Consider a closed-loop control system with unity negative feedback and 𝐾𝐺(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 ___________.
The point of interest is -1/K = -0.5 on the real axis. Counting the encirclements:
- Outer Loop: Clockwise, enclosing -0.5. (Count = +1)
- Left Inner Loop: Also clockwise, enclosing -0.5. (Count = +1)
- Right Inner Loop: Counter-clockwise, but -0.5 lies outside. (Count = 0)
Total Encirclements (N) = 2
Using the stability relation:
Hence, the closed-loop transfer function has 2 poles in the closed right-half plane.
Correct Option: (C) 2
Nyquist Stability Analysis
1. The Fundamental Principle
The number of closed-loop poles in the right-half plane (RHP), denoted as Z, is determined by the Nyquist Stability Criterion:
- N: Net number of clockwise (CW) encirclements of the critical point.
- P: Number of open-loop poles of G(s) in the RHP.
2. Identify Given Parameters
- Open-loop poles (P): The problem states G(s) has no poles in the RHP, so P = 0.
- Gain (K): Given as K = 2.
- Critical Point: Since the plot is for G(s), we look for encirclements of -1/K = -1/2 = -0.5.
3. Graphical Analysis of Encirclements (N)
The point of interest is -0.5 on the real axis (located between -0.4 and -0.8). Let's count the encirclements:
- Outer Loop: Tracing the largest circle (top arrow right, bottom arrow left), it moves Clockwise. Point -0.5 is inside. (Count = +1)
- Left Inner Lobe (between -0.4 and -0.8): Look at the small arrows. The top arrow points right and the bottom points left. This is also Clockwise. Point -0.5 is inside. (Count = +1)
- Right Inner Lobe (between 0 and -0.4): The top arrow points left and bottom points right. This is Counter-Clockwise, but point -0.5 is outside this loop. (Count = 0)
Total Clockwise Encirclements (N) = 1 + 1 = 2
4. Final Calculation
Plugging the values into the stability formula:
The number of poles of the closed-loop transfer function in the closed right-half of the complex plane is 2.