Consider the control system block diagram given in Figure (a). The loop transfer function $G(s)H(s)$ does not have any pole on the $j\omega$-axis. The counterclockwise contour with infinite radius, as shown in Figure (b), encircles two poles of $G(s)H(s)$. Choose the correct statement from the following options for closed loop stability of the system.
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The Nyquist stability criterion is crucial for determining the stability of closed-loop systems based on the open-loop transfer function. Remember the relationship $N = Z - P$, where $N$ is the number of counterclockwise encirclements of $-1 + j0$ by $G(s)H(s)$, $Z$ is the number of closed-loop poles in the RHP, and $P$ is the number of open-loop poles in the RHP. For stability, $Z$ must be 0.
The locus of $G(s)H(s)$ should encircle the origin twice in the counterclockwise direction
The locus of $1 + G(s)H(s)$ should encircle the origin twice in the clockwise direction
The locus of $G(s)H(s)$ should encircle the $-1 + j0$ point twice in the counterclockwise direction
The locus of $1 + G(s)H(s)$ should encircle the $-1 + j0$ point twice in the clockwise direction
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The Correct Option isB
Solution and Explanation
Step 1: Identify the number of open-loop poles in the RHP.
The problem states that the counterclockwise Nyquist contour encircles two poles of \(G(s)H(s)\). Since the Nyquist contour encloses the right-half \(s\)-plane (RHP), these two poles must be in the RHP. Thus, \(P = 2\). Step 2: Apply the Nyquist Stability Criterion for stability.
For the closed-loop system to be stable, the number of closed-loop poles in the RHP (\(Z\)) must be zero. The Nyquist Stability Criterion relates \(Z\), the number of open-loop poles in the RHP (\(P\)), and the number of counterclockwise encirclements of the \(-1 + j0\) point by the Nyquist plot of \(G(s)H(s)\) (\(N\)) as \(N = Z - P\).
For stability (\(Z = 0\)) and with \(P = 2\), we have \(N = 0 - 2 = -2\). A negative value of \(N\) indicates clockwise encirclements. Therefore, the Nyquist plot of \(G(s)H(s)\) must encircle the \(-1 + j0\) point twice in the clockwise direction for the closed-loop system to be stable. Step 3: Relate encirclements of \(-1 + j0\) by \(G(s)H(s)\) to encirclements of the origin by \(1 + G(s)H(s)\).
Consider the function \(F(s) = 1 + G(s)H(s)\). The Nyquist plot of \(F(s)\) is simply the Nyquist plot of \(G(s)H(s)\) shifted by \(+1\) along the real axis. Therefore, the number of encirclements of the origin by the Nyquist plot of \(1 + G(s)H(s)\) is the same as the number of encirclements of the \(-1 + j0\) point by the Nyquist plot of \(G(s)H(s)\), both in the same direction.
Since we need two clockwise encirclements of \(-1 + j0\) by \(G(s)H(s)\) for stability, we also need two clockwise encirclements of the origin by \(1 + G(s)H(s)\).
