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Performance Limitations of Linear Systems over Additive White Noise Channels


This thesis develops a framework to address the performance limits of feedback control systems with communication constraints modeled by additive white noise channels. By searching for the fundamental bounds on the control performance, we explore the relationship between the known limitations caused by the intrinsic properties of linear control systems and the characteristics of the communication channels. We analyze multiple-input multiple-output systems with the channel placed at either the uplink or downlink. We also study the stabilization conditions for single-input single-output systems when both channels are present in the closed loop.

For systems with uplink channels, we derive explicitly the analytical expressions for the necessary and sufficient conditions for stabilization and the best achievable performance under the channel input power constraint. The optimal tracking performance exhibits clear dependence on the power constraint and noise levels of the channel, and additionally on the unstable poles and nonminimum phase zeros of the plant. For systems with downlink channels, we derive a lower bound for the performance that incorporates the plant gain in the entire frequency range. Moreover, we use and optimize scaling as a method of channel compensation to exploit the channel and deal with the white noise. This simple strategy is shown to significantly improve the tracking performance. Also, we attempt to discover the optimal power allocation for each of the uplink parallel channels to achieve the best tracking performance. It is shown that, the optimal strategy is to allocate more power to a more problematic channel, in contrast to the widely-known ``water-filling'' solution, which is to maximize the capacity. Lastly, for first-order systems controlled over both uplink and downlink channels, we analyze the achievable region of the signal-to-noise ratios of the channels for stabilizability.

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