Joint Learning of Linear Time-Invariant Dynamical Systems

TL;DR

This paper introduces methods for jointly estimating multiple linear time-invariant systems' transition matrices, leveraging shared structures to improve accuracy and robustness over individual estimations.

Contribution

It proposes a novel joint learning framework for multiple systems with shared basis matrices, providing finite-time error bounds and robustness to model misspecification.

Findings

Pooling data across systems improves estimation accuracy.

The methods are robust against model misspecifications.

Finite-time error bounds depend on trajectory length, dimension, and number of systems.

Abstract

Linear time-invariant systems are very popular models in system theory and applications. A fundamental problem in system identification that remains rather unaddressed in extant literature is to leverage commonalities amongst related linear systems to estimate their transition matrices more accurately. To address this problem, the current paper investigates methods for jointly estimating the transition matrices of multiple systems. It is assumed that the transition matrices are unknown linear functions of some unknown shared basis matrices. We establish finite-time estimation error rates that fully reflect the roles of trajectory lengths, dimension, and number of systems under consideration. The presented results are fairly general and show the significant gains that can be achieved by pooling data across systems in comparison to learning each system individually. Further, they are shown to be robust against model misspecifications. To obtain the results, we develop novel techniques that are of interest for addressing similar joint-learning problems. They include tightly bounding estimation errors in terms of the eigen-structures of transition matrices, establishing sharp high probability bounds for singular values of dependent random matrices, and capturing effects of misspecified transition matrices as the systems evolve over time.