Question

Consider the block diagram in the figure with control input \( u \), disturbance \( d \), and output \( y \). For the feedforward controller, the ordered pair \( (K, \alpha / \beta) \) is: \includegraphics[width=0.5\linewidth]{q22 CE.PNG}

• A. (0.5, 2)
• B. (-0.5, 0.5)
• C. (-2, 2)
• D. (2, 0.5)

Hint:

For feedforward controller design, equate the product of the feedforward and disturbance transfer functions to the process transfer function for exact disturbance rejection.

Answer 1

The Correct Option is B

Step 1: Analyze the block diagram. The process transfer function is: \[ G(s) = \frac{2}{(0.5s + 1)^2}. \] The disturbance transfer function is: \[ G_d(s) = \frac{1}{(s + 1)^2}. \] The feedforward controller transfer function is: \[ F(s) = K \left( \frac{\alpha s + 1}{\beta s + 1} \right)^2. \] Step 2: Apply disturbance rejection condition. For complete disturbance rejection, the output \( y \) should be independent of \( d \). The feedforward controller \( F(s) \) must cancel the disturbance effect. This requires: \[ F(s) \cdot G_d(s) = G(s). \] Substitute \( G(s) \) and \( G_d(s) \): \[ K \left( \frac{\alpha s + 1}{\beta s + 1} \right)^2 \cdot \frac{1}{(s + 1)^2} = \frac{2}{(0.5s + 1)^2}. \] Step 3: Match the numerator and denominator. Equating denominators: \[ (\beta s + 1)^2 \cdot (s + 1)^2 = (0.5s + 1)^2. \] Expand and compare coefficients to solve for \( \beta \): \[ \beta = 0.5. \] Equating numerators: \[ K \cdot (\alpha s + 1)^2 = 2. \] Substitute \( \beta = 0.5 \) and solve for \( K \) and \( \alpha \): \[ K = -0.5, \quad \alpha = 0.5. \] Step 4: Conclusion. The ordered pair \( (K, \alpha / \beta) \) is \( (-0.5, 0.5) \).