Nonlinear System Model Research Articles

Optimal modeling of nonlinear systems Method of variable injections

Our work addresses a development and justification of the new approach to the modeling of nonlinear systems. Let $\f$ be an unknown input-output map of the system with a random input and output $\y$ and $\x$, respectively. It is assumed that $\y$ and $\x$ are available and covariance matrices formed from $\y$ and $\x$ are known. We determine a model of $\f$ so that an associated error is minimized. To this end, the model $\ttt_p$ is constructed as a sum of $p+1$ particular parts, in the form $\ttt_p (\y) = \sum_{j=0}^{p}G_j H_j Q_j(\vv_j)$ where $G_j$ and $ H_j$, for $j=0,\ldots, p$, are matrices to be determined, and $\vv_j$, for $j=1,\ldots,p$, is a special random vector called the injection. We denote $\vv_0=\y$. Injections $\vv_1,\ldots, \vv_p$ are aimed to diminish the error associated with the proposed model $\ttt_p$. Further, $Q_j$ is a special transform aimed to facilitate the numerical realization of model $\ttt_p$. It is determined in the way allowing us to optimally determine $G_j$ and $ H_j$ as a solution of $p+1$ separate error minimization problems which are simpler than the original minimization problem. The empirical determination of injections $\vv_1,\ldots, \vv_p$ is considered. The proposed method has several degrees of freedom to diminish the associated error. They are `degree' $p$ of $\ttt_p$, choice of matrices $G_0, H_0, \ldots,$ $G_p, H_p$, dimensions of matrices $G_0, H_0, \ldots,$ $G_p, H_p$ and injections $\vv_1,\ldots, \vv_p$, respectively. Four numerical examples are provided. At the end, the open problem is formulated.

Adaptive Fuzzy Sliding Mode Control and Dynamic Modeling of Flap Wheel Polishing Force Control System

Polishing force is one of the key process parameters in the polishing process of blisk blades, and its control accuracy will affect the surface quality and processing accuracy of the workpiece. The contact mechanism between the polishing surface and flap wheel was analyzed, and the calculation model of the polishing force and nonlinear dynamic model of the polishing force control system was established. Considering the influence of friction characteristics, parameter perturbation, and nonlinear dead zone on the control accuracy of the polishing force system, an adaptive fuzzy sliding mode controller (AFSMC) was designed. AFSMC uses a fuzzy system to adaptively approximate the nonlinear function terms in the sliding mode control law, adopts an exponential approach law in the switching control part of the sliding mode control (SMC), and designs the adaptive law for adjustable parameters in the fuzzy system based on the Lyapunov Theorem. Simulation and experimental results show that the designed AFSMC has a fast dynamic response, strong anti-interference ability, and high control accuracy, and it can reduce SMC high-frequency chatter. Polishing experiments show that compared with traditional PID, AFSMC can improve the form and position accuracy of the blade by 42% and reduce the surface roughness by 50%.

Comparative study of fractional Newell–Whitehead–Segel equation using optimal auxiliary function method and a novel iterative approach

This research explores the solution of the time-fractional Newell–Whitehead–Segel equation using two separate methods: the optimal auxiliary function method and a new iterative method. The Newell–Whitehead–Segel equation holds significance in modeling nonlinear systems, particularly in delineating stripe patterns within two-dimensional systems. Employing the Caputo fractional derivative operator, we address two case study problems pertaining to this equation through our proposed methods. Comparative analysis between the numerical results obtained from our techniques and an exact solution reveals a strong alignment. Graphs and tables illustrate this alignment, showcasing the effectiveness of our methods. Notably, as the fractional orders vary, the results achieved at different fractional orders are compared, highlighting their convergence toward the exact solution as the fractional order approaches an integer. Demonstrating both interest and simplicity, our proposed methods exhibit high accuracy in resolving diverse nonlinear fractional order partial differential equations.

Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems

Complex mechanical systems often exhibit strongly nonlinear behavior due to the presence of nonlinearities in the energy dissipation mechanisms, material constitutive relationships, or geometric/connectivity mechanics. Numerical modeling of these systems leads to nonlinear full-order models that possess an underlying Lagrangian structure. This work proposes a Lagrangian operator inference method enhanced with structure-preserving machine learning to learn nonlinear reduced-order models (ROMs) of nonlinear mechanical systems. This two-step approach first learns the best-fit linear Lagrangian ROM via Lagrangian operator inference and then presents a structure-preserving machine learning method to learn nonlinearities in the reduced space. The proposed approach can learn a structure-preserving nonlinear ROM purely from data, unlike the existing operator inference approaches that require knowledge about the mathematical form of nonlinear terms. From a machine learning perspective, it accelerates the training of the structure-preserving neural network by providing an informed prior (i.e., the linear Lagrangian ROM structure), and it reduces the computational cost of the network training by operating on the reduced space. The method is first demonstrated on two simulated examples: a conservative nonlinear rod model and a two-dimensional nonlinear membrane with nonlinear internal damping. Finally, the method is demonstrated on an experimental dataset consisting of digital image correlation measurements taken from a lap-joint beam structure from which a predictive model is learned that captures amplitude-dependent frequency and damping characteristics accurately. The numerical results demonstrate that the proposed approach yields generalizable nonlinear ROMs that exhibit bounded energy error, capture the nonlinear characteristics reliably, and provide accurate long-time predictions outside the training data regime.

Implementation of FOPID controller with modified harmony search optimization for precise modelling and auto-tuning of nonlinear systems

This research aims to optimize the fractional order proportional integral derivative (FOPID) controller for nonlinear process tests. Instead of relying on the traditional trial-and error method for dynamic parameter selection, this proposes a modified harmony search optimization method to compute the optimal dynamic parameters. Based on the physical parameters, using mathematical modelling techniques system transfer function is determined. Additionally, the delay of the control scheme is determined using the Open Loop Transfer Function (OLTF) response. The dynamic parameters of the FOPID controller are evaluated using the Harmonic Search Algorithm – Fractional Order Proportional Integral Derivative (HAS_FOPID) optimization technique, which aims to minimize the system's Integral Square Error (ISE). Time responses, including rise time, peak time, and peak overshoot, are obtained and optimized through the HAS_FOPID algorithm. To verify systems stability, the output response is analysed using various techniques such as bode plots, pole placement, and Nyquist plots. Furthermore, the HAS_FOPID optimization technique is compared with other natural optimization techniques to assess its effectiveness. The research also evaluates the system's robustness by supply/ load disturbances. The objective is to demonstrate that, the optimized HAS_FOPID technique, has significantly enhanced control performance, stability, and robustness of nonlinear systems compared to natural and alternative optimization approaches.

Modeling of a Jet Pipe Servovalve Considering Nonlinear Flow Forces Acting on the Spool

The design and analysis of jet pipe servovalves are mostly based on linear models. However, there are some nonlinear factors in this kind of electromechanical–hydraulic structure. The article deduces a linear rotation equation for the armature assembly and a linear flow equation for the control cavity. With the consideration of nonlinear hydraulic reaction forces in the second stage, the nonlinear dynamic equation of the main spool in an ideal jet pipe servovalve is derived. Based on the MATLAB (R2016a) software, the nonlinear model of a certain type of jet pipe servovalve is numerically investigated. The equilibrium points of the nonlinear system are calculated, the phase portraits are plotted, and the Hopf bifurcations caused by the flow-pressure coefficient as the control parameter and the period-doubling bifurcations caused by the variation of the input signal are analyzed. The vibration frequency of the time-domain response of the fifth-order system with a cosine signal as input is 242 Hz, which is similar to the experimental value of 233 Hz. The relative error between the two is 3.9%, verifying the validity of the nonlinear system model.

PD-Based Iterative Learning Control for the Nonlinear Low-Speed-Jitter Vibration of a Wind Turbine in Yaw Motion

Aiming at the nonlinear low-speed-jitter (LSJ) vibration suppression for a yaw system of a megawatt wind turbine, a kinematics mechanism of the yaw system is investigated from the perspective of tribology, and a kinematics model of the yaw system based on an equilibrium position is established. On the basis of the dynamic modeling of the yaw system, a nonlinear mathematical model of the LSJ system is deduced. Based on the two lead motors’ driving of the conventional yaw motion, an innovative design with a special installation of two auxiliary motors for yaw transmission is carried out, which is integrated with a matching centralized lubrication system (CLS). Based on open-loop proportional-derivative (PD) control and the iterative learning control methods of the time-varying continuous system, the stability control and jitter amplitude suppression of the yaw system are realized by using a combined driving torque provided by the lead and auxiliary gears. From the stability and convergence of the time-domain response and the convergence of the iterative error, the effectiveness of the iterative learning control method with the PD-based regulation is verified, and its advantages for engineering applications are shown based on the algorithm solver improvement. The feasibility of the physical realization and engineering application of the control methodology is verified by using controller-hardware-in-the-loop (C-HITL) simulation technology.

