A New Approach to Learning Linear Dynamical Systems
This solves a foundational problem in control theory for researchers and practitioners, enabling efficient system identification with theoretical guarantees.
The authors tackled the problem of learning linear dynamical systems from trajectory data, providing the first polynomial-time algorithm that achieves polynomial error under minimal assumptions of observability, controllability, and marginal stability, with statistical lower bounds for when these assumptions are violated.
Linear dynamical systems are the foundational statistical model upon which control theory is built. Both the celebrated Kalman filter and the linear quadratic regulator require knowledge of the system dynamics to provide analytic guarantees. Naturally, learning the dynamics of a linear dynamical system from linear measurements has been intensively studied since Rudolph Kalman's pioneering work in the 1960's. Towards these ends, we provide the first polynomial time algorithm for learning a linear dynamical system from a polynomial length trajectory up to polynomial error in the system parameters under essentially minimal assumptions: observability, controllability, and marginal stability. Our algorithm is built on a method of moments estimator to directly estimate Markov parameters from which the dynamics can be extracted. Furthermore, we provide statistical lower bounds when our observability and controllability assumptions are violated.