Routh-Hurwitz Criterion

The Routh criterion allows us to determine the stability of a polynomial without computing the roots.

Routh Table Construction

• The first two rows come directly from the coefficients of $a(s)$
• Each of the other rows is computed from its two preceding rows as

$$r_{ij}=-\frac{1}{r_{(i-1)0}}\det \left[\begin{array}{cc}{r}_{(i-2)0}& r_{(i-2)(j+1)}\\ r_{(i-1)0}& r_{(i-1)(j+1)}\end{array}\right]$$

• Whenever $r_{(i-1)(j+1)}$ is missing, let $r_{(i-1)(j+1)}=0$, then $r_{ij}=r_{(i-2)(j+1)}$.

Pole Locations from Routh Table

• Total number of poles = Degree of characteristic polynomial
• RHP poles: Number of sign changes in first column
• $j\omega$-axis poles: All-zero rows
• LHP poles: Total poles - RHP poles - $j\omega$-axis poles

Examples

Example 1

Example 2

Example 3

Example 4

Example 5