Time-Optimal Control Supported by PD in Real-Time*

The Magnetic Levitation System (MLS) is ideal for modelling because it is frictionless. With a model well-fitted to the system one can generate adjoint equations and solve both the state and adjoint differential sets numerically to obtain the time-optimal sequences corresponding to the desired initial and final positions of the levitating sphere. This action is repeated for a broad set of the sphere position differences. The resulting bang-bang controls are associated with the sphere levels and stored in the computer memory. Each sphere level is equipped with an appropriately tuned PD controller to maintain levitation. This means that the damping and elasticity coefficients are sustained at the values close to each other despite changes of the distance between the sphere and electromagnet. In this way the time-optimal, open-loop control applied for the position change aided by the PD controller at steady states becomes a variable control structure and appears to be the correct and rapid strategy to experiment with MLS in the real-time.