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# ME280 “Fractional Order Mechanics” Fractional Order Damping (Week-10)

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1. ME280 “Fractional Order Mechanics”Fractional Order Damping(Week-10) YangQuan Chen, Ph.D., Director, MESA (Mechatronics, Embedded Systems and Automation)Lab ME/EECS/SNRI/UCSolar, School of Engineering, University of California, Merced E: yqchen@ieee.org; or, yangquan.chen@ucmerced.edu T: (209)228-4672; O: SE1-254; Lab: Castle #22 (T: 228-4398) 10/29/2013 and 10/31/2013. Thursday 09:00-10:15, KL217

2. Fractional-Order Damping • Mass-spring-damper system F With control force: ME280 "Fractional Order Mechanics" @ UC Merced

3. Laplace transform Normalize it to have unity DC gain ME280 "Fractional Order Mechanics" @ UC Merced

4. We focus on Question: For step response, to make the ISE minimum, what is “a”? ME280 "Fractional Order Mechanics" @ UC Merced

5. ISE: integral of squared error ME280 "Fractional Order Mechanics" @ UC Merced

6. What care about this question of optimal damping coefficient “a”? • Answer – “Standard form” • Why standard form? controller e(t) r(t) x(t) G(s) H(s)=1 ME280 "Fractional Order Mechanics" @ UC Merced

7. Standard form • A way for model based controller design • T(s)=C(s)G(s)/[1+ C(s)G(s)]: • complementary sensitivity transfer function • Or, closed-loop transfer function • S(s)=1-T(s) • sensitivity transfer function • error transfer function • How to design C(s) • What do you know? What do you want? • We want T(s) to be in a standard form given G(s)! ME280 "Fractional Order Mechanics" @ UC Merced

8. Sensitivity function Complementary sensitivity function One degree-of-freedom configuration ME280 "Fractional Order Mechanics" @ UC Merced

9. Performance, good disturbance rejection • Performance, good command following • Mitigation of measurement noise on output • Small magnitude of input signals • Physical controller must be strictly proper • Nominal stability (stable plant) • Stabilization of unstable plant ME280 "Fractional Order Mechanics" @ UC Merced

10. So, controller C(s) is designed as • C(s)=T(s)/[G(s)(1-T(s))] = T(s)/[G(s)S(s)] - Simple! • What is a “desirable T(s)”? • LPF type, DC gain is one. • Suppose T(s) is 2nd order • What is best “a” to use? • Optimal damping concept. ME280 "Fractional Order Mechanics" @ UC Merced

11. ISE of the error of step response • step signal u(t)=1, when t≥0; u(t)=0 otherwise. • e(t)=u(t)-x(t); but • F(s)=1/s; (step) Parseval’s identity ME280 "Fractional Order Mechanics" @ UC Merced

12. The optimal answer isa=1 Note: We will use G(s) to replace T(s) in the following ME280 "Fractional Order Mechanics" @ UC Merced

13. What about ITSE? ME280 "Fractional Order Mechanics" @ UC Merced

14. Now let us assume fractional-order damping! Smaller JISE and JITSE? ME280 "Fractional Order Mechanics" @ UC Merced

15. IAE and ITAE • Integral Absolute Error (IAE) • Integral Time-Weighted Absolute Error (ITAE) ME280 "Fractional Order Mechanics" @ UC Merced

16. 10 years ago … in Chicago1st ASME FDTA • Tom T. Hartley and Carl F. Lorenzo (2003), Optimal Fractional-Order Damping, ASME 2003 Design Engineering Technical Conferences and Computer and Information in Engineering Conference ME280 "Fractional Order Mechanics" @ UC Merced

17. Results ME280 "Fractional Order Mechanics" @ UC Merced

18. How to find the optimal fractional-order damping? Calculate the integral of the squared error of the step response for all time function sum = JISE(a, q) step = 0.0025; low = -200; high = 200; sum = 0; for w = low:step:high if w==0, mag=0; else mag =((i*w)^2+(i*w).^q.*a)./((i*w)^3+(i*w).^(q+1).*a+(i*w)); end mag = (abs(mag)).^2; sum = sum + mag.*step; end sum = sum/2/pi; ME280 "Fractional Order Mechanics" @ UC Merced