A new paradigm for multi-fidelity continuation using parallel model refinement

Numerical path continuation is commonly applied to determine how limit states of dynamical systems evolve with a free parameter. For high-fidelity models of complicated nonlinear systems, the sequential nature inherent to continuation can become an unsurmountable obstacle. In the present work, we propose a new paradigm for obtaining the targeted solution curves in a parallelized way. The point of departure is a low-fidelity model, which is of lower order and/or neglects certain nonlinear/coupling terms. The simplifications must be as drastic as necessary to obtain an approximate solution curve with reasonable effort. A subset of relevant points along this approximate solution curve is then selected, and, departing there, points on the targeted solution branch of the high-fidelity model are then iteratively computed. It is important to note that the iterative correction towards the high-fidelity solution branch can be applied independently for each point, and is thus predestined for parallel computation. The proposed generic concept is exemplified for a selection of nonlinear vibration problems. Different types of system models, nonlinearities and analyses are considered, and the Harmonic Balance method is used in all cases to compute periodic limit states. In particular, it is shown that the proposed concept is applicable to modal and harmonic order refinement. Finally, it is shown that the concept is also interesting when instead of a model refinement a second system parameter (besides the continuation parameter) is varied, and it permits to robustly reach parameter ranges that are extremely difficult to obtain with conventional path continuation. The method is available, along with the presented numerical examples, as a branch PEACE (Parallelized Re-analysis Of Solution Curves) of the open source tool NLvib.

System Power Quality Analysis under Wind–Hydro Complementary Generation Mode

The highly random and characteristics of wind power generation challenge the power quality of the wind–hydro complementary generation system (WHCGS). Herein, the transient characteristics of power quality under the complementary generating mode are studied. First, a nonlinear hydropower system (HPS) model is innovatively established considering the dynamic characteristics of hydrounits under part load operation, and then a wind power system model including wind speed model, wind turbine and its controller, generator and converter is established. Finally, the applicability of WHCGS model in power quality evaluation is established and verified by IEEE‐9 node model. Combined with active power, frequency, and voltage power quality indicators, the effects of wind–hydro capacity ratio and voltage sag on the system are quantified. The results show that the increase in wind power penetration will deteriorate the power quality of WHCGS. Moreover, when the wind power capacity reaches 60 MW, it will damage the voltage and frequency stability of HPS. Besides, the larger the voltage sag amplitude, the worse the power quality of WHCGS. The longer the voltage sag lasts, the higher the risk of WHCGS instability. This article provides a technical reference for the safe and stable operation of WHCGS.

Transient synchronization stability of photovoltaics integration by singular perturbation analysis

The integration of large-scale photovoltaics (PVs) into the power grid has significantly altered the transient synchronization dynamics of traditional power systems dominated by synchronous generators (SGs) and posed great challenges to modeling and analysis of PVs integration. In this paper, the transient synchronization stability of the PV-SG system is studied using the singular perturbation technique. Firstly, a nonlinear model of a PV-SG system is established to reveal the multiscale transient synchronization characteristics. Further, the full system is decomposed into a slow subsystem and a fast subsystem by the singular perturbation technique. The fast subsystem containing the dynamics of the DC voltage control, terminal voltage control, and phase-locked loop, and the slow subsystem containing the dynamics of rotor motion can perfectly reflect the dynamics of the full system within the electromagnetic and electromechanical timescales, respectively. The proposed model provides a clearer physical picture of dynamics in the PV-SG system within the electromagnetic and electromechanical timescales. Subsequently, the stability of the slow and fast subsystems is investigated using the energy function and eigenvalue analysis methods, respectively. Meanwhile, the impacts of various operating, control, and structural parameters on the transient synchronization stability are uncovered. Different from the most existing research endeavors on the wide simulations of the PVs integration, the impact of PVs on the synchronization dynamics of SGs without considering the dynamical characteristics of the PV system, and the transient synchronization stability analyses of the PLL-based voltage source converter systems, it is the key contribution to study the transient synchronization dynamical characteristics of the PV system and its interaction with the SG under different timescales. All these are helpful and easy to extend to more complicated PV-SG systems. Finally, the analysis results are validated by extensive simulations.

