At the Karlsruhe Research Accelerator (KARA), an analytical online model of the orbit response matrix (ORM) has been developed and tested. The model, called the bilinear-exponential model with dispersion (BE+d model), is derived from the Mais-Ripken formalism describing coupled betatron motion. Compared to the standard approach of measuring the ORM, this method continuously adapts to changing beam optics without a dedicated measurement. It is especially useful for storage rings without turn-by-turn capable beam position monitors (BPMs) as the online model also gives access to estimates of the coupled optical functions. In the following, experimental orbit correction results and a comparison of fitted and simulated optical functions are presented

Andrian, Ivan

Assmann, Ralph

Bisoffi, Giovanni

Blomley, Edmund

Bründermann, Erik

Fabris, Alessandro

Koetter, Stephan

McIntosh, Peter

Mueller, Anke-Susanne

Santamaria Garcia, Andrea

Schuh, Marcel

Vinicola, Giulia

English

KITopen

ONLINE FIT OF AN ANALYTICAL RESPONSE MATRIX MODEL FORORBIT CORRECTION AND OPTICAL FUNCTION MEASUREMENTS. Kötter ∗, E. Blomley, E. Bründermann, A. Santamaria Garcia, M. Schuh, A-S. MüllerKarlsruhe Institute of Technology (KIT), Karlsruhe, GermanyAbstractAt the Karlsruhe Research Accelerator (KARA), an ana-lytical online model of the orbit response matrix (ORM)has been implemented. The model, called the bilinear-exponential model with dispersion (BE+d model), is derivedfrom the Mais-Ripken parameterization of coupled betatronmotion. The online fit continuously adapts the model tochanging beam optics without dedicated measurements us-ing only orbit correction results as input. This gives accessto an up-to-date ORM for orbit correction as well as esti-mates for the coupled beta function, betatron phase, and thetunes. After comparing such beta function fit results to anoptics simulation and evaluating orbit correction with themodel, problems of the approach are discussed.INTRODUCTIONKARA is a 2.5 GeV synchrotron light source and accelera-tor test facility at the Institute for Beam Physics and Technol-ogy (IBPT) of the Karlsruhe Institute of Technology (KIT).Here, a new orbit correction software is under development.The goal is a program that performs well with different exper-imental operation modes, such as negative alpha optics [1],and to investigate novel orbit correction approaches in gen-eral. A first iteration was derived from a program used at theDortmunder Elektronenspeicherring-Anlage (DELTA) [2].It relies on a conic solver for convex constrained optimizationfor calculating orbit corrections that allows orbit and steererstrength constraints, and can also correct the orbit lengthby modulating the frequency of the radio frequency (RF)accelerating cavity. At KARA, an analytical online model ofthe ORM based on the BE+d model that had been proposedin [3] was added to the software. Its ring buffer is loadedwith tuples of orbit and steerer strength changes resultingfrom orbit corrections, and gives access to estimates for thecoupled beta function, the betatron phase, and the tunes, aswell as an analytical representation of the ORM. Similarly tothe local optics from closed orbits (LOCO) approach [4], themethod measures the linear optics of the storage ring with-out turn-by-turn capable beam position monitors (BPMs).An advantage of the BE+d model fit is that it requires nodetailed lattice information to do so. Another usecase isorbit correction where it can be used as a replacement fora measured ORM. Compared to an online fit of the ORMitself, the fit of the analytical model is expected to requireless measurements for the same signal-to-noise ratio. Thereason is that the analytical model has only a fraction of thedegrees of freedom of the naive matrix.∗ stephan-robert.koetter@kit.eduFirst tests of the online fit of the model were done. To min-imize the required machine time, the program was trainedwith previously measured ORMs. Although these feature abetter signal-to-noise ratio than a buffer of orbit corrections,they allow to give a basic assessment of the program’s capa-bilities. Comparisons show that the linear-optics fit matchesthe OCELOT [5] optics model of the storage ring fairly well,and that the corresponding analytical ORM representationscan be used for orbit correction. However, problems arisefrom a non-linear dependence of the orbit on the steererstrengths.ONLINE MODELThe online model relies on the response set fit algorithm(RSFM) [6] to fit the product of an analytical representationof the ORM 𝑅𝑤𝑗𝑘 and the kick angle 𝜃𝑘𝑠 to a ring-bufferedset of measured orbit responses 𝑟𝑤𝑗𝑘𝑠 by solvingmin?