! How to design feedback filters! for a given system?!!

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1 How to design feedback filters for a given system? October 3, 3 Osamu Miyakawa Seminar at Kamioka Brach, October 3

2 Requirement : Assumption of Linearity Carrier Requirement : Control theory assumes linearity. Sidebands (probe for carrier) EOM:Modulation Unlocked Control area linear area (straight slope) Oscillator ~MHz ~ PD Mixer :Demodulation L Locked Signal [] ~nm +signal Lock point - signal Phase lock L [m] ~ Feedback Seminar at Kamioka Brach, October 3

3 Pendulum in Physics Γx damper m F kx d x dx F m + Γ + dt dt x x Ae iωt Arg(x) ω t π ft :angle dx iω x dt d x ω x dt : complex number A:amplitude kx Laplace transform L [ F] m L + Γ L + k L[ x] F ms x F x F / m ms s iω x F d x dt x + Γsx + kx + Γs + k ( s ( dx dt Γ F mω x + i Γωx + kx mω + i Γω + k Interpretation in Physics Γ 4mk m resonant anglar frequency : ω π f Transfer function fromforce to position : ))( s ( quality factor Q /energy loss: Q mω / Γ ω + iωω / Q + ω f Seminar at Kamioka Brach, October 3 3 st ( s) L[ f ( t)] f ( t) e dt k / m Γ+ Γ 4mk m ; poles ))

4 Bode plot : frequency vs. gain, phase Resonant frequency Γx Q Q Q Q/ Q corresponds to height of resonant peak. damper m F kx f - : poles x F / m ω + iωω / Q + ω if ω << ω x F / m ω :constant x if ω >> ω : f slope F / m ω x Q if ω ω : resonance F / m iω x if ω ω,q F / m : resonance,no damp if Q / cretical damping if Q < / over damping Seminar at Kamioka Brach, October 3 4

5 Phase [deg] Gain Phase [deg] Gain Basic concept : relationship between gain and phase pole: 9deg delay f Frequency log f [Hz] zero: 9deg advance pole: 8deg delay Phase [deg] Gain f Frequency log f [Hz] zero: 8deg advance Frequency log f [Hz] Frequency log f [Hz] Seminar at Kamioka Brach, October 3 5 Phase [deg] Gain f + f +

6 Actual pole and zero with phase 5.7deg pole/zero frequency 45deg 9deg 3 zero zero zero pole pole 3 pole 3 zero zero zero pole pole 3 pole Actual gain and phase slope change smoothly. Phase above pole (zero) frequency has N x 9degree phase delay (advance). Phase at pole (zero) frequency has N x 45degree phase delay (advance). Phase at times lower frequency of pole (zero) frequency has N x 5.7degree phase delay (advance). Seminar at Kamioka Brach, October 3 6

7 Laser Seismic noise EOM Modeling system: Single Fabry-Perot cavity Plant Pick off ITM Photo detector Suspension Feedback filter ETM Suspension Coil-magnet Actuator Seismic noise Actuator Mission: Make a feedback filter to keep distance between ITM and ETM constant N Oscillator Noise Mixer Sensor A Actuator Feedback filter P S Plant Sensor F P: flat around resonance S: flat F:?? A: suspension TF ~Hz f - Feedback filter Seminar at Kamioka Brach, October 3 7

8 Transfer function from Noise N to error signal out Outloop noise N Noise B A C Actuator D P S Plant Sensor F Feedback filter Error signal out A B C D OUT?? F using A P S OUT A N C D B OUT A Summary: Relationship between Error signal out and Outloop noise N can be written with Open loop transfer function G and transfer function P,S. OUT S P ( N A F OUT ) ( SPAF ) SP N + OUT SP SP OUT N N SPAF ( + ) ( + G) Open loop transfer function : G SPAF Seminar at Kamioka Brach, October 3 8

9 Inloop noise Transfer function from outloop noise to inloop noise C D Outloop noise N Noise B A Actuator P S Plant Sensor F Feedback filter out out SP C N N SP SP ( + G) ( + G) N if G >>, C << N; supression G if G <<, C N; no suppression A Summary: Outloop noise N is suppressed by Open loop transfer function G (if G>>) into Inloop noise N/(+G), then it is multiplied by transfer functions SP through output port. Seminar at Kamioka Brach, October 3 9

10 Outloop noise N Noise B Feedback signal Inloop noise C A Actuator A P Plant D F Feedback filter S Sensor Feed back signal out Summary: Outloop noise N is canceled out by Feedback signal G/(+G) N, then Inloop noise becomes N/(+G). More example; P. D + PSFA. B if if 3. N ( G) N N ( + G) ( + G) N ( + G) ( + G) ( + G) C ( + ) Seminar at Kamioka Brach, October 3 B G G G >>, B N N G G <<, B GN << N G N N N( + G) GN N + GN GN N G

11 Requirement : Negative feedback N Noise A Actuator P Plant F S Sensor Feedback filter G SPAF : Open loop transfer function Requirement : Phase delay of open loop transfer function at unity gain frequency (frequency at gain ) must be less than 8 degree. You must have negative feedback if total gain (OLG) is larger than. out Seminar at Kamioka Brach, October 3

12 Open Loop TF and Unity Gain Frequency P : flat x S : flat x F :?? x A : suspension system TF Hz f Gain Phase [deg] f - Not stable k k Frequency log f [Hz] Phase P : flat x S : flat x F : restore x A : suspension OLTF khz Hz f - Hz f +... UGF f zero@hz - 9 > -8deg pole@khz k k Frequency log f [Hz] Seminar at Kamioka Brach, October 3 Gain Phase [deg] f - f -

