Control - 3&4 - Feedback & Stability Analysis

Question 1

Closed loop transfer function derivation

Answer 1

Question 2

Routh Hurwitz method

Answer 2

• R-H rule 1:
  
  • If any a_i is negative, the characteristic equation has root(s) in the right half of the s-plane: Therefore the closed loop is unstable if any a_i is negative.
• R-H rule 2:
  
  • If any a_i is zero (i.e. missing), the characteristic equation either has root(s) in the right half of the s-plane or on the imaginary axis: Therefore the closed loop is unstable or marginal if any a_i is zero.
• R-H rule 3:
  
  • If all a_i are positive, the characteristic equation may or may not have root(s) in the right half of the s-plane: Therefore more investigation is needed if the ai are all positive.
• Further investigation involves the use of the Routh array:
  
  • fill up first rows with constants
    *

Question 3

Cauchy’s principle of the argument

Answer 3

Question 4

Nyquist Stability Criterion: Effect of poles and zeros on rotation

Answer 4

Circling a pole leads to counter clockwise rotation in the w-plane.

Circling a zero leads to clockwise rotation.

Similarly, if you look at cauchy’s principle of the argument if N is negative that means that the number of poles (P) is larger than zeros (Z).

Vice versa, if positive that means that there are more zeros than poles.

Question 5

If N=1 for a plot, what can we deduce for the contour plot?

Similarly, for N=-1?

Answer 5

That there is one more zero than pole.

That there is one more pole than zero.

Remember N = Z - P

or Z = N + P