Hilbert space. Asian J Control, 2016, 18: 779–783

    Article  MATH  Google Scholar 

  16. Guo Y X. Globally robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse control and time-varying delays. Ukr Math J, 2018, 69: 1220–1233

    Article  MATH  Google Scholar 

  17. Fu X Z, Zhu Q X, Guo Y X. Stabilization of stochastic functional differential systems with delayed impulses. Appl Math Comput, 2019, 346: 776–789

    MATH  Google Scholar 

  18. Zhu Q X, Wang H. Output feedback stabilization of stochastic feedforward systems with unknown control coefficients and unknown output function. Automatica, 2018, 87: 166–175

    Article  MATH  Google Scholar 

  19. Zhang T, Chen C L P, Chen L, et al. Design of highly nonlinear substitution boxes based on I-Ching operators. IEEE Trans Cybern, 2018, 48: 3349–3358

    Article  Google Scholar 

  20. Zhang T, Wang X H, Xu X M, et al. GCB-Net: graph convolutional broad network and its application in emotion recognition. IEEE Trans Affective Comput, 2019. doi: https://doi.org/10.1109/TAFFC.2019.2937768

  21. Chatterjee D, Liberzon D. Stabilizing randomly switched systems. SIAM J Control Optim, 2011, 49: 2008–2031

    Article  MATH  Google Scholar 

  22. Wang B, Zhu Q X. Stability analysis of semi-Markov switched stochastic systems. Automatica, 2018, 94: 72–80

    Article  MATH  Google Scholar 

  23. Jiang B P, Kao Y G, Karimi H R, et al. Stability and stabilization for singular switching semi-Markovian jump systems with generally uncertain transition rates. IEEE Trans Autom Control, 2018, 63: 3919–3926

    Article  MATH  Google Scholar 

  24. Kao Y G, Yang T S, Park J H. Exponential stability of switched Markovian jumping neutral-type systems with generally incomplete transition rates. Int J Robust Nonlinear Control, 2018, 28: 1583–1596

    Article  MATH  Google Scholar 

  25. Kim S H. Stochastic stability and stabilization conditions of semi-Markovian jump systems with mode transition-dependent sojourn-time distributions. Inf Sci, 2017, 385: 314–324

    Article  MATH  Google Scholar 

  26. Fang M, Wang L Q, Wu Z G. Asynchronous stabilization of Boolean control networks with stochastic switched signals. IEEE Trans Syst Man Cybern Syst, 2021, 51: 2425–2432

    Article  Google Scholar 

  27. Wang L Q, Fang M, Wu Z G, et al. Necessary and sufficient conditions on pinning stabilization for stochastic Boolean networks. IEEE Trans Cybern, 2020, 50: 4444–4453

    Article  Google Scholar 

  28. Hu T S, Teel A R, Zaccarian L. Anti-windup synthesis for linear control systems with input saturation: achieving regional, nonlinear performance. Automatica, 2008, 44: 512–519

    Article  MATH  Google Scholar 

  29. Li Y L, Lin Z L. Stability and Performance of Control Systems with Actuator Saturation. Berlin: Springer, 2018

    Book  MATH  Google Scholar 

  30. Liu H P, Boukas E K, Sun F C, et al. Controller design for Markov jumping systems subject to actuator saturation. Automatica, 2006, 42: 459–465

    Article  MATH  Google Scholar 

  31. Tarbouriech S, Garcia G, Silva J, et al. Stability and Stabilization of Linear Systems with Saturating Actuators. Berlin: Springer, 2011

    Book  MATH  Google Scholar 

  32. Zuo Z Q, Li Y Y, Wang Y J, et al. Event-triggered control for switched systems in the presence of actuator saturation. Int J Syst Sci, 2018, 49: 1478–1490

    Article  MATH  Google Scholar 

  33. Duan C, Wu F. Output-feedback control for switched linear systems subject to actuator saturation. Int J Control, 2012, 85: 1532–1545

    Article  MATH  Google Scholar 

  34. Benzaouia A, Saydy L, Akhrif O. Stability and control synthesis of switched systems subject to actuator saturation. In: Proceedings of American Control Conference, Boston, 2004

  35. Chen Y G, Wang Z D, Fei S M, et al. Regional stabilization for discrete time-delay systems with actuator saturations via a delay-dependent polytopic approach. IEEE Trans Autom Control, 2019, 64: 1257–1264

    Article  MATH  Google Scholar 

  36. Cao Y Y, Lin Z, Shamash Y. Set invariance analysis and gain-scheduling control for LPV systems subject to actuator saturation. Syst Control Lett, 2002, 46: 137–151

    Article  MATH  Google Scholar 

  37. Ghosh M K, Goswami A. Risk minimizing option pricing in a semi-Markov modulated market. SIAM J Control Optim, 2009, 48: 1519–1541

    Article  MATH  Google Scholar 

  38. Hu Z H, Mu X W. Stabilization for switched stochastic systems with semi-Markovian switching signals and actuator saturation. Inf Sci, 2019, 483: 419–431

    Article  MATH  Google Scholar 

  39. Hou Z T, Dong H L, Shi P. Asymptotic stability in the distribution of nonlinear stochastic systems with semi-Markovian switching. ANZIAM J, 2007, 49: 231–241

    Article  MATH  Google Scholar 

  40. Li F B, Wu L G, Shi P, et al. State estimation and sliding mode control for semi-Markovian jump systems with mismatched uncertainties. Automatica, 2015, 51: 385–393

    Article  MATH  Google Scholar 

  41. Shen Z X, Li C D, Li H F, et al. Estimation of the domain of attraction for discrete-time linear impulsive control systems with input saturation. Appl Math Comput, 2019, 362: 124502

    MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by Natural Science Foundation of Shandong Province of China (Grant No. ZR2017MA045). The first and the third authors would like to thank the National University of Singapore.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China

    Yingxin Guo

  2. Department of Electrical and Computer Engineering, National University of Singapore, Singapore, 117576, Singapore

    Shuzhi Sam Ge

  3. Chongqing Institutes of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing, 400714, China

    Jianting Fu

  4. University of Chinese Academy of Sciences, Beijing, 100049, China

    Jianting Fu

  5. Department of Control Science and Engineering, Zhejiang University, Hangzhou, 310027, China

    Chao Xu

Authors
  1. Yingxin Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shuzhi Sam Ge
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianting Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chao Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yingxin Guo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, Y., Ge, S.S., Fu, J. et al. Stability and stabilization of a class of switched stochastic systems with saturation control. Sci. China Inf. Sci. 64, 222201 (2021). https://doi.org/10.1007/s11432-020-3002-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11432-020-3002-7

Keywords

Search

Navigation