Data-Knowledge-Driven Self-Organizing Fuzzy Neural Network.

Fuzzy neural networks (FNNs) hold the advantages of knowledge leveraging and adaptive learning, which have been widely used in nonlinear system modeling. However, it is difficult for FNNs to obtain the appropriate structure in the situation of insufficient data, which limits its generalization performance. To solve this problem, a data-knowledge-driven self-organizing FNN (DK-SOFNN) with a structure compensation strategy and a parameter reinforcement mechanism is proposed in this article. First, a structure compensation strategy is proposed to mine structural information from empirical knowledge to learn the structure of DK-SOFNN. Then, a complete model structure can be acquired by sufficient structural information. Second, a parameter reinforcement mechanism is developed to determine the parameter evolution direction of DK-SOFNN that is most suitable for the current model structure. Then, a robust model can be obtained by the interaction between parameters and dynamic structure. Finally, the proposed DK-SOFNN is theoretically analyzed on the fixed structure case and dynamic structure case. Then, the convergence conditions can be obtained to guide practical applications. The merits of DK-SOFNN are demonstrated by some benchmark problems and industrial applications.

Nonlinear Modeling and Vibration Response Analysis of a Dual-Rotor System with an Inter-Shaft Graphite Seal

The inter-shaft graphite seal is a common structure in the dual-rotor system, and may generate nonlinear lateral excitation when there is large relative angular motion at the seal position. This paper represents the first attempt to study the nonlinear behaviors of a dual-rotor system with inter-shaft graphite seal excitations. First, an excitation model of the inter-shaft graphite seal is proposed by the analytical method. This model is then introduced into the beam finite element model of a dual rotor system, and the dynamic equations of the whole system are finally obtained. The vibration responses under the effect of the inter-shaft seal are clarified by analyzing 3D waterfall plots, rotor orbits, and time and frequency domain waveforms. The results show that the inter-shaft seal can lead to vibration coupling between HP and LP rotors. The axial spring stiffness and contact end face friction coefficient of the graphite seal have a significant effect on rotor vibration. When those two parameters are small, only coupling vibration phenomena can be observed, i.e., the rotation frequency of one rotor can be observed in another rotor vibration. With the increase in axial spring stiffness or contact end face friction coefficient, multiple modes of LP and HP rotors are excited, which dominates the vibration behaviors of the dual rotor system.

Investigation of Hydrodynamics Characteristics of Fluidized Bed with back propagation Artificial Neural Network (BPANN)

Correlations have also been developed with system parameters by using dimensional analysis and an artificial neural network approach. The paper describes an investigation for the thermal design of a fluidized bed cooler and prediction of heat transfer rate among the media categories. It is devoted to the thermal design of such equipment and their application in the industrial fields. In the present work, an extensive ANN by using back propagation (BP) has been carried out to correlate the expansion ratio, fluctuation ratio in gas-solid fluidized bed. Back propagation network is the most well-known and widely used among the current types of neural network system, several applications of ANN for modeling of nonlinear process systems and subsequent control were reported. In back-propagation, different ANN structures (I×H×O) with varying number of neurons in the hidden layer were used as a tool for training input and output data for prediction value of hydrodynamic characteristics of the bed system. It was noted that ring models are the best ones in reducing bed expansion ratio and fluctuation ratio around (25%) and (23.22%).

