Stabilizing A Linear System Using Phone Calls

We consider the problem of stabilizing an undisturbed, scalar, linear system over a "timing" channel, namely a channel where information is communicated through the times-tamps of the transmitted symbols. Each transmitted symbol is received at the controller subject to some to random delay. The sensor can encode messages in the holding 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 the receiver through a "ring" and, after a random time required to establish the connection, is aware of the "ring" being received. We show that for the state 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. In the case the symbol delays are exponentially distributed, we show a tight sufficient condition using a decoding strategy that successively refines the estimate of the decoded message every time a new symbol is received. These results extend our previous work on estimation over the timing channel to stabilization.