Improvement on reciprocally convex combination lemma and quadratic function negative-definiteness lemma

This paper is concerned with the stability problem for linear systems with a time-varying delay using the Lyapunov-Krasovskii (L-K) functional method. First, an a-polynomial reciprocally convex combination lemma (RCCL) is developed with an undetermined parameter m, which covers the well-known RCCLs. Second, the recently-reported necessary and sufficient negative-definiteness condition and quadratic-partitioning method are both extended to the general case when the low bound of the time-varying interval is not restricted to zero. Third, based on these new techniques, relaxed stability criteria are derived via an appropriately constructed L-K functional. Finally, three numerical examples are given to demonstrate the improvement and effectiveness of the proposed approach. (C) 2021 Published by Elsevier Ltd on behalf of The Franklin Institute.