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FO-LPF: Fractional Order Low-Pass Filter

FO-LPF: Fractional Order Low-Pass Filter. YangQuan Chen, Director Center for Self-Organizing and Intelligent Systems (CSOIS), Dept. of Electrical and Computer Engineering Utah State University E : yqchen@ieee.org; T : 1( 435)797-0148; F : 1(435)797-3054.

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FO-LPF: Fractional Order Low-Pass Filter

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  1. FO-LPF:Fractional Order Low-Pass Filter YangQuan Chen, Director Center for Self-Organizing and Intelligent Systems (CSOIS), Dept. of Electrical and Computer Engineering Utah State University E: yqchen@ieee.org; T: 1(435)797-0148; F: 1(435)797-3054 Thursday, Sept. 11, 2008 – 3:40-4:40 PM SIG FOC Weekly Meeting http://fractionalcalcuclus.googlepages.com/ http://mechatronics.ece.usu.edu/foc/yan.li/

  2. Outline • FO-LPFWhy • FO-LPF What • FO-LPF How • FO-LPF Code/Example • FODOB! SIG FOC Weekly Meeting

  3. Why • uspto.gov : 20010036026 Efficient sensorless rotational vibration and shock compensator for hard disk drives with higher TPI • YangQuan Chen*, Blas M. Vinagre and Igor Podlubny. "Fractional order disturbance observer for vibration suppression," (Kluwer) Nonlinear Dynamics, Vol. 38, Nos. 1-4, December 2004, pp. 355-367. SIG FOC Weekly Meeting

  4. Introduction to RVSC: Rotational Vibration/Shock Compensator (SEA9472.01) ABSTRACT: The present invention uses an observer scheme to estimate rotational vibration and shock in a disk drive servo system and provides a compensation signal. A filter with variable relative degree is used to filter this compensation signal and the filtered signal is used to attenuate the tracking error caused by shock, vibration and windup disturbances to the system. SIG FOC Weekly Meeting

  5. Motivations • Performance of HDD drives with higher TPI subject to RV/Shock effect - more dominant now! • Less change to SFB servo control architecture • Observe then compensate • RV/Shock-Sensorless Major ideas: 1. More attenuation in sensitivity function at the lower frequency range. 2. Rule based compensation based on disturbance amplitude. SIG FOC Weekly Meeting

  6. VCM RVSC block-diagram SFB DDD: digital double differentiator DDD Q-filter Rule-based decision SIG FOC Weekly Meeting

  7. More detailed block diagram Variable relative degree W/O RVSC PES SIG FOC Weekly Meeting

  8. Variable relative degree of Q-filter SIG FOC Weekly Meeting

  9. With RVSC • Without RVSC • DDD for Pn-1 : SIG FOC Weekly Meeting

  10. ETF and Gdy Blue: without RVSC Red: with RVSC SIG FOC Weekly Meeting

  11. ETF and Gdy (relative degree of Q-filter as a tuning knob) Red: w/o RVSC SIG FOC Weekly Meeting

  12. Correct setting of nd (delay steps in udac) clearly, nd should be set to 3. SIG FOC Weekly Meeting

  13. SHAKER RESULTS LEFT: PES one sigma versus shaker freq. for OD, MD and ID (Drive 3CR0024I). RIGHT: Averaged RV attenuation in terms of PES one sigma improvement in percentage versus shaker freq. for OD, MD and ID tracks (Drive 3CV003KK). Shaker RV level @ 26 rad/sec/sec. SIG FOC Weekly Meeting

  14. SHOCK Attenuation RESULTS PES Relative degree = 4 Relative degree = 1 Relative degree = 4, SFB redesigned SIG FOC Weekly Meeting

  15. SHOCK Attenuation RESULTS Relative degree = 4, Relative degree = 1, Relative degree = 4, and PES threshhold=19H Rule based shock compensation SIG FOC Weekly Meeting

  16. Conclusions: • RVSC is an easy-to-add block in servo control code without using high-order servo controller. This is especially useful in current SFB servo code. • RVSC is dedicated for low frequency shaping for the sensitivity function with tuning knobs easier to use; • Rule-based compensation is especially effective for shock disturbance; • The proposed scheme is effective and proved by drive level experiments. SIG FOC Weekly Meeting

