Abstract
This paper investigates the problem of adaptive finite-time stabilization of nonlinearly parameterized nonholonomic systems. By skilly using the parameter separation, input-state-scaling, and adding a power integrator techniques, an adaptive state feedback controller is obtained. Based on switching strategy to eliminate the phenomenon of uncontrollability, the proposed controller can guarantee that the system states globally finite-time converge to the origin, while other signals remain bounded. Simulation examples demonstrate the effectiveness and the robust features of the proposed approach.
Similar content being viewed by others
References
Komanovskyand, I., McClamroch, N.N.: Development in nonholonomic control problems. IEEE Control Syst. Mag. 15, 20–36 (1995)
Brockett, R.W.: Asymptotic stability and feedback stabilization. In: Differential Geometric Control Theory, pp. 181–208 (1983)
Murray, R.R., Sastry, S.S.: Nonholonomic motion planning: steering using sinusoids. IEEE Trans. Autom. Control 38, 700–716 (1993)
Astolfi, A.: Discontinuous control of nonholonomic systems. Syst. Control Lett. 27, 37–45 (1996)
Sun, Z.D., Ge, S.S., Huo, W., Lee, T.H.: Stabilization of nonholonomic chained systems via nonregular feedback linearization. Syst. Control Lett. 44, 279–289 (2001)
Tian, Y.P., Li, S.H.: Exponential stabilization of nonholonomic dynamic systems by smooth time-varying control. Automatica 38, 1139–1146 (2002)
Jiang, Z.P.: Robust exponential regulation of nonholonomic systems with uncertainties. Automatica 36, 189–209 (2000)
Xi, Z.R., Feng, G., Jiang, Z.P., Cheng, D.Z.: A switching algorithm for global exponential stabilization of uncertain chained systems. IEEE Trans. Autom. Control 48, 1793–1798 (2003)
Xi, Z.R., Feng, G., Jiang, Z.P., Cheng, D.Z.: Output feedback exponential stabilization of uncertain chained systems. J. Franklin Inst. 344, 36–57 (2007)
Do, K., Pan, J.: Adaptive global stabilization of nonholonomic systems with strong nonlinear drifts. Syst. Control Lett. 46, 195–205 (2002)
Ge, S.S., Wang, Z.P., Lee, T.H.: Adaptive stabilization of uncertain nonholonomic systems by state and output feedback. Automatica 39, 1451–1460 (2003)
Liu, Y.G., Zhang, J.F.: Output feedback adaptive stabilization control design for nonholonomic systems with strong nonlinear drifts. Int. J. Control 78, 474–490 (2005)
Wang, Q.D., Wei, C.L.: Robust adaptive control of nonholonomic systems withnonlinear parameterization. Acta Autom. Sin. 33, 399–403 (2007)
Zheng, X.Y., Wu, Y.Q.: Adaptive output feedback stabilization for nonholonomic systems with strong nonlinear drifts. Nonlinear Anal., Theory Methods Appl. 70, 904–920 (2009)
Gao, F.Z., Yuan, F.S., Yao, H.J.: Robust adaptive control for nonholonomic systems with nonlinear parameterization. Nonlinear Anal., Real World Appl. 11, 3242–3250 (2010)
Bhat, S., Bernstein, D.: Continuous finite-time stabilization of the translational and rotational double integrators. IEEE Trans. Autom. Control 43(5), 678–682 (1998)
Hong, Y.: Finite-time stabilization and stabilizability of a class of controllable systems. Syst. Control Lett. 46, 231–236 (2002)
Huang, X., Lin, W., Yang, B.: Global finite-time stabilization of a class of uncertain nonlinear systems. Automatica 41, 881–888 (2005)
Hong, Y., Wang, J., Cheng, D.: Adaptive finite-time control of nonlinear systems with parametric uncertainty. IEEE Trans. Autom. Control 51, 858–862 (2006)
Hong, Y., Wang, J., Xi, Z.: Stabilization of uncertain chained form systems within finite fettling time. IEEE Trans. Autom. Control 50, 1379–1384 (2005)
Qu, Z., Hull, R.A., Wang, J.: Globally stabilizing adaptive control design for nonlinearly parameterized systems. IEEE Trans. Autom. Control 51, 1073–1079 (2006)
Sun, M., Ge, S.S.: Adaptive repetitive control for a class of nonlinearly parametrized systems. IEEE Trans. Autom. Control 51, 1684–1688 (2006)
Hung, N.V.Q., Tuan, H.D., Narikiyo, T., Apkarian, P.: Adaptive control for nonlinearly parameterized uncertainties in robot manipulators. IEEE Trans. Control Syst. Technol. 16, 458–468 (2008)
Lin, W., Qian, C.: Adaptive control of nonlinearly parameterized systems: a nonsmooth feedback framework. IEEE Trans. Autom. Control 47, 757–774 (2002)
Acknowledgements
This work has been supported in part by National Nature Science Foundation of China under Grant 61073065 and Nature Science Foundation of Henan Province under Grant 092300410145.
The author would like to thank the editor and the anonymous reviewers for their constructive comments and suggestions for improving the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
Proof of Proposition 3
Recall that \(\xi_{i}=z_{i}^{1/q_{i}}-z_{i}^{*1/q_{i}}\) and \(z_{i}^{*}=\xi_{i-1}^{q_{i}}\beta_{i-1}\). By Lemma 3, for i=2,…,k,
Using (34), Lemma 4 and 0<q k <⋯<q 2<q 1=1, we have
where \(\bar{\gamma}_{k}\) is a nonnegative smooth function. □
Proof of Proposition 4
From Proposition 1, we have
Moreover, it follows from (34) that
where D k,1 is a nonnegative smooth function.
The equation \(z_{k}^{*1/q_{k}}=\xi_{k-1}\beta_{k-1}^{1/q_{k}}\) implies that \(z_{k}^{*1/q_{k}}\) is a smooth function and
Base on the Mean Value Theorem and Lemma 4, we arrive at
where D k,2 is a smooth nonnegative function.
Then, combining (69) and (71) yields
where \(\bar{\rho}_{k}\geq D_{k,1}+D_{k,2}\) is a smooth nonnegative function. □
Proof of Proposition 5
Since
Combining (73) and (74) yields
where \(\tilde{\phi}_{k}\geq D_{k,3}+(\bar{\gamma}_{k}+\bar{\rho}_{k})\eta_{k-1} \) is a nonnegative smooth function. □
Rights and permissions
About this article
Cite this article
Gao, F., Shang, Y. & Yuan, F. Robust Adaptive Finite-Time Stabilization of Nonlinearly Parameterized Nonholonomic Systems. Acta Appl Math 123, 157–173 (2013). https://doi.org/10.1007/s10440-012-9759-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-012-9759-2