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Transverse Dynamics - Measurements

[MCCPB, Chapter 2]

• some data analysis techniques• coherent oscillations & filamentation

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Data Analysis Techniques• measuring beam response to many steering corrections at

many BPMs, adjusting a large number of model parameters in a global fit to best reproduce the data (LOCO code, J. Safranek)

• singular-value decomposition (SVD); powerful, broad range of applications, e.g. reducing corrector strengths (V. Ziemann, 1992), orthogonal tuning knobs (P. Raimondi, 1998), BPM data analysis (J. Irwin, 1998)

• principal-axes transformations (‘eigenfit’); may be superior to least-squares fit (T. Lohse, P. Emma, 1989)

• model-independent analysis (MIA/PCA); extract temporal and spatial patterns from BPM data; identifies relevant physical variables (J. Irwin, C.-X. Wang, Y. Yan, 1998)

• independent component analysis (ICA); more robust w.r.t. noise, coupling, and nonlinearity; extract transverse

betatron phase & amplitude, dispersion, coupling, slices ( X. Huang, 2005; F. Wang, 2008; X. Pang, 2009);

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orbit response measurements N BPMsreading orbit change x

M dipolecorrectorsexciting deflectionangles

closedorbit in a storage ring

trajectory Iin part of a ring or transport line

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LOCO* – orbit response measurements and analysis

4

*J. Safranek, “Experimental determination of storage ring optics usins closed orbit response measurements”, Nucl. Inst. and Meth. A388, (1997), pg. 27.

closed orbit

trajectory

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LOCO cont’d

5

• measured R must be adjusted to match the model; • then optical functions are obtained from matched model- in a transfer line only the R matrix can be verified

The measured R also depends on the BPM and corrector calibrations:

1) Data preparation:

for all i and j

A vector holding the weighted difference between the measured and modelled response is built from all matrix elements :

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LOCO cont’d2) Local gradient:

• Evaluate the sensitivity w.r.t. parameters (BPM and corrector calibrations, strengths…).• Straightforward for calibrations, requires MADX runs for model parameters (quad strengths…) → linear approximation.

3) Least-square minimization:Solve the linearized equation for parameter changes Δc (based on SVD).

4) Iteration:

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CERN accelerator complex

7

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matrix sizes

8

For a ring /line with N BPMs and M correctors per plane, the minimum size of the gradient matrix G is :

(2 ×N ×M) ×(2 ×(N + M))…with only BPM and corrector calibrations as parameters for c.

SPS transfer line: N < 30, M < 30 1800 x 120 0.2 x 106 elementsTT10

SPS ring: N ~ 110 , M = 108 25000 x 220 6 x 106 elementsLHC: N ~ 500 , M ~ 250 250000 x 1500 375 x 106 elements

the complete LHC is tough to handle with all elements included:

RAM + precision + CPU time

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SPS example : before fit

9J. Wenninger

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SPS example : a few fit iterations later …

J. Wenninger

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example : TI8 - SPS to LHC transfer line - quadrupole with wrong setting

J. Wenninger

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Singular Value Decomposition

The singular value decomposition of a matrix is usually referred to as the SVD. This is the final and best factorization of any m × n X matrix whose entries are real numbers. The SVD factorization is of the form

X = U Vt

where U is an m × m orthogonal matrix, Σ is an m × n diagonal matrix with non-negative real numbers on the diagonal, and the n × n unitary matrix Vt denotes the transpose of the n × n orthogonal matrix V.Such a factorization is called a singular value decomposition of X.

The r (nonzero) diagonal entries σi of Σ are known as the singular values of X. A common convention is to list the singular values in descending order. In this case, the diagonal matrix Σ is uniquely determined by X (though the matrices U and V are not).v1, v2, ...vr is an orthonormal basis for the row space.

u1, u2, ...ur is an orthonormal basis for the column spacevr+1, ...vn is an orthonormal basis for the nullspaceur+1, ...um is an orthonormal basis for the left nullspace

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SVD - continued

eigenvector of XXt eigenvector of XtX

X = U Vt

X-1 = V -1 Ut“pseudo-inverse”

where -1 is a diagonal matrix with elements 1/s11,…

mxm mxn nxn

r is the rank of the matrix

OR

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model-independent analysis (MIA) original idea: no attempt to determine exact parameters of optics model,but use beam information and beam response to certain perturbation tomonitor stability, to identify misaligned linac rf structures, etc.

pmpp

m

m

bbb

bbb

bbb

B

...

......

...

...

