1 EEE480 Final Exam, Spring 2016 A.A. Rodriguez Rules: Calculators permitted, One 8.5 11 sheet, closed notes/books, open minds GWC 352, 965-3712 Problem 1 (Analysis of a Feedback System - Bode, Root Locus, Nyquist) Consider the feedback system defined by the open loop transfer function L = PK = 10(s [ ] 4 [ ] +1.93)(s2 +1.87s +2.591) 400 10 5 4 [ 10 7 ][ ] s s +10 7 s 2 (s 1) 2 s + 400 s +10 5 10 7 10 7 [ ][ with series compensator K = 10(s+1.93)(s 2 4 +1.87s+2.591) s s+400] and reference command pre-filter [ ][ ] 1.93 2.591 W = s +1.93 s 2. +1.87s +2.591 (a) Sketch approximate Bode magnitude and phase plots. (b) Determine the approximate 10 (±20 db) frequencies associated with L. (c) Sketch an approximate root locus plot. Compute angles of asymptotes. (d) Sketch a Nyquist plot (only partial polar plot is required). (e) Provide a complete stability summary - indicating Nyquist N cw values for open k i intervals. (f) Compute all margins ( GM, GM, PM, DM), their associated frequencies and bounds for peak S, T. EXTRA CREDIT: g, h, i (g) What is the approximate closed loop transfer function T ry and associated closed loop poles? (h) Approximate the output y when r = 10 1(t). What is (approximate) overshoot? settling time? (i) Determine r such that y ss =10+0.1sin(0.1t +10 )whend i =10sin(0.01t +10 ) + 30 cos(400t +40 ). Problem 2 (Control System Design: Design for Bandwidth and Phase Margin) Suppose [ ] [ 2Ω 1 s 0.5Ωs π P = s ] s 0.5 s +0.5Ω s s + 2Ωs π where Ω s = kω g (k>0) represents a (digital) sampling frequency. The first term in brackets represents an anti-aliasing filter. The second term in brackets represents a Pade approximation to a zero order hold half sample time delay. Suppose that k = 20. (a) Design a feedback control system such that the closed loop system (1) is stable, (2) exhibits constant steady state error to ramp input disturbances d i, (3) ω g = 2 rad/sec, (4) PM =60, (5) impact of high frequency sensor noise on controls is addressed, (6) overshoot to step commands is addressed. (b) Determine the approximate gain margins associated with your design.
2 Here is an mfile for Problem # 1 (Spring 2016 Final Exam). EEE480 SPRING 2016 FINAL A.A. Rodriguez (All rights reserved) 4th ORDER CLS L = ( c3 s^3 + c2 s^2 + c1 s + co ) / ( s^2 (s - p1) (s - p2) ) Phi_CL = s^4 + (c3 - p1 - p2)s^3 + (c2 +p1 p2) s^2 + c1 s + co Phi_CL_desired = (s^2 + a1 s + ao) (s^2 + b1 s + b0) = s^4 +(a1 + b1)s^3 + (ao + a1b1 + bo)s^2 + (ao b1 + a1 bo)s + aobo c3 = a1 + b1 + p1 + p2 c2 = ao + a1 b1 + bo - p1 p2 c1 = ao b1 + a1 bo co = ao bo PLANT P = 1 / ( (s - p1) (s - p1) ) NOTE: DATA GIVEN IN PROBLEM STATEMENT RESULTS IN THE FOLLOWING NICE DOMINANT CLOSED LOOP POLES! -1 +/- j 1-3 +/- j 4 PLEASE SET FLAG PROPERLY BEFORE RUNNING MFILE FLAG IS SET TO NOMINAL VALUE OF 1 - to get 1 unity gain crossover FLAG = 1 to get 1 unity gain crossover for nominal data given FLAG = 3 to get 3 unity gain crossovers for nominal data given PLEASE SET HIGH FREQUENCY POLE APPROXIMATION FLAG PROPERLY BEFORE RUNNING MFILE POLE APPROXIMATION FLAG IS CURRENTLY SET TO ITS NOMINAL VALUE FLAGAPPROX = 1 FLAGAPPROX = 0 NOMINAL: USE THIS TO APPROXIMATE HIGH FREQUENCY POLES AND GET NICER CLOSED LOOP POLE NUMBERS YOU SHOULD COMPUTE DURING EXAM USE THIS TO NOT APPROXIMATE HIGH FREQUENCY POLES PLANT DATA - PLANT = 1 / (s - p1) (s - p2)
