Learning the parameters of a linear time-invariant dynamical system (LTIDS) is a problem of current interest. In many applications, one is interested in jointly learning the parameters of multiple related LTIDS, which remains unexplored to date. To that end, we develop a joint estimator for learning the transition matrices of LTIDS that share common basis matrices. Further, we establish finite-time error bounds that depend on the underlying sample size, dimension, number of tasks, and spectral properties of the transition matrices. The results are obtained under mild regularity assumptions and showcase the gains from pooling information across LTIDS, in comparison to learning each system separately. We also study the impact of misspecifying the joint structure of the transition matrices and show that the established results are robust in the presence of moderate misspecifications.

翻译：学习线性时变动态系统(LTIDS)的参数是一个当前令人感兴趣的问题。在许多应用中,人们有兴趣共同学习多个相关LTIDS的参数,这些参数迄今尚未探索。为此,我们开发了一个联合估计器,用于学习共享共同基基矩阵的LTIDS过渡矩阵。此外,我们根据过渡矩阵的基本样本大小、尺寸、任务数量和光谱特性,设定了有限时间误差界限。这些结果是在温和的常规假设下取得的,并展示了将信息汇集到LTIDS的收益,与每个系统分别学习的情况相比较。我们还研究了错误预测过渡矩阵联合结构的影响,并表明在存在中度误差的情况下,既定结果是稳健的。