Solutions to Magnetic Bearing Lab #4 Lead Controller Design for a Magnetic Bearing System

Transcription

1 Solutions to Magnetic Bearing Lab #4 Lead Controller Design for a Magnetic Bearing System by Löhning, Matthias University of Calgary Department of Mechanical and Manufacturing Engineering 25 August 2004

2 Preliminary note: For the lead controller design, the number of unstable poles of the magnetic bearing system transfer function is essential. This can be obtained with the matlab code of Appix B preliminary note. Exercise 1: Figure 1 and 2 shows the Nyquist plot of the identified Magnetic Bearing open loop transfer function with the controller gain of 0.8. Normally the Nyquist plot show the transfer function for frequencies from minus infinity to infinity. Due to the taken measuring data the real frequency range is from minus 3100 Hz to 3100 Hz. The choice of the controller gain is described in Exercise 2. In figure 1, the non-filtered measuring data is taken. In figure 2, the average measuring data is taken. Figure 3 and 4 point up the curve progression over the increasing frequencies (only from zero to infinity). Figure 1: Nyquist plot of the identified magnetic bearing transfer function multiplied by 0.8 Page 2

3 Figure 2: Nyquist plot of the identified averaged magnetic bearing transfer function multiplied by 0.8 Figure 3: Nyquist plot (only positive frequencies) of the identified magnetic bearing transfer function multiplied by 0.8 Page 3

4 Figure 4: Nyquist plot (only positive frequencies) of the identified averaged magnetic bearing transfer function multiplied by 0.8 Exercise 2: For the controller design the gain, the desired phase margin and the factor between the two controller poles can be chosen. In order to get the controller values gain and time constant, the matlab code in Appix B Exercise 1 and 2 has to be run in an iterative fashion until the distance between the point (-1,0) and the open loop transfer function is big enough (see Figure 8). The open loop transfer function is the controller transfer function multiplied by the identified magnetic bearing transfer function. Figure 5 points out this approach by transforming the magnetic bearing control cycle into the standard control cycle. The * new input is r instead of r. Figure 5: transformation to a standard control cycle Page 4

5 The identified transfer function gaverage is chosen in order to get the lead controller values, due to the lower frequency noise. The following values dep on the identified measuring data and should only be considered as a reference point. Since gaverage is far enough to the left of the point (-1,0) for small frequencies, the gain of the lead controller k is set to 0.8,. The available phase margin of gaverage is rad (see black x in figure 4). With the desired phase margin of , rad is calculated as rad. Thus is The equation G j 0 * N j 0 * k gives 0 m of rad/sec (see red x in figure 1 4). The equation m gives T of msec. With the chosen factor 10, T the second pole of the lead controller is one decade greater than the other pole. Thus T 0 is one decade smaller than * T ; in this case msec. To summarize, the following controller is chosen: C l / f s *10 s 1 0.8* *10 s 1 * *10 s 1 Table 1 contains the Magnitude and the phase of the controller for selected frequencies. Figure 6 shows the Bode Diagram of the lead controller. f (Hz) Magnitude Phase (degree) Table 1: Magnitude and Phase of the controller Figure 6: Bode Diagram of the controller Page 5

6 Figure 7 and 8 show the Nyquist plot of the open loop transfer function for the original measuring data and the averaged data. Since there is one resulting counter clockwise encirclement of the point (-1,0) in both cases, the magnetic bearing system with this controller should theoretically be stable. This can be seen directly in figure 8: The negative frequencies have to be added to the plotted positive frequencies and are only reflected on the real axis. In figure 7, the same kind of encirclement exists; however, there is an additional one counter clockwise in the range of the first resonance frequency (see magenta part of figure 7). This encirclement will be compensated by the clockwise encirclement on the negative frequencies. Thus, it does not matter whether or not there is noise compared to figure 8. Figure 7: Nyquist plot (only positive frequencies) of the identified open loop transfer function C*G Page 6

