Nonlinear Dynamical Systems Research Articles

Flatness‐based control in successive loops for mechatronic motion transmission systems

AbstractMechatronic systems with nonlinear dynamics are met in motion transmission applications for vehicles and robots. In this article, the control problem for the nonlinear dynamics of mechatronic motion transmission systems is solved with the use of a flatness‐based control approach which is implemented in successive loops. The state‐space model of these systems is separated into a series of subsystems, which are connected between them in cascading loops. Each one of these subsystems can be viewed independently as a differentially flat system, and control about it can be performed with inversion of its dynamics as in the case of input–output linearized flat systems. In this chain of subsystems, the state variables of the subsequent ( )‐th subsystem become virtual control inputs for the preceding ‐th subsystem and so on. In turn, exogenous control inputs are applied to the last subsystem and are computed by tracing backwards the virtual control inputs of the preceding subsystems. The whole control method is implemented in successive loops, and its global stability properties are also proven through Lyapunov stability analysis. The validity of the control method is confirmed in the following two case studies: (a) control of a permanent magnet linear synchronous motor (PMLSM)‐actuated vehicle's clutch and (ii) control of a multi‐Degrees of Freedom (multi‐DOF) flexible joint robot.

Analytic solutions of alpha-beta -time- derivatives complex finance chaotic dynamical system: synchronization and extended center manifold. An explicit approach

The present study focuses on a real finance nonlinear dynamic system (FNLDS), which has been shown to exhibit chaotic behavior. The solutions for such nonlinear dynamical systems (NLDSs) have typically been derived using numerical techniques. The objective of this study aims to; firstly, derive approximate analytical solutions for the complex FNLDS (CFNLDS) by constructing the Picard iterative scheme. The convergence of this scheme is proven, and the error analysis shows good tolerance, indicating the efficiency of the technique. Second, a novel criterion for synchronizing the real and imaginary parts of the system is presented, based on a necessary condition. Thirdly, a new method for constructing the extended center manifold is introduced. The 3D portrait reveals a feedback scroll pattern, while the 2D portrait, representing the mutual components, shows multiple pools. The synchronization of the real and imaginary parts of the system is demonstrated graphically. The FNLDS is tested for sensitivity dependence against tiny variations in the initial conditions, and it is found that the system components are moderately sensitive. Furthermore, the Hamiltonian and the extended center manifold establish a two-fold structure. It is observed that the effect of the α-β derivative leads to a delay in the behavior of the solutions.

Nonlinear dynamics of postural control system under visual-vestibular habituation balance practice: evidence from EEG, EMG and center of pressure signals.

Human postural control system is inherently complex with nonlinear interaction among multiple subsystems. Accordingly, such postural control system has the flexibility in adaptation to complex environments. Previous studies applied complexity-based methods to analyze center of pressure (COP) to explore nonlinear dynamics of postural sway under changing environments, but direct evidence from central nervous system or muscular system is limited in the existing literature. Therefore, we assessed the fractal dimension of COP, surface electromyographic (sEMG) and electroencephalogram (EEG) signals under visual-vestibular habituation balance practice. We combined a rotating platform and a virtual reality headset to present visual-vestibular congruent or incongruent conditions. We asked participants to undergo repeated exposure to either congruent (n = 14) or incongruent condition (n = 13) five times while maintaining balance. We found repeated practice under both congruent and incongruent conditions increased the complexity of high-frequency (0.5-20 Hz) component of COP data and the complexity of sEMG data from tibialis anterior muscle. In contrast, repeated practice under conflicts decreased the complexity of low-frequency (<0.5 Hz) component of COP data and the complexity of EEG data of parietal and occipital lobes, while repeated practice under congruent environment decreased the complexity of EEG data of parietal and temporal lobes. These results suggested nonlinear dynamics of cortical activity differed after balance practice under congruent and incongruent environments. Also, we found a positive correlation (1) between the complexity of high-frequency component of COP and the complexity of sEMG signals from calf muscles, and (2) between the complexity of low-frequency component of COP and the complexity of EEG signals. These results suggested the low- or high-component of COP might be related to central or muscular adjustment of postural control, respectively.

