This question asks to identify the correct statements regarding the effects of feedback in control systems.

Statement A Analysis: Feedback Reduces System Error

Statement A suggests that feedback's primary function is to reduce system error. This is a core concept in control engineering. The system error, denoted mathematically as $ e(t) $, is the difference between the desired output value (setpoint) and the actual measured output value. Feedback mechanisms work by continuously measuring the system's output and comparing it to the setpoint. The resulting error signal, $ e(t) $, is then used by the controller to adjust the system's input. For negative feedback systems, the controller acts to minimize this error. As the feedback loop operates, it drives the actual output closer to the desired output, thus significantly reducing the steady-state error and improving the system's accuracy.

Statement B Analysis: Feedback's Gain-Frequency Relationship

Statement B claims that feedback increases system gain in one frequency range while decreasing it in another. While feedback does alter the frequency response of a system, this statement is often misleading or overly simplified, especially concerning negative feedback.

Typically, negative feedback is used to:

• Reduce the overall gain (especially the DC or low-frequency gain).
• Increase the system's bandwidth, meaning it can operate effectively over a wider range of frequencies.

Let $ A_{ol} $ be the open-loop gain and $ \beta $ be the feedback factor. The closed-loop gain $ A_{cl} $ for a negative feedback system is given by:

$$ A_{cl} = \frac{A_{ol}}{1 + \beta A_{ol}} $$

At low frequencies, where $ | \beta A_{ol} | \gg 1 $, the closed-loop gain approaches $ A_{cl} \approx \frac{1}{\beta} $. This value is constant and often lower than the peak open-loop gain $ A_{ol} $. At high frequencies, $ A_{ol} $ usually decreases rapidly. While the closed-loop system maintains a more constant gain over a broader frequency range (higher bandwidth), it doesn't necessarily *increase* the gain compared to the open-loop system's peak gain. It aims for stability and predictability across frequencies, often at the expense of maximum gain. Therefore, statement B is not universally correct in the way it's phrased, particularly the "increases the gain" part for negative feedback.

Statement C Analysis: Feedback Can Cause Instability

Statement C asserts that feedback can transform a stable system into an unstable one. This is a critical consideration in feedback system design. Although negative feedback is primarily used for stabilization, improper implementation can lead to instability. Instability often arises from excessive phase shifts in the feedback loop at higher frequencies. According to stability criteria (like the Nyquist stability criterion), if the phase shift around the loop reaches $ 180^\circ $ at a frequency where the loop gain magnitude is still greater than or equal to unity ($ | \beta A_{ol} | \ge 1 $), the negative feedback effectively acts as positive feedback. This positive feedback reinforces disturbances, causing the system output to oscillate with increasing amplitude or diverge, resulting in instability.

Conclusion on Correct Statements

Based on the analysis:

• Statement A is correct: Feedback is essential for reducing system error.
• Statement B is generally not correct as stated: Negative feedback typically reduces peak gain while increasing bandwidth, rather than increasing gain in specific ranges compared to the open-loop system.
• Statement C is correct: Poorly designed feedback loops can indeed lead to instability.

Therefore, the correct statements are A and C.