Identification Of Dynamic Systems Research Articles

On the Uncertainty Identification for Linear Dynamic Systems Using Stochastic Embedding Approach with Gaussian Mixture Models.

In control and monitoring of manufacturing processes, it is key to understand model uncertainty in order to achieve the required levels of consistency, quality, and economy, among others. In aerospace applications, models need to be very precise and able to describe the entire dynamics of an aircraft. In addition, the complexity of modern real systems has turned deterministic models impractical, since they cannot adequately represent the behavior of disturbances in sensors and actuators, and tool and machine wear, to name a few. Thus, it is necessary to deal with model uncertainties in the dynamics of the plant by incorporating a stochastic behavior. These uncertainties could also affect the effectiveness of fault diagnosis methodologies used to increment the safety and reliability in real-world systems. Determining suitable dynamic system models of real processes is essential to obtain effective process control strategies and accurate fault detection and diagnosis methodologies that deliver good performance. In this paper, a maximum likelihood estimation algorithm for the uncertainty modeling in linear dynamic systems is developed utilizing a stochastic embedding approach. In this approach, system uncertainties are accounted for as a stochastic error term in a transfer function. In this paper, we model the error-model probability density function as a finite Gaussian mixture model. For the estimation of the nominal model and the probability density function of the parameters of the error-model, we develop an iterative algorithm based on the Expectation-Maximization algorithm using the data from independent experiments. The benefits of our proposal are illustrated via numerical simulations.

Bandwidth Selection for Kernel Generalized Regression Neural Networks in Identification of Hammerstein Systems

Abstract This paper addresses the issue of data-driven smoothing parameter (bandwidth) selection in the context of nonparametric system identification of dynamic systems. In particular, we examine the identification problem of the block-oriented Hammerstein cascade system. A class of kernel-type Generalized Regression Neural Networks (GRNN) is employed as the identification algorithm. The statistical accuracy of the kernel GRNN estimate is critically influenced by the choice of the bandwidth. Given the need of data-driven bandwidth specification we propose several automatic selection methods that are compared by means of simulation studies. Our experiments reveal that the method referred to as the partitioned cross-validation algorithm can be recommended as the practical procedure for the bandwidth choice for the kernel GRNN estimate in terms of its statistical accuracy and implementation aspects.

Structural identification with physics-informed neural ordinary differential equations

This paper exploits a new direction of structural identification by means of Neural Ordinary Differential Equations (Neural ODEs), particularly constrained by domain knowledge, such as structural dynamics, thus forming Physics-informed Neural ODEs, aiming at governing equations discovery/approximation. Structural identification problems often entail complex setups featuring high-dimensionality, or stiff ODEs, which pose difficulties in the training and learning of conventional data-driven algorithms who seek to unveil the governing dynamics of a system of interest. In this work, Neural ODEs are re-casted as a two-level representation involving a physics-informed term, that stems from possible prior knowledge of a dynamical system, and a discrepancy term, captured by means of a feed-forward neural network. The re-casted format is highly adaptive and flexible to structural monitoring problems, such as linear/nonlinear structural identification, model updating, structural damage detection, driving force identification, etc. As an added step, for inferring an explainable model, we propose the adoption of sparse identification of nonlinear dynamical systems as an additional tool to distill closed-form expressions for the trained nets, that embed a more straightforward engineering interpretation. We demonstrate the framework on a series of numerical and experimental examples, with the latter pertaining to a structural system featuring highly nonlinear behavior, which is successfully learned by the proposed framework. The proposed structural identification with Physics-informed Neural ODEs comes with the benefits of direct approximation of the governing dynamics, and a versatile and flexible framework for discrepancy modeling in structural identification problems.

Nonlinear dynamical system identification using the sparse regression and separable least squares methods

This paper proposes a novel nonlinear dynamical system identification method based on the sparse regression algorithm and the separable least squares method. To effectively avoid solving the second derivative of the displacement signal and reduce the effect of noise, the Duhamel's integral is adopted to represent the dynamic relationship between the system input and output. In the expression form of Duhamel's integral, nonlinear dynamical system identification can be cast as a separable least squares problem. Thus, the separable least squares method is leveraged to separately identify the parameters of the linear subsystem and the coefficients corresponding to nonlinearities among the nonlinear dynamical system. During the identification process of nonlinear restoring forces, one complete set of nonlinear basis functions are used to represent the nonlinear restoring forces. Not all the candidate nonlinear terms are contributing, however, thus the sparse regression algorithm is adopted to select the actual contributing nonlinear components in the candidate nonlinear terms and eliminate the non-contributing nonlinear components, and then the corresponding parameters of contributing nonlinear components are estimated by the unbiased least squares method. Finally, one RKHS (Reproducing Kernel Hilbert Space)-based non-parametric de-noise method is further proposed to reduce the noise in the vibration displacement and obtain the noise-reduced velocity from the displacement signal. The numerical simulation about the identification of the rotating blade-casing system and the dynamic experiment of the HSLDS (high-static-low-dynamic stiffness) isolator system verify the effectiveness of the new identification method for nonlinear dynamical systems proposed in this paper.

