1. System type, steady state tracking, & Bode plot
R(s)
C(s)
Gp(s)
Y(s)
Type = N
At very low frequency:
gain plot slope = –20*N dB/dec.
phase plot value = –90*N 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 LF tangent line at ω = 1
= tangent line’s w0dB^2

6. 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

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

8. Example continued

9. 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,
If the input signal is a unit acceleration,

10. 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

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

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

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

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

15. System type on Nyquist plotKp

16. System relative order

17. Examples
System type =
Relative order =
System type =
Relative order =

18. In most cases, stability of this closed-loop
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)

21. 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.

22. Example:
Bode plot on next page.

24. Example:
Bode plot on next page.

26. Where does cross the –180° line Answer: __________ at ωpc, how much is
Closed-loop stability: __________

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

29. 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