Deep state space models (SSMs) are an actively researched model class for temporal models developed in the deep learning community which have a close connection to classic SSMs. The use of deep SSMs as a black-box identification model can describe a wide range of dynamics due to the flexibility of deep neural networks. Additionally, the probabilistic nature of the model class allows the uncertainty of the system to be modelled. In this work a deep SSM class and its parameter learning algorithm are explained in an effort to extend the toolbox of nonlinear identification methods with a deep learning based method. Six recent deep SSMs are evaluated in a first unified implementation on nonlinear system identification benchmarks.
University of Sheffield（谢菲尔德大学）；Technical University of Denmark（丹麦技术大学）
Identification of nonlinear dynamic systems remains a significant challenge across engineering. This work suggests an approach based on Bayesian filtering to extract and identify the contribution of an unknown nonlinear term in the system which can be seen as an alternative viewpoint on restoring force surface type approaches. To achieve this identification, the contribution which is the nonlinear restoring force is modelled, initially, as a Gaussian process in time. That Gaussian process is converted into a state-space model and combined with the linear dynamic component of the system. Then, …
Neural ordinary differential equations (NODE) have been recently proposed as a promising approach for nonlinear system identification tasks. In this work, we systematically compare their predictive performance with current state-of-the-art nonlinear and classical linear methods. In particular, we present a quantitative study comparing NODE's performance against neural state-space models and classical linear system identification methods. We evaluate the inference speed and prediction performance of each method on open-loop errors across eight different dynamical systems. The experiments show t…
University of Newcastle（纽卡斯尔大学）；Uppsala University（乌普萨拉大学）
This paper considers parameter estimation for nonlinear state-space models, which is an important but challenging problem. We address this challenge by employing a variational inference (VI) approach, which is a principled method that has deep connections to maximum likelihood estimation. This VI approach ultimately provides estimates of the model as solutions to an optimisation problem, which is deterministic, tractable and can be solved using standard optimisation tools. A specialisation of this approach for systems with additive Gaussian noise is also detailed. The proposed method is examin…
In recent years, identification of nonlinear dynamical systems from data has become increasingly popular. Sparse regression approaches, such as Sparse Identification of Nonlinear Dynamics (SINDy), fostered the development of novel governing equation identification algorithms assuming the state variables are known a priori and the governing equations lend themselves to sparse, linear expansions in a (nonlinear) basis of the state variables. In the context of the identification of governing equations of nonlinear dynamical systems, one faces the problem of identifiability of model parameters whe…
Jožef Stefan Institute（约瑟夫·斯特凡研究所）；University of Cambridge（剑桥大学）；University of Ljubljana（卢布尔雅那大学）；University of Nova Gorica（新戈里察大学）；Jožef Stefan International Postgraduate School（约瑟夫·斯特凡国际研究生院）
Equation discovery methods enable modelers to combine domain-specific knowledge and system identification to construct models most suitable for a selected modeling task. The method described and evaluated in this paper can be used as a nonlinear system identification method for gray-box modeling. It consists of two interlaced parts of modeling that are computer-aided. The first performs computer-aided identification of a model structure composed of elements selected from user-specified domain-specific modeling knowledge, while the second part performs parameter estimation. In this paper, recen…
Motivated by neuronal models from neuroscience, we consider the system identification of simple feedback structures whose behaviors include nonlinear phenomena such as excitability, limit-cycles and chaos. We show that output feedback is sufficient to solve the identification problem in a two-step procedure. First, the nonlinear static characteristic of the system is extracted, and second, using a feedback linearizing law, a mildly nonlinear system with an approximately-finite memory is identified. In an ideal setting, the second step boils down to the identification of a LTI system. To illust…
This text aims at providing a bird's eye view of system identification with special attention to nonlinear systems. The driving force is to give a feeling for the philosophical problems facing those that build mathematical models from data. Special attention will be given to grey-box approaches in nonlinear system identification. In this text, grey-box methods use auxiliary information such as the system steady-state data, possible symmetries, some bifurcations and the presence of hysteresis. The text ends with a sample of applications. No attempt is made to be thorough nor to survey such an e…
State-Space Kolmogorov Arnold Networks for Interpretable Nonlinear System Identification
Faculty of Engineering Technology, Vrije Universiteit Brussel（自由大学布鲁塞尔工程科技学院）；Department of Artificial Intelligence and Systems Engineering, Budapest University of Technology and Economics（布达佩斯技术与经济大学人工智能与系统工程系）
We consider the problem of identifying a dissipative linear model of an unknown nonlinear system that is known to be dissipative, from time domain input-output data. We first learn an approximate linear model of the nonlinear system using standard system identification techniques and then perturb the system matrices of the linear model to enforce dissipativity, while closely approximating the dynamical behavior of the nonlinear system. Further, we provide an analytical relationship between the size of the perturbation and the radius in which the dissipativity of the linear model guarantees loc…
Linear System Identification in a Nonlinear Setting - Nonparametric analysis of the nonlinear distortions and their impact on the best linear approximation
Institute for Computer Science and Control（计算机科学与控制研究所）
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…
