On the complexity of linear systems: an approach via rate distortion theory and emulating systems

Abstract

We define the complexity of a continuous-time linear system to be the minimum number of bits required to describe its forward increments to a desired level of fidelity, and compute this quantity using the rate distortion function of a Gaussian source of uncertainty in those increments. The complexity of a linear system has relevance in control-communications contexts requiring local and dynamic decision-making based on sampled data representations. We relate this notion of complexity to the design of attention-varying controllers, and demonstrate a novel methodology for constructing source codes via the endpoint maps of so-called emulating systems, with potential for non-parametric, data-based simulation and analysis of unknown dynamical systems.