The Routh array is a
tabular method permitting one to establish the
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 stabilit ...
of a system using only the coefficients of the characteristic
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 exa ...
. Central to the field of
control systems design, the
Routh–Hurwitz theorem
In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left half-plane. Polynomials with this property are called Hurwitz stable polynomials. The Routh-Hurwitz theorem is importan ...
and Routh array emerge by using the
Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...
and
Sturm's theorem
In mathematics, the Sturm sequence of a univariate polynomial is a sequence of polynomials associated with and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of loca ...
in evaluating
Cauchy indices.
The Cauchy index
Given the system:
:
Assuming no roots of
lie on the imaginary axis, and letting
:
= The number of roots of
with negative real parts, and
:
= The number of roots of
with positive real parts
then we have
:
Expressing
in polar form, we have
:
where
:
and
:
from (2) note that
:
where
:
Now if the i
th root of
has a positive real part, then (
using the notation y=(RE IM )
:
and
:
and
:
Similarly, if the i
th root of
has a negative real part,
:
and
:
and
:
From (9) to (11) we find that
when the i
th root of
has a positive real part, and from (12) to (14) we find that
when the i
th root of
has a negative real part. Thus,
:
So, if we define
:
then we have the relationship
:
and combining (3) and (17) gives us
:
and
Therefore, given an equation of
of degree
we need only evaluate this function
to determine
, the number of roots with negative real parts and
, the number of roots with positive real parts.
In accordance with (6) and Figure 1, the graph of
vs
, varying
over an interval (a,b) where
and
are integer multiples of
, this variation causing the function
to have increased by
, indicates that in the course of travelling from point a to point b,
has "jumped" from
to
one more time than it has jumped from
to
. Similarly, if we vary
over an interval (a,b) this variation causing
to have decreased by
, where again
is a multiple of
at both
and
, implies that