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Ch. 13 Frequency analysis

This document explores the frequency response of linear systems through sinusoidal input signals and their output responses. It discusses the concepts of amplitude ratio (AR) and phase shift (φ) when analyzing system transfer functions using complex numbers in the Laplace domain. Essential formulas are provided for calculating gain, phase margins, and using Bode plots for stability analysis. The document also covers the implications of time delays and integrators on system response and describes techniques to derive Bode plots in control systems, with specific examples for PI controllers.

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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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