Time to Failure of Quantized Control via a Binary Symmetric Channel

This paper considers the time-invariant, memorylessly quantized control of scalar plants over erroneous binary channels. As bit errors cannot be compensated for, plant stability is impossible. Thus, the simplest feasible objective is to ensure that the closed-loop is in some sense well-behaved for as long as possible. In this paper, a definition of time-to-failure is proposed for such systems, using the notions of conditional worst- and best-case states. It is shown that the stochastic evolution of this auxiliary system can be exactly characterized by a finite-state Markov chain. With this approach, explicit expressions for the time-to-failure distribution are derived for a range of non-trivial parameter values. These suggest that when the probability of bit error is small the mean time-to-failure is not an informative statistic, since it nearly coincides with the standard deviation. This motivates to instead use as a performance criterion the probability of failing after a specified time. We conclude by considering performance trade-offs when the total transmission and control power are finite