Discussion
Using the Routh Hurwitz stability criteria, identify whether the system with the given characteristic equation is stable or not?
\(s^4 + 2s^3 + 3s^2 + 4s + 5 = 0\)
Answer: B
Apply the Routh array to \(s^4 + 2s^3 + 3s^2 + 4s + 5 = 0\).
\[ \begin{array}{c|ccc} s^4 & 1 & 3 & 5 \\ s^3 & 2 & 4 & \\ s^2 & 1 & 5 & \\ s^1 & -6 & & \\ s^0 & 5 & & \end{array} \]
The \(s^2\) row is \((2 \times 3 - 4)/2 = 1\) and \((2 \times 5)/2 = 5\), and the \(s^1\) entry is \((1 \times 4 - 2 \times 5)/1 = -6\). The first column is \(1, 2, 1, -6, 5\), which has two sign changes, so two roots are in the right half plane and the system is unstable. The options stable, partially stable and stable at origin all need no sign change in the first column. The answer is Unstable, option B.
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