Typical Nonlinear System Research Articles

Suppression Limit Cycles in 2x2 Nonlinear Systems with Memory Type Nonlinearities

For several decades, the importance and weight-age of prediction of nonlinear self-sustained oscillations or Limit Cycles (LC) and their quenching by signal stabilization have been discussed, which is confined to Single Input and Single Output (SISO) systems. However, for the last five to six decades, the analysis of 2x2 Multi Input and Multi Output (MIMO) Nonlinear Systems gained importance in which a lot of literature available. In recent days’ people have started discussing suppression of LC which limits the performance of most of the physical systems in the world. It is a formidable task to suppress the limit cycles for 2x2 systems with memory type nonlinearity in particular. Backlash is one of the nonlinearities commonly occurring in physical systems that limit the performance of speed and position control in robotics, automation industry and other occasions like Load Frequency Control (LFC) in multi area power systems. The feasibility of suppression of such nonlinear self-oscillations has been explored in case of the memory type nonlinearities. Backlash is a common memory type nonlinearity which is an inherent Characteristic of a Governor, used for usual load frequency control of an inter-connected power system and elsewhere. Suppression LC using pole placement technique through arbitrary selection and optimal selection of feedback Gain Matrix K with complete state controllability condition and Riccati Equation respectively and is done through state feedback. The Governing equation is d/dt [X(t)] =(A-BK) X: which facilitates the determination of feedback gain matrix K for close loop Poles / Eigen values placement where the limit cycles are suppressed/eliminated in the general multi variable systems. The complexity involved in implicit non-memory type or memory type nonlinearities, it is extremely difficult to formulate the problem for 2x2 systems. Under this circumstance, the harmonic linearization/harmonic balance reduces the complexity considerably. Still the analytical expressions are so complex which loses the insight into the problem particularly for memory type nonlinearity in 2x2 system and the method is made further simpler assuming a 2x2 system exhibits the LC predominately at a single frequency. Hence in the proposed work an alternative attempt has been made to develop a graphical method for the prediction of Limit Cycling Oscillations in 2x2 memory type Nonlinear systems which not only reduces the complexity of formulations but also facilitates clear insight into the problem and its solution. The present techniques are well illustrated with an example and validated / substantiated by digital simulation (developed program using MATLAB codes) and use of SIMULINK Tool Box of MATLAB software. The present work has the brighter future scope of: Adapting the Techniques like Signal Stabilization and Suppression LC for 3x3 or higher dimensional nonlinear systems through an exhaustive analysis. Analytical/Mathematical methods may also be developed for signal stabilization using both deterministic and random signals based on Dual Input Describing function (DIDF) and Random Input Describing Function (RIDF) respectively. The phenomena of Synchronization and De-synchronization can be observed/identified analytically using Incremental Input Describing Function (IDF), which can also be validated by digital simulations.

Two-Step Fifth-Order Efficient Jacobian-Free Iterative Method for Solving Nonlinear Systems

This article introduces a novel two-step fifth-order Jacobian-free iterative method aimed at efficiently solving systems of nonlinear equations. The method leverages the benefits of Jacobian-free approaches, utilizing divided differences to circumvent the computationally intensive calculation of Jacobian matrices. This adaptation significantly reduces computational overhead and simplifies the implementation process while maintaining high convergence rates. We demonstrate that this method achieves fifth-order convergence under specific parameter settings, with broad applicability across various types of nonlinear systems. The effectiveness of the proposed method is validated through a series of numerical experiments that confirm its superior performance in terms of accuracy and computational efficiency compared to existing methods.

