Typical Nonlinear System Research Articles

Disturbance Observer-Based Terminal Sliding Mode Tracking Control for a Class of Nonlinear SISO Systems with Input Saturation

This paper focuses on the trajectory tracking control for general nonlinear single-input single-output (SISO) systems in which the output is not directly related to the control input. To address the tracking problem with the consideration of possible model uncertainty, external disturbance, and control input saturation, we employ the input-output feedback linearization technique and design a finite-time disturbance observer-based terminal sliding mode controller to improve the tracking performance and enhance the robustness. The stability analysis is carried out by using the Lyapunov method. To alleviate the chattering while achieving an acceptable control performance, a boundary layer method is adopted for the trade-off between the high-frequency control actions and the bounded unavoidable nonzero steady-state error. The proposed method is evaluated on the two typical nonlinear systems, which are fully linearizable and partially linearizable, respectively, and compared to the state-of-the-art method in terms of tracking and robustness through comprehensive numerical simulations. The results show that the proposed method not only renders the estimated disturbance error tends to be zero in finite time, but also has superiority in the fast reaction to disturbance and small tracking error without high-frequency chattering.

Vibration Drives with Two Impacting Pairs for Precise Robots

Such robots, when they consist of two separate sequentially connected robots, are of high precision. The first part – robot is not of a high precision adjusted to perform comparatively large displacements, while the second part – robot is meant to perform comparatively small displacements with very high precision if compared with the first part. Motors of all the motions are of vibrational type.
 First of all, vibrators of vibrational impact type with impacts of rigid bodies are investigated.
 Earlier it was shown that in some types of nonlinear systems between the hard and soft zones of excitation of vibrations optimal regimes according to the shape of motion exist. Here this fact is generalized for the vibrator with two symmetric impacting pairs of elements. According to the model of the system free vibrations of the system and their parameters are determined as well as forced harmonic vibrations and their specific features are determined. Recommendations for practical applications are provided.

외란관측기를 이용한 비선형 자기부상 시스템의 제어 성능 향상

Magnetic levitation system (MLS) is a typical nonlinear system that controls the position of a steel ball with the magnetic force of the electromagnetic actuator. Since disturbances, due to various external forces and modeling errors, may cause excessive vibration or poor command following, disturbance suppression is necessary to improve the control performance of the MLS. This paper presents a control performance improvement approach of an MLS with a disturbance observer (DOB). First, a mathematical model of the MLS was introduced and validated with the measured frequency response. The MLS steel ball was levitated with a proportional–integral–derivative (PID) controller and a DOB was designed based on the physical model of the MLS. Both disturbance rejection and command tracking performances of the MLS with the DOB were investigated with several design parameters such as PID gains and Q filter. The disturbance rejection and command tracking performances were improved by 76.1% and 64.7%, respectively by using DOB. Finally, the disturbance rejection and command-following performances of the MLS with the DOB were verified experimentally. The effectiveness and limitations of DOB were explained with measured closed-loop frequency responses.

Dependency of transition in thermocapillary convection on volume ratio in annular pools of large-Pr fluid in microgravity

Thermocapillary convection driven by the horizontal temperature gradient is a typical nonlinear dynamical system. As the applied temperature difference increases, the flow undergoes a series of transitions into turbulence. Volume ratio of the liquid layer Vr (liquid volume/volume of the container) is a critical parameter that affects transition behaviors of the convection. Direct numerical simulations on thermocapillary convection in annular liquid pools with various volume ratios are carried out. Characteristics of the oscillatory convection are detected by time series and frequency spectrum analyses. The dynamic mode decomposition (DMD) method is also adopted to investigate inherent structures of the flow field. Effect of Vr on oscillation characteristics under a wide range of temperature differences from 10K to 90K is discussed. Results show that in annular pools with lager volume ratios, the thermocapillary convection is chaotic under very large ΔT and its transition behaviors are more complicated.

A generalized method for the stationary probabilistic response of nonlinear dynamical system

Statistical properties of random response essentially depend on the system and excitation features. The mathematical expressions for the response statistical properties should include the information on the system and excitation. In order to adequately illustrate the dependence of random response on the system and excitation features, only the stationary response is concerned. The exponential-polynomial-closure (EPC) method is further generalized and explicitly includes this information in the response probability density function (PDF) approximation. The unknown coefficients in the EPC approximate solution, which are constants in the previous stationary expression, are generalized to be functions of the system and excitation parameters. With the consideration of the independence among these parameters, each unknown coefficient is finally expressed as the product of several functions of each parameter. Based on the simulated response PDF results, the function of each parameter was determined by the least-squares method. Such a generalization deeply depicts the essentials of the response statistical properties and uniquely determines the response PDF expression. Three typical nonlinear systems are taken as examples to verify the efficiency of the proposed method. Numerical results show that approximate results obtained by the proposed method agree well with the simulated or exact ones. In addition, the evolution of the response PDF to the system or excitation parameters is obtained, which can be utilized for stochastic sensitivity analysis.

