Gain scheduling control for cranes via parameter dependent Lyapunov functions

In this study, a method to design gain scheduling controller for a crane is proposed. A dynamical model of the crane is derived by the Euler-Lagrange equation. The model includes the rope length, its velocity and acceleration, that are time-varying parameters. A first characteristic of this study is that the velocity and the acceleration of the rope length are not ignored in the state equation, moreover they are used as gain scheduling parameters of the controller. A second characteristic is to construct an affine linear parameter varying system not ignoring a nonlinearity of the time-varying parameters. Using linear fractional transformation and descriptor representation, a state equation is led to be linear for the time-varying parameters. Then, the problem of the controller design is formulated by linear matrix inequality via parameter dependent Lyapunov function. The controller is constructed by a state feedback. The gains are given as a function of the time-varying parameters that are the rope length, its velocity and acceleration. Comparing with conventional methods a conservativeness of the proposed method is reduced. The effectiveness of the proposed method is illustrated by simulations.