MIMO-LTI Feedback Controller Design -Status report-

Transcription

1 MIMO-LTI Feedback Controller Design -Status report- Christian Schmidt Deutsches Elektronen Synchrotron Technische Universitaet Hamburg Harburg FLASH Seminar 4/1/28

2 Outline Current RF Feedback System MIMO controller structure System Identification black box modeling model validation Fixed order controller design H design with weighting filters Iterative Learning Control Outlook Christian Schmidt µ 2/21

3 Schematic View of the LLRF Control System DAC DAC ADC ADC Master ~ oscillator 1.3 Ghz Vector modulator Klystron Waveguide 8x cryomodule 1.3 Ghz + 25 khz LO 1.3 Ghz field probe 1.3 Ghz + 25 khz LO 1.3 Ghz field probe 25 khz 25 khz clock clock F =1MHz F =1MHz u I Feedforward f I uq u c, I FPGA System a b b a e I Calibration yi yq r I a b b a Setpoint K fq u c, Q e Q r Q Christian Schmidt µ 3/21

4 Controller Structure so far a decentralized P Controller is used GN_RAW_I GN_SCA_I SHIFT K e i x x u c, r e q x x u c,i GN_RAW_Q SHIFT K GN_SCA_Q Christian Schmidt µ 4/21

5 Controller Structure so far a decentralized P Controller is used new FPGA implemented controller is given by: GN_RAW_I GN_SCA_I MIMO-Controller SHIFT K 11 z K 21 z e i x x + u c, r x x e q x x + u c,i GN_RAW_Q SHIFT K 22 z K 12 z GN_SCA_Q K ij z =k ij a ij z 2 b ij z 1 1 c ij z 2 d ij z 1 1 tuning 2 parameters manually is not possible for users Christian Schmidt µ 4/21

6 Physical cavity model (LPV) can be described as a first order lowpass, where the envelope of the electric field is given by the real and imaginary part ( ẏi ẏ Q ) ( ) ( ) ( ) ω12 ω(t) yr ui = + R ω(t) ω L ω 12 y 1/2 i u Q ω 12 2π 216 Hz, ω = ω ω, R L Ω x 1 6 Ampltude [a.u] 1.5 u I u Q Amplitude [a.u] y I y Q t [µs] t [µs] Christian Schmidt µ 5/21

7 Open Loop measurements real system shows additional dynamics (Sources?) 2 y I y Q y I y Q Amplitude [a.u.] 1 Amplitude [a.u.] t [µs] 5 12 t [µs] disturbances generated from actuator or measurement system preamplifier, klystron, microphonics,... downconverter, ADC, VS-Calibration,... Christian Schmidt µ 6/21

8 Estimation of a black box model The physical model describes the ideal system, but: for higher order controller design it might be not sufficient enough disturbances and noise effects are hidden therefore black box modeling is used general state space system (LTI) ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) estimating system parameters A,B,C,D with subspace algorithm n4sid, provided by Matlabs System Identification Toolbox 1 1 The Mathworks,System Identification Toolbox User s Guide, The Mathworks, Inc., Natick, 24 Christian Schmidt µ 7/21

9 Open Loop Identification For Identification of the system dynamics special input disturbances need to excite the system ampl. 14 x Filling Input signals for Identification Flattop u I u Q wide bandwidth (rich enough) limited flattop time (max. 8µs) good signal to noise ratio (high exc. amplitude) excitation close to operation point (low exc. amplitude) t [µs] Sufficient model must be found to cover all real system dynamics and neglect artificial disturbances Christian Schmidt µ 8/21

10 Model validation Comparing the estimated models during flattop 1 2rd order (detrended) 1 3rd order (detrended) y I model meas t [µs] y I model meas t [µs] y Q 1 model meas y Q 1 model meas t [µs] t [µs] medium dynamic range can be modeled (cross validation) high frequency dynamics are estimated as disturbances 3rd order models tend to describe system sufficient Christian Schmidt µ 9/21

11 Models Pulse to Pulse Magnitude (db) ; Phase (deg) To: Out(1) To: Out(1) To: Out(2) To: Out(2) From: In(1) Bode Diagram From: In(2) Frequency (rad/sec) small variations in the frequency range of interest diagonal dominant system (detuning influence smaller) Christian Schmidt µ 1/21

12 MIMO-Feedback Control System DAC DAC ADC ADC Master ~ oscillator 1.3 Ghz Vector modulator Klystron Waveguide 8x cryomodule 1.3 Ghz + 25 khz LO 1.3 Ghz field probe 1.3 Ghz + 25 khz LO 1.3 Ghz field probe 25 khz 25 khz clock clock F =1MHz F =1MHz u I Feedforward f I fq uq u c, I u c, Q FPGA System a b b a MIMO -Controller a 11z b11 z 1 a12 z b12 z 1 K11 ( t) K 12( t) c11 z d11 z 1 c12 z d12 z a21z b21 z 1 a22 z b22 z 1 K21( t) K ( ) t 2 1 c21z d21z 1 c22 z d22z 1 e I e Q Calibration yi yq r I r Q a b b a Setpoint Christian Schmidt µ 11/21

13 Design Objectives RF field flatness during the flattop is the desired goal perfect tracking (reference = output) only possible for low frequencies due to system physics high loop gain (amplifies also disturbances) complementary sensitivity T (s) Christian Schmidt µ 12/21