Analysis of the Dynamics of a Cubic Nonlinear Five-Dimensional Memristive Chaotic System and the Study of Reduced-Dimensional Synchronous Masking

To improve the complexity of the chaotic system and achieve the effective transmission of image information, in this paper, a five-dimensional memristive chaotic system with cubic nonlinear terms is constructed, which has four pairs of symmetric coordinates. First, the cubic nonlinear memristive chaotic system is analyzed using the Lyapunov exponential map, bifurcation map, and attractor phase diagram. The experimental results show that under four pairs of symmetric coordinates, the system exists not only parameter-dependent symmetric rotational coexisting attractor and transient chaotic phenomena but also exists super-multistationary with alternating chaotic cycles dependent on the initial value of the memristor. Then, it is proposed to add a constant term to the linear state variable to explore the effect of the offset increment of the linear state variable on the system in four pairs of symmetric coordinates, while circuit simulation of the five-dimensional chaotic system is carried out using Simulink to verify its existence and realisability. Finally, the synchronization of the dimensionality reduction system and the confidential transmission of the image are achieved, using the control voltage of the system to replace the internal state variables of the memristor to achieve the one-dimensional reduction process, and an adaptive synchronization controller is designed to synchronize the system before and after the dimensionality reduction. Based on the above, an image to be transmitted is modulated into a one-dimensional array and then subjected to the fractional and cyclic operations and combined with the linear encryption and decryption functions and the chaotic masking technique, the simple encryption and decryption of the image processes are realized.

ModelFLOWs-app: Data-driven post-processing and reduced order modelling tools

This article presents an innovative open-source software named ModelFLOWs-app,1 written in Python, which has been created and tested to generate precise and robust hybrid reduced order models (ROMs) fully data-driven. By integrating modal decomposition and deep learning in diverse ways, the software uncovers the fundamental patterns in dynamic systems. This acquired knowledge is then employed to enrich the comprehension of the underlying physics, reconstruct databases from limited measurements, and forecast the progression of system dynamics. The hybrid ROMs produced by ModelFLOWs-app combine experimental and numerical databases, serving as highly accurate alternatives to numerical simulations. As a result, computational expenses are significantly reduced, and the models become powerful tools for optimization and control in various applications. The exceptional capability of ModelFLOWs-app in developing reliable data-driven hybrid ROMs has been demonstrated across a wide range of applications, making it a valuable resource for understanding complex nonlinear dynamical systems and providing insights in diverse domains. This article presents the mathematical background, as well as a review of some examples of applications. Program summaryProgram title: ModelFLOWs-appCPC Library link to program files:https://doi.org/10.17632/49tzcc8sf3.1Developer's repository link:github.com/modelflows/ModelFLOWs-appLicensing provisions: MIT licenseProgramming language: PythonSupplementary material: Tutorial, example datasets.Nature of problem: ModelFLOWs-app is an open-source Software for data post-processing, patterns identification and development of reduced order models using modal decomposition and deep learning architectures. ModelFLOWs-app provides its users with a user-friendly interface to efficiently identify patterns in data, reconstruct data by repairing or enhancing it, and make predictions based on the available data.Solution method: ModelFLOWs-app methodological framework is formed by two big modules: Module 1 uses modal decomposition methods, and Module 2 is formed by hybrid machine learning tools, which combine modal decomposition with deep learning architectures. Each module solves three different applications: (1) patterns identification, suitable to study the physics behind the data analysed; (2) data reconstruction, capable to reconstruct two- or three- dimensional databases from a set of selected points, using data from sensors, or repairing missing data; (3) data forecasting, which builds reduced order models (ROMs) to predict the spatio-temporal evolution of the signal analysed.Additional comments including restrictions and unusual features: ModelFLOWs-app offers an intuitive and user-friendly interface, making it accessible to users with varying levels of technical expertise. It is compatible with Windows, macOS, and Linux operating systems. Users can import their data in popular formats (.mat, .csv, .npy, .pkl, .h5), and the results can be exported in Matlab “.mat” or Numpy “.npy” for post-processing. The Deep Learning models have been developed using TensorFlow, and users have the option to search for the optimal hyperparameters for their database since hyperparameter optimization has been implemented for all models. See Refs. [1–10].

Nonlinear optimization filters for stochastic time‐varying convex optimization

AbstractWe look at a stochastic time‐varying optimization problem and we formulate online algorithms to find and track its optimizers in expectation. The algorithms are derived from the intuition that standard prediction and correction steps can be seen as a nonlinear dynamical system and a measurement equation, respectively, yielding the notion of nonlinear filter design. The optimization algorithms are then based on an extended Kalman filter in the unconstrained case, and on a bilinear matrix inequality condition in the constrained case. Some special cases and variations are discussed, notably the case of parametric filters, yielding certificates based on LPV analysis and, if one wishes, matrix sum‐of‐squares relaxations. Supporting numerical results are presented from real data sets in ride‐hailing scenarios. The results are encouraging, especially when predictions are accurate, a case which is often encountered in practice when historical data is abundant.

