SCIENCE CHINA Information Sciences, Volume 62 , Issue 9 : 192201(2019) https://doi.org/10.1007/s11432-018-9557-x

Stochastic stabilization using aperiodically sampled measurements

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  • ReceivedApr 1, 2018
  • AcceptedAug 3, 2018
  • PublishedJul 30, 2019



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.


[1] Fernholz R, Karatzas I. Relative arbitrage in volatility-stabilized markets. Ann Finance, 2005, 1: 149-177 CrossRef Google Scholar

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

[3] Teel A R, Subbaraman A, Sferlazza A. Stability analysis for stochastic hybrid systems: A survey. Automatica, 2014, 50: 2435-2456 CrossRef Google Scholar

[4] Hoshino K, Nishimura Y, Yamashita Y. Global Asymptotic Stabilization of Nonlinear Deterministic Systems Using Wiener Processes. IEEE Trans Automat Contr, 2016, 61: 2318-2323 CrossRef 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 CrossRef Google Scholar

[7] Mao X. Stochastic stabilization and destabilization. Syst Control Lett, 1994, 23: 279-290 CrossRef Google Scholar

[8] Appleby J A D, Mao X. Stochastic stabilisation of functional differential equations. Syst Control Lett, 2005, 54: 1069-1081 CrossRef Google Scholar

[9] Mao X, Yin G G, Yuan C. Stabilization and destabilization of hybrid systems of stochastic differential equations. Automatica, 2007, 43: 264-273 CrossRef Google Scholar

[10] Appleby J A D, Mao X, Rodkina A. Stabilization and Destabilization of Nonlinear Differential Equations by Noise. IEEE Trans Automat Contr, 2008, 53: 683-691 CrossRef Google Scholar

[11] Huang L. Stochastic stabilization and destabilization of nonlinear differential equations. Syst Control Lett, 2013, 62: 163-169 CrossRef Google Scholar

[12] Nishimura Y. Conditions for local almost sure asymptotic stability. Syst Control Lett, 2016, 94: 19-24 CrossRef Google Scholar

[13] Guo Q, Mao X, Yue R. Almost Sure Exponential Stability of Stochastic Differential Delay Equations. SIAM J Control Optim, 2016, 54: 1919-1933 CrossRef Google Scholar

[14] Mao X. Almost Sure Exponential Stabilization by Discrete-Time Stochastic Feedback Control. IEEE Trans Automat Contr, 2016, 61: 1619-1624 CrossRef Google Scholar

[15] Mao X. Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control. Automatica, 2013, 49: 3677-3681 CrossRef Google Scholar

[16] Mao X, Liu W, Hu L. Stabilization of hybrid stochastic differential equations by feedback control based on discrete-time state observations. Syst Control Lett, 2014, 73: 88-95 CrossRef Google Scholar

[17] Song G, Lu Z, Zheng B C. Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state. Sci China Inf Sci, 2018, 61: 70213 CrossRef Google Scholar

[18] Fridman E. A refined input delay approach to sampled-data control. Automatica, 2010, 46: 421-427 CrossRef 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. 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 CrossRef Google Scholar

[21] Chen W H, Yang W, Zheng W X. Adaptive impulsive observers for nonlinear systems: Revisited. Automatica, 2015, 61: 232-240 CrossRef Google Scholar

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

[23] Lofberg J. Pre- and Post-Processing Sum-of-Squares Programs in Practice. IEEE Trans Automat Contr, 2009, 54: 1007-1011 CrossRef Google Scholar

[24] Sturm J F. Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones. Optimization Methods Software, 1999, 11: 625-653 CrossRef Google Scholar

[25] Hu L, Mao X. Almost sure exponential stabilisation of stochastic systems by state-feedback control. Automatica, 2008, 44: 465-471 CrossRef 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 CrossRef Google Scholar

[27] Briat C, Seuret A. Convex Dwell-Time Characterizations for Uncertain Linear Impulsive Systems. IEEE Trans Automat Contr, 2012, 57: 3241-3246 CrossRef 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. Google Scholar

[29] Seuret A. A novel stability analysis of linear systems under asynchronous samplings. Automatica, 2012, 48: 177-182 CrossRef Google Scholar

  • Figure 3

    (Color online) Sample-path trajectories of stochastic system described in Example 4.2under sampled-data control law with $K=1$ and $\{t_k\}_{k\in\mathbb{N}_0}\in\mathcal{S}(0.3,1.48)$.

  • Table 1   The maximum values of sampling period $\sigma_{0}$ for different $N$
    Theorem$N$ or ${\rm~deg}(P(\sigma))$
    2 4 6 10 100
    Theorem 3.1 $0.061$ $0.086$ $0.086$ $0.086$ 0.086
    Theorem 3.4 $0.048$ $0.062$ $0.068$ $0.075$ $0.085$
  • Table 2   The maximum values of $\sigma_{1}$ for different approaches
    $\sigma_0$ Result
    Theorem 2 of [29] Theorem 3.4 Theorem 3.1
    $\sigma_0=0.21$ $0.43$ $0.60$ $0.72$
    $\sigma_0=0.40$ $1.25$ $1.64$ $1.82$
    $\sigma_0=1.25$ $1.57$ $1.96$ $2.02$