# Control Capacity

## TL;DR

This paper introduces the concept of control capacity, a fundamental limit on how quickly a controller can reduce uncertainty in a system, providing a new way to analyze stabilizability in control systems using information-theoretic principles.

## Contribution

It defines control capacity for scalar systems, offers a computable characterization, and connects control stabilizability with information capacity, extending classic results to arbitrary moments of stability.

## Key findings

- Control capacity determines stabilizability of scalar systems.
- For second-moment stability, recovers classic uncertainty threshold.
- Framework parallels Shannon capacity, enabling computation of side-information value.

## 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.