Multi-Array Electron Beam Stabilization using Block-Circulant Transformation and Generalized Singular Value Decomposition

Abstract

We introduce a novel structured controller design for the electron beam stabilization problem of the UK's national synchrotron light source. Because changes to the synchrotron will not allow the application of existing control approaches, we develop a novel method to diagonalize the multi-input multi-output (MIMO) system. A generalized singular value decomposition (GSVD) is used to simultaneously diagonalize the actuator response matrices, which is applicable to an arbitrary number of actuator dynamics in a cross-directional setting. The resulting decoupled systems are regulated using mid-ranged control and the controller gains derived as a function of the generalized singular values. In addition, we exploit the inherent block-circulant symmetry of the system. The performance of our controller is demonstrated using simulations that involve machine data.

A Higher-Order Generalized Singular Value Decomposition for Rank-Deficient Matrices

“…Applications. The standard (HO-)GSVD has already been applied in various fields such as bioinformatics [23,30,31], medicine [17], acoustics [26], and control theory [14]. On one hand, the (HO-)GSVD can be used as an exact matrix decomposition to simplify or decouple problems involving several matrices sharing the same column or row dimension, such as generalized eigenvalue or generalized total least squares problems [3].…”

Model Predictive Control for Electron Beam Stabilization in a Synchrotron

“…with (•) = {s,f} do not share the same matrix of left singular vectors U (•) . In this case, the orbit response matrices can be simultaneously decomposed using alternative methods [4], [7]. Other approaches introduce a frequency deadband between slow and fast actuators and setup two independent control loops [6], which is a suboptimal approach because it prevents control action in the frequency deadband.…”

“…The coefficient matrix S ∈ R nu+ny×2nu has more columns than rows and the Moore-Penrose pseudoinverse S † = S T S 1 S T can be used to solve for xk and ūk . Note the zeros in the lefthand side vector of (7), so that in practice, only the last n y columns of S † need to be considered.…”

“…After finding a solution to (12), the set U N must therefore be updated as well as the vector q on the arrival of a new measurement. In practice, we substitute (7) in (13b) to avoid computing the setpoints x and ū.…”

A Higher-Order Generalized Singular Value Decomposition for Rank Deficient Matrices

“…Control Theory: In [8], the GSVD has been used to diagonalize the orbital dynamics of electrons in a synchrotron with N = 2 actuator (magnet) arrays. The actuator matrices typically have full row-rank and fewer rows than columns.…”