RouthProof2 - 1306 IEEE TRANSACTIONS ON AUTOMATIC CONTROL...

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1306 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-
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