Non-parametric identification of homogeneous dynamical systems

The aim of this study is to design a non-parametric identifier for homogeneous systems based on a class of artificial neural networks with continuous dynamics. The identification algorithm is developed for input-affine systems with uncertain gains and diverse degrees of homogeneity. One of the main contributions of this study is the extension of the universal approximation property of neural networks for continuous homogeneous systems. Another contribution is the development of a differential non-parametric identifier based on the novel concept of homogeneous neural networks. The adjustment laws for the weights are obtained from a Lyapunov stability analysis taking homogeneity properties of the system into account. The ultimate boundedness of the origin for the identification error is demonstrated using the persistent excitation condition. The effectiveness of the proposed identifier is verified by the simulation of the three-tank homogeneous model. In this example, the proposed identification scheme is compared with a classical ANN identifier, and we present a statistical analysis of such comparison. It is shown in simulations that the identification error of the proposed homogeneous algorithm has faster convergence and less oscillations.

Identification of nonlinear dynamical systems with time delay

This paper proposes a technique to identify nonlinear dynamical systems with time delay. The sparse optimization algorithm is extended to nonlinear systems with time delay. The proposed algorithm combines cross-validation techniques from machine learning for automatic model selection and an algebraic operation for preprocessing signals to filter the noise and for removing the dependence on initial conditions. We further integrate the bootstrapping resampling technique with the sparse regression to obtain the statistical properties of estimation. We use Taylor expansion to parameterize time delay. The proposed algorithm in this paper is computationally efficient and robust to noise. A nonlinear Duffing oscillator is simulated to demonstrate the efficiency and accuracy of the proposed technique. An experimental example of a nonlinear rotary flexible joint is presented to further validate the proposed method.

Fitting of Generic Process Models by an Asymmetric Short Relay Feedback Experiment—The n-Shifting Method

This paper presents the generalization of the shifting method for relay feedback identification of dynamic systems of any order. The original shifting method enables the fitting of a maximum of five parameters of a transfer function model from the information obtained from a short relay test and without prior knowledge of the process to identify. The generalization, known as n-shifting, allows the estimation of the parameters of transfer functions of any order by applying one short relay test to the process to identify. Without loss of generality, the n-shifting approach is applied to fit an n-order plus time delay (n-OPTD) model but the approach can be also developed to identify models with other structures (non-minimum phase, unstable, integrators). Some examples of estimations are presented.

Frequency Response From Solar PV: A Dynamic Equivalence Closed-Loop System Identification Approach

The analysis of the frequency response of integrated transmission-distribution networks with deep penetration of solar photovoltaic (PV) generation faces major challenges due to the complexity emerging from the dynamic models of the numerous and diverse PV units involved. This article proposes converter-based dynamic equivalent models for both distributed (distribution network-connected) and large-scale (transmission network-connected) PV units, which take into account practical issues, such as measurement and coordination delays. Different from the previous work that adopted an open-loop identification, the unknown model parameters are identified here through a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">closed-loop</i> identification process based on the least-square minimization. This allows capturing the continuous interaction between system and PV responses, thus improving the outcome of the overall frequency response model. The proposed models are validated with the real data from the August 2018 separation event in Australia. The results demonstrate the excellent performance of the proposed models in determining the frequency response from PV in both transmission and distribution networks, hence paving the way to its adoption in the frequency stability analysis in low-carbon grids dominated by frequency-responsive renewables.

Numerically stable algorithm for identification of linear dynamical systems by extended instrumental variables

Instrumental variables are widely used to identify linear dynamical systems. The advantages of instrumental variables include low computational complexity, as well as the possibility of identification for different models of color noise. Often the method of instrumental variables leads to ill-conditioned problems, which significantly limits the application of this method. The paper proposes a solution to the problem of extended instrumental variables based on augmented normal equations. Test examples showed a high accuracy of the proposed approach.

