Nonlinear System Model Research Articles

Collaborative Integrated Navigation for Unmanned Aerial Vehicle Swarms Under Multiple Uncertainties

UAV swarms possess unique advantages in performing various tasks and have been successfully applied across multiple scenarios. Accurate navigation serves as the foundation and prerequisite for executing these tasks. Unlike single UAV localization, swarms enable the sharing and propagation of precise positioning information, which enhances overall swarm localization accuracy but also introduces the issue of uncertainty propagation. To address this challenge, this paper proposes an integrated navigation and positioning method that models, propagates, and mitigates uncertainties. To tackle the issue of uncertainty in information quality caused by outliers in external correction data, a robust integrated navigation method for nonlinear systems is derived based on a normal gamma distribution model. Considering uncertainty propagation, a statistical linearization model for nonlinear systems is developed. Building upon this model, an augmented measurement nonlinear least squares positioning method is applied, achieving further improvements in localization accuracy. Simulation experiments demonstrate that the proposed method, which thoroughly accounts for the effects of multiple uncertainties, can achieve robust tracking and provide relatively accurate positioning results.

Large-scale-aware data augmentation for reduced-order models of high-dimensional flows

Convolutional autoencoders have proven to be an adequate tool to perform reduced-order modeling for high-dimensional nonlinear dynamical systems. Their goal is to reduce dimensionality strongly while preserving the most characteristic features of the system. Here, we show that these models rely sensitively on the completeness of the provided data. This is particularly challenging for fully turbulent flows with their coherent structures ranging from large-scale superstructures to dissipative eddies over orders of magnitude in time and space. As a result, an unrealistically large number of data snapshots would be required to properly cover all the essential dynamics, whereas the features on small time and length scales require only a small number of snapshots of the respective flow, especially the long lasting large-scale structures that are difficult to characterize either numerically or experimentally. We demonstrate for three types of flows that a missing representation of large-scale turbulent structures leads to failures in the training process. We suggest a method to mitigate this shortcoming. This includes the transformation of data samples to new large-scale structures, which enhance the data. Furthermore, we skip augmentations that are more detrimental to the model performance. We evaluate our method for three datasets, two from numerical simulations of turbulent Rayleigh–Bénard convection flows and one from laboratory experiment for the flow past an array of cylinders. We show that the method can substantially improve model utility for high-dimensional data. In this way, we avoid an intensive grid search through possible augmentation combinations without further knowledge about the underlying system.

Neural-Network-Based Incremental Backstepping Sliding Mode Control for Flying-Wing Aircraft

The nonlinear trajectory tracking control problem is studied for a flying-wing aircraft. Starting from a nonlinear dynamics model of the flying-wing aircraft, the trajectory tracking control is decomposed into multiple loops of position control, flight path control, and attitude control. An incremental backstepping sliding mode control is proposed to implement attitude control, while an incremental nonlinear dynamic inversion and a nonlinear dynamic inversion design are used to deal with the nonlinear system model for the flight path and position control, respectively. In addition, a radial basis function neural-network-based extended state disturbance observer is proposed to deal with model uncertainties, gust disturbances, and unknown faults of the aircraft. The closed-loop control system is proved to be stable using Lyapunov theory. The performance of the proposed disturbance-observer-based incremental backstepping sliding mode control is demonstrated in simulation through a set of three-dimensional tracking scenarios. Compared with both backstepping control and backstepping sliding mode control, tracking performance measured by settling time, tracking error, and overshoot are improved by the proposed design when realistic trajectory tracking missions are executed.

Asymptotic stability of fractional order switching nonlinear system based on short memory principle

AbstractFractional order derivatives have memory effects and are widely used in real world applications. However, they require large storage space and lead to low computational efficiency. Therefore, fractional order systems based on the short memory principle have gradually attracted the scholars' attention. In this paper, the asymptotic stability of Caputo fractional order switching nonlinear systems is investigated based on the Markov process and short memory principle. Firstly, a model of Caputo fractional order Markovian switching nonlinear systems (CFMNSs) based on the short memory principle is constructed so that the lower bound initial time and the corresponding initial state values are updated synchronously with switching. Secondly, the stability of the system is investigated based on the probabilistic analysis method and stochastic multi‐Lyapunov functions and the sufficient conditions for the asymptotic stability of the system are given. Using a similar method, we also study the asymptotic stability of CFMNs with variable fractional order. Finally, the simulation results show that the proposed stability scheme is effective and reasonable.

