A New Way to Compute the Lyapunov Characteristic Exponents for Non-Smooth and Discontinues Dynamical Systems

One of the most important problems of nonlinear dynamics is related to the development of methods concerning the identification of the dynamical modes of the corresponding systems. The classical method is related to the calculation of the Lyapunov characteristic exponents ( LCEs ). Usually, to implement the classical algorithms for the LCEs calculation the smoothness of the right-hand sides of the corresponding equations is required. In this work, we propose a new algorithm for the LCEs computation in systems with strong nonlinearities (these nonlinearities can not be linearized ) including the hysteresis. This algorithm uses the values of the Jacobi matrix in the vicinity of singularities of the right-hand sides of the corresponding equations. The proposed modification of the algorithm is also can be used for systems containing such design hysteresis nonlinearity as the Preisach operator (thus, this modification can be used for all members of the hysteresis family). Moreover, the proposed algorithm can be successfully applied to the well-known chaotic systems with smooth nonlinearities . Examples of dynamical systems with hysteresis nonlinearities , such as the Lorentz system with hysteresis friction and the van der Pol oscillator with hysteresis block, are considered. These examples illustrate the efficiency of the proposed algorithm.

Experimental Research and Numerical Simulation of Single Soil-Arc-Grounding-Fault in Distribution Networks

Among the distribution network faults, single-phase grounding faults have the greatest probability. The faults are often accompanied by arcs in the grounding point soil. This type of fault current has a small amplitude and seldom can obtain field record data. A soil arc grounding fault is tested on a realistic-distribution-network-experimental-platform (RDNEP), and it is concluded that the soil-arc-grounding-fault (SAGF) has three main characteristics: hysteresis, nonlinearity, and asymmetry. By comparing with the characteristics of common arc models, it is pointed out that common arc models cannot accurately fit the characteristics of SAGF. This paper proposes and establishes a double exponential function arc model. Through the comparison of simulation waveforms with experimental data, it is verified that the numerical simulation method proposed in this paper can simulate the development process of SAGF more accurately. Furthermore, the equivalence of RDNEP is verified on the real distribution network system (RDNS). On this basis, analyzed the arc characteristic changes of different SAGF development cycles. Finally, by studying the applicability of the proposed model in simulating ground faults in grass and gravel roads, it is verified that the model proposed in this paper has a strong generalization capability. The research has laid a theoretical foundation for a detection algorithm that is based on the characteristics of SAGF.

Modeling and Compensation of Dynamic Hysteresis with Force-Voltage Coupling for Piezoelectric Actuators

Piezoelectric actuators are widely used in the field of micro- and nanopositioning due to their high frequency response, high stiffness, and high resolution. However, piezoelectric actuators have hysteresis nonlinearity, which severely affects their positioning accuracy. As the driving frequency increases, the performance of piezoelectric actuators further degrades. In addition, the impact of force on piezoelectric actuators cannot be ignored in practical applications. Dynamic hysteresis with force-voltage coupling makes the hysteresis phenomenon more complicated when force and driving voltage are both applied to the piezoelectric actuator. Existing hysteresis models are complicated, or inaccurate in describing dynamic hysteresis with force-voltage coupling. To solve this problem, a force-voltage-coupled Prandtl–Ishlinskii (FVPI) model is proposed in this paper. First, the influence of driving frequency and dynamic force on the output displacement of the piezoelectric actuators are analyzed. Then, the accuracy of the FVPI model is verified through experiments. Finally, a force integrated direct inverse (F-DI) compensator based on the FVPI model is designed. The experimental results from this study show that the F-DI compensator can effectively suppress dynamic hysteresis with force-voltage coupling of piezoelectric actuators. This model can improve the positioning accuracy of piezoelectric actuators, thereby improving the working accuracy of the micro- or nano-operating system.

Construction of a Nonlinear Model of Tourism Economy Forecast Based on Wireless Sensor Network from the Perspective of Digital Economy