A novel study of analytical solutions of some important nonlinear fractional differential equations in fluid dynamics

The space-time fractional Burger-like equation and the space-time coupled Boussinesq equation with conformable derivative, both of which are significant in fluid dynamics, are investigated in this work using the improved [Formula: see text] method. The process works effectively and produces soliton solutions. The method was successfully and consistently implemented with Maple, a symbolic computing tool. The solutions also contain a few of graphics. Numerous novel exact solutions to these equations, distinct from those found earlier with the proposed approach, have been provided. The study’s findings add to the body of knowledge by offering insightful justifications for various types of nonlinear systems. The results demonstrated the value of the proposed method as a mathematical tool and the ease, dependability, and speed increases that result from carrying out these tasks using a symbolic computing program. Notably, it applies to many nonlinear evolution problems in mathematical physics.

From breather soliton molecules to chaos in a laser cavity: the scenario of intermittent transitions.

Intermittency is widely observed in various nonlinear dynamical systems as an intriguing transient dynamic far from equilibrium. The internal dynamics formed by a pair of interacting optical solitons are often analogized to typical nonlinear systems. However, whether intermittency exists within the intramolecular motion remains to be investigated. Here, we study the intermittent dynamics of soliton molecules in ultrafast lasers, employing balanced optical cross-correlation techniques with sub-femtosecond temporal resolution. We demonstrate the occurrence of the bursting phase of intense variations of pulse separation within regular breather rhythms. In addition, we discover the intermittent transitions route to chaotic soliton molecules, facilitated by gain control. A series of analysis methods are used to assess the chaotic signals, providing compelling experimental evidence that soliton molecules can be analogized to their matter molecule counterparts. Our experimental findings shed light on the non-equilibrium intramolecular dynamics, providing insight into the transition of the attractors within interacting dissipative solitons in laser and fiber resonators.

A novel LSTM-integrated non-intrusive ROM for reliability analysis of hysteretic systems with large stochastic dimension

Failure probability estimation of hysteretic systems subjected to random process excitation appears in many practical engineering applications, such as structures subjected to earthquake and vibration control. This task is computationally challenging due to the need for repeated solutions of a high-resolution model. This cost grows significantly with the stochastic dimension — characterized by the number of random variables, in this case, arising from the discretization of the random excitation. To address this issue, a novel non-intrusive reduced order model (ROM) is proposed in this paper. Accordingly, the complexity due to spatial discretization is first reduced using proper orthogonal decomposition. However, interpolation among response time histories in the resulting reduced solution space is computationally infeasible due to large stochastic dimensionality. An existing ROM developed by the authors is now improved by adopting a long short-term memory (LSTM) network for this interpolation. Furthermore, this adoption required two additional modifications in the mathematical structure of the ROM related to data compression and nonlinear coupling. The proposed methodology can be seamlessly integrated with black-box nonlinear solvers. Through detailed numerical studies on a beam on Winkler foundation and a multi-storied building frame subjected to stationary and non-stationary excitations, the proposed ROM is found to be accurate, efficient, and robust with respect to the frequency band of interest. The ROM has shown reduction of two orders of magnitude in the computational time. Furthermore, the proposed ROM is inherently parallelizable and holds promise to be scaled to other types of nonlinear systems with large stochastic dimensionality.

Reliability control of nonlinear stochastic dynamical system based on discrete data

Stochastic optimal control based on reliability plays a crucial role in mitigating structural failure and ensuring safe operation. For general stochastic vibration systems, a data-driven method for reliability-based optimal control is proposed based on random state data. Firstly, the feedback control is split into conservative and dissipative components in coherence with physical intuition, and each component is expanded upon using pre-selected basic functions, resulting in a modified system with undetermined coefficients. Then, from discrete random samples, the expressions for the probability densities of the first-passage time and the stationary response can be identified, explicitly including system and excitation parameters, as well as initial and safety boundary conditions. Finally, a performance index, combining system reliability and control cost, is constructed and can be explicitly expressed in terms of the undetermined coefficients of the control forces. The optimal control problem for determining the optimal control forces is reformulated as an optimization problem for determining the coefficients that minimize the performance index. This unconstrained optimization problem can be easily solved. Three typical nonlinear stochastic systems, a controlled Duffing oscillator, a controlled nonlinear hysteretic system, and a controlled 2-DOF nonlinear dynamical system, are given to illustrate the validity and accuracy of the proposed data-driven method. Additionally, the control performance of the optimal control strategy in improving the system reliability is also discussed.

