Identification Of Dynamic Systems Research Articles

Existence of n-cycles and border-collision bifurcations in piecewise-linear continuous maps with applications to recurrent neural networks

Piecewise linear recurrent neural networks (PLRNNs) form the basis of many successful machine learning applications for time series prediction and dynamical systems identification, but rigorous mathematical analysis of their dynamics and properties is lagging behind. Here, we contribute to this topic by investigating the existence of n-cycles (nge 3) and border-collision bifurcations in a class of m-dimensional piecewise linear continuous maps which have the general form of a PLRNN. This is particularly important as for one-dimensional maps the existence of 3-cycles implies chaos. It is shown that these n-cycles collide with the switching boundary in a border-collision bifurcation, and parametric regions for the existence of both stable and unstable n-cycles and border-collision bifurcations will be derived theoretically. We then discuss how our results can be extended and applied to PLRNNs. Finally, numerical simulations demonstrate the implementation of our results and are found to be in good agreement with the theoretical derivations. Our findings thus provide a basis for understanding periodic behavior in PLRNNs, how it emerges in bifurcations, and how it may lead into chaos.

Data-driven modeling of the chaotic thermal convection in an annular thermosyphon

dentifying accurate and yet interpretable low-order models from data has gained a renewed interest over the past decade. In the present work, we illustrate how the combined use of dimensionality reduction and sparse system identification techniques allows us to obtain an accurate model of the chaotic thermal convection in a two-dimensional annular thermosyphon. Taking as guidelines the derivation of the Lorenz system, the chaotic thermal convection dynamics simulated using a high-fidelity computational fluid dynamics solver are first embedded into a low-dimensional space using dynamic mode decomposition. After having reviewed the physical properties the reduced-order model should exhibit, the latter is identified using SINDy, an increasingly popular and flexible framework for the identification of nonlinear continuous-time dynamical systems from data. The identified model closely resembles the canonical Lorenz system, having a similar structure and exhibiting the same physical properties. Finally, extensions to other flow configurations with or without control are discussed.

Experimental and Numerical Study on Dynamics of Two Footbridges with Different Shapes of Girders

The paper presents the experimental and numerical results of the dynamic system identification and verification of the behavior of two footbridges in Poland. The experimental part of the study involved vibration testing under different scenarios of human-induced load, impulse load, and excitations induced by vibration exciter. Based on the results obtained, the identification of dynamic parameters of the footbridges was performed using the peak-picking method. With the impulse load applied to both structures, determination of their natural vibration frequencies was possible. Then, based on the design drawings, detailed finite element method (FEM) models were developed, and the numerical analyses were carried out. The comparison between experimental and numerical results obtained from the modal analysis showed a good agreement. The results also indicated that both structures under investigation have the first natural bending frequency of the deck in the range of human-induced excitation. Therefore, the risk of excessive structural vibrations caused by pedestrian loading was then analysed for both structures. The vibration comfort criteria for both footbridges were checked according to Sétra guidelines. In the case of the first footbridge, the results showed that the comfort criteria are fulfilled, regardless of the type of load. For the second footbridge, it was emphasized that the structure meets the assumptions of the guidelines for vibration severability in normal use; nevertheless, it is susceptible to excitations induced by synchronized users, even in the case of a small group of pedestrians.

On sparse identification of complex dynamical systems: A study on discovering influential reactions in chemical reaction networks

A wide variety of real life complex networks are prohibitively large for modeling, analysis and control. Understanding the structure and dynamics of such networks entails creating a smaller representative network that preserves its relevant topological and dynamical properties. While modern machine learning methods have enabled identification of governing laws for complex dynamical systems, their inability to produce white-box models with sufficient physical interpretation renders such methods undesirable to domain experts. In this paper, we introduce a hybrid black-box, white-box approach for the sparse identification of the governing laws for complex, highly coupled dynamical systems with particular emphasis on finding the influential reactions in chemical reaction networks for combustion applications, using a data-driven sparse-learning technique. The proposed approach identifies a set of influential reactions using species concentrations and reaction rates, with minimal computational cost without requiring additional data or simulations. The new approach is applied to analyze the combustion chemistry of H2 and C3H8 in a constant-volume homogeneous reactor. The influential reactions determined by the sparse-learning method are consistent with the current kinetics knowledge of chemical mechanisms. Additionally, we show that a reduced version of the parent mechanism can be generated as a combination of the significantly reduced influential reactions identified at different times and conditions and that for both H2 and C3H8 fuel, the reduced mechanisms perform closely to the parent mechanisms as a function of the ignition delay time over a wide range of conditions. Our results demonstrate the potential of the sparse-learning approach as an effective and efficient tool for dynamical system analysis and reduction. The uniqueness of this approach as applied to combustion systems lies in the ability to identify influential reactions in specified conditions and times during the evolution of the combustion process. This ability is of great interest to understand chemical reaction systems.