A credible and adjustable load resource trading system based on blockchain networks

In recent years, the residents’ demands for power supply is increasing. The load resource trading system responds to the demands through intelligent scheduling, which can effectively relieve the severe power load pressure. The load resource trading system is a type of nonlinear system because the trading price is adjustable with user’s credit, instead of being linear to the power trading volume. The adjustable load resource trading is faced with the challenges of large demands, strong user autonomy, and secure and tamper-proof transaction data. The blockchain technology has been widely concerned by industrial and academic domains due to its decentralization, strong encryption of account information, and traceability of transaction behaviour. In this paper, we propose a credible and adjustable load resource trading framework based on blockchain, which uses blockchain to achieve credible grid load resource trading. Firstly, we propose a two-layer blockchain architecture based on the alliance chain. The main chain maintains all the data of the system, and the station-area nodes constitute the alliance chain. We design a distributed trading processing mechanism based on hybrid consensus and sharding technologies, which promotes the speed of cross-station transaction consensus. Next, we propose a two-level bidding model, which determines the trading price of load resources based on the maximum benefit of users and the lowest cost of grid companies. The results of extensive experiments shows that our proposed framework can achieve the promising result.

An integral sliding‐mode parallel control approach for general nonlinear systems via piecewise affine linear models

AbstractThe fundamental problem of stabilizing a general nonaffine continuous‐time nonlinear system is investigated via piecewise affine linear models (PALMs) in this article. A novel integral sliding‐mode parallel control (ISMPC) approach is developed, where an uncertain piecewise affine system (PWA) is constructed to model a nonaffine continuous‐time nonlinear system equivalently on a compact region containing the origin. A piecewise sliding‐mode parallel controller is designed to globally stabilize the PALM and, consequently, to semiglobally stabilize the original nonlinear system. The proposed scheme enjoys three favorable features: (i) some restrictions on the system input channel are eliminated, thus the developed method is more relaxed compared with the published approaches; (ii) it is convenient to be used to deal with both matched and unmatched uncertainties of the system; and (iii) the proposed piecewise parallel controller generates smooth control signals even around the boundaries between different subspaces, which makes the developed control strategy more implementable and reliable. Moreover, we provide discussions about the universality analysis of the developed control strategy for two kinds of typical nonlinear systems. Simulation results from two numerical examples further demonstrate the performance of the developed control approach.

The generalized EPC method for the non-stationary probabilistic response of nonlinear dynamical system

It is well known that the non-stationary response probability density function (PDF) plays an important role in the reliability and failure analysis. With current approximate or numerical methods, the non-stationary approximate PDF is generally obtained in terms of the ones at the discrete time instants. Repeated computation must be conducted for the ones at other time instants since the response PDF at only one time instant can be obtained after performing the solution procedure. Thus, the computational efficiency suffers a major setback. In this paper, the exponential polynomial closure (EPC) method is further improved and enhanced to obtain the completely non-stationary PDF solution, which is distributed continuously in the time domain. It takes the temporal base function into the PDF approximation. The unknown coefficients in the EPC solution are generalized to be explicit functions of the time parameter. With the least squares method, the explicit time functions can be determined based on the few simulated response PDF values. This implementation makes the continues PDF distribution in the time domain available, which greatly improves the computational efficiency without the repeating computation. Two typical nonlinear systems under stationary and non-stationary random excitations are taken as examples to illustrate the efficiency of the proposed method. Numerical results show that the results obtained by the proposed method agree well with the simulated results. In addition, the relationship between the explicit time function and the modulate function is discussed.

An Alternating Efficient Approach for Determination of the Non-Stationary Responses of Strongly Nonlinear Systems Driven by Random Excitations

Abstract An alternating efficient approach for predicting non-stationary response of randomly excited nonlinear systems is proposed by a combination of radial basis function neural network (RBFNN) and stochastic averaging method (SAM). First, the n-degree-of-freedom quasi-non-integrable-Hamiltonian (QNIH) system is reduced to a one-dimensional averaged Itô differential equation within the framework of SAM for QNIH. Subsequently, the associated Fokker–Planck–Kolmogorov (FPK) equation is solved with the RBFNN. Specifically, the solution of the associated FPK equation is expressed in a linear combination of a series of basis functions with time-correlation weights. These time-depended weights are solved by minimizing a loss function, which involves the residual of the differential equations and the constraint conditions. Three typical nonlinear systems are studied to verify the applicability of the developed scheme. Comparisons to the data generated by simulation technique indicate that the approach yields reliable results with high efficiency.

