Abstract
In this paper, an output feedback tracking control scheme is put forwarded for a class of stochastic nonlinear systems, whose dynamics involve not only unknown parameters but also unmeasured states multiplied by output nonlinearities. A type of reduced-order observer is first developed. By adding some output related items in the observer, the estimation error realize global asymptotic convergence under disturbance free condition, and global bounded convergence when considering disturbance. Besides, the dimension of the closed-loop system is reduced, and the update law of this observer gain is beneficial for steady tracking. After the observer was established, the controller is constructed by employing the adaptive backstepping approach, and a smooth nonsingular robust item is proposed to handle the influence of stochastic disturbance. All the signals in the closed system is proved to be globally bounded in probability. Moreover the output tracking error converges to an arbitrary small neighborhood of the origin by proper choosing of the design parameters. The simulation results based on current control scheme and the comparison with the previous method illustrate that the proposed output feedback scheme realizes good tracking performance and strong ability on stochastic disturbance attenuation.
Similar content being viewed by others
References
Andrieu, V., Praly, L.: A unifying point of view on output feedback designs for global asymptotic stabilization. Automatica 45, 1789–1798 (2009)
Marino, R., Tomei, P.: Global adaptive output-feedback control of nonlinear systems. In: Proceedings of the IEEE Conference on Decision and Control, Brighton, pp. 1077–1081 (1991).
Pomet, J.-B., Hirschorn, R., Cebuhart, W.: Dynamic output feedback regulation for a class of nonlinear systems. Math. Control Signals Syst. 6, 106–124 (1993)
Andrieu, V., Praly, L.: Global asymptotic stabilization for nonminimum phase nonlinear systems admitting a strict normal form. IEEE Trans. Autom. Control 53(5), 1120–1132 (2008)
Yang, B., Lin, W.: Homogeneous observers, iterative design, and global stabilization of high-order nonlinear systems by smooth output feedback. IEEE Trans. Autom. Control 49(7), 1069–1080 (2004)
Qian, C.-J., Lin, W.: Recursive observer design, homogeneous approximation, and nonsmooth output feedback stabilization of nonlinear systems. IEEE Trans. Autom. Control 51(9), 1457–1471 (2006)
Polendo, J., Qian, C.-J.: A generalized homogeneous domination approach for global stabilization of inherently nonlinear systems via output feedback. Int. J. Robust Nonlinear Control 17, 605–629 (2007)
Andrieu, V., Praly, L., Astolfi, A.: Homogeneous approximation, recursive observer design and output feedback. SIAM J. Control Optim. 47(4), 1814–1850 (2008)
Praly, L., Astolfi, A.: Global asymptotic stabilization by output feedback under a state norm detectability assumption. In: Proceedings of the IEEE Conference on Decision and Control and the European Control Conference, Seville, pp. 2634–2639 (2005)
Zhang, Z.-Q., Xu, S.-Y., Shen, H.: Reduced-order observer-based output-feedback tracking control of nonlinear systems with state delay and disturbance. Int. J. Robust Nonlinear Control 20, 1723–1738 (2010)
Krishnamurthy, P., Khorrami, F., Jiang, Z.-P.: Global output feedback tracking for nonlinear systems in generalized output-feedback canonical form. IEEE Trans. Autom. Control 47(5), 814–819 (2002)
Krishnamurthy, P., Khorrami, F.: Dynamic high-gain scaling: state and output feedback with application to systems with ISS appended dynamics driven by all states. IEEE Trans. Autom. Control 49(12), 2219–2239 (2004)
Marino, R., Tomei, P.: A class of globally output feedback stabilizable nonlinear nonminimum phase systems. IEEE Trans. Autom. Control 50(12), 2097–2101 (2005)
Krishnamurthy, P., Khorrami, F., Jiang, Z.-P.: Global output feedback tracking for nonlinear systems in generalized output-feedback canonical form. IEEE Trans. Autom. Control 47(5), 814–819 (2002)
Zhang, Z.-Q., Chen, W.-S.: Adaptive output feedback control of nonlinear systems with actuator failures. Inf. Sci. 179, 4249–4260 (2009)
Zhang, Z.-Q., Xu, S.-Y., Wang, B.: Adaptive output feedback control of nonlinear systems with actuator failures. IET Control Theory Appl. 5(16), 1859–1867 (2011)
Kanellakopoulos, I., Kokotovic, P., Morse, A.: Adaptive output-feedback control of a class of nonlinear systems. In: Proceedings of the Conference on Decision and Control, Brighton, 1082–1087 (1991)
Li, Y.M., Ren, C.E., Tong, S.C.: Adaptive fuzzy backstepping output feedback control for a class of MIMO time delay nonlinear systems based on high-gain observer. Nonlinear Dyn. 67(2), 1175–1191 (2012)
Li, Y., Ren, C.E., Tong, S.: Adaptive fuzzy backstepping output feedback control of nonlinear uncertain time-delay systems based on high-gain filters. Nonlinear Dyn. 69(3), 781–792 (2012)
