Exponential Stability and Robust Stability for Linear Time-Varying Singular Systems of Second Order Difference Equations

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In this paper, solvability, stability, and robust stability of linear time-varying singular systems of second order difference equations are studied. The leading coefficient is allowed to be singular, i.e., the system does not generate an explicit recursion. By transforming the system into an appropriate form, the existence and uniqueness of solutions are established under the so-called strangeness-free assumption. Consistent initial conditions are also explicitly constructed. Then, some criteria for exponential stability and a Bohl--Perron-type theorem are presented. Finally, we investigate the robust stability when the system coefficients are subject to structured perturbations. Examples are also given for illustration. The approach presented here can be extended to the analysis of singular systems of high order and delay difference equations.