  17. U6 RVSC Results Results for different (fixed) relative degrees (RD) of Q-filter RVSC: Rotational Vibration/Shock Compensator NOTE: Waterbed effect! SIG FOC Weekly Meeting

  18. U6 RVSC Results Detailed block diagram of RVSC Variable relative degree W/O RVSC PES SIG FOC Weekly Meeting

  19. U6 RVSC Results Detailed block diagram of RVSC with a NEW scheme PES Variable relative degree NEW scheme using filtered PES! PES_Q LPF W/O RVSC SIG FOC Weekly Meeting

  20. U6 RVSC Results Results for the proposed NEW scheme - IMPRESSIVE 16%trk=41count=29HEX SIG FOC Weekly Meeting

  21. U6 RVSC Results Results for the proposed NEW scheme See the BLACK line when PES_threshold is reduced by half SIG FOC Weekly Meeting

  22. U6 EM4:Inject 10 mV=0.559 G U6 RVSC Results Results for the proposed NEW scheme RVSC’s behavior changes when the RV/Shock size differs. Our RVSC is nonlinear! SIG FOC Weekly Meeting

  23. U6 RVSC Results Results for the proposed NEW scheme Clearly, using a filtered PES as a decision variable to auto-change the RD of Q-filter in RVSC is a clever choice! We pay less!! SIG FOC Weekly Meeting

  24. U6 RVSC Results Concluding Remarks: 1. RVSC outperforms HOC or H_inf shaping etc. due to its clear physical meaning in attacking RV/Shock 2. The tuning knobs are easy to handle by non-experts. 3. U6 drive level results are encouraging. 4. It’s important to use filtered PES signal for RD decision. (Patent Pending. Original idea due to kk) PES LPF PES_Q W/O RVSC Variable relative degree NEW scheme using filtered PES! SIG FOC Weekly Meeting

  25. U6 RVSC Results Some time domain traces via DSA inject sine. • Inject sine at a fixed frequency. Amplitude = 17.8mVrms (equiv. to 1G) • In each slides two pictures at the same frequency. • Two schemes (Left: Old RVSC scheme; Right: New RVSC scheme) • In each plot, there are four (4) curves • Trace A. No RVSC • Trace B. RVSC on , fixed RD=4 • Trace C. RVSC on, PES_threshold=29H (16%trk) • Trace D. RVSC on, PES_threshold=15H (16%trk /2) SIG FOC Weekly Meeting

  26. Frequency=20Hz New scheme U6 RVSC Results Old scheme No big deal here SIG FOC Weekly Meeting

  27. Frequency=300Hz New scheme U6 RVSC Results Old scheme No big deal here SIG FOC Weekly Meeting

  28. Frequency=400Hz New scheme U6 RVSC Results Old scheme No big deal here SIG FOC Weekly Meeting

  29. Frequency=1350Hz New scheme U6 RVSC Results Old scheme New one is better SIG FOC Weekly Meeting

  30. Review this Bode plot again U6 RVSC Results Results for the proposed NEW scheme Clearly, using a filtered PES as a decision variable to auto-change the RD of Q-filter in RVSC is a clever choice! We pay less!! SIG FOC Weekly Meeting

  31. Outline • FO-LPF Why • FO-LPFWhat • FO-LPF How • FO-LPF Code/Example • FODOB! SIG FOC Weekly Meeting

  32. See ASME FDTA 2003 Slides SIG FOC Weekly Meeting

  33. Questions • Can we write the analytical expression of the impulse response of • Sure. (Not known previously!) SIG FOC Weekly Meeting

  34. GMLF – Generalized Mittag-Leffler Function SIG FOC Weekly Meeting

  35. GMLF definition T.R.Prabhakar,A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19(1971) 7–15. SIG FOC Weekly Meeting

  36. G-GMLF? A.K. Shukla ∗, J.C. Prajapati. On a generalization of Mittag-Leffler function and its properties. J. Math. Anal. Appl. 336 (2007) 797–811 SIG FOC Weekly Meeting

