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Ch. 13 Frequency analysis

Ch. 13 Frequency analysis. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A. Force linear system with input x(t) = A sin t . Here is the output y(t):. Chapter 14. u(t) = A sin( ! t). u. y. As t ! 1 :

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Ch. 13 Frequency analysis

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  1. Ch. 13 Frequency analysis TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAA

  2. Force linear system with input x(t) = A sin t . Here is the output y(t): Chapter 14

  3. u(t) = A sin(!t) u y As t!1: y(t) = AR¢ A sin(!t + Á) General (VERY SIMPLE): AR = |G(j!)| Á = Å G(j!)

  4. 1 0.8 0.6 0.4 0.2 1 0 0.8 -0.2 -0.4 0.6 1 -0.6 0.4 0.8 -0.8 0.2 -1 0 2 4 6 8 10 12 14 16 18 20 0.6 0 -0.2 0.4 -0.4 0.2 -0.6 1 -0.8 0 0.8 -1 0 2 4 6 8 10 12 14 16 18 20 0.6 1 -0.2 0.4 0.8 0.2 0.6 -0.4 0.4 0 0.2 -0.2 -0.6 0 -0.4 -0.2 -0.8 -0.6 -0.4 -0.8 -0.6 -1 -1 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 -0.8 -1 0 2 4 6 8 10 12 14 16 18 20 SINUSOIDAL RESPONSE OF FIRST-ORDER SYSTEM k = 1, ¿ = 1 s y(t) = AR¢ sin(!t + Á) u(t) = sin(!t) Plots: VARY ! from 0.1 to 30 rad/s w=0.3; tau=1; t = linspace(0,20,1000); u = sin(w*t); AR = 1/sqrt((w*tau)^2+1) phi = - atan(w*tau), phig=phi*180/pi, dt=-phi/w y = AR*sin(w*t+phi); plot(t,y,t,u)

  5. Chapter 14

  6. Mathematics. Complex numbers, j2=-1 Im(G) G(j!)=R+jI I Á=Å G R Re(G) Chapter 14

  7. Polar form Multiply complex numbers: Multiply magnitudes and add phases Chapter 14

  8. Simple method to find sinusoidal response of system G(s) • Input signal to linear system: u = u0 sin(! t) • Steady-state (“persistent”, t!1) output signal: y = y0 sin(! t + Á) • What is AR = y0/u0 and Á? • Solution (extremely simple!) • Find system transfer function, G(s) • Let s=j! (imaginary number, j2=-1) and evaluate G(j!) = R + jI (complex number) • Then (“believe it or not!”) • AR = |G(j!)| (magnitude of the complex number) • Á = Å G(j!) (phase of the complex number) Im(G) G(j!)=R+jI I R Re(G) |G| Å G

  9. Example 13.1: 1. 2. Chapter 14 R I 3. Gain and phase shift of sinusoidal response! SIMPLER: Polar form; see board

  10. Chapter 14

  11. Chapter 14 -1 rad = -57o at !µ = 1 =-!µ Figure 14.4 Bode diagram for a time delay, e-qs.

  12. Peak goes to infinity when ³! 0 Phase increases for LHP zero Oops! Phase drops for RHP zero

  13. Bode Diagram 40 30 Magnitude (dB) 20 10 0 90 45 Phase (deg) 0 -45 -90 -4 -3 -2 -1 0 1 2 10 10 10 10 10 10 10 Frequency (rad/s) Electrical engineers (and Matlab) use decibel for gain • |G| (dB) = 20 log10|G| s=tf('s') g = 10*(100*s+1)/[(10*s+1)*(s+1)] bode(g) % gives AR in dB * *To change magnitude from dB to abs: Right click + properties + units Other way: |G| = 10|G|(dB)/20 GM=2 is same as GM = 6dB

  14. ASYMPTOTES Frequency response of term (Ts+1): set s=j!. Asymptotes: (j!T + 1) ¼ 1 for !T ¿ 1 (j!T+ 1) ¼j!Tfor !T À1 Note: Integrator (1/s) has one “asymptote”: =1/(j!)= -j!-1 at all frequencies Rule for asymptotic Bode-plot, L = k(Ts+1)/(¿s+1)….. : Start with low-frequency asymptote (a) If constant (L=k): Gain=k and Phase=0o (b) If integrator (L=k/s): Phase: -90o. Gain: Has slope -1 (on log-log plot) and gain=1 at !=k 2. Break frequencies:

  15. Chapter 14

  16. Figure 14.6 Bode plots of ideal parallel PID controller and series PID controller with derivative filter (α = 0.1). Ideal parallel: Series with Derivative Filter: Chapter 14

