Interaction between beam control and rf feedback loops for high Q cavities an heavy beam loading. Revision A

An open-loop state space model of all the major low-level rf feedback control loops is derived. The model has control and state variables for fast-cycling machines to apply modern multivariable feedback techniques. A condition is derived to know when exactly we can cross the boundaries between time-varying and time-invariant approaches for a fast-cycling machine like the Low Energy Booster (LEB). The conditions are dependent on the Q of the cavity and the rate at which the frequency changes with time. Apart from capturing the time-variant characteristics, the errors in the magnetic field are accounted in the model to study the effects on synchronization with the Medium Energy Booster (MEB). The control model is useful to study the effects on beam control due to heavy beam loading at high intensities, voltage transients just after injection especially due to time-varying voltages, instability thresholds created by the cavity tuning feedback system, cross coupling between feedback loops with and without direct rf feedback etc. As a special case we have shown that the model agrees with the well known Pedersen model derived for the CERN PS booster. As an application of the model we undertook a detailed study of the cross coupling between the loops by considering all of them at once for varying time, Q and beam intensities. A discussion of the method to identify the coupling is shown. At the end a summary of the identified loop interactions is presented