Achievable performance of zero-delay variable-rate coding in rate-constrained networked control systems with channel delay

This paper presents an upper bound on the minimum data rate required to achieve a prescribed closed-loop performance level in networked control systems (NCSs). The considered feedback loop includes a linear time-invariant (LTI) plant with single measurement output and single control input. Moreover, in this NCS, a causal but otherwise unconstrained feedback system carries out zero-delay variable-rate coding, and control. Between the encoder and decoder, data is exchanged over a rate-limited noiseless digital channel with a known constant time delay. Here we propose a linear source-coding scheme that with the use of entropy-coded dithered quantizers (ECDQs), attains each quadratic performance level with a rate that exceeds the lower bound in [1] by at most (approximately) 1.254 bits per sample. The upper bound obtained by ECDQ is demonstrated, via simulations, to be an increasing function of the channel time delay at any given performance. In other words, attaining a specific performance level necessitates achieving a higher data rate when the channel time delay grows. The theoretical framework is demonstrated via an illustrative example.