Learning-Based Modeling and Predictive Control for Unknown Nonlinear System With Stability Guarantees

Learning-Based Modeling and Predictive Control for Unknown Nonlinear System With Stability Guarantees

An experimental approach to estimating the current oil recovery coefficient using a complex of stochastic and numerical computer modeling methods

This paper presents an experimental study devoted to determining the prospects for the integrated application of stochastic and numerical computer modeling methods to solve one of the fundamental problems of field development – assessing current oil recovery based on various parameters that reflect the properties of productive formations and their saturating fluids. The object of the study was the long-term deposits of the Volga–Ural oil and gas province. As part of the initial stage, a discriminant analysis was carried out according to the tectonic-stratigraphic criterion, as a result of which three clusters of homogeneous objects were formed based on seventeen parameters. When interpreting the distribution of deposits in the axes of canonical discriminant functions, the unstable parameter of the total thickness of the reservoir was replaced by one of the variable indicators selected using the digital implementation of the uniform search method and its interpretation using known approaches. In order to reduce the influence factor of the uncertainty zone that arises when considering objects in the axes of canonical discriminant functions, the method of constructing structured grids was used. The reliability and scope of the obtained dependence were established using the Broyden-Fletcher-Goldfarb-Shanno optimization algorithm. The conclusion is made about the possibility of using the developed scientific and methodological approach to forecasting oil recovery at different values of the well grid density. Keywords: oil recovery coefficient; modeling of nonlinear systems; deposits of the Volga-Ural oil and gas province; tectonic and stratigraphic confinement of objects; stochastic and numerical methods of computer analysis and data processing; development of oil fields, geological and field parameters.

Fuzzy Neural Network Applications in Biomass Gasification and Pyrolysis for Biofuel Production: A Review

The increasing demand for sustainable energy has spurred interest in biofuels as a renewable alternative to fossil fuels. Biomass gasification and pyrolysis are two prominent thermochemical conversion processes for biofuel production. While these processes are effective, they are often influenced by complex, nonlinear, and uncertain factors, making optimization and prediction challenging. This study highlights the application of fuzzy neural networks (FNNs)—a hybrid approach that integrates the strengths of fuzzy logic and neural networks—as a novel tool to address these challenges. Unlike traditional optimization methods, FNNs offer enhanced adaptability and accuracy in modeling nonlinear systems, making them uniquely suited for biomass conversion processes. This review not only highlights the ability of FNNs to optimize and predict the performance of gasification and pyrolysis processes but also identifies their role in advancing decision-making frameworks. Key challenges, benefits, and future research opportunities are also explored, showcasing the transformative potential of FNNs in biofuel production.

Feature analysis and ensemble-based fault detection techniques for nonlinear systems

AbstractMachine learning approaches play a crucial role in nonlinear system modeling across diverse domains, finding applications in system monitoring, anomaly/fault detection, control, and various other areas. With technological advancements, today such systems might include hundreds or thousands of sensors that generate large amounts of multivariate data streams. This inevitably results in increased model complexity. In response, feature selection techniques are widely employed as a means to reduce complexity, avoid the curse of high dimensionality, decrease training and inference times, and eliminate redundant features. This paper introduces a sensitivity-inspired feature analysis technique for regression tasks. Leveraging the energy distance on the model prediction errors, this approach performs both feature ranking and selection. Additionally, this paper introduces an ensemble-based unsupervised fault detection methodology that incorporates homogeneous units, specifically long short-term memory (LSTM) predictors and cumulative sum-based detectors. The proposed predictors utilize a variant of the teacher forcing (TF) algorithm during both the training and inference phases. Additionally, predictors are used to model the normal behavior of the system, whereas detectors are used to identify deviations from normality. The detector decisions are aggregated using a majority voting scheme. The validity of the proposed approach is illustrated on the two representative datasets, where numerous experiments are performed for feature selection and fault detection evaluation. Experimental assessment reveals promising results, even compared to well-established techniques. Nevertheless, the results also demonstrate the need to perform additional experiments with datasets originating from both simulators and real systems. Further possible refinements of the detection ensemble include the addition of heterogeneous units and other decision fusion techniques.

