Inverse modeling of circular lattices via orbit response measurements in the presence of degeneracy D. Vilsmeier

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Inverse modeling of circular lattices via orbit response

measurements in the presence of degeneracy

D. Vilsmeier∗

Johann Wolfgang Goethe-University Frankfurt,

60323 Frankfurt am Main, Germany

R. Singh

GSI Helmholtz Centre for Heavy Ion Research,

64291 Darmstadt, Planckstr. 1, Germany

M. Bai

SLAC National Accelerator Laboratory,

2575 Sand Hill Rd, Menlo Park, CA 94025, USA

(Dated: October 27, 2022)

Abstract

The number and relative placement of BPMs and steerers with respect to the quadrupoles in

a circular lattice can lead to degeneracy in the context of inverse modeling of accelerator optics.

Further, the measurement uncertainties introduced by beam position monitors can propagate by

the inverse modeling process in ways that prohibit the successful estimation of model errors. In this

contribution, the inﬂuence of BPM and steerer placement on the conditioning of the inverse problem

is studied. An analytical version of the Jacobian, linking the quadrupole gradient errors along with

BPM and steerer gain errors with the orbit response matrix, is derived. It is demonstrated that

this analytical version of the Jacobian can be used in place of the numerically obtained Jacobian

during the ﬁtting procedure. The approach is ﬁrst tested with simulations and the ﬁndings are

veriﬁed by measurement data taken on SIS18 synchrotron at GSI. The results are crosschecked

with the standard numerical Jacobian approach. The quadrupole errors causing tune discrepancies

observed at SIS18 are identiﬁed.

∗d.vilsmeier@gsi.de

arXiv:2210.14779v1 [physics.acc-ph] 26 Oct 2022

CONTENTS

I. Introduction 3

II. Orbit response matrix 5

III. Degeneracy 7

A. Analytical derivative of orbit response 8

B. Pure degeneracy 10

1. Global degeneracy 12

C. Quasi-degeneracy 14

D. Example 17

E. Counteracting quasi-degeneracy 19

1. Placement of BPMs/steerers 20

IV. Fitting of the orbit response matrix 21

V. Experiment 23

A. Measured data 23

B. Mitigation of quasi-degeneracy 26

1. SVD cutoﬀ 27

2. ∆K1Lweights 27

3. Leaving out T-quadrupoles 28

4. Comparison 28

VI. Conclusions 30

A. Derivative of orbit response with respect to quadrupole strength 32

B. Derivative of orbit response with respect to quadrupole strength for beamlines 35

C. Proof: S,Q3,B Jacobian is rank deﬁcient 35

D. Proof: Sh,Sv,Q6,Bh,Bv Jacobian is rank deﬁcient 42

References 43

I. INTRODUCTION

Precise knowledge of the lattice’s optics elements is crucial for optimal operation of any

circular accelerator. It is especially important for the ﬂexible and fast ramping synchrotron

like SIS18 where transient eﬀects can change the lattice properties cycle to cycle as well as

during the energy ramp. Inability to identify these changes or model errors in general can

lead to beam emittance dilution or beam losses. Linear Optics from Closed Orbits (LOCO)

is a common method for machine model estimation which relies only on the measurement

of the orbit response matrix (ORM). First detailed discussion of LOCO can be found in

[1] and since then the technique has experienced frequent usage at diﬀerent institutes [2–5].

Typically, LOCO takes a measured ORM and varies all relevant lattice parameters in a

multi-dimensional optimization problem to match the simulated with the measured ORM.

Based on the outcome of the optimization procedure, machine parameters are adjusted to

reach the design values.

Beam position monitor (BPM) errors are unavoidable during measurement and will cast

an uncertainty on the measured ORM. This uncertainty then propagates through the inverse

modeling process and inﬂuences the precision of derived parameters. Depending on the

lattice and the optics, the eﬀect of BPM errors can be more or less problematic for the

accuracy of inverse modeling results. In some cases, the inﬂuence of BPM errors can even

hinder the successful reconstruction of quadrupole errors. An improvement of the eﬃciency

was introduced in [6] by adding speciﬁc constraints for the ﬁtting parameters. A related

approach for improving the eﬃciency was introduced in [7].

Because measurement of the ORM typically varies one steerer at a time it can take

signiﬁcant amount of machine time. There have been eﬀorts to reduce the time and impact

of the measurement, for example by sine-wave excitation of multiple steerers at diﬀerent

frequencies simultaneously [8]. Another approach used the data obtained from closed-orbit

feedback correction to continuously update an estimate of the ORM; for suﬃcient number of

iterations, this will converge to the true ORM [9]. In addition to the measurement time, the

inverse modeling process itself contributes to the required time until results are available.

