Post LS2 implementations of the beam phase, frequency and synchro loops

• Total applied frequency correction based on the sum of the three
• Discrete equations of the closed-loop system
  • Integral equations for the SL
  • 2 equations for each loop -> matrix equation of 6 variables
    • Want to feedback only on frequency and have phase correction = 0
    • Linear response of 3 loops
    • Can find a suitable matrix with the Linear Quadratic Regulator (LQR) and solve numerically
    • Feedback design: find best regulator and state-cost matrices
  • Want to check closed-loop system with external perturbation and verify loop stability
  • Obtaining the transfer function for the 3 loops

Implementation in BLonD

• Calculate the LQR gains e.g. every 5000 turns, then interpolate to turn-by-turn values for tracking 
  • Some entries are defined dynamically
  • The K matrix stores the 6 gains along the cycle
• Dependencies: python control library
• Option for Bode plot plotting of transfer functions
• Optional: tracking of gain solutions and some observables that can be compared with BLonD tracking results
• Loops ON
  • Can import Q11 also from measurements
  • Delay turns: when the feedback will act: after 3 turns, for instance
  • Initialize BeamFeedback and pass to tracker
• BeamFeedback SPS entry
  • Total frequency correction from PL + FL + SL
• Modifications to previous methods listed
• Correction of phase set point for collective effects
• Benchmarks
  • Good agreement btw python and Mathematica for time-evolution of matrices
  • Good agreement with impulse responses as calculated by Philippe
  • Matrix tracking vs BLonD tracking 
    • Impulse response not exactly equivalent, as the frequency correction is an output for BLonD and input for the matrix tracking, but good enough agreement
• Performance
  • 450000 turns tracking, precalculation of gains every 5000 turns before the simulation (8 s total)
  • BeamFeedback 1.4 s/turn, called every turn vs kick+drift ~60 s/turn
  • NO performance bottleneck