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

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.

[1]  A. Packard,et al.  Optimal control of perturbed linear static systems , 1996, IEEE Trans. Autom. Control..

[2]  Stephen R. Duncan,et al.  Design of an electron beam stabilisation controller for a synchrotron , 2014 .

[3]  Stephen Duncan,et al.  Uncertainty modeling and robust stability analysis of a synchrotron electron beam stabilisation control system , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[4]  Paul J. Goulart,et al.  Symmetry Exploitation in Orbit Feedback Systems of Synchrotron Storage Rings , 2020 .

[5]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[6]  F. R. Elder,et al.  Radiation from Electrons in a Synchrotron , 1947 .

[7]  D. Rose.  Matrix identities of the fast fourier transform , 1980 .

[8]  Junqiang Fan.  Model predictive control for multiple cross-directional processes : analysis, tuning, and implementation , 2003 .

[9]  Stephen Duncan,et al.  Discrete-time anti-windup compensation for synchrotron electron beam controllers with rate constrained actuators , 2016, Autom..

[10]  William P. Heath,et al.  An Internal Model Control Approach to Mid-Ranging Control , 2009 .

[11]  R. Ursic,et al.  Fast closed orbit control in the SLS storage ring , 1999, Proceedings of the 1999 Particle Accelerator Conference (Cat. No.99CH36366).

[12]  Paul J. Goulart,et al.  Alternating Direction of Multipliers Method for Block Circulant Model Predictive Control , 2019, 2019 IEEE 58th Conference on Decision and Control (CDC).

[13]  Stephen Duncan,et al.  Design of multi-array controllers for electron beam stabilisation on synchrotrons , 2013, 2013 American Control Conference.

[14]  M. Saunders,et al.  Towards a Generalized Singular Value Decomposition , 1981 .

[15]  Carlos E. Garcia,et al.  Internal model control. 2. Design procedure for multivariable systems , 1985 .

[16]  G. Decker,et al.  Closed orbit correction using singular value decomposition of the response matrix , 1993, Proceedings of International Conference on Particle Accelerators.

[17]  O. Alter,et al.  A Higher-Order Generalized Singular Value Decomposition for Comparison of Global mRNA Expression from Multiple Organisms , 2011, PloS one.

[18]  Optimal Control of Perturbed Static Systems for Synchrotron Electron Beam Stabilisation , 2017 .

[19]  G. Dumont,et al.  Automatic tuning of paper machines cross-direction controllers via linear matrix inequalities , 2015 .

[20]  Gene H. Golub,et al.  Tikhonov Regularization and Total Least Squares , 1999, SIAM J. Matrix Anal. Appl..

[21]  S. R. Duncan.  The design of robust cross-directional control systems for paper making , 1995, Proceedings of 1995 American Control Conference - ACC'95.