Stability from Nyquist plot G(s) Get complete Nyquist plot Obtain the # of encirclement of “-1” # (unstable poles of closed-loop) Z = # (unstable poles of open-loop) P + # encirclement N To have closed-loop stable: need Z = 0, i.e. N = –P

Here we are counting only poles with positive real part as “unstable poles” jw-axis poles are excluded Completing the NP when there are jw-axis poles in the open-loop TF G(s): If jwo is a non-repeated pole, NP sweeps 180 degrees in clock-wise direction as w goes from wo- to wo+. If jwo is a double pole, NP sweeps 360 degrees in clock-wise direction as w goes from wo- to wo+.

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)

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

System type, steady state tracking C(s) Gp(s)

Type 0: magnitude plot becomes flat as w  0 phase plot becomes 0 deg as w  0 Kv = 0, Ka = 0 Kp = flat magnitude height near w  0

Type 1: magnitude plot becomes -20 dB/dec as w  0 Asymptotic straight line Type 1: magnitude plot becomes -20 dB/dec as w  0 phase plot becomes -90 deg as w  0 Kp = ∞, Ka = 0 Kv = height of asymptotic line at w = 1 = w at which asymptotic line crosses 0 dB horizontal line

Type 2: magnitude plot becomes -40 dB/dec as w  0 Asymptotic straight line Ka Sqrt(Ka) Type 2: magnitude plot becomes -40 dB/dec as w  0 phase plot becomes -180 deg as w  0 Kp = ∞, Kv = ∞ Ka = height of asymptotic line at w = 1 = w2 at which asymptotic line crosses 0 dB horizontal line

Prototype 2nd order system frequency response For small zeta, resonance freq is about wn BW ranges from 0.5wn to 1.5wn For good z range, BW is 0.8 to 1.1 wn So take BW = wn z=0.1 0.2 0.3 No resonance for z <= 0.7 Mr=1dB for z=0.6 Mr=3dB for z=0.5 Mr=7dB for z=0.4

0.2 z=0.1 0.3 0.4 wgc In the range of good zeta, wgc is about 0.65 times to 0.8 times wn w/wn

In the range of good zeta, PM is about 100*z z=0.1 0.2 0.3 0.4 w/wn

Important relationships Prototype wn, open-loop wgc, closed-loop BW are all very close to each other When there is visible resonance peak, it is located near or just below wn, This happens when z <= 0.6 When z >= 0.7, no resonance z determines phase margin and Mp: z 0.4 0.5 0.6 0.7 PM 44 53 61 67 deg ≈100z Mp 25 16 10 5 %

Important relationships wgc determines wn and bandwidth As wgc ↑, ts, td, tr, tp, etc ↓ Low frequency gain determines steady state tracking: L.F. magnitude plot slope/(-20dB/dec) = type L.F. asymptotic line evaluated at w = 1: the value gives Kp, Kv, or Ka, depending on type High frequency gain determines noise immunity

Desired Bode plot shape