We consider federated learning of linearly-parameterized nonlinear systems. We establish theoretical guarantees on the effectiveness of federated nonlinear system identification compared to centralized approaches, demonstrating that the convergence rate improves as the number of clients increases. Although the convergence rates in the linear and nonlinear cases differ only by a constant, this constant depends on the feature map $ϕ$, which can be carefully chosen in the nonlinear setting to increase excitation and improve performance. We experimentally validate our theory in physical settings where client devices are driven by i.i.d. control inputs and control policies exhibiting i.i.d. random perturbations, ensuring non-active exploration. Experiments use trajectories from nonlinear dynamical systems characterized by real-analytic feature functions, including polynomial and trigonometric components, representative of physical systems including pendulum and quadrotor dynamics. We analyze the convergence behavior of the proposed method under varying noise levels and data distributions. Results show that federated learning consistently improves convergence of any individual client as the number of participating clients increases.

The state space dynamics representation is the most general approach for nonlinear systems and often chosen for system identification. During training, the state trajectory can deform significantly leading to poor data coverage of the state space. This can cause significant issues for space-oriented training algorithms which e.g. rely on grid structures, tree partitioning, or similar. Besides hindering training, significant state trajectory deformations also deteriorate interpretability and robustness properties. This paper proposes a new type of space-filling regularization that ensures a favorable data distribution in state space via introducing a data-distribution-based penalty. This method is demonstrated in local model network architectures where good interpretability is a major concern. The proposed approach integrates ideas from modeling and design of experiments for state space structures. This is why we present two regularization techniques for the data point distributions of the state trajectories for local affine state space models. Beyond that, we demonstrate the results on a widely known system identification benchmark.

We tackle the problem of informative input design for system identification, where we select inputs, observe the corresponding outputs from the true system, and optimize the parameters of our model to best fit the data. We propose a methodology that is compatible with any system and parametric family of models. Our approach only requires input-output data from the system and first-order information from the model with respect to the parameters. Our algorithm consists of two modules. First, we formulate the problem of system identification from a Bayesian perspective and propose an approximate iterative method to optimize the model's parameters. Based on this Bayesian formulation, we are able to define a Gaussian-based uncertainty measure for the model parameters, which we can then minimize with respect to the next selected input. Our method outperforms model-free baselines with various linear and nonlinear dynamics.

Deep-learning-based nonlinear system identification has shown the ability to produce reliable and highly accurate models in practice. However, these black-box models lack physical interpretability, and often a considerable part of the learning effort is spent on capturing already expected/known behavior due to first-principles-based understanding of some aspects of the system. A potential solution is to integrate prior physical knowledge directly into the model structure, combining the strengths of physics-based modeling and deep-learning-based identification. The most common approach is to use an additive model augmentation structure, where the physics-based and the machine-learning (ML) components are connected in parallel. However, such models are overparametrized, training them is challenging, potentially causing the physics-based part to lose interpretability. To overcome this challenge, this paper proposes an orthogonal projection-based regularization technique to enhance parameter learning, convergence, and even model accuracy in learning-based augmentation of nonlinear baseline models.

The Fisher Information Matrix (FIM) provides a way for quantifying the information content of an observable random variable concerning unknown parameters within a model that characterizes the variable. When parameters in a model are directly linked to individual features, the diagonal elements of the FIM can signify the relative importance of each feature. However, in scenarios where feature interactions may exist, a comprehensive exploration of the full FIM is necessary rather than focusing solely on its diagonal elements. This paper presents an end-to-end black box system identification approach that integrates the FIM into the training process to gain insights into dynamic importance and overall model structure. A decision module is added to the first layer of the network to determine the relevance scores using the entire FIM as input. The forward propagation is then performed on element-wise multiplication of inputs and relevance scores. Simulation results demonstrate that the proposed methodology effectively captures various types of interactions between dynamics, outperforming existing methods limited to polynomial interactions. Moreover, the effectiveness of this novel approach is confirmed through its application in identifying a real-world industrial system, specifically the PH neutralization process.

Nonlinear system identification remains an important open challenge across research and academia. Large numbers of novel approaches are seen published each year, each presenting improvements or extensions to existing methods. It is natural, therefore, to consider how one might choose between these competing models. Benchmark datasets provide one clear way to approach this question. However, to make meaningful inference based on benchmark performance it is important to understand how well a new method performs comparatively to results available with well-established methods. This paper presents a set of ten baseline techniques and their relative performances on five popular benchmarks. The aim of this contribution is to stimulate thought and discussion regarding objective comparison of identification methodologies.

In this paper, we propose an approach for identifying linear and nonlinear discrete-time state-space models, possibly under $\ell_1$- and group-Lasso regularization, based on the L-BFGS-B algorithm. For the identification of linear models, we show that, compared to classical linear subspace methods, the approach often provides better results, is much more general in terms of the loss and regularization terms used, and is also more stable from a numerical point of view. The proposed method not only enriches the existing set of linear system identification tools but can be also applied to identifying a very broad class of parametric nonlinear state-space models, including recurrent neural networks. We illustrate the approach on synthetic and experimental datasets and apply it to solve the challenging industrial robot benchmark for nonlinear multi-input/multi-output system identification proposed by Weigand et al. (2022). A Python implementation of the proposed identification method is available in the package \texttt{jax-sysid}, available at \url{https://github.com/bemporad/jax-sysid}.

To benefit from the modeling capacity of deep models in system identification, without worrying about inference time, this study presents a novel training strategy that uses deep models only at the training stage. For this purpose two separate models with different structures and goals are employed. The first one is a deep generative model aiming at modeling the distribution of system output(s), called the teacher model, and the second one is a shallow basis function model, named the student model, fed by system input(s) to predict the system output(s). That means these isolated paths must reach the same ultimate target. As deep models show a great performance in modeling of highly nonlinear systems, aligning the representation space learned by these two models make the student model to inherit the approximation power of the teacher model. The proposed objective function consists of the objective of each student and teacher model adding up with a distance penalty between the learned latent representations. The simulation results on three nonlinear benchmarks show a comparative performance with examined deep architectures applied on the same benchmarks. Algorithmic transparency and structure efficiency are also achieved as byproducts.

This study designs and evaluates multiple nonlinear system identification techniques for modeling the UAV swarm system in planar space. learning methods such as RNNs, CNNs, and Neural ODE are explored and compared. The objective is to forecast future swarm trajectories by accurately approximating the nonlinear dynamics of the swarm model. The modeling process is performed using both transient and steady-state data from swarm simulations. Results show that the combination of Neural ODE with a well-trained model using transient data is robust for varying initial conditions and outperforms other learning methods in accurately predicting swarm stability.

We present a unifying view of discrete-time operator models used in the context of finite word length linear signal processing. Comparisons are made between the recently presented gamma operator model, and the delta and rho operator models for performing nonlinear system identification and prediction using neural networks. A new model based on an adaptive bilinear transformation which generalizes all of the above models is presented.