Force-displacement hysteresis characteristics of PAM based on improved generalised Prandtl-Ishlinskii model

As an advanced actuator that simulates the characteristics of biological muscle, the internal structure and working principle of pneumatic artificial muscle imply significant complexity and highly nonlinear characteristics, which are particularly prominent in the theoretical modelling process, and pose many challenging problems for researchers. The classical Prandtl-Ishlinskii model fails to provide ideal mathematical descriptions and accurate models when dealing with the nonlinear characteristics of pneumatic artificial muscle, resulting in the construction of models for such nonlinear systems that still require further research. Based on the above, this paper proposes a generalised Prandtl-Ishlinskii model based on the improved force-displacement hysteresis characteristics of pneumatic artificial muscle, which employs an improved particle swarm optimisation to identify the unknown parameters of the model, and applies the identified parameters to construct the force-displacement hysteresis curves of pneumatic artificial muscle. Comparing the model fitting results with the experimental data, it is found that the maximum error of the improved generalised Prandtl-Ishlinskii model is reduced by more than 50% and the average error is reduced by more than 75% compared with the classical Prandtl-Ishlinskii model when the internal pressure of the pneumatic artificial muscle is 0.35, 0.4 and 0.45 MPa. The improved generalised Prandtl-Ishlinskii model can better characterise the asymmetric hysteresis property, which improves the model accuracy of the force-displacement coupling of pneumatic artificial muscle, and provides a certain theoretical basis for the later realisation of the precise control of the force-displacement coupling of pneumatic artificial muscle.

Data-free non-intrusive model reduction for nonlinear finite element models via spectral submanifolds

The theory of spectral submanifolds (SSMs) has emerged as a powerful tool for constructing rigorous, low-dimensional reduced-order models (ROMs) of high-dimensional nonlinear mechanical systems. A direct computation of SSMs requires explicit knowledge of nonlinear coefficients in the equations of motion, which limits their applicability to generic finite-element (FE) solvers. Here, we propose a non-intrusive algorithm for the computation of the SSMs and the associated ROMs up to arbitrary polynomial orders. This non-intrusive algorithm only requires system nonlinearity as a black box and hence, enables SSM-based model reduction via generic finite-element software. Our expressions and algorithms are valid for systems with up to cubic-order nonlinearities, including velocity-dependent nonlinear terms, asymmetric damping, and stiffness matrices, and hence work for a large class of mechanics problems. We demonstrate the effectiveness of the proposed non-intrusive approach over a variety of FE examples of increasing complexity, including a micro-resonator FE model containing more than a million degrees of freedom.

Real-time Error Compensation Transfer Learning with Echo State Networks for Enhanced Wind Power Prediction

Accurate wind power forecasting is essential for efficient energy management and grid stability, enabling energy providers to balance supply and demand, optimize renewable energy integration, reduce operational costs, and enhance power grid reliability. The Echo State Network (ESN) is widely used for modeling nonlinear dynamic systems due to its simple and rapid training process. However, ESNs can be prone to system errors, leading to inaccurate models when handling high-order nonlinear complexities. To overcome this, we developed the Error Compensation Transfer Learning Echo State Network (ETL-ESN), which combines a computing layer based on ESN and a compensation layer using transfer learning. Our model identifies error auto-correlation as a key factor that increases variance in ESN predictions, and addresses this with an error compensation layer to reduce system errors. We further leverage transfer learning to prevent overfitting within the error domain. Extensive experiments using real-world wind power data demonstrate that the ETL-ESN model reduces training time from 65 s to 2 s compared to LSTM, while lowering MAE by up to 95%. The ETL-ESN achieves a 95% to 98% improvement in prediction accuracy across different turbines compared to traditional models. The code and datasets used in this study are available at GitHub repository https://github.com/zhuyingqin/Error-Transfer-ESN for further research and replication.

