Linear dynamical systems are the foundational statistical model upon which control theory is built. Both the celebrated Kalman filter and the linear quadratic regulator require knowledge of the system dynamics to provide analytic guarantees. Naturally, learning the dynamics of a linear dynamical system from linear measurements has been intensively studied since Rudolph Kalman's pioneering work in the 1960's. Towards these ends, we provide the first polynomial time algorithm for learning a linear dynamical system from a polynomial length trajectory up to polynomial error in the system parameters under essentially minimal assumptions: observability, controllability, and marginal stability. Our algorithm is built on a method of moments estimator to directly estimate Markov parameters from which the dynamics can be extracted. Furthermore, we provide statistical lower bounds when our observability and controllability assumptions are violated.

翻译：线性动态系统是建立控制理论的基础统计模型。 著名的卡尔曼过滤器和线性二次调节器都需要对系统动态的了解才能提供分析保证。 当然, 自1960年代鲁道夫·卡尔曼的开创性工作以来,通过线性测量对线性动态系统的动态进行了深入研究。 为达到这些目的,我们提供了第一个多元时间算法,用于从多元长度轨迹中学习线性动态系统,直到系统参数中的多元长度错误,其基本假设为:可观察性、可控性和边际稳定性。我们的算法建立在直接估计马尔科夫参数的时空估计法上,可以从中提取动态参数。此外,当我们的可观察性和可控性假设被违反时,我们提供了较低的统计界限。