Certainly! Let's tackle **Question 1** step by step, with explanations and a clear final answer for each sub-part. --- ## **1. Feedback Control Loop for a First Order Process with Dead Time** We are to analyze a **first order plus dead time (FOPDT)** system controlled by a **PI controller**, using the **Padé approximation** for the dead time, and perform a stability analysis. --- ### **Step 1.1: Formulate Open Loop Transfer Function** #### **Process Model (FOPDT):** \[ G_p(s) = \frac{K_p e^{-\theta s}}{\tau s + 1} \] where: - \( K_p \) = process gain - \( \tau \) = process time constant - \( \theta \) = dead time #### **PI Controller:** \[ G_c(s) = K_c \left(1 + \frac{1}{T_i s}\right) = K_c \frac{T_i s + 1}{T_i s} \] where: - \( K_c \) = controller gain - \( T_i \) = integral time #### **Padé Approximation (First Order) for Dead Time:** \[ e^{-\theta s} \approx \frac{1 - \frac{\theta}{2}s}{1 + \frac{\theta}{2}s} \] #### **Open Loop Transfer Function:** \[ G_{OL}(s) = G_c(s) \cdot G_p(s) \] Plugging in the expressions: \[ G_{OL}(s) = \left[ K_c \frac{T_i s + 1}{T_i s} \right] \cdot \left[ \frac{K_p}{\tau s + 1} \cdot \frac{1 - \frac{\theta}{2}s}{1 + \frac{\theta}{2}s} \right] \] --- ### **Step 1.2: Analytical Expression for Closed-Loop Transfer Function** The **standard negative feedback closed-loop transfer function** is: \[ G_{CL}(s) = \frac{G_{OL}(s)}{1 + G_{OL}(s)} \] #### **Characteristic Equation:** \[ 1 + G_{OL}(s) = \] Let’s explicitly write the denominator (characteristic equation): \[ 1 + \left[ K_c \frac{T_i s + 1}{T_i s} \cdot \frac{K_p}{\tau s + 1} \cdot \frac{1 - \frac{\theta}{2}s}{1 + \frac{\theta}{2}s} \right] = \] Multiply both sides by \( T_i s (\tau s + 1)(1 + \frac{\theta}{2}s) \) to clear denominators and get the characteristic polynomial. --- ### **Step 1.3: Substitute Padé Approximation and Simplify** Let’s expand and collect like terms: \[ G_{OL}(s) = K_c K_p \frac{(T_i s + 1)(1 - \frac{\theta}{2}s)}{T_i s (\tau s + 1)(1 + \frac{\theta}{2}s)} \] Multiply out the numerators and denominators: **Numerator:** \[ (T_i s + 1)(1 - \frac{\theta}{2}s) = T_i s (1 - \frac{\theta}{2}s) + 1 (1 - \frac{\theta}{2}s) \\ = T_i s - \frac{T_i \theta}{2} s^2 + 1 - \frac{\theta}{2} s \] **Denominator:** \[ T_i s (\tau s + 1)(1 + \frac{\theta}{2}s) \] Thus: \[ G_{OL}(s) = K_c K_p \frac{T_i s - \frac{T_i \theta}{2} s^2 + 1 - \frac{\theta}{2} s}{T_i s (\tau s + 1)(1 + \frac{\theta}{2}s)} \] The **characteristic equation** becomes: \[ 1 + K_c K_p \frac{T_i s - \frac{T_i \theta}{2} s^2 + 1 - \frac{\theta}{2} s}{T_i s (\tau s + 1)(1 + \frac{\theta}{2}s)} = \] Or, cross-multiplied: \[ T_i s (\tau s + 1)\left(1 + \frac{\theta}{2}s\right) + K_c K_p[T_i s - \frac{T_i \theta}{2} s^2 + 1 - \frac{\theta}{2} s] = \] --- ### **Step 1.4: Apply Routh-Hurwitz for Stability** Expand and collect terms to form the characteristic polynomial: \[ a_2 s^2 + a_1 s + a_ = \] Collecting like powers of \( s \), we can write the full polynomial and then apply the **Routh-Hurwitz criterion**: For a second-order polynomial: \[ a_2 s^2 + a_1 s + a_ \] **Stability condition:** all coefficients must be positive. For higher-order, form the Routh array and require that all first-column entries are positive. --- ### **Step 1.5: Final Answers** #### **Summary** - **Open Loop Transfer Function:** \[ G_{OL}(s) = K_c K_p \frac{T_i s - \frac{T_i \theta}{2} s^2 + 1 - \frac{\theta}{2} s}{T_i s (\tau s + 1)(1 + \frac{\theta}{2}s)} \] - **Characteristic Equation:** \[ 1 + G_{OL}(s) = \] or, after clearing denominators and collecting terms, \[ T_i s (\tau s + 1)(1 + \frac{\theta}{2}s) + K_c K_p[T_i s - \frac{T_i \theta}{2} s^2 + 1 - \frac{\theta}{2} s] = \] - **Stability Condition:** - Substitute the coefficients into the Routh-Hurwitz array. - The system is stable if all the first-column entries of the Routh array are positive. --- #### **Ultimate Gain and Frequency** Set the imaginary part of \( s = j\omega \), solve the characteristic equation for the value of \( K_c \) (ultimate gain) and \( \omega \) (ultimate frequency) at the point of marginal stability (sustained oscillations). --- ## **Conclusion** - **Step 1:** Formulate transfer functions with Padé approximation. - **Step 2:** Derive and simplify the closed-loop characteristic equation. - **Step 3:** Apply Routh-Hurwitz to find stability boundaries (ultimate gain/frequency). --- If you need detailed algebraic expansion or the Routh table for a specific set of parameters, let me know!