Lattice calibration with turn-by-turn BPM data X. Huang 3/17/2010 3/17/2010 IUCF Workshop -- X. Huang 1
Lattice calibration methods Outline Orbit response matrix LOCO Turn-by-turn BPM data MIA, ICA, etc. Transfer matrix from turn-by-turn BPM data Model fitting with turn-by-turn data Simple case General case Simulation results 3/17/2010 IUCF Workshop -- X. Huang 2
Benefits of lattice calibration Lattice calibration: set machine optics/nonlinearity to the ideal case (model) usually symmetric and periodic. Benefit of lattice calibration Reliable model for ring parameter evaluations Reduce resonance driving terms. Reduce maximum beta function and dispersion. Discover human errors wrong cabling, wrong setpoint, etc Calibrated rings tend to have increased injection efficiency, lifetime and stability. 3/17/2010 IUCF Workshop -- X. Huang 3
Linear optics from closed orbit (LOCO) The most successful method and the most widely used. Measure closed orbit response matrix Fit the model so that it reproduces the measured response matrix BPM gains, rolls and corrector gains are included in fitting. It had limitations quadrupole errors sometimes too large to be true. But it was fixed with improved algorithm* Slow down the convergence, express most chi2 with minimum changes to magnets to avoid unconstrained patterns in solution. The new algorithm is integrated to LOCO and has cured many machines Soleil, Diamond, CLS, SNS, etc *X. Huang et al, ICFA Newsletter 44 3/17/2010 IUCF Workshop -- X. Huang 4
Methods based on turn-by-turn BPM data Analyze simultaneous turn-by-turn BPM data MIA, principal component analysis or SVD ICA, independent component analysis Other methods (Sussix?, harmonic analysis?, and more) Can obtain precise phase advance measurements, but beta functions are coupled with BPM gains. Fit the model for phase advance, beta and dispersion. Fitting beta functions and phase advances is not the most convenient Cannot include BPM gains in fitting Loss of information in case of coupled motion and nonlinear motion. 3/17/2010 IUCF Workshop -- X. Huang 5
Nonlinear LOCO Calibration of nonlinearity Orbit response with large corrector kicks Fitting nonlinear resonance driving terms obtained from turn-by-turn BPMs (R. Bartolini) ICA for sextupoles (X. Pang) Identify an independent mode driven by sextupoles. 3/17/2010 IUCF Workshop -- X. Huang 6
A new method for lattice calibration Derive 4D phase space coordinates from two BPMs. Fit turn-by-turn orbit data directly by comparing measurements to tracking. Suppose BPM 1, 2 are separated by a drift space with length L. 3/17/2010 IUCF Workshop -- X. Huang 7
Fit for the transfer matrix 4 4 matrix M, 4 N matrix X Not symplectic Fit for the symplectic 4D transfer matrix, 10 free parameters. Minimize 3/17/2010 IUCF Workshop -- X. Huang 8
A simulated case Tracking data with 50 µm noise Fitted matrix Matrix calculated from model Normal mode coordinates obtained with fitted matrix Differences between fitted and calculated matrices are from BPM noises and lattice nonlinearities. 3/17/2010 IUCF Workshop -- X. Huang 9
Calibration of one quadrupole The simplest case Derive (x, x, y, y ) with BPMs 0 and 1, then with data from many passes. 3/17/2010 IUCF Workshop -- X. Huang 10
A general accelerator section Do not consider gains and rolls for BPM 0, 1 Consider gains and rolls for other BPMs: BPM reading Actual coordinates Fit turn-by-turn data BPM reading Predictions from tracking Use the usual iterative least-square fitter: Define residual vector r Jacobian matrix Solve for 3/17/2010 IUCF Workshop -- X. Huang 11
Simulation with SPEAR3 model Generate simulated data: Plant quadrupole and skew quadrupole errors to the model as target. Track 200 turns with 2 mm X/Y offset at middle of rf straight section. Apply artificial gain and roll errors to all BPMs except BPM 0 and 1. Add 50 µm noise to all BPMs 6 1 57 BPMs 72 quadrupole parameters 13 skew quadrupole parameters 4 BPM 1 BPM 0 6 3/17/2010 IUCF Workshop -- X. Huang 12
Quadrupole results Two iterations with constraints Error bars obtained from 10 random seeds. 3/17/2010 IUCF Workshop -- X. Huang 13
BPM parameters 3/17/2010 IUCF Workshop -- X. Huang 14
Including sextupoles Tracking data with initial offset 5 mm horizontal and 3 mm vertical. 4 sextupole parameters. Only 1 BPM per cell from the arc (between sextupoles) is used. There are systematic errors. More BPMs between quads may help. 3/17/2010 IUCF Workshop -- X. Huang 15
Application to real machine No difficulty is expected for linear optics. Waiting for data from Soleil. For sextupole parameters, need large oscillation amplitude. But large BPM readings are affected by BPM nonlinearities. linear model x 0 + x (mm) 15 10 5 0-5 -10 Measurement and modeling for SPEAR3 BPMs. Measured nonlinear response may be used for calibration. SVD/ICA may be used to reduce random noises. -15-20 -10 0 10 20 actual beam position x 0 + x (mm) 3/17/2010 IUCF Workshop -- X. Huang 16
Summary The new method fit turn-by-turn BPM directly to model. Advantages: Fast data acquisition. Can fit BPM gains and rolls. Natural choice for beam motion with linear coupling and nonlinear response. Disadvantages: Gain and roll errors, noises in BPM 0 and 1 propagate to tracking data. 3/17/2010 IUCF Workshop -- X. Huang 17