Question

For the transfer function \( G(s) = 1 + \frac{2s - 1}{s^3 + 5s^2 + 3s + 22} \), the number of zeros lying in the left half of the \( s \)-plane is

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

The Routh-Hurwitz criterion is a powerful tool for determining the number of roots of a polynomial in the right-half $s$-plane without explicitly finding the roots. The number of sign changes in the first column of the Routh array equals the number of roots in the RHP.

Answer 1

The Correct Option is D

Step 1: Find the numerator of the transfer function \(G(s)\).
$$G(s) = \frac{s^3 + 5s^2 + 3s + 22 + (2s - 1)}{s^3 + 5s^2 + 3s + 22} = \frac{s^3 + 5s^2 + 5s + 21}{s^3 + 5s^2 + 3s + 22}$$
The zeros of \(G(s)\) are the roots of the numerator polynomial \(N(s) = s^3 + 5s^2 + 5s + 21 = 0\).
Step 2: Apply the Routh-Hurwitz criterion to \( N(s) \).
Routh array:
\[ \begin{array}{c|cc} s^3 & 1 & 5 \\ s^2 & 5 & 21 \\ s^1 & \dfrac{5 \times 5 - 1 \times 21}{5} = \dfrac{4}{5} & 0 \\ s^0 & 21 & \end{array} \]
Step 3: Analyze the first column of the Routh array.
The first column is \(1, 5, 4/5, 21\). There are no sign changes in the first column.
Step 4: Determine the location of the roots.
Since there are no sign changes in the first column of the Routh array, there are no roots of the polynomial \(N(s)\) in the right half of the \(s\)-plane. The polynomial is of degree 3, so it has 3 roots. Since there are no roots on the \(j\omega\)-axis (indicated by no row of zeros), all 3 roots must lie in the left half of the \(s\)-plane.
Final Answer: (D)