美国麻省理工大学：《航空系统的估计与控制》教学资源（讲义，英文版）Handout 2：Gain and Phase margins

Handout 2: Gain and Phase margins Eric feron Feb6.2004 Nyquist plots and Cauchys principle 土S Let H(s be a transfer function. eg H(s= (s+1)(s+3 Evaluate H on a contour in the s-plane.(your plots here)

Handout 2: Gain and Phase margins Eric Feron Feb 6, 2004 Nyquist plots and Cauchy’s principle s2 + s + 1 Let H(s) be a transfer function. eg H(s) = (s + 1)(s + 3) Evaluate H on a contour in the s­plane. (your plots here) 1

s2+s+1 (s+3)(s-3) Evaluate H on another contour of the s-plane(your plots here) 2

s2 + s + 1 H = (s + 3)(s − 3) Evaluate H on another contour of the s­plane (your plots here) 2

Cauchy's Principle Control application: Given KG(s, we encircle the entire to get the contour evaluation of Closed-loop roots are poles of They are zeros of If there are no RHPs, then 1+ KG encirclement of 0 means With no rHP poles, KG encirclement of-1 means 3

Cauchy’s Principle: Control application: Given KG(s), we encircle the entire to get the contour evaluation of Closed­loop roots are poles of They are zeros of If there are no RHPs, then 1 + KG encirclement of 0 means With no RHP poles, KG encirclement of ­1 means 3

With right half plane open-loop poles a clockwise contour enclosing a zero of 1+ KG(s) will result in A clockwise contour enclosing a pole of 1+KG(s will result in Nyquist plot rules 1. Plot KG(s for s=-joo to +joo 2. Count number of 3. Determine number of Nunber of unstable closed-loop roots is

With right half plane open­loop poles A clockwise contour enclosing a zero of 1 + KG(s) will result in A clockwise contour enclosing a pole of 1 + KG(s) will result in Nyquist plot rules 1. Plot KG(s) for s = −j∞ to +j∞ 2. Count number of 3. Determine number of 4. Nunber of unstable closed­loop roots is 4

Example: G(s) Bode plot 5

1 Example: G(s) = s2 + 3s + 1 Bode plot Nyquist plot 5

Example: G(s) Bode plot Nyquist plot

1 Example: G(s) = s(s + 1)2 Bode plot Nyquist plot 6

Gain and Phase margins Nyquist plot for G(s) Gain Margin is Phase margin is 7

Gain and Phase margins Nyquist plot for G(s). Gain Margin is Phase Margin is 7

G(s) 1 (s+1)2 8

1 G(s) = s2 + 3s + 1 1 G(s) = 2 s(s + 1) 8