Stability Domains in the Feedback Gains Space for an Overhead Crane with Flexible Cable

The problem of stabilization of desired crane position and load at the flexible cable tip is considered. The stabilizing control is linear feedback. This control law depends on the platform position, velocity and integral of this position. It contains also the information about the angle between the vertical and the cable at the point of its connection to the platform and angular velocity, about the cable deformation. The time delay in the control loop is taken into account. Mathematical model of the system consists of differential equations with partial and ordinary derivatives. The general equations describing boundaries of asymptotic stability domain are obtained. In the space of the feedback gains, the regions of asymptotic stability are designed for different particular cases. With feedback coefficients from these domains the desired equilibrium of the system is asymptotically stable.