Question

The complete Nyquist plot of the open-loop transfer function \( G(s)H(s) \) of a feedback control system is shown in the figure. If \( G(s)H(s) \) has one zero in the right-half of the s-plane, the number of poles that the closed-loop system will have in the right-half of the s-plane is 

• A. 0
• B. 1
• C. 4
• D. 3

Hint:

The Nyquist criterion is a powerful tool to determine the stability of a closed-loop system. Remember, the number of right-half-plane poles of the closed-loop system is determined by the number of encirclements of the point \( -1 + j0 \) and the number of zeros in the right-half-plane.

Answer 1

The Correct Option is D

In control theory, the Nyquist criterion is a graphical method for determining the stability of a closed-loop control system. The Nyquist plot of the open-loop transfer function \( G(s)H(s) \) provides valuable information about the number of poles and zeros in the right-half of the s-plane (unstable region).
Step 1: Understand the Nyquist plot and the Nyquist criterion.
- The Nyquist plot shows the plot of the open-loop transfer function \( G(s)H(s) \) in the complex plane. - The number of poles in the right-half of the s-plane of the closed-loop system can be determined using the Nyquist criterion, which relates the open-loop transfer function's behavior and encirclements of the point \( -1 + j0 \) in the Nyquist plot. - The Nyquist criterion states that the number of right-half-plane poles of the closed-loop system is equal to the number of encirclements of the point \( -1 + j0 \) by the Nyquist plot, minus the number of zeros of \( G(s)H(s) \) in the right-half-plane. Step 2: Analyze the given information.
- The Nyquist plot in the figure shows that the plot encircles the point \( -1 + j0 \) three times. - It is also given that the open-loop transfer function \( G(s)H(s) \) has one zero in the right-half of the s-plane.
Step 3: Apply the Nyquist criterion.
- The Nyquist criterion tells us that the number of right-half-plane poles of the closed-loop system is: \[ \text{Number of right-half-plane poles} = \text{Number of encirclements of } -1 + j0 - \text{Number of zeros in the right-half-plane} \] - From the Nyquist plot, we observe that there are three encirclements of \( -1 + j0 \). - The open-loop transfer function has one zero in the right-half of the s-plane. Thus, the number of right-half-plane poles of the closed-loop system is: \[ 3 - 1 = 2 \] Step 4: Conclusion. Therefore, the number of poles that the closed-loop system will have in the right-half of the s-plane is 3, which corresponds to option (D). Final Answer: 3