\(\mathcal{GKL}\)-finite-time stability via comparison principles for stochastic impulsive systems. (English) Zbl 07892450
Summary: This paper studies the finite-time stability (FTS) for stochastic impulsive systems (SIS). The notions of FTS including \(\mathcal{GKL}\)-FTS and event-\(\mathcal{GKL}\)-FTS are proposed for SIS. The comparison principles for \(\mathcal{GKL}\)-FTS and event-\(\mathcal{GKL}\)-FTS are established, respectively. Then the FTS criteria with the settling time estimates are derived. The criteria of \(\mathcal{GKL}\)-FTS and event-\(\mathcal{GKL}\)-FTS relax the requirements on stabilizing flow and jump for FTS in the literature. Finally, two examples are presented to demonstrate the obtained results.
© 2023 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
© 2023 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
MSC:
| 93-XX | Systems theory; control |
Keywords:
comparison principle; finite-time stability (FTS); GKL-FTS; Ito’s formula; stochastic impulsive systems (SIS)References:
| [1] | G.Kamenkov, On stability of motion over a finite interval of time, J. Appl. Math. Mech.17 (1953), 529-540. · Zbl 0055.32101 |
| [2] | S.Bhat and D.Bernstein, Continuous finite‐time stabilization of the translational and rotational double integrator, IEEE Trans. Automat. Control43 (1998), 678-682. · Zbl 0925.93821 |
| [3] | S.Bhat and D.Bernstein, Finite‐time stability of continuous autonomous systems, SIAM, J. Control Optim.38 (2000), 751-766. · Zbl 0945.34039 |
| [4] | Y.Hong, Y.Xu, and J.Huang, Finite‐time control for robot manipulators, Syst. Control Lett.46 (2002), 243-253. · Zbl 0994.93041 |
| [5] | E.Moulay and W.Perruquetti, Finite time stability of differential inclusions, IMA J. Math. Control Inf.22 (2005), 465-475. · Zbl 1115.93351 |
| [6] | E.Moulay and W.Perruquetti, Finite time stability and stabilization of a class of continuous systems, J. Math. Anal. Appl.323 (2006), 1430-1443. · Zbl 1131.93043 |
| [7] | M.Galicki, Finite‐time control of robotic manipulators, Automatica51 (2015), 49-54. · Zbl 1309.93109 |
| [8] | E.Moulay, M.Dambrine, N.Yeganefar, and W.Perruquetti, Finite‐time stability and stabilization of time‐delay systems, Syst. Control Lett.57 (2008), 561-566. · Zbl 1140.93447 |
| [9] | W.Chen and L.Jiao, Finite‐time stability theorem of stochastic nonlinear systems, Automatica46 (2010), 2105-2108. · Zbl 1401.93213 |
| [10] | J.Yin, S.Khoo, Z.Man, and X.Yu, Finite‐time stability and instability of stochastic nonlinear systems, Automatica47 (2011), 2671-2677. · Zbl 1235.93254 |
| [11] | X.Yu, J.Yin, and S.Khoo, Generalized Lyapunov criteria on finite‐time stability of stochastic systems, Automatica107 (2019), 183-189. · Zbl 1429.93411 |
| [12] | W.Haddad and A.Afflitto, Finite‐time stabilization and optimal feedback control, IEEE Trans. Automat. Control61 (2016), 1069-1074. · Zbl 1359.93366 |
| [13] | Z.Yan, X.Zhou, G.Chang, and Z.Gao, Finite‐time annular domain stability and stabilization of stochastic systems with semi‐Markovian switching, IEEE Trans. Automat. Control, DOI 10.1109/TAC.2022.3228202. · Zbl 07794468 |
| [14] | Y.Su, C.Zheng, and P.Mercorelli, Global finite‐time stabilization of a class of perturbed planar systems with actuator saturation and disturbances, Asian J. Control24 (2022), no. 3, 1497-1502. · Zbl 07887070 |
| [15] | Z.He, B.Wu, Y.Wang, and M.He, Finite‐time event‐triggered ([H {}_{\infty } \]\) filtering of the continuous‐time switched time‐varying delay linear systems, Asian J. Control24 (2022), no. 3, 1197-1208. · Zbl 07887046 |
| [16] | C.Yin, D.Zhao, J.Zhang, S.Wang, X.Xu, X.Sun, and D.Shi, Body height robust control of automotive air suspension system using finite‐time approach, Asian J. Control24 (2022), no. 2, 859-871. · Zbl 07887018 |
