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Stochastic stabilization using aperiodically sampled measurements

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Abstract

This paper addresses the stabilization problem of sector-bounded nonlinear systems with sampled measurements via discrete-time stochastic feedback control. Unlike the previous studies, the closed-loop system is modeled as an impulsive stochastic differential equation. By developing a quasi-periodic polynomial Lyapunov function and sampling-time-dependent Lyapunov function based methods, two sufficient conditions for almost sure exponential stability are derived in terms of differential matrix inequalities (DMIs) and linear matrix inequalities (LMIs). It is shown that the DMI-based conditions can be formulated as a sum of squares (SOSs). Moreover, the obtained results are adapted to sampled-data stochastic/deterministic systems. The numerical examples illustrate the theoretical results.

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References

  1. Fernholz R, Karatzas I. Relative arbitrage in volatility-stabilized markets. Ann Finance, 2005, 1: 149–177

    Article  MATH  Google Scholar 

  2. Mao X R. Stochastic Differential Equations and Applications. 2nd ed. Chichester: Howrwood, 2007

    MATH  Google Scholar 

  3. Teel A R, Subbaraman A, Sferlazza A. Stability analysis for stochastic hybrid systems: a survey. Automatica, 2014, 50: 2435–2456

    Article  MathSciNet  MATH  Google Scholar 

  4. Hoshino K, Nishimura Y, Yamashita Y, et al. Global asymptotic stabilization of nonlinear deterministic systems using wiener processes. IEEE Trans Autom Control, 2016, 61: 2318–2323

    Article  MathSciNet  MATH  Google Scholar 

  5. Has’minskii R Z. Stochastic Stability of Differential Equations. Gronigen: Sijthoff and Noordhoff, 1981

    Google Scholar 

  6. Arnold L, Crauel H, Wihstutz V. Stabilization of linear systems by noise. SIAM J Control Optim, 1983, 21: 451–461

    Article  MathSciNet  MATH  Google Scholar 

  7. Mao X R. Stochastic stabilization and destabilization. Syst Control Lett, 1994, 23: 279–290

    Article  MathSciNet  MATH  Google Scholar 

  8. Appleby J A D, Mao X R. Stochastic stabilisation of functional differential equations. Syst Control Lett, 2005, 54: 1069–1081

    Article  MathSciNet  MATH  Google Scholar 

  9. Mao X R, Yin G G, Yuan C. Stabilization and destabilization of hybrid systems of stochastic differential equations. Automatica, 2007, 43: 264–273

    Article  MathSciNet  MATH  Google Scholar 

  10. Appleby J A D, Mao X R, Rodkina A. Stabilization and destabilization of nonlinear differential equations by noise. IEEE Trans Autom Control, 2008, 53: 683–691

    Article  MathSciNet  MATH  Google Scholar 

  11. Huang L R. Stochastic stabilization and destabilization of nonlinear differential equations. Syst Control Lett, 2013, 62: 163–169

    Article  MathSciNet  MATH  Google Scholar 

  12. Nishimura Y. Conditions for local almost sure asymptotic stability. Syst Control Lett, 2016, 94: 19–24

    Article  MathSciNet  MATH  Google Scholar 

  13. Guo Q, Mao X R, Yue R X. Almost sure exponential stability of stochastic differential delay equations. SIAM J Control Optim, 2016, 54: 1919–1933

    Article  MathSciNet  MATH  Google Scholar 

  14. Mao X R. Almost sure exponential stabilization by discrete-time stochastic feedback control. IEEE Trans Autom Control, 2016, 61: 1619–1624

    Article  MathSciNet  MATH  Google Scholar 

  15. Mao X R. Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control. Automatica, 2013, 49: 3677–3681

    Article  MathSciNet  MATH  Google Scholar 

  16. Mao X R, Liu W, Hu L J, et al. Stabilization of hybrid stochastic differential equations by feedback control based on discrete-time state observations. Syst Control Lett, 2014, 73: 88–95

    Article  MathSciNet  MATH  Google Scholar 

  17. Song G F, Lu Z Y, Zheng B C, et al. Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state. Sci China Inf Sci, 2018, 61: 070213

    Article  MathSciNet  Google Scholar 

  18. Fridman E. A refined input delay approach to sampled-data control. Automatica, 2010, 46: 421–427

    Article  MathSciNet  MATH  Google Scholar 

  19. Mohammed S E A, Scheutzow M K R. Lyapunov exponents of linear stochastic functional differential equations. Part II. Examples and case studies. Ann Prob, 1997, 25: 1210–1240

    Article  MATH  Google Scholar 

  20. Briat C. Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints. Automatica, 2013, 49: 3449–3457

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen W H, Yang W, Zheng W X. Adaptive impulsive observers for nonlinear systems: revisited. Automatica, 2015, 61: 232–240

    Article  MathSciNet  MATH  Google Scholar 

  22. Papachristodoulou A, Anderson J, Valmorbida G, et al. SOSTOOLS: Sum of Squares Optimization Toolbox for MATLAB v3.00, 2013

  23. Lofberg J. Pre- and post-processing sum-of-squares programs in practice. IEEE Trans Autom Control, 2009, 54: 1007–1011

    Article  MathSciNet  MATH  Google Scholar 

  24. Sturm J F. Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optim Method Softw, 1999, 11: 625–653

    Article  MathSciNet  MATH  Google Scholar 

  25. Hu L J, Mao X R. Almost sure exponential stabilisation of stochastic systems by state-feedback control. Automatica, 2008, 44: 465–471

    Article  MathSciNet  MATH  Google Scholar 

  26. Chen W H, Zheng W X, Lu X. Impulsive stabilization of a class of singular systems with time-delays. Automatica, 2017, 83: 28–36

    Article  MathSciNet  MATH  Google Scholar 

  27. Briat C, Seuret A. Convex dwell-time characterizations for uncertain linear impulsive systems. IEEE Trans Autom Control, 2012, 57: 3241–3246

    Article  MathSciNet  MATH  Google Scholar 

  28. Chesi G. On the complexity of SOS programming: formulas for general cases and exact reductions. In: Proceedings of SICE International Symposium on Control Systems (SICE ISCS), Okayama, 2017

  29. Seuret A. A novel stability analysis of linear systems under asynchronous samplings. Automatica, 2012, 48: 177–182

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 61873099, 61733008, 61573156), and Scholarship from China Scholarship Council (Grant No. 201806150120). The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

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Authors and Affiliations

  1. School of Automation Science and Engineering, South China University of Technology, Guangzhou, 510640, China

    Shixian Luo, Feiqi Deng, Xueyan Zhao & Zhipei Hu

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  1. Shixian Luo
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  2. Feiqi Deng
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  3. Xueyan Zhao
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  4. Zhipei Hu
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Correspondence to Shixian Luo or Feiqi Deng.

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Luo, S., Deng, F., Zhao, X. et al. Stochastic stabilization using aperiodically sampled measurements. Sci. China Inf. Sci. 62, 192201 (2019). https://doi.org/10.1007/s11432-018-9557-x

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  • DOI: https://doi.org/10.1007/s11432-018-9557-x

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