Question

Given the process transfer function \[ G_P = \frac{20}{s - 2}, \] and controller transfer function \[ G_C = K_C, \] and assuming the transfer function of all other elements in the control loop are unity, what is the range of \( K_C \) for which the closed-loop response will be stable?

• A. \( K_C<\frac{1}{10} \)
• B. \( K_C<\frac{1}{100} \)
• C. \( \frac{1}{100}<K_C<\frac{1}{10} \)
• D. \( K_C>\frac{1}{10} \)

Hint:

For stability in control systems, ensure that the poles of the closed-loop transfer function lie in the left half-plane. In this case, the condition \( K_C < \frac{1}{10} \) keeps the poles in the stable region, ensuring the system remains stable.

Answer 1

The Correct Option is A

- For stability, the open-loop transfer function must not have poles with positive real parts. 
- The characteristic equation for this system is \( 1 + K_C G_P = 0 \), which simplifies to \[ 1 + \frac{K_C \cdot 20}{s - 2} = 0. \] - The system will be stable if the magnitude of \( K_C \) keeps the poles of the closed-loop system in the left half-plane, which gives the condition \( K_C < \frac{1}{10} \).
Conclusion: 
The range of \( K_C \) for which the closed-loop response will be stable is \( K_C < \frac{1}{10} \), as given by