Stability of connected and automated vehicles platoon considering communications failures

As a form of a future traffic system, a connected and automated vehicle (CAV) platoon is a typical nonlinear physical system. CAVs can communicate with each other and exchange information. However, communication failures can change the platoon system status. To characterize this change, a dynamic topology-based car-following model and its generalized form are proposed in this work. Then, a stability analysis method is explored. Finally, taking the dynamic cooperative intelligent driver model (DC-IDM) for example, a series of numerical simulations is conducted to analyze the platoon stability in different communication topology scenarios. The results show that the communication failures reduce the stability, but information from vehicles that are farther ahead and the use of a larger desired time headway can improve stability. Moreover, the critical ratio of communication failures required to ensure stability for different driving parameters is studied in this work.

Numerical study of a nonlinear fractional chaotic Chua's circuit

<abstract> <p>As an exponentially growing sensitivity to modest perturbations, chaos is pervasive in nature. Chaos is expected to provide a variety of functional purposes in both technological and biological systems. This work applies the time-fractional Caputo and Caputo-Fabrizio fractional derivatives to the Chua type nonlinear chaotic systems. A numerical analysis of the mathematical models is used to compare the chaotic behavior of systems with differential operators of integer order versus systems with fractional differential operators. Even though the chaotic behavior of the classical Chua's circuit has been extensively investigated, our generalization can highlight new aspects of system behavior and the effects of memory on the evolution of the chaotic generalized circuit.</p> </abstract>

Aperiodic resonance of a nonlinear system excited by aperiodic binary signal or <i>M</i>-ary signal

The aperiodic resonance of a typical nonlinear system that excited by a single aperiodic binary or <i>M</i>-ary signal and its measuring method are studied. The focus is on exploring aperiodic resonance caused by the system parameter. A response amplitude gain index suitable for aperiodic excitation is proposed to measure the effect of aperiodic resonance, and the research is carried out by combining the cross-correlation coefficient index and bit error rate index. The results show that the cross-correlation coefficient can better describe the synchronization and waveform similarity between the system output and the input aperiodic signal, but cannot describe the situation whether the signal is amplified after passing through the nonlinear system. The response amplitude gain can better describe the amplification of signal amplitude after passing through the nonlinear system, but cannot reflect the synchronization and waveform similarity between the system output and the input aperiodic signal. The aperiodic resonance occurs at the valley corresponding to the cross-correlation coefficient and the peak corresponding the response amplitude gain. The aperiodic resonance locations reflected on both the cross-correlation coefficient and the response amplitude gain curves are the same. The bit error rate can describe the synchronization between the system output and the input signal at appropriate thresholds, as well as the degree to which the aperiodic signal is amplified after passing through the nonlinear system. The bit error rate curve can directly indicate the resonance region of the aperiodic resonance. The aperiodic resonance can occur in a nonlinear system excited by a single aperiodic binary or <i>M</i>-ary signal, and its aperiodic resonance effect needs to be measured by combining the cross-correlation coefficient, response amplitude gain, bit error rate and other indices together.

A New Control Strategy for Greenhouse Environment Control System Based on Inverse Model

The greenhouse environment control system is a type of non-linear system since the temperature and humidity of the system are highly coupled. Besides, the time lag of the temperature and humidity control process is large, so it’s quite difficult to linearize and decouple the temperature and humidity of the system. To cope with this issue, this paper proposed a novel control strategy for greenhouse environment control system based on Back Propagation Neural Network (BPNN) and inverse model, the proposed method can perform inverse identification on the temperature and humidity control system to attain higher accuracy. Then, the inverse model and the original system were connected in series to form a pseudo linear system to realize the decoupled control of temperature and humidity. After that, aiming at the impact of some non-linear factors on the greenhouse environment system, this paper adopted the adaptive fuzzy Proportion Integration Differentiation (PID) controller to enhance the adaptability of the system, thereby reducing control error and the interference caused by non-linear factors of the temperature and humidity control system. At last, the experimental results showed that, the temperature error of the system could be controlled within 1.2℃ and the error of relative humidity was less than 2.5%. The proposed method can improve the control effect of the greenhouse environment to a certain extent, and it provides a novel approach of greenhouse control.

Attitude Estimation of Four-Rotor UAV Based on Extended Kalman Filter

Quadrotor unmanned aerial vehicle (UAV) is a typical multi-input multi-output (MIMO), nonlinear and strong coupling underactuated system. In the working process of the system, it is necessary to perform information fusion on the attitude detected by the sensor to achieve accurate measurement of attitude angle and angular velocity. Accurate and efficient measurement of UAV attitude angle is the basis of UAV flight control. In this paper, the extended Kalman filter (EKF) algorithm is used to estimate the attitude information of the four-rotor UAV. Firstly, a four-rotor UAV simulation model is established on Simulink in Matlab, and then the attitude information of the UAV is measured and estimated. The results show that the extended Kalman filter algorithm can effectively estimate the attitude information of UAV.

