Lattice calibration with turn-by-turn BPM dataphysics.indiana.edu/~shylee/ap/workshop_10/LatCal_TBT_IUworksho… · 3/17/2010 1 Lattice calibration with turn -by-turn BPM data X.

3/17/2010 1

Lattice calibration with turn-by-turn BPM data

X. Huang3/17/2010

IUCF Workshop -- X. Huang

Outline• Lattice calibration methods

– 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

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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

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Calibrated rings tend to have increased injection efficiency, lifetime and stability.

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

3/17/2010 IUCF Workshop -- X. Huang 4*X. Huang et al, ICFA Newsletter 44

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.

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Calibration of nonlinearity• Nonlinear LOCO

– 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.

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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.

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Suppose BPM 1, 2 are separated by a drift space with length L.

Fit for the transfer matrix

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4×4 matrix M, 4 ×N matrix X

Not symplectic

Fit for the symplectic 4D transfer matrix, 10 free parameters.

Minimize

A simulated case

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Fitted matrix

Matrix calculated from model

Differences between fitted and calculated matrices are from BPM noises and lattice nonlinearities.

Tracking data with 50 µm noise

Normal mode coordinates obtained with fitted matrix

Calibration of one quadrupole

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The simplest case

Derive (x, x’, y, y’) with BPMs 0 and 1, then

with data from many passes.

A general accelerator section

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Do not consider gains and rolls for BPM 0, 1Consider gains and rolls for other BPMs:

BPM readingActual coordinates

Fit turn-by-turn data BPM reading Predictions from tracking

Use the usual iterative least-square fitter:

Jacobian matrixDefine residual vector r

Solve for

Simulation with SPEAR3 model

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BPM 0BPM 1

57 BPMs72 quadrupole parameters13 skew quadrupole parameters

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

1

4

6

6

Quadrupole results

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Two iterations with constraintsError bars obtained from 10 random seeds.

BPM parameters

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Including sextupoles

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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.

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.

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-20 -10 0 10 20-15

-10

-5

0

5

10

15

actual beam position x0+∆ x (mm)

linear

model x

0+∆

x (

mm

)

Measurement and modeling for SPEAR3 BPMs.Measured nonlinear response may be used for calibration.

SVD/ICA may be used to reduce random noises.

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.

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