3 p1 = 1; 1 Nominal plant pole p2 = 1; 2 Nominal plant pole plant = tf(1, [1 -(p1+p2) p1*p2 ]) zpk(plant) w = logspace(-2,3,2000); Vector of frequencies [plant_mag, plant_phase] = bode(plant, w); figure(10) semilogx(w,20*log10(plant_mag(1,:)) ) title( PLANT: P ) figure(11) semilogx(w,plant_phase(1,:) ) title( PLANT: P ) ylabel( Phase (degrees) ) DESIRED CLOSED LOOP CHARACTERISTIC EQUATION Phi_CL_desired = (s^2 + a1 s + ao) (s^2 + b1 s + b0) = s^4 +(a1 + b1)s^3 + (ao + a1b1 + bo)s^2 + (ao b1 + a1 bo)s + aobo SET NOMINAL CLOSED LOOP POLES a1 = 2 ao = 2 s^2 + 2s + 2 - -1\pm j 1 if FLAG == 1 b1 = 6; 6 bo = 25; 225/16 s^2 + 6s + 25 = 0 yields faster CLPs = -2 \pm j 3/2 Yields 1 unity gain crossover for nominal data given! end if FLAG == 1 b1 = 12 bo = 225/4 s^2 + 12s + 225/16 = 0 yields faster CLPs = -6 \pm j 9/2 Yields 1 unity gain crossover for nominal data given! end if FLAG == 1 b1 = 1 bo = 0.5 s^2 + s + 0.5 = 0 yields CLPs = -0.5 \pm j 0.5 Yields 1 unity gain crossover for nominal data given! end
4 if FLAG==3 b1 = 2 bo = 25/16 s^2 + 2s + 25/16 = 0 yields slower CLPs = -1 \pm j 3/4 yields 3 unity gain crossovers for nominal data given! YUK!!!! end COMPENSATOR: K = ( c3 s^3 + c2 s^2 + c1 s + co ) / s^2 c3 = a1 + b1 + p1 + p2 c2 = ao + a1*b1 + bo - p1*p2 c1 = ao*b1 + a1*bo co = ao*bo hfp = 400 70 High frequency poles for roll off in controller Actual value of pole used in problem statement if FLAGAPPROX ==1 hfp = hfp + 10000 High frequency poles for roll off in controller USE THIS TO GET NICER CLOSED LOOP POLE NUMBERS; i.e. approximation that should be made during the exam SET FLAGAPPROX = 0 TO GET NUMBERS IN PROBLEM STATEMENT end comp_num = [ c3 c2 c1 co] comp_zeros = roots(comp_num) Compensator zeros comp_den = [ 1 0 0] comp = tf (comp_num,comp_den) roll_off = tf(hfp, [1 hfp] ) num_rolloff_terms = 4; Number of Roll Off Terms for end rolloff_counter = 1:1: num_rolloff_terms comp = series(comp, roll_off) add high frequency roll off to comp NOTE: This just includes 5 roll off terms. Does NOT include all pass term in Plant P given on Exam#2 Spring 2016 Problem 1 statement. PM computed in this mfile will therefore be bigger!...off by about 2*atan(7/100)*180/pi = 8.0083 degrees zpk(comp) [comp_mag, comp_phase] = bode(comp, w); figure(20)
