On the Monotonic Condition for Schur Stability of Real Polynomials

Abstract

It is well known that an nth-order real polynomial D(z) = Sigma(n)(i=0) d(i)z(i) is Schur stable if its coefficients satisfy the monotonic condition, i.e., d(n) > d(n-1) > ... > d(1) > d(0) > 0. In this letter it is shown that even if the monotonic condition is violated by one coefficient (say d(k)), D(z) is still Schur stable if the deviation of d(k) from d(k+1) or d(k-1) is not too large. More precisely we derive upper bounds for the admissible deviations of d(k) from d(k+1) or d(k-1) to ensure the Schur stability of D(z). It is also shown that the results obtained in this letter always yield the larger stability range for d(k) than an existing result.