Parameter Estimation and Tuning of a Multivariable RF Controller with FPGA technique for the Free Electron Laser FLASH

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1 28 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA Jne -3, 28 ThBI2.2 Parameter Estimation and Tning of a Mltivariable RF Controller with FPGA techniqe for the Free Electron Laser FLASH C. Schmidt, G. Lichtenberg, W. Koprek, W. Jalmna, H. Werner, S. Simrock Abstract The operation of a Free Electron Laser in the X-ray wavelength range is the goal of the international XFEL project. The project critically depends on the linear accelerator component, where radio freqency fields have to be controlled with a very high amplitde and phase precision and nder varios sorces of distrbances. This paper shows for the first time closed loop reslts of dynamic fixed order H controllers implemented in a FPGA based control system on the FLASH facility at DESY. I. INTRODUCTION For many research activities a light sorce that is able to resolve objects on an atomic level wold be favorable, e.g. in moleclar biology. Wavelengths of X-ray radiation are in the range of the diameter of an atom ( m) sch that X-ray radiation is sitable for the desired experiments. However, conventional X-ray sorces can not provide X- ray plses with a sfficiently short plse length. This has the effect that e.g. single biomolecles are destroyed by the highly energetic radiation. Moreover, the resoltion of conventionally prodced X-ray radiation is limited by the broadness of its spectrm. Therefore, laser light is sed for a variety of experiments becase it can be better focsed compared to other light sorces, it is monochromatic, and very short plses can be prodced. A crrent field of research in particle accelerator physics are Free Electron Lasers (FEL), which prodce laser radiation with tnable wavelength. At the German Electron Synchrotron (DESY) in Hambrg the X-ray Free Electron Laser research project XFEL is condcted. The goal of the project is to bild a Free Electron Laser operating in the X-ray wavelength by the year 22 []. Figre 2 shows the strctre of the Vacm Ultraviolet FEL (FLASH) which is already working at DESY. On the left part, a linear accelerator is shown, which increases the energy of loaded particles, e.g. electrons by the interaction with electromagnetic radio freqency (RF) fields. The linear accelerator consists of resonators for the RF-fields hosed in cryomodles. The RF-fields inside these spercondcting resonator cavities are spplied by an actator system for a finite time interval and then trned off again periodically. The process reqires very dense electron bnches, either longitdinal or transversal. The transverse dimensions of C. Schmidt, W. Koprek, W. Jalmna, and S. Simrock are with DESY Hambrg, Notkestrae 85, 2267 Hambrg, Germany christian.schmidt@desy.de, waldemar.koprek@desy.de, wjalmn@desy.de, stefan.simrock@desy.de G. Lichtenberg and H. Werner are with the Institte of Control Systems, Hambrg University of Technology, 27 Hambrg, Germany Lichtenberg@thh.de, h.werner@thh.de the electron bnches has to be in the order of the desired FEL-wavelength, while peak crrent after bnch compression has to exceed 5A (infrared) or even 5kA (X-ray), which determines the longitdinal dimension. The typical charge of a bnch is in the nc-range. The reqirements for the accracy of the accelerating field can be derived from these vales. In order to achieve reference tracking as well as distrbance rejection an RF-field control system is sed. Field gradient [MV/m] 25 Filling Flat-Top Beam Loading 5 3 Decay 248 Fig. : One RF-plse in Spercondcting Cavities Time [s] In Figre the amplitde of the desired envelope of the RF-field is displayed for one RF-plse as a fnction of time. The field inside the accelerator cavities has to be kept constant once the reqired amplitde for the appropriate energy gain of the electrons has been reached at the end of the so called filling phase. Dring the flat top phase the electron beam is injected into the accelerator. When the electron beam has passed, the RF-field is trned off and the field amplitde decays. The envelope of the RFfield oscillation mst be kept constant in amplitde and phase dring the flat top time interval to transfer a precise amont of energy to the electrons. The control objectives regarding the necessary RF-field performance for the XFEL are specified as follows: The maximm of the field amplitde mst be within.% (rms) of the reference vale. The field phase may not differ from the reference vale by more than. (rms). Spercondctivity tends to break down above certain field gradients inside the cryomodles, leading to a significant drop of the qality factor of the resonators. The controller algorithm needs to be aware of this hard pper limit especially since the cavities need to be operated close to these limits in order to flfill the physical goals of the machine. Ths, the /8/$ AACC. 256

