The question asks about the range of the damping factor (\(\xi\)) required for most practical control systems. The concept of damping factor is crucial in the analysis and design of control systems, particularly in determining the stability and performance of system responses.

Theoretical Background:

• The damping factor, also known as the damping ratio, is an important parameter in control systems. It indicates how oscillations in a system decay after a disturbance.
• It is defined as \(\xi = \frac{c}{2\sqrt{mk}}\), where \(c\) is the damping coefficient, \(m\) is the mass, and \(k\) is the spring constant in a mechanical system.
• A system can be classified as underdamped, critically damped, or overdamped based on the value of \(\xi\).
  • \(0 \lt \xi \lt 1\) represents an underdamped system, leading to oscillatory responses.
  • \(\xi = 1\) represents a critically damped system, where the system returns to equilibrium as quickly as possible without oscillating.
  • \(\xi \gt 1\) represents an overdamped system, which returns to equilibrium without oscillating but slower than the critically damped system.

Explanation of Options:

• \(0 \lt \xi \lt 0.1\) and \(0 \lt \xi \lt 0.09\): These represent very low damping ratios, resulting in highly oscillatory responses. Such low values are usually not suitable for practical control systems.
• \(0 \lt \xi \lt 0.7\): Although this range includes suitable damping factors, it also includes very low values (<0.28) that are not ideal.
• \(0.28 \lt \xi \lt 0.7\): This is the typical range used in practical systems. It provides a balance where the system is neither too oscillatory nor too slow in its response, thus ensuring a smooth return to equilibrium.

Conclusion: The correct answer is \(0.28 \lt \xi \lt 0.7\) as it represents a practical range of damping factors for most control systems, providing a good compromise between response speed and stability.