The Effect of Time Delay on the Average Data Rate and Performance in Networked Control Systems

This article studies the performance of a feedback control loop closed via an error-free digital communication channel with transmission delay. The system comprises a discrete-time noisy linear time-invariant plant, whose single measurement output is mapped into its single control input by a causal, but otherwise arbitrary, coding and control scheme. We consider a single-input multiple-output channel between the encoder–controller and the decoder–controller, which is lossless and imposes random time delay. We derive a lower bound on the minimum average feedback data rate that guarantees achieving a certain level of average quadratic performance over all possible realizations of the random delay. For the special case of a constant channel delay, we obtain an upper bound by proposing linear source-coding schemes that attain desired performance levels with rates that are at most 1.254 b per sample greater than the lower bound. We supplement our results with a numerical experiment demonstrating that the obtained bounds and operational rates are increasing functions of the constant delay.