Question

A unity feedback control system is characterized by the open-loop transfer function
$G\left(s\right)=\frac{10K(s+2)}{s^3+3s^2+10}.$
The Nyquist path and the corresponding Nyquist plot of $G\left(s\right)$ are shown in the figures below.


If $0<K<1$, then the number of poles of the closed-loop transfer function that lie in the right -half of the s-plane is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is C $2$

Given that, the open loop transfer function of a unity feedback system is,
$G\left(s\right)=\frac{10k(s+2)}{s^3+3s^2+10}$
From the given Nyquist plot , for $0<k<1$, the encirclement about the point $(-1+j0)$ is,
$N=0$
$N=P-Z$
P= number ofopen loop poles in the right half od s-plane
Z= number of closed loop poles in the right half of s-plane
To determine the value of $P$:
Applying R-H criteria to $G\left(s\right)$,
$G\left(s\right)=\frac{10k(s+2)}{s^3+3s^2+10}$
$\begin{array}{ccc}{s}^3& 1& 0\\ s^2& 3& 10\\ s^1& -10/3& 0\\ s^0& 10& 0\end{array}$
Two sign changes are there in the first column if the RH table. So, two open loop poles arethere in right half of s-plane.
So, $P=2$
$N=P-Z$
$0=2-Z$
$Z=2$
$Z=2$ indicates, there are two closed loop poles in right half of s-plane.