An implicit gain-scheduling controller for cranes

To improve the efficiency of cargo handling with cranes it is necessary to control the crane trolley position so that the swing of the hanging load is minimized. In this paper we consider a linear parameter-varying model of the crane, where the time-varying parameter is the length of the suspending rope. We consider the set of models given by frozen values of the rope length and show how all these models can be reduced to a single time-invariant model using a suitable time scaling. The time scaling relation can be used to derive a control law for the time-varying system that implements an implicit gain scheduling. Using a Lyapunov-like theorem, it is also possible to find relative upper bounds for the rate of change of the varying parameter that ensure the stability of the time-varying system.