Abstract
Smooth observers are able to converge asymptotically to the actual value of the state, in the case where no measurement noise and no persistently acting perturbations are present. Under the same conditions continuous observers can converge in finite time. However, they are unable to converge if a perturbation/ uncertainty is present. In order to achieve finite time and exact convergence in the presence of perturbations, it is necessary to use discontinuous injection terms. In this chapter, some recent developments in this direction for second order systems will be presented and the results will be illustrated by means of simple examples. It will be also shown that by including non globally Lipschitz injection terms the convergence time of the observers can be made independent of the initial condition. The restriction to the two dimensional case is due to the fact that all proofs are done by means of Lyapunov functions, that are only available for planar systems. However, this has as advantage that the treatment is mainly tutorial, and provides on the one side an easy introduction to the topic, and on the other side it presents in the simplest case the main results that are (probably) valid for the general case. We hope to be able to provide a similar treatment of the general case in the near future.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
References
Andrieu, V., Praly, L., Astolfi, A.: Homogeneous approximation, recursive observer design and output feedback. SIAM Journal of Control and Optimization 47(4), 1814–1850 (2008)
Marco Tulio, A.-B., Moreno, J.A., Fridman, L.M.: Optimal Gain for the Super-Twisting Differentiator in the Presence of Measurement Noise. In: The 2012 American Control Conference (ACC 2012), Montréal, Canada, June 27-29, pp. 6154–6159 (2012)
Besançon, G.: An Overview on Observer Tools for Nonlinear Systems. In: Besançon, G. (ed.) Nonlinear Observers and Applications. LNCIS, vol. 363, pp. 1–33. Springer, Heidelberg (2007)
Baccioti, A., Rosier, L.: Lyapunov functions and stability in control theory, 2nd edn. Springer, New York (2005)
Cruz-Zavala, E., Moreno, J.A., Fridman, L.: Uniform Robust Exact Differentiator. IEEE Trans. on Automatic Control 56(11), 2727–2733 (2011), doi:10.1109/TAC.2011.2160030
Davila, J., Fridman, L., Levant, A.: Second-Order Sliding- Modes Observer for Mechanical Systems. IEEE Transactions on Automatic Control 50(11), 1785–1789 (2005)
Esfandiari, F., Khalil, H.K.: Output feedback stabilization of fully linearizable systems. Int. J. Control 56, 1007–1037 (1992)
Filippov, A.F.: Differential equations with discontinuous right-hand side, 304 p. Kluwer, Dordrecht (1988)
Fridman, L., Levant, A.: Higher order sliding modes. In: Barbot, J.P., Perruquetti, W. (eds.) Sliding Mode Control in Engineering, pp. 53–101. Marcel Dekker, New York (2002)
Gauthier, J.-P., Bornard, G.: Observability for any u(t) of a class of nonlinear systems. IEEE Trans. Aut. Cont. 26, 922–926 (1981)
Gauthier, J.-P., Hammouri, H., Othman, S.: A simple observer for nonlinear systems. Applications to bioreactors. IEEE Trans. Automatic Control 37, 875–880 (1992)
Gauthier, J.-P., Kupka, I.: Deterministic Observation Theory and Applications. Cambridge University Press, Cambridge (2001)
Hautus, M.L.J.: Strong detectability and observers. Linear Algebra and its Applications 50, 353–368 (1983)
Khalil, H.: High-Gain Observers in Nonlinear Feedback Control. In: Nijmeijer, H., Fossen, T. (eds.) New Directions in Nonlinear Observer Design. LNCIS, vol. 244, pp. 249–268. Springer, Heidelberg (1999)
Khalil, H.K.: Nonlinear Systems, 3rd edn., 750 p. Prentice–Hall, Upsaddle River (2002)
Levant, A.: Sliding order and sliding accuracy in sliding mode control. International Journal of Control 58(6), 1247–1263 (1993)
Levant, A.: Robust Exact Differentiation via Sliding Mode Technique. Automatica 34(3), 379–384 (1998)
Levant, A.: Homogeneity approach to high-order sliding mode design. Automatica (41), 823–830 (2005)
Moreno, J.A., Osorio, M.: A Lyapunov approach to second-order sliding mode controllers and observers. In: 47th IEEE Conference on Decision and Control, CDC 2008, pp. 2856–2861 (2008)
Moreno, J.A., Osorio, M.: Strict Lyapunov functions for the Super-Twisting Algorithm. IEEE Trans. on Automatic Control 57(4), 1035–1040 (2012), doi:10.1109/TAC.2012.2186179
Moreno, J.A.: A Linear Framework for the Robust Stability Analysis of a Generalized Supertwisting Algorithm. In: Proc. 6th Int. Conf. Elect. Eng., Comp. Sci. and Aut. Cont (CCE 2009), Mexico, November 10-13, pp. 12–17 (2009)
Moreno, J.A., Alvarez, J., Rocha-Cozatl, E., Diaz-Salgado, J.: Super-Twisting Observer-Based Output Feedback Control of a Class of Continuous Exothermic Chemical Reactors. In: 2010 IFAC 9th International Symposium on Dynamics and Control of Process Systems, DYCOPS 2010. Leuven, Belgium, July 5-7 (2010)
Moreno, J.A.: Lyapunov Approach for Analysis and Design of Second Order Sliding Mode Algorithms. In: Fridman, L., Moreno, J., Iriarte, R. (eds.) Sliding Modes. LNCIS, vol. 412, pp. 113–149. Springer, Heidelberg (2011)
Moreno, J.A.: A Lyapunov Approach to Output Feedback Control using Second Order Sliding Modes. IMA Journal of Mathematical Control and Information (2012), doi:10.1093/imamci/dnr036 (published on line January 2, 2012)
Moreno, J.A., Dochain, D.: Global observability and detectability analysis of uncertain reaction systems and observer design. International Journal of Control 81, 1062–1070 (2008)
Vasiljevic, L.K., Khalil, H.K.: Error bounds in differentiation of noisy signals by high-gain observers. Systems & Control Letters 57, 856–862 (2008)
Utkin, V., Guldner, J., Shi, J.: Sliding Mode Control in Electro-Mechanical Systems, 2nd edn. CRC Press, Taylor & Francis (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Moreno, J.A. (2013). On Discontinuous Observers for Second Order Systems: Properties, Analysis and Design. In: Bandyopadhyay, B., Janardhanan, S., Spurgeon, S. (eds) Advances in Sliding Mode Control. Lecture Notes in Control and Information Sciences, vol 440. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36986-5_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-36986-5_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36985-8
Online ISBN: 978-3-642-36986-5
eBook Packages: EngineeringEngineering (R0)