Hurwitz polynomial

In mathematics, a Hurwitz polynomial, named after 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.^[1] 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 axis (i.e., a Hurwitz stable polynomial).^[2][3]

A polynomial function P(s) of a complex variable s is said to be Hurwitz if the following conditions are satisfied:

1. P(s) is real when s is real.

2. The roots of P(s) 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 the following:

$x^2 + 2x + 1.$

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

$(x+1)^2.$

In general, all second-degree 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 determinant b^2-4ac is less than zero, then the polynomial will have to complex-conjugate solutions with real part -b/a, which is negative for positive a and b. If it is equal to zero, there will be to coinciding real solutions in -b/a. Finally, if the determinant 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 second-degree 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.

The properties of Hurwitz polynomials are:

1. All the poles and zeros are in the left half plane or on its boundary, the imaginary axis.
2. Any poles and zeros on the imaginary axis are simple (have a multiplicity of one).
3. Any poles on the imaginary axis have real strictly positive residues, and similarly at any zeros on the imaginary axis, the function has a real strictly positive derivative.
4. Over the right half plane, the minimum value of the real part of a PR function occurs on the imaginary axis (because the real part of an analytic function constitutes a harmonic function over the plane, and therefore satisfies the maximum principle).
5. The polynomial should not have missing powers of s.

References

1. Kuo, Franklin F. (1966). Network Analysis and Synthesis, 2nd Ed. John Wiley & Sons. pp. 295–296. ISBN 0471511188.
2. Weisstein, Eric W (1999). "Hurwitz polynomial". Wolfram Mathworld. Wolfram Research. Retrieved July 3, 2013.
3. Reddy, Hari C. (2002). "Theory of two-dimensional Hurwitz polynomials". The Circuits and Filters Handbook, 2nd Ed. CRC Press. pp. 260–263. ISBN 1420041401. Retrieved July 3, 2013.

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