Hurwitz polynomial

In mathematics, a Hurwitz polynomial (named after German mathematician Adolf Hurwitz) is a polynomial whose roots (zeros) are located in the left half-plane of the complex plane or on the imaginary axis, that is, the real part of every root is zero or negative. Such a polynomial must have coefficients that are positive real numbers. The term is sometimes restricted to polynomials whose roots have real parts that are strictly negative, excluding the imaginary axis (i.e., a Hurwitz stable polynomial).

A polynomial function Template:Math of a complex variable Template:Mvar is said to be Hurwitz if the following conditions are satisfied:

1. Template:Math is real when Template:Mvar is real.
2. The roots of Template:Math have real parts which are zero or negative.

Hurwitz polynomials are important in control systems theory, because they represent the characteristic equations of stable linear systems. Whether a polynomial is Hurwitz can be determined by solving the equation to find the roots, or from the coefficients without solving the equation by the Routh–Hurwitz stability criterion.

Examples

A simple example of a Hurwitz polynomial is:

$x^2+2x+1.$

The only real solution is −1, because it factors as

$(x+1{)}^2.$

In general, all quadratic polynomials with positive coefficients are Hurwitz. This follows directly from the quadratic formula:

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$

where, if the discriminant b²−4ac is less than zero, then the polynomial will have two complex-conjugate solutions with real part −b/2a, which is negative for positive a and b. If the discriminant is equal to zero, there will be two coinciding real solutions at −b/2a. Finally, if the discriminant is greater than zero, there will be two real negative solutions, because $\sqrt{b^2-4ac}<b$ for positive a, b and c.

Properties

For a polynomial to be Hurwitz, it is necessary but not sufficient that all of its coefficients be positive (except for quadratic polynomials, which also imply sufficiency). A necessary and sufficient condition that a polynomial is Hurwitz is that it passes the Routh–Hurwitz stability criterion. A given polynomial can be efficiently tested to be Hurwitz or not by using the Routh continued fraction expansion technique.

References

• Wayne H. Chen (1964) Linear Network Design and Synthesis, page 63, McGraw Hill.