We consider the problem of learning a realization of a partially observed bilinear dynamical system (BLDS) from noisy input-output data. Given a single trajectory of input-output samples, we provide a finite time analysis for learning the system's Markov-like parameters, from which a balanced realization of the bilinear system can be obtained. Our bilinear system identification algorithm learns the system's Markov-like parameters by regressing the outputs to highly correlated, nonlinear, and heavy-tailed covariates. Moreover, the stability of BLDS depends on the sequence of inputs used to excite the system. These properties, unique to partially observed bilinear dynamical systems, pose significant challenges to the analysis of our algorithm for learning the unknown dynamics. We address these challenges and provide high probability error bounds on our identification algorithm under a uniform stability assumption. Our analysis provides insights into system theoretic quantities that affect learning accuracy and sample complexity. Lastly, we perform numerical experiments with synthetic data to reinforce these insights.

翻译：本文研究从含噪声的输入输出数据中学习部分观测双线性动力系统(BLDS)实现的问题。给定输入输出样本的单条轨迹，我们对学习系统的类马尔可夫参数进行了有限时间分析，基于这些参数可获得双线性系统的平衡实现。我们的双线性系统辨识算法通过将输出回归到高度相关、非线性且重尾的协变量来学习系统的类马尔可夫参数。此外，BLDS的稳定性取决于用于激励系统的输入序列。这些部分观测双线性动力系统特有的性质，对我们学习未知动力学算法的分析构成了重大挑战。我们通过建立均匀稳定性假设，解决了这些挑战，并为辨识算法提供了高概率误差界。我们的分析揭示了影响学习精度与样本复杂度的系统理论量。最后，我们通过合成数据进行了数值实验以验证这些理论认识。