  17. CLOSED-LOOP STABILITY • L = gcgm = loop transfer function with negative feedback • Bode’s stability condition: |L(!180)|<1| • Limitations • Open-loop stable (L(s) stable) • Phase of L crosses -180o only once • More general: Nyquist stability condition: Locus of L(j!) should encircle the (-1)-point P times in the anti-clockwise direction (where P = no. of unstable poles in L). Stable plant (P=0): Closed-loop stable if L has no encirclements of -1 (=Bode’s stability condition)

  18. |L(j!)| = c 180 Chapter 14 Å L(j!)= c 180 Sigurd’s preferred notation in red  Time delay margin, ¢µ = PM[rad]/!c

  19. EXAMPLE

  20. Slope=-1 -2 -1 GM=1/0 = 1 -2 Slope Help lines -2 -1 !=0.5 !=0.01 !=0.05 -90o -90o PM=57o -180o -180o L(s): SIMC PI-control with ¿c=4 for g(s) = 1/(100s+1)(2s+1) !c = 0.19 rad/s !180 = 1 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

  21. Slope=-1 With added delay, e-µs with µ=2 No change in gain -2 -1 GM=1/0,4=2.5 -2 !=0.5 !=0.01 !=0.05 -90o -90o With added delay, e-µs with µ=2. Contribution to phase is: -5.7o at !=0.1/µ = 0.05 -57o at !=1/µ = 0.5 PM=35o = 0.61 rad -180o -180o !180 = 0.4 !c = 0.19

  22. Example. PI-control of integrating process with delay • g(s) =k’e-µs/s • Derive: Pu = 4µ and Ku = (¼/2)/(k’µ) • PI-controller, c(s) = Kc (1+1/¿Is) Task: Compare Bode-plot (L=gc), robustness and simulations (use k’=1, µ=1).

  23. Bode Diagram Bode Diagram Gm = 2.96 (at 1.49 rad/s) , Pm = 46.9 deg (at 0.515 rad/s) Gm = 1.87 (at 1.35 rad/s) , Pm = 24.9 deg (at 0.76 rad/s) 4 4 10 10 2 2 10 10 Magnitude (abs) Magnitude (abs) 0 0 10 10 -2 -2 10 10 0 0 -180 -180 Phase (deg) Phase (deg) -360 -360 -540 -540 -720 -720 -2 -1 0 1 -2 -1 0 1 10 10 10 10 10 10 10 10 Frequency (rad/s) SIMC-PI Ziegler-Nichols PI GM GM PM PM SIMC is a lot more robust: ¢µ = PM[rad]/!c ZN: ¢µ = 24.9*(3.14/180)/0.76 = 0.572s SIMC: ¢µ = 46.9*(3.14/180)/0.515 = 1.882s s=tf('s') g = exp(-s)/s Kc=0.707, taui=3.33 c = Kc*(1+1/(taui*s)) L1 = g*c figure(1), margin(L1) % Bode-plot with margins % To change magnitude from dB to abs: Right click + properties + units Kc=0.5, taui=8 c = Kc*(1+1/(taui*s)) L2 = g*c figure(2), margin(L2)

  24. 3 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 0 5 10 15 20 25 30 35 40 SIMC OUTPUT, y ZN Simulink file: tunepid4 s=tf('s') g = exp(-s)/s Kc=0.707, taui=3.33, taud=0 % ZN sim('tunepid4') plot(Tid,y,'red',Tid,u,'red') Kc=0.5, taui=8, taud=0 % SIMC sim('tunepid4') plot(Tid,y,'blue',Tid,u,'blue') INPUT, u t=0: setpoint change, t=20: input disturbance • Conclusion: Ziegler-Nichols (ZN) responds faster to the input disturbance, • but is much less robust. • ZN goes unstable if we increase delay from 1s to 1.57s. • SIMC goes unstable if we increase delay from 1s to 2.88s.

  25. 4 3 2 1 0 -1 -2 0 5 10 15 20 25 30 35 40 SIMC OUTPUT, y ZN INPUT, u t=0: setpoint change, t=20: input disturbance ZN is almost unstable when the delay is increased from 1s to 1.5s. SIMC does not change very much

  26. 1 10 0 10 -1 10 -2 10 -3 10 -2 -1 0 1 10 10 10 10 Closed-loop frequency response e SIMC: Ms=1.70 ZN: Ms = 2.93 SIMC ZN w = logspace(-2,1,1000); [mag1,phase]=bode(1/(1+L1),w); [mag2,phase]=bode(1/(1+L2),w); figure(1), loglog(w,mag1(:),'red',w,mag2(:),'blue',w,1,'-.') axis([0.01,10,0.001,10]) NO EFFECT BAD Control: GOOD

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