Less than 1-Bit Control of an Unstable AR Process with 1-Bit Quantizers

Consider the problem of controlling an unstable process while using as little information as possible. There is a fundamental trade-off between the amount of information used per sample and the controller's performance. Indeed, very recently, Kostina et al. 2022 considered the specific case of an unstable AR(1) process with bounded noise in [-B, B] and showed that with a fixed rate per sample control, an unstable process with a gain α > 1 has a fundamental limit of ⌊α⌋+1 quantization levels per sample. E.g., with 1 < α < 2, there is a fundamental limit of 1 bit per sample. We revisit the problem assuming an average rate per sample constraint. We show that not only the 1 bit per sample bound is pessimistic, one can control an unstable AR(1) process at a negligible rate for reasonable α, with a simple interleaved application of a 1-bit quantizer. In this case, we derive a new converse result for average rate control and show it is much lower than that with a fixed rate. The achievable scheme we suggest asymptotically matches the lower bound. We give a practical and simple controller with a bit-rate close to the theoretical limit shown in [l]. For example, with α=1+ϵ, we prove that a rate of log2(1+ϵ)=ϵ/ln(2)+-O(ϵ2) bits per sample is necessary, yet a rate 1/⌊/in(2)/ϵ⌋is achievable.