5 semilogx(w,20*log10(comp_mag(1,:)) ) title( CONTROLLER: K ) figure(21) semilogx(w,comp_phase(1,:),w, -180, r-. ) title( CONTROLLER: K ) ylabel( Phase (degrees) ) OPEN LOOP: L = PK loop = series(plant,comp) zpk(loop) [loop_mag, loop_phase] = bode(loop, w); figure(30) semilogx(w,20*log10(loop_mag(1,:)) ) title( LOOP: PK ) figure(31) semilogx(w,loop_phase(1,:), w, -180, r-. ) title( LOOP: PK ) ylabel( Phase (degrees) ) figure(32) bode(loop, w) allmargin(loop) SENSITIVITY: S = 1 / (1 + PK) sen = feedback(1,loop) Form sensitivity eig(sen) closed loop poles [sen_mag, sen_phase] = bode(sen, w); figure(40) semilogx(w,20*log10(sen_mag(1,:)),w, 6, r-- ) title( SENSITIVITY: S = 1 / (1 + L) ) axis([0.01 100-160 7])
6 COMPLEMENTARY SENSITIVITY: T = L / (1 + PK) compsen = feedback(loop,1) Form complementary sensitivity eig(compsen) closed loop poles [compsen_mag, compsen_phase] = bode(compsen, w); figure(50) semilogx(w,20*log10(compsen_mag(1,:)),w, 6, r-- ) title( COMPLEMENTARY SENSITIVITY: T = L / (1 + L) ) axis([0.1 100-40 7]) KS ksen = feedback(comp,plant) Form KS eig(ksen) closed loop poles [ksen_mag, ksen_phase] = bode(ksen, w); figure(60) semilogx(w,20*log10(ksen_mag(1,:)), w, 20, r--, w, 40, r-- ) title( KS ) axis([0.1 100-40 6]) SP psen = feedback(plant, comp) Form SP eig(psen) closed loop poles [psen_mag, psen_phase] = bode(psen, w); figure(70) semilogx(w,20*log10(psen_mag(1,:)), w,-40, r-- ) title( SP ) axis([0.1 100-40 6]) CLOSED LOOP STEP RESPONSE figure(100)
7 t = 0:.001:10; Vector of time points num_tpoints = size(t)*[0 1] ref_command = ones(1, num_tpoints); prefilter = tf(co/c3, [ 1 c2/c3 c1/c3 co/c3 ] ) zpk(prefilter) filtref_command = lsim(prefilter,ref_command, t); Filtered Reference Command y = lsim(compsen,ref_command, t); u = lsim(ksen, ref_command, t); y_filt = lsim(compsen,filtref_command, t); u_filt = lsim(ksen, filtref_command, t); plot(t,y,t,y_filt) title( Output Response to Step Reference (Filtered and Unfiltered) ) xlabel( Time (sec) ) ylabel( Output: y ) figure(101) plot(t,u,t,u_filt) title( Control Response to Step Reference (Filtered and Unfiltered) ) xlabel( Time (sec) ) ylabel( Control: u ) axis([0 10-20 20]) ROOT LOCUS figure(200) k = 0:.001:20; rlocus(loop,k) axis([-1 2.5 0 4]) figure(210) rlocus(loop,k) axis([-10 2.5 0 5]) ************************************************************************** NOMINAL LOOP AT LOW FREQUENCIES compnom_num = [ c3 c2 c1 co] compnom_den = [ 1 0 0]
8 compnom = tf (compnom_num,compnom_den) [compnom_mag, compnom_phase] = bode(compnom, w); figure(20) semilogx(w,20*log10(comp_mag(1,:)), w,20*log10(compnom_mag(1,:)) ) title( CONTROLLER: K ) figure(21) semilogx(w,comp_phase(1,:),w, -180, r-., w,compnom_phase(1,:) ) title( CONTROLLER: K ) ylabel( Phase (degrees) ) OPEN LOOP: L = PK loopnom = series(plant, compnom); zpk(loopnom) allmargin(loopnom) damp(eig(1/(1+loopnom))) [loopnom_mag, loopnom_phase] = bode(loopnom, w); figure(30) semilogx(w,20*log10(loop_mag(1,:)), w,20*log10(loopnom_mag(1,:)), w, -20, r--, w, 0, r--, w, 20, r-- title( LOOP: PK ) figure(31) semilogx(w,loop_phase(1,:), w,loopnom_phase(1,:), w, -180, r-. ) title( LOOP: PK ) ylabel( Phase (degrees) ) SENSITIVITY: S = 1 / (1 + PK) sennom = feedback(1,loopnom) Form sensitivity eig(sennom) closed loop poles
