Gerben I. Beintema

Lucian Cristian Iacob, Máté Szécsi, Gerben Izaak Beintema et al.

This paper presents a novel identification approach of Koopman models of nonlinear systems with inputs under rather general noise conditions. The method uses deep state-space encoders based on the concept of state reconstructability and an efficient multiple-shooting formulation of the squared loss of the prediction error to estimate the dynamics and the lifted state only from input-output data. Furthermore, the Koopman model structure includes an innovation noise term that is used to handle process and measurement noise. It is shown that the proposed approach is statistically consistent (estimation error tends to zero when the number of data points goes to infinity) and computationally efficient due to the multiple-shooting formulation, by which the prediction error of the model can be calculated on multiple subsections of the data in parallel. The latter allows for efficient batch optimization of the network parameters and, at the same time, excellent long-term prediction capabilities of the obtained models. The performance of the approach is illustrated by nonlinear benchmark examples and experimental data from a Crazyflie 2.1 quadcopter.

Gerben I. Beintema, Maarten Schoukens, Roland Tóth

Using Artificial Neural Networks (ANN) for nonlinear system identification has proven to be a promising approach, but despite of all recent research efforts, many practical and theoretical problems still remain open. Specifically, noise handling and models, issues of consistency and reliable estimation under minimisation of the prediction error are the most severe problems. The latter comes with numerous practical challenges such as explosion of the computational cost in terms of the number of data samples and the occurrence of instabilities during optimization. In this paper, we aim to overcome these issues by proposing a method which uses a truncated prediction loss and a subspace encoder for state estimation. The truncated prediction loss is computed by selecting multiple truncated subsections from the time series and computing the average prediction loss. To obtain a computationally efficient estimation method that minimizes the truncated prediction loss, a subspace encoder represented by an artificial neural network is introduced. This encoder aims to approximate the state reconstructability map of the estimated model to provide an initial state for each truncated subsection given past inputs and outputs. By theoretical analysis, we show that, under mild conditions, the proposed method is locally consistent, increases optimization stability, and achieves increased data efficiency by allowing for overlap between the subsections. Lastly, we provide practical insights and user guidelines employing a numerical example and state-of-the-art benchmark results.

Gerben I. Beintema, Maarten Schoukens, Roland Tóth

Continuous-time (CT) modeling has proven to provide improved sample efficiency and interpretability in learning the dynamical behavior of physical systems compared to discrete-time (DT) models. However, even with numerous recent developments, the CT nonlinear state-space (NL-SS) model identification problem remains to be solved in full, considering common experimental aspects such as the presence of external inputs, measurement noise, latent states, and general robustness. This paper presents a novel estimation method that addresses all these aspects and that can obtain state-of-the-art results on multiple benchmarks with compact fully connected neural networks capturing the CT dynamics. The proposed estimation method called the subspace encoder approach (SUBNET) ascertains these results by efficiently approximating the complete simulation loss by evaluating short simulations on subsections of the data, by using an encoder function to estimate the initial state for each subsection and a novel state-derivative normalization to ensure stability and good numerical conditioning of the training process. We prove that the use of subsections increases cost function smoothness together with the necessary requirements for the existence of the encoder function and we show that the proposed state-derivative normalization is essential for reliable estimation of CT NL-SS models.

Rishi Ramkannan, Gerben I. Beintema, Roland Tóth et al.

The SUBNET neural network architecture has been developed to identify nonlinear state-space models from input-output data. To achieve this, it combines the rolled-out nonlinear state-space equations and a state encoder function, both parameterised as neural networks The encoder function is introduced to reconstruct the current state from past input-output data. Hence, it enables the forward simulation of the rolled-out state-space model. While this approach has shown to provide high-accuracy and consistent model estimation, its convergence can be significantly improved by efficient initialization of the training process. This paper focuses on such an initialisation of the subspace encoder approach using the Best Linear Approximation (BLA). Using the BLA provided state-space matrices and its associated reconstructability map, both the state-transition part of the network and the encoder are initialized. The performance of the improved initialisation scheme is evaluated on a Wiener-Hammerstein simulation example and a benchmark dataset. The results show that for a weakly nonlinear system, the proposed initialisation based on the linear reconstructability map results in a faster convergence and a better model quality.

Max D. Champneys, Gerben I. Beintema, Roland Tóth et al.

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.

Sarvin Moradi, Gerben I. Beintema, Nick Jaensson et al.

Hamiltonian neural networks (HNNs) represent a promising class of physics-informed deep learning methods that utilize Hamiltonian theory as foundational knowledge within neural networks. However, their direct application to engineering systems is often challenged by practical issues, including the presence of external inputs, dissipation, and noisy measurements. This paper introduces a novel framework that enhances the capabilities of HNNs to address these real-life factors. We integrate port-Hamiltonian theory into the neural network structure, allowing for the inclusion of external inputs and dissipation, while mitigating the impact of measurement noise through an output-error (OE) model structure. The resulting output error port-Hamiltonian neural networks (OE-pHNNs) can be adapted to tackle modeling complex engineering systems with noisy measurements. Furthermore, we propose the identification of OE-pHNNs based on the subspace encoder approach (SUBNET), which efficiently approximates the complete simulation loss using subsections of the data and uses an encoder function to predict initial states. By integrating SUBNET with OE-pHNNs, we achieve consistent models of complex engineering systems under noisy measurements. In addition, we perform a consistency analysis to ensure the reliability of the proposed data-driven model learning method. We demonstrate the effectiveness of our approach on system identification benchmarks, showing its potential as a powerful tool for modeling dynamic systems in real-world applications.

