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Nonlinear Model Predictive Control of a Distributed Parameter System with Time-Varying Soft Constraints

Due to the flexible structure of stacker cranes (STCs), dynamic motions cause undesirable mast vibrations. This effect reduces the stability and positioning accuracy of these cranes. To address this problem, this paper proposes a nonlinear model predictive control (NMPC) approach for both optimal trajectory planning and tracking control. At each sampling instant, a finite horizon open-loop optimal control problem is firstly solved to obtain the desired trajectories, then in the closed-loop these trajectories are coped as a tracking reference. Therefore, for STC both tracking of desired trajectories and an active vibration damping are guaranteed. Additionally, this paper discusses a possibility of treating the undesired resonance frequencies without shrinking the operational bandwidth. The corresponding resonance frequencies are analyzed and modeled as bounded subsets. The enforcement of the resonance frequencies as hard state constraints is overly conservative and even infeasible. Therefore, in combination with other physical hard constraints the resonance frequencies are addressed as time-varying soft constraints in the frame of NMPC using slack variables. In fact, this approach can be extended to contain not only resonance, but any selective frequency. Simulation results are used to illustrate the good performance and convergence behavior of the proposed method.