9 [sennom_mag, sennom_phase] = bode(sennom, w); figure(40) semilogx(w,20*log10(sen_mag(1,:)),w, 6, r--, w,20*log10(sennom_mag(1,:)), w, -20, r-- ) title( SENSITIVITY: S = 1 / (1 + L) ) axis([0.01 100-160 7]) COMPLEMENTARY SENSITIVITY: T = L / (1 + PK) compsennom = feedback(loopnom,1) Form complementary sensitivity eig(compsennom) closed loop poles [compsennom_mag, compsennom_phase] = bode(compsennom, w); figure(50) semilogx(w,20*log10(compsen_mag(1,:)),w, 6, r--, w,20*log10(compsennom_mag(1,:)), w, -20, r-- ) title( COMPLEMENTARY SENSITIVITY: T = L / (1 + L) ) axis([0.1 100-40 7]) [Gm,Pm,Wcg,Wcp] = margin(loop) figure(55) semilogx(w,20*log10(sen_mag(1,:)),w,20*log10(compsen_mag(1,:)), w, 20*log10(Gm), r--, w,20*log10(s title( SENS AND COMP SENSITIVITY: S = 1 / (1 + L), T = L / (1 + L) ) axis([0.1 100-40 7]) KS ksennom = feedback(compnom,plant) Form KS eig(ksennom) closed loop poles [ksennom_mag, ksennom_phase] = bode(ksennom, w); figure(60) semilogx(w,20*log10(ksen_mag(1,:)), w, 20, r--, w, 40, r--, w,20*log10(ksennom_mag(1,:)) ) title( KS ) axis([0.1 100-40 6])
10 SP psennom = feedback(plant, compnom) Form SP eig(psennom) closed loop poles [psennom_mag, psennom_phase] = bode(psennom, w); figure(70) semilogx(w,20*log10(psen_mag(1,:)), w,20*log10(psennom_mag(1,:)), w,-40, r-- ) title( SP ) axis([0.1 100-40 6]) CLOSED LOOP STEP RESPONSE ynom = lsim(compsennom,ref_command, t); unom = lsim(ksennom, ref_command, t); CANT SIMULATE WITHOUT ROLLOFF IN K ynom_filt = lsim(compsennom,filtref_command, t); unom_filt = lsim(ksennom, filtref_command, t); CANT SIMULATE WITHOUT ROLLOFFIN K figure(100) plot(t,y,t,y_filt, t,ynom,t,ynom_filt) title( Output Response to Step Reference (Filtered and Unfiltered) ) xlabel( Time (sec) ) ylabel( Output: y ) figure(101) plot(t,u,t,u_filt) title( Control Response to Step Reference (Filtered and Unfiltered) ) xlabel( Time (sec) ) ylabel( Control: u ) axis([0 10-20 20]) DESIGN DATA zpk(loop) damp(comp_zeros) allmargin(loop) eig(sen) damp(eig(sen)) FLAG FLAGAPPROX c3 c2
11 c1 co zpk(loopnom) allmargin(loopnom) damp(eig(1/(1+loopnom))) ************************************************************************* ************************************************************************* ************************************************************************* EEE480 FINAL SPRING 2016 - PROBLEMS 1-4 DID NOT APPROXIMATED HIGH FREQUENCY POLES LOOP Zero/pole/gain: 256000000000 (s+1.93) (s^2 + 1.87s + 2.591) ------------------------------------------- s^2 (s+400)^4 (s-1)^2 = 10 (s+1.93) (s^2 + 1.87s + 2.591) (400)^4 ------------------------------------- ------ s^2 (s-1)^2 (s+400)^4 CONTROLLER ZEROS Eigenvalue Damping Freq. (rad/s) -1.93e+000 1.00e+000 1.93e+000-9.35e-001 + 1.31e+000i 5.81e-001 1.61e+000-9.35e-001-1.31e+000i 5.81e-001 1.61e+000 ALLMARGINS GainMargin: [0 0.376634431495105 21.843066002611838] GMFrequency: [0 3.779835541711269 1.614932025841673e+002] PhaseMargin: 51.030070599309326 PMFrequency: 9.989275258297823 DelayMargin: 0.089159896634470 DMFrequency: 9.989275258297823 Stable: 1 CLOSED LOOP POLES 1.0e+002 * -5.467129030297774