Data-driven dynamic state estimation in power systems via sparse regression unscented Kalman filter

This paper proposes a novel data-driven modeling and dynamic state-estimation approach for nonlinear power and energy systems, highlighting the critical role of a known dynamic model for accurate state estimation in the face of uncertainty and complex models. The proposed framework consists of a two-phase approach: data-driven model identification and state-estimation. During the model identification phase, which spans a relatively short time interval, state feedback is collected to identify the dynamics of the nonlinear systems in the power grid using a novel density-guided sparse identification algorithm. Unlike conventional sparse regression, which relies on a large library of linear and nonlinear functions to fit data, our proposed algorithm iteratively updates a relatively small initial library by adding higher-order nonlinear functions if the coefficients of the current functions are dense. Following the identification of the model’s dynamics, the estimation phase addresses the challenge of incomplete state measurements. By implementing an unscented Kalman filter, the state variables of the system are dynamically estimated by measuring the noisy output. Finally, simulation results on an IEEE 30-bus system are presented to illustrate the effectiveness of the density-guided sparse regression unscented Kalman filter compared to a physics-based unscented Kalman filter with model uncertainty. This study contributes to the fields of data-driven modeling techniques, machine learning for power systems, and computational intelligence in smart grids. It emphasizes the use of advanced sparse regression and unscented Kalman filter methods for state estimation, enhancing the robustness and accuracy of monitoring and control in electrical and energy systems.

A dynamical systems model for the total fission rate in Drp1-dependent mitochondrial fission.

Mitochondrial hyperfission in response to cellular insult is associated with reduced energy production and programmed cell death. Thus, there is a critical need to understand the molecular mechanisms coordinating and regulating the complex process of mitochondrial fission. We develop a nonlinear dynamical systems model of dynamin related protein one (Drp1)-dependent mitochondrial fission and use it to identify parameters which can regulate the total fission rate (TFR) as a function of time. The TFR defined from a nondimensionalization of the model undergoes a Hopf bifurcation with bifurcation parameter [Formula: see text] where [Formula: see text] is the total concentration of mitochondrial fission factor (Mff) and k+ and k- are the association and dissociation rate constants between oligomers on the outer mitochondrial membrane. The variable μ can be thought of as the maximum build rate over the disassembling rate of oligomers. Though the nondimensionalization of the system results in four dimensionless parameters, we found the TFR and the cumulative total fission (TF) depend strongly on only one, μ. Interestingly, the cumulative TF does not monotonically increase as μ increases. Instead it increases with μ to a certain point and then begins to decrease as μ continues to increase. This non-monotone dependence on μ suggests interventions targeting k+, k-, or [Formula: see text] may have a non-intuitive impact on the fission mechanism. Thus understanding the impact of regulatory parameters, such as μ, may assist future therapeutic target selection.

PID control algorithm based on multistrategy enhanced dung beetle optimizer and back propagation neural network for DC motor control.

Traditional Proportional-Integral-Derivative (PID) control systems often encounter challenges related to nonlinearity and time-variability. Original dung beetle optimizer (DBO) offers fast convergence and strong local exploitation capabilities. However, they are limited by poor exploration capabilities, imbalance between exploration and exploitation phases, and insufficient precision in global search. This paper proposes a novel adaptive PID control algorithm based on enhanced dung beetle optimizer (EDBO) and back propagation neural network (BPNN). Firstly, the diversity of exploration is increased by incorporating a merit-oriented mechanism into the rolling behavior. Then, a sine learning factor is introduced to balance the global exploration and local exploitation capabilities. Additionally, a dynamic spiral search strategy and adaptive [Formula: see text]-distribution disturbance are presented to enhance search precision and global search capability. The BPNN is employed to fine-tune both PID and network parameters, leveraging its powerful generalization and learning ability to model nonlinear system dynamics. In the simplified motor experiments, the proposed controller achieved the lowest overshoot (0.5%) and the shortest response time (0.012s), with a settling time of 0.02s and a steady-state error of just 0.0010. In another set of experiments, the proposed controller recorded an overshoot and response time of 0.7% and 0.0010s, across five DC motor tests. These results demonstrate the proposed adaptive PID control algorithm has superior performance in optimizing control system parameters, as well as improving system robustness and stability.

