×

zbMATH — the first resource for mathematics

Stochastic nonlinear stabilization. I: A backstepping design. (English) Zbl 0902.93049
Summary: While the current robust nonlinear control toolbox includes a number of methods for systems affine in deterministic bounded disturbances, the problem when the disturbance is an unbounded stochastic noise has hardly been considered. We present a control design which achieves global asymptotic (Lyapunov) stability in probability for a class of strict-feedback nonlinear continuous-time systems driven by white noise. In a companion paper, we develop inverse optimal control laws for general stochastic systems affine in the noise input, and for strict-feedback systems. A reader of this paper needs no prior familiarity with techniques of stochastic control.

MSC:
93C99 Model systems in control theory
93E15 Stochastic stability in control theory
93D15 Stabilization of systems by feedback
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Başar, T.; Bernhard, P., H∞-optimal control and related minimax design problems: A dynamic game approach, (1995), Birkhäuser Boston, MA · Zbl 0835.93001
[2] Deng, H.; Krstić, M., Stochastic nonlinear stabilization — II: inverse optimality, Systems control lett., 32, 151-159, (1997), this issue · Zbl 0902.93050
[3] Florchinger, P., A universal formula for the stabilization of control stochastic differential equations, Stochastic anal. appl., 11, 155-162, (1993) · Zbl 0770.60058
[4] Florchinger, P., Lyapunov-like techniques for stochastic stability, SIAM J. control optim., 33, 1151-1169, (1995) · Zbl 0845.93085
[5] Florchinger, P., Global stabilization of cascade stochastic systems, (), 2185-2186
[6] Freeman, R.A.; Kokotović, P.V., Robust nonlinear control design, (1996), Birkhäuser Boston, MA · Zbl 0863.93075
[7] James, M.R.; Baras, J.; Elliott, R.J., Risk-sensitive control and dynamic games for partially observed discrete-time nonlinear systems, IEEE trans. automat. control, 39, 780-792, (1994) · Zbl 0807.93067
[8] Khalil, H.K., Nonlinear systems, (1996), Prentice-Hall Englewood Cliffs, NJ · Zbl 0626.34052
[9] Khas’minskii, R.Z., Stochastic stability of differential equations, (1980), S & N International Publisher Rockville, MD · Zbl 0441.60060
[10] Krstić, M.; Kanellakopoulos, I.; Kokotović, P.V., Nonlinear and adaptive control design, (1995), Wiley New York · Zbl 0763.93043
[11] Kushner, H.J., Stochastic stability and control, (1967), Academic New York · Zbl 0178.20003
[12] Nagai, H., Bellman equations of risk-sensitive control, SIAM J. control optim., 34, 74-101, (1996) · Zbl 0856.93107
[13] Øksendal, B., Stochastic differential equations — an introduction with applications, (1995), Springer New York · Zbl 0841.60037
[14] Pan, Z.; Başar, T., Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion, SIAM J. control optim., (1996), submitted
[15] Runolfsson, T., The equivalence between infinite horizon control of stochastic systems with exponential-of-integral performance index and stochastic differential games, IEEE trans. automat. control, 39, 1551-1563, (1994) · Zbl 0930.93084
[16] Sontag, E.D., A ‘universal’ construction of Artstein’s theorem on nonlinear stabilization, Systems control lett., 13, 117-123, (1989) · Zbl 0684.93063
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.