12-4.136002656422188 + 1.541116624754088i -4.136002656422188-1.541116624754088i -2.156182156991147-0.032354419873433 + 0.041642041468861i -0.032354419873433-0.041642041468861i -0.009987330059916 + 0.010034066632910i -0.009987330059916-0.010034066632910i CLOSED LOOP POLES Eigenvalue Damping Freq. (rad/s) -5.47e+002 1.00e+000 5.47e+002-4.14e+002 + 1.54e+002i 9.37e-001 4.41e+002-4.14e+002-1.54e+002i 9.37e-001 4.41e+002-2.16e+002 1.00e+000 2.16e+002-3.24e+000 + 4.16e+000i 6.14e-001 5.27e+000-3.24e+000-4.16e+000i 6.14e-001 5.27e+000-9.99e-001 + 1.00e+000i 7.05e-001 1.42e+000-9.99e-001-1.00e+000i 7.05e-001 1.42e+000 FLAG = 1 FLAGAPPROX = 0 CONTROLER COEFFICIENTS c3 = 10 c2 = 38 c1 = 62 co = 50 ************************************************************************* ************************************************************************* ************************************************************************* EEE480 FINAL SPRING 2016 NOMINAL LOOP Zero/pole/gain: 10 (s+1.93) (s^2 + 1.87s + 2.591) --------------------------------- s^2 (s-1)^2 ALL MARGINS GainMargin: [0 0.367583634323156] GMFrequency: [0 3.691435930821560]
13 PhaseMargin: 56.794037323225666 PMFrequency: 10.001839060974213 DelayMargin: 0.099106068468801 DMFrequency: 10.001839060974213 Stable: 1 NOMINAL CLOSED LOOP POLES Eigenvalue Damping Freq. (rad/s) -3.00e+000 + 4.00e+000i 6.00e-001 5.00e+000-3.00e+000-4.00e+000i 6.00e-001 5.00e+000-1.00e+000 + 1.00e+000i 7.07e-001 1.41e+000-1.00e+000-1.00e+000i 7.07e-001 1.41e+000 ************************************************************************* ************************************************************************* ************************************************************************* EEE480 FINAL SPRING 2016 APPROXIMATED HIGH FREQUENCY POLES - using BIG HIGH FREQ POLE NUMBERS LOOP Zero/pole/gain: 116985856000000000 (s+1.93) (s^2 + 1.87s + 2.591) ------------------------------------------------- s^2 (s+1.04e004)^4 (s-1)^2 10 (s+1.93) (s^2 + 1.87s + 2.591) (10400)^4 = ----------------------------------- -------------- s^2 (s-1)^2 (s+10400)^4 CONTROLLER ZEROS Eigenvalue Damping Freq. (rad/s) -1.93e+000 1.00e+000 1.93e+000-9.35e-001 + 1.31e+000i 5.81e-001 1.61e+000-9.35e-001-1.31e+000i 5.81e-001 1.61e+000 ALL MARGINS GainMargin: [0 0.367688943954142 5.900680656291955e+002] GMFrequency: [0 3.692470146180528 4.302229412475644e+003] PhaseMargin: 56.573521413151354 PMFrequency: 10.001806563822662 DelayMargin: 0.098721587125919 DMFrequency: 10.001806563822662 Stable: 1