Nonlinear vibration characteristics of an aeroengine cantilever flexible rotor considering interference fit and unbalance

Aeroengine power turbine rotor is a typical nonlinear rotor system, in which the interference connection between long shaft and short shaft has a prominent influence on the coupling vibration of the system, especially at high speed, the nonlinear stiffness change of interference contact is more obvious. Therefore, the paper considers the change of the contact characteristics of the long shaft and the short shaft of the turbine rotor, and the nonlinear restoring force of the bearing to analyze the dynamic characteristics of the rotor. Firstly, according to the theory of elasticity, the contact parameters under different speed states are calculated, the contact stiffness model under interference fit is established, and the relationship between contact stiffness and speed is analyzed. Secondly, according to the Hertz contact theory, a bearing nonlinear restoring force model considering time-varying characteristics is established. Finally, the dynamic model of the bearing-flexible rotor coupling considering interference contact is established. The influence of contact parameters and unbalance parameters on the nonlinear vibration characteristics of the rotor is analyzed by amplitude-frequency curve, bifurcation diagram, time-domain waveform, spectrum diagram, trajectory diagram and Poincaré diagram. In addition, the simulation results are compared with the experiment results of the rotor test rig to verify the accuracy of the established dynamic model. By analyzing the influence of different parameters such as interference magnitude, unbalance position, unbalance magnitude and unbalance couple on the nonlinear vibration of the rotor, the basis for the design and vibration control of the cantilever turbine rotor is provided.

Adaptive integral sliding-mode finite-time control with integrated extended state observer for uncertain nonlinear systems

For the typical uncertain nonlinear systems subjected to controller gain variation and disturbances, an integrated adaptive robust controller, called integral sliding-mode finite-time controller with an integrated extended state observer (ISFC-IESO), is proposed. The designed controller mainly consists of an integrated ESO (IESO) unit and an integral sliding-mode control (ISMC) unit. The IESO unit is composed of improved linear and nonlinear ESOs which are designed to automatically switch each other, with the aim of enhancing disturbance estimation capability. Moreover, the ISMC unit produces an integral sliding-mode control law (SMCL) based on the newly designed integral sliding-mode surface function (SMSF) and sliding-mode reaching law (SMRL). The integral SMSF is developed based on the n-th estimated state and a designed convergence function of tracking error to suppress the influence of the disturbance estimation error. On this basis, SMRL is further presented by combining integral SMSF with a specially designed variable boundary layer thickness saturation function, for the purpose of eliminating chattering and enhancing robustness. Finally, fourteen control methods are compared to illustrate the effectiveness and superiority of the proposed ISFC-IESO.

Enhancing grid-connected inverter performance under non-ideal grid conditions: a multi-functional multiplexing control strategy

Compared to the traditional thermal power generation, new energy sources, such as wind and photovoltaic systems, are more vulnerable to the effects of non-ideal power grids due to their limited capacity. This susceptibility can jeopardize the safe operation of power equipment, degrade power output quality, and lead to non-compliance with grid-connected specifications. The LCL-type grid-connected inverter is a typical nonlinear system that weakens the controllability of the grid-connected energy. To address these challenges, this study employs feedback linearization theory to transform the inverter into a standard linear system. Subsequently, it utilizes linear system methodologies to develop robust control laws, ultimately introducing a multi-functional multiplexing control strategy for grid-connected inverters based on feedback linearization and Hamilton-Jacobi-Issacs inequality. Simulation results demonstrate that this multi-functional strategy outperforms traditional grid-connected inverter control schemes, effectively mitigating issues related to low short-circuit ratios, voltage fluctuations, imbalances, harmonics, and other non-ideal grid conditions. Furthermore, it significantly expands the system’s adaptability to varying weak grid impedances.

