Stabilizing a Linear System Using Phone Calls When Time is Information

We study stabilizing a scalar linear system over a “timing” channel, where information is communicated via the timestamps of transmitted symbols. Each symbol, sent from a sensor to a controller in a closed-loop system, is received after a random delay. The sensor encodes information in the waiting times between transmissions, and the controller decodes it using the inter-reception times of symbols. This setup resembles a telephone system: a transmitter signals a call with a “ring,” and the receiver becomes aware of the “ring” after a random connection delay. With no data payload exchanged, this setup provides an abstraction for performing event-triggering control with zero-payload rate. We show that for system stabilization (state converging to zero in probability), the timing capacity of the channel must be at least as large as the entropy rate of the system. For exponentially distributed symbol delays, we provide an “almost” tight sufficient condition using a coding strategy that refines the message estimate with each new symbol. Our results generalize previous zero-payload event-triggering control strategies, revealing a fundamental limit for using timing information in stabilization, independent of the transmission strategy.