Controllability of time-varying fractional dynamical systems with distributed delays in control

Abstract The objective of this study is to assess controllability results in the realm of Caputo fractional derivatives, specifically focusing on time-varying linear and nonlinear fractional dynamical systems with distributed control delays. In the setting of a linear system, using the Grammian matrix's positive definiteness makes it possible to ascertain both the necessary and sufficient conditions. It is possible to determine, for nonlinear systems, the sufficient conditions for the presence of a solution by making use of the Schauder fixed point theorem. In addition to that, the paper discusses the controllability of a nonlinear integrodifferential system for a particular case. A few examples and the graphs that correspond to them are shown here to make it easier to conceptualize the conclusions of the theoretical discussion.

An approach to nonlinear optimisation via set stabilisation of dynamical systems with an application to synchronous machines

An approach to constraint nonlinear optimisation is proposed, where the variables of the objective function and of the constraint function belong to the state of a nonlinear dynamical system. A nonlinear closed-loop feedback system is designed, whose solutions converge to a target set consisting of the optimal points of the underlying optimisation problem. The feedback law for the control input of the nonlinear dynamical system is designed by exact linearisation of a dynamical formulation of a first-order optimality condition. For the problem formulation and the stability analysis of the nonlinear closed-loop feedback system, the notion of V-detectability is considered. An application to energy-optimal feed-forward torque control for different types of synchronous machines is presented, which demonstrates the usefulness of the method. In this regard, anisotropic permanent magnet synchronous machines and electrically excited synchronous machines are covered.

Data-Driven Identification of Crane Dynamics Using Regularized Genetic Programming

The meaningful problem of improving crane safety, reliability, and efficiency is extensively studied in the literature and targeted via various model-based control approaches. In recent years, crane data-driven modeling has attracted much attention compared to physics-based models, particularly due to its potential in real-time crane control applications, specifically in model predictive control. This paper proposes grammar-guided genetic programming with sparse regression (G3P-SR) to identify the nonlinear dynamics of an underactuated crane system. G3P-SR uses grammars to bias the search space and produces a fixed number of candidate model terms, while a local search method based on an l0-regularized regression results in a sparse solution, thereby also reducing model complexity as well as reducing the probability of overfitting. Identification is performed on experimental data obtained from a laboratory-scale overhead crane. The proposed method is compared with multi-gene genetic programming (MGGP), NARX neural network, and Takagi-Sugeno fuzzy (TSF) ARX models in terms of model complexity, prediction accuracy, and sensitivity. The G3P-SR algorithm evolved a model with a maximum mean square error (MSE) of crane velocity and sway prediction of 1.1860 × 10−4 and 4.8531 × 10−4, respectively, in simulations for different testing data sets, showing better accuracy than the TSF ARX and MGGP models. Only the NARX neural network model with velocity and sway maximum MSEs of 1.4595 × 10−4 and 4.8571 × 10−4 achieves a similar accuracy or an even better one in some testing scenarios, but at the cost of increasing the total number of parameters to be estimated by over 300% and the number of output lags compared to the G3P-SR model. Moreover, the G3P-SR model is proven to be notably less sensitive, exhibiting the least deviation from the nominal trajectory for deviations in the payload mass by approximately a factor of 10.

Approximate controllability of fractional differential equations driven by Markovian switching and Lévy noise with infinite delay

Motivated by applications of nonlinear dynamical systems and based on some existing works, we introduce a general class of Atangana-Baleanu nonlinear fractional neutral differential equations driven by Markovian switching and Lévy noise with infinite delay. In this paper, we aim to derive new results on the existence of solutions as well as approximate controllability for the corresponding Atangana-Baleanu equations in Banach spaces. Under suitable conditions with the help of semigroup theory, fixed point theorem as well as stopping time argument, we prove the main results. As an application, we present two examples with numerical simulations to support our theory. The results obtained in this paper improve and extend some related conclusions on this topic.

Nonlinear system dynamics of calcium and nitric oxide due to cell memory and superdiffusion in neurons

The integer-order interdependent calcium ([Ca2+]) and nitric oxide (NO) systems are unable to shed light on the influences of the superdiffusion and memory in triggering Brownian motion (BM) in neurons. Therefore, a mathematical model is constructed for the fractional-order nonlinear spatiotemporal systems of [Ca2+] and NO incorporating reaction-diffusion equations in neurons. The two-way feedback process between [Ca2+] and NO systems through calcium feedback on NO production and NO feedback on calcium through cyclic guanosine monophosphate (cGMP) with plasmalemmal [Ca2+]-ATPase (PMCA) was incorporated in the model. The Crank–Nicholson scheme (CNS) with Grunwald approximation along spatial derivatives and L1 scheme along temporal derivatives with Gauss–Seidel (GS) iterations were employed. The numerical outcomes were analyzed to get insights into superdiffusion, buffer, and memory exhibiting BM of [Ca2+] and NO systems. The conditions, events and mechanisms leading to dysfunctions in calcium and NO systems and causing different diseases like Parkinson’s were explored in neurons.

