Understanding the Concept:
The Routh-Hurwitz stability criterion provides a method to determine the number of roots of a characteristic polynomial that lie in the right-half of the $s$-plane (RHP). The number of RHP roots is exactly equal to the number of sign changes in the first column of the Routh array. Alternatively, we can analyze the polynomial by factoring it directly.
Step 1: Direct Factorization Method.
Let us examine the structure of the given characteristic equation:
\[
q(s) = s^3 + 3s^2 + 3s + 1 = 0
\]
This matches the standard algebraic expansion for a perfect cube:
\[
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\]
Setting $a = s$ and $b = 1$, we can rewrite the polynomial as:
\[
(s + 1)^3 = 0
\]
Solving for the roots:
\[
s_1 = -1, \quad s_2 = -1, \quad s_3 = -1
\]
All three roots are located at $s = -1$, which lies in the left-half of the $s$-plane (LHP). Therefore, zero poles lie in the right-half plane.
Step 2: Verification using the Routh Array.
Let us construct the Routh array for the polynomial coefficients to verify our result:
s^3 &: 1 3
s^2 &: 3 1
s^1 &: \frac{(3 \times 3) - (1 \times 1)}{3} = \frac{9 - 1}{3} = \frac{8}{3}
s^0 &: 1
Let us check the signs of the terms in the first column:
- Element 1: $+1$ (Positive)
- Element 2: $+3$ (Positive)
- Element 3: $+\frac{8}{3}$ (Positive)
- Element 4: $+1$ (Positive)
Since there are zero sign changes in the first column, no roots lie in the right-half plane.