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.
lie on the imaginary axis, and letting 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 positive real part, and from (12) to (14) we find that
Thus, So, if we define then we have the relationship and combining (3) and (17) gives us Therefore, given an equation of
, the number of roots with negative real parts and
, the number of roots with positive real parts.
In the case where the starting point is on an incongruity (i.e.
In this case, we can achieve this same index (difference in positive and negative jumps) by shifting the axes of the tangent function by
Thus, our index is now fully defined for any combination of coefficients in
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.
, of negative and positive jumping incongruities encountered while traversing
To derive Routh's criterion, first we'll use a different notation to differentiate between the even and odd terms of
odd, making it the proper index in this latter case.
odd: Lo and behold we are evaluating the same Cauchy index for both:
, then: A sequence satisfying these requirements is obtained using the Euclidean algorithm, which is as follows: Starting with
, and so on, we obtain the relationships: or in general where the last non-zero remainder,
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 (29), through the use of the Euclidean algorithm above that the Routh matrix is formed.
, 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
, 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 (20).
An observation worthy of note is that in the regular case the polynomials
Note now, that in determining the signs of the members of the sequence of polynomials
will be the first term of each of these polynomials, and thus only these coefficients corresponding to the highest powers of
and have derived Routh's theorem - The number of roots of a real polynomial
is equal to the number of changes of sign in the first column of the Routh scheme.
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.