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.
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
and
Similarly, if the ith root of has a negative real part,
and
and
From (9) to (11) we find that when the ith root of has a positive real part, and from (12) to (14) we find that when the ith 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 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.
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 (17) (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 (15) 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 (23), for even:
and from (19) and (24), 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.