Question

A unity-gain negative-feedback control system has a loop-gain \( L(s) \) given by \[ L(s) = \frac{6}{s(s-5)} \] The closed-loop system is _________

• A. Causal and stable
• B. Causal and unstable
• C. Non-causal and stable
• D. Non-causal and unstable

Hint:

In control systems, the system is stable if all poles lie in the left half of the s-plane. If any pole lies in the right half, the system is unstable.

Answer 1

The Correct Option is B

Step 1: Understanding the system.
The system has a transfer function that is a rational function of \( s \). The poles of the system can be found by setting the denominator equal to zero: \[ s(s-5) = 0 \quad \Rightarrow \quad s = 0, 5 \] Since the pole at \( s = 0 \) is in the right half of the s-plane, the system is unstable.
Step 2: Analyzing the options.
- (A) Incorrect, the system is unstable due to the pole at \( s = 0 \).
- (B) Correct, the system is causal (since it has no future values depending on past inputs) but unstable due to the pole at \( s = 0 \).
- (C) Incorrect, the system is not non-causal, and it is unstable.
- (D) Incorrect, the system is not non-causal. Step 3: Conclusion.
Thus, the correct answer is (B) Causal and unstable.