Let's solve the block diagram step by step to find the transfer function \( T(s) = \frac{C(s)}{R(s)} \). ### **Step 1: Label Each Signal** - Let the output of the first summing point (leftmost) be \( X(s) \). - Output after \(\frac{s+1}{s^2}\) block is \( Y(s) \). - Output after the feedback \( s \) block is \( Z(s) \). - Output after the second summing point is \( W(s) \). - Output after the final block \(\frac{s}{s^2+2}\) is \( C(s) \). ### **Step 2: Write Equations for Each Block** #### **First Summing Point:** \[ X(s) = R(s) - Z(s) \] #### **First Block:** \[ Y(s) = \frac{s+1}{s^2} X(s) \] #### **Second Summing Point:** \[ W(s) = Y(s) + R(s) \] #### **Final Block:** \[ C(s) = \frac{s}{s^2+2} W(s) \] #### **Feedback Path:** \[ Z(s) = s Y(s) \] --- ### **Step 3: Substitute Back** #### **Substitute \( Z(s) \) into \( X(s) \):** \[ X(s) = R(s) - s Y(s) \] #### **Substitute \( X(s) \) into \( Y(s) \):** \[ Y(s) = \frac{s+1}{s^2} [R(s) - s Y(s)] \] \[ Y(s) = \frac{s+1}{s^2} R(s) - \frac{s+1}{s^2} s Y(s) \] \[ Y(s) + \frac{(s+1)s}{s^2} Y(s) = \frac{s+1}{s^2} R(s) \] \[ Y(s) \left[1 + \frac{s(s+1)}{s^2}\right] = \frac{s+1}{s^2} R(s) \] \[ Y(s) \left[\frac{s^2 + s(s+1)}{s^2}\right] = \frac{s+1}{s^2} R(s) \] \[ Y(s) \left[\frac{s^2 + s^2 + s}{s^2}\right] = \frac{s+1}{s^2} R(s) \] \[ Y(s) \left[\frac{2s^2 + s}{s^2}\right] = \frac{s+1}{s^2} R(s) \] \[ Y(s) \left[2 + \frac{1}{s}\right] = \frac{s+1}{s^2} R(s) \] \[ Y(s) = \frac{s+1}{s^2} \cdot \frac{1}{2 + \frac{1}{s}} R(s) \] \[ Y(s) = \frac{s+1}{s^2} \cdot \frac{s}{2s + 1} R(s) \] \[ Y(s) = \frac{(s+1)s}{s^2(2s+1)} R(s) \] \[ Y(s) = \frac{s+1}{s(2s+1)} R(s) \] --- ### **Step 4: Substitute \( Y(s) \) into \( W(s) \):** \[ W(s) = Y(s) + R(s) = \frac{s+1}{s(2s+1)} R(s) + R(s) \] \[ W(s) = \left[\frac{s+1 + s(2s+1)}{s(2s+1)}\right] R(s) \] \[ W(s) = \left[\frac{s+1 + 2s^2 + s}{s(2s+1)}\right] R(s) \] \[ W(s) = \left[\frac{2s^2 + 2s + 1}{s(2s+1)}\right] R(s) \] --- ### **Step 5: Find \( C(s) \):** \[ C(s) = \frac{s}{s^2+2} W(s) \] \[ C(s) = \frac{s}{s^2+2} \cdot \frac{2s^2 + 2s + 1}{s(2s+1)} R(s) \] \[ C(s) = \frac{2s^2 + 2s + 1}{(s^2 + 2)(2s + 1)} R(s) \] --- ### **Step 6: Write the Overall Transfer Function** \[ \boxed{ T(s) = \frac{C(s)}{R(s)} = \frac{2s^2 + 2s + 1}{(s^2 + 2)(2s + 1)} } \] --- ## **Final Answer:** \[ \boxed{ T(s) = \frac{C(s)}{R(s)} = \frac{2s^2 + 2s + 1}{(s^2 + 2)(2s + 1)} } \]