14 CLOSED LOOP POLES 1.0e+004 * -1.216092126195456-1.047819122683930 + 0.181924163347982i -1.047819122683930-0.181924163347982i -0.847268006735989-0.000300815648326 + 0.000400568591274i -0.000300815648326-0.000400568591274i -0.000099995202021 + 0.000100013158378i -0.000099995202021-0.000100013158378i CLOSED LOOP POLES (NICE "EXAM" NUMBERS!!!!) Eigenvalue Damping Freq. (rad/s) -1.22e+004 1.00e+000 1.22e+004-1.05e+004 + 1.82e+003i 9.85e-001 1.06e+004-1.05e+004-1.82e+003i 9.85e-001 1.06e+004-8.47e+003 1.00e+000 8.47e+003-3.01e+000 + 4.01e+000i 6.00e-001 5.01e+000-3.01e+000-4.01e+000i 6.00e-001 5.01e+000-1.00e+000 + 1.00e+000i 7.07e-001 1.41e+000-1.00e+000-1.00e+000i 7.07e-001 1.41e+000 FLAG = 1 FLAGAPPROX = 1 CONTROLER COEFFICIENTS c3 = 10 c2 = 38 c1 = 62 co = 50 ************************************************************************* ************************************************************************* ************************************************************************* EEE480 EXAM#2 SPRING 2016 - PROBLEMS 1-4 APPROXIMATED HIGH FREQUENCY POLES - using BIG HIGH FREQ POLE NUMBERS LOOP
Zero/pole/gain: 735707035070000010000 (s+1.304) (s^2 + 1.128s + 1.901) ------------------------------------------------------ s^3 (s+1.01e004)^5 (s-1) 7 (s+1.304) (s^2 + 1.128s + 1.901) (1.01e004)^5 = ----------------------------------- ---------- s^3 (s-1) (s+1.01e004)^5 COMPENSATOR ZEROS Eigenvalue Damping Freq. (rad/s) -1.30e+000 1.00e+000 1.30e+000-5.64e-001 + 1.26e+000i 4.09e-001 1.38e+000-5.64e-001-1.26e+000i 4.09e-001 1.38e+000 ALL MARGINS GainMargin: [0.3879 601.5336 Inf] GMFrequency: [2.3126 3.2781e+003 Inf] PhaseMargin: 61.0411 PMFrequency: 6.8648 DelayMargin: 0.1552 DMFrequency: 6.8648 Stable: 1 CLOSED LOOP POLES 1.0e+004 * -1.1970 + 0.1298i -1.1970-0.1298i -0.9474 + 0.2290i -0.9474-0.2290i -0.7605-0.0002 + 0.0002i -0.0002-0.0002i -0.0001 + 0.0001i -0.0001-0.0001i CLOSED LOOP POLES Eigenvalue Damping Freq. (rad/s) -1.20e+004 + 1.30e+003i 9.94e-001 1.20e+004-1.20e+004-1.30e+003i 9.94e-001 1.20e+004-9.47e+003 + 2.29e+003i 9.72e-001 9.75e+003-9.47e+003-2.29e+003i 9.72e-001 9.75e+003-7.60e+003 1.00e+000 7.60e+003-2.01e+000 + 1.50e+000i 8.01e-001 2.51e+000 15
16-2.01e+000-1.50e+000i 8.01e-001 2.51e+000-9.99e-001 + 1.33e+000i 6.00e-001 1.67e+000-9.99e-001-1.33e+000i 6.00e-001 1.67e+000 FLAG = 1 FLAGAPPROX = 1 CONTROLLER COEFFICIENTS c3 = 7 c2 = 17.0278 c1 = 23.6111 co = 17.3611 ************************************************************************* ************************************************************************* ************************************************************************* EEE480 EXAM#2 SPRING 2016 - PROBLEMS 1-4 DID NOT APPROXIMATE HIGH FREQUENCY POLES LOOP Zero/pole/gain: 70000000000 (s+1.304) (s^2 + 1.128s + 1.901) -------------------------------------------- s^3 (s+100)^5 (s-1) 7 (s+1.304) (s^2 + 1.128s + 1.901) (100)^5 = -------------------------------------------- -------- s^3 (s-1) (s+100)^5 COMPENSATOR ZEROS Eigenvalue Damping Freq. (rad/s) -1.30e+000 1.00e+000 1.30e+000-5.64e-001 + 1.26e+000i 4.09e-001 1.38e+000-5.64e-001-1.26e+000i 4.09e-001 1.38e+000 ALL MARGINS GainMargin: [0.4100 5.3159 Inf] GMFrequency: [2.4851 29.9777 Inf] PhaseMargin: 41.4842 PMFrequency: 6.7833 DelayMargin: 0.1067 DMFrequency: 6.7833 Stable: 1