7 Figure 8: Nyquist plot (only positive frequencies) of the identified averaged open loop transfer function C*G Exercise 3: With the help of a matlab m-file (see Appix B Exercise 3), C 1, C 2 and C 4 are set so that R 1 and R 2 are in the range of 1 kω and 10 kω, R 3 is in the range of 1 kω and 1 MΩ and R 4 is in the range of 10 kω and 100 kω. Given the assumptions C 1 1 F, C 100nF 2 and C 1 1 nf and the equations T R *C, T R *C, R2 * R T0 R4 *C4 and k, the solution is: R1 * R3 R k, R k, R k, R k Exercise 4: For the same reasons described in Lab3 solution Exercise 6, the controller is implemented in matlab as a quasi continuous controller. As a consequence, there has to be no adjustment for the resistances. Page 7

8 Exercise 5: Appix A shows the used set-up for the below exercises. The controller values in the used presentation in simulink can be obtained with the matlab code of Appix B Exercise 5. The test of the controller obtained in Exercise 2 shows unstable oscillations, which cause the rotor to contact the touchdown bushings. Exercise 6: The adjustments of the gain to 0.7 and the zero to 1/0.003 make the system stable. Exercise 7: The adjustments of Exercise 6 are sufficient that no instability occurs even with disturbances like component heating up or gentle shaft turning. Exercise 8: The following tests can be run to test the robustness of the implemented controller: - stronger turning of the shaft - change of the turning direction - change of the point of contact to speed up the shaft - impulse disturbance in the middle of the shaft (first mode excitation) with different taps strengths - impulse disturbance in one fourth of the shaft length (second mode excitation) with different taps strengths No instability occurs in any of these tests. The implemented controller can also be tested at the horizontal left magnetic bearing. Different system parameter exists (for example different sensor adjustments). Furthermore, it is possible to test the controller on the right and left vertical part of the magnetic bearing. The adjustments above are sufficient to stay still in a stable condition. If the system is stable in all cases, the controller can be called robust. The measurement of the resistors and capacitors will not be necessary if the controller is implemented in simulink. With the controller s -3 3*10 s 1 0.7* 5 C, l / f the final values are: *10 s 1 * *10 s 1 - k T 3*10 sec T *10 sec 0 Page 8

9 Appix A: Simulink model of the notch filter and the controller: Matlabcode Preliminary note: % to get the poles of the fitting model of lab2 Appix B: load lab2ex9g_ver3_sys3 [asys,bsys,csys,dsys]=unpck(sys); [num,den]=ss2tf(asys,bsys,csys,dsys); gf=tf(num,den); r=roots(den); % load the fitting model sys of lab2 % generate the state space matrices % generate the numerator and denominator % to get the transfer function of the fitting model % to get the roots of the denominator Matlabcode Exercise 1 and 2: % lab 4 exercise 1 and 2 % this values have to be chosen k=0.8; % the gain of the controller kk=101; % average filter: kk values were put together; it has to be odd a=0.01; % the range if the unit circle where the phase margin is chosen aaverage=0.1; % the range if the unit circle where the phase margin is chosen secpole=10; % the factor between the first pole and the second pole dpm=-160; % desired phase margin in degree disp(['k=',num2str(k),'; desired phase margin=',num2str(dpm),'; factor between poles=',num2str(secpole)]); % load data load lab2ex9g_ver3_rpsg % load the measured data from lab 2 lengthg=length(g); % length of g Page 9

