Derivation of the Routh array
Encyclopedia
The Routh array is a tabular method permitting one to establish the stability
of a system using only the coefficients of the characteristic polynomial
. Central to the field of control systems design
, the Routh–Hurwitz theorem
and Routh array emerge by using the Euclidean algorithm
and Sturm's theorem
in evaluating Cauchy indices
.
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...
of a system using only the coefficients of the characteristic 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...
. Central to the field of control systems design
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...
, the Routh–Hurwitz theorem
Routh–Hurwitz theorem
In mathematics, Routh–Hurwitz theorem gives a test to determine whether a given polynomial is Hurwitz-stable. It was proved in 1895 and named after Edward John Routh and Adolf Hurwitz.-Notations:...
and Routh array emerge by using the Euclidean algorithm
Euclidean algorithm
In mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
and Sturm's theorem
Sturm's theorem
In mathematics, Sturm's theorem is a symbolic procedure to determine the number of distinct real roots of a polynomial. It was named for Jacques Charles François Sturm...
in evaluating Cauchy indices
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...
.
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 ith root of
has a positive real part, then (using the notation y=(RE[y],IM[y]))
-
and
Similarly, if the ith root of
has a negative real part,
and
Therefore,
when the ith root of 
has a positive real part, and 
when the ith root of 
has a negative real part. Alternatively,
and
So, if we define
then we have the relationship
and combining (3) and (16) 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.
.jpg)
Figure 1 
versus 
Equations (13) and (14) show that at
, 
is an integer multiple of 
. Note now, in accordance with (6) and Figure 1, the graph of 
vs 
, that 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 
has jumped from 
to 
one more time than it has jumped from 
to 
as 
was varied over the said interval.
Thus,
is 
times the difference between the number of points at which 
jumps from 
to 
and the number of points at which 
jumps from 
to 
as 
ranges over the interval 
provided that at 
, 
is defined.
.svg.png)
Figure 2 
versus 
In the case where the starting point is on an incongruity (i.e.
, i = 0, 1, 2, ...) the ending point will be on an incongruity as well, by equation (16) (since 
is an integer and 
is an integer, 
will be an integer). In this case, we can achieve this same index (difference in positive and negative jumps) by shifting the axes of the tangent function by 
, through adding 
to 
. Thus, our index is now fully defined for any combination of coefficients in 
by evaluating 
over the interval (a,b) = 
when our starting (and thus ending) point is not an incongruity, and by evaluating
over said interval when our starting point is at an incongruity.
This difference,
, of negative and positive jumping incongruities encountered while traversing 
from 
to 
is called the Cauchy Index of the tangent of the phase angle, the phase angle being 
or 
, depending as 
is an integer multiple of 
or not.
The Routh criterion
To derive Routh's criterion, first we'll use a different notation to differentiate between the even and odd terms of
:
Now we have:
-
Therefore, if
is even,
and if
is odd:
-
Now observe that if
is an odd integer, then by (3) 
is odd. If 
is an odd integer, then 
is odd as well. Similarly, this same argument shows that when 
is even, 
will be even. Equation (13) shows that if 
is even, 
is an integer multiple of 
. Therefore, 
is defined for 
even, and is thus the proper index to use when n is even, and similarly 
is defined for 
odd, making it the proper index in this latter case.
Thus, from (6) and (22), for
even:
and from (18) and (23), for
odd:
Lo and behold we are evaluating the same Cauchy index for both:
Sturm's theorem
Sturm gives us a method for evaluating
. His theorem states as follows:
Given a sequence of polynomials
where:
1) If
then 
, 
, and 
2)
for 
and we define
as the number of changes of sign in the sequence 
for a fixed value of 
, then:
A sequence satisfying these requirements is obtained using the Euclidean algorithm, which is as follows:
Starting with
and 
, and denoting the remainder of 
by 
and similarly denoting the remainder of 
by 
, and so on, we obtain the relationships:
or in general
where the last non-zero remainder,
will therefore be the highest common factor of 
. It can be observed that the sequence so constructed will satisfy the conditions of Sturm's theorem, and thus an algorithm for determining the stated index has been developed.
It is in applying Sturm's theorem (28) to (26), through the use of the Euclidean algorithm above that the Routh matrix is formed.
We get
and identifying the coefficients of this remainder by
, 
, 
, 
, and so forth, makes our formed remainder
where
Continuing with the Euclidean algorithm on these new coefficients gives us
where we again denote the coefficients of the remainder
by 
, 
, 
, 
,
making our formed remainder
and giving us
The rows of the Routh array are determined exactly by this algorithm when applied to the coefficients of (19). An observation worthy of note is that in the regular case the polynomials
and 
have as the highest common factor 
and thus there will be 
polynomials in the chain 
.
Note now, that in determining the signs of the members of the sequence of polynomials
that at 
the dominating power of 
will be the first term of each of these polynomials, and thus only these coefficients corresponding to the highest powers of 
in 
, and 
, which are 
, 
, 
, 
, ... determine the signs of 
, 
, ..., 
at 
.
So we get
that is, 
is the number of changes of sign in the sequence 
, 
, 
, ... which is the number of sign changes in the sequence 
, 
, 
, 
, ... and 
; that is 
is the number of changes of sign in the sequence 
, 
, 
, ... which is the number of sign changes in the sequence 
, 
, 
, 
, ...
Since our chain
, 
, 
, 
, ... will have 
members it is clear that 
since within 
if going from 
to 
a sign change has not occurred, within
going from 
to 
one has, and likewise for all 
transitions (there will be no terms equal to zero) giving us 
total sign changes.
As
and 
, and from (17) 
, we have that 
and have derived Routh's theorem -
The number of roots of a real polynomial
which lie in the right half plane 
is equal to the number of changes of sign in the first column of the Routh scheme.
And for the stable case where
then 
by which we have Routh's famous criterion:
In order for all the roots of the polynomial
to have negative real parts, it is necessary and sufficient that all of the elements in the first column of the Routh scheme be different from zero and of the same sign.
-
-
-
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