Roland Tóth

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.

Pepijn Bastiaan Cox, Roland Tóth

Different representations to describe noise processes and finding connections or equivalence between them have been part of active research for decades, in particular for linear time-invariant case. In this paper the linear parameter-varying (LPV) setting is addressed; starting with the connection between an LPV state-space (SS) representation with a general noise structure and the LPV-SS model in an innovation structure, i.e., the Kalman filter. More specifically, the considered LPV-SS representation with general noise structure has static, affine dependence on the scheduling signal; however, we show that its companion innovation structure has a dynamic, rational dependency structure. Following, we would like to highlight the consequences of approximating this Kalman gain by a static, affine dependency structure. To this end, firstly, we use the "fading memory" effect of the Kalman filter to reason how the Kalman gain can be approximated to depend only on a partial trajectory of the scheduling signal. This effect is shown by proving an asymptotically decreasing error upper bound on the covariance matrix associated to the innovation structure in case the covariance matrix is subjected to an incorrect initialization or disturbance. Secondly, we show by an example that an LPV-SS representation that has dynamical, rational dependency on the scheduling signal can be transformed into static, affinely dependent representation by introducing additional states. Therefore, an approximated Kalman gain can, in some cases, be represented by a static, affine Kalman gain at the cost of additional states.

Julius P. J. Krebbekx, Roland Tóth, Amritam Das

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of nonlinear systems. There have been recent efforts to generalize SRG analysis to Multiple-Input Multiple-Output (MIMO) systems. However, these attempts yielded only results for square systems, due to the inherent Hilbert space structure of the SRG. In this paper, we develop an SRG analysis method that accommodates non-square operators. The key element is the embedding of operators to a space of operators acting on a common Hilbert space, while restricting the input space to the original input dimension, to avoid conservatism. We generalize SRG interconnection rules to restricted input spaces and develop stability theorems to guarantee causality, well-posedness and (incremental) $L_2$-gain bounds for the overall interconnection. We show utilization of the proposed theoretical concepts on the analysis of nonlinear systems in a Linear Fractional Representation (LFR) form, which is a rather general class of systems, and the LFR is directly utilizable for control. Moreover, we provide formulas for the computation of MIMO SRGs of stable LTI operators and diagonal and non-square static nonlinear operators. Finally, we demonstrate the advantages of our embedding approach on several examples.

Kristóf Floch, Amon Lahr, Roland Tóth et al.

This paper presents a unified framework that connects sequential quadratic programming (SQP) and the iterative linear-parameter-varying model predictive control (LPV-MPC) technique. Using the differential formulation of the LPV-MPC, we demonstrate how SQP and LPV-MPC can be unified through a specific choice of scheduling variable and the 2nd Fundamental Theorem of Calculus (FTC) embedding technique and compare their convergence properties. This enables the unification of the zero-order approach of SQP with the LPV-MPC scheduling technique to enhance the computational efficiency of robust and stochastic MPC problems. To demonstrate our findings, we compare the two schemes in a simulation example. Finally, we present real-time feasibility and performance of the zero-order LPV-MPC approach by applying it to Gaussian process (GP)-based MPC for autonomous racing with real-world experiments.

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.

Chris Verhoek, Ruigang Wang, Roland Tóth

This paper presents two direct parameterizations of stable and robust linear parameter-varying state-space (LPV-SS) models. The model parametrizations guarantee a priori that for all parameter values during training, the allowed models are stable in the contraction sense or have their Lipschitz constant bounded by a user-defined value $γ$. Furthermore, since the parametrizations are direct, the models can be trained using unconstrained optimization. The fact that the trained models are of the LPV-SS class makes them useful for, e.g., further convex analysis or controller design. The effectiveness of the approach is demonstrated on an LPV identification problem.

T. Bloemers, I. Proimadis, Y. Kasemsinsup et al.

The present report summarizes the work conducted during the internship on Feedforward Control of the Magnetic Levitation Setup. Different feedforward strategies, specifically tailored for this setup, are developed and reviewed. These feedforward methods explicitly take the intrinsic position-dependent behavior of the magnetic levitation setup into account. Additionally, closed-loop stability of the given setup is assessed. All investigations are carried out under the rigid-body assumption of the structure. Analysis and simulation show the potential performance improvement obtained with such feedforward strategies.

