Closed-loop Stability Analysis of a Gantry Crane with Heavy Chain

TL;DR

This paper presents a stability analysis of a boundary control method for a gantry crane with a heavy chain, demonstrating conditions for asymptotic and exponential stability of the system.

Contribution

The paper provides a rigorous mathematical analysis of a backstepping-based boundary control for gantry cranes, establishing stability conditions and exponential stability proof.

Findings

Solutions form a contraction semigroup under derived conditions.

System is asymptotically stable with the proposed control.

Exponential stability is achieved under certain coefficient conditions.

Abstract

In this paper, we analyze a systematically designed and easily tunable backstepping-based boundary control concept developed by Thull, Wild, and Kugi (2006) 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 $C_0$-semigroup 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.