During acceleration in SIS18/SIS100 at GSI/FAIR longitudinal beam-oscillations are expected to occur. To reduce emittance blow-up, dedicated LLRF beam feedback systems are planned. To date longitudinal beam oscillations have been damped in machine experiments with a finite-impulse-response (FIR) filter controller with 3 filter taps. An alternative approach implementing the FIR filter as a derivative estimator controller is simulated and tested. This approach shares the same controller topology and can therefore be easily integrated in the system. It exploits the fact that the sampling rate of the feedback hardware is considerably higher than the frequency of the beam oscillations. It is therefore capable of damping oscillations without overshoot within one oscillation period

Domont-Yankulova, Dilyana

Klingbeil, Harald

Reichardt, Benjamin R.

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LONGITUDINAL BEAM STABILIZATION AT FAIR BY MEANS OF ADERIVATIVE ESTIMATION∗B. Reichardt†, RMR, TU Darmstadt, GermanyD. Lens, GSI, Darmstadt, GermanyH. Klingbeil, TEMF, TU Darmstadt; GSI, Darmstadt, GermanyD. Domont-Yankulova, TEMF, TU Darmstadt; RMR, TU Darmstadt, GermanyAbstractDuring acceleration in SIS18/SIS100 at GSI/FAIR longi-tudinal beam-oscillations are expected to occur. To reduceemittance blow-up, dedicated LLRF beam feedback sys-tems are planned. To date longitudinal beam oscillationshave been damped in machine experiments with a finite-impulse-response (FIR) filter controller with 3 filter taps[1].An alternative approach implementing the FIR filter as aderivative estimator controller is simulated and tested. Thisapproach shares the same controller topology and can there-fore be easily integrated in the system. It exploits the factthat the sampling rate of the feedback hardware is consider-ably higher than the frequency of the beam oscillations. It istherefore capable of damping oscillations without overshootwithin one oscillation period.INTRODUCTION+-Figure 1: Control loop.Figure 1 shows the simplified topology of the beam phasecontrol loop used in the experiment and in the macro particlesimulations. Starting from beam dynamics the beam phaseΔϕB is measured with respect to a reference DDS signal.The phase of this DDS signal is not influenced by the feed-back. In this paper, the measurement noise will be modeledas white noise. The beam phase is processed by an FIRfilter, which is realised as delayed derivative estimator. Inaddition, there is a transport delay from the processing DSPsystem to the cavity. In Fig. 1, the correction signal Δϕcorr issubtracted from a set value Δϕset. The result is forwarded tothe block cavity dynamics, which models the phase responseof the cavity including the so-called cavity synchronisationloop. A first order low-pass transfer function with a timeconstant of T1 = 20 μs is used to model this block. Thesample time of the DSP-system is TS = 3.22 μs and the trans-port delay time is TD = 10 μs. The range of frequencies of∗ Work supported by GSI† benjamin.reichardt@rmr.tu-darmstadt.dethe beam phase oscillations range from fsyn,min = 10 Hz tofsyn,max = 6000 Hz.FIR FILTER AS DERIVATIVEESTIMATORThe applied derivative estimation approach in this pa-per is a discrete first order delayed derivative. The currentmeasurement value, as well as past values are needed.The filter is designed with 10 consecutive taps. The deriva-tive estimation approach proposed in [2] implements thefilter to perform a linear regression. The derivative esti-mation with a FIR filter with current measurement value i,filter length N ≥ 2 and N measurement values ri−N+1, ..., riis obtained byxi =∑Nn=1(−N + 2n − 1) · ri−N+nTs∑Nn=1n(n−1)2,where the sum in the numerator is the actual filter and thesum in the denominator is a normalizing factor dependingon the filter length. There is a delay of Td =N−12· Ts, asone estimates the derivative in the middle of a series ofmeasurements.Simulations of the filter on a constant signal with whitenoise show a noise level dependence proportional to aboutN−1.5. Larger filters regard more measurement points forderivative estimation which increases statistics as well as alarger time window which increases the signal to noise ratio.On the downside there is additional delay time and errorsdue to not estimating nonlinearities correctly.PARAMETER STUDIES FOR CONTROLPERFORMANCEIn this section an estimation is given on how strong amismatch for the feedback gain influences the control perfor-mance. In stationary operation an ideal reference particle isat the zero crossing of the RF-cavity voltage in the longitudi-nal phase space. Particles