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：线性时不变系统是系统理论及应用中的经典模型。现有文献中尚未充分探讨的一个系统辨识基础性问题，是如何利用相关线性系统间的共性来更精确地估计其转移矩阵。针对该问题，本文研究了联合估计多个系统转移矩阵的方法。我们假设各系统的转移矩阵是某些未知共享基矩阵的未知线性函数，并建立了有限时间估计误差率，该误差率完整反映了轨迹长度、维度和系统数量等因素的作用。所提出结果具有较强普适性，充分展示了相较于独立学习各系统，跨系统数据聚合所能带来的显著增益。此外，这些结果对模型误设具有鲁棒性。为获得上述结论，我们发展了解决类似联合学习问题的新技术，具体包括：基于转移矩阵特征结构对估计误差进行紧界约束，建立依赖随机矩阵奇异值的尖锐高概率界，以及刻画随时间演化的误设转移矩阵的影响。