Adaptive iterative learning control for high‐order nonlinear systems with different types of uncertainties

AbstractThe present work aims at investigating the adaptive iterative learning control (AILC) design for uncertain high‐order fully actuated nonlinear systems. In order to show the design principles, three types of nonlinear systems are considered, including systems with just parametric uncertainty, systems with both parametric and input distribution uncertainties as well as systems with both parametric uncertainty and an unknown control gain matrix. For different systems, the corresponding AILC schemes are proposed with different techniques in dealing with various system uncertainties, for which the convergence analysis are conducted rigorously based on the composite energy function. The effectiveness of the proposed AILC strategies is verified through numerical simulations.

On the Newton-Broyden method for solving systems of nonlinear equations

The Newton-Broyden method for solving systems of nonlinear equations with operator decomposition is studied in this paper. A local convergence analysis is provided under classical Lipschitz conditions and results of numerical experiments are presented. The effectiveness of this method in solving various types of nonlinear systems is demonstrated.

Adaptive integral alternating minimization method for robust learning of nonlinear dynamical systems from highly corrupted data

This paper proposes an adaptive integral alternating minimization method (AIAMM) for learning nonlinear dynamical systems using highly corrupted measured data. This approach selects and identifies the system directly from noisy data using the integral model, encompassing unknown sparse coefficients, initial values, and outlier noisy data within the learning problem. It is defined as a sparse robust linear regression problem. An adaptive threshold parameter selection method is proposed to constrain model fitting errors and select appropriate threshold parameters for sparsity. The robustness and accuracy of the proposed AIAMM are demonstrated through several numerical experiments on typical nonlinear dynamical systems, including the van der Pol oscillator, Mathieu oscillator, Lorenz system, and 5D self-exciting homopolar disc dynamo. The proposed method is also compared to several advanced methods for sparse recovery, with the results indicating that the AIAMM demonstrates superior performance in processing highly corrupted data.

Quasi-zero stiffness interval optimization design and dynamics analysis of a new bi-directional horizontal isolation system

The quasi-Zero Stiffness (QZS) isolation system is a typical nonlinear isolation system, which has high static stiffness characteristics to improve the bearing capacity and deformation resistance, and its low dynamic stiffness characteristics can reduce the inherent frequency of the system to realize a wider vibration isolation frequency band, so it has a more excellent low-frequency vibration isolation performance compared with the linear system. In this paper, a horizontal QZS isolator is constructed based on a pre-compressed oblique spring system, and two sets of the horizontal QZS isolators are mechanically assembled to work together to achieve a bi-directional horizontal QZS isolation system. Considering that the construction of two sets of horizontal vibration isolators is approximately the same, we focus on one of the isolators for the static analysis and derive the parameter conditions to be satisfied for achieving quasi-zero stiffness characteristics. We considered the effect of each adjustable parameter of the horizontal vibration isolator on the mechanical performance of the QZS system, and sought to optimize the QZS interval to improve the vibration isolation performance. A comparative study of the dynamical properties between the QZS system with ordinary and optimized parameters, as well as the equivalent linear system, is further carried out by means of nonlinear approximate analysis. Finally, the dynamic characteristics of the bi-directional horizontal QZS system and the equivalent linear system are compared by shaker tests on the prototype device, which proved the excellent vibration isolation performance of the proposed QZS isolation system.

