Backstepping controller design for a class of stochastic nonlinear systems with Markovian switching. (English) Zbl 1162.93044

Summary: A more general class of stochastic nonlinear systems with irreducible homogeneous Markovian switching are considered in this paper. As preliminaries, the stability criteria and the existence theorem of strong solutions are first presented by using the inequality of mathematic expectation of a Lyapunov function. The state-feedback controller is designed by regarding Markovian switching as constant such that the closed-loop system has a unique solution, and the equilibrium is asymptotically stable in probability in the large. The output-feedback controller is designed based on a quadratic-plus-quartic-form Lyapunov function such that the closed-loop system has a unique solution with the equilibrium being asymptotically stable in probability in the large in the unbiased case and has a unique bounded-in-probability solution in the biased case.


93E15 Stochastic stability in control theory
60J75 Jump processes (MSC2010)
93B51 Design techniques (robust design, computer-aided design, etc.)
93B52 Feedback control
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: DOI Link


[1] Arnold, L., Stochastic differential equations: theory and applications, (1972), Wiley New York
[2] Basak, G.K.; Bisi, A.; Ghosh, M.K., Stability of a random diffusion with linear drift, Journal of mathematical analysis and applications, 202, 604-622, (1996) · Zbl 0856.93102
[3] Friedman, A., Stochastic differential equations and their applications, (1976), Academic Press New York
[4] Ji, Y.; Chizeck, H.J., Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE transactions on automatic control, 35, 777-788, (1990) · Zbl 0714.93060
[5] Jiang, Z.P., A combined backstepping and small-gain approach to adaptive output feedback control, Automatica, 35, 1131-1139, (1999) · Zbl 0932.93045
[6] Jiang, Z.P.; Praly, L., Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties, Automatica, 34, 825-840, (1998) · Zbl 0951.93042
[7] Khas’minskii, R.Z., Stochastic stability of differential equations, (1980), S & N International Publisher Rockville, MD · Zbl 0441.60060
[8] Krstić, M.; Deng, H., Stability of nonlinear uncertain systems, (1998), Springer New York
[9] Krstić, M.; Kanellakopoulos, I.; Kokotović, P., Nonlinear and adaptive control design, (1995), John Wiley New York · Zbl 0763.93043
[10] 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
[11] Liu, Y.G.; Zhang, J.F., Practical output-feedback risk-sensitive control for stochastic nonlinear systems with stable zero-dynamics, SIAM journal of control and optimization, 45, 885-926, (2006) · Zbl 1117.93067
[12] Mao, X., Stochastic differential equations and applications, (1997), Horwood New York · Zbl 0874.60050
[13] Mao, X., Stability of stochastic differential equations with Markovian switching, Stochastic processes and their applications, 79, 45-67, (1999) · Zbl 0962.60043
[14] Mao, X.; Yuan, C., Stochastic differential equations with Markovian switching, (2006), Imperial College Press London · Zbl 1126.60002
[15] Marino, R.; Tomei, P., Nonlinear control design: geometric, adaptive and robust, (1995), Prentice Hall New Jersey · Zbl 0833.93003
[16] Pan, Z.G.; Başar, T., Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion, SIAM journal of control and optimization, 37, 957-995, (1999) · Zbl 0924.93046
[17] Ross, S.M., Stochasic processes, (1996), John Wiley & Sons, Inc New York
[18] Shi, P.; Boukas, E.K., H-infinity control for Markovian jumping linear systems with parametric uncertainties, Journal of optimization theory and applications, 95, 75-99, (1997) · Zbl 1026.93504
[19] Shi, P.; Boukas, E.K.; Agarwal, R., Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay, IEEE transactions on automatic control, 44, 2139-2144, (1999) · Zbl 1078.93575
[20] Shi, P.; Xia, Y.Q.; Liu, G.P.; Rees, D., On designing of sliding-mode control for stochastic jump systems, IEEE transactions on automatic control, 51, 97-103, (2006) · Zbl 1366.93682
[21] Skorohod, A.V., Asymptotic methods in the theory of stochastic differential equations, (1989), American Mathematical Society Providence, RI
[22] Tian, J.; Xie, X.J., Adaptive state-feedback stabilization for high-order stochastic non-linear systems with uncertain control coefficients, International journal of control, 79, 1635-1646, (2006) · Zbl 1124.93057
[23] Tong, S.; Li, Y., Direct adaptive fuzzy backstepping control for a class of nonlinear systems, International journal of innovative computing, information and control, 3, 887-896, (2007)
[24] Wu, Z.J.; Xie, X.J.; Zhang, S.Y., Robust decentralized adaptive stabilization for a class of interconnected systems with unmodeled dynamics, International journal of systems science, 35, 389-405, (2004)
[25] Wu, Z.J.; Xie, X.J.; Zhang, S.Y., Adaptive backstepping controller design using stochastic small-gain theorem, Automatica, 43, 608-620, (2007) · Zbl 1114.93104
[26] Xie, X.J.; Tian, J., State-feedback stabilization for high-order stochastic nonlinear systems with stochastuic inverse dynamics, International journal of robust and nonlinear control, 17, 1343-1362, (2007) · Zbl 1127.93354
[27] Yuan, C.; Mao, X., Robust stability and controllability of stochastic differential delay equations with Markovian switching, Automatica, 40, 343-354, (2004) · Zbl 1040.93069
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.