21

22221

11211 m BPMsp measurements

form huge matrix

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decompose matrix with SVD

)( tD VWB

W eigenvectorsform basis oftemporal patterns

V eigenvectorsform basis ofspatial patterns

Singular values computed forthe SLAC linac, using 5000 pulsesand 130 BPMs

only about 6-15 independent variables affecting the beammotion

a few high-resolution BPMsBPM noise floor

“signals”

16

independent-component analysis (ICA)

ICA calculated l source signals, and determines (unknown) mixing matrix A ;

source signal independence is related to “diagonality” of covariance matrices

time-lagged source signal covariance matrix

modified time-lagged co-variance matrix variance

apply SVD:

where W is the unitary demixing matrix and Dk is a diagonal matrix

apply SVD:

independent-component analysis (ICA) – cont’d

91*J.F. Cardoso and A. Souloumiac, SIAM J. Matrix Anal. Apl. 17, 161 (1996)

Simple description of ICADespite its wide range of applicability, ICA can be understood in terms of the classic ‘cocktail party’ problem, which ICA solves in an ingenious manner. Consider a cocktail party where many people are talking at the same time. If a microphone is present then its output is a mixture of voices. When given such a mixture, ICA identifies those individual signal components of the mixture that are unrelated. Given that the only unrelated signal components within the signal mixture are the voices of different people, this is precisely what ICA finds. ICA does not incorporate any knowledge specific to speech signals; in order to work, it requires simply that the individual voice signals are unrelated.

18

TRENDS in Cognitive Sciences Vol.6 No.2 February 2002

19

coherent tune shift along a long bunchICA MIA/PCA

X. Pang et al, Phys. Rev. ST - Accel. Beams 15, 112802 (2012)

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coherent oscillations & filamentationA) response to a kick: coherent damping

2

' 22 xxxJ

...111

HTSR

)(Re1

zeffbHT

ZN

2)(cos0 tJexx

t

exp. decay

amplitude-dependentoscillation frequency

head-tail damping

synchrotron-radiation damping

bunch population

chromaticity

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1/

1/(100 turns)

1/SR

Ib

Q’=2.7

Q’=14

LEP 45.63 GeV,damping rate1/vs. Ibunch

for different chromaticities[A.-S. Muller]

xrev

SR

JE

fU

0

0

2

11

horizontal damping partition number

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LEP: tune change during dampingQ

turns[A.-S. Muller]

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LEP: detuning with amplitude from single kickQ

2 J[A.-S. Muller]

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B) another response to a kick: filamentation

21

1~ :/1

:1 21

tt

et tc

)(cos1

2

22

122

ttet

Zx t

tZ

x

R. Meller et al.,

SSC-N -360, 1987

both different from e-t/

Z: kick in =(20)2

Q=Q0-a2

amplitude in

ex.:

turns1000after

1 :10 4

t

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amplitude decoherence factor vs. turn number

5 kick

1/5 kick

oscillationamplitude

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Horizontal phase space generated with MAD at the pick-up in section 10. Two chains of stable islands (period 4 and 5) are clearly visible. The inner chain is the one studied during the measurement sessions.

Upper part: Horizontal beam position vs. time measured with a pick-up for the few turns right after the kick (left) and for the last 1500 turns (right). Bottom part: Normalised phase space reconstructed from two pick-ups.

measurement in the CERN PS (A.-S. Muller et al., EPAC 2002) simulated phase space

a fraction of particles trapped in the resonance island

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C) 3rd response to a kick: decoherence due to chromaticity

>0<0

(Q’>0)

t=0 t=Ts/2 t=Ts

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)(cos

2/sin'2 2

2

22

tteZxt

Q

Q

x

ss

R. Meller et al.,SSC-N -360, 1987

x

time

Ts

large

small

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measurement in the CERN PS (A.-S. Muller at al., EPAC 2002)

Measurements of de- and recoherence for two synchrotron frequencies. Both plots are composed of several individual datasets with measurements started at different times after the kick (the current setup only allows the acquisition of about 2000 turns).

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Summary

data analysis techniques LOCO fitsSVDMIA, ICA

beam response to kick excitationcoherent dampingfilamentationchromaticity

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Computer lab, Wednesday 17 June 2015 BCM

consider a simple storage ring consisting of a defocusing quadrupoleand a focusing quadrupole and two bends (or drifts)

10

1 L

1

01

KK

1

01

K

10

1 L

compute tune and at focusing quadrupolecompute tune shift as a function of quadrupole strength change K

over a range -0.1…+0.1compute as and plot vs. K

compute as and plot vs K

use L=1 m, K=0.5 m-1

QQQK

2sin2cos12cot2

0

K

Q

4

hints: 2cos(2Q)=R11+R22

R12=sin(2Q)

1