19. Calculate the integral of the time-weighted squared error of the step response for all time function sum = JITSE(a, q) step = 0.01; low = -500; high = 500; sum = 0; prime=phase(((i*(low))^2+(i*(low)).^q.*a)./((i*(low))^3+(i*(low)).^(q+1).*a+(i*(low)))); for w = low:step:high if w==-step/2, mag=0; else mag =((i*(w+step/2))^2+(i*(w+step/2)).^q.*a)./((i*(w+step/2))^3+(i*(w+step/2)).^(q+1).*a+(i*(w+step/2))); end if w == -step late = 0; else late=phase(((i*(w+step))^2+(i*(w+step)).^q.*a)./((i*(w+step))^3+(i*(w+step)).^(q+1).*a+(i*(w+step)))); end dphase=abs(late-prime); prime=late; mag = (abs(mag)).^2; sum = sum + mag.*dphase; end sum = sum/2/pi;

20. By choosing different values of a and q, calculate the integral of the squared error of the step response for all time, then plot the contours for different valued of the integral of the squared error and find the minimum. The value of a and q when choosing minimum is the optimal damping.a=0.8791 q=0.8459 >>[X,Y] = meshgrid(.6:.01:1.2,.6:.01:1.2); Z=JISE(X,Y); contour(X,Y,Z,30)

21. ME280 "Fractional Order Mechanics" @ UC Merced

22. By using The ISE Optimal Form, for fractional-order optimal damping and integer-order optimal damping, compare their step response and find that the fractional-order optimal damping is more optimal than integer-order optimal damping >>G1=fotf([1 1 1],[2 1 0],1,0); G2=fotf([1 0.8791 1],[2 0.8459 0],1,0); t1=0:0.001:30; t2=0:0.001:30; y1=step(G1,t1); y2=step(G2,t2); plot(t1,y1,t2,y2,'--') grid xlabel('time(sec)') ylabel('step response') title('Time response for optimal systems')

23. ME280 "Fractional Order Mechanics" @ UC Merced

24. By using the ISE Optimal Form, for fractional-order optimal damping and integer-order optimal damping, compare their frequency response and also find that the fractional-order optimal damping is more optimal than integer-order optimal damping >>G1=fotf([1 1 1],[2 1 0],1,0); G2=fotf([1 0.8791 1],[2 0.8459 0],1,0); H1=bode(G1); H2=bode(G2); bode(H1,H2,'--g') xlim([0.1 10]) grid

25. ME280 "Fractional Order Mechanics" @ UC Merced

26. The following is the simulink models to imitate the optimal damping ME280 "Fractional Order Mechanics" @ UC Merced

27. ME280 "Fractional Order Mechanics" @ UC Merced

28. Then using these models, compare their step responses >>step(fractionalISE,integerISE) grid xlim([0 16]) xlabel('time') ylabel('step response') title('Time responses for optimal systems') legend('fractioanl-order ISE optimal','Integer-order ISE optimal')

29. ME280 "Fractional Order Mechanics" @ UC Merced

30. Then using these models, compare their frequency responses >>bode(fractionalISE,integerISE) grid xlim([0.1 10]) xlabel('frequency') title('frequency responses for optimal systems') legend('fractioanl-order ISE optimal','Integer-order ISE optimal') ME280 "Fractional Order Mechanics" @ UC Merced

31. This bode figure is different with slide 9 ME280 "Fractional Order Mechanics" @ UC Merced

32. By choosing different values of a and q, calculate the integral of the time-weighted squared error of the step response for all time, then plot the contours for different valued of the integral of the time-weighted squared error and find the minimum. The value of a and q when choosing minimum is the optimal damping.a=1.1204 q=0. 9611 >>[X,Y] = meshgrid(.6:.01:1.2,.6:.01:1.2); Z=JITSE(X,Y); contour(X,Y,Z,30)

33. ME280 "Fractional Order Mechanics" @ UC Merced

34. By using The ITSE Optimal Form, for fractional-order optimal damping and integer-order optimal damping, compare their step response and find that the fractional-order optimal damping is more optimal than integer-order optimal damping >>G1=fotf([1 1.1892 1],[2 1 0],1,0); G2=fotf([1 1.1204 1],[2 0.9611 0],1,0); t1=0:0.001:16; t2=0:0.001:16; y1=step(G1,t1); y2=step(G2,t2); plot(t1,y1,t2,y2,'--') grid xlabel('time(sec)') ylabel('step response') title('Time response for optimal systems')

