Consider a negative unity feedback system with forward path transfer function$G\left(s\right)=\frac{K}{\left(s+a\right)\left(s-b\right)\left(s+c\right)}$ , where K, a, b, c are positive real numbers. For a Nyquist path enclosing the entire imaginary axis and right half of the s-plane is the clockwise direction, the Nyquist plot of (1 + G(s)), encircles the origin (1 + G(s)) –plane once in the clockwise direction and never passes through this origin for a certain value of K. then, the number of poles $\frac{G\left(s\right)}{1+G\left(s\right)}$of lying in the open right half of the s-plane is ______.

The correct answer is 2.

Step 1: Understand the Nyquist Stability Criterion

According to the Nyquist stability criterion, the relation between clockwise encirclements of the origin by the plot of $1+G\left(s\right)$, open-loop poles in the right half of the s-plane ($P$), and closed-loop poles in the right half of the s-plane ($Z$) is given by:

$N=P-Z$

Where:

• $N$ = Number of counter-clockwise encirclements of the origin by $1+G\left(s\right)$ (or of the critical point $-1+j0$ by $G\left(s\right)$).

• $P$ = Number of open-loop poles lying in the open right-half of the s-plane (RHS).

• $Z$ = Number of closed-loop poles (poles of $\frac{G\left(s\right)}{1+G\left(s\right)}$) lying in the open right-half of the s-plane (RHS).

Step 2: Identify the Open-Loop Poles ($P$)

The forward path transfer function is given as:

$G\left(s\right)=\frac{K}{(s+a)(s-b)(s+c)}$

Since $a,b,c$ are positive real numbers, the poles of $G\left(s\right)$ are at:

$s=-a$ (Left-half of s-plane)

$s=+b$ (Right-half of s-plane)

$s=-c$ (Left-half of s-plane)

Therefore, there is only 1 open-loop pole in the open right-half of the s-plane:

$P=1$

Step 3: Determine the Encirclements ($N$)

The problem states that the plot of $1+G\left(s\right)$ encircles the origin once in the clockwise direction.

Since $N$ is defined as the number of counter-clockwise encirclements:

$N=-1$

Step 4: Calculate the Closed-Loop Poles in RHS ($Z$)

Rearranging the Nyquist equation to solve for $Z$:

$Z=P-N$

Substitute the values of $P=1$ and $N=-1$:

$Z=1-(-1)=2$

Thus, the number of closed-loop poles lying in the open right-half of the s-plane is 2.