Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach

Abstract

This paper deals with the uniform exponential stabilization of a hybrid PDE–ODE system which describes an overhead crane with flexible cable. A previous linear boundary feedback law (see d'Andréa-Novel, Boustany, Conrad & Rao (1994). MCSS Journal, 1, 1–22) depending on the platform position and velocity and on the angular displacement of the cable at the connection point to the platform, led to asymptotic stabilization but could not provide an exponential decay (see Rao (1993). European Journal of Applied Mathematics, 4, 303–319). Taking advantage of the “cascaded” structure of the hybrid system, we propose here a back-stepping approach leading to a linear boundary feedback which “naturally” depends in addition, on the angular velocity of the cable. We prove that this boundary feedback law produces uniform exponential stability and illustrative simulations are displayed. In d'Andréa-Novel & Coron ((1997). Proceedings of the IFAC SYROCO ’97 Conference, Nantes) this result has been established under a small gain condition on the feedback stabilizing the subsystem made of the PDE. Here, by using a result of Datko (see Datko (1970). Journal of Mathematical Analysis and Application, 32, 610–616) we show that this condition can be relaxed.