Hybrid approach to the dynamic systems identification based on the self-configuring genetic programming algorithm and the differential evolution method

This paper considers a hybrid approach to the identification of dynamical systems based on a self-configuring genetic programming algorithm and a differential evolution method. The value of this approach is in the automatic determination of the order, structure and parameters of differential equation, i.e., a model of a dynamic system. The application of the differential evolution method can significantly increase the accuracy of the resulting model confirmed by the results of the experiments presented in this paper.

An RF-PCE Hybrid Surrogate Model for Sensitivity Analysis of Dams

Quantification of structural vibration characteristics is an essential task prior to perform any dynamic health monitoring and system identification. Anatomy of vibration in concrete arch dams (especially tall dams with un-symmetry shape) is very complicated and requires special techniques to solve the eigenvalue problem. The situation becomes even more complicated if the material distribution is assumed to be heterogeneous within the dam body (as opposed to conventional isotropic homogeneous relationship). This paper proposes a hybrid Random Field (RF)–Polynomial Chaos Expansion (PCE) surrogate model for uncertainty quantification and sensitivity assessment of dams. For different vibration modes, the most sensitive spatial locations within dam body are identified using both Sobol’s indices and correlation rank methods. Results of the proposed hybrid model is further validated using the classical random forest regression method. The outcome of this study can improve the results of system identification and dynamic analysis by properly determining the vibration characteristics.

Sparse identification of nonlinear dynamical systems via reweighted [formula omitted]-regularized least squares

This work proposes an iterative sparse-regularized regression method to recover governing equations of nonlinear dynamical systems from noisy state measurements. The method is inspired by the Sparse Identification of Nonlinear Dynamics (SINDy) approach of Brunton et al. (2016), which relies on two main assumptions: 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. The aim of this work is to improve the accuracy and robustness of SINDy in the presence of state measurement noise. To this end, a reweighted ℓ1-regularized least squares solver is developed, wherein the regularization parameter is selected from the corner point of a Pareto curve. The idea behind using weighted ℓ1-norm for regularization – instead of the standard ℓ1-norm – is to better promote sparsity in the recovery of the governing equations and, in turn, mitigate the effect of noise in the state variables. We also present a method to recover single physical constraints from state measurements. Through several examples of well-known nonlinear dynamical systems, we demonstrate empirically the accuracy and robustness of the reweighted ℓ1-regularized least squares strategy with respect to state measurement noise, thus illustrating its viability for a wide range of potential applications.

Gray-Box system modeling using symbolic regression and nonlinear model predictive control of a semibatch polymerization

A nonlinear model, predictive control approach, based on a gray-box model is introduced in this contribution. The gray-box model is built by combining white-box and black-box modeling elements. The white-box elements consist of relevant mass and energy balances, including algebraic relations that describe different phenomena of the system behavior (e.g., reaction kinetics, heat transfer phenomena, thermodynamic assumptions). To fill in or compensate for modeling gaps (e.g., missing kinetics information, heat transfer efficiency, inaccurate thermodynamic assumptions), black-box model elements trained from historical process data are introduced into the model. An algorithm for Sparse Identification of Nonlinear Dynamical Systems (SINDy) that identifies the most promising regressors from a pool of possible explanatory functions is used to generate the model’s black-box part. The proposed gray-box modeling approach and its application to model predictive control were tested for performance and robustness using a benchmark problem, and the approach shows promising results.

Study on fault identification of mechanical dynamic nonlinear transmission system

Abstract To solve the problems of large mechanical powertrain such as complex structure, serious accident, strong nonlinear characteristics of running state, bad operating environment, non-Gaussian noise, and various uncertain factors, it is difficult to make an accurate fault diagnosis. This paper proposes a method for dealing with nonlinear characteristics using nuclear waves, as well as a system, deeply conducted nuclear base fault feature extraction, classification, and decision making, such as nuclear base state trend prediction technology research, focusing on exploring and improving the accuracy of fault diagnosis under nonlinear conditions, technical method, and way to state prediction accuracy. It offers effective technical assistance for the advancement and use of mechanical power train monitoring and diagnosis technology. A fault detection method based on kernel method is proposed. Based on the characteristics of this method in dealing with nonlinear problems, the research on kernel feature extraction, kernel fault classification and decision making, and kernel state trend prediction are carried out systematically. The experimental results show that the simulation analysis of typical chaotic time series prediction and the application of the operation state prediction of a certain ship main steam turbine unit have achieved good results, among which the average relative error of the single-step prediction of the unit state is 1.7881%, and the average relative error of the 30-step prediction is 3.3983%. Proved that the nuclear methods systematically applied to mechanical power transmission system fault diagnosis and state prediction, effectively enhancing some traditional methods and techniques dealing with nonlinear feature extraction, the nonlinear prediction capability for fault identification, and nonlinear state, to deal with nonlinear fault diagnosis problems of engineering practice, a large number of explored effective solution.

