Fast orbit feedback using the GSVD for systems with multiple slow corrector arrays

Advances in detector speed and resolution at 4th generation light sources make electron beam stability a critical requirement. At Diamond-II, the fast orbit feedback (FOFB) will stabilise the beam using 252 beam position monitors and two actuator arrays of 252 slow and 138 fast correctors. We previously proposed an approach based on the generalised singular value decomposition (GSVD), a two-matrix factorisation, to decouple the system into two-input modes controlled by both slow and fast correctors, and single-input modes controlled by slow correctors alone. This approach assumed identical dynamics for all correctors within each array. However, recent developments have shown that variations in vessel geometries and cooling channels introduce significant differences in corrector bandwidths, particularly for the slow correctors in the horizontal plane. Specifically, Diamond-II will have at least three distinct types of slow correctors in the horizontal plane. To address this, we extend the GSVD-based approach with balancing input filters to incorporate multiple slow corrector arrays with different dynamics. We also introduce a new regularisation matrix that preserves controller properties between the original and mode spaces for any choice of regularisation parameter. We analyse the resulting control system and present simulation results from preliminary Diamond-II data, demonstrating that the control specifications are met.