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Minimal data rates and entropy in digitally networked systems

In digitally networked systems the assumption of classical control theory that information can be transmitted instantaneously, lossless and with arbitrary precision is violated. This raises the question about the smallest data rate above which a control task can be solved. For a single control loop and the problem to make a set Q of states invariant, the minimal data rate can be described by an entropy‐like quantity, the so‐called invariance entropy. Under some controllability and hyperbolicity assumptions, the invariance entropy can be expressed in terms of Lyapunov exponents. Furthermore, one can show that for making Q invariant with a data rate close to the smallest, no strategies more complicated than stabilization at periodic trajectories are necessary. For a network with n subsystems, which can all communicate with each other, there are different ways to formulate the question about the smallest data rate for the invariance problem, but also in this setting entropy‐like quantities yield important information. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)