Petar Bevanda, Bas Driessen, Lucian Cristian Iacob et al.

This paper presents a novel Koopman composition operator representation framework for control systems in reproducing kernel Hilbert spaces (RKHSs) that is free of explicit dictionary or input parametrizations. By establishing fundamental equivalences between different model representations, we are able to close the gap of control system operator learning and infinite-dimensional regression, enabling various empirical estimators and the connection to the well-understood learning theory in RKHSs under one unified framework. Consequently, our proposed framework allows for arbitrarily accurate finite-rank approximations in infinite-dimensional spaces and leads to finite-dimensional predictors without apriori restrictions to a finite span of functions or inputs. To enable applications to high-dimensional control systems, we improve the scalability of our proposed control Koopman operator estimates by utilizing sketching techniques. Numerical experiments demonstrate superior prediction accuracy compared to bilinear EDMD, especially in high dimensions. Finally, we show that our learned models are readily interfaced with linear-parameter-varying techniques for model predictive control.

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.

Bendegúz M. Györök, Jan H. Hoekstra, Johan Kon et al.

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 a considerable part of the learning effort is often spent on capturing already expected/known behavior of the system, that can be accurately described by first-principles laws of physics. A potential solution is to directly integrate such prior physical knowledge 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, i.e., additively. 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 and even model accuracy in learning-based augmentation of nonlinear baseline models.

Alberto Bemporad, Roland Tóth

We present a general system identification procedure capable of estimating of a broad spectrum of state-space dynamical models, including linear time-invariant (LTI), linear parameter-varying} (LPV), and nonlinear (NL) dynamics, along with rather general classes of noise models. Similar to the LTI case, we show that for this general class of model structures, including the NL case, the model dynamics can be separated into a deterministic process and a stochastic noise part, allowing to seamlessly tune the complexity of the combined model both in terms of nonlinearity and noise modeling. We parameterize the involved nonlinear functional relations by means of artificial neural-networks (ANNs), although alternative parametric nonlinear mappings can also be used. To estimate the resulting model structures, we optimize a prediction-error-based criterion using an efficient combination of a constrained quasi-Newton approach and automatic differentiation, achieving training times in the order of seconds compared to existing state-of-the-art ANN methods which may require hours for models of similar complexity. We formally establish the consistency guarantees for the proposed approach and demonstrate its superior estimation accuracy and computational efficiency on several benchmark LTI, LPV, and NL system identification problems.

G. J. E. van Otterdijk, S. Moradi, S. Weiland et al.

Port-Hamiltonian neural networks (pHNNs) are emerging as a powerful modeling tool that integrates physical laws with deep learning techniques. While most research has focused on modeling the entire dynamics of interconnected systems, the potential for identifying and modeling individual subsystems while operating as part of a larger system has been overlooked. This study addresses this gap by introducing a novel method for using pHNNs to identify such subsystems based solely on input-output measurements. By utilizing the inherent compositional property of the port-Hamiltonian systems, we developed an algorithm that learns the dynamics of individual subsystems, without requiring direct access to their internal states. On top of that, by choosing an output error (OE) model structure, we have been able to handle measurement noise effectively. The effectiveness of the proposed approach is demonstrated through tests on interconnected systems, including multi-physics scenarios, demonstrating its potential for identifying subsystem dynamics and facilitating their integration into new interconnected models.

Sarvin Moradi, Nick Jaensson, Roland Tóth et al.