35. ME280 "Fractional Order Mechanics" @ UC Merced

36. By using The ITSE Optimal Form, for fractional-order optimal damping and integer-order optimal damping, compare their frequency response and also find that the fractional-order optimal damping is more optimal than integer-order optimal damping >>G1=fotf([1 1.1892 1],[2 1 0],1,0); G2=fotf([1 1.1204 1],[2 0.9611 0],1,0); H1=bode(G1); H2=bode(G2); bode(H1,H2,'--g') xlim([0.1 10]) grid

37. ME280 "Fractional Order Mechanics" @ UC Merced

38. The following is the simulink models to imitate the optimal ITSE damping ME280 "Fractional Order Mechanics" @ UC Merced

39. ME280 "Fractional Order Mechanics" @ UC Merced

40. Then using these models, compare their step responses >>step(fractionalITSE,integerITSE) grid xlim([0 16]) xlabel('time') ylabel('step response') title('Time responses for optimal systems') legend('fractioanl-order ITSE optimal','Integer-order ITSE optimal')

41. ME280 "Fractional Order Mechanics" @ UC Merced

42. Then using these models ,compare their frequency responses (figure7) >>bode(fractionalITSE,integerITSE) grid xlim([0.1 10]) xlabel('frequency') title('frequency responses for optimal systems') legend('fractioanl-order ITSE optimal','Integer-order ITSE optimal') ME280 "Fractional Order Mechanics" @ UC Merced

43. ME280 "Fractional Order Mechanics" @ UC Merced

44. The unity-resonance is to force the frequency response to have a magnitude at resonance of unity; to obtain the unity-resonance, q is sampled, and at each sample, a is varied until the criterion is satisfied. The fractional-order damping with the smallest damping coefficients is chosen as the optimal unity-resonance. ME280 "Fractional Order Mechanics" @ UC Merced

45. The maximally flat frequency response is to push the resonance down so that the slope of the magnitude response is either always negative (before and after the resonant point) or zero (at the resonant point); to obtain the maximally flat frequency response, q is sampled, and at each sample, a is varied until the criterion is satisfied. The fractional-order damping with the smallest damping coefficients is chosen as the optimal maximally flat frequency response. ME280 "Fractional Order Mechanics" @ UC Merced

46. Compare the step responses of optimal fractional-order maximally-flat damping (a=1.1712,q=0.8736)and integer-order Butterworth damping (The Butterworth optimal forms are well known to provide “nice” step responses that have fast rise- times, small overshoots, and rapid settling on point, a=1.4142). Find that optimal fractional-order maximally flat step response is more optimal than integer-order Butterworth. >>G1=fotf([1 1.4142 1],[2 1 0],1,0); G2=fotf([1 1.1712 1],[2 0.8736 0],1,0); t1=0:0.001:16; t2=0:0.001:16; y1=step(G1,t1); y2=step(G2,t2); plot(t1,y1,t2,y2,'--') grid xlabel('time(sec)') ylabel('step response') title('Time response for optimal systems')

47. ME280 "Fractional Order Mechanics" @ UC Merced

48. :Compare the step responses of optimal fractional-order unity-resonance damping (a=1.0000,q=0.7539)and integer-order Butterworth (The Butterworth optimal forms are well known to provide “nice” step responses that have fast rise- times, small overshoots, and rapid settling on point, a=1.4142). Find that optimal fractional-order unity-resonance step response is more optimal than integer-order Butterworth. >>G1=fotf([1 1.4142 1],[2 1 0],1,0); G2=fotf([1 1 1],[2 0.7539 0],1,0); t1=0:0.001:16; t2=0:0.001:16; y1=step(G1,t1); y2=step(G2,t2); plot(t1,y1,t2,y2,'--') grid xlabel('time(sec)') ylabel('step response') title('Time response for optimal systems')

49. ME280 "Fractional Order Mechanics" @ UC Merced

50. Compare the frequency responses of optimal fractional-order maximally-flat damping (a=1.1712,q=0.8736)and integer-order Butterworth damping (The Butterworth optimal forms are well known to provide “nice” step responses that have fast rise- times, small overshoots, and rapid settling on point, a=1.4142). Find that optimal fractional-order maximally flat step response is more optimal than integer-order Butterworth. >>G1=fotf([1 1.4142 1],[2 1 0],1,0); G2=fotf([1 1.1712 1],[2 0.8736 0],1,0); H1=bode(G1); H2=bode(G2); bode(H1,H2,'--g') xlim([0.1 10]) title('Frequency response for optimal systems') grid