Damping of longitudinal modes in heavy-ion synchrotrons by RF-feedback

The stability of RF (radio frequency) control systems is an essential part of particle accelerator technology. In accelerators, such as heavy-ion synchrotrons, RF feedback systems are used to damp longitudinal oscillation modes of the particle beam. Longitudinal modes have been analyzed in detail by accelerator physicists and there exists a closed theory that describes most of the observed phenomena in synchrotrons. However, from the point of view of the control engineer who wants to design the RF controls, there is a lack of convenient models to study the RF control loop dynamics analytically. Recently, it was shown that the interaction of the RF and the first two longitudinal modes can be described by a bilinear model. In this paper, it is shown that the measurement of the first harmonic of the beam current signal is sufficient to calculate the model output vector. This result is used to design nonlinear controls for the second longitudinal oscillation mode. The control design is based on the sum-of- squares decomposition. The controls are compared using particle tracking simulations.