⃗?∑𝑠𝑗𝑤(𝑅𝑤𝑗𝑘𝜃𝑘𝑠 − 𝑟𝑤𝑗𝑘𝑠)2 . (1)Here 𝑤 indexes the horizontal or vertical plane, 𝑗 referencesthe BPM, 𝑘 the steering magnet, and 𝑠 is the number of thesample in the ring buffer. The analytical ORM representationis the BE+d model [7]𝑅𝑤𝑗𝑘 = 𝑅𝑤𝑗𝑘 (𝛽𝑚𝑤𝑗, Φ𝑚𝑤𝑗, 𝑞𝑚, 𝐷𝑤𝑗𝛿𝑘) , (2)which is based on a complex variant [8] of the Mais-Ripkenparameterization [9] that describes the trajectories in a linearstorage ring as a superposition of two modes of betatronmotion indexed by 𝑚. As such, the model depends on thecoupled beta function 𝛽𝑚𝑤𝑗, the coupled betatron phaseΦ𝑚𝑤𝑗 and the mode tune 𝑞𝑚, as well as a dispersion term𝐷𝑤𝑗𝛿𝑘. Together these quantities make up the vector of fitvariables ?⃗?.For 𝑊 = 2 planes, 𝑀 = 2 betatron modes, 𝐽 = 40 −1 = 39 BPMs [10] (one BPM is not in use), and 𝐾 = 44horizontal and vertical steering magnets [11] that KARA isequipped with, the BE+d model has [12]2𝑀𝑊𝐽 + 2𝑀𝐾 + 𝑀 + 𝑊𝐽 + 𝐾 − (2𝑀 + 1) = 607 (3)degrees of freedom. The measured ORM has about fivetimes more 2𝐽𝐾 = 3432. Recording an ORM at KARAtherefore requires at least 44 distinct orbit measurementswith all steering magnets perturbed sequentially, while theRSFM-based online model only requires 607/(2𝐽) < 8 mea-surements with all steerers perturbed simultaneously. Thelatter should be seen as a bare minimum though. A previousstudy at DELTA recommends twice this number [6].20 40 60 80 100𝑠 (m)0.02.55.07.510.012.515.017.520.0𝛽𝑥and𝛽𝑥𝑥(m)20 40 60 80 100𝑠 (m)051015202530𝛽𝑦and𝛽𝑦𝑦(m)Figure 1: Comparison of the horizontal (top) and verticalbeta function (bottom) from OCELOT (blue) and the RSFM(red). The standard deviations of the measurement were sosmall that they are not shown (see text).BETA FUNCTION COMPARISONThe coupled beta function estimates resulting from aBE+d model fit of an ORM recorded after injection at0.5 GeV with open insertion devices 𝛽𝑥𝑥𝑗 and 𝛽𝑦𝑦𝑗 are com-pared to the beta function values calculated with an opticsmodel of KARA in OCELOT [5] 𝛽𝑥(𝑠) and 𝛽𝑦(𝑠) in Fig. 1.Here, 𝛽𝑥𝑥𝑗 is the the horizontal projection of the primarilyhorizontal betatron mode at BPM 𝑗 and 𝛽𝑦𝑦𝑗 the vertical pro-jection of the primarily vertical mode. The fitted tunes are𝑄𝑥 ≈ 6.767 as well as 𝑄𝑦 ≈ 2.785 and the fractional tunesmeasured by the bunch-by-bunch feedback are 𝑞𝑥 = 0.771 aswell as 𝑞𝑦 = 0.791. While the fitted tunes and the measuredones are very similar, the RSFM beta function estimates andthe optics calculation only match fairly well. The fit wastherefore repeated 100 times in a Monte-Carlo analysis witha uniformly distributed noise of 5 mm/mrad on the inputdata to give an estimate of the statistical uncertainties of thetune and beta function estimates. Of these fits, one divergedand was, on the condition of max(𝛽𝑥𝑦) > 0.1 ⋅ max(𝛽𝑦𝑦)and max(𝛽𝑦𝑥) > 0.1 ⋅ max(𝛽𝑥𝑥) (KARA has practically nocoupling), excluded from the analysis. The standard devi-ations across the remaining 99 fits were so small that theyare not shown here. The deviation of the fitted peak betafunction values from the optics calculation therefore mustbe a systematic error (for example due to non-linearities inthe ring buffer measurements).ORBIT AND RF FREQUENCYCORRECTION EVALUATIONThe analytical ORM representation for an ORM recordedin user operation at 2.5 GeV with closed insertion deviceswas evaluated for orbit correction. In general, this workedas well as any measured ORM but it also inherited a prob-lem. In user operation at KARA, a method for RF frequencycorrection [13] is usually used to maintain a small orbitRMS even if the orbit lengthens or shortens due to thermaleffects. The correction of the RF frequency is calculatedby adding the corresponding orbit response as a column tothe ORM and solving the orbit correction problem for themodified matrix [14]. While using a simulated ORM anddispersion function works fairly well, utilizing measureddata leads to oscillations such as shown in Fig. 2. Althoughthese can be suppressed via regularization, selecting a suit-able singular-value cut-off is neither straight forward nornecessarily always possible. This problem persisted whenusing the analytical ORM.0 5 10 15 20t (min)−0.6−0.4−0.20.00.2∆f(kHz)0.5960.5980.6000.6020.6040.606RMS(mm)Figure 2: Correction of RF frequency (blue) and horizontalorbit RMS over all BPMs (red) vs time. After about 12 min,the singluar value cut-off is changed. The oscillations shiftslightly but persist anyways.NON-LINEAR BPM READINGSThe orbit response of a storage ring is only