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

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

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**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**Fractional-Order Damping**• Mass-spring-damper system F With control force: ME280 "Fractional Order Mechanics" @ UC Merced**Laplace transform**Normalize it to have unity DC gain ME280 "Fractional Order Mechanics" @ UC Merced**We focus on**Question: For step response, to make the ISE minimum, what is “a”? ME280 "Fractional Order Mechanics" @ UC Merced**ISE: integral of squared error**ME280 "Fractional Order Mechanics" @ UC Merced**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**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**Sensitivity**function Complementary sensitivity function One degree-of-freedom configuration ME280 "Fractional Order Mechanics" @ UC Merced**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**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**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**The optimal answer isa=1**Note: We will use G(s) to replace T(s) in the following ME280 "Fractional Order Mechanics" @ UC Merced**What about ITSE?**ME280 "Fractional Order Mechanics" @ UC Merced**Now let us assume fractional-order damping!**Smaller JISE and JITSE? ME280 "Fractional Order Mechanics" @ UC Merced**IAE and ITAE**• Integral Absolute Error (IAE) • Integral Time-Weighted Absolute Error (ITAE) ME280 "Fractional Order Mechanics" @ UC Merced**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**Results**ME280 "Fractional Order Mechanics" @ UC Merced**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**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;**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)**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')**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**The following is the simulink models to imitate the optimal**damping ME280 "Fractional Order Mechanics" @ UC Merced**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')**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**This bode figure is different with slide 9**ME280 "Fractional Order Mechanics" @ UC Merced**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)**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')**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**The following is the simulink models to imitate the optimal**ITSE damping ME280 "Fractional Order Mechanics" @ UC Merced**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')**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**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**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**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')**: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')**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