Modeling and Control of an Elastic Ship-Mounted Crane Using Variable-Gain Model-Based Controller
This work deals with mathematical modeling and control of elastic ship-mounted cranes which have the Maryland Rigging. The developed model contains three independent inputs to control the vibrations in the plane of the boom; the luff angle is utilized to ensure the controllability of the elastic boom, and the total length of the upper cable in conjunction with the position of its lower suspension point are used to guarantee the controllability of the payload. The disturbance acting on the ship due to sea motions is represented by the rolling displacement of the ship about its center of gravity. The full nonlinear model of the crane is developed and Taylor series is utilized to expand the nonlinear terms about the current equilibrium point which vary with the luff angle and the length of the upper cable. This has led to a linear model with additive nonlinear terms (higher order terms) collected in a separate column vector. Simulation results show that, within a considerable range of pendulation displacements of the payload, the nonlinear model and the linear one obtained by neglecting the nonlinear terms from consideration reflect nearly equivalent responses. Consequently, the linear model is used to design the control system of the crane. The coefficient matrices of this linear model are calculated at the current (instantaneous) equilibrium point, which vary with the luff angle and the length of the upper cable, therefore, a variable-model problem is created and accordingly a variable-gain observer and a variable-gain controller are designed to cover the operation of the crane for all possible equilibrium points in the working space of the crane. The switching between these gains takes place automatically according to the output of a region finder, which uses the measurements of the luff angle and the length of the upper cable to detect the current operating region. A PI-Observer is used to estimate the states and the unknown disturbance force or forces acting directly on the payload; this guarantees that the estimated states converge to their true values even though a nonzero disturbance force acts on the payload. The controller uses the estimated states and the measured roll angle to create the required damping and to compensate for the rolling action of the ship. Stability and performance robustness of the system are ensured for the total working space and also for the expected range of the payload mass. Simulation and experimental results show that the observer can estimate the states and the unknown disturbance acting on the payload very well and the controller can reduce the payload pendulations significantly.