Explicit machine learning-based model predictive control of nonlinear processes via multi-parametric programming

Machine learning-based model predictive control (ML-MPC) has been developed to control nonlinear processes with unknown first-principles models. While ML models can capture nonlinear dynamics of complex systems, the complexity of ML models leads to increased computation time for real-time implementation of ML-MPC. To address this issue, in this work, we propose an explicit ML-MPC framework for nonlinear processes using multi-parametric programming. Specifically, a self-adaptive approximation algorithm is first developed to obtain a piecewise linear affine function that approximates the behaviors of ML models. Then, multi-parametric quadratic programming (mpQP) problems are formulated to generate the solution map for states in discretized state–space. Furthermore, to accelerate the implementation of explicit ML-MPC, a neighbor-first search algorithm is developed. Finally, an example of a chemical reactor is used to demonstrate the effectiveness of the explicit ML-MPC.

Empirical study of robust/developed PID control for nonlinear time-delayed dynamical system in discrete time domain

In addition to the high nonlinearity of liquid dynamics inside a tank, a study was conducted to overcome loading/unloading liquid storage tank control problems. In this study, the author developed a nonlinear control system to conquer both nonlinearity and the significant time delay arising from using a pressure-difference-based level sensor. To this end, this study proposes the implementation of nonlinear state dependent (SDP-PID+) control using the SDP transfer function model as a class of nonlinear descriptions of dynamical systems. By incorporating additional robust (plus) compensators alongside the traditional P-, I-, and D-compensators, the robust SDP-PID controller utilizes the complete state feedback to create a time-varying state variable feedback (SDP-SVF) control law. This approach effectively mitigates the effects of the discrete-time SDP-TF. It introduces the pole placement tuning approach, which renders significant performance using the laboratory test rig TPS8.2.2.4, for automatic liquid level control.

A vibration analysis of the permanent magnet synchronous motor under the effect of proportional derivative control

In industrial environments, excessive vibration can pose a safety risk since it can weaken the structural integrity of PMSMs and adjacent equipment. Vibration levels can be understood and controlled to help assure safe operation and avoid mishaps or equipment breakdowns. Friction and mechanical resistance are examples of vibration-induced losses that can lower PMSMs’ overall efficiency. Through the optimization of motor design and operation, engineers may minimize vibration and increase energy efficiency while lowering operating costs. Since an electric motor’s tendency to overheat after a brief period of use is linked to oscillating vibration, this study employs a proportional derivative control (PD control) to demonstrate the potential effectiveness of a permanent magnet synchronous motor (PMSM). The external force is a component of the non-linear dynamical system. The equation of the (PMSM) system is explained with two-degree-of-freedom (2dof) differential coupled equations, consisting of the major body (motor current) and cutting drive system. The present study uses the approximate method of multiple scales perturbation technique (MSPT) to get an approximate solution, showing the response equation before using (PD) and to achieve the highest effect on the control system so as to ensure the highest efficiency at the lowest possible cost. Different numerical tests have been conducted (NPDCVF, NVC, PPF, PD, delay velocity) and showed that (PD) has the best effect and is able to control the vibrations with more accuracy than others. Then, the vibration value of the system has been studied numerically before and after applying the control method of (MSPT). By combining the analysis of the resonance situation via both the phase plane techniques and the frequency response equation ω1≅ω2,ω≅ω2, within the utilization of Runge–Kutta of the fourth-order, the stability of the numerical solution has been investigated. Then, by employing the MATLAB tool, the impact of various parameters on model performance has been examined numerically. Lastly, a comparison is made with earlier research and other techniques used in prior work using other types of control mechanisms for other systems.

Hidden attractors and nonlocal oscillations in gene networks models

Abstract We study periodic trajectories of nonlinear dynamical systems considered as models of the simplest molecular repressilator. In the phase portraits of these systems, we find hidden attractors and nonlocal oscillations. The cases of nonuniqueness of cycles in these portraits are described as well.