Dual-Sliding-Surface Robust Control for the PEMFC Air-Feeding System Based on Terminal Sliding Mode Algorithm

The proton exchange membrane fuel cell (PEMFC) is the most widely used fuel cell, but it also has some limitations. One of the research pain points is controlling the oxygen content in PEMFCs. A moderate excess of oxygen boosts electrochemical reaction efficiency, while an appropriate oxygen content ensures system stability. In this paper, a fourth-order nonlinear mathematical model of a PEMFC stack air supply system is established to solve the problem of optimal oxygen excess ratio (OER) control under dynamic load conditions. Based on the model, a nonsingular terminal sliding mode controller (NTSMC) based on a sliding mode observer (SMO) is proposed. The NTSM exhibits superior robustness and performance compared to other sliding mode structures. Meanwhile, the SMO accurately predicts system states, facilitating precise control actions. Additionally, the dual sliding mode surfaces enhance system stability against parameter uncertainties and external disturbances. Our results demonstrate that the proposed controller outperforms traditional ones in terms of robustness and performance, which significantly enhances PEMFC system efficiency and stability.

Research on lateral–torsional coupled vibration induced by substantial imbalance in an overhung flexible rotor

Substantial imbalance is a real possibility in aero–engines and is usually caused by blade off, bird strikes, etc. A substantially unbalanced rotor system will exhibit extremely complicated nonlinear lateral–torsional behavior, and the related dynamics are critical to the safe design of a rotor system. In this paper, a comprehensive nonlinear model of an imbalanced flexible rotor system is proposed, in which translational–torsional coupling and angular–torsional coupling are introduced simultaneously. Based on the model, the lateral–torsional coupling features are discussed, and the lateral as well as the torsional vibration behaviors are further analyzed in detail. The results show that the lateral–torsional coupling factors only exist under the nonsynchronous whirl condition. Thus, if the imbalance is small, the lateral vibration exhibits the linear feature of a traditional unbalanced response, and the torsional vibration does not occur in steady state. However, under substantially unbalanced conditions, the vibration responses are significantly changed. In lateral vibration, many combined frequency components between rotation frequency (fu) and torsional natural frequency (ft1 and ft2) such as fu-ft1, fu-2ft1 and fu-ft2 can be induced, which can lead to combined resonance phenomena and many additional resonance peaks. The lateral–torsional coupling effect is strong at that time. Both the lateral translational–torsional coupling excitation and lateral angular–torsional coupling excitation exhibits the multiple harmonics whose frequencies are closely related to ft1 and ft2. Consequently, the torsional vibration in those combined resonance regions can be very high. The corresponding vibration exhibits the steady harmonic feature with vibration frequency being ft1 or ft2.

Pattern‐moving‐based dynamic description and optimal control for non‐Newtonian mechanical systems with generalized cell mapping

AbstractThis article focuses on the problem of modelling and control for a non‐Newtonian mechanical system that may be subject to the statistical law instead of the traditional Newtonian principles mechanics. A discrete method, synthesizing the generalized cell mapping (GCM) together with dynamic programming (DP), with inverse search strategy, using pattern moving theory (PMT) framework, was proposed. The basic idea is to describe and control the system's dynamic peculiarity utilizing pattern class variable in ‘pattern moving space’. First, a few prior concepts were reviewed, including PMT, cell mapping, and optimal control for this kind system. Then, we articulated a cross‐mapping method to analyze the system dynamic behaviour, which takes into account the computational and statistical properties of pattern class variable simultaneously. For system optimal control, the improved GCM and cost function were performed to determine the discrete optimal control table (DOCT) in accordance with dynamic programming and inverse search. Finally, simulation results of two cases demonstrate the efficiency and practicality of the proposed approach. The study has resulted in a solution of describing and controlling based on pattern class variable for non‐Newtonian mechanical systems, and its main objective was to give a different perspective in term of research into non mechanistic principles modelling and application of nonlinear systems.