S-synchronization Structural Identifiability and Identification of Nonlinear Dynamic Systems

An approach to the structural identifiability analysis of nonlinear dynamic systems under uncertainty is proposed. We have shown that S-synchronization is the necessary condition for the structural identifiability of a nonlinear system. Conditions are obtained for the design of a model which identifies the nonlinear part of the system. The method is proposed for the obtaining of a set which contains the information on the nonlinear part. A class of geometric frameworks which reflect the state of the system nonlinear part is introduced. Geometrical frameworks are defined on the synthesized set. The conditions are given for the structural indistinguishability of geometric frameworks on the set of S-synchronizing inputs. Local identifiability conditions are obtained for the nonlinear part. We are shown that a non-synchronizing input gives an insignificant geometric framework. This leads to a structural non-identifiability of the system nonlinear part. The method is proposed for the estimation of the structural identifiability the nonlinear part of the system. Conditions for parametric identifiability of the system linear part are obtained. We show that the structural identifiability is the basis for the structural identification of the system. The hierarchical immersion method is proposed for the estimation of nonlinear system structural parameters. The method is used for the structural identification of a system with Bouc-Wen hysteresis.

Flight dynamics modeling of a flexible wing unmanned aerial vehicle

This paper shows the results of traditional aerodynamic analysis and the flexible flight dynamics modeling including the effects of structural motion for a flexible wing unmanned aerial vehicle. The influence of some design parameters such as wing flexibility, horizontal/vertical tail aerodynamics is investigated for aeroelasticity and flight dynamics of flexible aircraft. The research platform is an Unmanned Aerial Vehicle (UAV), made of composite material. Its wing span is 4 m and reaches a high aspect ratio, whose value is 18.9. For a traditional analysis, the Vortex Lattice Method (VLM) was used to obtain conventional aerodynamic and control derivatives. The flexible flight dynamics model is based on the work of Waszack and Schmidt. In this approach, it is used the mean-axes reference system and it is assumed that structural deformations is small and described by a set of eigenmodes. The dynamics model incorporates the first normal modes obtained by Ground Vibration Test (GVT) campaigns. A focus of the paper lies on providing a useful model for dynamics system identification and in-flight aeroelastic testing. A developed platform for in-flight system identification and the acquisition system is described. A flight path reconstruction process from a flight test campaign results is shown.

PySINDy: A Python package for the sparse identification of nonlinear dynamical systems from data

Scientists have long quantified empirical observations by developing mathematical models that characterize the observations, have some measure of interpretability, and are capable of making predictions. Dynamical systems models in particular have been widely used to study, explain, and predict system behavior in a wide range of application areas, with examples ranging from Newton’s laws of classical mechanics to the Michaelis-Menten kinetics for modeling enzyme kinetics. While governing laws and equations were traditionally derived by hand, the current growth of available measurement data and resulting emphasis on data-driven modeling motivates algorithmic approaches for model discovery. A number of such approaches have been developed in recent years and have generated widespread interest, including Eureqa (Schmidt & Lipson, 2009), sure independence screening and sparsifying operator (Ouyang, Curtarolo, Ahmetcik, Scheffler, & Ghiringhelli, 2018), and the sparse identification of nonlinear dynamics (SINDy) (Brunton, Proctor, & Kutz, 2016). Maximizing the impact of these model discovery methods requires tools to make them widely accessible to scientists across domains and at various levels of mathematical expertise.

A Note on the Numerical Solutions of Kernel-Based Learning Problems

In the last decade, kernel-based learning approaches typically employed for classification and regression have shown outstanding performance also in dynamic system identification. The typical way to compute the solution of this learning problem subsumes the inversion of the kernel matrix. However, due to limited machine precision, this might not be possible in many practical applications. In this article, we analyze the aforementioned problem and show that the typical estimate is just one of the possible infinite solutions that can be leveraged, considering both the supervised and the semisupervised settings. We show under which conditions the infinite solutions are equivalent, and if it is not the case, we provide a bound on the mismatch between two generic solutions. Then, we propose two specific solutions that are particularly suited to boost sparsity or performance.

