Time-Delayed Feedback Control of Electron Beam Energy Using ${H} _{\infty }$ -Optimal Fractional Delay

In the two-bunch operation mode of the Swiss Free Electron Laser (SwissFEL) machine, two electron bunches, separated by 28 ns, are fired and accelerated every 10 ms. The electron beam is passed through several accelerating cavities that are energized by high-power radio frequency (RF) pulses. The RF voltage generates a specific time-dependant accelerating gradient profile that results in different energy gains for the two bunches. It is often required that the two electron beams experience the same accelerating gradient through the cavities. This can be achieved by shifting the RF pulse in time to change the relative energy of the two beams. However, in the digital RF system used here, the minimum shift is limited by the sampling time of the digital-to-analog converters. In this brief, we develop a control scheme for the energy difference control, which finely tunes the RF pulse timing by a fractional delay filter. The method has been successfully tested at the SwissFEL injector test facility.