This article introduces the Tensor Network B-spline model for the regularized identification of nonlinear systems using a nonlinear autoregressive exogenous (NARX) approach. Tensor network theory is used to alleviate the curse of dimensionality of multivariate B-splines by representing the high-dimensional weight tensor as a low-rank approximation. An iterative algorithm based on the alternating linear scheme is developed to directly estimate the low-rank tensor network approximation, removing the need to ever explicitly construct the exponentially large weight tensor. This reduces the computa…
Target Parameterization in Diffusion Models for Nonlinear Spatiotemporal System Identification
Université Marie et Louis Pasteur, SUPMICROTECH, CNRS, institut FEMTO-ST（玛丽-路易·巴斯蒂安大学，SUPMICROTECH，法国国家科学研究中心，FEMTO-ST研究所）；LLF, CNRS, Université Paris Cité（LLF，法国国家科学研究中心，巴黎cité大学）
This paper concerns the problem of adaptive output regulation for multivariable nonlinear systems in normal form. We present a regulator employing an adaptive internal model of the exogenous signals based on the theory of nonlinear Luenberger observers. Adaptation is performed by means of discrete-time system identification schemes, in which every algorithm fulfilling some optimality and stability conditions can be used. Practical and approximate regulation results are given relating the prediction capabilities of the identified model to the asymptotic bound on the regulated variables, which b…
This paper presents a discrete-time nonlinear system identification method while satisfying the stability and safety properties of the system with high probability. An Extreme Learning Machine (ELM) is used with a Gaussian assumption on the function reconstruction error. A quadratically constrained quadratic program (QCQP) is developed with probabilistic safety and stability constraints that are only required to be satisfied at sampled points inside the invariant region. The proposed method is validated using two simulation examples: a two degrees-of-freedom (DoF) robot manipulator with constr…
This paper concerns identification of uncontrolled or closed loop nonlinear systems using a set of trajectories that are generated by the system in a domain of attraction. The objective is to ensure that the trajectories of the identified systems are close to the trajectories of the real system, as quantified by an error bound that is prescribed a priori. A majority of existing methods for nonlinear system identification rely on techniques such as neural networks, autoregressive moving averages, and spectral decomposition that do not provide systematic approaches to meet pre-defined error boun…
In this paper we consider the joint problems of state estimation and model identification for a class of continuous-time nonlinear systems in output-feedback canonical form. An adaptive observer is proposed that combines an extended high-gain observer and a discrete-time identifier. The extended observer provides the identifier with a data set permitting the identification of the system model and the identifier adapts the extended observer according to the new estimated model. The design of the identifier is approached as a system identification problem and sufficient conditions are presented …
Wadhwani School of Data Science and AI（瓦德哈尼数据科学与人工智能学校）；Department of Electrical Engineering（电气工程系）；Center for Responsible AI（负责任人工智能中心）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；AI Innovation Institute（人工智能创新研究院）；Stony Brook University（石溪大学）
Many autonomous control systems are frequently exposed to attacks, so methods for attack identification are crucial for a safe operation. To preserve the privacy of the subsystems and achieve scalability in large-scale systems, identification algorithms should not require global model knowledge. We analyze a previously presented method for hierarchical attack identification, that is embedded in a distributed control setup for systems of systems with coupled nonlinear dynamics. It is based on the exchange of local sensitivity information and ideas from sparse signal recovery. In this paper, we …
The purpose of this article is to discuss some aspects of the convergence analysis performed in the paper [Design of momentum fractional LMS for Hammerstein nonlinear system identification with application to electrically stimulated muscle model, Eur. Phys. J. Plus (2019) \textbf{134}: 407]. It is highlighted that the way the authors prove convergence suffers a lack of correct and valid mathematical justifications.
For model-based control, an accurate and in its complexity suitable representation of the real system is a decisive prerequisite for high and robust control quality. In a structured step-by-step procedure, a model predictive control (MPC) scheme for a Schunk PowerCube robot is derived. Neweul-M$^2$ provides the necessary nonlinear model in symbolical and numerical form. To handle the heavy online computational burden involved with the derived nonlinear model, a linear time-varying MPC scheme is developed based on linearizing the nonlinear system concerning the desired trajectory and the a prio…
Models can be built directly from input and output data trough a process known as system identification. The Nonlinear AutoRegressive with eXogenous inputs (NARMAX) models are among the most used mathematical representations in the area and has many successful applications on data-driven modeling in different fields. Such models become extremely large when they have high degree of non-linearity and long-term dependencies. Hence, a structure selection process must be performed to make them parsimonious. In the present paper, it is introduced a toolbox in Python that performs the structure selec…
Balancing the model complexity and the representation capability towards the process to be captured remains one of the main challenges in nonlinear system identification. One possibility to reduce model complexity is to impose structure on the model representation. To this end, this work considers the linear fractional representation framework. In a linear fractional representation the linear dynamics and the system nonlinearities are modeled by two separate blocks that are interconnected with one another. This results in a structured, yet flexible model structure. Estimating such a model dire…
We consider the problem of safe control design for a class of nonlinear, control-affine systems subject to an unknown, additive, nonlinear disturbance. Leveraging recent advancements in the application of Koopman operator theory to the field of system identification and control, we introduce a novel fixed-time identification scheme for the infinitesimal generator of the infinite-dimensional, but notably linear, Koopman dynamical system analogous to the nonlinear system of interest. That is, we derive a parameter adaptation law that allows us to recover the unknown, residual nonlinear dynamics …
