Question

Let \( G(s) = \frac{1}{10s^2} \) be the transfer function of a second-order system. A controller \( M(s) \) is connected to the system \( G(s) \) in the configuration shown below.

Consider the following statements.

• (i) There exists no controller of the form \( M(s) = \frac{K_I}{s} \), where \( K_I \) is a positive real number, such that the closed-loop system is stable.
• (ii) There exists at least one controller of the form \( M(s) = K_p + sK_D \), where \( K_p \) and \( K_D \) are positive real numbers, such that the closed-loop system is stable.

Which one of the following options is correct?

 

• A. \( (i) \) is TRUE and \( (ii) \) is FALSE
• B. \( (i) \) is FALSE and \( (ii) \) is TRUE
• C. Both \( (i) \) and \( (ii) \) are FALSE
• D. Both \( (i) \) and \( (ii) \) are TRUE

Hint:

For systems with integrators or low-frequency poles, a PD controller can be used to stabilize the system by introducing a zero, improving phase margin and overall system stability.

Answer 1

The Correct Option is D

(i) There exists no controller of the form \( M(s) = \frac{K_I}{s} \), where \( K_I \) is a positive real number, such that the closed-loop system is stable:
The given transfer function is \( G(s) = \frac{1}{10s^2} \).
If the controller \( M(s) = \frac{K_I}{s} \) is used, the closed-loop transfer function becomes:
\[ \text{Closed-loop transfer function} = \frac{M(s) \cdot G(s)}{1 + M(s) \cdot G(s)} = \frac{\frac{K_I}{s} \cdot \frac{1}{10s^2}}{1 + \frac{K_I}{s} \cdot \frac{1}{10s^2}}. \] This results in a pole at \( s = 0 \), which leads to instability due to the division by \( s \). Therefore, no such controller of the form \( \frac{K_I}{s} \) can stabilize the system.
Thus, statement (i) is TRUE.

(ii) There exists at least one controller of the form \( M(s) = K_p + sK_D \), where \( K_p \) and \( K_D \) are positive real numbers, such that the closed-loop system is stable:
The controller \( M(s) = K_p + sK_D \) is a PD controller.
For a PD controller, the closed-loop transfer function becomes: \[ \text{Closed-loop transfer function} = \frac{(K_p + sK_D) \cdot G(s)}{1 + (K_p + sK_D) \cdot G(s)} = \frac{(K_p + sK_D) \cdot \frac{1}{10s^2}}{1 + (K_p + sK_D) \cdot \frac{1}{10s^2}}. \] By carefully selecting appropriate values for \( K_p \) and \( K_D \), it is possible to stabilize the system. Specifically, a PD controller can add phase lead and improve the stability of the system.
Thus, statement (ii) is TRUE.

Therefore, the correct answer is (D), as both statements (i) and (ii) are true.