Stability analysis of networked linear control systems with direct-feedthrough terms

We consider networked control systems (NCSs) composed of a linear plant and a linear controller interconnected by packet-based communication channels with communication constraints. We are interested in the setup where direct-feedthrough terms are present in the plant and/or in the controller, a case that is largely ignored in the literature due to its inherent complexity and counterintuitive results in the analysis despite its relevance for important classes of controllers including Proportional-Integral (PI) regulators. This setup calls for a novel stability analysis, for which we take a renewed look at the concept of uniformly globally exponentially stable (UGES) scheduling protocols that turned out to be instrumental in earlier approaches. We provide a generalization of the UGES property, called (DP,DC)-UGES with DP∕DC being the direct-feedthrough matrices of the plant/controller, respectively, and we present generic conditions on these direct-feedthrough terms DP∕DC such that the classical UGES property of scheduling protocols implies (DP,DC)-UGES. This allows us to derive conditions leading to a maximally allowable transmission interval (MATI) such that stability of the overall NCS is guaranteed. In addition, it is shown that it is possible to get more tailored results for the well-known sampled-data (SD), round-robin (RR), and try-once-discard (TOD) protocols leading to less conservative conditions on the direct-feedthrough terms than the generic ones. We also introduce new (DP,DC)-UGES scheduling protocols, designed to handle the direct-feedthrough terms in a more effective way than existing protocols. Our results are illustrated using the example of a batch reactor.