Question

Consider a unity feedback configuration with a plant and a PID controller as shown in the figure. \( G(s) = \dfrac{1}{(s+1)(s+3)} \text{  and  } C(s) = \dfrac{K(s+3-j)(s+3+j)}{s} \) with \( K \) being scalar. The closed loop is 

• A. only stable for \( K > 0 \)
• B. only stable for \( K \) between -1 and +1
• C. only stable for \( K < 0 \)
• D. stable for all values of \( K \)

Hint:

For stability analysis in control systems, always check the pole locations using the characteristic equation. For this system, stability depends on the gain \( K \) being positive.

Answer 1

The Correct Option is A

The closed-loop transfer function for the given system is derived by using the standard feedback control loop formula: \[ T(s) = \frac{C(s)G(s)}{1 + C(s)G(s)} \] where \( C(s) = \frac{K(s+3-j)(s+3+j)}{s} \) and \( G(s) = \frac{1}{(s+1)(s+3)} \). Substituting these into the equation for the closed-loop transfer function: \[ T(s) = \frac{\frac{K(s+3-j)(s+3+j)}{s}}{1 + \frac{K(s+3-j)(s+3+j)}{s(s+1)(s+3)}} \] Now, we need to analyze the stability of this system. For stability, we use the Routh-Hurwitz criterion or pole location analysis, which involves finding the poles of the closed-loop transfer function. The poles are determined by the characteristic equation: \[ 1 + C(s)G(s) = 0 \] If the poles are in the left half of the complex plane, the system is stable. From the analysis, we find that the system is only stable when \( K > 0 \). Therefore, the correct answer is option (A). Final Answer: only stable for \( K > 0 \)