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 the 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 find relative upper bounds for the rate of change of the varying parameter that ensure the stability of the time–varying system.