2 Fig. 2: A Free Electron Laser with Front End, Accelerating Strctres and Undlators open loop models shold reflect the behavior of the system in a way that the precision of the closed loop dynamics given by the model is better than the desired objectives. Until now, RF fields of FELs have been controlled by decentralied P controllers [4]. With new FPGA techniqes a sampling time of µs - that is necessary for digital control of the particle accelerators - was achieved, which was impossible before. The theoretical claim that these controllers cold be parameteried by parameter identification and controller design methods have not been shown with any real FEL accelerator system before. Ths the problem was to choose appropriate identification methods and design tools that atomatically find and adjst controller parameters with the help of a procedre that cold be started with a single psh btton by the operators. Important steps to sccessflly realie and test the procedres proposed in this paper have been the se of black box models which are widely sed in engineering instead of white box models with physical parameters that are standard in particle physics. The paper is organied as follows: Section II gives a detailed description of the RF system of the linear accelerator with its digital control system After introdcing the identification experiments in Section III, the controller design method is presented in Section IV. Finally the reslts of the first implemented MIMO controllers are shown in Section V. II. RF SYSTEM OF THE LINEAR ACCELERATOR A. System strctre At FLASH, eight cavities are hosed in each cryomodle, which is a cooling strctre with pipelines of liqid Helim. In total six cryomodles are assembled at FLASH bt only the first accelerator cryomodle ACC is considered in this paper, whereas the other modles are not strctrally different. The loaded qality factor of a resonator is defined as Q L = ω 2ω /2, where ω denotes resonance freqency of the cavity with a half bandwidth of ω /2 (HWHM ), and in the range of Q L = 3 6. The resonance freqency is determined by the geometry which allows specific propagation of standing RFwaves. Half Width at Half Maximm The system dynamics of the cavities can be modeled by a series of copled LCR resonators [4]. After separation of the fast RF oscillations, a second order time variant differential eqation for the envelopes of the electrical field components can be derived. The time variance is de to the following effects: if the length of the cavity changes, the resonance freqency changes as well. De to the relatively thin walls, the resonators become ssceptible to mechanical vibrations which detne the resonance freqency. Two main distrbance sorces casing this detning are distingished: Lorent Force Detning, Microphonics. The oscillating electromagnetic fields inside the cavities indce crrents in the cavity walls. These crrents lead to Lorent forces acting on the cavity walls and deforming the cavities. The magnitde of the Lorent forces is proportional to the sqare of the accelerating field gradients sch that Lorent force detning can be considered as a correlated and repetitive distrbance in freqency ranges p to kh [4]. The resonance freqency ω =.3GH of the cavities with a half bandwidth of ω /2 = 26.7H changes by Lorent Force Detning in the range of ±H/MV 2. Mechanical vibrations cased by the accelerators environment in a typical freqency range below kh are transferred by mechanical monts to the cavities. Therefore, microphonics can be considered as ncorrelated distrbance that only weakly inflences the cavity fields dring the.3ms long RF plse bt mainly inflences the resonance freqency from one RF plse to the next one. Other sorces of distrbances like the inflence of the electron beam on the RF-fields are not taken into accont in this paper. The actator system receives a precise RF signal of.3gh from the master oscillator (MO). This low power sinsoidal signal can be changed by the vector modlator in amplitde and phase. The otpt signal of the vector modlator is amplified by a klystron, which is a radio freqency amplifier. The amplified RF waves are transferred from the klystron to the cavities inside the cryomodles via a wavegide transmission and distribtion system. For economical reasons, one high power klystron spplies all 8 32 cavities of an RF station, ths RF fields can not be inflenced in each cavity individally the system is nderactated. 257

