System type steady state tracking Bode plot Rs

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 d. B/dec. phase plot value = – 90 N deg

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

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

Back to general theory N = 2, type = 2 Bode gain plot has – 40 d. B/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 = ws 0 d. B^2

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

Example Ka ws 0 d. B=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, ess would be = If the input signal is a unit acceleration, ess would be =

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

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

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

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

Type 2 system, N=2 Ka=lims 0 s 2 G(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 =

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

If never cross 0 d. B line (always below 0 d. B 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 d. B several times, then there are several PM’s.

Example: Bode plot on next page.

Example: Bode plot on next page.

1. Where does cross the – 180° line Answer: _____ at ωpc, how much is 2. Closed-loop stability: _____

1. crosses 0 d. B at _____ at this freq, 2. Does cross – 180° line? ____ 3. 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