Consider a closed-loop control system with unity negative feedback and $K G(s)$ in the forward path, where the gain $K = 2$. The complete Nyquist plot of the transfer function $G(s)$ is shown in the figure. Note that the Nyquist contour has been chosen to have the clockwise sense. Assume $G(s)$ has no poles on the closed right-half of the complex plane. The number of poles of the closed-loop transfer function in the closed right-half of the complex plane is ____________.
Show Hint
When $G(s)$ has no right-half-plane poles, the stability of the closed-loop system depends only on whether the Nyquist plot encircles $-1$. No encirclement means a stable system.
The Nyquist stability criterion states:
\[
N = Z - P,
\]
where
$N =$ number of clockwise encirclements of the point $-1 + 0j$,
$P =$ number of poles of $G(s)$ in the right-half plane,
$Z =$ number of closed-loop poles in the right-half plane. Step 1: Determine $P$.
The question states that $G(s)$ has no poles in the closed right-half plane. Thus,
\[
P = 0.
\]
Step 2: Determine $N$ from the Nyquist plot.
The Nyquist plot (clockwise sense) is shown. Gain $K = 2$ shifts the critical point to $-1$.
Inspecting the plot, the Nyquist curve does not encircle the point $-1$.
Thus,
\[
N = 0.
\]
Step 3: Solve for $Z$.
Using the Nyquist equation:
\[
N = Z - P \quad \Rightarrow \quad 0 = Z - 0 \Rightarrow Z = 0.
\]
Thus, there are no closed-loop poles in the right-half plane, meaning the system is stable. Final Answer: 0
