Non-homogeneous observer-based-output feedback scheme for the double integrator

https://doi.org/10.58571/CNCA.AMCA.2022.071

Resumen: In this paper we propose a non-homogeneous observer-based-output feedback scheme for the double integrator system. The control scheme is composed by two non-homogeneous structures. It resembles a non-homogeneous state feedback controller, where the unmeasurable state is replaced by the that provided by a non-homogeneous observer. Compared to other reported schemes, the closed-loop system is not homogeneous of negative degree, so it is not possible to use the results established for homogeneous systems. To overcome this problem, the finite-time stability analysis is carried out by means of (local) strict Lyapunov functions.

¿Cómo citar?

Emmanuel Cruz-Zavala, Emmanuel Nuño & Jaime A. Moreno. Non-homogeneous observer-based-output feedback scheme for the double integrator. Memorias del Congreso Nacional de Control Automático, pp. 268-273, 2022. https://doi.org/10.58571/CNCA.AMCA.2022.071

Palabras clave

Control de Sistemas No Lineales; Control Basado en pasividad; Otros Tópicos Afines

• Andrieu, V., Praly, L., and Astolfi, A. (2008). Homogeneous approximation, recursive observer design and output feedback. SIAM J. Contr. Optim., 47(4), 1814–1850.
• Bacciotti, A. and Rosier, L. (2005). Lyapunov functions and stability in control theory. Springer-Verlag, New York, 2nd edition.
• Bernuau, E., Perruquetti,W., Efimov, D., and Moulay, E. (2014). Robust finite-time output feedback stabilisation of the double integrator. Int. J. Control, 88(3), 451–460.
• Bhat, S. and Bernstein, D. (1997). Finite-time stability of homogeneous systems. In Proc. Amer. Contr. Conf., 2513–2514. Albuquerque, NM.
• Bhat, S. and Bernstein, D. (1998). Continuous finite-time stabilization of the translational and rotational double integrators. IEEE Trans. Autom. Control, 43, 678–682.
• Cruz-Zavala, E., Nuño, E., and Moreno, J. (2021). Nonhomogeneous finite-time controllers for double integrator dynamics. In Memorias de Congreso Nacional de Control Automático, 261–266. Guanajuato, México.
• Cruz-Zavala, E., Sanchez, T., Moreno, J.A., and Nuño, E. (2018). Strict Lyapunov functions for homogeneous finite-time second-order systems. In 2018 56th IEEE Conf. Decision and Control (CDC), 1530–1535.
• Haimo, V.T. (1986). Finite time controllers. SIAM J. Contr. Optim., 24, 760–770.
• Hestenes,M.R. (1966). Calculus of variations and optimal control theory. John Wiley & Sons, New York.
• Hong, Y., Huang, J., and Xu, Y. (2001). On an output feedback finite-time stabilization problem. IEEE Trans. Autom. Control, 46(2), 305–309.
• Hong, Y., Xu, Y., and Huang, J. (2002). Finite-time control for robot manipulators. Syst. Control Lett., 46, 243–253.
• Mercado-Uribe, A. and Moreno, J.A. (2017). Discontinuous integral control for systems in controller form. In Memorias del Congreso Nacional de Control Automático 2017, 630–635.
• Michel, A., Hou, L., and Liu, D. (2008). Stability of Dynamical Systems: Continuous, Discontinuous, and Discrete Systems. Systems & Control: Foundations & Applications. Birkhauser, Boston.
• Moreno, J.A. (2014). On strict Lyapunov functions for some non-homogeneous super-twisting algorithms. J.
• Franklin Inst., 351(4), 1902–1919. Special Issue on 2010-2012 Advances in Variable Structure Systems and Sliding Mode Algorithms.
• Moulay, E. and Perruquetti, W. (2006). Finite-Time Stability and Stabilization: State of the Art, 23–41.
• Springer Berlin Heidelberg, Berlin, Heidelberg. Nakamura, N., Nakamura, H., Yamashita, Y., and Nishitani, H. (2009). Homogeneous stabilization for input affine homogeneous systems. IEEE Trans. Autom. Control., 54(9), 2271–2275.
• Orlov, Y. (2009). Discontinuos Systems: Lyapunov analysis and robust synthesis under uncertainty conditions. Springer.
• Orlov, Y., Aoustin, Y., and Chevallereau, C. (2011). Finite-time stabilization of a perturbed double integrator-part I: Continuous sliding mode-based output feedback synthesis. IEEE Trans. Autom. Control, 56(3), 614–618.
• Polyakov, A., Orlov, Y., Oza, H., and Spurgeon, S. (2015). Robust finite-time stabilization and observation of a planar system revisited. In 2015 54th IEEE Conf. Decision and Control (CDC), 5689–5694.
• Qian, C. and Lin,W. (2003). Nonsmooth output feedback stabilization of a class of genuinely nonlinear systems in the plane. IEEE Trans. Autom. Control, 48(10), 1824–1829.
• Su, Y. and Zheng, C. (2015). Robust finite-time output feedback control of perturbed double integrator. Automatica, 60, 86–91.
• Venkataraman, S. and Gulati, S. (1993). Terminal slider control of robot systems. J. Intell. Robot. Syst., 7, 31–55.
• Yu, S.H., Yu, X.H., Shirinzadeh, B., and Man, Z. (2005). Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica, 41(11), 1957–1964.
• Zavala-Rio, A. and Fantoni, I. (2014). Global finitetime stability characterized through a local notion of homogeneity. IEEE Trans. Autom. Control, 59(2), 471–477.
• Zavala-Rio, A., Sanchez, T., and Zamora-G´omez, G.I. (2022). On the continuous finite-time stabilization of the double integrator. SIAM J. Contr. Optim., 60(2), 699–719.