Using a robust mu-synthesis based controller to eliminate the adverse effects of uncertainties and external disturbances in nonlinear 3D overhead cranes with hoisting mechanism

In this paper, a robust \(\mu\)-optimal controller is developed for a three-dimensional overhead crane system with hoisting mechanism using the \(\mu\)-synthesis method. The 3D overhead crane system is modeled as an underactuated five degrees of freedom (5DOFs) system with uncertain parameters. The system receives only three input signals: Two forces that move the trolley along the \(x\) and \(y\) axes, and the hoisting force that moves the payload along the rope. In the first step to design the \(\mu\)-optimal controller for the 3D overhead crane system, nonlinear equations of the system are linearized around the equilibrium point to obtain the transfer functions. Next, to ensure that the system performs well and is robust against uncertainties, efficient weight functions for both performance and uncertainty are calculated. Finally, the \(\mu\)-optimal robust controller is designed using MATLAB’s Robust Control Toolbox, implementing the D-K iteration algorithm, and analyzing \(\mu\)-plots. It is shown that not only does the proposed controller provide nominal stability and performance, but it also ensures robust stability and performance. The proposed controller is applied to the original nonlinear system and simulation results demonstrate that this controller satisfies the control objectives well and is also robust to severe parametric uncertainties and external disturbances. Moreover, this controller provides better results compared to a conventional sliding mode controller (SMC) and a second-order SMC, by applying much less control forces. Another advantage of the proposed controller is that, unlike the other two controllers, it does not need feedback from states at the speed level. Therefore, in practice, the proposed robust controller needs fewer and cheaper sensors.