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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 6, JUNE 1999
A Simple Proof of the Routh Test
A. Ferrante, A. Lepschy, and U. Viaro
Abstract—
An elementary proof of the classic Routh method for count-
ing the number of left half-plane and right half-plane zeroes of a real
coefficient polynomial
of degree
is given. Such a proof refers to
the polynomials
of degree
formed from the entries of the
rows of order
and
of the relevant Routh array. In particular, it is
based on the consideration of an auxiliary polynomial
, linearly
dependent on a real parameter
, which reduces to either polynomial
or to polynomial
for particular values of
. In this way,
it is easy to show that
zeroes of
lie in the same half-plane
as the zeroes of
, and the remaining zero lies in the left or in
the right half-plane according to the sign of the ratio of the leading
coefficients of
and
. By successively applying this property
to all pairs of polynomials in the sequence, starting from
and
, the standard rule for determining the zero distribution of
is immediately derived.
Index Terms—
Continuous curves, Routh test, stability, zeroes of poly-
nomials.
I.
I
NTRODUCTION
Most undergraduate control courses, and the related textbooks,
include the Routh–Hurwitz stability criterion. A recent survey [1],
conducted by the Working Group on Curriculum Development of the
IEEE Control Systems Society, shows that this topic is covered in
93% of Electrical Engineering departments and 89% of Mechanical
Engineering departments. The proof of the criterion, however, is very
often omitted, or it is limited to the “stability” condition, stating
that the sign of the leading elements of all rows in the Routh array
must be the same; very seldomly is it proved that the number of
the left half-plane (LHP) zeroes of the considered polynomial is
equal to the number of sign permanencies between consecutive entries
encountered along the first column of the table and that the number
of the right half-plane (RHP) zeroes to that of the sign variations.
The classical proofs available in the literature are diverse; the main
approaches are based on Cauchy’s indexes and Sturm sequences, on
continued fraction expansions, on the Hermite–Biehler theorem, and
on the second method of Lyapunov (cf., e.g., [2]–[4]). The importance
of the topic and the complexity of the mathematical tools required in
most of the above proofs motivate the continual interest in the subject
(see, e.g., [5]). Furthermore, the critical cases in the construction of
the Routh table are usually treated in a fashion that bears no relation
to the proof.
Another reason why the proofs are often omitted in standard
textbooks is that they give little insight into the nature of the
algorithm.
This paper presents an alternative proof of the Routh rules for
determining the zero distribution (with respect to the imaginary axis)
of a polynomial with real coefficients. The advantage of the suggested
approach consists of the simplicity of the mathematical tools required:
Manuscript received March 19, 1997; revised November 5, 1997. Recom-