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...

 is said to be stable if either:

• all its roots lie in the open
  Open set
  The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
  
   left half-plane, or
• all its roots lie in the open
  Open set
  The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
  
   unit disk.

The first condition defines 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...

 (or continuous-time) stability and the second one Schur

Schur

Schur is a German or Jewish surname and may refer to:* Alexander Schur , German footballer* Dina Feitelson-Schur , Israeli educator* Fritz Schur , Danish businessman* Gustav-Adolf Schur , German cyclist...

 (or discrete-time) stability. Stable polynomials arise in various mathematical fields, for example in control theory

Control theory

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

 and differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s. Indeed, a linear, time-invariant system

Time-invariant system

A time-invariant system is one whose output does not depend explicitly on time.This property can be satisfied if the transfer function of the system is not a function of time except expressed by the input and output....

 (see LTI system theory

LTI system theory

Linear time-invariant system theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and time-invariant...

) is said to be BIBO stable

BIBO stability

In electrical engineering, specifically signal processing and control theory, BIBO stability is a form of stability for linear signals and systems that take inputs. BIBO stands for Bounded-Input Bounded-Output...

 if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

 bounded inputs produce bounded outputs; this is equivalent to requiring that the denominator of its transfer function

Transfer function

A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...

 (which can be proven to be rational) is stable. The denominator is required to be Hurwitz stable if the system is in continuous-time and Schur stable if it is in discrete-time. Stable polynomials are sometimes called Hurwitz polynomial

Hurwitz polynomial

In mathematics, a Hurwitz polynomial, named after Adolf Hurwitz, is a polynomial whose coefficients are positive real numbers and whose zeros are located in the left half-plane of the complex plane, that is, the real part of every zero is negative...

s and Schur polynomial

Schur polynomial

In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of...

s.

Properties

• The Routh-Hurwitz theorem provides an algorithm for determining if a given polynomial is Hurwitz stable.
• To test if a given polynomial P (of degree d) is Schur stable, it suffices to apply this theorem to the transformed polynomial

obtained after the Möbius transformation  which maps the left half-plane to the open unit disc: P is Schur stable if and only if Q is Hurwitz stable.

• Necessary condition: a Hurwitz stable polynomial (with real coefficients) has coefficients of the same sign (either all positive or all negative).

• Sufficient condition: a polynomial with (real) coefficients such that:

is Schur stable.

• Product rule: Two polynomials f and g are stable (of the same type) if and only if the product fg is stable.

Examples

• is Schur stable because it satisfies the sufficient condition;
• is Schur stable (because all its roots equal 0) but it does not satisfy the sufficient condition;
• is not Hurwitz stable (its roots are -1,2) because it violates the necessary condition;
• is Hurwitz stable (its roots are -1,-2).
• The polynomial (with positive coefficients) is neither Hurwitz stable nor Schur stable. Its roots are the four primitive fifth roots of unity
  Root of unity
  In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
  
  

Note here that

It is a "boundary case" for Schur stability because its roots lie on the unit circle. The example also shows that the necessary (positivity) conditions stated above for Hurwitz stability are not sufficient.

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