This paper considers the problem of system identification (ID) of linear and nonlinear non-autonomous systems from noisy and sparse data. We propose and analyze an objective function derived from a Bayesian formulation for learning a hidden Markov model with stochastic dynamics. We then analyze this objective function in the context of several state-of-the-art approaches for both linear and nonlinear system ID. In the former, we analyze least squares approaches for Markov parameter estimation, and in the latter, we analyze the multiple shooting approach. We demonstrate the limitations of the optimization problems posed by these existing methods by showing that they can be seen as special cases of the proposed optimization objective under certain simplifying assumptions: conditional independence of data and zero model error. Furthermore, we observe that our proposed approach has improved smoothness and inherent regularization that make it well-suited for system ID and provide mathematical explanations for these characteristics' origins. Finally, numerical simulations demonstrate a mean squared error over 8.7 times lower compared to multiple shooting when data are noisy and/or sparse. Moreover, the proposed approach can identify accurate and generalizable models even when there are more parameters than data or when the underlying system exhibits chaotic behavior.
翻译:本文探讨了从繁琐和稀少的数据中找出线性和非线性非自主系统(ID)系统的问题。我们提议和分析一种来自巴伊西亚配方的客观功能,用于学习隐蔽的Markov模型,并带有随机动态。然后,我们结合线性和非线性系统ID的几种最先进的方法分析这一客观功能。在前者,我们分析Markov参数估计的最小方位方法,而在后者,我们分析多射击方法。我们通过表明这些现有方法带来的优化问题的局限性,表明它们在某些简化假设(数据有条件独立和零模型错误)下可以被视为拟议的优化目标的特殊案例。此外,我们注意到,我们拟议的方法改善了光滑和内在的规范化,使之适合于系统标识,并为这些特征的起源提供了数学解释。最后,数字模拟表明,在数据噪音和/或稀疏散时,其平均正方位错误比多射击低8.7倍。此外,拟议的方法可以确定准确和一般的模型,即使参数大于数据参数或系统混乱行为。