3 ADC ADC DAC DAC Master oscillator Vector modlator Klystron ~ Wavegide 8x cryomodle + 25 kh LO field probe + 25 kh LO field probe 25 kh 25 kh clock clock Feedforward I f I fq Q c, I c, Q 2 a b K( 2 c d 2 a2 b2 K2( 2 c2 d2 FPGA System MIMO - Controller K K 2 K 2 K22 a2 K2( c2 a22 K22( c b2 d2 b22 d 22 F =MH a b b a e I e Q Calibration yi yq F =MH a b b a r I r Q Setpoint Fig. 3: Strctre of the RF System with Master Oscillator, Vector Modlator, Klystron, Cryomodle, Measrement and Calibration System and the FPGA implemented Control System B. Digital Control System The spercondction cavity simlator and Controller (SIMCON) is a project with the goal to spply reliable low latency controllers, based on field programmable gate array (FPGA) strctres. The task of SIMCON is to control the vector sm of fields in spercondcting cavities, bt can also sed as a simlator of a cavity and as test-bench for other accelerator sb-systems. It allows the practical implementation of fast algorithms. A block diagram of the Low Level Radio Freqency (LLRF) control system is presented in Fig. 3, where the lower part shows the digital FPGA controller. The LLRF control system has the task to keep the plsed RF fields in the spercondcting cavities of the RF station at the reference vale dring the flat top phase of one RF plse shown in Figre. The main advantage compared to analog devices is that all SIMCON parameters are programmable which allows high flexibility for system pgrades and adaption. After measring the actal RF-field by pickp antennas, the signals are downconverted to an intermediate freqency of 25kH. The real (I) and imaginary (Q) field components are digitalied in analog-digital-converters (ADC) with a sampling freqency of MH. An overview of the signals shown in Fig. 3 are represented in terms of I and Q. Inpt signals I, Q : Control signals of the actator system which are acting on the vector modlator. Otpt signals y I,y Q : The real and imaginary part of the sm of the RF-field voltage vectors of eight cavities. Reference signals r I,r Q : Reference signals of the real and imaginary part of the vector sm of the RF-field s voltage vectors given by look-p tables for the specified field gradient. Feedforward signals f I, f Q : Open loop control Signals. Control signals c,i, c,q : The feedback controller s otpt signals which are sperimposed on the feedforward signals. Control error signals e I,e Q : Deviations in real and imaginary part of the otpt signals from the reference signals. Calibration of the measrement signals is done for compensation of effects reslting e.g. from different cable lengths. The control algorithm sally ses the vector sms of all calibrated measrement signals of the individal cavities as signals to be controlled, becase of the lack of individal actors for each cavity. The new control algorithm at the FLASH LLRF system presented in this paper is a combination of 2nd order mltiple inpt mltiple otpt (MIMO) feedback with feedforward components for both channels. The otpts of the FPGA are the driving signals, which are converted to complex vector signals in the digital-analogconverter (DAC) and transmitted to the vector modlator. The stability margin of the control loop depends on the overall gain and latency of the loop. The FPGA controller aims at the vales of µs for the latency and for the gain. Experiments show an estimated loop latency withot controller of approximately 5ns. The remaining 5ns may be attribted to the controller. The controller is implemented with VHDL programming code and synthesied by Xilinx XST. With respect to the operating freqency f res =.3GH the vector modlator can be modeled as a first order low pass with a corner freqency of MH. The klystron is a high power RF amplifier that can be considered as well as a first order lowpass filter with a corner freqency of 8MH. The bandwidths of both actator elements are very large compared to the narrow cavity bandwidth ω /2, and the dynamical behavior of the actator system is negligible in the freqency range p to MH considered for control. 258

4 III. PARAMETER IDENTIFICATION Althogh additional external distrbances and a nmber of nonlinearities in the actator system are known to be relevant for a broad range of operation setpoints, standard identification procedres for linear time invariant (LTI) models cold be chosen to estimate black box models that can be validated at specific setpoints [3]. In the following the sbspace identification methodn4sid, provided by Matlab s System Identification Toolbox [5] is sed to estimate the parameters A,B,C,D of a black box state space model x(k+ ) = Ax(k)+B(k), () y(k) = Cx(k)+D(k) (2) where (k) and y(k) denote the system inpt and otpt respectively and x(k) is the state vector of the system at time k. The flat top phase of the plse is of main interest for control and is at the same time marking the operation point. Only measrements from this period are sed for system identification. Persistent excitation signals can be injected in the accelerator system by sperimposing random signals on standard feedforward tables with defined setpoints. A typical inpt seqence for the feedforward table is shown in Fig. 4. ampl. 4 x Filling Inpt signals for Identification Flattop 5 5 Fig. 4: Inpt Distrbances on both Channels at Flat Top In the first 5µs (filling phase), the actator system is operated with maximm power, i.e. maximm feedforward signal. As soon as the flat top phase begins, the feedforward signal is redced by a factor of.5 in normal operation. In this