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Adaptive backstepping controller design using stochastic small-gain theorem. (English) Zbl 1114.93104

Summary: A more general class of stochastic nonlinear systems with unmodeled dynamics and uncertain nonlinear functions are considered in this paper. With the concept of input-to-state practical stability (ISpS) and nonlinear small-gain theorem being extended to stochastic case, by combining stochastic small-gain theorem with backstepping design technique, an adaptive output-feedback controller is proposed. It is shown that the closed-loop system is practically stable in probability. A simulation example demonstrates the control scheme.

MSC:

93E35 Stochastic learning and adaptive control
93B52 Feedback control
93C40 Adaptive control/observation systems
93D25 Input-output approaches in control theory
93E03 Stochastic systems in control theory (general)
93C15 Control/observation systems governed by ordinary differential equations
93C10 Nonlinear systems in control theory
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[1] Deng, H.; Krstić, M., Stochastic nonlinear stabilization part I: A backstepping design, System and control letters, 32, 143-150, (1997) · Zbl 0902.93049
[2] Deng, H.; Krstić, M., Stochastic nonlinear stabilization part II: inverse optimality, System and control letters, 32, 151-159, (1997) · Zbl 0902.93050
[3] Deng, H.; Krstić, M., Output-feedback stochastic nonlinear stabilization, IEEE transactions on automatic control, 44, 328-333, (1999) · Zbl 0958.93095
[4] Deng, H.; Krstić, M., Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance, System and control letters, 39, 173-182, (2000) · Zbl 0948.93053
[5] Deng, H.; Krstić, M.; Williiams, R., Stabilization of stochastic nonlinear driven by noise of unknown covariance, IEEE transactions on automatic control, 46, 1237-1253, (2001) · Zbl 1008.93068
[6] Desoer, C.A.; Vidyasagar, M., Feedback systems: input-output properties, (1975), Academic Press New York · Zbl 0327.93009
[7] Florchinger, P., A universal formula for the stabilization of control stochastic differential equations, Stochastic analysis and applications, 11, 155-162, (1993) · Zbl 0770.60058
[8] Florchinger, P., Lyapunov-like techniques for stochastic stability, SIAM journal of control and optimization, 33, 1151-1169, (1995) · Zbl 0845.93085
[9] Florchinger, P., Feedback stabilization of affine in the control stochastic differential systems by the control Lyapunov function method, SIAM journal of control and optimization, 35, 500-511, (1997) · Zbl 0874.93092
[10] Gihman, I.I.; Skorohod, A.V., Stochastic differential equations, (1972), Springer New York · Zbl 0242.60003
[11] Hill, D.; Moylan, P., General instability results for interconnected systems, SIAM journal of control and optimization, 21, 256-279, (1983) · Zbl 0505.93052
[12] Isidori, A., Nonlinear control systems, (1995), Springer New York · Zbl 0569.93034
[13] Jiang, Z.P., A combined backstepping and small-gain approach to adaptive output feedback control, Automatica, 35, 1131-1139, (1999) · Zbl 0932.93045
[14] Jiang, Z.P.; Lin, Y.; Wang, Y., Nonlinear small-gain theorems for discrete-time feedback systems and applications, Automatica, 40, 2129-2136, (2004) · Zbl 1077.93034
[15] Jiang, Z.P.; Mareels, I.M.Y.; Hill, D.J., Robust control of uncertain nonlinear systems via measurement feedback, IEEE transactions on automatic control, 44, 807-812, (1999) · Zbl 0957.93072
[16] Jiang, Z.P.; Mareels, I.M.Y.; Wang, Y., A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems, Automatica, 32, 1211-1215, (1996) · Zbl 0857.93089
[17] Jiang, Z.P.; Praly, L., Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties, Automatica, 34, 825-840, (1998) · Zbl 0951.93042
[18] Jiang, Z.P.; Teel, A.; Praly, L., Small-gain theorem for ISS systems and applications, Mathematics of control signals and systems, 7, 95-120, (1994) · Zbl 0836.93054
[19] Jiang, Z.P.; Wang, Y., Input-to-state stability for discrete-time nonlinear systems, Automatica, 37, 857-869, (2001) · Zbl 0989.93082
[20] Khalil, H.K., Nonlinear systems, (1996), Prentice-Hall Englewood Cliffs, NJ · Zbl 0626.34052
