Abstract
Starting from states near to a closed set S we want to steer S and to stay always close to S. Unfortunately, open-loop controls are very sensitive to disturbances and can lead to very bad practical results. For that reason, we propose an approach for constructing a discontinuous feedback control law that asymptotically stabilizes the system in a neighborhood of the set S.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bacciotti, A.: Some remarks on generalized solutions of discontinuous differential equations. Int. J. Pure Appl. Math. 10, 257–266 (2004)
Brockett, R.: Asymptotic stability and feedback stabilization. In: Brockett, R., Millmann, R., Sussmann, H. (eds.) Differential Geometric Control Theory, Progr. Math., vol. 27, pp. 181–191. Birkhäuser, Basel-Boston (1983)
Clarke, F., Ledyaev, Y., Sontag, E., Subbotin, A.: Asymptotic controllability omplies feedbackstabilizatuion. IEEE Trans. Automat. Control 42, 1394–1407 (1997)
Clarke, F., Ledyaev, Y., Stern, R., Wolenski, P.: Nonsmooth analysis and control theory. Graduate Text in Mathematics, vol. 178. Springer, New York (1998)
Coron, J.-M.: Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems 5, 295–312 (1992)
Coron, J.-M.: Links between local controllability and local continuous stabilization. In: Fliess, M. (ed.) IFAC Nonlinear Control Systems Design, Bordeaux, pp. 165–171 (1992)
Coron, J.-M.: Stabilization in finite time of locally controllable systems by means of continuous time-varying feedback laws. SIAM J. Control Optim. 33, 804–833 (1995)
Hermes, H.: On the synthesis of a stabilizing feedback control via Lie algebraic methods. SIAM Journal on Control and Optimization 16, 715–727 (1978)
Hermes, H.: Lie algebras of vector fields and local approximation of attainable sets. SIAM Journal on Control and Optimization 18, 352–361 (1980)
Krasovskii, N., Subbotin, A.: Game-Theoretical Control Problems. Springer, New York (1988)
Krastanov, M., Quincampoix, M.: Local small-time controllability and attainability of a set for nonlinear control systems. ESAIM: Control. Optim. Calc. Var. 6, 499–516 (2001)
Krastanov, M.I., Veliov, V.M.: On the controllability of switching linear systems. Automatica 41, 663–668 (2005)
Michalska, H., Torres-Torriti, M.: Feedback stabilization of strongly nonlinear systems using the CBH formula. Int. J. Control 77, 562–571 (2004)
Sontag, E.: Mathematical Control Theory: Deterministic Finite Dimensional Systems. In: Texts in Applied Mathematics, vol. 6. Springer, New York (1990)
Sontag, E., Sussmann, H.: Remarks on continuous feedback. In: Proc. of the 19-th IEEE Conf. Decision and Control (Albuquerque), pp. 916–921. IEEE Press, Piscataway (1980)
Veliov, V., Krastanov, M.: Controllability of piecewise linear systems. Systems & Control Letters 7, 335–341 (1986)
Veliov, V.: On the controllability of control constrained linear systems. Math. Balk. New Ser. 2, 147–155 (1988)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Krastanov, M.I. (2006). On the Synthesis of a Stabilizing Feedback Control. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2005. Lecture Notes in Computer Science, vol 3743. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11666806_26
Download citation
DOI: https://doi.org/10.1007/11666806_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-31994-8
Online ISBN: 978-3-540-31995-5
eBook Packages: Computer ScienceComputer Science (R0)