In the context of the
characteristic polynomial of a
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
or
difference equation, a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
is said to be stable if either:
* all its roots lie in the
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' ( ...
left
half-plane, or
* all its roots lie in the
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' ( ...
unit disk.
The first condition provides
stability
Stability may refer to:
Mathematics
*Stability theory, the study of the stability of solutions to differential equations and dynamical systems
** Asymptotic stability
** Linear stability
** Lyapunov stability
** Orbital stability
** Structural sta ...
for
continuous-time linear systems, and the second case relates to stability
of
discrete-time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
linear systems. A polynomial with the first property is called at times a
Hurwitz polynomial and with the second property a
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 ...
. Stable polynomials arise in
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
and in mathematical theory
of differential and difference equations. A linear,
time-invariant system
In control theory, a time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is ...
(see
LTI system theory
LTI can refer to:
* '' LTI – Lingua Tertii Imperii'', a book by Victor Klemperer
* Language Technologies Institute, a division of Carnegie Mellon University
* Linear time-invariant system, an engineering theory that investigates the response o ...
) is said to be
BIBO stable if every bounded input produces bounded output. A linear system is BIBO stable if its characteristic polynomial 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. In practice, stability is determined by applying any one of several
stability criteria.
Properties
* The
Routh–Hurwitz theorem
In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all root of a function, roots of a given polynomial lie in the left half-plane. Polynomials with this property are called stable polynomial, Hurwitz stable polynomials. ...
provides an algorithm for determining if a given polynomial is Hurwitz stable, which is implemented in the
Routh–Hurwitz and
Liénard–Chipart tests.
* 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 and
. For higher degree polynomials the extra computation involved in this mapping can be avoided by testing the Schur stability by the Schur-Cohn test, the
Jury test or the
Bistritz test.
* Necessary condition: a Hurwitz stable polynomial (with
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
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.
*Hadamard product: The Hadamard (coefficient-wise) product of two Hurwitz stable polynomials is again Hurwitz 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 and 2) because it violates the necessary condition;
*
is Hurwitz stable (its roots are −1 and −2).
* The polynomial
(with positive coefficients) is neither Hurwitz stable nor Schur stable. Its roots are the four primitive fifth
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
::
: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.
See also
*
Kharitonov region A Kharitonov region is a concept in mathematics. It arises in the study of the stability of polynomials.
Let D be a simply-connected set in the complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a ...
*
Stability criterion In control theory, and especially stability theory, a stability criterion establishes when a system is stable. A number of stability criteria are in common use:
* Circle criterion
* Jury stability criterion
*Liénard–Chipart criterion
* Nyquis ...
*
Stability radius
In mathematics, the stability radius of an object (system, function, matrix, parameter) at a given nominal point is the radius of the largest ball, centered at the nominal point, all of whose elements satisfy pre-determined stability condition ...
References
{{Reflist
External links
Mathworld page
Stability theory
Polynomials
fr:Polynôme de Hurwitz