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

Dominik Stürzer,Anton Arnold,Andreas Kugi

doi:10.1080/00207179.2017.1335439

Abstract

ABSTRACTIn 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.

Highlights

This paper deals with the rigorous stability analysis of a control concept presented by Thull, Wild, and Kugi (2006) applied to the infinite-dimensional model of a gantry crane with heavy chain and payload
We show an even stronger result, namely the exponential stability of the semi-group {T(t)}t ࣙ 0, i.e. we prove that every solution of the initial value problem (2.23) tends to zero exponentially
In Thull et al (2006), a backstepping-based controller was proposed for the infinite-dimensional model of a gantry crane with heavy chain and payload

Summary

Introduction

This paper deals with the rigorous stability analysis of a control concept presented by Thull, Wild, and Kugi (2006) applied to the infinite-dimensional model of a gantry crane with heavy chain and payload. In contrast to the (simple) passive controller presented by Conrad and Mifdal (1998), the (damping) controller design by Thull et al (2006) is based on the idea to influence the energy flow between the cart and the chain, which is represented by the collocated variables cart velocity ࢚tw(t, 0) = v(t, 0) and internal force in the pivot bearing carrying the chains Fi = P(0)࢚xw(t, 0), forming an energy port, see right-hand side of Figure 1. If v(t, were the (virtual) control input, the control law v(t, 0) ࣎ χ 1Fi, with the controller parameter χ 1 > 0, would render the closed-loop system passive This would ensure a good damping of the chain vibrations, but the cart position w(t, 0).

Formulation as a Dissipative Evolution Equation

Asymptotic stability

Exponential stability

Conclusions