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System type, steady state tracking, & Bode plot

System type, steady state tracking, & Bode plot. C(s). G p (s). R(s). Y(s). Type = N. At very low frequency: gain plot slope = –20N dB/dec. phase plot value = –90N deg. Type 0: gain plot flat at very low frequency phase plot approached 0 deg. K v = 0 K a = 0

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System type, steady state tracking, & Bode plot

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  1. System type, steady state tracking, & Bode plot C(s) Gp(s) R(s) Y(s) Type = N At very low frequency: gain plot slope = –20N dB/dec. phase plot value = –90N deg

  2. Type 0: gain plot flat at very low frequency phase plot approached 0 deg Kv = 0 Ka = 0 Low freq phase = 0o

  3. Type 1: gain plot -20dB/dec at very low frequency phase plot approached 90 deg Low frequency tangent line Kp = ∞ Ka = 0 Low freq phase = -90o =Kv

  4. Back to general theory N = 2, type = 2 Bode gain plot has –40 dB/dec slope at low freq. Bode phase plot becomes flat at –180° at low freq. Kp= DC gain → ∞ Kv = ∞ also Ka = value of straight line at ω = 1 = ws0dB^2

  5. Type 1: gain plot -40dB/dec at very low frequency phase plot approached 180 deg Low frequency tangent line Kp = ∞ Kv = ∞ Low freq phase = -180o

  6. Example Ka ws0dB=Sqrt(Ka) How should the phase plot look like?

  7. Example continued

  8. Example continued Suppose the closed-loop system is stable: If the input signal is a step, ess would be = If the input signal is a ramp, ess would be = If the input signal is a unit acceleration, ess would be =

  9. System type, steady state tracking, & Bode plot At very low frequency: gain plot slope = –20N dB/dec. phase plot value = –90N deg If LF gain is flat, N=0, Kp = DC gain, Kv=Ka=0 If LF gain is -20dB/dec, N=1, Kp=inf, Kv=wLFg_tan_c , Ka=0 If LF gain is -40dB/dec, N=2, Kp=Kv=inf, Ka=(wLFg_tan_c)2

  10. System type, steady state tracking, & Nyquist plot C(s) Gp(s) As ω → 0

  11. Type 0 system, N=0 Kp=lims0 G(s) =G(0)=K Kp w0+ G(jw)

  12. Type 1 system, N=1 Kv=lims0sG(s) cannot be determined easily from Nyquist plot winfinity w0+ G(jw)  -j∞

  13. Type 2 system, N=2 Ka=lims0 s2G(s) cannot be determined easily from Nyquist plot winfinity w0+ G(jw)  -∞

  14. System type on Nyquist plot Kp

  15. System relative order

  16. Examples System type = Relative order = System type = Relative order =

  17. Margins on Bode plots In most cases, stability of this closed-loop can be determined from the Bode plot of G: • Phase margin > 0 • Gain margin > 0 G(s)

  18. If never cross 0 dB line (always below 0 dB line), then PM = ∞. If never cross –180° line (always above –180°), then GM = ∞. If cross –180° several times, then there are several GM’s. If cross 0 dB several times, then there are several PM’s.

  19. Example: Bode plot on next page.

  20. Example: Bode plot on next page.

  21. Where does cross the –180° lineAnswer: __________at ωpc, how much is • Closed-loop stability: __________

  22. crosses 0 dB at __________at this freq, • Does cross –180° line? ________ • Closed-loop stability: __________

  23. Margins on Nyquist plot Suppose: • Draw Nyquist plot G(jω) & unit circle • They intersect at point A • Nyquist plot cross neg. real axis at –k

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