A Characterization of the Minimal Average Data Rate That Guarantees a Given Closed-Loop Performance Level

This paper studies networked control systems closed over noiseless digital channels. We focus on noisy linear time-invariant (LTI) plants with stationary Gaussian disturbances, Gaussian initial state, scalar-valued control inputs and sensor outputs. For this set-up, we show that the absolute minimal directed information rate that allows one to achieve a prescribed level of performance (not necessarily stationary), over all combinations of encoder-controller-decoder, is achieved when the decoder output is jointly Gaussian with the other signals in the system. This directed information rate lower bounds the achievable operational data rates. When restricting our attention to encoder-controller-decoders which make the random processes in the loop (strongly) asymptotically wide-sense stationary, this bound can be expressed in terms of their asymptotic power spectral densities. Then we show that the directed information rate and stationary performance of any such scheme can be achieved when the concatenated encoder, channel, controller and decoder behave as an AWGN channel with LTI filters. We also present a simple coding scheme that allows one to achieve (operational) average data rates that are at most (approximately) 1.254 bits away from the derived lower bound, while satisfying the performance constraint. A numerical example is presented to illustrate our findings.