Generalized quadratic embeddings for nonlinear dynamics using deep learning

The engineering design process often relies on mathematical modeling that can describe the underlying dynamic behavior. In this work, we present a data-driven methodology for modeling the dynamics of nonlinear systems. To simplify this task, we aim to identify a coordinate transformation that allows us to represent the dynamics of nonlinear systems using a common, simple model structure. The advantage of a common simple model is that customized design tools developed for it can be applied to study a large variety of nonlinear systems. The simplest common model – one can think of – is linear, but linear systems often fall short in accurately capturing the complex dynamics of nonlinear systems. In this work, we propose using quadratic systems as the common structure, inspired by the lifting principle. According to this principle, smooth nonlinear systems can be expressed as quadratic systems in suitable coordinates without approximation errors. However, finding these coordinates solely from data is challenging. Here, we leverage deep learning to identify such lifted coordinates using only data, enabling a quadratic dynamical system to describe the system’s dynamics. Additionally, we discuss the asymptotic stability of these quadratic dynamical systems. We illustrate the approach using data collected from various numerical examples, demonstrating its superior performance with the existing well-known techniques.

Detection of myocardial infarction using Shannon energy envelope, FA-MVEMD and deterministic learning

AbstractMyocardial infarction (MI) poses a significant clinical challenge, necessitating expeditious and precise detection to mitigate potentially fatal outcomes. Current MI diagnosis predominantly relies on electrocardiography (ECG); however, it is fraught with limitations, including inter-observer variability and a reliance on expert interpretation. This study introduces an automated MI detection framework that capitalizes on hybrid signal processing methodologies and deterministic learning theory. The initial step involves the extraction of the Shannon energy envelope (SEE) and its derivative from a single-lead ECG. Integration of the SEE into the ECG’s phase portrait provides a means to capture the underlying nonlinear system dynamics. Subsequently, the application of fast and adaptive multivariate empirical mode decomposition (FA-MVEMD) yields discriminative features originating from the most energetically dominant intrinsic mode components (IMFs) within the SEE. Profound dissimilarities are discernible between ECG signals recorded from healthy subjects and those afflicted with MI. In the subsequent phase, deterministic learning theory, implemented through neural networks, is employed to facilitate the classification of ECG signals into two distinct groups. The method’s efficacy is meticulously evaluated using the PTB diagnostic ECG database, resulting in a noteworthy average classification accuracy of 99.21$$\%$$ % within a tenfold cross-validation framework. In summation, the findings affirm that the proposed features not only complement conventional ECG attributes but also align closely with the underlying dynamics of the ECG system, ultimately fortifying the automatic detection of MI. The imperative requirement for early and accurate MI diagnosis is addressed through our approach, offering a robust and dependable means to fulfill this pivotal clinical need.

Fuzzy fractional delay differential inclusions driven by hemivariational inequalities in Banach spaces

This paper investigates a novel class of nonlinear dynamical fuzzy systems referred to as fuzzy fractional delay differential hemivariational inequalities (FFDDHVIs) in Banach spaces. These systems integrate fractional fuzzy differential inclusions with delay and hemivariational inequalities. The existence theorem for the HVIs is established based on the KKM theorem. Moreover, specific propositions are proved, covering the superpositional measurability and the upper semicontinuity for the HVIs. Next, by using the fixed points theorem, we establish the existence and compactness of mild solution sets for the FFDDHVIs under certain mild conditions. Finally, As an illustrative application, we investigate a frictional quasistatic contact problem for viscoelastic materials, in which the friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals.

Approximate Noether theorem and its inverse for nonlinear dynamical systems with approximate nonstandard Lagrangian

The approximate Noether theorem and its inverse theorem for the nonlinear dynamical systems with approximate exponential Lagrangian and approximate power-law Lagrangian are investigated. For each case, the approximate differential equations of motion for the nonlinear dynamical systems with approximate nonstandard Lagrangian are established, the generalized Noether identities are given. The relationship between the approximate Noether symmetries and approximate conserved quantities for the system with approximate nonstandard Lagrangian are established, and the approximate Noether theorems and their inverse theorems are obtained. Two examples are given to illustrate the application of the results.

Analyzing Observability of Fractional Dynamical Systems Employing ψ-Caputo Fractional Derivative

In this present article, we inquire into the observability of linear and nonlinear fractional dynamical systems in terms of ψ-Caputo fractional derivative. The observability Grammian matrix, which is positive definite and expressed by the Mittag-Leffler functions, is used to obtain the necessary and sufficient conditions of observability of linear fractional dynamical systems, and Banach’s fixed point theorem is used to get the sufficient conditions for the observability of nonlinear fractional systems. Three numerical examples are given to demonstrate the applicability of theoretical results for linear and nonlinear cases.