Li, D.J.: Adaptive output feedback control of uncertain nonlinear chaotic systems based on dynamic surface control technique. Nonlinear Dyn. 68(1–2), 235–243 (2012)
Zhang, X., Lin, Y.: Adaptive output feedback tracking for a class of nonlinear systems. Automatica 48, 2372–2376 (2012)
Zhang, X., Lin, Y.: A new approach to global asymptotic tracking for a class of low-triangular nonlinear systems via output feedback. IEEE Trans. Autom. Control 57(12), 3192–3196 (2012)
Deng, H., Krstic’, M.: Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance. Syst. Control Lett. 39(3), 173–182 (2000)
Deng, H., Krstic’, M., Williams, R.J.: Stabilization of stochastic nonlinear systems driven by noise of unknown covariance. IEEE Trans. Autom. Control 46(8), 1237–1253 (2001)
Deng, H., Krstic’, M.: Stochastic nonlinear stabilization, Part I: A backstepping design. Syst. Control Lett. 32(3), 143–150 (1997)
Wen, S.-P., Zeng, Z.-G., Huang, T.-W., et al.: Observer-based HN control of a class of mixed delay systems with random data losses and stochastic nonlinearities. ISA Trans. 52, 207–214 (2013)
Wen, S.-P., Zeng, Z.-G., Huang, T.-W., et al.: Exponential adaptive lag synchronization of memristive neural networks via fuzzy method and applications in pseudo random number generators. IEEE Trans. Fuzzy Syst. (2013). doi:10.1109/TFUZZ.2013.2294855
Wen, S.-P., Zeng, Z.-G., Huang, T.-W.: Observer-based \(H_\infty \)control of discrete time-delay systems with random communication packet losses and multiplicative noises. Appl. Math. Comput. 219, 6484–6493 (2013)
Wen, S.-P., Zeng, Z.-G., Huang, T.-W.: Robust\(H_\infty \)output tracking control for fuzzy networked systems with stochastic sampling and multiplicative noise. Nonlinear Dyn. 70, 1061–1077 (2012)
Arslan, G., Basar, T.: Risk sensitive adaptive trackers for strict-feedback systems with output measurements. IEEE Trans. Autom. Control 49(11), 1754–1758 (2002)
Fan, H., Ge, S.-S.: Adaptive state feedback control for a class of stochastic nonlinear systems. In: Proceedings of the IEEE Conference on Decision and Control, Atlantis, pp. 2996–3000 (2004)
Fan, H., Xie, W.: Decentralized adaptive tracking of stochastic nonlinear interconnected systems by output feedback. In: Proceedings of the IEEE conference on industrial electronics and applications, Taichung, pp. 137–142 (2010)
Liu, S.-J., Zhang, J.-F., Jiang, Z.-P.: Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems. Automatica 43(2), 238–251 (2007)
Chen, W.-S., Jiao, L.-C., Li, J., et al.: Adaptive NN backstepping output-feedback control for stochastic nonlinear strict-feedback systems with time-varying delays. IEEE Trans. Syst. Man Cybern. Part B. Cybern. 40(3), 939–950 (2010)
Fan, H.-J., Han, L.-X., Wen, C.-Y., et al.: Decentralized adaptive output-feedback controller design for stochastic nonlinear interconnected systems. Automatica 48, 2866–2873 (2012)
Xie, X.-J., Tian, J.: State-feedback stabilization for high-order stochastic nonlinear systems with stochastic inverse dynamics. Int. J. Robust Nonlinear Control 17(14), 1343–1362 (2007)
Pan, Z., Basa, T.: Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion. SIAM J. Control Optim. 37(3), 957–995 (1999)
Ji, H.-B., Xi, H.-S.: Adaptive output-feedback tracking of stochastic nonlinear systems. IEEE Trans. Autom. Control 51(2), 355–360 (2006)
Krstic, M., Deng, H.: Stabilization of Nonlinear Uncertain Systems. Springer, London (1998)
Hardy, G., Littlewood, J.-E., Polya, G.: Inequalities. Cambridge Univ. Press, Cambridge (1989)
Has’minskii, R.Z.: Stochastic Stability of Differential Equations. Kluwer Academic Publishers, Norwell (1980)
Higham, D.-J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001)
Wen, S.-P., Zeng, Z.-G., Huang, T.-W., et al.: \(H_\infty \)filtering for neutral systems with mixed delays and multiplicative noises. IEEE Trans. Circuits Syst. II Exp. Briefs 59(11), 820–824 (2012)
Zhang, X.-F., Liu, L., Feng, G., et al.: Output feedback control of large-scale nonlinear time-delay systems in lower triangular form. Automatica 49(11), 3476–3483 (2013)
Wen, S.-P., Zeng, Z.-G., Huang, T.-W., et al.: Passivity and passification of stochastic impulsive memristor-based piecewise linear system with mixed delays. Int. J. Robust Nonlinear Control (2013). doi:10.1002/rnc.3112
Acknowledgments
This work is supported by the National Natural Science Foundation of China under grant 61273083 and 61374012.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, Q., Hou, D., Dong, C. et al. Reduced-order observer-based backstepping tracking control for a class of stochastic nonlinear systems. Nonlinear Dyn 78, 939–956 (2014). https://doi.org/10.1007/s11071-014-1487-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-014-1487-x