Question

If the open-loop transfer function $G(s)H(s)$ has one pole in the right-half of the $s$-plane, for the closed-loop system to be stable, the Nyquist plot must encircle the $(-1, j0)$ point _______.

• A. once in the clockwise direction
• B. twice in the counter-clockwise direction
• C. once in the counter-clockwise direction
• D. zero times

Hint:

Using the Nyquist criterion formula $N_{ccw} = P - Z$: To stabilize an open-loop system that has 1 unstable pole ($P=1$), we require $Z=0$. This dictates $N_{ccw} = 1$, meaning exactly one counter-clockwise encirclement of the critical point $(-1, j0)$ is required.

Answer 1

The Correct Option is C

Concept: The Nyquist stability criterion establishes a relationship between the open-loop poles and the closed-loop stability using the principle of argument mapping. The core formula is: \[ N = Z - P \] where:
• $N$ = Number of counter-clockwise (CCW) encirclements of the critical point $(-1, j0)$ by the Nyquist plot.
• $Z$ = Number of zeros of the characteristic equation $1 + G(s)H(s)$ in the right-half plane (which correspond to unstable closed-loop poles).
• $P$ = Number of poles of the open-loop transfer function $G(s)H(s)$ located in the right-half plane.

Step 1: Identify given stability constraints.
The problem states the following conditions: 1. The open-loop transfer function has exactly one pole located in the right-half plane: \[ P = 1 \] 2. For the resulting closed-loop system to be stable, it must have zero unstable closed-loop poles, meaning no roots of the characteristic equation can reside in the right-half plane: \[ Z = 0 \]

Step 2: Calculate the required number of encirclements ($N$).
Substitute these parameters directly into the Nyquist relation: \[ N = Z - P \quad \Rightarrow \quad N = 0 - 1 = -1 \] According to the sign convention for this formulation: - A positive value ($N > 0$) indicates counter-clockwise encirclements. - A negative value ($N < 0$) indicates clockwise encirclements. *Alternative Convention:* If the formula is written as $N_{cw} = P - Z$, where $N_{cw}$ represents clockwise encirclements: \[ N_{cw} = 1 - 0 = 1 \] This means the plot must encircle the critical point $(-1, j0)$ exactly once in the counter-clockwise direction (or $-1$ times clockwise).