Question

Consider a standard negative feedback configuration with \(G\left( s \right) = \frac{1}{{\left( {s + 1} \right)\left( {s + 2} \right)}}\) and \(H\left( s \right) = \frac{{s + \alpha }}{s}\). For the closed loop system to have poles on the imaginary axis, the value of α should be equal to (up to one decimal place) _____.

Answer 1

\(G\left( s \right) = \frac{1}{{\left( {s + 1} \right)\left( {s + 2} \right)}}\)

\(H\left( s \right) = \frac{{s + \alpha }}{s}\)

1 + G(s) H(s) = 0

\(\Rightarrow 1 + \frac{1}{{\left( {s + 1} \right)\left( {s + 2} \right)}}.\frac{{s + \alpha }}{s} = 0\)

⇒ s³ + 3s² + 2s + s + α = 0

⇒ s³ + 3s² + 3s + α = 0

\(\begin{array}{*{20}{c}} {{s^3}}\\ {{s^2}}\\ {\begin{array}{*{20}{c}} {{s^1}}\\ {s^\circ } \end{array}} \end{array}\begin{array}{*{20}{c}} 1&3&0\\ 3&\alpha &0\\ {\begin{array}{*{20}{c}} {\frac{{9 - \alpha }}{3}}\\ \alpha \end{array}}&{\begin{array}{*{20}{c}} 0\\ 0 \end{array}}&{\begin{array}{*{20}{c}} 0\\ 0 \end{array}} \end{array}\)

Closed loop poles will be on the imaginary axis if any row in Routh’s array became zero.

for \(\frac{{9 - \alpha }}{3} = 0\), the s¹ row will become zero.

⇒ 9 – α = 0 ⇒ α = 9.