Stabilizing a linear system using phone calls
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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 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 setup 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 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. In the case the symbol delays are exponentially distributed, we show a tight sufficient condition using a random-coding strategy. Our results generalize previous event-triggering control approaches, revealing the limitation of using timing information for stabilization, independent of any transmission strategy.
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