Quantifying uncertainty in neural network predictions of forced vibrations

AbstractThe prediction of forced vibrations in nonlinear systems is a common task in science and engineering, which can be tackled using various methodologies. A classical approach is based on solving differential (algebraic) equations derived from physical laws ('first principles'). Alternatively, Artificial Neural Networks (ANNs) may be applied, which rely on learning the dynamics of a system from given data. However, a fundamental limitation of ANNs is their lack of transparency, making it difficult to understand and trust the model's predictions. In this contribution, we follow a hybrid modelling approach combining a data‐based prediction using a stabilised Autoregressive Neural Network (s‐ARNN) with a priori knowledge from first principles. Moreover, aleatoric and epistemic uncertainty is quantified by a combination of mean‐variance estimation (MVE) and deep ensembles. Validating this approach for a classical Duffing oscillator suggests that the MVE ensemble is the most accurate and reliable method for prediction accuracy and uncertainty quantification. These findings underscore the significance of understanding uncertainties in deep ANNs and the potential of our method in improving the reliability of predictive nonlinear system modelling. We also demonstrate that including partially known dynamics can further increase accuracy, highlighting the importance of combining ANNs and physical laws.

Physics-informed deep sparse regression network for nonlinear dynamical system identification

This paper presents a novel physics-informed deep sparse regression network for nonlinear dynamical system identification. The fundamental concept is to employ implicit governing equations to guide neural network training, thereby constraining the solution space and inducing an interpretable model. Firstly, inspired by sparse regression methods, a portable sparse regression layer with a function library is developed to characterize system nonlinearity. Secondly, three Hybrid-LSTM networks are connected in parallel with state dependency constraints to construct the Hybrid-LSTM3 network. This configuration enables accurate full-state predictions even from partial measurements. Finally, the sparse regression layer and the Hybrid-LSTM3 network are synthesized to constitute the physics-informed deep sparse regression network, yielding full-state outputs and explicit closed-form dynamical formulations simultaneously. An alternate optimization method is developed to sequentially optimize the two components. The term “physics-informed” herein denotes the incorporation of state dependency constraints and residual loss from learned governing equations via the sparse regression layer. Through this fusion strategy, the proposed framework holds promise to deliver a physically interpretable model for nonlinear dynamical systems from partial noisy measurements. The effectiveness, robustness, and applicability of the proposed method are demonstrated through numerical simulations and experimental studies.

Workpiece temperature control in friction stir welding of Inconel 718 through integrated numerical analysis and process control

Friction stir welding (FSW) offers significant advantages over fusion welding, particularly for high-strength alloys like Inconel 718. However, achieving optimal surface quality in Inconel 718 FSW remains challenging due to its sensitivity to temperature fluctuations during welding. This study integrates finite element simulations, statistical analysis, and advanced control methodologies to enhance weld surface quality through adequate thermal management. High-fidelity simulations of the FSW process were conducted using a validated 3D transient COMSOL Multiphysics model, producing a comprehensive dataset correlating process parameters (rotational speed, axial force, and welding speed) with workpiece temperature. This dataset facilitated statistical analysis and parameter optimization through Analysis of variance (ANOVA) method, leading to a deeper understanding of process variables. A nonlinear state-space system model was subsequently developed using experimental data and the system identification toolbox in Matlab, incorporating domain-specific insights. This model was rigorously validated with an independent dataset to ensure predictive accuracy. Utilizing the validated model, tailored control strategies, including proportional-integral-derivative (PID) and model predictive control (MPC) in both single and multivariable configurations, were designed and evaluated. These control strategies excelled in maintaining welding temperatures within optimal ranges, demonstrating robustness in response times and disturbance handling. This precision in thermal management is poised to significantly refine the FSW process, enhancing both surface integrity and microstructural uniformity. The strategic implementation of these controls is anticipated to substantially improve the quality and consistency of welding outcomes.