Control capacity

This paper presents a notion of “control capacity” that gives a fundamental limit on the control of a system through an unreliable actuation channel. It tells us how fast we can reliably actively dissipate uncertainty in a system through that actuation channel. We give a computable single-letter characterization for scalar systems with memoryless stationary multiplicative actuation channels. The sense of control capacity is tight for answering questions of stabilizability for scalar linear systems - a system is stabilizable through an actuation channel if and only if the control capacity of that actuation channel is larger than the log of the unstable open-loop eigenvalue. For second-moment senses of stability, our result recovers the classic uncertainty-threshold principle result. However, our formulation can also deal with any other moment. The limits of higher and higher moment senses of stability correspond to a “zero-error” sense of control capacity and taking the limit to weaker-andweaker moments corresponds to a “Shannon” sense of control capacity.