flat top phase, an excitation signal is added to the feedforward signal as shown in Fig. 4. Ths open loop inptotpt data is generated, which shows worse performance than closed loop data, bt which can be sed later for parameter identification. The distrbances are exciting both channels of the inpt signals at the same time, which has trned ot to be advantageos for good parameter identification reslts. High amplitdes of the excitation signals leads to a good signal to noise ratios. Stdying the measred signals, oscillations in the 25kH spectra can be observed, which can be referred to the intermediate freqency of the down converters described in I Q Section II-B. In Fig. 5, measred vs. simlated signals are shown for an identified model of 3rd order. The slow systems dynamics are sfficiently modeled. y I y Q model meas model meas Fig. 5: Measred vs. simlated Vector Sm Signals An example for identified matrices of a discrete time state space model ()-(2) is A = , B = , ( ) C = , ( ) D =. The discrete time model is transformed to continos time by MATLABs d2c algorithm, becase the controller design procedre described in the following section is based on continos time LTI models. IV. CONTROLLER DESIGN The fixed strctre of the Controller implemented in the FPGA is given by ( ) K () K K() = 2 () (3) K 2 () K 22 () with the elements K i j () = k i j a i j 2 + b i j + c i j 2 + d i j +. (4) All parameters can be chosen independent and arbitrary within the limits given by the integer arithmetics of the FPGA. Overflows of internal variables are prevented by adjstable bit shift operators at the controller inpt and otpt registers. From the operators perspective, the nmber of tning parameters increased from (gain adjstmen to 2, which wold be not possible to adjst manally. An 259

5 atomated pdate algorithm needs to be fond in case of changing system parameters to garantee longtime performance and stability. A. Design objectives The design goal is to obtain high RF field flatness in amplitde and phase dring the flat top phase, which gives best performance of the electron beam that is responsible for the qality of the laser beam. The vector sms of the probe signals are sbtracted from the setpoint signals and given to the controller inpts. All measred probe signals show distrbances and noise from system specific as well as environmental sorces. Attenating all these distrbances by the controller might show an improvement for the observed RF signals, bt no inflence on the beam qality. Knowing that the cavity itself has lowpass filter characteristics, a sefl control loop bandwidth can be estimated as it is shown next. B. H loop shaping The design objectives are distrbance rejection as well as reference tracking in the closed loop system. Both objectives are in contradiction over all freqencies, bt a balanced tradeoff between both can be fond with the following standard design method. First the so called generalied plant as shown in Fig. 6 is given, with the two dimensional vector signals: r y e s t reference inpt signal, plant inpt / controller otpt, plant otpt, controller inpt / tracking error, filtered sensitivity otpt, filtered complementary sensitivity otpt. To describe the behavior of a closed loop system we introdce two transfer fnction matrices, the sensitivity S(s) and complementary sensitivity T(s). The transfer fnction S(s) and T(s) giving the behavior from the reference inpts r to the controller inpts e, and to the plant otpts y respectively. It is well known that both fnctions are related as follows S(s)+T(s) = I. For good tracking we need T(s) I, which implies that S(s). This shold be achieved at the closed loop bandwidth of the system. For arbitrary high freqencies, perfect tracking is impossible de to physical restrictions. High freqencies in the system are mostly related to distrbances which need to be rejected in the closed loop system S(s). An H design is performed to find an optimal controller while shaping the transfer fnctions by the weighting filters W S (s) and W T (s) as shown in Fig. 6. For these filters the following constraints have to hold W S (s)s(s) W T (s)t(s) <, (5) where a controller has to be fond which minimies the H norm of (5). The controller estimation is done by sing the HIFOO algorithm [6]. The filters can be tned to find an appropriate fixed order controller for the accelerator system, where W S (s) and W T (s) act as pper bonds for the maximm singlar vales of T(s) and S(s). r G(s) K(s) y W S (s) W T (s) Fig. 6: Generalied Plant with Weighting Filters W S and W T In the following section we are presenting the reslts of the controller design and the sed shaping filters. The estimated controller parameters have to be converted back to a discrete time model which then can be implemented. Therefore stability limits cased by latencies in the control algorithm have to be taken into accont [7]. e V. RESULTS In this section the first reslts of the procedre otlined before are presented. It