On the variable structure control approach with sliding modes to robust finite-time consensus problems: A methodological overview based on nonsmooth analysis

When controlling cyber–physical systems via consensus algorithms, the robustness issue is of paramount importance because of model mismatching and/or disturbances that generally modify the Laplacian flow dynamics associated to the overall network, compromising the possibility to achieve any expected, orchestrated emergent behavior. To face this issue, the Variable Structure Control (VSC) approach has recently been shown to be an effective tool in designing control protocols over networks, providing robustness and often yielding finite-time convergence. Moreover, VSC further enables the decoupling of simultaneous control objectives where reaching a consensus state is only part of a more complex task, for instance as it happens in distributed optimization. Thus motivated, this paper overviews, from a tutorial perspective, some selected recent advances in the application of VSC with Sliding Modes to design robust consensus controllers in both the leader-less and the leader-following settings. Efforts are also made to unify the notation and to discuss the theoretical foundations of the nonsmooth analysis tools at the basis of their design for a wide readership unfamiliar with these problems and formal tools. To this aim, examples supporting the treatment, and numerical simulations, are given and discussed in detail. Finally, hints for future investigations along with some current open problems are provided.