System type, steady state tracking, & Bode plot R(s) C(s) Gp(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 Kv = 0 Ka = 0 Low freq phase = 0o

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

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

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

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

Example continued

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,

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

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

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

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

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

System type on Nyquist plot Kp

System relative order

Examples System type = Relative order = System type = Relative order =

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)

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.

Example: Bode plot on next page.

Example: Bode plot on next page.

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

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

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