17 CLOSED LOOP POLES 1.0e+002 * -1.4498 + 0.3001i -1.4498-0.3001i -0.8830 + 0.5668i -0.8830-0.5668i -0.2405-0.0327 + 0.0110i -0.0327-0.0110i -0.0093 + 0.0129i -0.0093-0.0129i CLOSED LOOP POLES Eigenvalue Damping Freq. (rad/s) -1.45e+002 + 3.00e+001i 9.79e-001 1.48e+002-1.45e+002-3.00e+001i 9.79e-001 1.48e+002-8.83e+001 + 5.67e+001i 8.42e-001 1.05e+002-8.83e+001-5.67e+001i 8.42e-001 1.05e+002-2.40e+001 1.00e+000 2.40e+001-3.27e+000 + 1.10e+000i 9.48e-001 3.45e+000-3.27e+000-1.10e+000i 9.48e-001 3.45e+000-9.30e-001 + 1.29e+000i 5.86e-001 1.59e+000-9.30e-001-1.29e+000i 5.86e-001 1.59e+000 FLAG = 1 FLAGAPPROX = 0 CONTROLLER COEFFICIENTS c3 = 7 c2 = 17.0278 c1 = 23.6111 co = 17.3611 ************************************************************************* ************************************************************************* ************************************************************************* EEE480 EXAM#2 FALL 2015 Zero/pole/gain: 41334300000 (s+1) (s^2 + 1.214s + 0.9286)
18 ----------------------------------------- s^2 (s+90)^5 (s^2-2s + 2) 7 (s+1) (s^2 + 1.214s + 0.9286) 90^5 = -------------------------------- --------- s^2 (s^2-2s + 2) (s+90)^5 Eigenvalue Damping Freq. (rad/s) -1.00e+000 1.00e+000 1.00e+000-6.07e-001 + 7.48e-001i 6.30e-001 9.64e-001-6.07e-001-7.48e-001i 6.30e-001 9.64e-001 ans = GainMargin: [0 0.4406 4.5495 Inf] GMFrequency: [0 3.1147 26.0544 Inf] PhaseMargin: 32.8955 PMFrequency: 6.9360 DelayMargin: 0.0828 DMFrequency: 6.9360 Stable: 1 ans = 1.0e+002 * -1.3123 + 0.2750i -1.3123-0.2750i -0.7930 + 0.5196i -0.7930-0.5196i -0.2014-0.0290 + 0.0386i -0.0290-0.0386i -0.0050 + 0.0051i -0.0050-0.0051i Eigenvalue Damping Freq. (rad/s) -1.31e+002 + 2.75e+001i 9.79e-001 1.34e+002-1.31e+002-2.75e+001i 9.79e-001 1.34e+002-7.93e+001 + 5.20e+001i 8.36e-001 9.48e+001-7.93e+001-5.20e+001i 8.36e-001 9.48e+001-2.01e+001 1.00e+000 2.01e+001-2.90e+000 + 3.86e+000i 6.01e-001 4.82e+000-2.90e+000-3.86e+000i 6.01e-001 4.82e+000-5.01e-001 + 5.06e-001i 7.04e-001 7.12e-001-5.01e-001-5.06e-001i 7.04e-001 7.12e-001 FLAG =
19 1 FLAGAPPROX = 0 c3 = 7 c2 = 15.5000 c1 = 15 co = 6.5000 ************************************************************************* ************************************************************************* ************************************************************************* EEE480 FINAL SPRING 2015 Zero/pole/gain: 16807000000 (s+0.8211) (s^2 + 1.035s + 0.8563) ---------------------------------------------- s^2 (s+70)^5 (s-2) (s-1) 10 (s+0.8211) (s^2 + 1.035s + 0.8563) 70^5 = -------------------------------------- -------- s^2 (s+70)^5 (s-2) (s-1) (s+70)^5 Eigenvalue Damping Freq. (rad/s) -5.18e-001 + 7.67e-001i 5.59e-001 9.25e-001-5.18e-001-7.67e-001i 5.59e-001 9.25e-001-8.21e-001 1.00e+000 8.21e-001 ans =
