In
mathematics, the Routh–Hurwitz theorem gives a test to determine whether all
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of a given
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 ...
lie in the left half-plane.
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 ...
s with this property are called
Hurwitz stable polynomials. The Routh-Hurwitz theorem is important in
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s and
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 ...
, because the characteristic polynomial of the
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 ...
s of a
stable linear system
In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator.
Linear systems typically exhibit features and properties that are much simpler than the nonlinear case.
As a mathematical abstractio ...
has roots limited to the left half plane (negative eigenvalues). Thus the theorem provides a test to determine whether a linear dynamical system is stable without solving the system.
The Routh–Hurwitz theorem was proved in 1895, and it was named after
Edward John Routh
Edward John Routh (; 20 January 18317 June 1907), was an English mathematician, noted as the outstanding coach of students preparing for the Mathematical Tripos examination of the University of Cambridge in its heyday in the middle of the ninet ...
and
Adolf Hurwitz.
Notations
Let ''f''(''z'') be a polynomial (with
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
coefficients) of
degree ''n'' with no roots on the
imaginary axis
An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . Fo ...
(i.e. the line ''Z'' = ''ic'' where ''i'' is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
and ''c'' is a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
). Let us define
(a polynomial of degree ''n'') and
(a nonzero polynomial of degree strictly less than ''n'') by
, respectively the
real and imaginary parts
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
of ''f'' on the imaginary line.
Furthermore, let us denote by:
* ''p'' the number of roots of ''f'' in the left
half-plane (taking into account multiplicities);
* ''q'' the number of roots of ''f'' in the right half-plane (taking into account multiplicities);
*
the variation of the argument of ''f''(''iy'') when ''y'' runs from −∞ to +∞;
* ''w''(''x'') is the number of variations of the
generalized Sturm chain obtained from
and
by applying the
Euclidean algorithm;
*
is the
Cauchy index In mathematical analysis, the Cauchy index is an integer associated to a real rational function over an interval. By the Routh–Hurwitz theorem, we have the following interpretation: the Cauchy index of
:''r''(''x'') = ''p''(''x'')/''q''(''x'')
...
of the
rational function ''r'' over the
real line.
Statement
With the notations introduced above, the Routh–Hurwitz theorem states that:
:
From the first equality we can for instance conclude that when the variation of the argument of ''f''(''iy'') is positive, then ''f''(''z'') will have more roots to the left of the imaginary axis than to its right.
The equality ''p'' − ''q'' = ''w''(+∞) − ''w''(−∞) can be viewed as the complex counterpart of
Sturm's theorem. Note the differences: in Sturm's theorem, the left member is ''p'' + ''q'' and the ''w'' from the right member is the number of variations of a Sturm chain (while ''w'' refers to a generalized Sturm chain in the present theorem).
Routh–Hurwitz stability criterion
We can easily determine a stability criterion using this theorem as it is trivial that ''f''(''z'') is
Hurwitz-stable iff ''p'' − ''q'' = ''n''. We thus obtain conditions on the coefficients of ''f''(''z'') by imposing ''w''(+∞) = ''n'' and ''w''(−∞) = 0.
References
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*
*
*
External links
Mathworld entry
{{DEFAULTSORT:Routh-Hurwitz theorem
Theorems about polynomials
Theorems in complex analysis
Theorems in real analysis