Question

A PD controller with transfer function \( G_c \) is used to stabilize an open-loop unstable process with transfer function \( G_p \), where: \[ G_C = K_C \frac{\tau_D s + 1}{\tau_D s}, \quad G_P = \frac{1}{(s - 1)(10s + 1)}, \] and time is in minutes. From the necessary conditions for closed-loop stability, the maximum feasible value of \( \tau_D \), in minutes, rounded off to 1 decimal place, is \_\_\_\_\_\_.

Hint:

For stability analysis, use root-locus or frequency-response methods to determine the maximum allowable derivative time constant \( \tau_D \).

Answer 1

Given: \[ G_C = \frac{K_C(s + 2)}{s}, \quad G_P = \frac{1}{(s - 3)(10s + 1)}. \] Using Routh Stability Criteria: Step 1: Write the characteristic equation: \[ 1 + G_C G_P = 0. \] \[ 1 + \frac{K_C(s + 2)}{s} \cdot \frac{1}{(s - 3)(10s + 1)} = 0. \] Simplify: \[ 1 + \frac{K_C(s + 2)}{s(s - 3)(10s + 1)} = 0. \] Multiply through by the denominator to obtain: \[ (10s^3) + s^2(200 - 9\tau_D) + s(20K_C \tau_D - \tau_D - 180) + (20K_C \tau_D) = 0. \] Routh's Array Construction: The characteristic polynomial is: \[ 10s^3 + s^2(200 - 9\tau_D) + s(20K_C\tau_D - \tau_D - 180) + (20K_C\tau_D). \] Routh's array is: \[ \begin{array}{c|c|c} s^3 & 10 & 20K_C\tau_D - \tau_D - 180
s^2 & 200 - 9\tau_D & 20K_C\tau_D
s^1 & \frac{(200 - 9\tau_D)(20K_C\tau_D) - 10(20K_C\tau_D - \tau_D - 180)}{200 - 9\tau_D} & 0
s^0 & 20K_C\tau_D &
\end{array} \] Step 2: Stability Condition: For closed-loop stability, the first column must be positive: \[ 200 - 9\tau_D>0. \] Solve for \( \tau_D \): \[ \tau_D<\frac{200}{9} \approx 22.22. \] Maximum Feasible Value: The maximum feasible value of \( \tau_D \) is: \[ \tau_D = 22.22. \] Final Answer: The system is stable if: \[ \tau_D \leq 22.22. \]