This article investigates the controllability of networked sampled-data systems with various time delays on both control and transmission channels. Necessary and sufficient controllability conditions are first derived for systems with a single delay and then extended to systems with multiple delays. It is found that delays in control signals have no effects on the overall controllability. For a networked system whose topology matrix has only zero eigenvalues, delays of neither control nor transmission signals will affect the overall controllability. It is proved that an uncontrollable mode 1 of such a networked sampled-data system cannot be altered by arbitrary delays. Finally, the networked sampled-data system with first-order holders is discussed, which is modeled as a variant of time-delayed system, and some easy-to-verify algebraic conditions on the controllability are given based on matrix rank checking.

The controllability analysis of networked systems is challenging due to their high dimensionality and complex structure. The influence of sampling on network controllability is rarely studied, making it an important topic to explore. In this article, the state controllability of multilayer networked sampled-data systems is studied, considering the deep network structure, multidimensional node dynamics, various inner couplings, and sampling patterns. Necessary and/or sufficient controllability conditions are proposed and validated by numerical and practical examples, requiring less computation than the classic Kalman criterion. Single-rate and multirate sampling patterns are analyzed, showing that adjusting the sampling rate of local channels can affect the controllability of the overall system. It is shown that the pathological sampling of single-node systems can be eliminated by an appropriate design of interlayer structures and inner couplings. In the case of systems with drive-response mode, the overall system may not lose controllability even when the response layer is uncontrollable. The results demonstrate that mutually coupled factors collectively affect the controllability of the multilayer networked sampled-data system.