Abstract
In this paper, (reverse) mode-dependent average dwell time (MDADT) method combined with multiple incremental Lyapunov functions is utilized to investigate the stochastic incremental input-to-state stability (SIISS) of stochastic switched systems. The sufficient conditions in terms of the (reverse) MDADT scheme are extracted to ensure SIISS, while it is shown that the SIISS can still be achieved even if all subsystems are not stochastically incrementally input-to-state stable. In particular, an incremental supply rate is introduced to obtain sufficient conditions to ensure SIISS for stochastic feedback interconnected switched systems. Specifically, when the stochastic switched systems are composed of some SIISS and non-SIISS subsystems, the (reverse) MDADT method is employed to establish a relationship between these two kinds of subsystems to ensure SIISS for the stochastic switched systems. Two examples are presented to demonstrate the effectiveness of the results.
Similar content being viewed by others
References
D. Angeli, Further results on incremental input-to-state stability. IEEE Trans. Autom. Control 54(6), 1386–1391 (2009)
C. Briat, Stability analysis and stabilization of stochastic linear impulsive switched and sampled-data systems under dwell-time constraints. Automatica 74, 279–287 (2016)
W. Chen, W. Zheng, Input-to-state stability and integral input-to-state stability of nonlinear impulsive systems with delays. Automatica 45(6), 1481–1488 (2009)
H. Chen, P. Shi, C. Lim, Stability analysis for neutral stochastic delay systems with Markovian switching. Syst. Control Lett. 110, 38–48 (2017)
P. Cheng, S. He, J. Cheng, X. Luan, F. Liu, Asynchronous output feedback control for a class of conic-type nonlinear hidden Markov jump systems within a finite-time interval. IEEE Trans. Syst. Man Cybern. Syst. (2020). https://doi.org/10.1109/TSMC.2020.2980312
S. Dashkovskiy, A. Mironchenko, Input-to-state stability of nonlinear impulsive systems. SIAM J. Control Optim. 51(3), 1962–1987 (2013)
W. Feng, J. Tian, P. Zhao, Stability analysis of switched stochastic systems. Automatica 47(1), 148–157 (2011)
A.S.R. Ferreira, M. Arcak, E.D. Sontag, Stability certification of large scale stochastic systems using dissipativity. Automatica 48(11), 2956–2964 (2012)
W.H. Haddad, T. Rajpurohit, X. Jin, Energy-based feedback control for stochastic port-controlled Hamiltonian systems. Automatica 97, 134–142 (2018)
P. Jagtap, M. Zamani, Backstepping design for incremental stability of stochastic Hamiltonian systems with jumps. IEEE Trans. Autom. Control 63(1), 255–261 (2018)
M. Li, F. Deng, Moment exponential input-to-state stability of non-linear switched stochastic systems with Lévy noise. IET Control Theory Appl. 12(9), 1208–1215 (2018)
X. Li, P. Li, Q. Wang, Input/output-to-state stability of impulsive switched systems. Syst. Control Lett. 116, 1–7 (2018)
H. Liang, X. Guo, Y. Pan, T. Huang, Event-triggered fuzzy bipartite tracking control for network systems based on distributed reduced-order observers. IEEE Trans. Fuzzy Syst. (2020). https://doi.org/10.1109/TFUZZ.2020.2982618
M.A. Müller, D. Liberzon, Input/output-to-state stability and state-norm estimators for switched nonlinear systems. Automatica 48, 2029–2039 (2012)
C.D. Persis, R.D. Santis, A.S. Morse, Switched nonlinear systems with state-dependent dwell-time. Syst. Control Lett. 50(4), 291–302 (2003)
A. Patel, B. Kosko, Stochastic resonance in continuous and spiking neuron models with levy noise. IEEE Trans. Neural. Netw. 19(12), 1993–2008 (2008)
Y. Pan, P. Du, H. Xue, H.K. Lam, Singularity-free fixed-time fuzzy control for robotic systems with user-defined performance. IEEE Trans. Fuzzy Syst. (2020). https://doi.org/10.1109/TFUZZ.2020.2999746
W. Ren, J. Xiong, Stability and stabilization of switched stochastic systems under asynchronous switching. Syst. Control Lett. 97, 184–192 (2016)
W. Ren, J. Xiong, Stability analysis of impulsive stochastic nonlinear systems. IEEE Trans. Autom. Control 62(9), 4791–4797 (2017)
Y. Ren, W. Wang, Y. Wang, Incremental \(H_\infty \) control for switched nonlinear systems. Appl. Math. Comput. 331, 251–263 (2018)
Y. Ren, W. Wang, Y. Wang, M. Shen, Incremental \(H_{\infty }\) performance for a class of stochastic switched nonlinear systems. J. Frank. Inst. 355(15), 7134–7157 (2018)
W. Ren, J. Xiong, Vector Lyapunov function based input-to-state stability of stochastic impulsive switched time-delay systems. IEEE Trans. Autom. Control 64(2), 654–669 (2019)
C. Ren, S. He, Finite-time stabilization for positive Markovian jumping neural networks. Appl. Math. Comput. 365, 124631 (2020)
