Finite Sample Identification of Bilinear Dynamical Systems
This work addresses the challenge of system identification for bilinear systems, which are common in various domains, providing theoretical guarantees for data efficiency.
The paper tackles the problem of learning bilinear dynamical systems from a single trajectory of states and inputs, establishing optimal sample complexity and statistical error rates under mild stability assumptions, with results validated by numerical experiments.
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.