| [17] | F.Amato, R.Ambrosino, M.Ariola, and C.Cosentino, Finite‐time stability of linear time‐varying systems with jumps, Automatica45 (2009), 1354-1358. · Zbl 1162.93375 |
| [18] | F.Amato, R.Ambrosino, M.Ariola, and G.De Tommasi, Robust finite‐time stability of impulsive dynamical linear systems subject to norm‐bounded uncertainties, Int. J. Robust. Nonlinear Control21 (2021), 1080-1092. · Zbl 1225.93087 |
| [19] | J.Xu and J.Sun, Finite‐time stability of nonlinear switched impulsive systems, Int. J. Syst. Sci.44 (2013), no. 5, 889-895. · Zbl 1278.93231 |
| [20] | G. P.Chen and Y.Yang, Robust finite‐time stability of fractional order linear time‐varying impulsive systems, Circ. Syst. Sig. Process.34 (2015), 1325-1341. · Zbl 1341.93060 |
| [21] | X. D.Li, D. W. C.Ho, and J.Cao, Finite‐time stability and settling‐time estimation of nonlinear impulsive systems, Automatica99 (2019), 361-368. · Zbl 1406.93260 |
| [22] | Z.Ai and G.Zong, Finite‐time stochastic input‐to‐state stability of impulsive switched stochastic nonlinear systems, Appl. Math. Comput.245 (2014), 462-473. · Zbl 1335.93136 |
| [23] | Q.Gao, B.Liu, and D.Liu, Global finite‐time stability for stochastic impulsive systems via comparison approach, Proc. of the 40th Chinese Control Conference, Shanghai, China, July, 26-28, 2021. |
| [24] | B.Liu, Stability of solution for stochastic impulsive system via comparison approach, IEEE Trans. Automat. Control53 (2008), no. 9, 2128-2133. · Zbl 1367.93523 |
| [25] | B.Liu and D. J.Hill, Comparison principle and stability of discrete‐time impulsive hybrid systems, IEEE Trans. Circuits Syst. I Reg. Papers56 (2009), 233-245. |
| [26] | J. L.Mancilla‐Aguilar and H.Haimovich, Uniform input‐to‐state stability for switched and time‐varying impulsive systems, IEEE Trans. Automat. Control65 (2020), 5028-5042. · Zbl 1536.93742 |
| [27] | X.Mao, G.Yin, and C.Yuan, Stability and destabilization of hybrid systems of stochastic differential equations, Automatica43 (2007), 264-273. · Zbl 1111.93082 |
| [28] | B.Liu, D. J.Hill, and Z. J.Sun, Stabilisation to input‐to‐state stability for continuous‐time dynamical systems via event‐triggered impulsive control with three levels of events, IET Control Theory Appl.12 (2018), no. 9, 1167-1179. |
| [29] | B.Liu, B.Xu, and T.Liu, Almost sure contraction for stochastic switched impulsive systems, IEEE Trans. Automat. Control66 (2021), no. 11, 5393-5400. · Zbl 1536.93852 |
| [30] | B.Niu, D.Wang, N. D.Alotaibi, and E.Fuad, Alsaadi, Adaptive neural state‐feedback tracking control of stochastic nonlinear switched systems: an average dwell‐time method, IEEE Trans. Neural Netw. Learn. Syst.30 (2019), no. 4, 1076-1087. |
| [31] | B.Liu and D. J.Hill, Stability via hybrid‐event‐time Lyapunov function and impulsive stabilization for discrete‐time delayed switched systems, SIAM J. Control Optim.52 (2014), no. 2, 1338-1365. · Zbl 1292.93103 |
| [32] | B.Liu, D. J.Hill, and Z. J.Sun, Input‐to‐state‐KL‐stability with criteria for a class of hybrid dynamical systems, Appl. Math. Comput.326 (2018), 124-140. · Zbl 1426.93294 |
| [33] | W.Lu, X.Liu, and T.Chen, A note on finite‐time and fixed‐time stability, Neural Netw.81 (2016), 11-15. · Zbl 1417.34123 |
| [34] | F. H.Clarke, Yu. S.Ledyaev, and R. J.Stern, Asymptotic stability and smooth Lyapunov function, J. Differ. Equ.149 (1998), 69-114. · Zbl 0907.34013 |
| [35] | Z.Cao, Y.Niu, J.Lam, and X.Song, A hybrid sliding mode control scheme of Markovian jump systems via transition rates optimal design, IEEE Trans. Syst. Man Cybern. Syst.51 (2021), 7752-7763. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
