• Organisation

A usual and effective way to realize closed-loop controllers is to use cascaded SISO feedback and to rely on some kind of linear PID structure with parameters adjusted manually in simulations or experiments. Such a control may not reach optimal performance if the system is coupled or non-linear. Regarding intense beams, longitudinal beam loading can be compensated by detuning. But the coupling between phase and amplitude (or I and Q component) highly depends on the tuning, that is on the resonant frequency of the cavity. It is derived that cavity and beam dynamics thus show bi-linear nature, i.e. belong to a well investigated class of non-linear systems with appropriate control strategies available. Different controller designs are compared in terms of performance but also design transparency, the need of previous knowledge like the expected magnitude of beam loading and adaptability to different conditions, e.g. during acceleration or if applied to the full range of ion species as at GSI. The performance evaluation is based on macro-particle tracking simulations. In particular avail and limits of an optimal (quadratic cost) MIMO controller for bi-linear systems are shown.