Xie, Xue-Jun; Tian, Jie Adaptive state-feedback stabilization of high-order stochastic systems with nonlinear parameterization. (English) Zbl 1154.93427 Automatica 45, No. 1, 126-133 (2009). Summary: This paper investigates the adaptive state-feedback stabilization of high-order stochastic systems with nonlinear parameterization. By using the parameter separation lemma in [W. Lin and C. Qian, Adaptive control of nonlinearly parameterized systems: A nonsmooth feedback framework. IEEE Trans. Autom. Control 47, 757–774 (2002)] and some flexible algebraic techniques, and choosing an appropriate Lyapunov function, a smooth adaptive state-feedback controller is designed, which guarantees that the closed-loop system has an almost surely unique solution for any initial state, the equilibrium of interest is globally stable in probability, and the state can be regulated to the origin almost surely. Cited in 84 Documents MSC: 93D21 Adaptive or robust stabilization 93E15 Stochastic stability in control theory 93E03 Stochastic systems in control theory (general) 93E12 Identification in stochastic control theory Keywords:high-order stochastic systems; nonlinear parameterization; adaptive state-feedback stabilization PDFBibTeX XMLCite \textit{X.-J. Xie} and \textit{J. Tian}, Automatica 45, No. 1, 126--133 (2009; Zbl 1154.93427) Full Text: DOI References: [1] Annaswamy, A. M.; Skantze, F. P.; Loh, A. P., Adaptive control of continuous time systems with convex/concave parameterization, Automatica, 34, 33-49 (1998) · Zbl 0910.93049 [2] Cao, C.; Annaswamy, A. M., A hierarchical discretized-parameter polynomial adaptive estimator for non-linearly parameterized systems, International Journal of Control, 79, 831-844 (2006) · Zbl 1162.93406 [3] Deng, H.; Krstić, M., Stochastic nonlinear stabilization, part i: A backstepping design, Systems and Control Letters, 32, 143-150 (1997) · Zbl 0902.93049 [4] Deng, H.; Krstić, M., Stochastic nonlinear stabilization, part ii: Inverse optimality, Systems and Control Letters, 32, 151-159 (1997) · Zbl 0902.93050 [5] Deng, H.; Krstić, M., Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance, Systems and Control Letters, 39, 173-182 (2000) · Zbl 0948.93053 [6] Deng, H.; Krstić, M.; Williams, R. J., Stabilization of stochastic nonlinear driven by noise of unknown covariance, IEEE Transactions on Automatic Control, 46, 1237-1253 (2001) · Zbl 1008.93068 [7] Has’minskii, R. Z., Stochastic stability of diffrential equations (1980), Kluwer Academic Publishers: Kluwer Academic Publishers Massachusetts [8] Kojić, A.; Annaswamy, A. M.; Loh, A. P.; Lozano, R., Adaptive control of a class of nonlinear systems with convex/concave parameterization, IEEE Transactions on Automatic Control, 37, 267-274 (1999) · Zbl 0942.93016 [9] Krstić, M.; Deng, H., Stabilization of uncertain nonlinear systems (1998), Springer: Springer New York · Zbl 0906.93001 [10] Kushner, H. J., Stochastic stability and control (1967), Academic Press: Academic Press New York · Zbl 0178.20003 [11] Lin, W.; Pongvuthithum, R., Nonsmooth adaptive stabilization of cascade systems with nonlinear parameterization via partial-state feedback, IEEE Transactions on Automatic Control, 48, 1809-1816 (2003) · Zbl 1364.93711 [12] Lin, W.; Qian, C., Adding one power integrator: A tool for global stabilization of high-order lower-triangular systems, Systems and Control Letters, 39, 339-351 (2000) · Zbl 0948.93056 [13] Lin, W.; Qian, C., Adaptive control of nonlinearly parameterized systems: A nonsmooth feedback framework, IEEE Transactions on Automatic Control, 47, 757-774 (2002) · Zbl 1364.93400 [14] Lin, W.; Qian, C., Adaptive control of nonlinearly parameterized systems: The smooth feedback case, IEEE Transactions on Automatic Control, 47, 1249-1266 (2002) · Zbl 1364.93399 [15] Liu, Y.; Pan, Z. G.; Shi, S., Output feedback control design for strict-feedback stochastic nonlinear systems under a risk-sensitivie cost, IEEE Transactions on Automatic Control, 48, 509-514 (2003) [16] Liu, Y.; Zhang, J., Reduced-order obserer-based control design for nonlinear stochastic systems, Systems and Control Letters, 52, 123-135 (2004) · Zbl 1157.93538 [17] Liu, Y.; Zhang, J., 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 [18] 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 [19] Loh, A. P.; Annaswamy, A. M.; Skantze, F. P., Adaptation in the presence of a general nonlinear parameterization: An error model approach, IEEE Transactions on Automatic Control, 44, 1634-1652 (1999) · Zbl 0958.93051 [20] Marino, R.; Tomei, P., Global adaptive output feedback control nonlinear systems, part ii: Nonlinear parameterization, IEEE Transactions on Automatic Control, 38, 33-48 (1993) · Zbl 0799.93023 [21] Marino, R.; Tomei, P., Nonlinear control design: Geometric, adaptive, and robust (1995), Prentice Hall: Prentice Hall London · Zbl 0833.93003 [22] Pan, Z. G.; Basar, T., Adaptive controller design for tracking and disturbance attenuation in parametric strict-feedback nonlinear systems, IEEE Transactions on Automatic Control, 43, 1066-1083 (1998) · Zbl 0957.93046 [23] Pan, Z. G.; Basar, T., Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion, SIAM Journal on Control and Optimization, 37, 957-995 (1999) · Zbl 0924.93046 [24] Pan, Z. G.; Ezal, K.; Krener, A.; Kokotović, P. V., Backstepping design with local optimality matching, IEEE Transactions on Automatic Control, 46, 1014-1027 (2001) · Zbl 1007.93025 [25] Pan, Z. G.; Liu, Y.; Shi, S., Output feedback stabilization for stochastic nonlinear systems in observer canonical form with stable zero-dynamics, Science in China, 44, 292-308 (2001) · Zbl 1125.93489 [26] 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 [27] Wu, Z. J.; Xie, X. J.; Zhang, S. Y., Stochastic adaptive backstepping controller design by introducing dynamic signal and changing supply function, International Journal of Control, 79, 1635-1646 (2006) · Zbl 1124.93057 [28] 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 [29] Ye, X., Global adaptive control of nonlinearly parameterized systems, IEEE Transactions on Automatic Control, 48, 169-173 (2003) · Zbl 1364.93724 [30] Ye, X., Switching adaptive output-feedback control of nonlinearly parameterized systems, Automatica, 41, 983-989 (2005) · Zbl 1091.93016 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.