E.D. Sontag, Smooth stabilization implies coprime factorization. IEEE Trans. Autom. Control 34(4), 435–443 (1989)
E.D. Sontag, On the input-to-state stability property. Eur. J. Control 1(1), 24–36 (1995)
E.D. Sontag, On characterizations of the input-to-state stability property. Syst. Control Lett. 24(5), 351–359 (1995)
M. Shen, C. Fei, W. Fei, X. Mao, Boundedness and stability of highly nonlinear hybrid neutral stochastic systems with multiple delays. Sci. China Inf. Sci. 62(10), 202205 (2019)
J. Tsinias, Stochastic input-to-state stability and applications to global feedback stabilization. Int. J. Control 71(5), 907–930 (1998)
A.R. Teel, J.P. Hespanha, A. Subbaraman, Equivalent characterizations of input-to-state stability for stochastic discrete-time systems. IEEE Trans. Autom. Control 59(2), 516–522 (2014)
L. Vu, D. Chatterjee, D. Liberzon, Input-to-state stability of switched systems and switching adaptive control. Automatica 43, 639–646 (2007)
X. Wu, Y. Tang, W. Zhang, Stability analysis of stochastic delayed systems with an application to multi-agent systems. IEEE Trans. Autom. Control 63(12), 4143–4149 (2016)
X. Wu, Y. Tang, W. Zhang, Input-to-state stability of impulsive stochastic delayed systems under linear assumptions. Automatica 66, 195–204 (2016)
X. Wu, Y. Tang, J. Cao, Input-to-state stability of time-varying switched systems with time-delays. IEEE Trans. Autom. Control 64(6), 2537–2544 (2019)
P. Wang, X. Wang, H. Su, Stability analysis for complex-valued stochastic delayed networks with Markovian switching and impulsive effects. Commun. Nonlinear Sci. Numer. Simul. 73, 35–51 (2019)
W. Wang, H. Liang, Y. Pan, T. Li, Prescribed performance adaptive fuzzy containment control for nonlinear multiagent systems using disturbance observer. IEEE Trans. Cybern. (2020). https://doi.org/10.1109/TCYB.2020.2969499
W. Xie, C. Wen, Z. Li, Input-to-state stabilization of switched nonlinear systems. IEEE Trans. Autom. Control 46(7), 1111–1116 (2001)
G. Yang, D. Liberzon, Input-to-state stability for switched systems with unstable subsystems: a hybrid Lyapunov construction, in Proceedings of the 53rd IEEE Conference on Decision and Control. Los Angeles, California, USA, pp. 6240–6245 (2014)
G. Yang, D. Liberzon, A Lyapunov-based small-gain theorem for interconnected switched systems. Syst. Control Lett. 78, 47–54 (2015)
Y. Yin, X. Zhao, X. Zheng, New stability and stabilization conditions of switched systems with mode-dependent average dwell time. Circuits Syst. Signal Process 36(1), 82–98 (2016)
L. Zhang, P. Shi, Stability \(l_2\)-gain and asynchronous \(H_\infty \) control of discrete-time switched systems with average dwell time. IEEE Trans. Autom. Control 54(9), 2193–2200 (2009)
L. Zhang, H. Gao, Asynchronously switched control of switched linear systems with average dwell time. Automatica 46(5), 953–958 (2010)
X. Zhao, L. Zhang, P. Shi, M. Liu, Stability and stabilization of switched linear systems with mode-dependent average dwell time. IEEE Trans. Autom. Control 57(7), 1809–1815 (2012)
P. Zhao, W. Feng, Y. Kang, Stochastic input-to-state stability of switched stochastic nonlinear systems. Automatica 48(10), 2569–2576 (2012)
X. Zhao, Y. Yin, B. Niu, X. Zheng, Stabilization for a class of switched nonlinear systems with novel average dwell time switching by T–S fuzzy modeling. IEEE Trans. Cybern. 46(8), 1952–1957 (2016)
X. Zhao, P. Shi, Y. Yin, S.K. Nguang, New results on stability of slowly switched systems: a multiple discontinuous Lyapunov function approach. IEEE Trans. Autom. Control 62(7), 3502–3509 (2017)
T. Zhang, J. Li, W. Xu, X. Li, Stability and \(L_2\)-gain analysis for impulsive switched systems. Commun. Nonlinear Sci. Numer. Simul. 78, 1048–1054 (2019)
Z. Zhu, Y. Pan, Q. Zhou, C. Lu, Event-triggered adaptive fuzzy control for stochastic nonlinear systems with unmeasured states and unknown backlash-like hysteresis. IEEE Trans. Fuzzy Syst. (2020). https://doi.org/10.1109/TFUZZ.2020.2973950
Acknowledgements
This work was jointly supported by National Natural Science Foundation of China (Grant No. 61603188), Talents Introduction Project of Hebei Agricultural University (Grant No. YJ2020036) and Fundamental and Frontier Research Project of Chongqing (Grant No. cstc2018jcyjAX0144).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Data Availability
The authors declare that the manuscript has no associated data.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ren, Y., Wang, W., Zhou, W. et al. Stochastic Incremental Input-to-State Stability of Nonlinear Switched Systems with Brownian Motions. Circuits Syst Signal Process 40, 2242–2266 (2021). https://doi.org/10.1007/s00034-020-01574-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-020-01574-9