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