Routh–Hurwitz theorem


In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left half-plane. Polynomials with this property are called Hurwitz stable polynomials. The Routh-Hurwitz theorem is important in dynamical systems and control theory, because the characteristic polynomial of the differential equations of a stable linear system has roots limited to the left half plane. Thus the theorem provides a test for whether a linear dynamical system is stable. The Routh–Hurwitz theorem was proved in 1895, and it was named after Edward John Routh and Adolf Hurwitz.

Notations

Let f be a polynomial of degree n with no roots on the imaginary line. Let us define and by, respectively the real and imaginary parts of f on the imaginary line.
Furthermore, let us denote by:
With the notations introduced above, the Routh–Hurwitz theorem states that:
From the first equality we can for instance conclude that when the variation of the argument of f is positive, then f will have more roots to the left of the imaginary axis than to its right.
The equality p − q = w − w can be viewed as the complex counterpart of Sturm's theorem. Note the differences: in Sturm's theorem, the left member is p + q and the w from the right member is the number of variations of a Sturm chain.

Routh–Hurwitz stability criterion

We can easily determine a stability criterion using this theorem as it is trivial that f is Hurwitz-stable iff p − q = n. We thus obtain conditions on the coefficients of f by imposing w = n and w = 0.