Deep Learning-based State and Parameter Estimation in Nonlinear Dynamical Systems

QC 20251106

Abstract

The tasks of state estimation and parameter estimation are closely related and often treated jointly by researchers in the areas of signal processing, control, and system identification. In recent literature, there is a growing trend of applying machine learning and deep learning techniques to tackle the above problems. The thesis follows a similar line of thought and is primarily concerned with the development of deep learning-based approaches for addressing state estimation and parameter estimation, under modeling constraints. 

The first main contribution of the thesis concerns deep learning approaches for addressing Bayesian state estimation in a model-free scenario. We introduce DANSE -- a Data-driven Nonlinear State Estimation method where the underlying dynamical model is unknown. The core of the DANSE framework consists of directly modeling the prior distribution of the state using a recurrent neural network. We assume knowledge of the measurement system, which is considered noisy, linear, and additive. This assumption offers attractive advantages in Bayesian state estimation. Firstly, it enables DANSE to provide a closed-form posterior distribution of the state. Secondly, the training of DANSE can be carried out in an unsupervised manner as long as the noisy measurements contain sufficient information about the unknown state (either when we have a full-rank measurement system or an over-determined measurement system). In the case of under-determined measurements, the state estimation problem becomes challenging. To meet this challenge, we modify the existing unsupervised learning framework of DANSE by using a limited amount of labelled training data. This modification results in a semi-supervised learning framework, and the resulting method is named semi-supervised DANSE (SemiDANSE). Finally, ideas from particle filters and importance sampling are borrowed for handling noisy and nonlinear measurements. This results in a particle-based DANSE (pDANSE). pDANSE approximates the unknown posterior distribution using the first two statistical moments, while requiring no special knowledge of the underlying dynamic model. We employ nonlinear dynamical systems, such as chaotic attractors, in our experimental study. Such attractors are a popular choice for benchmarking state estimation performances. We compare the empirical performance of DANSE and its extensions, SemiDANSE and pDANSE, against those of classical, model-driven approaches such as the extended Kalman filter (EKF), the unscented Kalman filter (UKF), the particle filter (PF), data-driven, and recent hybrid approaches like KalmanNet.

The second main contribution of the thesis concerns deep learning-based parameter estimation. Challenges associated with classical parameter estimation techniques include computationally intensive inference and dependency on parameter initialization. For addressing these challenges, we propose DeepBayes -- a method for constructing an estimator using recurrent neural networks. For training the recurrent neural networks, a simulable model is used to generate a large quantity of synthetic, labelled data for offline training using parameters sampled from a non-informative uniform prior. The result is that once trained, DeepBayes can provide rapid inference without any special consideration regarding initialisation. Furthermore, we empirically demonstrate that DeepBayes can provide comparable parameter estimation performance compared to the mean-squared-error optimal Bayes estimator and relevant maximum-likelihood based estimators on popular nonlinear dynamical system benchmarks. 

urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-372376