D.B. Hassibi, Professor and Executive Officer of Electrical Engineering
California Institute of Technology, Pasadena, CA
Two theories that have served as the underpinning of many of the technological advancements of the past six decades are information theory and control theory. The former spawned much of today's telecommunications and information technologies and deals with how to reliably transmit information over unreliable channels. It does so by ignoring real-time constraints and allowing for arbitrary long delays in encoding and decoding at the transmitter and receiver. The latter is responsible for advancements in the space age, guidance, etc., and developed independently, ostensibly because it had to deal with real-time constraints upfront and since controllers must take action using information only currently available. This approach works fine when the plant, controller and observer are collocated. However, there are ever-increasing applications where these entities are distributed in different locations and where they exchange information (measurements and control signals) over unreliable channels and networks. It is not hard to show that, in such settings, a purely information-theoretic or purely control-theoretic approach does not work: one needs to deal with both the real-time constraints and the underlying unreliability in a simultaneous and systematic way.
We will review the works of Schulman and Sahai, developed over the past two decades, that study such problems and that introduce the notions of "tree codes" and "anytime capacity", respectively. Tree codes are a new construct in coding theory that allows real-time control over unreliable channels. While their existence was shown by Schulman in 1994, the field has largely remained dormant because to date there have been no explicit constructions of tree codes and no efficient encoding and decoding schemes. We will show the existence with "high probability" of "linear" tree codes and, for the first time, construct explicit codes with efficient encoding and decoding for the erasure channel. We show the efficacy of the method by stabilizing example unstable plants over erasure channels, and argue how this enables the solution of several problems which were hitherto beyond reach.