] [ 200. ] 3 [ 10 4 s. [ ] s + 10 [ P = s [ 10 8 ] 3. s s (s 1)(s 2) series compensator ] 2. s command prefilter [ 0.


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1 EEE480 Exam 2, Spring 204 A.A. Rodriguez Rules: Calculators permitted, One 8.5 sheet, closed notes/books, open minds GWC 352, Problem (Analysis of a Feedback System) Consider the feedback system defined by the plant series compensator command prefilter and sensor dynamics [ P = (s )(s 2) ] [ 200 s ] 3 [ 0 4 s s ] [ ] s [ [ ] [ ] (s ) K = 7.2 s s W = H = [ ] 2 s [ 0 8 ] 3 s s s (a) What is the approximate closed loop transfer function T ry? What are the dominant closed loop poles, the associated time constant τ, settling time t s, damping factor ζ, overshoot to a step command M p, time to peak for a step command t p, and rise time t r? (b) Approximate the output y when r = 0 (t). (c) Approximate the steady state output y ss when r = 0 + sin(0.0t + 30 ) and d i = cos(0.0t 60 ). (d) Sketch and interpret magnitude plots for T diy, T doy. (Note: Neglect all high frequency terms to do this.) (e) Determine r such that y ss = + 0. sin(0.0t 20 ) when d i = 2 cos(0.0t 65 ). Is this r unique? Problem 2 (Bode Magnitude and Phase) Sketch Bode magnitude and phase plots for the open loop transfer function L = P KH in Problem. Identify the upward gain margin GM, downward gain margin GM, phase margin P M, and associated frequencies on your plots. Compute all margins (including delay margin DM) and the associated frequencies. Problem 3 (Root Locus) (a) Construct a root locus for the feedback system defined in Problem. Determine the number of asymptotes, the angle of each asymptote, the center of gravity, and all imaginary crossovers. Carefully label all important features on your plot. Provide a stability summary. Problem 4 (Nyquist Plot, Sensitivity Bounds) (a) Sketch a Nyquist plot for the open loop transfer function L = P KH in Problem. Identify the upward gain margin GM, downward gain margin GM, and phase margin P M on your plot. (b) Provide a stability summary. (c) Given the downward gain margin GM, phase margin P M, and upward gain margin GM, provide lower bounds for the peak sensitivity and complementary sensitivity. For each, give sufficient conditions when the lower bound is greater than ; greater than 2. (d) Given bounds on the peak sensitivity and complementary sensitivity, provide bounds on the downward gain margin GM, phase margin P M, and upward gain margin GM. Problem 5 (Control System Design) (a) Pole Placement. Suppose that we have a plant P = P o = (s + 4)(20 s) (s + 2) 2 [ ] s [ ] [ s s ] 3 ] 4 [ 0 0 ] s s Design a feedback control system such that the closed loop system () is stable, (2) exhibits constant steady state error to parabolic input disturbances d i, (3) exhibits a settling time to step commands r of t s 0, (4) overshoot close to 0 when a step command is issued. Support your design with a (simple) root locus
2 2 plot. (b) Bandwidth and Robustness. Now suppose that P = P o [ 5ωg s+5ω g ] 3 e s ( π 0ωg ). Design a feedback control system such that the closed loop system () is stable, (2) exhibits zero steady state error to input disturbances d i = 5t 2, (3) unity gain crossover of ω g 2 rad/sec, and (4) phase margin P M 60. Your controller must suitably address control action as well as overshoot in the system output. Discuss any inherent problems/flaws with your design methodology. Problem 6 (Control System Design: Pole [ Placement) ] [ ] [ (a) Consider the following LTI plant P = (s 2)(s 4) 000 s s s+000 ]. Design a feedback control system such that the closed loop system () is stable, (2) exhibits zero steady state error to step input