Reservoir computing decoupling memory–nonlinearity trade-off

Reservoir computing (RC), a variant recurrent neural network, has very compact architecture and ability to efficiently reconstruct nonlinear dynamics by combining both memory capacity and nonlinear transformations. However, in the standard RC framework, there is a trade-off between memory capacity and nonlinear mapping, which limits its ability to handle complex tasks with long-term dependencies. To overcome this limitation, this paper proposes a new RC framework called neural delayed reservoir computing (ND-RC) with a chain structure reservoir that can decouple the memory capacity and nonlinearity, allowing for independent tuning of them, respectively. The proposed ND-RC model offers a promising solution to the memory–nonlinearity trade-off problem in RC and provides a more flexible and effective approach for modeling complex nonlinear systems with long-term dependencies. The proposed ND-RC framework is validated with typical benchmark nonlinear systems and is particularly successful in reconstructing and predicting the Mackey–Glass system with high time delays. The memory–nonlinearity decoupling ability is further confirmed by several standard tests.

Robust control for a tracked mobile robot based on a finite-time convergence zeroing neural network.

Since tracked mobile robot is a typical non-linear system, it has been a challenge to achieve the trajectory tracking of tracked mobile robots. A zeroing neural network is employed to control a tracked mobile robot to track the desired trajectory. A new fractional exponential activation function is designed in this study, and the implicit derivative dynamic model of the tracked mobile robot is presented, termed finite-time convergence zeroing neural network. The proposed model is analyzed based on the Lyapunov stability theory, and the upper bound of the convergence time is given. In addition, the robustness of the finite-time convergence zeroing neural network model is investigated under different error disturbances. Numerical experiments of tracking an eight-shaped trajectory are conducted successfully, validating the proposed model for the trajectory tracking problem of tracked mobile robots. Comparative results validate the effectiveness and superiority of the proposed model for the kinematical resolution of tracked mobile robots even in a disturbance environment.

Improved SSA‐RBF neural network‐based dynamic 3‐D trajectory tracking model predictive control of autonomous underwater vehicles with external disturbances

AbstractThis paper studies the three‐dimensional (3‐D) dynamic trajectory tracking control of an autonomous underwater vehicle (AUV). As AUV is a typical nonlinear system, each degree of freedom is strongly coupled, so the traditional control method based on the nominal model of AUV cannot guarantee the accuracy of the control system. To solve this problem, we first propose a prediction model based on a radial basis function neural network (RBF‐NN). The nonlinearity of AUV is learned and modeled offline by RBF‐NN based on previous data. This model can reflect the time sequence state and control variables of AUV. Secondly, to avoid the overfitting problem in network training based on the traditional gradient descent method, a new adaptive chaotic sparrow search algorithm (ACSSA) is proposed to optimize the network parameters, to improve the full approximation ability of RBF‐NN to nonlinear systems. To eliminate the steady‐state error caused by external interference during AUV trajectory tracking, a nonlinear optimizer is designed by updating the deviation of the NN model output layer. In each sampling period, the predictive control law is calculated online according to the deviation between the predicted value and the actual value. In addition, the stability analysis based on the Lyapunov method proves the asymptotic stability of the controller. Finally, the 3‐D dynamic trajectory tracking the performance of AUV under different external disturbances is verified by MATLAB/Simulink, and the results show that the proposed controller is more efficient and robust than the standard model predictive controller (MPC) controller and the standard NN model predictive controller (NNPC).

Asymptotic output tracking in a class of non‐minimum phase nonlinear systems via learning‐based inversion

AbstractAsymptotic output tracking of non‐minimum phase (NMP) nonlinear systems has been a popular topic in control theory and applications. Many approaches have focused on finding solutions under minimal assumptions either in the target system or desired trajectories, as there is no general solution available. In this article, we propose a practical and simple solution for cases where the reference trajectory is periodic in time. Our approach employs a learning‐based scheme to iteratively determine the desired feedforward input. Unlike previous learning‐based frameworks, our method only requires the output tracking error to update the feedforward input iteratively and can be applicable to NMP systems. Our method retains the key advantages of the learning‐based framework, including robustness to parameter uncertainties and periodic disturbances. We evaluate the effectiveness of our algorithm using simulation results with an inverted pendulum on a cart, a typical NMP nonlinear system.