only oscillating in close distance(small phase deviations) of the zero crossing, experiencingonly the linear part of the accelerating voltage behave asharmonic oscillators. The corresponding frequency is thelinear synchrotron frequency. Particles oscillating in a largerdistance experience also the nonlinear parts of the accelerat-ing voltage and perform slower oscillations which is callednonlinear synchrotron frequency. The linear synchrotronfrequency is an upper limit of the nonlinear synchrotronfrequency. If there are coherent oscillations, such as dipoleProceedings of IPAC2017, Copenhagen, Denmark TUPIK04806 Beam Instrumentation, Controls, Feedback and Operational AspectsT05 Beam Feedback SystemsISBN 978-3-95450-182-31795 Copyright©2017CC-BY-3.0andbytherespectiveauthors200 250 300 3501.01.52.0Gain [a.u.]Settlingtimewith1%boundary[Tsyn]422 Hz401 Hz380 Hz358 Hz338 Hz316 Hz200 250 300 35005101520Gain [a.u.]Overshoot[%]Figure 2: Single particle simulation of beam phase controlloop with an initial excitation of 10◦ for different frequencies.oscillations, the different oscillation frequencies lead to a de-phasing, also known as filamentation. This leads to a naturaldamping of the beam phase ΔϕB and also to an increase inRMS-emittance. The frequency of the coherent dipole oscil-lation will be referred to as coherent synchrotron frequencyand is lower than the linear synchrotron frequency.For the estimation of the dynamics in the experimentalsetup where fsyn,lin = 422 Hz, a single particle simulationis performed for the control loop. Typical macro particlesimulations show a coherent synchrotron frequency in therange of 75 % to 90 % of the linear synchrotron frequency de-pending on the bunch size. Figure 2 shows the results of thesingle particle simulation with linear synchrotron frequen-cies 422 Hz, 401 Hz, 380 Hz, 358 Hz, 338 Hz and 316 Hzfrom bottom to top in the right picture (color code is thesame in the left part). In the left part is the settling time ofthe beam phase, in the right part the overshoot time againstthe gain. The black borders mark the gain values for ape-riodic behaviour for the frequencies of 422 Hz and 317 Hz,which is a gain interval from 212 to 287. There are steps inthe left picture when the gain value is below threshold foraperiodic behaviour because there are oscillations aroundthe set value. Each half oscillation leaving the 1% boundaryaround the set value leads to an additional step.Tuning for higher synchrotron frequencies would lead toovershoot, whereas tuning for lower synchrotron frequencieswould lead to supercritical behaviour and therefore longersettling times. Nevertheless, 10 % overshoot or 1.75 oscilla-tion periods is still a very good performance.In this simulation, an undamped harmonic oscillator wasused as a model for the beam phase dynamics. Dampingtends to result in lower synchrotron frequencies and somepredamping of the beam phase. Damping requires less gainfor aperiodic behaviour whereas a lower synchrotron fre-quency requires more gain. The uncertainties cancel outeach other partially.Parameter uncertainties have an impact on the controlperformance. Nevertheless the whole plotted range in Fig. 2250 Hz, 1%/s250 Hz, 2%/s600 Hz, 1%/s600 Hz, 2%/s0 2 4 6 8 100.00.20.40.60.81.03 measurement noise [±°]Normalizedgain[a.u.]Figure 3: Macro particle simulation to obtain borders for crit-ical normalized gain which leads to certain RMS-emittancegrowth at a given frequency while using a filter with N = 9filter coefficients.shows reasonable results with settling times below 2.5 oscil-lation periods and overshoots of about 20 % maximum forlarge gain mismatches, compared to about 40 % overshootin [1].NOISE LIMITATIONS FOR CONTROLPERFORMANCEAnother important point to consider is measurement noise.The use of the derivative estimation controller should notdecrease the beam quality, if there is only measurment noisebut no distortion. Macro particle simulations indicate that anoutput noise of the cavity below a threshold of 3σcav = ±5◦does not increase the RMS-emittance of the beam.Figure 3 shows at which gain value - dependend on thenoise level - a given emittance growth rate in the macroparticle simulation is exceeded. The normalized gain value isthe gain for aperiodic behaviour for an undamped harmonicoscillator at the linear synchrotron frequency fsyn,lin. Criticalvalues below the normalized gain 1 are plotted. The dashedextensions of the curves are worst case assumptions, as thegrowth rate limit has not been exceeded at gain 1.The frequencies of 600 Hz and 250 Hz enclose the ex-pected synchrotron frequency in the experiment. As