Effects of Time Delay on Feedback Stabilisation Over Signal-to-Noise Ratio Constrained Channels

Abstract

The expanding integration of control and communication networks in recent years has generated an increasing interest in control problems with feedback over a communication channel. Significant research activity has concentrated on stabilisation in face of channel effects such as quantisation and data-rate limits. In a recent paper, the authors have studied the problem of feedback stabilisation over a communication channel with a constraint on the admissible signal-to-noise ratio (SNR). It has been shown therein that for a delay-free, linear time-invariant feedback loop, a SNR constraint in the feedback channel imposes fundamental limitations in the ability to achieve closed-loop stability. The present paper extends these results by introducing a time-delay in the loop, and shows that the lowest SNR required for closed-loop stability increases by a factor that may grow exponentially with the time-delay and the unstable open loop poles of the system. This result contributes to the quantification of performance tradeoffs in integrated control and communication environments.

Fundamental limitations in control over a communication channel

“…For the problem of feedback stabilisation, such limitations are quantified by the optimal H 2 norm of the closed-loop transfer function between the channel noise input and the channel signal input, and are expressed as a function of the plant unstable poles, NMP zeros, and time-delay. It is also shown in Braslavsky et al (2007) that for state feedback, or for a minimum-phase, delay-free plant, the infimal SNR condition matches the exact same requirement on channel transmission data rate derived in the general formulation by Nair and Evans (2004).…”

“…It is shown in Braslavsky et al (2007) that the presence of time-delay increases the SNR required to stabilise an unstable system. In Braslavsky et al (2005), time-delays are treated in continuous time, with a similar conclusion, albeit with the considerably greater technical complexity required to treat infinite dimensional systems. It follows from Braslavsky et al (2005) that for a system with one unstable pole the increase in SNR is exponential in the magnitude of the time-delay and that of the unstable pole.…”

“…A framework for the study of control over a communication channel with a constraint on its signal-to-noise ratio (SNR) has been developed in Braslavsky et al (2007). In that paper, the authors derive the infimal SNR required to stabilise an unstable plant by feedback over an additive white Gaussian noise (AWGN) communication channel.…”

“…In Braslavsky et al (2007) time-delays are treated only in discrete-time, wherein they are simpler because they appear as an increase in relative degree of a finite dimensional transfer function. It is shown in Braslavsky et al (2007) that the presence of time-delay increases the SNR required to stabilise an unstable system.…”

“…In Braslavsky et al (2007) time-delays are treated only in discrete-time, wherein they are simpler because they appear as an increase in relative degree of a finite dimensional transfer function. It is shown in Braslavsky et al (2007) that the presence of time-delay increases the SNR required to stabilise an unstable system. In Braslavsky et al (2005), time-delays are treated in continuous time, with a similar conclusion, albeit with the considerably greater technical complexity required to treat infinite dimensional systems.…”