Signal-to-noise ratio fundamental limitations in the discrete-time domain

Fundamental limitations in feedback control are a well established area of research. In recent years it has been extended to the study of limitations imposed by the consideration of a communication channel in the control loop. Previous results characterised these limitations in terms of a minimal data transmission rate necessary for stabilisation. In this paper a signal-to-noise ratio (SNR) approach is used to obtain a tight condition for the linear time invariant output feedback stabilisation of a discrete-time, single-input-single-output (SISO) unstable, non-minimum phase (NMP) plant with arbitrary relative degree over an additive Gaussian coloured noise (ACGN) communication channel with memory. The obtained result gives a guideline in estimating the severity of the fundamental SNR limitation imposed by the plant unstable poles, NMP zeros and relative degree as well as the channel NMP zeros, bandwidth, and noise colouring. We then characterise the output feedback sensitivity function for the infimal SNR solution and follow up by quantifying the extra SNR imposed by suboptimal solutions (for example due to plant modelling errors).