In order to make data-driven models of physical systems interpretable and reliable, it is essential to include prior physical knowledge in the modeling framework. Hamiltonian Neural Networks (HNNs) implement Hamiltonian theory in deep learning and form a comprehensive framework for modeling autonomous energy-conservative systems. Despite being suitable to estimate a wide range of physical system behavior from data, classical HNNs are restricted to systems without inputs and require noiseless state measurements and information on the derivative of the state to be available. To address these challenges, this paper introduces an Output Error Hamiltonian Neural Network (OE-HNN) modeling approach to address the modeling of physical systems with inputs and noisy state measurements. Furthermore, it does not require the state derivatives to be known. Instead, the OE-HNN utilizes an ODE-solver embedded in the training process, which enables the OE-HNN to learn the dynamics from noisy state measurements. In addition, extending HNNs based on the generalized Hamiltonian theory enables to include external inputs into the framework which are important for engineering applications. We demonstrate via simulation examples that the proposed OE-HNNs results in superior modeling performance compared to classical HNNs.

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.

Zechen Liu, Zizhang Wu, Roland Tóth

Estimating 3D orientation and translation of objects is essential for infrastructure-less autonomous navigation and driving. In case of monocular vision, successful methods have been mainly based on two ingredients: (i) a network generating 2D region proposals, (ii) a R-CNN structure predicting 3D object pose by utilizing the acquired regions of interest. We argue that the 2D detection network is redundant and introduces non-negligible noise for 3D detection. Hence, we propose a novel 3D object detection method, named SMOKE, in this paper that predicts a 3D bounding box for each detected object by combining a single keypoint estimate with regressed 3D variables. As a second contribution, we propose a multi-step disentangling approach for constructing the 3D bounding box, which significantly improves both training convergence and detection accuracy. In contrast to previous 3D detection techniques, our method does not require complicated pre/post-processing, extra data, and a refinement stage. Despite of its structural simplicity, our proposed SMOKE network outperforms all existing monocular 3D detection methods on the KITTI dataset, giving the best state-of-the-art result on both 3D object detection and Bird's eye view evaluation. The code will be made publicly available.

Dhruv Khandelwal, Maarten Schoukens, Roland Tóth

Model structure and complexity selection remains a challenging problem in system identification, especially for parametric non-linear models. Many Evolutionary Algorithm (EA) based methods have been proposed in the literature for estimating model structure and complexity. In most cases, the proposed methods are devised for estimating structure and complexity within a specified model class and hence these methods do not extend to other model structures without significant changes. In this paper, we propose a Tree Adjoining Grammar (TAG) for stochastic parametric models. TAGs can be used to generate models in an EA framework while imposing desirable structural constraints and incorporating prior knowledge. In this paper, we propose a TAG that can systematically generate models ranging from FIRs to polynomial NARMAX models. Furthermore, we demonstrate that TAGs can be easily extended to more general model classes, such as the non-linear Box-Jenkins model class, enabling the realization of flexible and automatic model structure and complexity selection via EA.

Jurre Hanema, Mircea Lazar, Roland Tóth

This paper presents a stabilizing tube-based MPC synthesis for LPV systems. We employ terminal constraint sets which are required to be controlled periodically contractive. Periodically (or finite-step) contractive sets are easier to compute and can be of lower complexity than "true" contractive ones, lowering the required computational effort both off-line and on-line. Under certain assumptions on the tube parameterization, recursive feasibility of the scheme is proven. Subsequently, asymptotic stability of the origin is guaranteed through the construction of a suitable terminal cost based on a novel Lyapunov-like metric for compact convex sets containing the origin. A periodic variant on the well-known homothetic tube parameterization that satisfies the necessary assumptions and yields a tractable LPV MPC algorithm is derived. The resulting MPC algorithm requires the on-line solution of a single linear program with linear complexity in the prediction horizon. The properties of the approach are demonstrated by a numerical example.

Mert Bastug, Mihaly Petreczky, Roland Toth et al.

We present a novel algorithm for reducing the state dimension, i.e. order, of linear parameter varying (LPV) discrete-time state-space (SS) models with affine dependence on the scheduling variable. The input-output behavior of the reduced order model approximates that of the original model. In fact, for input and scheduling sequences of a certain length, the input-output behaviors of the reduced and original model coincide. The proposed method can also be interpreted as a reachability and observability reduction (minimization) procedure for LPV-SS representations with affine dependence.