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Abstract-There has recently been significant interest in feedback stabilization problems with communication constraints including constraints on the available data rate. Signal-to-noise ratio (SNR) constraints are one way in which data-rate limits arise, and are the focus of this paper. In both continuous and discrete-time settings, we show that there are limitations on the ability to stabilize an unstable plant over a SNR constrained channel using finite-dimensional linear time invariant (LTI) feedback. In the case of state feedback, or output feedback with a delay-free, minimum phase plant, these limitations in fact match precisely those that might have been inferred by considering the associated ideal Shannon capacity data rate over the same channel. In the case of LTI output feedback, additional limitations are shown to apply if the plant is nonminimum phase. In this case, we show that for a continuous-time nonminimum phase plant, a periodic linear time varying feedback scheme with fast sampling may be used to recover the original SNR requirement at the cost of robustness properties. The proposed framework inherently captures channel noise effects in a simple formulation suited to conventional LTI control performance and robustness analysis, and has potential to handle time delays and bandwidth constraints in a variety of control over communication links problems.

Structural controllability has been proposed as an analytical framework for making predictions regarding the control of complex networks across myriad disciplines in the physical and life sciences (Liu et al., Nature:473(7346):167–173, 2011). Although the integration of control theory and network analysis is important, we argue that the application of the structural controllability framework to most if not all real-world networks leads to the conclusion that a single control input, applied to the power dominating set, is all that is needed for structural controllability. This result is consistent with the well-known fact that controllability and its dual observability are generic properties of systems. We argue that more important than issues of structural controllability are the questions of whether a system is almost uncontrollable, whether it is almost unobservable, and whether it possesses almost pole-zero cancellations.

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.

There has recently been significant interest in feedback stabilization problems with communication constraints including constraints on the available data rate. Signal-to-noise ratio (SNR) constraints are one way in which data-rate limits arise, and are the focus of this paper. In both continuous and discrete-time settings, we show that there are limitations on the ability to stabilize an unstable plant over a SNR constrained channel using finite-dimensional linear time invariant (LTI) feedback. In the case of state feedback, or output feedback with a delay-free, minimum phase plant, these limitations in fact match precisely those that might have been inferred by considering the associated ideal Shannon capacity data rate over the same channel. In the case of LTI output feedback, additional limitations are shown to apply if the plant is nonminimum phase. In this case, we show that for a continuous-time nonminimum phase plant, a periodic linear time varying feedback scheme with fast sampling may be used to recover the original SNR requirement at the cost of robustness properties. The proposed framework inherently captures channel noise effects in a simple formulation suited to conventional LTI control performance and robustness analysis, and has potential to handle time delays and bandwidth constraints in a variety of control over communication links problems.

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