On Strong Stability and Robust Strong Stability of Linear Difference Equations with Two Delays

This paper provides a necessary and sufficient condition for guaranteeing exponential 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 $n\times n$ square matrices, in terms of a linear matrix inequality (LMI) of size $\left( k+1\right) n\times \left( k+1\right) n$ where $k\geq1$ is some integer. Different from an existing condition where the coefficients $\left( A,B\right)$ appear as highly nonlinear functions, the proposed LMI condition involves matrices that are linear functions of $\left( A,B\right) .$ Such a property is further used to deal with the robust stability problem in case of norm bounded uncertainty and polytopic uncertainty, and the state feedback stabilization problem. Solutions to these two problems are expressed by LMIs. A time domain interpretation of the proposed LMI condition in terms of Lyapunov-Krasovskii functional is given, which helps to reveal the relationships among the existing methods. Numerical example demonstrates the effectiveness of the proposed method.