Bearing fault feature extraction based on MOMEDA and CS-Wood–Saxon stochastic resonance

Since rolling bearing is of great significance to ensure the safe and stable operation of rotating machinery, bearing fault feature extraction then demonstrates a hot topic of general interest in industry. In this work, we applied Multipoint Optimal Minimum Entroy Deconvolution Adjusted preprocessing algorithm to deal with the large amount of background noise containing in the collected bearing fault original signal. Then, the Wood–Saxon stochastic resonance nonlinear system model is adopted to solve the bearing fault feature extraction problem, which avoids the frequency interval and system parameters disadvantages in bistable stochastic resonance system. Furthermore, the parameter step and scale transform factor in the Wood–Saxon stochastic resonance nonlinear system is optimized adaptively by Cuckoo Search algorithm, in which way the output signal-to-noise of bearing fault signal is improved significantly. Therefore, the bearing fault feature can be extracted more effectively compared with the classical bistable stochastic resonance system model. Simulation and examples demonstrated that the proposed method can effectively reduce the noise in the signal and enhance the weak feature in bearing fault signal, so as to realize the accurate early bearing fault diagnosis.

Robust H2 nonlinear fuzzy decentralized control design for a VSC‐HVDC link

SummaryA Robust Nonlinear Fuzzy Decentralized Controller ( RNFDC) is proposed to improve the overall robustness and tracking ability of a VSC‐HVDC link. The studied system is considered as composed by two overlapping nonlinear subsystems. The nonlinear interaction between the two subsystems is treated as a disturbance rejection problem by fuzzy control techniques. First, the Takagi Sugeno (TS) fuzzy model is adopted for fuzzy modeling of the uncertain nonlinear system. Next, new stability conditions for a generalized class of uncertain HVDC systems are derived from robust control techniques based on Linear Matrix Inequalities (LMIs). The design method employs the so‐called Parallel Distributed Compensation (PDC) to obtain RNFDC gains based on LMIs.The efficiency and robustness of the proposed controllers are analytically proven and tested through validation simulations. The main contributions of this paper are: (i) resilience: in case of failure of one converter or loss of measures or controls, the control of the other converter is not affected; (ii) robustness is improved in order to provide good responses in case of network variations and HVDC line parameters changes; (iii) fuzzy techniques are adopted to handle nonlinearities and changes of operating conditions. The proposed RNFDC is compared with Fuzzy H Decentralized State Feedback Control ( NFDC) and Linear Decentralized Control ( LDC) in simulation to illustrate the control synthesis and its effectiveness.

Model predictive control for automatic operation of space nuclear reactors: Design, simulation, and performance evaluation

Space nuclear reactors possess the inherent benefits of independence from solar energy and the capability to withstand intricate external circumstances. Distinguishing themselves from terrestrial reactors, space nuclear reactors function within the realm of outer space, operating autonomously throughout their entire operational cycles. Consequently, the establishment of a dependable automatic control system assumes utmost significance. In this paper, a high-fidelity nonlinear space thermionic nuclear reactor model is established, encompassing neutron physics, thermal–hydraulic, and thermoelectric conversion system models. An automatic control system with a model predictive controller (MPC) is designed based on the system. A simplified state-space model is established based on system identification as the internal model for the MPC controller, demonstrating good agreement with the dynamic response of the nonlinear system model. The deployment of the MPC controller for system control simulations exhibits excellent control performance under various operating conditions. In the case of a nuclear power step response, the MPC controller achieves an overshoot of 0.07 % and a settling time of 0.65 s. For an electric power step response, the MPC controller yields an overshoot of 0.28 % and a settling time of 36.6 s. These control performances are significantly better than those achieved by the PID controller. In power tracking control, the MPC controller demonstrates virtually error-free tracking performance. When external reactivity disturbances are introduced, the MPC controller outperforms the PID controller in swiftly compensating for the disturbances, maintaining the system's operating state at the desired level.

Nonlinear Dynamic Analysis of a Spur Gear Pair System with Wear Considering the Meshing Position

In this paper, a nonlinear dynamic model of a parallel shaft gear system consisting of two involute spur gears is developed to investigate the coupling effect between the gradual surface wear of gear teeth over time and nonlinear dynamic characteristics. A uniform wear model that accounts for how the volumetric wear of the gear teeth affect their meshing position, backlash, and stiffness is proposed. Additionally, a nonlinear dynamic model with six degrees of freedom is described that considers friction, time-varying gear backlash, and time-varying meshing stiffness. The proposed model significantly changes the mesh stiffness, not only in terms of value but also in terms of contact ratio. Furthermore, the nonlinear dynamic characteristics of the gear system vary significantly. It is found that the gradual wear of gear teeth affects the meshing position and further has a significant impact on the nonlinear dynamic characteristics of the spur gear system. This paper provides a basis for studying the nonlinear dynamic characteristics of the spur gear system as it experiences the gradual wear of teeth over time.