20 GainMargin: [0 0.4252 2.2546 Inf] GMFrequency: [0 3.5525 18.8082 Inf] PhaseMargin: 22.4240 PMFrequency: 9.3034 DelayMargin: 0.0421 DMFrequency: 9.3034 Stable: 1 ans = 1.0e+002 * -1.0589 + 0.2382i -1.0589-0.2382i -0.6110 + 0.4531i -0.6110-0.4531i -0.0457 + 0.0984i -0.0457-0.0984i -0.0289-0.0050 + 0.0051i -0.0050-0.0051i Eigenvalue Damping Freq. (rad/s) -1.06e+002 + 2.38e+001i 9.76e-001 1.09e+002-1.06e+002-2.38e+001i 9.76e-001 1.09e+002-6.11e+001 + 4.53e+001i 8.03e-001 7.61e+001-6.11e+001-4.53e+001i 8.03e-001 7.61e+001-4.57e+000 + 9.84e+000i 4.21e-001 1.08e+001-4.57e+000-9.84e+000i 4.21e-001 1.08e+001-2.89e+000 1.00e+000 2.89e+000-5.00e-001 + 5.09e-001i 7.01e-001 7.14e-001-5.00e-001-5.09e-001i 7.01e-001 7.14e-001 FLAG = 1 FLAGAPPROX = 0 Zero/pole/gain: 240100000 (s+0.8211) (s^2 + 1.035s + 0.8563) -------------------------------------------- s^2 (s+70)^4 (s-2) (s-1)
10 (s+0.8211) (s^2 + 1.035s + 0.8563) 70^4 = --------------------------------------- ---------- s^2 (s-2) (s-1) (s+70)^4 Eigenvalue Damping Freq. (rad/s) -5.18e-001 + 7.67e-001i 5.59e-001 9.25e-001-5.18e-001-7.67e-001i 5.59e-001 9.25e-001-8.21e-001 1.00e+000 8.21e-001 ans = GainMargin: [0 0.4109 3.2109] GMFrequency: [0 3.3868 25.1076] PhaseMargin: 29.9875 PMFrequency: 9.3844 DelayMargin: 0.0558 DMFrequency: 9.3844 Stable: 1 ans = 1.0e+002 * -1.0817-0.7472 + 0.4023i -0.7472-0.4023i -0.0759 + 0.0868i -0.0759-0.0868i -0.0321-0.0050 + 0.0051i -0.0050-0.0051i Eigenvalue Damping Freq. (rad/s) -1.08e+002 1.00e+000 1.08e+002-7.47e+001 + 4.02e+001i 8.81e-001 8.49e+001-7.47e+001-4.02e+001i 8.81e-001 8.49e+001-7.59e+000 + 8.68e+000i 6.58e-001 1.15e+001-7.59e+000-8.68e+000i 6.58e-001 1.15e+001-3.21e+000 1.00e+000 3.21e+000-5.00e-001 + 5.07e-001i 7.02e-001 7.12e-001-5.00e-001-5.07e-001i 7.02e-001 7.12e-001 FLAG = 1 FLAGAPPROX = 21
22 0 Zero/pole/gain: 102829537440100000 (s+0.8211) (s^2 + 1.035s + 0.8563) ----------------------------------------------------- s^2 (s+1.007e004)^4 (s-2) (s-1) = 10 (s+0.8211) (s^2 + 1.035s + 0.8563) (1.007e004)^4 --------------------------------------- ----------------- s^2 (s-2) (s-1) (s+1.007e004)^4 Eigenvalue Damping Freq. (rad/s) -5.18e-001 + 7.67e-001i 5.59e-001 9.25e-001-5.18e-001-7.67e-001i 5.59e-001 9.25e-001-8.21e-001 1.00e+000 8.21e-001 ans = GainMargin: [0 0.3741 571.5113] GMFrequency: [0 2.9420 4.1665e+003] PhaseMargin: 61.3860 PMFrequency: 9.7447 DelayMargin: 0.1099 DMFrequency: 9.7447 Stable: 1 ans = 1.0e+004 * -1.1788-1.0147 + 0.1776i -1.0147-0.1776i -0.8188-0.0003 + 0.0002i -0.0003-0.0002i -0.0001 + 0.0001i -0.0001-0.0001i Eigenvalue Damping Freq. (rad/s) -1.18e+004 1.00e+000 1.18e+004-1.01e+004 + 1.78e+003i 9.85e-001 1.03e+004-1.01e+004-1.78e+003i 9.85e-001 1.03e+004-8.19e+003 1.00e+000 8.19e+003-3.01e+000 + 2.25e+000i 8.01e-001 3.76e+000-3.01e+000-2.25e+000i 8.01e-001 3.76e+000-5.00e-001 + 5.00e-001i 7.07e-001 7.07e-001
-5.00e-001-5.00e-001i 7.07e-001 7.07e-001 FLAG = 1 FLAGAPPROX = 1 23