Closed-loop stability analysis of a gantry crane with heavy chain and payload

In this paper, we analyse a systematically designed and easily tunable backstepping-based boundary control concept for a gantry crane with heavy chain and payload. The corresponding closed-loop system is formulated as an abstract evolution equation in an appropriate Hilbert space. Non-restrictive conditions for the controller coefficients are derived, under which the solutions are described by a C₀-semi-group of contractions, and are asymptotically stable. Moreover, by applying Huang's theorem we can finally even show that under these conditions the controller renders the closed-loop system exponentially stable.