1. Y.S. Ledyaev and E.D. Sontag. A Lyapunov characterization of robust stabilization. Nonlinear Anal., 37(7, Ser. A: Theory Methods):813-840, 1999. [PDF] Keyword(s): nonlinear control, feedback stabilization. Abstract:

   One of the fundamental facts in control theory (Artstein's theorem) is the equivalence, for systems affine in controls, between continuous feedback stabilizability to an equilibrium and the existence of smooth control Lyapunov functions. This equivalence breaks down for general nonlinear systems, not affine in controls. One of the main results in this paper establishes that the existence of smooth Lyapunov functions implies the existence of (in general, discontinuous) feedback stabilizers which are insensitive to small errors in state measurements. Conversely, it is shown that the existence of such stabilizers in turn implies the existence of smooth control Lyapunov functions. Moreover, it is established that, for general nonlinear control systems under persistently acting disturbances, the existence of smooth Lyapunov functions is equivalent to the existence of (possibly) discontinuous) feedback stabilizers which are robust with respect to small measurement errors and small additive external disturbances.




2. Y.S. Ledyaev and E.D. Sontag. A notion of discontinuous feedback. In Control using logic-based switching (Block Island, RI, 1995), volume 222 of Lecture Notes in Control and Inform. Sci., pages 97-103. Springer, London, 1997.



3. F. H. Clarke, Y.S. Ledyaev, E.D. Sontag, and A.I. Subbotin. Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Control, 42(10):1394-1407, 1997. [PDF] Abstract:

   It is shown that every asymptotically controllable system can be stabilized by means of some (discontinuous) feedback law. One of the contributions of the paper is in defining precisely the meaning of stabilization when the feedback rule is not continuous. The main ingredients in our construction are: (a) the notion of control-Lyapunov function, (b) methods of nonsmooth analysis, and (c) techniques from positional differential games.

1. Y.S. Ledyaev and E.D. Sontag. Stabilization under measurement noise: Lyapunov characterization. In Proc. American Control Conf., Philadelphia, June 1998, pages 1658-166, 1998.



2. Y.S. Ledyaev and E.D. Sontag. A remark on robust stabilization of general asymptotically controllable systems. In Proc. Conf. on Information Sciences and Systems (CISS 97), Johns Hopkins, Baltimore, MD, March 1997, pages 246-251, 1997. [PDF] Abstract:

   We showned in another recent paper that any asymptotically controllable system can be stabilized by means of a certain type of discontinuous feedback. The feedback laws constructed in that work are robust with respect to actuator errors as well as to perturbations of the system dynamics. A drawback, however, is that they may be highly sensitive to errors in the measurement of the state vector. This paper addresses this shortcoming, and shows how to design a dynamic hybrid stabilizing controller which, while preserving robustness to external perturbations and actuator error, is also robust with respect to measurement error. This new design relies upon a controller which incorporates an internal model of the system driven by the previously constructed feedback.




3. F.H. Clarke, Y.S. Ledyaev, E.D. Sontag, and A.I. Subbotin. Asymptotic controllability and feedback stabilization. In Proc. Conf. on Information Sciences and Systems (CISS 96)Princeton, NJ, pages 1232-1237, 1996. Keyword(s): control-Lyapunov functions, feedback stabilization.

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