Abstract
The problem of robust stabilization is considered for a class of nonlinear systems in the presence of structure uncertainties, external disturbances, and unknown time-varying virtual control coefficients. It is supposed that the upper bounds of the external disturbances and the virtual control coefficients are unknown. The unknown structural uncertainties are approximated by using neural networks (NNs). In particular, the prior knowledge about the weights and approximation errors of NNs is not required. The improved adaptation laws with \(\sigma \)-modification are employed to estimate the unknown parameters, which contain the upper bounds of the external disturbances and the virtual control coefficients, and the norm forms of weights and approximate errors of the NNs. Then, by making use of the updated values of these unknown parameters, a class of backstepping approach-based continuous adaptive robust state feedback controllers is synthesized. It is also shown that the proposed adaptive robust backstepping controller can guarantee the uniform asymptotic stability of such uncertain dynamical systems. Finally, two numerical examples are given to demonstrate the effectiveness of the proposed controller.
Similar content being viewed by others
References
Brogliato, B., Neto, A.T.: Practical stabilization of a class of nonlinear systems with partially known uncertainties. Automatica 31(1), 145–150 (1995)
Wu, H.: Continuous adaptive robust controllers guaranteeing uniformly ultimate boundedness for uncertain nonlinear systems. Int. J. Control 72(2), 115–122 (1999)
Khargonekar, P.P., Petersen, I.R., Zhou, K.: Robust stabilization of uncertain linear systems: quadratic stabilizability and \(H_\infty \) control theory. IEEE Trans. Autom. Control 35(3), 356–361 (1990)
Xie, L., Souza, C.E.: Robust \(H_\infty \) control for linear systems with norm-bounded time-varying uncertainty. IEEE Trans. Autom. Contr. 37(8), 1188–1191 (1992)
Li, H.X., Tong, S.C.: A hybrid adaptive fuzzy control for a class of nonlinear MIMO systems. IEEE Trans. Fuzzy Syst. 11(1), 24–34 (2003)
Tong, S.C., Li, H.X.: Fuzzy adaptive sliding-mode control for MIMO nonlinear systems. IEEE Trans. Fuzzy Syst. 11(3), 354–360 (2003)
Freeman, R.A., and Kokotovic, P.V.: Backstepping design of robust controllers for a class of nonlinear systems. In Proceedings of the IFAC NOLCOS’92 Symposium, Bordeaux, France, pp. 307–312 (1992)
Jiang, Z.P., Praly, L.: Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties. Automatica 34(7), 825–840 (1998)
Stotsky, A., Hedrick, J., Yip, P.: The use of sliding modes to simplify the backstepping control method. In Proceedings of the American Control Conference, Albuquerque, NM, USA, pp. 1703–1708 (1997)
Koshkouei, A.J., Zinober, A.S.I., Burnham, K.J.: Adaptive sliding mode backstepping control of nonlinear systems with unmatched uncertainty. Asian J. Control 6(4), 447–453 (2004)
Hsu, C.F., Lin, C.M., Lee, T.T.: Wavelet adaptive backstepping control for a class of nonlinear systems. IEEE Trans. Neural Netw. 17(5), 1175–1183 (2006)
Tong, S., Li, Y.: Observer-based fuzzy adaptive control for strict-feedback nonlinear systems. Fuzzy Sets Syst 160(12), 1749–1764 (2009)
Tong, S.C., He, X.L., Zhang, H.G.: A combined backstepping and small-gain approach to robust adaptive fuzzy output feedback control. IEEE Trans. Fuzzy Syst. 17(5), 1059–1069 (2009)
Knanellakopoulos, I., Kokotovic, P.V., Morse, A.S.: Systematic design of adaptive controllers for feedback linearizable systems. IEEE Trans. Autom. Control 36(11), 1241–1253 (1991)
Wu, H.: Adaptive robust stabilization for a class of uncertain nonlinear systems with external disturbances. In IECON 36th Annual Conference IEEE Industrial Electronics Society, Glendale, AZ, USA, pp. 53–58 (2010)
Wang, S., Chang, Y., Li, X., Zhang, J.: Parameter identification for a class of nonlinear chaotic and hyperchaotic flows. Nonlinear Anal. Real World Appl. 11(1), 423–431 (2010)
Li, X., Leung, A.C., Liu, X., Han, X., Chu, Y.: Adaptive synchronization of identical chaotic and hyper-chaotic systems with uncertain parameters. Nonlinear Anal. Real World Appl. 11(4), 2215–2223 (2010)
