TY - GEN

T1 - Less than 1-Bit Control of an Unstable AR Process with 1-Bit Quantizers

AU - Bonen, Rachel

AU - Cohen, Asaf

N1 - Publisher Copyright:
© 2024 IEEE.

PY - 2024/1/1

Y1 - 2024/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85202838659&partnerID=8YFLogxK

U2 - 10.1109/ISIT57864.2024.10619331

DO - 10.1109/ISIT57864.2024.10619331

M3 - Conference contribution

AN - SCOPUS:85202838659

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 3243

EP - 3248

BT - 2024 IEEE International Symposium on Information Theory, ISIT 2024 - Proceedings

PB - Institute of Electrical and Electronics Engineers

T2 - 2024 IEEE International Symposium on Information Theory, ISIT 2024

Y2 - 7 July 2024 through 12 July 2024

ER -