disturbances, (3) has dominant poles at 5 ± j5, 0. Address control action as well as overshoot in the system output. Hint: Pick a controller structure with suitable degrees of freedom. Solve for all controller coefficients. (b) Repeat for dominant poles at: 0 ± j0, 5. PLANT  P = gp / ( s  pp) (s  pp2) PID CONTROLLER  K = gk ( s^2 + bk s + bko ) / s Actual Characteristic Equation: s( s  pp) (s  pp2)+ gp gk ( s^2 + bk s + bko ) = s^3 + ( gp gk  ( pp + pp2 ) ) s^2 + ( gp gk bk + pp pp2 )s + gp gk bko Desired Characteristic Equation: (s^2 + 2 sigmao s + sigmao^2 + omegao^2 ) (s + c) = s^3 + (c + 2sigmao) s^2 + ( 2sigmao c + sigmao^2 + omegao^2 ) s + (sigmao^2 + omegao^2) c (s  clp) ( s  clp2) (s  clp3) = s^3  (clp + clp2 + clp3 ) s^2 + ( clp clp2 + (clp + clp2)clp3 ) s  clp clp2 clp3 Solution: gk = (pp + pp2 + c + 2 sigmao) / gp = ( (clp + clp2 + clp3 ) + ( pp + pp2 )) / gp bk = ( ( 2sigmao c + sigmao^2 + omegao^2 )  pp pp2 ) / gp gk = ( ( clp clp2 + (clp + clp2)clp3 )  pp pp2 ) / gp gk bko = (sigmao^2 + omegao^2) c / gp gk =  clp clp2 clp3 / gp gk PLANT gp = plant high frequency gain pp = plant pole pp2 = 2 plant pole 2 plant = tf (gp, [ (pp + pp2) pp*pp2 ]) Form Plant DESIRED CLPs clp = CLP
3 3 clp2 = 2 +j*2 CLP 2 clp3 = 2 j*2 CLP 3 above produces the following controller: (s ) (s ) (000)^ s (s+000)^2 clp = 2 CLP clp2 =  +j* CLP 2 clp3 =  j* CLP 3 PID CONTROLLER gk = ( (clp + clp2 + clp3 ) + ( pp + pp2 ) ) / gp controller h bk = real( ( ( clp * clp2 + (clp + clp2)*clp3 )  pp * pp2 ) / ( gp*gk ) ) controller c bko = ( clp * clp2 * clp3 ) / ( gp*gk ) controller c hfpolem = 000 controller h controller = tf(hfpolem*hfpolem, [ 2*hfpolem hfpolem*hfpolem]) Form control controller = series(controller, tf (gk*[ bk bko], [ 0 ]) ) Form PID Con zpk(controller) tzero(controller) controller = tf(7.2*[ 2* *0.4724], [ 0]) loop = series(plant,controller) zpk(loop) allmargin(loop) sen = /( + loop); eig(sen) damp(eig(sen)) figure(000) bode(loop) figure(00) margin(loop) x = 0:e3:0; x = :e5:2.2289; y = 2*atan(x/0.4724)*80/pi (80  atan(x)*80/pi)  (80  atan(x/2)*80/pi); figure(200) plot(x,y,x,80, r ) w = logspace(2,2,000); LOOP [loop_mag, loop_phase] = bode(loop, w); figure(300)
4 4 semilogx(w,20*log0(loop_mag(,:)), w, 20, r, w, 0, r, w, title( Open Loop Magnitude ) 20, r ) figure(400) semilogx(w,loop_phase(,:), w, 80, r ) title( Open Loop Phase ) ylabel( degrees ) SENSITIVITY [sen_mag, sen_phase] = bode(sen, w); figure(400) semilogx(w,20*log0(sen_mag(,:)), w, 20, r ) title( Sensitivity ) COMPLEMENTARY SENSITIVITY compsen = feedback(loop,) zpk(compsen) damp(eig(compsen)) [compsen_mag, compsen_phase] = bode(compsen, w); figure(500) semilogx(w,20*log0(compsen_mag(,:)), w, 20, r ) title( Complementary Sensitivity ) COMMAND PREFILTER prefilter = tf([0.472*0.472], [ 2* *0.472]) zpk(prefilter) REFERENCE TO OUTPUT tr2y = series(prefilter, compsen); [tr2y_mag, tr2y_phase] = bode(tr2y, w); figure(500) semilogx(w,20*log0(tr2y_mag(,:)), w, 20, r )
5 5 title( Reference to Output ) RESPONSE TO STEP REFERENCE t = 0:.0:25; figure(500) step(compsen,t) title( Output Response to Step Reference (Unfiltered) ) xlabel( time (sec) ) figure(600) step(tr2y,t) title( Output Response to Step Reference (Filtered) ) xlabel( time (sec) ) RESULTING DATA >> zpk(plant) Zero/pole/gain: (s2) (s) >> zpk(controller) Zero/pole/gain: 7.2 (s )^ s >> zpk(loop) Zero/pole/gain: 7.2 (s )^ s (s2) (s) >> allmargin(loop) ans = GainMargin: GMFrequency: PhaseMargin: PMFrequency:
6 6 DelayMargin: DMFrequency: Stable: >> eig(sen) damp(eig(sen)) ans = i i Eigenvalue Damping Freq. (rad/s) 2.00e e+000i 7.07e e e e+000i 7.07e e e00.00e e00
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