State Vector Identification of Hybrid Model of a Gas Turbine by Real-Time Kalman Filter

A model and real-time simulation of a gas turbine engine (GTE) by real-time tasks (RTT) is presented. A Kalman filter is applied to perform the state vector identification of the GTE model. The obtained algorithms are recursive and multivariable; for this reason, ANSI C libraries have been developed for (a) use of matrices and vectors, (b) dynamic memory management, (c) simulation of state-space systems, (d) approximation of systems using equations in matrix finite difference, (e) computing the mean square errors vector, and (f) state vector identification of dynamic systems through digital Kalman filter. Simulations were performed in a Single Board Computer (SBC) Raspberry Pi 2® with a real-time operating system. Execution times have been measured to justify the real-time simulation. To validate the results, multiple time plots are analyzed to verify the quality and convergence time of the mean square error obtained.

Dynamic System Identification and Prediction Using a Self-Evolving Takagi–Sugeno–Kang-Type Fuzzy CMAC Network

This study proposes a Self-evolving Takagi-Sugeno-Kang-type Fuzzy Cerebellar Model Articulation Controller (STFCMAC) for solving identification and prediction problems. The proposed STFCMAC model uses the hypercube firing strength for generating external loops and internal feedback. A differentiable Gaussian function is used in the fuzzy hypercube cell of the proposed model, and a linear combination function of the model inputs is used as the output of the proposed model. The learning process of the STFCMAC is initiated using an empty hypercube base. Fuzzy hypercube cells are generated through structure learning, and the related parameters are adjusted by a gradient descent algorithm. The proposed STFCMAC network has some advantages that are summarized as follows: (1) the model automatically selects the parameters of the memory structure, (2) it requires few fuzzy hypercube cells, and (3) it performs identification and prediction adaptively and effectively.

System Identification with Time-Aware Neural Sequence Models

Established recurrent neural networks are well-suited to solve a wide variety of prediction tasks involving discrete sequences. However, they do not perform as well in the task of dynamical system identification, when dealing with observations from continuous variables that are unevenly sampled in time, for example due to missing observations. We show how such neural sequence models can be adapted to deal with variable step sizes in a natural way. In particular, we introduce a ‘time-aware’ and stationary extension of existing models (including the Gated Recurrent Unit) that allows them to deal with unevenly sampled system observations by adapting to the observation times, while facilitating higher-order temporal behavior. We discuss the properties and demonstrate the validity of the proposed approach, based on samples from two industrial input/output processes.

Dynamic property test and system identification of model aircraft actuators

Dynamic property test and system identification of model aircraft actuators

The Use of Kalman Filter Techniques for Ship Track Estimation

Unscented Kalman Filter (UKF) is a technique used in non-linear applications and dynamic systems identification (e.g. tracking marine vessels and ships) that require state and parameter estimation. This paper studies Kalman Filter (KF) based techniques for tracking ships using Global Positioning System (GPS) data. The present work proposes to exploit information from GPS sensors in order to track a ship in real-time. The absence and presence problem of a ship is handled by a applying KF theory to analyze GPS coordinates and compare current marine vessel routes to previously recorded ones. To study tracking performance, the system was implemented in C++ and simulation results demonstrate the feasibility and high accuracy of the proposed tracking method

Stallу Estimation of the Initial Trajectory of Motion in the Problem of Identification of Dynamic Systems Based on the Quasilinearization Method

Stallу Estimation of the Initial Trajectory of Motion in the Problem of Identification of Dynamic Systems Based on the Quasilinearization Method

Identification of the stick and slip motion between contact surfaces using artificial neural networks

The paper presents work related to nonlinear system parameters identification. The research is focused on systems with hysteretic stiffness characteristics. The identification procedure is developed with use of artificial neural networks. The presented method assumes two separate clusters of neural networks, which are supported by additional signal processing block. Such approach gives an advantage over the conventional identification methods due to its small restrictions. The validation process considers structural responses in time and frequency domains as well as the restoring force plane of the dynamic structure. First, verification of the identification method is performed on the numerical simulation of the system with hysteretic stiffness. Next, the identification of the real dynamic system with contact-related nonlinearity is carried out. The steel samples with contacting surfaces were used in the experiment. Electromagnetic shaker was used to excite the structure and enforce a relative shear motion between surfaces in contact. The system response was recorded using the Polytec laser vibrometer.