  37. Laplace transform of G-GMLF Special Case: SIG FOC Weekly Meeting

  38. Other special cases  SIG FOC Weekly Meeting

  39. Outline • FO-LPF Why • FO-LPF What • FO-LPFHow • FO-LPF Code/Example • FODOB! SIG FOC Weekly Meeting

  40. >> help prony • PRONY Prony's method for time-domain IIR filter design. • [B,A] = PRONY(H, NB, NA) finds a filter with numerator order • NB, denominator order NA, and having the impulse response in • vector H. The IIR filter coefficients are returned in • length NB+1 and NA+1 row vectors B and A, ordered in • descending powers of Z. H may be real or complex. • If the largest order specified is greater than the length of H, • H is padded with zeros. • See also stmcb, lpc, butter, cheby1, cheby2, ellip, invfreqz. • Reference page in Help browser • doc prony Reference: [1] T.W. Parks and C.S. Burrus, Digital Filter Design, John Wiley and Sons, 1987, p226. SIG FOC Weekly Meeting

  41. Impulse response can be obtained analytically for FO-LPF using GMLF!! • Fitting the ideal impulse response, we can obtain a finite dimensional approximate discretization! • Prony technique (check >> help prony) SIG FOC Weekly Meeting

  42. Outline • FO-LPF Why • FO-LPF What • FO-LPF How • FO-LPFCode/Example • FODOB! SIG FOC Weekly Meeting

  43. I’m #131 in MatlabCentral • http://www.mathworks.com/matlabcentral/fileexchange/loadAuthor.do?objectId=1093726&objectType=author • Impulse response invariant discretization of fractional order low-pass filters (#21365) • Generalized Mittag-Leffler function (#20849) SIG FOC Weekly Meeting

  44. gml_fun.m • function f=gml_fun(a,b,c,x,eps0) gamma_c=1.0/gamma(c); if nargin<5, eps0=eps; end f=0; fa=1; j=0; while norm(fa,1)>=eps0 fa=(gamma(c+j)*gamma_c)/gamma(j+1)/gamma(a*j+b) *x.^j; f=f+fa; j=j+1; end • Surprisingly simple – due to Prof. Dingyu Xue’s efforts! SIG FOC Weekly Meeting

  45. irid_folpf(tau,r,Ts,norder) • function [sr]=irid_folpf(tau,r,Ts,norder) • wmax0=2*pi/Ts/2; % rad./sec. Nyquist frequency • L=abs(tau)*4/Ts; % the number of points of the impulse response function h(n) • Taxis=[0:L-1]*Ts; ha0=(7.0*Ts/8)^r; n=1:L-1; n=n*Ts ; • h1=gml_fun(1,r,r,-1/tau *n); • h2=(1/tau)^r*h1.*(n.^(r-1)); • h0= ha0*(1/(tau+ (7.0*Ts/8)))^r; h=[h0,h2*Ts]; %% [ha0, (Ts^r)*(n.^(r-1))/gamma(r)]; • q=norder;p=norder; [b,a]=prony((h),q,p); sr=tf(b,a,Ts); SIG FOC Weekly Meeting

  46. sr=irid_folpf(0.4,0.5,0.001,4) >> sr=irid_folpf(0.4,0.5,0.001,4) Transfer function: 0.04672 z^4 - 0.09233 z^3 + 0.05376 z^2 - 0.006988 z - 0.0008313 ---------------------------------------------------------------- z^4 - 2.579 z^3 + 2.279 z^2 - 0.7728 z + 0.07276 Sampling time: 0.001 SIG FOC Weekly Meeting

  47. SIG FOC Weekly Meeting

  48. figure;pzmap(sr) SIG FOC Weekly Meeting

  49. sr=irid_folpf(0.4,0.5,0.001,7) >> sr=irid_folpf(0.4,0.5,0.001,7) Transfer function: 0.04672 z^7 - 0.1739 z^6 + 0.2534 z^5 - 0.1797 z^4 + 0.06152 z^3 - 0.007705 z^2 - 0.0003173 z + 7.239e-005 -------------------------------------------------------------------------------- z^7 - 4.325 z^6 + 7.603 z^5 - 6.934 z^4 + 3.461 z^3 - 0.9081 z^2 + 0.1072 z - 0.003578 Sampling time: 0.001 SIG FOC Weekly Meeting

  50. SIG FOC Weekly Meeting

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