10 % % Nyquist plot of g figure; plot(-1,0,'ko'); % attach the point -1 hold on; plot(real(k*g),imag(k*g),'r'); % attach the transfer function plot(real(k*g),-imag(k*g),'m'); % attach the transfer function for i=1:1000:lengthg plot(real(k*g(i)),imag(k*g(i)),'bx'); % plots an x for every 1000 values leg('(-1,0) point','frequencies from 0 to infinity','frequencies from -infinity to 0','every 1000 value',4) title(['nyquist plot of ', num2str(k),'*g']); xlabel(['controller gain k = ',num2str(k)]); hold off; % gaverage calculation and Nyquist plot of gaverage rpsaverage=0; aa=1; gaverage=0; for m = ((kk+1)/2):kk:(lengthg-(kk+1)/2) rpsaverage(aa,1)=rps(m,1); gaverage(aa,1)=sum(g(m-(kk-1)/2:m+(kk-1)/2,1))/kk; aa=aa+1; lengthgaverage=length(gaverage); figure; plot(-1,0,'ko'); hold on; plot(real(k*gaverage),imag(k*gaverage),'r'); plot(real(k*gaverage),-imag(k*gaverage),'m'); for i=1:10:lengthgaverage % initialization of rpsaverage % to count the loops of the for loop % initialization of gaverage % kk values of g were put together as one value gaverage % length of gaverage % attach the point -1 % attach the transfer function % attach the transfer function plot(real(k*gaverage(i)),imag(k*gaverage(i)),'bx') % plots an x for every 10 values leg('(-1,0) point','frequencies from 0 to infinity','frequencies from -infinity to 0','every 10 value',4) title(['nyquist plot of ', num2str(k),'*g_a_v_e_r_a_g_e']); xlabel(['controller gain k = ',num2str(k)]); hold off; % % to get the phase margin of g b=0; % initialization for the ph vector ph=0; % initialization of ph rpsnew=0; % initialization of rpsnew gnew=0; % initialization of gnew for i=1:1:9800 if abs(abs(k*g(i,1))-1)<a b=b+1; ph(b,1)=angle(k*g(i,1)); % the ph vector include all k*g(i) in the chosen range of the unit circle rpsnew(b,1)=rps(i,1); % to get the frequencies which correspond to the phase gnew(b,1)=g(i,1); % to get the transfer function value correspond to the phase c=0; phasemargin=0; % initialization of the phase margin for b=1:1:length(ph) if abs(ph(b,1))>abs(phasemargin) c=b; % to get the rps and g value next to (-1,0) Page 10

11 phasemargin=ph(b,1); % take the current biggest phase margin disp(['the phase margin of g is ',num2str(phasemargin),' (in rad)']); disp(['the frequency of the maximum phase of the controller for g is ',num2str(rpsnew(c,1)),' (in rad/sec)']); % to get the phase margin of gaverage baverage=0; phaverage=0; rpsaveragenew=0; gaveragenew=0; for i=1:1:lengthgaverage if abs(abs(k*gaverage(i,1))-1)<aaverage baverage=baverage+1; phaverage(baverage,1)=angle(k*gaverage(i,1)); rpsaveragenew(baverage,1)=rpsaverage(i,1); gaveragenew(baverage,1)=gaverage(i,1); caverage=0; phasemarginaverage=0; for baverage=1:1:length(phaverage) if abs(ph(baverage,1))>abs(phasemarginaverage) % initialization for the phaverage vector % initialization of phaverage % initialization of rpsaveragenew % initialization of gaveragenew % the phaverage vector include all k*gaverage(i) in the chosen range of the unit circle % to get frequencies which correspond to the phase % to get the transfer function value correspond to the phase % initialization of the phase margin caverage=baverage; % to get the rps and g value next to (-1,0) phasemarginaverage=ph(baverage,1); % take the current biggest phase margin disp(['the phase margin of gaverage is ',num2str(phasemarginaverage),' (in rad)']); disp(['the frequency of the maximum phase of the controller for gaverage is ',num2str(rpsaveragenew (caverage,1)),' (in rad/sec)']); % % calculate the controller desiredphasemargin=dpm/180*pi; phi=desiredphasemargin-phasemarginaverage; alpha=(1-sin(phi))/(1+sin(phi)); d=1; for i=2:1:lengthgaverage if abs(abs(gaverage(i,1))-sqrt(alpha))<abs(abs(gaverage(d,1))-sqrt(alpha)) d=i; omega0=rpsaverage(d,1); % to get the frequencies omega0 T=1/(sqrt(alpha)*omega0); alphat=alpha*t; T0=alpha*T/secpole; disp(['alpha=', num2str(alpha),'; w0=',num2str(omega0),'; T=',num2str(T),'; alpha*t=',num2str(alphat),'; T0=',num2str(T0),]); % % Half Nyquist plot of g (only positive frequencies) figure; plot(-1,0,'ko'); hold on; f1=round(rps(1,1)/(2*pi)); plot(real(k*g(1:400,1)),imag(k*g(1:400,1)),'k'); f2=round(rps(400,1)/(2*pi)); % attach the point -1 % attach the first points of the transfer function in the plot Page 11