Characteristics of a Low-Pressure Carbon-Dioxide Surface Dielectric Barrier Discharge Plasma Ion-Wind Thruster

This study investigates the discharge and dynamic characteristics of surface dielectric barrier discharge ion-wind thrusters made of polymethyl methacrylate, quartz, and polyvinylidene difluoride on a Martian surface filled with carbon 4 dioxide. Surface dielectric barrier discharge is a typical dissipative nonlinear spatiotemporal dynamical system. Space-charge accumulation and electron backflow are the main causes of gas multipulse asymmetric discharge. The heat generated by high-velocity particle bombardment and polarization loss in the filamentary discharge channel results in the partial breakdown of polymethyl methacrylate and polyvinylidene difluoride, reducing the electrical capacity of both dielectrics. The space-charge and shielding effects of the discharge plasma region are the main internal mechanisms influencing the thrust and thrust-to-power ratio. The maximum surface flow velocities of the three ion-wind thrusters are 22.1, 23.2, and , respectively. The maximum thrust reaches approximately 84.48, 97.48, and , respectively.The maximum thrust-to-power ratio is 15.18, 17.03, and , respectively. The change in the thrust and thrust-to-power ratio spans four orders of magnitude and can be adjusted in a broad range.

An efficient multilayer adaptive self-organizing modeling methodology for improving the generalization ability of organic Rankine cycle (ORC) data-driven model

The efficient and accurate model construction of an organic Rankine cycle (ORC) system is the key to its analysis, prediction, and optimization. As a typical multidisturbance nonlinear dynamic system, the ORC system always operates in a nonstationary state. Under uncertain disturbance, accurately identifying the thermal power conversion process state and quickly realizing the accurate association of various mappings are the key considerations of constructing the data-driven model of the ORC system. From the perspective of data selection, parameter association, and structural design, this study proposes a methodology for the efficient multilayer adaptive self-organizing modeling of ORC systems. This methodology can realize efficient autonomous modeling in the whole design process of the data-driven model of the ORC system. Moreover, the proposed methodology can minimize the structural risk of the model by balancing the empirical risk and structural complexity. By taking real data as the test base, the generalization ability and time cost of the data self-selection layer, information self-correlation layer and adaptive self-organizing part of the structure layer are evaluated. Compared with the direct construction of the ORC system data model, the proposed methodology can reduce the model construction time cost by 75.54% and improve the generalization ability by 61.88%. In addition, maximizing the generalization capability with minimum structural risk is an important part of data-driven model construction of ORC systems. In this study, a data-driven model structural reliability assessment approach for ORC systems is proposed. Then, the proposed adaptive self-organizing optimization methodology is verified on the basis of the structural reliability assessment model. The multilayer adaptive self-organizing modeling methodology proposed in this study can provide new ideas and necessary theoretical guidance.

Water nanolayer facilitated solitary-wave-like blisters in MoS2 thin films

Solitary waves are unique in nonlinear systems, but their formation and propagation in the nonlinear fluid-structure interactions have yet to be further explored. As a typical nonlinear system, the buckling of solid thin films is fundamentally related to the film-substrate interface that is further vulnerable to environments, especially when fluids exist. In this work, we report an anomalous, solitary-wave-like blister (SWLB) mode of MoS2 thin films in a humid environment. Unlike the most common telephone-cord and web buckling deformation, the SWLB propagates forward like solitary waves that usually appear in fluids and exhibits three-dimensional expansions of the profiles during propagation. In situ mechanical, optical, and topology measurements verify the existence of an interfacial water nanolayer, which facilitates a delamination of films at the front side of the SWLB and a readhesion at the tail side owing to the water nanolayer-induced fluid-structure interaction. Furthermore, the expansion morphologies and process of the SWLB are predicted by our theoretical model based on the energy change of buckle propagation. Our work not only demonstrates the emerging SWLB mode in a solid material but also sheds light on the significance of interfacial water nanolayers to structural deformation and functional applications of thin films.