14 Design Objectives RF field flatness during the flattop is the desired goal perfect tracking (reference = output) only possible for low frequencies due to system physics high loop gain (amplifies also disturbances) complementary sensitivity T (s) disturbance rejection high frequency noise is filtered by lowpass characteristics good suppression demands small feedback gain sensitivity S(s) but... S(s) + T (s) = I therefore weighting filters are introduces to shape the closed loop behavior Christian Schmidt µ 12/21

15 Generalized Plant with Weighting filters using shaping filters to restrict closed loop behavior on fictions outputs of the generalized plant W S (s) = W T (s) = 1 (s + ω S1 )(s + ω S2 ) M S (s + ω S3 )(s + ω S4 ) 1 (s + ω T 1 )(s + ω T 2 ) M T (s + ω T 3 )(s + ω T 4 ) r u G(s) K(s) y e W S(s) W T(s) parameter estimation can be found by solving norm-optimal H with HIFOO 1 W S(s) S(s) W T (s) T (s) < 1 z S z T 1 J. V. Burke, D. Henrion, A. S. Lewis, M. L. Overton,HIFOO - A matlab package for fixed-order controller design and H optimization, Proceedings of 5th IFAC, 26 Christian Schmidt µ 13/21

16 Tuning Weighting filters Singular Values (db) Singular Values (db) Shape closed loop system to desired performance Singular Values T(s) 1/W T (s) Frequency (rad/sec) Singular Values S(s) 1/W S (s) Frequency (rad/sec) pulse duration 7µs 1kHz lower limit intermediate frequency 25kHz upper limit C.Schmidt, G.Lichtenberg, W.Koprek, W.Jalmuzna, H.Werner, S. Simrock, Parameter Estimation and Tuning of a Multivariable RF Controller with FPGA technique for the Free Electron Laser FLASH, Proceedings of ACC, 28 Christian Schmidt µ 14/21

17 φ / φ E / E First implemented controllers 2 x slow approach (C) to estimated controller gain Amplitude 1% 2% 5% 7% 9% 95% 1% t/µs Phase.2 1% 2%.1 5% 7% 9% 95% 1% t/µs E 1.5 x ( K11 (z) K K(z) = C 21 (z) K 12 (z) K 22 (z) Energy spread t in µs highest performance reached so far (1st order) instability problems with full order controller parameters limited measurement time restricts online tests ) 9% 95% 1% Christian Schmidt µ 15/21

18 Measurement Procedure and Problems 1 Reprogramming the FPGA to use MIMO-Controller set back to operation mode takes time!, parallel system no DOOCS Interface, direct writing into FPGA! testing MIMO-Controller implementation, DONE 2 System Identification algorithms work, investigation of excitation signals 3 Controller parameter estimation unstable higher order controllers, bug found but not verified automatic controller design (under investigation) long term stability proof must be performed 4 Performance test with beam not tested so far because of instability correct gradient/phase settings must be checked (no DOOCS!) Christian Schmidt µ 16/21

19 Iterative Learning Control Adaptive feedforward 3 solving the minimum-norm optimization problem u k+1 = arg min u k+1 {J k+1 (u k+1 ) : e k+1 = r y k+1, y k+1 = G u k+1 } using the estimated system model (A,B,C) G K is the solution of Riccati-equation K(t) = A T K(t + 1) A + C T W 1 (t + 1) C [A T K(t + 1)B {B T K(t + 1) B + W 2 (t + 1)} 1 B T K(t + 1)A] W 1 2 (t) B T } 1 W 1 and W 2 are tuning matrices (tradeoff) 3 N. Amann, D.H. Owens and E. Rogers,Iterative learning control for discrete-time systems with exponential rate of convergence, IEEE Proceedings - Control Theory and Applications, , 143, 1996 Christian Schmidt µ 17/21

20 Iterative Learning Control algorithm computing update input signal as: u k+1 (t) = u k + W 1 2 (t) B T ξ k+1 (t) ξ k+1 (t) = α(t)a T ξ k+1 (t + 1) + α(t)c T W 1 (t + 1)e k (t) α(t) = {I + K(t) B W 1 2 (t) B T } 1 solving non-causal update input updates ideally between two pulses based on same model ampl. [a.u.] 1 5 k trial k+1 (k+1) k t [µs] Christian Schmidt µ 18/21

21 First open loop measurements show promising results inp I [a.u.] x time [µs] Sum I [a.u.] 2.5 x time [µs].5 1 x x 14 2 inp Q [a.u.] Sum Q [a.u.] time [µs] time [µs] Christian Schmidt µ 19/21

22 Further developments on......iterative Learning Control currently the updating law is computed with Matlab (SLOW) using C for programming the algorithm tests with beam must be performed in next measurements exception handling while missing beam! closed loop measurements with MIMO or P-Controller submitted to CDC 28: S.Kirchhoff, C.Schmidt, G.Lichtenberg, H.Werner, An Iterative Learning Algorithm for Control of an Accelerator based Free Electron Laser, 28 Christian Schmidt µ 2/21

23 Summary first MIMO-Controller parameters can be estimated self updating identification procedure self adapting controller parameter Iterative Learning controller based on same model Automation! More measurement time needed to test the controller performance for stability with beam! Special thanks to: G.Lichtenberg, W.Koprek, W. Jalmuzna, S. Kirchhoff, A. Popov... Christian Schmidt µ 21/21