  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.
