Input-to-state-stability-type comparison principles and input-to-state stability for discrete-time dynamical networks with time delays

This paper investigates the input-to-state stability (ISS) issue for discrete-time dynamical networks (DDNs) with time delays. Firstly, a general comparison principle for solutions of DDNs is proposed. Then, based on this general comparison principle, three kinds of ISS-type comparison principles for DDNs are established, including the comparison principle for input-to-state Κβ-stability, ISS, and exponential ISS. The ISS-type comparison principles are then used to investigate stability properties related to ISS for three kinds (linear, affine, and nonlinear) of DDNs. It shows that the ISS property of a DDN can be derived by comparing it with a linear or lower-dimension DDN with known ISS property. By using methods such as variation of parameters, uniform M-matrix, and the ISS-type comparison principle, conditions of global exponential ISS for time-varying linear DDNs with time delays are derived. Moreover, the obtained ISS results for DDNs are extended to the hybrid DDNs with time delays. As one application, the synchronization within an error bound in the sense of ISS is achieved for DDNs with coupling time delays and external disturbances. Finally, two examples are given to illustrate the results.