zbMATH — the first resource for mathematics

Small-gain control method for stochastic nonlinear systems with stochastic iISS inverse dynamics. (English) Zbl 1218.93089
Summary: This paper investigates the small-gain type conditions on Stochastic iISS (SiISS) systems and makes full use of these conditions in the design and analysis of the controller. The contributions are as follows: (1) A new proof of the stochastic LaSalle theorem is provided; (2) The small-gain type conditions on SiISS are developed and their relationship is discussed; (3) Based on the stochastic LaSalle theorem and SiISS small-gain type conditions, the adaptive controllers are designed to guarantee that all of the closed-loop signals are bounded almost surely and the stochastic closed-loop systems are globally (asymptotically) stable in probability.

93E03 Stochastic systems in control theory (general)
93C10 Nonlinear systems in control theory
93E15 Stochastic stability in control theory
93C40 Adaptive control/observation systems
Full Text: DOI
[1] Deng, H.; Krstić, M.; Williams, R.J., Stabilization of stochastic nonlinear systems driven by noise of unknown covariance, IEEE transactions on automatic control, 46, 1237-1253, (2001) · Zbl 1008.93068
[2] Ito, H., State-dependent scaling problems and stability of interconnected iiss and ISS systems, IEEE transactions on automatic control, 51, 1626-1643, (2006) · Zbl 1366.93604
[3] Ito, H.; Jiang, Z.P., Necessary and sufficient small gain conditions for integral input-to-state stable systems: a Lyapunov perspective, IEEE transactions on automatic control, 54, 2389-2404, (2009) · Zbl 1367.93598
[4] Jiang, Z.P.; Mareels, I.; Hill, D.J.; Huang, J., A unifying framework for global regulation via nonlinear output feedback: from ISS to iiss, IEEE transactions on automatic control, 49, 549-562, (2004) · Zbl 1365.93177
[5] Jiang, Z.P.; Mareels, I.; Wang, Y., A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems, Automatica, 32, 1211-1215, (1996) · Zbl 0857.93089
[6] Jiang, Z.P.; Teel, A.R.; Praly, L., Small-gain theorem for ISS systems and applications, Mathematics of control, signal, and systems, 7, 95-120, (1994) · Zbl 0836.93054
[7] Khas’minskii, R.Z., Stochastic stability of differential equations, (1980), Kluwer Academic Publishers Norwell, Massachusetts · Zbl 0441.60060
[8] Khalil, H.K., Nonlinear systems, (2002), Prentice-Hall New Jersey · Zbl 0626.34052
[9] Krstić, M.; Deng, H., Stabilization of uncertain nonlinear systems, (1998), Springer New York · Zbl 0906.93001
[10] LaSalle, J.P., Stability theory for ordinary differential equations, Journal of differential equations, 4, 57-65, (1968) · Zbl 0159.12002
[11] Liu, S.J.; Jiang, Z.P.; Zhang, J.F., Global output-feedback stabilization for a class of stochastic non-minimum-phase nonlinear systems, Automatica, 44, 1944-1957, (2008) · Zbl 1283.93230
[12] Liu, Y.G.; Zhang, J.F., Practical output-feedback risk-sensitive control for stochastic nonlinear systems with stable zero-dynamics, SIAM journal on control and optimization, 45, 885-926, (2006) · Zbl 1117.93067
[13] Liu, S.J.; Zhang, J.F.; Jiang, Z.P., Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems, Automatica, 43, 238-251, (2007) · Zbl 1115.93076
[14] Mao, X.R., Stochastic versions of the Lasalle theorem, Journal of differential equations, 153, 175-195, (1999) · Zbl 0921.34057
[15] Mao, X.R., A note on the Lasalle-type theorems for stochastic differential delay equations, Journal of mathematical analysis and applications, 268, 125-142, (2002) · Zbl 0996.60064
[16] Pan, Z.G.; Liu, Y.G.; Shi, S.J., Output feedback stabilization for stochastic nonlinear systems in observer canonical form with stable zero-dynamics, Science in China (series F), 44, 292-308, (2001) · Zbl 1125.93489
[17] Shen, Y.; Luo, Q.; Mao, X., The improved Lasalle-type theorems for stochastic functional differential equations, Journal of mathematical analysis and applications, 318, 134-154, (2006) · Zbl 1090.60059
[18] Sontag, E.D., Smooth stabilization implies coprime factorization, IEEE transactions on automatic control, 34, 435-443, (1989) · Zbl 0682.93045
[19] Sontag, E.D., Comments on integral variants of ISS, Systems and control letters, 34, 93-100, (1998) · Zbl 0902.93062
[20] Sontag, E.D.; Teel, A., Changing supply functions in input/state stable systems, IEEE transactions on automatic control, 40, 1476-1478, (1995) · Zbl 0832.93047
[21] Spiliotis, J.; Tsinias, J., Notions of exponential robust stochastic stability, ISS and their Lyapunov characterizations, International journal of robust and nonlinear control, 13, 173-187, (2003) · Zbl 1049.93086
[22] Tang, C., & Basar, T. (2001). Stochastic stability of singularly perturbed nonlinear systems. In Proceedings of the 40th IEEE conference on decision and control. Orlando, Florida, USA (pp. 399-404).
[23] Teel, A.R., A nonlinear small gain theorem for the analysis of control systems with saturation, IEEE transactions on automatic control, 41, 1256-1270, (1996) · Zbl 0863.93073
[24] Tsinias, J., Stochastic ISS and applications to global feedback stabilization, International journal of control, 71, 907-930, (1998) · Zbl 0953.93073
[25] Tsinias, J., The concept of ‘exponential input to state stability’ for stochastic systems and applications to feedback stabilization, Systems and control letters, 36, 221-229, (1999) · Zbl 0913.93067
[26] Z.J., Wu; Xie, X.J.; Zhang, S.Y., Adaptive backstepping controller design using stochastic small-gain theorem, Automatica, 43, 608-620, (2007) · Zbl 1114.93104
[27] Xie, X.J.; Tian, J., State-feedback stabilization for high-order stochastic nonlinear systems with stochastic inverse dynamics, International journal of robust and nonlinear control, 17, 1343-1362, (2007) · Zbl 1127.93354
[28] Yu, X.; Xie, X.J., Output feedback regulation of stochastic nonlinear systems with stochastic iiss inverse dynamics, IEEE transactions on automatic control, 55, 304-320, (2010) · Zbl 1368.93584
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.