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 (negative eigenvalues). Thus the theorem provides a mathematical test, the Routh–Hurwitz stability criterion, to determine whether a linear dynamical system is stable without solving the system. The Routh–Hurwitz theorem was proved in 1895, and it was named after Edward John Routh and Adolf Hurwitz.

Notations
Let $f(z)$ be a polynomial (with complex coefficients) of degree $n$ with no roots on the imaginary axis (i.e. the line $z = ic$ where $i$ is the imaginary unit and $c$ is a real number). Let us define real polynomials $P_{0}(y)$ and $P_{1}(y)$ by $f(iy) = P_{0}(y) + iP_{1}(y)$, respectively the real and imaginary parts of $f$ on the imaginary line.

Furthermore, let us denote by:
 * $p$ the number of roots of $f$ in the left half-plane (taking into account multiplicities);
 * $q$ the number of roots of $f$ in the right half-plane (taking into account multiplicities);
 * $Δ arg f(iy)$ the variation of the argument of $f(iy)$ when $y$ runs from $−∞$ to $+∞$;
 * $w(x)$ is the number of variations of the generalized Sturm chain obtained from $P_{0}(y)$ and $P_{1}(y)$ by applying the Euclidean algorithm;
 * $I+∞ −∞ r$ is the Cauchy index of the rational function $r$ over the real line.

Statement
With the notations introduced above, the Routh–Hurwitz theorem states that:


 * $$p-q=\frac{1}{\pi}\Delta\arg f(iy)= \left.\begin{cases} +I_{-\infty}^{+\infty}\frac{P_0(y)}{P_1(y)} & \text{for odd degree} \\[10pt] -I_{-\infty}^{+\infty}\frac{P_1(y)}{P_0(y)} & \text{for even degree} \end{cases}\right\} = w(+\infty)-w(-\infty).$$

From the first equality we can for instance conclude that when the variation of the argument of $f(iy)$ is positive, then $f(z)$ 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 (while $w$ refers to a generalized Sturm chain in the present theorem).

Routh–Hurwitz stability criterion
We can easily determine a stability criterion using this theorem as it is trivial that $f(z)$ is Hurwitz-stable if and only if $p − q = n$. We thus obtain conditions on the coefficients of $f(z)$ by imposing $w(+∞) = n$ and $w(−∞) = 0$.