Hierarchical multi-objective optimization of proton exchange membrane fuel cell with parameter uncertainty

It is important to improve the performance indexes including the net power and system efficiency for polymer electrolyte membrane fuel cell (PEMFC), especially considering the parameter uncertainty. To this end, this paper proposes a hierarchical multi-objective optimization framework including the off-line and on-line optimization. Firstly, a steadystate non-linear PEMFC system model is established as the base model. Secondly, an adaptive multi-objective particle swarm optimization (AMPSO) algorithm scheme is proposed in the off-line optimization procedure for the exact optimization of operation parameters. Meanwhile, an adaptive flight parameter strategy based on the particle dispersity (PD) information is proposed to balance the convergence and diversity of Pareto solutions. Finally, to decrease the influence caused by parameter uncertainty, the interval optimization method is proposed in the on-line optimization layer based on the results of AMPSO. The convergence condition of the proposed optimization scheme is verified by theory analysis. The proposed AMPSO is compared with different algorithms in the numerical simulation and hardware-in-loop (HIL) experiments. Meanwhile, the performance of the proposed method is tested on four representative benchmark problems. These results demonstrate that the PEMFC system with the proposed optimization scheme performs better than the base model and classical optimization algorithms in terms of the net power and system efficiency indexes, revealing the success of this hierarchical optimization approach in solving the accurate optimization and parameter uncertainty problems. By using statistical test methods, the proposed algorithm performs better hypervolume (HV) metric from the Pareto solution distribution.

An improved WINGS-ISM approach with consensus model in group decision-making for addressing critical success factors of collaborative emergency management

Since the variety and destructiveness of unexpected natural disasters and health emergencies, and the weakness of emergency systems around the world, studies on collaborative emergency management (CEM) are receiving increasingly attention recently. The existing literature has proved that with respect to the constraint of limited resources, it’s usually unrealistic for CEM practitioners to push forward all influential factors that affect the comprehensive performance of CEM simultaneously. Thus, identifying the most critical success factors (CSFs) is of great significance to promote the effectiveness of CEM practices. To achieve this intention, we proposed a novel multi-granularity extended probabilistic linguistic weighted influence nonlinear gauge system and interpretative structural modeling (WINGS-ISM) approach with a consensus model optimized by the convolutional neural network algorithm. The new technique specializes in addressing the CSFs of episodes involving CEM characterized by the combination of uncertainty and complexity in group decision-making process. To validate its feasibility and robustness, we mainly investigate 10 CSFs to conduct an illustrative CEM case, the result shows that ‘Applicable emergency response plan and regulations’, ‘Government unity of leadership and coordination’ and ‘Reasonable organizational structure and clear functions’ are identified out of 10 influential factors. Besides, the cause-effect and hierarchal relationship digraphs indicate that the CSFs of CEM are interconnected, and this finding can support CEM practitioners to devote more attention to the most important factors, which may yield more improvements in CEM practices. Finally, some research implications are provided for better improving the effectiveness of CEM practices based on the findings of this study.

Uncertainty Evaluation of a Gas Turbine Model Based on a Nonlinear Autoregressive Exogenous Model and Monte Carlo Dropout.

Gas turbines are thermoelectric plants with various applications, such as large-scale electricity production, petrochemical industry, and steam generation. In order to optimize the operation of a gas turbine, it is necessary to develop system identification models that allow for the development of studies and analyses to increase the system's reliability. Current strategies for modeling complex and non-linear systems can be based on artificial intelligence techniques, using autoregressive neural networks of the NARX and LSTM type. In this context, this work aims to develop a model of a gas turbine capable of estimating the rotation speed of the turbine and simultaneously estimating the uncertainty associated with the estimation. These methodologies are based on artificial neural networks and the Monte Carlo dropout simulation method. The results were obtained from experimental data from a 215 MW gas turbine, getting the best model with a MAPE of 0.02% and an uncertainty associated with the turbine rotation speed of 2.2 RPM.