Title: Control Capacity,
Abstract: Feedback control actively dissipates uncertainty from a dynamical system by
means of actuation. We develop a notion of "control capacity" that gives a
fundamental limit (in bits) on the rate at which a controller can dissipate the
uncertainty from a system, i.e. stabilize to a known fixed point. We give a
computable single-letter characterization of control capacity for memoryless
stationary scalar multiplicative actuation channels. Control capacity allows us
to answer questions of stabilizability for scalar linear systems: a system with
actuation uncertainty is stabilizable if and only if the control capacity is
larger than the log of the unstable open-loop eigenvalue.
For second-moment senses of stability, we recover the classic uncertainty
threshold principle result. However, our definition of control capacity can
quantify the stabilizability limits for any moment of stability. Our
formulation parallels the notion of Shannon's communication capacity, and thus
yields both a strong converse and a way to compute the value of
side-information in control. The results in our paper are motivated by
bit-level models for control that build on the deterministic models that are
widely used to understand information flows in wireless network information
theory.