13 Low-pass filter to avoid feed back noise Phase P : flat x S : flat Restore x F : + x A : suspension OLTF Low-pass f - khz Hz f + f - zero@hz pole@khz,khz Hz f - Gain Phase [deg] f - UGF f -3 k k Frequency log f [Hz] Too high feedback signal at high frequency will be not only noise but also saturation, not desired mechanical resonance, then can break control Seminar at Kamioka Brach, October 3 3

14 Notch filter to avoid mechanical resonance Phase Suspension Restore + P : flat x S : flat x F : + x A : Mechnical OLTF Low-pass f resonance - + notch Hz Hz f + khz f - zero@hz pole@khz,khz notch@hz, khz Mechanical resonance can make another UGF which has more than 8 degree phase delay. Notch filter has big phase delay and advance in very narrow band, so it can be used both above and below UGF, but not too close to UGF. Hz Seminar at Kamioka Brach, October 3 4 f - khz Gain Phase [deg]. f - UGF f -3 k k Frequency log f [Hz]

15 Boost filter; engaged after lock acquisition Signal [] Control area Hz f - -9 m Lock point Phase Restore Suspension + P : flat x S : flat x F : x A : + OLTF Mechnical resonance f -4 Low-pass + Notch + boost khz Hz f + f - zero@hz,hz,hz pole@hz,hz,khz,khz Hz Hz khz N G ( + ) G 5 m < ( + G) > -5 m Seminar at Kamioka Brach, October 3 5 f - Gain Phase [deg] 4 k k Frequency log f [Hz] 9 f - UGF f -3 m@hz

16 Example of Feedback filter Notch for bounce mode Notch for iolin mode F : Phase Restore + Low-pass + notch Hz f + 4kHz f - 75degree phase advance zero@hz pole@4khz,4khz notch@6hz, 44Hz Target UGF Seminar at Kamioka Brach, October 3 6

17 Example of Boost Notch for bounce mode Notch for iolin mode F : Phase Restore + Boost 4kHz 3Hz f + f - f - zero@hz,3hz,3hz pole@hz,hz,4khz,4khz degree phase delay by boost Target UGF Seminar at Kamioka Brach, October 3 7

18 Advanced Boost using complex pole/zero Notch for bounce mode Notch for iolin mode F : Phase Restore + Boost 5degree phase recovered by complex zero 4kHz 3Hz f + f - f - zero@hz complex zero@3hz pole@hz,hz,4khz,4khz Target UGF Seminar at Kamioka Brach, October 3 8

19 How to adjust feedback gain? High gain: limited by phase delay due to slow circuit, DAC/ADC time delay etc., or by feedback noise, or by saturation Low gain : limited by phase delay of boost, or by too low DC gain. f -4 Gain... 9 f - f -3 Once you get stable operation, it is very important to measure open loop TF to see how stable Phase [deg] k k Frequency log f [Hz] Seminar at Kamioka Brach, October 3 9

20 How to measure Open Loop TF? N Noise A Actuator P Plant F Feedback filter S Sensor. Use closed transfer function: C Measure IN EXC ( + G) ( + G) IN EXC IN Calculate G G C. Measure OLTF directly Measure IN FAPS IN G Seminar at Kamioka Brach, October 3

21 Example of measured Open loop TF f - slope UGF Generally, phase margin should be more than 3degree at least. Phase margin 45degree Seminar at Kamioka Brach, October 3

22 Why low frequency measurement is dirty? N Noise P Plant S Sensor out A Actuator F Feedback filter IN EXC IN Excitation signal is suppressed a lot with very high gain G>> at low frequency. IN EXC << Noise + G ( ) Seminar at Kamioka Brach, October 3

23 What is Coherence? Coherence: coh(f) How much related between input and output coh( h) W W xy xx ( f ) :cross spectrum of ( f ), W yy W xx W xy ( f ) ( f ) W yy ( f ) x and y ( f ) :power spectrum of x and y < coh(f) < Coherence is sometimes convenient to estimate whether the measurement is reliable. Generally, If coherence is smaller than.8 the measurement is not good. Seminar at Kamioka Brach, October 3 3

24 N Noise A Actuator What does Closed Loop TF mean?. C :used to estimate gain oscillation IN Measure + G Gain oscillation is caused by small phase margin at UGF.. C : Measure P Plant F Feedback filter EXC IN EXC IN S Sensor ( ) G + ( G) EXC IN out There are two definitions for closed loop TF C Gain Gain. ( + G) UGF G + ( G). k k Frequency log f [Hz] Seminar at Kamioka Brach, October 3 4 C k k Frequency log f [Hz] UGF

25 Example of measured Closed loop TF Gain oscillation 6dB Generally, gain oscillation should be less than db at most. Seminar at Kamioka Brach, October 3 5

26 Example of measured Closed loop TF Seminar at Kamioka Brach, October 3 6

27 Summary of designing feedback filters. Look at target system carefully and estimate each transfer function even if you cannot measure it directly.. Design a feedback filter by putting pole and zero to have f - slope around UGF. 3. Close loop with a designed feedback filter 4. Adjust gain of feedback filter if system is not stable. 5. If system is stable, measure open loop transfer function in order to see how stable (don t forget to see closed loop TF and coherence function also). 6. Put boost, notch, low-pass filters if you need. Seminar at Kamioka Brach, October 3 7