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 complex 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 Template:Math be a polynomial (with complex coefficients) of degree Template:Math with no roots on the imaginary axis (i.e. the line Template:Math where Template:Math is the imaginary unit and Template:Math is a real number). Let us define real polynomials Template:Math and Template:Math by Template:Math, respectively the real and imaginary parts of Template:Math on the imaginary line.

Furthermore, let us denote by:

• Template:Math the number of roots of Template:Math in the left half-plane (taking into account multiplicities);
• Template:Math the number of roots of Template:Math in the right half-plane (taking into account multiplicities);
• Template:Math the variation of the argument of Template:Math when Template:Math runs from Template:Math to Template:Math;
• Template:Math is the number of variations of the generalized Sturm chain obtained from Template:Math and Template:Math by applying the Euclidean algorithm;
• Template:Math is the Cauchy index of the rational function Template:Math over the real line.

Statement

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

$p-q=\frac{1}{\pi }\mathrm{\Delta }\arg f(iy)=\{\begin{array}{cc}+I_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{P_0(y)}{P_1(y)}& \text{for odd degree}\\ -I_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{P_1(y)}{P_0(y)}& \text{for even degree}\end{array}\}=w(+\mathrm{\infty })-w(-\mathrm{\infty }).$

From the first equality we can for instance conclude that when the variation of the argument of Template:Math is positive, then Template:Math will have more roots to the left of the imaginary axis than to its right. The equality Template:Math can be viewed as the complex counterpart of Sturm's theorem. Note the differences: in Sturm's theorem, the left member is Template:Math and the Template:Math from the right member is the number of variations of a Sturm chain (while Template:Math refers to a generalized Sturm chain in the present theorem).

Routh–Hurwitz stability criterion

Template:Main We can easily determine a stability criterion using this theorem as it is trivial that Template:Math is Hurwitz-stable if and only if Template:Math. We thus obtain conditions on the coefficients of Template:Math by imposing Template:Math and Template:Math.

See also

• Plastic ratio

References

• Template:Cite book
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• Explaining the Routh–Hurwitz Criterion (2020)

External links

• Mathworld entry