In this paper, we review how Shannon's classical notion of capacity is not enough to characterize a noisy communication channel if the channel is intended to be used as part of a feedback loop to stabilize an unstable scalar linear system. While classical capacity is not enough, another sense of capacity (parametrized by reliability) called "anytime capacity" is necessary for the stabilization of an unstable process. The required rate is given by the log of the unstable system gain and the required reliability comes from the sense of stability desired. A consequence of this necessity result is a sequential generalization of the Schalkwijk-Kailath scheme for communication over the additive white Gaussian noise (AWGN) channel with feedback. In cases of sufficiently rich information patterns between the encoder and decoder, adequate anytime capacity is also shown to be sufficient for there to exist a stabilizing controller. These sufficiency results are then generalized to cases with noisy observations, delayed control actions, and without any explicit feedback between the observer and the controller. Both necessary and sufficient conditions are extended to continuous time systems as well. We close with comments discussing a hierarchy of difficulty for communication problems and how these results establish where stabilization problems sit in that hierarchy. Index Terms-Anytime decoding, control over noisy channels, error exponents, feedback, real-time information theory, reliability functions, sequential coding. 2 The differing roles of the past and future are made clear in [6] .
Instead of the classical notion of Shannon capacity, a new notion of "anytime capacity" is proposed for NCS with a noisy communication link in REF .
The Necessity and Sufficiency of Anytime Capacity for Stabilization of a Linear System Over a Noisy Communication Link&#8212;Part I: Scalar Systems
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Computer Science Mathematics