Determine the stability of the system whose characteristics equation is:

.

Solution

All coefficients are positive and non-zero; therefore, the necessary condition for stability is satisfied. Let’s write the Routh array:



At this stage, we see that the top row corresponding to can be divided by two to make the calculation a little bit easier. So, we go ahead and divide that row by two:

*** QuickLaTeX cannot compile formula:
\begin{array}{l | c c c} s^5 &1 & 1 & 1 \\ s^4 &3 & 1 & 2 \\ s^3 &\\ s^2 &\\s^1 &\\s^0\end{array

*** Error message:
File ended while scanning use of \end .
Emergency stop.

Let’s continue writing the Routh table:

*** QuickLaTeX cannot compile formula:
\begin{array}{l | c c c}s^5 &1 & 1 & 1 \\ s^4 &3 & 1 & 2 \\ s^3 &\frac{2}{3} &\frac{1}{3} & \\ s^2 &-\frac{1}{2} & 2 & \\s^1 &3 & \\s^0 $ \frac{3 \times 2 -0 \times -\frac{1}{2}}{-\frac{1}{2}}\end{array}

*** Error message:
Missing $ inserted.
leading text: ...mes 2 -0 \times -\frac{1}{2}}{-\frac{1}{2}}
Missing $ inserted.
leading text: ...mes 2 -0 \times -\frac{1}{2}}{-\frac{1}{2}}
Missing $ inserted.
leading text: ...mes 2 -0 \times -\frac{1}{2}}{-\frac{1}{2}}
Extra }, or forgotten $.
leading text: ...mes 2 -0 \times -\frac{1}{2}}{-\frac{1}{2}}
Missing } inserted.
leading text: ...imes -\frac{1}{2}}{-\frac{1}{2}}\end{array}
Extra }, or forgotten $.
leading text: ...imes -\frac{1}{2}}{-\frac{1}{2}}\end{array}
Missing } inserted.
leading text: ...imes -\frac{1}{2}}{-\frac{1}{2}}\end{array}
Extra }, or forgotten $.
leading text: ...imes -\frac{1}{2}}{-\frac{1}{2}}\end{array}
Missing } inserted.
leading text: ...imes -\frac{1}{2}}{-\frac{1}{2}}\end{array}
Extra }, or forgotten $.

Since there are two sign changes in the first column, the characteristic equation has two roots with negative real parts. Therefore, the system is unstable.