New solution bounds for unified algebraic Lyapunov equation and robust stability in delta operator system

Abstract

In this paper, we utilize matrix transformations and inequalities to derive a novel upper bound and two lower bounds to solve the unified algebraic Lyapunov matrix equation (UALE). We then review existing bounds for the UALE and compare them with our new bounds, highlighting that our upper bound is the least restrictive among current results. The restrictions of our newly established lower bound are either weaker than the existing lower bounds or consistent with them. Our upper and lower bounds demonstrate increased accuracy over existing results through some numerical examples. As an application to linear systems, we illustrate how our upper bound can be employed to analyze the robust stability of the unified system based on the delta operator. Finally, we validate the effectiveness and superiority of our results through a series of numerical examples.