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A decentralized proportional-integral-differential (PID) control structure has been always the first option for magnetic levitation systems due to its model-free nature as well as the simplicity. It is very challenging to tune PID parameters for each suspension point to ensure that the closed-loop system is less sensitive to uncertainties/disturbances. Even though, by a large number of experiments, a reasonably good PID control parameter set can be obtained to levitate a single suspension point for a given set point, these parameters are very sensitive when the set point changes or the decentralized PID controller is applied to the half bogie where coupling exists, strong oscillations will be observed and retuning of parameters is needed. This paper utilizes the extremum seeking (ES) method to tune PID parameters on-line to improve the steady-state performance of the system with few oscillations. A “sampled-data”structure is used to obtain a moving window approximation of the steady-state behavior, leading to a discrete-time ES tuning for the decentralized PID parameters. By using projection method, the tuning parameters are constrained in a compact set, in which the stability can be guaranteed and the optimal performance can be achieved. Experiments have shown the effectiveness of the proposed method.
Chen et al. (Mon,) studied this question.