Hierarchical Bayesian operational modal analysis: Theory and computations

This paper presents a hierarchical Bayesian modeling framework for the uncertainty quantification in modal identification of linear dynamical systems using multiple vibration data sets. This novel framework integrates the state-of-the-art Bayesian formulations into a hierarchical setting aiming to capture both the identification precision and the variability prompted due to modeling errors. Such developments have been absent from the modal identification literature, sustained as a long-standing problem at the research spotlight. Central to this framework is a Gaussian hyper probability model, whose mean and covariance matrix are unknown, encapsulating the uncertainty of the modal parameters. Detailed computation of this hierarchical model is addressed under two major algorithms using Markov chain Monte Carlo (MCMC) sampling and Laplace asymptotic approximation methods. Since for a small number of data sets the hyper covariance matrix is often unidentifiable, a practical remedy is suggested through the eigenbasis transformation of the covariance matrix, which effectively reduces the number of unknown hyper-parameters. It is also proved that under some conditions the maximum a posteriori (MAP) estimation of the hyper mean and covariance coincide with the ensemble mean and covariance computed using the optimal estimations corresponding to multiple data sets. This interesting finding addresses relevant concerns related to the outcome of the mainstream Bayesian methods in capturing the stochastic variability from dissimilar data sets. Finally, the dynamical response of a prototype structure tested on a shaking table subjected to Gaussian white noise base excitation and the ambient vibration measurement of a cable footbridge are employed to demonstrate the proposed framework.

Evolutionary understanding of the conditions leading to estimation of behavioral properties through system dynamics

One of the basic approaches in science views behavioral products as a process within a dynamic system. The mechanism might be seen as a representation of many instances of centralized control in real time. Many real systems, however, exhibit autonomy by denying statically treated mechanisms. This study addresses the issues related to the identification of dynamic systems and suggests how determining the basic principles of a collective structure may be the key to understanding complex behavioral processes. A fundamental model is derived to assess the advantages of this perspective using a basic methodology. The connection between perspective and technique demonstrates certain aspects within their actual context while also clearly including the framework of actual dynamic system identification.

A new multiple kernel-based regularization method for identification of delay linear dynamic systems

Model parameter estimation, model order selection, variable selection and time delay estimation are four important issues that receive increasing interests in dynamic system identification. However, previous work did not solve the four issues simultaneously. Motivated by the multiple kernel-based regularization method (MKRM), this paper proposes a new multiple kernel-based regularization method for joint model parameter estimation, model order selection, variable selection and time delay estimation for delay linear dynamic systems, referred to as the MKRM-D. Then, an efficient iterative reweighted algorithm is derived to solve the resulting difference of convex functions programming (DCP) problem. In addition, by exploiting the structure of the objective function in each iteration of this algorithm, the alternating direction method of multipliers (ADMM) is employed to decompose the centralized problem into a series of independent subproblems with lower variable dimension, which can be solved in a parallel and distributed manner. The performance of the proposed method is demonstrated by numerical experiments using both synthetic and real data.

Identification of Dynamical Systems Using a Broad Neural Network and Particle Swarm Optimization

System identification plays an important role in improving the structure and parameters of a system, but there are many problems encountered in actual operation. The identification of dynamic systems is not as simple as it is for static systems; thus, choosing effective model structures and parameters is the key to solving this problem. This paper proposes a novel algorithm based on a combination of a broad learning system (BLS) and particle swarm optimization (PSO) to identify nonlinear dynamical systems. The proposed method first uses the dimension expansion of the data set as the input of the BLS and then optimizes the model weight by the PSO algorithm. To verify the effectiveness of our proposal, we use four second-order systems for simulation experiments. The simulation results clearly show the efficiency and anti-interference ability of the proposed method.

Nonlinear dynamic system identification with a cooperative population-based algorithm featuring a restart metaheuristic

Dynamic system identification is commonly reduced to an optimization problem which is complex and multimodal. To find a global optimum of this problem, evolutionary algorithms are often applied. However, as it was shown in many studies, conventional evolution-based algorithms do not demonstrate the acceptable performance for this class of problems, therefore, some effective modifications have been proposed so far. In our study, we combine two approaches which were previously used in linear dynamic system identification and allowed their accurate identification. More specifically, we present a cooperative evolutionary algorithm with a restart metaheuristic and apply it for the parameter identification of a nonlinear cascaded system. The experimental results prove the effectiveness of the proposed evolution-based identification compared to other known solutions of this problem.