Fundamental limitations in control over a communication channel

Abstract

Fundamental limitations in feedback control is 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 continuous-time, unstable, non minimum phase (NMP) plant with time-delay over an additive Gaussian coloured noise communication channel. By working on a linear setting the infimal SNR for stabilisability is defined as the infimal achievable H 2 norm between the channel noise input and the channel signal input. The result gives a guideline in estimating the severity of the fundamental SNR limitation imposed by the plant unstable poles, NMP zeros, time-delay as well as the channel NMP zeros, bandwidth, and channel noise colouring.

Channel signal-to-noise ratio constrained feedback control: performance and robustness

“…The proof of (7) is given in [17] and is included in the Appendix 1 for completeness. Formula (7) presents explicitly the main obstacles to feedback stability in terms of a limitation in the channel SNR, that is: unstable poles, NMP zeros and time delay.…”

“…The analysis introduced in [15,16] considers an additive white Gaussian noise (AWGN) channel model, casting the stabilisability problem of a linear time invariant (LTI) unstable plant as one of lower bounding the channel signal-to-noise ratio (SNR). In [17] we extended the formulation of [15,16] to the case of an additive coloured Gaussian noise (ACGN) channel with a bandwidth limitation. Such bandwidth limitation may be imposed, for example, to avoid interference between different channels, while the coloured noise assumption is more realistic for a general communication channel.…”

Stabilization over frequency‐selective channels subject to transmission delay and signal‐to‐noise ratio limitations

“…In these papers, the minimum signal-to-noise ratio (SNR) of Correspondence to: Reza Mahboobi Esfanjani, E-mail: mahboobi@sut.ac.ir channel required for stabilizability of the system was obtained in terms of the plant's unstable poles and nonminimum phase (NMP) zeros. Similar problem was solved in [17], where the communication medium is an additive colored Gaussian noise channel with finite bandwidth. Utilizing an output feedback controller, it was shown that the bandwidth limitation of channel leads to greater minimum SNR required for stabilization than the one obtained in [16].…”

“…Owing to using a frequencyselective notion of SNR, the problem was solved in a simpler way than the similar one in [17]. The presented stabilizability threshold includes NMP zeros of the channel besides its other factors and consequently the effect of these zeros on system stabilizability was shown for the first time.…”

Noise attenuation over additive Gaussian channels

“…Among several research directions on this topic, a great deal of attention has been given to the case when the communication channels can be modeled as additive white gaussian noise (AWGN) channels [1]. The issue of time delay has been addressed in [2], the encoder/decoder design has been addressed in [3], and the disturbance rejection has been studied in [4].…”