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. 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).
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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:
The only real solution is −1, as it factors to
In general, all second-degree polynomials with positive coefficients are Hurwitz. This follows directly from the quadratic formula:
where, if the determinant b^2-4ac 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 it is equal to zero, there will be two coinciding real solutions at -b/2a. Finally, if the determinant is greater than zero, there will be two real negative solutions, because
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 doesn't 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:
- All the poles and zeros are in the left half plane or on its boundary, the imaginary axis.
- Any poles and zeros on the imaginary axis are simple (have a multiplicity of one).
- 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.
- 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).
- The polynomial should not have missing powers of s.