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

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'Andrea-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'Andrea-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.

[1]  Brigitte d'Andréa-Novel,et al.  Stabilization of a rotating body beam without damping , 1998, IEEE Trans. Autom. Control..

[2]  R. Datko.  Extending a theorem of A. M. Liapunov to Hilbert space , 1970 .

[3]  Miroslav Krstic,et al.  Nonlinear and adaptive control de-sign , 1995 .

[4]  Ö. Morgül,et al.  On the stabilization of a cable with a tip mass , 1994, IEEE Trans. Autom. Control..

[5]  Brigitte d'Andréa-Novel,et al.  Feedback stabilization of a hybrid PDE-ODE system: Application to an overhead crane , 1994, Math. Control. Signals Syst..

[6]  A. Isidori,et al.  New results and examples in nonlinear feedback stabilization , 1989 .

[7]  Marshall Slemrod.  Feedback stabilization of a linear control system in Hilbert space with ana priori bounded control , 1989, Math. Control. Signals Syst..

[8]  John Tsinias,et al.  Sufficient lyapunov-like conditions for stabilization , 1989, Math. Control. Signals Syst..

[9]  Abdelkrim Mifdal,et al.  Stabilisation uniforme d'un système hybride , 1997 .

[10]  D. Russell.  Decay rates for weakly damped systems in Hilbert space obtained with control-theoretic methods☆ , 1975 .

[11]  M. Slemrod,et al.  Asymptotic behavior of nonlinear contraction semigroups , 1973 .

[12]  C. Samson,et al.  Time-varying exponential stabilization of a rigid spacecraft with two control torques , 1997, IEEE Trans. Autom. Control..

[13]  B. Rao.  Decay estimates of solutions for a hybrid system of flexible structures , 1993, European Journal of Applied Mathematics.

[14]  J. L. Webb.  OPERATEURS MAXIMAUX MONOTONES ET SEMI‐GROUPES DE CONTRACTIONS DANS LES ESPACES DE HILBERT , 1974 .

[15]  I. Kanellakopoulos,et al.  A toolkit for nonlinear feedback design , 1992 .

[16]  H. Brezis.  Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert , 1973 .

[17]  Brigitte d'Andréa-Novel,et al.  Exponential Stabilization of an Overhead Crane with Flexible Cable Via the Cascade Approach 1 , 1997 .

[18]  J. Coron,et al.  Adding an integrator for the stabilization problem , 1991 .