Question

The characteristic equation of a closed-loop system is s(s+1) (s+3) + K(s+2) = 0, K>0. Which of the following statements is true?

• A. Its roots are always real.
• B. It cannot have a breakway point in the range -1 <Re[s] < 0.
• C. Two of its roots tend to infinity along the asymptotes Re[s] = -1.
• D. It may have complex roots in the right half plane.

Answer 1

The correct option is C Two of its roots tend to infinity along the asymptotes Re[s] = -1.

Characteristic equation,

s(s+1)(s+3) + K(s+2) = 0

$1+\frac{K(s+2)}{s(s+1)(s+3)}=0$

Comparing with 1 + G(s) H(s) = 0

G(s) H(s) = open-loop transfer function

$\left(OLTF\right)=\frac{K(s+2)}{s(s+1)(s+2)}$

Number of zeros = Z = 1 zero at -2

Number of poles = P = 3 poles at 0, -1 and -3

Number of branches terminating at infinity

= P - Z = 3 - 1 = 2

Angle of asymptotes

$=\frac{(2k+1){180}^0}{P-Z}=\frac{(2k+1){180}^0}{2}$

$=(2k+1)\times {90}^0={90}^{0}and{270}^0$

$Centroid=\frac{\sum poles-\sum zeros}{P-Z}$

$=\frac{0-1-3-(-2)}{2}=-1$



Breakway points lies in the range $-1<Re\left[s\right]<0$ and two branches terminates at infinity along the asymptotes Re(s) = -1.