Jonas Weigand, Gerben I. Beintema, Jonas Ulmen et al.

The importance of proper data normalization for deep neural networks is well known. However, in continuous-time state-space model estimation, it has been observed that improper normalization of either the hidden state or hidden state derivative of the model estimate, or even of the time interval can lead to numerical and optimization challenges with deep learning based methods. This results in a reduced model quality. In this contribution, we show that these three normalization tasks are inherently coupled. Due to the existence of this coupling, we propose a solution to all three normalization challenges by introducing a normalization constant at the state derivative level. We show that the appropriate choice of the normalization constant is related to the dynamics of the to-be-identified system and we derive multiple methods of obtaining an effective normalization constant. We compare and discuss all the normalization strategies on a benchmark problem based on experimental data from a cascaded tanks system and compare our results with other methods of the identification literature.

Lucian Cristian Iacob, Gerben Izaak Beintema, Maarten Schoukens et al.

The present paper treats the identification of nonlinear dynamical systems using Koopman-based deep state-space encoders. Through this method, the usual drawback of needing to choose a dictionary of lifting functions a priori is circumvented. The encoder represents the lifting function to the space where the dynamics are linearly propagated using the Koopman operator. An input-affine formulation is considered for the lifted model structure and we address both full and partial state availability. The approach is implemented using the the deepSI toolbox in Python. To lower the computational need of the simulation error-based training, the data is split into subsections where multi-step prediction errors are calculated independently. This formulation allows for efficient batch optimization of the network parameters and, at the same time, excellent long term prediction capabilities of the obtained models. The performance of the approach is illustrated by nonlinear benchmark examples.

G. Rödönyi, G. I. Beintema, R. Tóth et al.

Automated driving applications require accurate vehicle specific models to precisely predict and control the motion dynamics. However, modern vehicles have a wide array of digital and mechatronic components that are difficult to model, manufactures do not disclose all details required for modelling and even existing models of subcomponents require coefficient estimation to match the specific characteristics of each vehicle and their change over time. Hence, it is attractive to use data-driven modelling to capture the relevant vehicle dynamics and synthesise model-based control solutions. In this paper, we address identification of the steering system of an autonomous car based on measured data. We show that the underlying dynamics are highly nonlinear and challenging to be captured, necessitating the use of data-driven methods that fuse the approximation capabilities of learning and the efficiency of dynamic system identification. We demonstrate that such a neural network based subspace-encoder method can successfully capture the underlying dynamics while other methods fall short to provide reliable results.

Gerben Izaak Beintema, Roland Toth, Maarten Schoukens

Identifying systems with high-dimensional inputs and outputs, such as systems measured by video streams, is a challenging problem with numerous applications in robotics, autonomous vehicles and medical imaging. In this paper, we propose a novel non-linear state-space identification method starting from high-dimensional input and output data. Multiple computational and conceptual advances are combined to handle the high-dimensional nature of the data. An encoder function, represented by a neural network, is introduced to learn a reconstructability map to estimate the model states from past inputs and outputs. This encoder function is jointly learned with the dynamics. Furthermore, multiple computational improvements, such as an improved reformulation of multiple shooting and batch optimization, are proposed to keep the computational time under control when dealing with high-dimensional and large datasets. We apply the proposed method to a video stream of a simulated environment of a controllable ball in a unit box. The study shows low simulation error with excellent long term prediction capability of the model obtained using the proposed method.

Gerben Beintema, Roland Toth, Maarten Schoukens

Nonlinear state-space identification for dynamical systems is most often performed by minimizing the simulation error to reduce the effect of model errors. This optimization problem becomes computationally expensive for large datasets. Moreover, the problem is also strongly non-convex, often leading to sub-optimal parameter estimates. This paper introduces a method that approximates the simulation loss by splitting the data set into multiple independent sections similar to the multiple shooting method. This splitting operation allows for the use of stochastic gradient optimization methods which scale well with data set size and has a smoothing effect on the non-convex cost function. The main contribution of this paper is the introduction of an encoder function to estimate the initial state at the start of each section. The encoder function estimates the initial states using a feed-forward neural network starting from historical input and output samples. The efficiency and performance of the proposed state-space encoder method is illustrated on two well-known benchmarks where, for instance, the method achieves the lowest known simulation error on the Wiener--Hammerstein benchmark.

Gerben Beintema, Alessandro Corbetta, Luca Biferale et al.

Thermal convection is ubiquitous in nature as well as in many industrial applications. The identification of effective control strategies to, e.g., suppress or enhance the convective heat exchange under fixed external thermal gradients is an outstanding fundamental and technological issue. In this work, we explore a novel approach, based on a state-of-the-art Reinforcement Learning (RL) algorithm, which is capable of significantly reducing the heat transport in a two-dimensional Rayleigh-Bénard system by applying small temperature fluctuations to the lower boundary of the system. By using numerical simulations, we show that our RL-based control is able to stabilize the conductive regime and bring the onset of convection up to a Rayleigh number $Ra_c \approx 3 \cdot 10^4$, whereas in the uncontrolled case it holds $Ra_{c}=1708$. Additionally, for $Ra > 3 \cdot 10^4$, our approach outperforms other state-of-the-art control algorithms reducing the heat flux by a factor of about $2.5$. In the last part of the manuscript, we address theoretical limits connected to controlling an unstable and chaotic dynamics as the one considered here. We show that controllability is hindered by observability and/or capabilities of actuating actions, which can be quantified in terms of characteristic time delays. When these delays become comparable with the Lyapunov time of the system, control becomes impossible.