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.
Let be a polynomial (with complex coefficients) of degree with no roots on the imaginary axis (i.e. the line where is the imaginary unit and is a real number). Let us define real polynomials and by, respectively the real and imaginary parts of on the imaginary line.
Furthermore, let us denote by:
With the notations introduced above, the Routh–Hurwitz theorem states that:
| p-q= | 1 |
| \pi |
\Delta\argf(iy)=\left.\begin{cases}
| +infty | |
| +I | |
| -infty |
| P0(y) | |
| P1(y) |
&forodddegree\\[10pt]
| +infty | |
| -I | |
| -infty |
| P1(y) | |
| P0(y) |
&forevendegree\end{cases}\right\}=w(+infty)-w(-infty).
From the first equality we can for instance conclude that when the variation of the argument of is positive, then will have more roots to the left of the imaginary axis than to its right.The equality can be viewed as the complex counterpart of Sturm's theorem. Note the differences: in Sturm's theorem, the left member is and the from the right member is the number of variations of a Sturm chain (while refers to a generalized Sturm chain in the present theorem).
See main article: Routh–Hurwitz stability criterion. We can easily determine a stability criterion using this theorem as it is trivial that is Hurwitz-stable if and only if . We thus obtain conditions on the coefficients of by imposing and .
. Felix Gantmacher . 2005 . 1959 . Applications of the Theory of Matrices . Dover . New York . 226–233 . 0-486-44554-2.