Question

A satellite attitude control system, as shown below, has a plant with transfer function: \[ G(s) = \frac{1}{s^2}, \] cascaded with a compensator: \[ C(s) = \frac{K(s + \alpha)}{s + 4}, \] \vspace{0.5cm} \begin{center} \includegraphics[width=8cm]{38.png} \end{center} where \(K\) and \(\alpha\) are positive real constants. In order for the closed-loop system to have poles at \(-1 \pm j\sqrt{3}\), the value of \(\alpha\) must be \(\_\_\_\_\).

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

For pole placement, expand the characteristic equation and match its coefficients with the desired polynomial to determine unknown parameters.

Answer 1

The Correct Option is B

Step 1: Derive the closed-loop characteristic equation.
The open-loop transfer function is: \[ T(s) = \frac{C(s)G(s)}{1 + C(s)G(s)} = \frac{\frac{K(s+\alpha)}{(s+4)s^2}}{1 + \frac{K(s+\alpha)}{(s+4)s^2}}. \] The characteristic equation for the closed-loop system is: \[ s^2(s+4) + K(s+\alpha) = 0. \] Step 2: Apply the desired pole placement.
The desired closed-loop poles are given as \(-1 \pm j\sqrt{3}\). These correspond to the quadratic term: \[ (s - (-1 + j\sqrt{3}))(s - (-1 - j\sqrt{3})) = s^2 + 2s + 4. \] Step 3: Match coefficients to solve for \(\alpha\).
Expand the characteristic equation: \[ s^2(s+4) + K(s+\alpha) = s^3 + 4s^2 + Ks + K\alpha. \] Equate this to the desired polynomial: \[ s^3 + 2s^2 + 4s. \] Comparing coefficients of \(s\), we find: \[ K = 4. \] Comparing the constant terms gives: \[ K\alpha = 4 \quad \Rightarrow \quad 4\alpha = 4 \quad \Rightarrow \quad \alpha = 1. \] Final Answer: \[\boxed{{(2) } 1}\]