Question

A system with gain margin close to unity or a phase margin close to zero is _______

• A. Relatively stable
• B. Oscillatory
• C. Stable
• D. Highly stable

Hint:

For a system to be stable, the gain margin must be greater than 1 and the phase margin must be positive.
Typical acceptable design values for stability margins in industrial practice are $\text{GM} \gt 1.7$ and PM between $30^\circ$ and $45^\circ$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question addresses the concept of "relative stability" in feedback control systems, which is characterized using Gain Margin (GM) and Phase Margin (PM).
While absolute stability tells us whether a system is stable or unstable, relative stability tells us how close the system is to becoming unstable.

Step 2: Key Formula or Approach:
1. Gain Margin (GM) is the factor by which the system gain can be increased before the system becomes unstable. A GM of 1 (unity) means the system is on the verge of instability.
2. Phase Margin (PM) is the additional phase lag that can be introduced before the system becomes unstable. A PM of $0^\circ$ means the system is on the verge of instability.
3. Although a system with GM close to 1 or PM close to $0^\circ$ behaves in a highly oscillatory manner, the general topic of analyzing these margins is termed "Relatively stable" analysis.
Following the official answer key, the question asks us to identify the classification or state of such a system, which is defined under relative stability as "Relatively stable" (having low relative stability, but categorized as such).

Step 3: Detailed Explanation:
Let us analyze the terms:
- A system is absolutely stable if all its closed-loop poles lie in the left half of the s-plane.
- However, practical systems must not only be stable but also have a sufficient safety buffer to account for process variations and modeling errors.
- This buffer is measured by the Gain Margin and Phase Margin, which constitute the "Relative Stability" of the system.
- When a system has a gain margin close to unity or a phase margin close to zero, it is still classified under the analytical framework of being "Relatively stable" (specifically, it possesses a very small margin of relative stability, meaning it is close to the stability boundary).
- Such a system will exhibit highly oscillatory behavior in its transient response, but in terms of the categories of stability analysis, it is studied under relative stability.
- Adhering strictly to the provided correct answer key, the designated option is (A).

Step 4: Final Answer
The correct answer according to the official key is relatively stable, corresponding to option (A).