Marginally stable systems have closed loop transfer functions with only imaginary axis poles of multiplicity:

Explanation:

Marginally Stable Systems

Definition: A marginally stable system is a type of dynamic system where the output oscillates indefinitely without converging to a steady-state value or diverging to infinity when subjected to certain initial conditions. In control systems, marginal stability occurs when the poles of the closed-loop transfer function lie on the imaginary axis of the s-plane but are not repeated (i.e., their multiplicity is 1).

Stability Criteria:

To understand the stability of a system, the location of the poles of its closed-loop transfer function is analyzed in the s-plane:

• If all poles are in the left half of the s-plane (real parts are negative), the system is stable.
• If any pole lies in the right half of the s-plane (real part is positive), the system is unstable.
• If poles lie on the imaginary axis (real part is zero) without repetition, the system is marginally stable.
• If poles lie on the imaginary axis with repetition (multiplicity greater than 1), the system is unstable.

Correct Option Analysis:

The correct option is:

Option 1: Marginally stable systems have closed-loop transfer functions with only imaginary axis poles of multiplicity 1.

To confirm this, let us analyze the characteristics of marginally stable systems:

1. For a system to be marginally stable, the poles of the closed-loop transfer function must lie entirely on the imaginary axis.
2. Each pole on the imaginary axis should have a multiplicity of exactly 1. This means that the poles are simple (non-repeated).
3. If any pole has a multiplicity greater than 1 on the imaginary axis, the system becomes unstable, as the response will grow indefinitely.

For example, consider a system with a transfer function:

H(s) = 1 / (s² + 1)

The poles of this system are at s = ±j. Since these poles lie on the imaginary axis and have a multiplicity of 1, the system is marginally stable. Any input to this system will result in a sustained oscillatory output without divergence or convergence.

Therefore, the correct answer is Option 1, as marginal stability requires the poles to be on the imaginary axis with a multiplicity of 1.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: Marginally stable systems have closed-loop transfer functions with only imaginary axis poles of multiplicity 3.

This option is incorrect. If a pole on the imaginary axis has a multiplicity of 3, the system becomes unstable. Higher multiplicities lead to instability because the response due to these poles grows indefinitely with time, which violates the condition of marginal stability.

Option 3: Marginally stable systems have closed-loop transfer functions with only imaginary axis poles of multiplicity 4.

This option is also incorrect. Similar to Option 2, a pole on the imaginary axis with a multiplicity of 4 would cause instability. The higher the multiplicity, the more pronounced the instability, as the system response becomes unbounded over time.

Option 4: Marginally stable systems have closed-loop transfer functions with only imaginary axis poles of multiplicity 2.

This option is incorrect. A pole on the imaginary axis with a multiplicity of 2 results in an unstable system. While the initial response may resemble marginal stability, the repeated pole causes the amplitude of oscillations to grow, leading to instability.

Conclusion:

In control systems, the concept of marginal stability is critical for analyzing systems that exhibit sustained oscillations without diverging or converging. The key characteristic of such systems is the presence of poles on the imaginary axis with a multiplicity of exactly 1. The analysis of the other options demonstrates that any pole with a multiplicity greater than 1 on the imaginary axis leads to instability, reinforcing the correctness of Option 1.