Identification of nonlinear dynamic system using machine learning techniques

Identification of nonlinear systems finds extensive applications in control design and stability analysis. To identify complex nonlinear systems, the neural network has drawn the attention of many researchers due to its broad application area. In this paper, an improved identification method based on robust regularised exponentially extended random vector functional link network (RERVFLN) has been proposed for nonlinear system identification. The input is extended using trigonometric expansion which increases the accuracy of the algorithm. To verify the accuracy of the proposed model, some benchmark Monte Carlo simulations are carried out through simulation study and the obtained results are compared with some established techniques such as original RVFLN, ELM, and LMS. Prediction accuracy of the proposed method RERVFLN is higher than the normal RVFLN for different nonlinear systems which is clear from the performance evaluation section.

Identification of Nonlinear Dynamical System Based on Raised-Cosine Radial Basis Function Neural Networks

In this paper, we present and investigate a new type of radial basis function (RBF) neural networks mechanism using raised-cosine (RC) function to identify nonlinear dynamic system. In this design, the RBF neural networks mechanism utilizes RC function to replace Gaussian function, which is called RCRBF. An N-dimensional RC function has the constant interpolation property, which is benefit for the function approximating errors analysis in the neural networks. Based on multivariable RC function approximation theory, we develop how to select the updated parameters and the distance of adjacent nodes in lattice points. Therefore, the proposed networks can uniformly approximate nonlinear dynamical function. As persistency excitation (PE) plays an important part in neural networks learning system, how does PE condition behave in input sequences is formulated by RC function analysis. The weights updating and errors convergence are concluded by Lyapunov function analysis. To illustrate the effectiveness of the proposed RCRBF method, Van Der Pol and Rossler dynamical system are used as test examples, in comparison with GRBF mechanism. The results show that the proposed method has better accurate identification and approximating effect than that of GRBF mechanism.

Identification of linear dynamic systems of fractional order with errors in variables based on an augmented system of equations

Уравнения с производными и разностями дробного порядка находят широкое применение для описания различных процессов и явлений. В настоящее время активно развиваются методы идентификации систем, описываемых уравнениями с разностями дробного порядка. Статья посвящена идентификации дискретных динамических систем, описываемых уравнениями с разностями дробного порядка с ошибками в переменных. Задачи идентификации систем с ошибками в переменных часто бывают плохо обусловленными. В статье предложен алгоритм, использующий представление нормальной смещенной системы в виде расширенной эквивалентной системы. Данное представление позволяет уменьшить число обусловленности решаемой задачи. Тестовые примеры показали, что предложенный алгоритм обладает более высокой точностью по сравнению с алгоритмами на основе разложения Холецкого и минимизации обобщенного отношения Релея.

Identification of nonlinear dynamic system using machine learning techniques

Identification of nonlinear dynamic system using machine learning techniques

The Loewner framework for nonlinear identification and reduction of Hammerstein cascaded dynamical systems

AbstractWe present an algorithm for data‐driven identification and reduction of nonlinear cascaded systems with Hammerstein structure. The proposed algorithm relies on the Loewner framework (LF) which constitutes a non‐intrusive algorithm for identification and reduction of dynamical systems based on interpolation. We address the following problem: the actuator (control input) enters a static nonlinear block. Then, this processed signal is used as an input for a linear time‐invariant system (LTI). Additionally, it is considered that the orders of the linear transfer function and of the static nonlinearity are not a priori known.

On-line identification methods as part of the tunnel ventilation control algorithm

The paper presents algorithms of on-line methods of discrete parametric identification of pollutant concentrations in a tunnel tube and comparison of their properties. It describes the characteristics of an individual methods and a detailed evaluation in order to select the most suitable method for the identification of dynamical systems with a significant stochastic component and changing parameters. The aim of the paper is to propose discrete parametric identification methods for systems that have variable parameters. It is a choice of model structure and design of an algorithm that is able to monitor changes in system parameters. The article also describes ways to implement identification algorithms in PLC (Programmable Logic Controller) by programming the target device directly from the simulation environment MATLAB Simulink using the toolbox "B&R Automation Studio Target for Simulink". From a practical point of view, these algorithms serve to create an apparatus for optimizing the ventilation control in the tunnel and thus ensure the safe passage of vehicles through the tunnel.