theexperiment has a measurement noise of about 3σ = ±0.5◦which is four times lower than the critical noise of the bluecurve for the normalized gain of 1, there should be an emit-tance growth much lower than 1%s.SIMULATION AND EXPERIMENTALRESULTSIn this section the measurement data of a beam experi-ment which took place on October 18th 2014 is comparedto the results of a macro particle simulation with the samederivative estimation controller and the former used 3 tapfilter approach. The beam was accelerated to a kinetic beamenergy of Ekin = 300MeVuand a dipole oscillation was inten-tionally excited by shifting the gap voltage by ϕset = 2◦. TheTUPIK048 Proceedings of IPAC2017, Copenhagen, DenmarkISBN 978-3-95450-182-31796Copyright©2017CC-BY-3.0andbytherespectiveauthors06 Beam Instrumentation, Controls, Feedback and Operational AspectsT05 Beam Feedback SystemsTable 1: Parameter Specifications for Simulation and Exper-imentParameter ValueIon type 238U73+Number of ions ca. 4 · 108Kinetic energy 300MeVuGap voltage 2.18 kVLinear synchrotron frequency 422 HzPhase shift for dipole oscillation 2◦Harmonic number h = 2open-loop operation is shown in Fig. 4, while the results inthe closed-loop case are shown in Fig. 5. The accelerationwas done in open-loop mode, whereas the feedback-loopwas closed as soon as flattop was reached. Table 1 gives anoverview over the parameter specifications in the experimentand the simulations.Figure 5 shows that both the experimental measurement(blue) and the macro particle simulation with the FIR filteras derivative estimator (red) are in good agreement. Theexcited dipole oscillation is damped within one oscillationperiod and there is almost no overshoot. The gain is 186,which is lower than the estimated interval from 212 to 287.Figure 4 shows that there is damping in open-loop case,which leads to less required gain for aperiodic behaviourin the closed-loop case. For comparison there is also thesimulated case where the FIR filter is used as a bandpassfilter [1]. An overshoot occurs and the oscillation can beconsidered as damped after at 4.875 s.This measurement is a proof of principle. The good agree-ment of experimental measurement data and the macro par-ticle simulation indicates that there are no major effectsmissing in the modeling of the whole control loop.4.76 4.77 4.78 4.79 4.80-7-6-5-4-3-2-10Time [s]PhaseDeviation[°]Figure 4: Phase shift ϕset applied in the experiment in theopen-loop case.4.87 4.88 4.89 4.90-12-10-8-6-4-2Time [s]PhaseDeviation[°]Figure 5: Phase shift ϕset applied in experiment and simu-lations with arbitrary offsets: Derivative estimation experi-ment, Derivative estimation simulation, 3 tap FIR bandpasssimulation.CONCLUSION AND OUTLOOKTheoretical studies of a derivative estimation feedbackapproach as well as macro particle simulations regardingperformance limitations caused by measurement noise forsynchrotron frequencies around 400Hz were performed. Itwas demonstrated in an experiment that longitudinal dipoleoscillations can be damped within one oscillation periodwith nearly no overshoot by using an FIR filter feedbackcontroller implemented as derivative estimator. The beambehaviour in the closed-loop case is simulated accurately.Further studies also show that the controller should be usablewith either more noise or higher synchrotron frequenciesup to fsyn,lin = 6kHz and a measurement noise level up to3σ = ±10◦ regarding the measurement precision of thebeam phase. Future work will consider beam phase controlalso during acceleration of the beam, as well as for dual-harmonic operation.ACKNOWLEDGMENTThe authors would like to thank K. Groß, U. Hartel,U. Laier, S. Schäfer, B. Zipfel, D. Ondreka and H. Lieber-mann for their contribution to the machine experiment.REFERENCES[1] H. Klingbeil, B. Zipfel, M. Kumm, P. Moritz, “A Digital Beam-Phase Control System for Heavy-Ion Synchrotrons”, IEEETransactions on Nuclear Science, vol. 54, no. 6, pp. 2604-2610,Dec. 2007.[2] P. Mai, C. Hillermeier, “Least Squares based Derivative Estima-tion: Theory and Tuning Rules for the Practical Application”,at-Automatisierungstechnik, vol. 56, no. 10, pp. 530-538, Oct.2008.Proceedings of IPAC2017, Copenhagen, Denmark TUPIK04806 Beam Instrumentation, Controls, Feedback and Operational AspectsT05 Beam Feedback SystemsISBN 978-3-95450-182-31797 Copyright©2017CC-BY-3.0andbytherespectiveauthors

Longitudinal Beam Stabilization at FAIR by Means of a Derivative Estimation

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Longitudinal Beam Stabilization at FAIR by Means of a Derivative Estimation

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