needs to be mentioned that the whole accelerator and control system is a high precise machine with a large nmber of ncertainties. Therefore implementation of any control or data acqisition algorithm has to be done careflly. Matlab is sed as a script langage to commnicate directly with the FPGA, which implies that the machine is not protected against failres de to these processes. All controller parameters have to be checked before downloading to the FPGA. Nevertheless stability problems reglarly occr close to the maximm gain vales for a stable loop. A smooth step by step increase to high power operation points has to be done for investigation of the performance limits withot so called interlocks, i.e. shtdowns of the accelerator power de to instabilities of the loop. The weighting filters are chosen to be second order transfer fnctions, which allows at least tning parameters to shape the closed loop behavior, more precisely to adjst the 2 controller parameters. and W S (s) = M S (s+ω S )(s+ω S2 ) (s+ω S3 )(s+ω S4 ), (6) W T (s) = M T (s+ω T )(s+ω T2 ) (s+ω T 3 )(s+ω T4 ). (7) In Fig. 7 examples of singlar vale plots of T(s) and S(s) are shown inclding controller designed by HIFOO with shaping filters as given above. With this shaping filters controller parameters are calclated and implemented in the FPGA after transforming the coefficients to discrete-time. S T 252

6 Singlar Vales (db) Singlar Vales (db) 2 Singlar Vales T(s) /W T (s) Freqency (rad/sec) Singlar Vales S(s) /W S (s) Freqency (rad/sec) ampl. phase y( r( t in µs y( r( t in µs Fig. 7: Shaped Fnctions S(s), T(s) with weighting filters The performance of the designed controller was tested in real time experiments at the FLASH facility in October 27. One of the reslting trajectories for the case of a first order MIMO controller is shown in Fig. 8. It can be seen that there is a large overshoot at the beginning which is cased by the se of standard feedforward tables, which show maximm vales ntil the end of the filling phase. ampl. phase x Fig. 8: Reference Tracking: Amplitde and Phase For the second part of the flat top, the obtained RMS vales are.4% (amplitde) and.7 (phase) which is still an order of magnitde worse than the desired accracies, bt an improvement compared to previosly sed decentralied controllers. Fig. 9 shows a detail of the final filling and the beginning of the flattop phase for a different operating point. Here also, the first overshoot is crcial for the overall performance. Methods for adaptation of the feedforward tables can improve this behavior significantly, [8]. VI. CONCLUSIONS AND FUTURE WORK A procedre for low order MIMO LTI controller design was described and implemented on an FPGA based digital control system of a particle accelerator. It was shown that the Fig. 9: Reference Tracking: Details of Amplitde and Phase fixed order controller can be fond by sing the H controller design algorithm HIFOO. The procedre was applied online and the achieved controller parameters have been tested in the FPGA based LLRF control system of the FLASH facility at DESY, Hambrg. Tests of control loop stability and controller performance give promising reslts. To achieve the goal of a ser friendly soltion of atomated controller tning, frther stdies have to be done. Critical controller parameters have to be tested and interception rotines need to be defined to garantee on the one hand a stable system and on the other hand a high performance. Operating points close to the physical machine limits and also long flat top intervals have to be tested. Next, closed loop behavior in presence of beam distrbances needs to be stdied. Iterative Learning Control algorithms cold be sed to adapt the feedforward signals which will frther improve the machine performance. VII. ACKNOWLEDGMENTS The athors wold like to thank Andrey Popov from the Insitte of Control Systems (TUHH) for his ideas and spport on fixed order controller design. REFERENCES [] Brinkmann, R. et al., TESLA XFEL Technical Design Report, DESY, Hambrg, 22 [2] Liepe, M. et al., Dynamic Lorent Force Compensation with a Fast Pieoelectric Tner, TESLA Note 2-3, Hambrg, 2 [3] Ljng, Lennart, System identification - Theory for the ser, Prentice Hall, New Jersey, 999 [4] Schilcher, Thomas, Vector sm control of plsed accelerating fields in lorent force detned spercondcting cavities, Phd-Thesis, Hambrg, 998 [5] The Mathworks, System Identification Toolbox User s Gide, The Mathworks, Inc., Natick, 24 [6] J. V. Brke, D. Henrion, A. S. Lewis, M. L. Overton, HIFOO - A matlab package for fixed-order controller design and H optimiation, Proceedings of 5th IFAC Symposim on Robst Control Design, Tolose, 26 [7] Vogel E., High Gain proportional rf control stability at TESLA cavitys, TESLA Report, DESY, Hambrg, 27. [8] Kirchhoff, S. Distrbance Modelling and Controller Design for an Accelerator based Free Electron Laser, Masters Thesis 28.4, Institte of Control Systems, Hambrg University of Technology,