Finite Sample Identification of Bilinear Dynamical Systems

TL;DR

This paper establishes optimal sample complexity and error bounds for identifying bilinear dynamical systems from a single trajectory under stability assumptions, using martingale small-ball techniques.

Contribution

It provides the first finite-sample analysis for bilinear system identification with optimal rates, applicable to unstable systems.

Findings

Sample complexity is optimal and scales with system dimension and input size.

Error rates do not worsen with increased instability.

Numerical experiments confirm theoretical predictions.

Abstract

Bilinear dynamical systems are ubiquitous in many different domains and they can also be used to approximate more general control-affine systems. This motivates the problem of learning bilinear systems from a single trajectory of the system's states and inputs. Under a mild marginal mean-square stability assumption, we identify how much data is needed to estimate the unknown bilinear system up to a desired accuracy with high probability. Our sample complexity and statistical error rates are optimal in terms of the trajectory length, the dimensionality of the system and the input size. Our proof technique relies on an application of martingale small-ball condition. This enables us to correctly capture the properties of the problem, specifically our error rates do not deteriorate with increasing instability. Finally, we show that numerical experiments are well-aligned with our theoretical results.