12 plot(real(k*g(400:600,1)),imag(k*g(400:600,1)),'g'); f3=round(rps(600,1)/(2*pi)); plot(real(k*g(600:1500,1)),imag(k*g(600:1500,1)),'c'); f4=round(rps(1500,1)/(2*pi)); plot(real(k*g(1500:9801,1)),imag(k*g(1500:9801,1)),'b'); f5=round(rps(9801,1)/(2*pi)); plot(real(k*g(9801:9920,1)),imag(k*g(9801:9920,1)),'m'); f6=round(rps(9920,1)/(2*pi)); plot(real(k*g(9920:30000,1)),imag(k*g(9920:30000,1)),'r'); % attach the next points of the transfer function % attach the first points of the transfer function in the plot % attach the next points of the transfer function % attach the first points of the transfer function in the plot % attach the next points of the transfer function f7=round(rps(30000,1)/(2*pi)); plot(real(k*g(30000:lengthg,1)),imag(k*g(30000:lengthg,1)),'y'); % attach the first last points of the transfer function f8=round(rps(lengthg,1)/(2*pi)); plot(real(k*gnew(c)),imag(k*gnew(c)),'kx'); % plot a "x" where the phase margin is taken title(['nyquist plot of ', num2str(k),'*g (only positive frequencies)']); leg('(-1,0) point',['lowest f (',num2str(f1),' to ',num2str(f2),' Hz)'],['second lowest f (',num2str(f2),' to ',num2str(f3),' Hz)'],['lower middle f (',num2str(f3),' to ',num2str(f4),' Hz)'],['middle f (',num2str(f4),' to ',num2str (f5),' Hz)'],['higher middle f (',num2str(f5),' to ',num2str(f6),' Hz)'],['second highest f (',num2str(f6),' to ',num2str(f7),' Hz)'],['highest f (',num2str(f7),' to ',num2str(f8),' Hz)'],'read off phase margin',4) xlabel(['controller gain k = ',num2str(k)]); hold off; % Half Nyquist plot of gaverage (only positive frequencies) figure; plot(-1,0,'ko'); % attach the point -1 hold on; f1a=round(rpsaverage(1,1)/(2*pi)); plot(real(k*gaverage(1:10,1)),imag(k*gaverage(1:10,1)),'k'); % attach the first points of the transfer function f2a=round(rpsaverage(10,1)/(2*pi)); plot(real(k*gaverage(10:50,1)),imag(k*gaverage(10:50,1)),'g'); % attach the next points of the transfer function f3a=round(rpsaverage(50,1)/(2*pi)); plot(real(k*gaverage(50:100,1)),imag(k*gaverage(50:100,1)),'c'); % attach the first points of the transfer function f4a=round(rpsaverage(100,1)/(2*pi)); plot(real(k*gaverage(100:258,1)),imag(k*gaverage(100:258,1)),'b'); % attach the next points of the transfer function f5a=round(rpsaverage(258,1)/(2*pi)); plot(real(k*gaverage(258:290,1)),imag(k*gaverage(258:290,1)),'m'); % attach the first points of the transfer function f6a=round(rpsaverage(290,1)/(2*pi)); plot(real(k*gaverage(290:350,1)),imag(k*gaverage(290:350,1)),'r'); % attach the next points of the transfer function f7a=round(rpsaverage(350,1)/(2*pi)); plot(real(k*gaverage(350:lengthgaverage,1)),imag(k*gaverage(350:lengthgaverage,1)),'y'); % attach the first last points of the transfer function f8a=round(rpsaverage(lengthgaverage,1)/(2*pi)); plot(real(k*gaveragenew(caverage)),imag(k*gaveragenew(caverage)),'kx'); % plot a "x" where the phase plot(real(k*gaverage(d)),imag(k*gaverage(d)),'rx'); margin is taken % plot a "x" where the controller has to be the maximum phase title(['nyquist plot of ', num2str(k),'*g_a_v_e_r_a_g_e (only positive frequencies)']); leg('(-1,0) point',['lowest f (',num2str(f1a),' to ',num2str(f2a),' Hz)'],['second lowest f (',num2str(f2a),' to ',num2str(f3a),' Hz)'],['lower middle f (',num2str(f3a),' to ',num2str(f4a),' Hz)'],['middle f (',num2str(f4a),' to ',num2str(f5a),' Hz)'],['higher middle f (',num2str(f5a),' to ',num2str(f6a),' Hz)'],['second highest f (',num2str Page 12