approximatelylinear in a small parameter range, outside which the effectsof sextupoles, BPM geometry, and hysteresis introduce anon-linear behavior. At KARA, this was now investigated byramping-up the currents of the horizontal steering magnets(HSMs) in steps of Δ𝐼 = 4 mA until a maximum orbit devi-ation of ±6 mm was reached at a particular BPM (shown inthe top plot of Fig. 3). Each power source was then rampeddown until the orbit reached the maximum orbit deviationfor a second time before it was ramped up again to returnto its initial setting. Each data series was then fitted with athird-order polynomial𝑓 (𝐼) = 𝑎𝐼3 + 𝑏𝐼2 + 𝑐𝐼 + 𝑑, (4)where 𝐼 is the steerer current. Examples of this measure-ment and the fitted polynomials for two BPM-HSM pairsare given in Fig. 3. While the bottom plot presumably showsthe effect of a sextupole magnet where the linear increase insteerer strength is firstly off-set and then reversed by the sex-tupole field strength that quadratically depends on the orbitdeviation (large 𝑏), the top measurement displays the typicalthird-order symmetry of the non-linear behavior introducedby the BPM geometry (large 𝑎). Both data series also showthe effect of hysteresis.−0.1 0.0 0.1 0.2 0.3 0.4 0.5𝐼 (A) at HSM:S1:01−0.34−0.32−0.30−0.28−0.26−0.24−0.22−0.20𝑥(mm)atBPMX:S1:08𝑓 (𝐼) = 0.33𝐼3 − 1.4𝐼2 + 0.42𝐼 − 0.23−0.75 −0.50 −0.25 0.00 0.25 0.50 0.75𝐼 (A) at HSM:S1:02−6−4−20246𝑥(mm)atBPMX:S4:08𝑓 (𝐼) = −1.8𝐼3 − 0.35𝐼2 + 7.8𝐼 + 0.26Figure 3: Orbit measurement (red) and polynomial fit (blue)for different steerer strengths at different BPM-HSM pairs.The polynomial fit of the data was repeated for each BPM-HSM pair after transforming the orbit readings and changesof the steerer currents to a uniform interval [−1, 1]. A com-parison of the second- and third-order coefficients of all pairsis given in Fig. 4. Unsurprisingly, many show some formof third-order distortion (bottom plot) as the BPM geome-try affects all BPM readings. Large quadratic coefficientsare considerably less prevalent (top plot) and mostly appearwhen then the amplitude of betatron motion is small.CONCLUSIONThe RSFM analysis built into the online model producesreliable beta function and tune estimates and gives accessto an analytical ORM representation that can be used fororbit correction. The deviation of the fitted beta functionestimates from an OCELOT optics model in the peaks andoscillations appearing while correcting the RF frequencycan probably be attributed to a non-linear dependence ofthe transverse orbit measurement on the steerer strengths.S4:09S1:03S1:06S1:09S2:02S2:05S2:08S3:01S3:04S3:07S3:10S4:03S4:06BPMXS1:01S1:04S1:08S2:03S2:07S3:02S3:06S4:01S4:04S4:08HSM−101𝑏(mm/A2)S4:09S1:03S1:06S1:09S2:02S2:05S2:08S3:01S3:04S3:07S3:10S4:03S4:06BPMXS1:01S1:04S1:08S2:03S2:07S3:02S3:06S4:01S4:04S4:08HSM−0.2−0.10.00.10.2𝑎(mm/A3)Figure 4: Second-order coefficients 𝑏 (top) and third-ordercoefficients 𝑎 (bottom) of the polynomial fit according toEq. (4) for orbit and steerer-currents normalized to uniformintervals for all BPM-HSM pairs.Most BPM-HSM pairs show either a second- or third-orderdependence.OUTLOOKThe problems arising from the linear assumption inher-ent to matrix-based orbit correction approaches are usuallycountered with regularization. As cutting of singular valuesdoes not work sufficiently well in our case, Thikonov reg-ularization could be tried. However, it might be advisableto switch to a non-linear orbit response model instead aswas shown in Refs. [15] and [16]. Such an approach wouldprobably not only remove the problem of the oscillationsduring RF frequency correction but would also work betterwith non-linear and experimental optics such as negativealpha optics [1].ACKNOWLEDGEMENTSS. K. thanks all scientists and technicians supporting op-erations at KARA. This work was in part supported by theHelmholtz Innovation Pool project ACCLAIM.REFERENCES[1] P. Schreiber, “Negative momentum compaction operationand its effect on the beam dynamics at the accelerator testfacility KARA,” Dissertation, Karlsruhe Institute of Technol-ogy (KIT), 2022. doi:10.5445/IR/1000148354[2] S. Kötter, “Introducing two Energy-Correction Schemes atDELTA,” in Proc. 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Online fit of an analytical response matrix model for orbit correction and optical function measurement

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Online fit of an analytical response matrix model for orbit correction and optical function measurement

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