While diﬀerent optimizers need diﬀerent number of iterations until convergence, Jacobian-

based optimizers use by far the fewest number of iterations since the Jacobian contains lots

of information about where the minima lies. However, signiﬁcant time is spent to compute

the Jacobian via ﬁnite-diﬀerence approximation. One aspect of the presented work is to

reduce the Jacobian’s computation time.

In this contribution, we derive an analytical version of the Jacobian relating the ORM

and the quadrupole strength errors along with BPM and steerer gain errors. This Jacobian

matrix is used by the optimizer, e.g. Levenberg-Marquardt, in order to improve the current

best guess of lattice errors during an iterative process. We have studied the properties

of this analytical Jacobian with respect to conditioning of the inverse problem. We show

that the analytical Jacobian highlights all relevant properties of the model error estimation

problem. Rank deﬁciency of the Jacobian implies a degeneracy of the inverse problem while

small eigenvalues of the Jacobian suggest quasi-degeneracy for some patterns of quadrupole

errors. These patterns are more susceptible to measurement uncertainty. We further use the

analytical version of the Jacobian, obtained from the lattice’s Twiss data, during the ﬁtting

procedure and show that it reaches convergence similar to using the numerically obtained

Jacobian. The analytical Jacobian is obtained quickly since it requires only a single Twiss

computation for the lattice.

In light of the evaluated Jacobian and its discussed properties, inverse modeling of the

SIS18 synchrotron is performed for the ﬁrst time. We have identiﬁed and diagnosed several

model errors for SIS18. A notable error is the tune oﬀset of 0.02 units in the horizontal

plane which was a known discrepancy for several years in the SIS18 machine model.

In this process, we also devised a general iterative method for automatic and online

correction of quadrupolar errors simply based on the analytical Jacobian and measured

ORM. This method has similarities with iterative closed orbit correction.

In the following, the structure of the paper is described. In section II we introduce

the lattice used throughout this contribution and the concept of orbit response matrix.

Section III explains the inverse problem with regard to degeneracy of its solutions. The

analytical derivation of the Jacobian is presented. Also, the inﬂuence of BPM and steerer

placement on the degeneracy is shown. Section IV discusses the ﬁtting procedure by using

the Jacobian as well as discusses the convergence properties for diﬀerent approaches. In

section V the experimental results are presented.

II. ORBIT RESPONSE MATRIX

The orbit change xbat BPM bwhen changing the steerers indexed with sby a kick δs, is

given by [10]:

xb=X

δs

√βbβs

2 sin(πQ)cos(πQ − |µb−µs|)−DbDs

1

γ2−1

γ2

tC

(1)

where βb,s and µb,s denote, respectively, the beta functions and the phase advances at

BPM and steerer position, and Qis the betatron tune. In the second term, Db,s denotes the

the dispersion at BPM and steerer position and Cis the circumference of the synchrotron;

γand γtdenote, respectively, the beam energy and transition energy of the lattice (γ=E

E0).

This term is only relevant for synchrotrons operating near transition energy.

Hence, the orbit change is a linear function in the applied kick and it encodes the optics

via the lattice functions βand µ. The orbit response rbs at BPM breacting to a single

steerer sis deﬁned as:

rbs =xb

δs

(2)

The orbit response matrix (ORM) arranges the orbit responses for all BPM/steerer pairs

in a matrix form: rbs where bis the row index and refers to BPMs and sis the column index

and refers to steerers.

The exemplary lattice of SIS18, which is used throughout this contribution, consists of

12 sections. An overview is presented in Fig. 1. Each section contains three quadrupoles,

labeled F,Dand T, and the placement and strength of these quadrupoles is identical in

each of the sections. This triplet structure is utilized to increase the transverse acceptance

during beam injection. The strength of T-quadrupoles is gradually decreased by one order

of magnitude during the ramp, resulting in a small strength during extraction optics. The

36 quadrupoles are connected with ﬁve distinct power supplies, separating the quadrupoles

into the following families:

•6 F-quads from odd numbered sections,

•6 F-quads from even numbered sections,

•6 D-quads from odd numbered sections,

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InversemodelingofcircularlatticesviaorbitresponsemeasurementsinthepresenceofdegeneracyD.VilsmeierJohannWolfgangGoethe-UniversityFrankfurt,60323FrankfurtamMain,GermanyR.SinghGSIHelmholtzCentreforHeavyIonResearch,64291Darmstadt,Planckstr.1,GermanyM.BaiSLACNationalAcceleratorLaboratory,2575SandHillRd,...

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Inverse modeling of circular lattices via orbit response measurements in the presence of degeneracy D. Vilsmeier

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