Signal-to-Noise Ratio Stability Analysis Subject to Variable Transmission Delay

In this letter we study the effect of an additive white noise (AWN) channel, with a variable transmission time delay, on the stability of a discrete time linear time invariant control feedback loop. The analysis proposes first the use of a buffer, for the received transmissions, of length equal to the maximum expected variable transmission delay, when each transmission can be time stamped. Alternatively, we propose modeling the AWN channel variable transmission delay as a sequence of erasure processes, representing the probability of one sample delay, two samples delay, etc., up to the maximum expected transmission delay, when the time stamping of each transmission is not possible. The overall stability of the control feedback loop is then analyzed, for both cases, in terms of the resulting communication channel signal-to-noise ratio (SNR). The second scenario, for treating the variable transmission time delay, results in a more demanding SNR for stability.