13 (f6a),' to ',num2str(f7a),' Hz)'],['highest f (',num2str(f7a),' to ',num2str(f8a),' Hz)'],'read off phase margin','maximum phase of the controller',4) xlabel(['controller gain k = ',num2str(k)]); hold off; % % Half Nyquist plot of the identified magnetic bearing system with the controller Clf=k*(T*j*rps+1)./((alpha*T*j*rps+1).*(T0*j*rps+1)); % the controller vector for the frequencies defined in rps Fo=Clf.*g; figure; plot(-1,0,'ko'); % attach the point -1 hold on; plot(real(fo(1:400,1)),imag(fo(1:400,1)),'k'); % attach the first points of the transfer function plot(real(fo(401:600,1)),imag(fo(401:600,1)),'g'); % attach the next points of the transfer function plot(real(fo(601:1500,1)),imag(fo(601:1500,1)),'c'); % attach the first points of the transfer function in the plot plot(real(fo(1501:9801,1)),imag(fo(1501:9801,1)),'b'); % attach the next points of the transfer function plot(real(fo(9801:9920,1)),imag(fo(9801:9920,1)),'m'); % attach the first points of the transfer function in plot(real(fo(9920:30000,1)),imag(fo(9920:30000,1)),'r'); the plot % attach the next points of the transfer function plot(real(fo(30001:lengthg,1)),imag(fo(30001:lengthg,1)),'y'); % attach the first last points of the transfer function title('nyquist plot of the identified magnetic bearing system with controller (only positive frequencies)'); leg('(-1,0) point',['lowest f (',num2str(f1),' to ',num2str(f2),' Hz)'],['second lowest f (',num2str(f2),' to ',num2str(f3),' Hz)'],['lower middle f (',num2str(f3),' to ',num2str(f4),' Hz)'],['middle f (',num2str(f4),' to ',num2str (f5),' Hz)'],['higher middle f (',num2str(f5),' to ',num2str(f6),' Hz)'],['second highest f (',num2str(f6),' to ',num2str(f7),' Hz)'],['highest f (',num2str(f7),' to ',num2str(f8),' Hz)'],4) xlabel(['k=',num2str(k),'; desired phase margin=',num2str(dpm),'; factor between poles=',num2str(secpole)]); hold off; % Half Nyquist plot of the identified averaged magnetic bearing system with the controller Clf=k*(T*j*rpsaverage+1)./((alpha*T*j*rpsaverage+1).*(T0*j*rpsaverage+1)); % the controller vector for the frequencies defined in rpsaverage Fo=Clf.*gaverage; figure; plot(-1,0,'ko'); % attach the point -1 hold on; plot(real(fo(1:10,1)),imag(fo(1:10,1)),'k'); % attach the first points of the transfer function plot(real(fo(10:50,1)),imag(fo(10:50,1)),'g'); % attach the next points of the transfer function plot(real(fo(50:100,1)),imag(fo(50:100,1)),'c'); % attach the first points of the transfer function in the plot plot(real(fo(100:258,1)),imag(fo(100:258,1)),'b'); % attach the next points of the transfer function plot(real(fo(258:290,1)),imag(fo(258:290,1)),'m'); % attach the first points of the transfer function in plot(real(fo(290:350,1)),imag(fo(290:350,1)),'r'); the plot % attach the next points of the transfer function plot(real(fo(350:lengthgaverage,1)),imag(fo(350:lengthgaverage,1)),'y'); % attach the first last points of the transfer function title('nyquist plot of the identified averaged magnetic bearing system with controller (only positive frequencies)'); leg('(-1,0) point',['lowest f (',num2str(f1a),' to ',num2str(f2a),' Hz)'],['second lowest f (',num2str(f2a),' to ',num2str(f3a),' Hz)'],['lower middle f (',num2str(f3a),' to ',num2str(f4a),' Hz)'],['middle f (',num2str(f4a),' to Page 13

