A Linear Control Theory Analysis of Transverse Coherent Bunch Instabilities Feedback Systems (The Control Theory Approach to Hill's Equation)
- Author(s): Bengtsson, Johan
- Briggs, Don
- Portmann, Gregory
- et al.
There is an on-going effort to build a feedback system for transverse coherent bunch instabilities for ALS [1,2,3]. The beam dynamics issues were already addressed in the conceptual design report [ 4] and more detailed studies have been earned out . On-going work is the development of a general simulation code including the full 6-dirnensional dynamics for coherent bunch instabilities (by using Taylor series maps) as well as related feedback systems . Recently, there has been some confusion about how to choose the gain matrix in the feedback loop. In particular, the current analytical formulas were found (from numerical simulations by D. Briggs using the newly developed simulation code) to only be valid if ak = 0 at the kicker. This motivated us to perform a more careful design study of the transverse feedback system based on linear control theory. This paper presents the general formulas for tuning the system. Also, by a careful analytical study of the performance of the system, based on linear accelerator theory combined with linear control theory for sampled systems, we discovered that the performance of the system can be dramatically improved by slightly changing one of the two coefficients in the gain matrix.