3. Routh-Hurwitz Criterion:

An alternative to Schur criterion consists of applying a transformation which maps the interior of the unit circle into the left hand half plane, and then appealing to the well-known Routh-Hurwitz criterion, which gives necessary and sufficient condition for the roots of a polynomial to have negative real parts. The appropriate transformation is ; this maps the circle into the imaginary axis the interior of the circle into the half plane , and the point into . Under this transformation, the stability polynomial becomes

.

On multiplying through by this becomes a polynomial equation of degree , which we write

(8.37)

where, we assume without loss of generality that . The necessary and sufficient condition for the roots of (8.37) to lie in the half plane , that is, for the roots of to lie within the circle , is that all the leading principal minors of be positive, where is the matrix defined by

and where if . It can be shown that this condition implies . Thus the positivity of the coefficients in (8.37) is a necessary but not sufficient condition for absolute stability. For the necessary and sufficient condition for absolute stability given by this criterion are as follows,

.

K = 3 :