Observer-controller design for three dimensional overhead cranes using time-scaling

In this paper we address the design of an observer-controller for a three degrees of freedom overhead crane. We consider a linear model of the crane where the length of the suspending rope is a time-varying parameter. The set of models given by frozen values of the rope length can be reduced to a single time-invariant reference model using suitable time-scalings. We construct a controller and an observer for the reference model assigning the desired closed loop eigenvalues for both system and estimation error. The time-scaling relations can be used to derive a control law for the time-varying system that implements an implicit gain-scheduling [6]. A second gain-scheduling is used to choose suitable closed-loop eigenvalues for different values of the load and lifting/lowering operations. Using a Lyapunov-like theorem, it is also possible to rnd relative upper bounds for the rate of change of the varying parameter that ensure the stability of the time-varying system.