With the outbreak of the new crown epidemic, the world economy has been severely tested, making predictions more difficult. Wireless sensors have the advantages of low cost, ease of use, high reliability, and high safety and have been widely used in the tourism economy. In order to understand the ability of wireless sensors to predict the regional economy, this article uses an example to construct a nonlinear model of wireless sensors to predict the regional economy. With the continuous development of the concept of circular economy, circular economy has gradually been recognized by Chinese scholars and practitioners. After domestic scholars continue to study the theory of circular economy, practicing the concept of circular economy and taking the road of sustainable development have become one of the important directions of the development of industrial theory. Literature analysis and other methods were used to conduct research on databases such as CNKI, Wan fang Database, and SSCI. Literature was collected, and GIS spatial analysis technology was used to analyze different areas and finally get a prediction model. The phenomenon is nonlinearity (such as saturation nonlinearity in the magnetic circuit), and some are caused by the nonlinear relationship between system variables (such as linear resistance and squared nonlinearity between current and power) and some artificially introduced nonlinear links (such as the hysteresis nonlinearity of relays). Experiments have proved that there is a certain error between the prediction model and the actual result; the error value is about 9%, which is less than the value of other prediction models. This shows that the output results of the nonlinear model of wireless sensor regional economic prediction should be processed reasonably. This result has a certain reference value, and its output should be combined with the actual situation. Related research found that under the nonlinear model, the more accurate and comprehensive the input value is, the closer the output result is to the actual value.

Hysteresis modelling and compensation for piezoelectric actuator using Jaya-BP neural network

This paper proposes a new training algorithm using a hybrid Jaya-back propagation algorithm (called H-Jaya) to optimize the neural network weights, which is applied to identify the nonlinear hysteresis Piezoelectric actuator based on the experimental input-output data. The identified H-Jaya-neural model will be used to design an advanced feed-forward (FF) controller for compensating the hysteresis nonlinearity. Furthermore as to improve the tracking performance, a feed-forward-feedback control scheme is conducted. To evaluate the effectiveness of the proposed approach, firstly, it is tested through identifying the nonlinear hysteresis of Piezoelectric (PZT) actuator and compared with other meta-heuristic techniques, including differential evolution (DE), particle swarm optimization (PSO), and Jaya. Then, the accuracy of the hysteresis model-based compensator is evaluated under various control experiments using the piezoelectric actuator. The results of experiments executed on PZT actuator configured with a PZS001 from Thorlabs prove that the proposed approach obtains an excellent performance in hysteresis modeling and compensation.

Hysteresis Modeling and Compensation of Fast Steering Mirrors with Hysteresis Operator Based Back Propagation Neural Networks

Fast steering mirrors (FSMs), driven by piezoelectric ceramics, are usually used as actuators for high-precision beam control. A FSM generally contains four ceramics that are distributed in a crisscross pattern. The cooperative movement of the two ceramics along one radial direction generates the deflection of the FSM in the same orientation. Unlike the hysteresis nonlinearity of a single piezoelectric ceramic, which is symmetric or asymmetric, the FSM exhibits complex hysteresis characteristics. In this paper, a systematic way of modeling the hysteresis nonlinearity of FSMs is proposed using a Madelung’s rules based symmetric hysteresis operator with a cascaded neural network. The hysteresis operator provides a basic hysteresis motion for the FSM. The neural network modifies the basic hysteresis motion to accurately describe the hysteresis nonlinearity of FSMs. The wiping-out and congruency properties of the proposed method are also analyzed. Moreover, the inverse hysteresis model is constructed to reduce the hysteresis nonlinearity of FSMs. The effectiveness of the presented model is validated by experimental results.

Sliding Mode Active Disturbance Rejection Control for Magnetorheological Impact Buffer System

In the magnetorheological (MR) impact buffer system, the internal or external disturbance of the MR damper is one of the main factors that affect the buffer performance of the system. This study aims to suppress or eliminate the influence of the disturbance of the MR damper. The continuous terminal sliding mode control (CTSMC) strategy with a high gain has a strong antidisturbance ability. However, the high gain may cause fluctuation of the damping force of the system. Therefore, a composite control strategy of sliding mode active disturbance rejection control (ADRC) based on an extended state observer (ESO) is proposed in this study. The total disturbance of the system is estimated by the ESO in real time, and the estimated disturbance is used as a feedforward compensation to the controller to reduce the influence of disturbance on the system. The gain of the CTSMC law of the closed-loop system can be reduced. In addition, the Lyapunov stability criterion is used to ensure the stability of the proposed controller. In order to verify the performance of the proposed CTSMC controller on response speed, overshoot, and hysteresis suppression ability, the window function, square wave function, and multistep function are given as the inputs of the control system. To verify the performance of the proposed sliding mode ADRC for the MR impact buffer system, the mechanical model and the control model are established and simulated using MATLAB/Simulink. The simulation results show that the CTSMC controller has the fastest response time and no overshoot and can suppress the hysteresis nonlinearity of the MR device compared with the open-loop control, PID control, and fractional order PID control. The MR impact buffer system with the sliding mode ADRC obtained the minimum peak value of 4350N within the permitted buffer displacement range compared with the other three traditional control methods. That means the proposed control method in this study has the advantage on buffer performance for the MR impact buffer system.