Stochastic input-to-state stability of switched stochastic nonlinear systems. (English) Zbl 1271.93171

Zhao, Ping; Feng, Wei; Kang, Yu

Automatica 48, No. 10, 2569-2576 (2012).

Summary: In this paper, global asymptotic stability in probability (GASiP) and stochastic input-to-state stability (SISS) for nonswitched stochastic nonlinear (nSSNL) systems and switched stochastic nonlinear (SSNL) systems are investigated. For the study of GASiP, the definition which we considered is not the usual notion of asymptotic stability in probability (stability in probability plus attractivity in probability); it can depict the properties of the system quantitatively. Correspondingly, based on this definition, some sufficient conditions are provided for nSSNL systems and SSNL systems. Furthermore, the definition of SISS is introduced and corresponding criteria are provided for nSSNL systems and SSNL systems. In the proof of the above results, to overcome the difficulties coming with the appearance of switching and the stochastic property at the same time, we generalize the past comparison principle and fully use the properties of the functions which we constructed. In terms of the average dwell-time of the switching laws, a sufficient SISS condition is obtained for SSNL systems. Finally, some examples are provided to demonstrate the applicability of our results.

MSC:

93E15	Stochastic stability in control theory
93C10	Nonlinear systems in control theory
93C30	Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)

Keywords:

stochastic systems; nonlinear systems; switched systems; stochastic input-to-state stability; SISS-Lyapunov function

References:

[1]	Angeli, D., Input-to-state stability of PD-controlled robotic systems, Automatica, 35, 1285-1290 (1999) · Zbl 0949.93072
[2]	Angeli, D., A Lyapunov approach to incremental stability properties, IEEE Transactions on Automatic Control, 47, 410-422 (2002) · Zbl 1364.93552
[3]	Angeli, D.; Sontag, E. D.; Wang, Y., A Lyapunov characterization of integral input-to-state stability, IEEE Transactions on Automatic Control, 45, 6, 1082-1097 (2000) · Zbl 0979.93106
[4]	Arslan, G.; Basar, T., Disturbance attenuating controller design for strict-feedback systems with structurally unknown dynamics, Automatica, 37, 1175-1188 (2001) · Zbl 0983.93043
[6]	Deng, H.; Krstić, H., Stochastic nonlinear stabilization-I: a backstepping design, Systems & Control Letters, 32, 143-150 (1997) · Zbl 0902.93049
[7]	Deng, H.; Kristić, M.; Williams, R. J., Stabilization of stochastic nonlinear systems driven by noise of unknown covariance, IEEE Transactions on Automatic Control, 46, 8, 1237-1253 (2001) · Zbl 1008.93068
[8]	Feng, W.; Zhang, J. F., Input-to-state stability of switched nonlinear systems, Science in China (Series F: Information Sciences), 51, 12, 1992-2004 (2008) · Zbl 1291.93284
[10]	Khalil, H. K., Nonlinear systems (1996), Prentice Hall, Inc. · Zbl 0626.34052
[11]	Khas’minskii, R. Z., Stochastic stability of differential equations (1980), S&N International publisher: S&N International publisher Rockville, Maryland · Zbl 0441.60060
[12]	Khas’minskii, R. Z.; Zhu, C.; Yin, G., Stability of regime-switching diffusions, Stochastic Processes and Their Applications, 117, 1037-1051 (2007) · Zbl 1119.60065
[13]	Krstić, M.; Deng, H., Stabilization of nonlinear uncertain systems (1998), Springer-Verlag: Springer-Verlag London · Zbl 0906.93001
[14]	Liberzon, D.; Morse, A. S.; Sontag, E. D., Output-input stability and minimum-phase nonlinear systems, IEEE Transactions on Automatic Control, 47, 3, 422-436 (2002) · Zbl 1364.93727
[15]	Lin, Y. D.; Sontag, E. D.; Wang, Y., Input to state stabilizability for parameterized families of systems, International Journal of Robust Nonlinear Control, 5, 3, 187-205 (1995) · Zbl 0830.93060
[16]	Lin, Y. D.; Sontag, E. D.; Wang, Y., A smooth converse Lyapunov theorem for robust stability, SIAM Journal on Control and Optimization, 34, 124-160 (1996) · Zbl 0856.93070
[17]	Liu, S. J.; Zhang, J. F.; Jiang, Z. P., A notion of stochastic input-to-state stability and its application to stability of cascaded stochastic nonlinear systems, Acta Mathematicae Applicatae Sinica, English Series, 24, 1, 1-16 (2008)
[18]	Mancilla-Aguilar, J. L.; Garcia, R. A., On converse Lyapunov theorems for ISS and iISS switched nonlinear systems, Systems and Control Letters, 42, 47-53 (2001) · Zbl 0985.93052
[20]	Praly, L.; Wang, Y., Stabilization in spite of matched unmodelled dynamics and an equivalent definition of input-to-state stability, Mathematics of Control, Signals, and Systems, 9, 1, 1-33 (1996) · Zbl 0869.93040
[21]	Sontag, E. D., Smooth stabilization implies coprime factorization, IEEE Transactions on Automatic Control, 34, 4, 435-443 (1989) · Zbl 0682.93045
[23]	Sontag, E. D.; Wang, Y., On characterization of the input-to-state stability property, Systems & Control Letters, 24, 5, 351-359 (1995) · Zbl 0877.93121
[24]	Sontag, E. D.; Wang, Y., New characterizations of input to state stability, IEEE Transactions on Automatic Control, 41, 9, 1283-1294 (1996) · Zbl 0862.93051
[26]	Sontag, E. D.; Wang, Y., Lyapunov characterizations of input to output stability, SIAM Journal on Control and Optimization, 39, 1, 226-249 (2001) · Zbl 0968.93076
[27]	Tsinias, J., Stochastic input-to-state stability and applications to global feedback stabilization, International Journal of Control, 71, 5, 907-930 (1998) · Zbl 0953.93073
[28]	Vu, L.; Chatterjee, D.; Liverzon, D., Input-to-state stability of switched systems and switching adaptive control, Automatica, 43, 639-646 (2007) · Zbl 1261.93049
[29]	Wu, Z. J.; Cui, M. Y.; Xie, X. J.; Shi, P., Theory of stochastic dissipative systems, IEEE Transactions on Automatic Control, 56, 7, 1650-1655 (2011) · Zbl 1368.93613
[30]	Wu, Z. J.; Xie, X. J.; Shi, P.; Xia, Y. Q., Backstepping controller design for a class of stochastic nonlinear systems with Markovian switching, Automatica, 45, 997-1004 (2009) · Zbl 1162.93044
[31]	Yin, G.; Zhu, C., Hybrid switching diffusions: properties and applications (2010), Springer: Springer New York · Zbl 1279.60007
[32]	Zhou, M. Q., Theory of real variable functions (2001), Peking University Press: Peking University Press Beijing, (in Chinese)