We consider the problem of stabilizing an undisturbed, scalar, linear system over a timing channel, namely a channel where information is communicated through the timestamps of the transmitted symbols. Each symbol transmitted from a sensor to a controller in a closed-loop system is received subject to some to random delay. The sensor can encode messages in the waiting times between successive transmissions and the controller must decode them from the inter-reception times of successive symbols. This set-up is analogous to a telephone system where a transmitter signals a phone call to a receiver through a ring and, after the random delay required to establish the connection, the receiver is aware of the ring being received. Since there is no data payload exchange between the sensor and the controller, the set-up provides an abstraction for performing event-triggering control with zero payload rate. We show the following requirement for stabilization: for the state of the system to converge to zero in probability, the timing capacity of the channel should be at least as large as the entropy rate of the system. Conversely, in the case the symbol delays are exponentially distributed, we show a tight sufficient condition using a coding strategy that refines the estimate of the decoded message every time a new symbol is received. Our results generalize previous event-triggering control approaches, revealing a fundamental limit in using timing information for stabilization, independent of any transmission strategy.
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