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.

翻译：在系统理论和应用中,线性时间变量系统是非常流行的模式。在现有的文献中,系统识别方面一个尚未解决的基本问题是利用相关线性系统之间的共同点来更准确地估计其过渡矩阵。为解决这一问题,本文件调查了联合估计多个系统过渡矩阵的方法。假设过渡矩阵是某些未知的共享基基矩阵的未知线性功能。我们建立了充分反映出轨距长度、尺寸和审议中的系统数目等作用的有限时间估计错误率。提出的结果相当笼统,表明通过将数据汇集到各系统之间,与每个系统单独学习相比,可以取得重大收益。此外,还表明它们能够有力地应对模型的误差。为了获得结果,我们开发了处理类似联合学习问题的新技术。这些方法包括:从某些未知的共享基矩阵的埃根结构上严格捆绑估计错误,为依赖性随机矩阵的单值设定高概率界限,并随着系统的发展而捕捉取错误描述过渡矩阵的影响。