In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Hurwitz polynomial, named after Adolf Hurwitz

Adolf Hurwitz

Adolf Hurwitz was a German mathematician.-Early life:He was born to a Jewish family in Hildesheim, former Kingdom of Hannover, now Lower Saxony, Germany, and died in Zürich, in Switzerland. Family records indicate that he had siblings and cousins, but their names have yet to be confirmed...

, is a polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

 whose coefficients are positive real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s and whose zeros are located in the left half-plane of the complex plane

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

, that is, the real part of every zero is negative. One sometimes uses the term Hurwitz polynomial simply as a (real or complex) polynomial with all zeros in the left-half plane (i.e., a Hurwitz stable polynomial

Stable polynomial

A polynomial is said to be stable if either:* all its roots lie in the open left half-plane, or* all its roots lie in the open unit disk.The first condition defines Hurwitz stability and the second one Schur stability. Stable polynomials arise in various mathematical fields, for example in...

).
A polynomial 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.

• Note: Here P(s) is any polynomial in s.

Examples

A simple example of a Hurwitz polynomial is the following:


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

Properties

For a polynomial to be Hurwitz, it is necessary but not sufficient that all of its coefficients be positive. For all of a polynomial's roots to lie in the left half-plane, it is necessary and sufficient that the polynomial in question pass the Routh-Hurwitz stability criterion

Routh-Hurwitz stability criterion

The Routh–Hurwitz stability criterion is a necessary and sufficient method to establish the stability of a single-input, single-output , linear time invariant control system. More generally, given a polynomial, some calculations using only the coefficients of that polynomial can lead to the...

.
A given polynomial can be tested to be Hurwitz or not by using the continued fraction expansion technique.

1. All the poles and zeros of a function are in the left half plane or on its boundary the imaginary axis.
2. Any poles and zeroes 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 zeroes 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. there have no any missing term of 's'but it possible after the testing the prf stability