Relaxed Monotonic Conditions for Schur Stability of Real Polynomials

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Sufficient Schur stability conditions of polynomials are still of much use, although the exact conditions of several forms are available. For example, a preliminary simple check method would be preferable to the intensive computation requirement of the exact stability tests. So far, a number of sufficient conditions for the Schur stability of real polynomials has been studied in [1], [2], [4]-[9]. Among these conditions, monotonic condition [2] is a typical one for Schur stability of a real polynomial. The condition is so simple that it can be checked at a glance. The monotonic condition also possesses another feature: it requires the positivity of all the coefficients of a polynomial. These two features, simplicity and positivity, could fit in some specific control applications, where controller and controlled variables are invariably non-negative [11]. A typical amongst them is a controller design of certain communication networks in which controlled variable is queue length, a positive number [10]. The motivation of the present work originates from that design scheme, where relaxation of the strict inequalities in the monotomc condition was strongly desired. This leads to the results of this note. It is pointed out that by looking into the original proof in [2] closely, the required strict inequalities can be relaxed in such a way that equalities are allowed in all the inequalities but two which are located at appropriate positions. These results will be proven with several comments in Sect. 3 after stating the Schur stability test by Jury as a preliminary lemma in Sect. 2. Section 4 concludes the note.
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2007-10-01

Relaxed Monotonic Conditions for Schur Stability of Real Polynomials

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