Multi-Task System Identification of Similar Linear Time-Invariant Dynamical Systems

This paper presents a system identification framework -- inspired by multi-task learning -- to estimate the dynamics of a given number of linear time-invariant (LTI) systems jointly by leveraging structural similarities across the systems. In particular, we consider LTI systems that model networked systems with similar connectivity, or LTI systems with small differences in their matrices. The system identification task involves the minimization of the least-squares (LS) fit for individual systems, augmented with a regularization function that enforces structural similarities. The proposed method is particularly suitable for cases when the recorded trajectories for one or more LTI systems are not sufficiently rich, leading to ill-conditioning of LS methods. We analyze the performance of the proposed method when the matrices of the LTI systems feature a common sparsity pattern (i.e., similar connectivity), and provide simulations based on real data for the estimation of the brain dynamics. We show that the proposed method requires a significantly smaller number of fMRI scans to achieve similar error levels of the LS.