On the Monotonic Condition for Schur Stability of Real Polynomials

Abstract

It is well known that an nth-order real polynomial D(z)=∑_i=0ⁿd_izⁱ is Schur stable if its coefficients satisfy the monotonic condition, i.e., d_n>d_n-1>…>d₁>d₀>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.