On strong stability and robust strong stability of linear difference equations with two delays

Abstract

This note studies strong stability of the linear difference equation x(t)=Ax(t−a)+Bx(t−b) where a>0,b>0 are constants and A,B are n×n square matrices. A necessary and sufficient condition in terms of a linear matrix inequality (LMI), where the coefficients A,B appear as a linear function, is introduced. Such a condition is further used to deal with the robust strong stability problem with norm bounded uncertainty, whose solutions are expressed by LMIs. A time-domain interpretation of the LMI condition in terms of integral Lyapunov–Krasovskii functional is given, which helps to reveal the relationships among the existing methods. A numerical example demonstrates the effectiveness of the proposed method.