A novel approach of full state tendency measurement for complex systems based on information causality and PageRank: A case study of a hydropower generation system

The hydropower generation system is a typical complex nonlinear system with hybrid state responses. The interaction between the state responses of the system is affected closely by the coupling of hydraulic, mechanical, and electromagnetic factors and the frequent changes in working conditions during operation. However, the precise roles of the interaction relationship are unknown. Here, we show that this interaction depends on the causal coupling between subsystems, and use this relationship to propose a time series data mining and data prediction strategy based on the information causality and the PageRank algorithm. A nonlinear model is used to prove that the proposed prediction strategy can effectively reduce the dimension of auxiliary variables. Finally, the strategy is validated with a 250 MW hydropower unit. Our results show that the information causal coupling between variables is cross-scale with definite Markov orders of time series, and the prediction accuracy can be improved by considering the information transfer sequences between the prediction object variable and the causal variable in the absence of state auxiliary variables. Furthermore, the proposed method can be also applied to data mining of other complex systems and variable selection of prediction models and builds a bridge between mechanism- and data-driven approaches, which has a high engineering application value.

Stability of equilibria of exponential type system of three differential equations under stochastic perturbations

A system of three differential equations with exponential nonlinearity is considered. It is shown that the considered system has both the zero and nonzero (positive or negative) equilibria. It is supposed that the system is exposed to stochastic perturbations that are directly proportional to the deviation of a system state from an equilibrium. Via the general method of Lyapunov functionals construction and the method of linear matrix inequalities sufficient conditions of stability in probability for an each equilibrium are obtained. Numerical simulations and figures are presented to demonstrate the obtained results. The proposed investigation method can be applied for many other types of nonlinear systems.

Complex dynamics of passively Q-switched microchip laser

The laser-diode-pumped 1.5 μm multimode passively Q-switched Er,Yb:YAl3(BO3)4 microchip laser with a saturable absorber of Co2+:MgAl2O4 was studied experimentally. The maxim repetition rate of 81.25 kHz was obtained with output energy of 2.99 μJ and pulse duration of 10.8 ns. The complex dynamics was found with the increase of pumping power: from 3 W to 9.4 W, five regions with complex multimode dynamics and five others with simple dynamics were found, and they appeared alternately. In the regions with multimode dynamics, the clear antiphase phenomenon of pulses with different modes was observed at the beginning, and then the dynamics of laser pulse train tends to be more complex. The phenomenon is very similar to the self-similarity of typical nonlinear dynamical systems.

About Stability of Nonlinear Stochastic Differential Equations with State-Dependent Delay

A nonlinear stage-structured population model with a state-dependent delay under stochastic perturbations is investigated. Delay-independent and delay-dependent conditions of stability in probability for two equilibria of the considered system are obtained via the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). The model under consideration is not the aim of the work and was chosen only to demonstrate the proposed research method, which can be used for the study of other types of nonlinear systems with a state-dependent delay.

Fault Diagnosis of Hydraulic Systems Based on Deep Learning Model With Multirate Data Samples.

Hydraulic systems are a class of typical complex nonlinear systems, which have been widely used in manufacturing, metallurgy, energy, and other industries. Nowadays, the intelligent fault diagnosis problem of hydraulic systems has received increasing attention for it can increase operational safety and reliability, reduce maintenance cost, and improve productivity. However, because of the high nonlinear and strong fault concealment, the fault diagnosis of hydraulic systems is still a challenging task. Besides, the data samples collected from the hydraulic system are always in different sampling rates, and the coupling relationship between the components brings difficulties to accurate data acquisition. To solve the above issues, a deep learning model with multirate data samples is proposed in this article, which can extract features from the multirate sampling data automatically without expertise, thus it is more suitable in the industrial situation. Experiment results demonstrate that the proposed method achieves high diagnostic and fault pattern recognition accuracy even when the imbalance degree of sample data is as large as 1:100. Moreover, the proposed method can increase about 10% diagnosis accuracy when compared with some state-of-the-art methods.

Time series difference approach for evaluating sensitivity of nonlinear dynamic systems

This research is aimed to establish a novel approach for assessing sensitivities of nonlinear systems to initial conditions and system parameters via an evaluation of Time Series Difference. An evaluation method is proposed for measuring the differences of two trajectories representing the solutions of nonlinear systems, in responding to different initial conditions and/or system parameters. Recurrence relations are established for numerically evaluating the time series differences. Various nonlinear responses are evaluated with the approach proposed. A typical nonlinear dynamic system the Duffing system are considered for demonstrating the application of the approach in numerically and graphically assessing the sensitivities. The approach shown effectiveness in the assessment and can a useful tool for scientists and engineers in evaluating the initial-condition and system-parameter dependent sensitivities.