14 ',num2str(f5a),' Hz)'],['higher middle f (',num2str(f5a),' to ',num2str(f6a),' Hz)'],['second highest f (',num2str (f6a),' to ',num2str(f7a),' Hz)'],['highest f (',num2str(f7a),' to ',num2str(f8a),' Hz)'],4) xlabel(['k=',num2str(k),'; desired phase margin=',num2str(dpm),'; factor between poles=',num2str(secpole)]); hold off; % % Bode Diagram of the controller and the truth table s=tf('s'); Cs=k*(T*s+1)/((alpha*T*s+1)*(T0*s+1)); % transfer function of the controller freq = logspace(1, 5, 31000); % generate logarithmically spaced vector "freq" in rad/sec sys=ss(cs); % transfer to matlab system sys [num,den] = tfdata(sys); % calculate the numerator and denominator [mag,phase]=bode(num,den,freq); % calculate the magnitude and the phase figure; subplot(211); loglog(freq/(2*pi),mag); title(['bode Diagram of the controller; k=',num2str(k),'; desired phase margin=',num2str(dpm),'; factor between poles=',num2str(secpole)]); ylabel('magnitude (log scale)'); xlim([ ]); subplot(212); semilogx(freq/(2*pi),phase); xlabel('frequency (Hz)'); ylabel('phase (deg)'); xlim([ ]); f1c=10; f2c=100; f3c=300; f4c=700; f5c=1000; f6c=3100; % the chosen frequencies for the following exercise in Hz fc=[f1c*2*pi,f2c*2*pi,f3c*2*pi,f4c*2*pi,f5c*2*pi,f6c*2*pi]; % frequency vector in rad/sec [magtab,phasetab]=bode(num,den,fc); disp(['the frequencies are ',num2str(fc(1,1)),' ',num2str(fc(1,2)),' ',num2str(fc(1,3)),' ',num2str(fc(1,4)),' ',num2str(fc(1,5)),' ',num2str(fc(1,6))]); disp(['the magnitudes are ',num2str(magtab(1,1)),' ',num2str(magtab(2,1)),' ',num2str(magtab(3,1)),' ',num2str(magtab(4,1)),' ',num2str(magtab(5,1)),' ',num2str(magtab(6,1))]); disp(['the phases are ',num2str(phasetab(1,1)),' ',num2str(phasetab(2,1)),' ',num2str(phasetab(3,1)),' ',num2str(phasetab(4,1)),' ',num2str(phasetab(5,1)),' ',num2str(phasetab(6,1))]); save lab4ex2controller_table fc magtab phasetab; save lab4ex2controller_values k T alphat T0; % save the table in a mat-file % save the controller values in a mat-file Matlabcode Exercise 3: % lab4 exercise 3 % chose the values of the capacitors C1=1000*10^(-9); C2=100*10^(-9); C4=1*10^(-9); load lab4ex2controller_values % C1 is 1 microf % C2 is 100 nf % C4 is 1 nf % load k, T, alphat and T0 Page 14

15 R1=T/C1/1000; R2=alphaT/C2/1000; R4=T0/C4/1000; R3=R2*R4/(R1*k); % R1 in kohm % R2 in kohm % R4 in kohm % R3 in kohm disp(['c1=',num2str(c1*10^9),' nf; C2=',num2str(C2*10^9),' nf; C4=',num2str(C4*10^9),' nf']); disp(['r1=',num2str(r1),' kohm; R2=',num2str(R2),' kohm; R3=',num2str(R3),' kohm; R4=',num2str(R4),' kohm']); Matlabcode Exercise 5: % to transformate the controller to the presentation used in simulink load lab4ex2controller_values s=tf('s'); Cs=k*(T*s+1)/((alphaT*s+1)*(T0*s+1)) % load k, T, alphat and T0 % transfer function of the controller Page 15