Tractable Razumikhin-type conditions for input-to-state stability analysis of delay difference inclusions

For the stability analysis of time-delay systems, the Razumikhin approach provides (at the cost of some conservatism) a set of conditions that are relatively easy to verify when compared to the Krasovskii approach. Unfortunately, currently, for linear delay difference inclusions (DDIs) verification of these conditions is only possible by solving a bilinear matrix inequality (BMI). To obtain a tractable stability analysis method for DDIs, an alternative set of Razumikhin-type conditions is proposed in this paper, which are based on a technique that was developed for interconnected systems in Willems (1972). In particular, via the proper selection of storage and supply functions, these conditions can be used to establish input-to-state stability (ISS) for general DDIs. When linear DDIs and quadratic functions are considered, ℓ2ℓ2-disturbance attenuation can be established by solving a single linear matrix inequality (LMI). Moreover, this LMI is shown to be less conservative than the BMI corresponding to the existing Razumikhin-type conditions for linear DDIs.