Li, X., Leung, A.C., Han, X., Liu, X., Chu, Y.: Complete (anti-)synchronization of chaotic systems with fully uncertain parameters by adaptive control. Nonlinear Dyn. 63(1–2), 263–275 (2011)
Slotine, J.J.E., Li, W.: Applied nonlinear control. Prentice-Hall, Englewood Cliffs (1991)
Wang, W.Y.: Stable adaptive fuzzy controllers with application to inverted pendulum tracking. IEEE Trans. Syst. Man Cybern. B 26(5), 677–691 (1996)
Koofigar, H.R., Hosseinnia, S., Sheikholeslam, F.: Robust adaptive nonlinear control for uncertain control-affine systems and its applications. Nonlinear Dyn. 56(1–2), 13–22 (2009)
Zhang, T., Ge, S.S., Hang, C.C.: Adaptive neural network control for strict-feedback nonlinear systems using backstepping design. Automatica 36(12), 1835–1846 (2000)
Ge, S.S., Wang, C.: Adaptive NN control of uncertain nonlinear pure-feedback systems. Automatica 38(4), 671–682 (2002)
Ge, S.S., Hong, F., Lee, T.H.: Adaptive neural control of nonlinear time-delay systems with unknown virtual control coefficients. IEEE Trans. Syst. Man Cybern. B 34(1), 499–516 (2004)
Bechlioulis, C.P., Rovithakis, G.A.: Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems. Automatica 45(2), 532–538 (2009)
Yu, Y., Zhong, Y.S.: Robust backstepping output tracking control for SISO uncertain nonlinear systems with unknown virtual control coefficients. Int. J. Control 83(6), 1182–1192 (2010)
Yu, Y., Zhong, Y.: Semi-global robust output tracking for non-linear uncertain systems in strict-feedback form. IET Control Theory Appl. 6(5), 751–759 (2012)
Ge, S.S., Yang, C., Lee, T.H.: Adaptive robust control of a class of nonlinear strict-feedback discrete-time systems with unknown control directions. Syst. Control Lett. 57(11), 888–895 (2008)
Chen, B., Liu, X., Liu, K., Lin, C.: Direct adaptive fuzzy control of nonlinear strict-feedback systems. Automatica 45(6), 1530–1535 (2009)
Ren, C., Tong, S., Li, Y.: Fuzzy adaptive high-gain-based observer backstepping control for SISO nonlinear systems with dynamical uncertainties. Nonlinear Dyn. 67(2), 941–955 (2012)
Ge, S.S., Wang, J.: Robust adaptive tracking for time-varying uncertain nonlinear systems with unknown control coefficients. IEEE Trans. Autom. Control 48(8), 1463–1469 (2003)
Yang, Y., Zhou, C.: Adaptive fuzzy \(H_{\infty }\) stabilization for strict-feedback canonical nonlinear systems via backstepping and small-gain approach. IEEE Trans. Fuzzy Syst. 13(1), 104–114 (2005)
Ferrari, S., Stengel, R.F.: Smooth function approximation using neural networks. IEEE Trans. Neural Netw. 16(1), 24–38 (2005)
Zainuddin, Z., Pauline, O.: Function approximation using artificial neural networks. WSEAS Trans. Math. 7(6), 333–338 (2008)
Huang, G.B., Saratchandran, P., Sundararajan, N.: A generalized growing and pruning RBF (GGAP-RBF) neural network for function approximation. IEEE Trans. Neural Netw. 16(1), 57–67 (2005)
Yang, H., Li, Z.: Adaptive backstepping control for a class of semistrict feedback nonlinear systems using neural networks. J. Control Theory Appl. 9(2), 220–224 (2011)
Liu, Y., Jia, Y.M.: Adaptive \(H_{\infty }\) control for a class of non-linear systems using neural networks. IET Control Theory Appl. 3(7), 813–822 (2009)
Tong, S.C., Li, Y.M., Zhang, H.G.: Adaptive neural network decentralized backstepping output-feedback control for nonlinear large-scale systems with time delays. IEEE Trans. Neural Netw. 22(7), 1073–1086 (2011)
Leu, Y.G., Lee, T.T., Wang, W.Y.: Observer-based adaptive fuzzy-neural control for unknown nonlinear dynamical systems. IEEE Trans. Syst. Man Cybern. B 29(5), 583–591 (1999)
Wu, H.: Adaptive robust tracking and model following of uncertain dynamical systems with multiple time delays. IEEE Trans. Autom. Control 49(4), 611–616 (2004)
Isidori, A.: Nonlinear control systems. Springer, Berlin (1995)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, Y., Wu, H. Adaptive robust backstepping control for a class of uncertain dynamical systems using neural networks. Nonlinear Dyn 81, 1597–1610 (2015). https://doi.org/10.1007/s11071-015-2093-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-015-2093-2