[21] Khas’minskii, R.Z., Stochastic stability of differential equations, (1980), S&N International Publisher Rockville, MD · Zbl 0441.60060
[22] Kokotović, P.V.; Arcak, M., Constructive nonlinear control: A historical perspective, Automatica, 37, 637-662, (2001) · Zbl 1153.93301
[23] Krstić, M.; Deng, H., Stability of nonlinear uncertain systems, (1998), Springer New York
[24] Krstić, M.; Kanellakopoulos, I.; Kokotović, P., Nonlinear and adaptive control design, (1995), Wiley New York · Zbl 0763.93043
[25] Liu, S. J., & Zhang, J. F. (2005). Global output feedback stabilization for stochastic nonlinear systems with stochastic input-to-state stable zero-dynamics. In Proceedings of the 24th Chinese control conference (pp. 93-98), Guang Zhou, PR China.
[26] Liu, Y.G.; Pan, Z.G.; Shi, S.J., Output feedback control design for strict-feedback stochastic nonlinear systems under a risk-sensitive cost, IEEE transactions on automatic control, 48, 509-513, (2003) · Zbl 1364.93283
[27] Liu, Y.G.; Zhang, J.F., Minimal-order observer deign and output-feedback stabilizing control of stochastic nonlinear systems, Science in China series F, 47, 527-544, (2004) · Zbl 1186.93065
[28] Liu, Y.G.; Zhang, J.F., Reduced-order observer-based control design for nonlinear stochastic systems, System and control letters, 52, 123-135, (2004) · Zbl 1157.93538
[29] Liu, Y. G., & Zhang, J. F. (2006). Practical output-feedback risk-sensitive control for stochastic nonlinear systems with stable zero-dynamics. SIAM Journal of Control and Optimization, 45, 885-926. · Zbl 1117.93067
[30] Mareels, I.M.Y.; Hill, D.J., Monotone stability of nonlinear feedback systems, Journal of mathematical systems estimation control, 2, 275-291, (1992) · Zbl 0776.93039
[31] Pan, Z.G.; Basar, 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
[32] Pan, Z. G., Liu, Y. G., & Shi, S. J. (2002). Output feedback stabilization for stochastic nonlinear systems in observer canonical form with stable zero-dynamics. In Proceedings of the 41st IEEE conference on decision and control (pp. 1392-1397), Las Vegas, NV, USA.
[33] Safonov, M.G., Stability and robustness of multivariable feedback systems, (1980), MIT Press Cambridge, MA · Zbl 0552.93002
[34] Sandberg, I.W., A frequency domain condition for the stability of systems containing a single time-varying nonlinear element, The Bell system technical journal, 43, 1601-1638, (1964) · Zbl 0131.31704
[35] Sandberg, I.W., On the \(\mathcal{L}_2\)-boundedness of solutions of nonlinear functional equations, The Bell system technical journal, 43, 1581-1599, (1964) · Zbl 0156.15803
[36] Sontag, E.D., Smooth stabilization implies coprime factorization, IEEE transactions on automatic control, 34, 435-443, (1989) · Zbl 0682.93045
[37] Sontag, E.D., Further facts about input-to-state stabilization, IEEE transactions on automatic control, 35, 473-476, (1990) · Zbl 0704.93056
[38] Sontag, E.D., On the input-to-state stability property, European journal of control, 1, 24-36, (1995) · Zbl 1177.93003
[39] 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
[40] Tsinias, J., Stochastic input-to-state stability and applications to global feedback stabilization (special issue on breakthrough in the control of nonlinear systems), International journal of control, 71, 907-930, (1998) · Zbl 0953.93073
[41] Wen, C.Y.; Zhang, Y.; Soh, Y.C., Robustness of an adaptive backstepping controller without modification, System and control letters, 36, 87-100, (1999) · Zbl 0924.93042
[42] Zames, G., On the input-output stability of time-varying nonlinear feedback systems—parts I and II, IEEE transactions on automatic control, 11, 228-238, (1966)
[43] Zhang, Y.; Wen, C.Y.; Soh, Y.C., Discrete-time robust adaptive control for nonlinear time-varying systems, IEEE transactions on automatic control, 45, 1749-1755, (2000) · Zbl 0990.93062
[44] Zhang, Y.; Wen, C.Y.; Soh, Y.C., Robust adaptive control of nonlinear discrete-time systems by backstepping without overparameterization, Automatica, 37, 551-558, (2001) · Zbl 0990.93063
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