Minimal data rates and entropy for control problems

In the theory of networked control systems, the assumption of classical control theory that information can be transmitted within control loops instantaneously, lossless and with arbitrary precision is no longer satisfied. This raises the question about the smallest possible rate of information above which a given control task can be solved.

In the theory of networked control systems, the assumption of classical control theory that information can be transmitted within control loops instantaneously, lossless and with arbitrary precision is no longer satisfied. This raises the question about the smallest possible rate of information above which a given control task can be solved. Though networked control systems can have a complicated topology, consisting of multiple sensors, controllers and actuators, a first step towards understanding the problem of minimal data rates is to analyze the simplest possible network topology, consisting of one controller and one dynamical system connected by a digital channel. In this setting, the notion of invariance entropy provides a measure for the minimal data rate associated with the control objective of rendering a subset of the state space invariant. In my talk, I will explain this concept and present results which show that the invariance entropy can be estimated in terms of dynamical quantities of the control system such as Lyapunov exponents and escape rates. In particular, I will present examples, in which an explicit formula for the entropy is available, that has some similarity with the well-known integral formula for the topological entropy of smooth maps on compact manifolds.

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