Heun Functions Describing Fermions Evolving in Parallel Electric and Magnetic Fields

The present work deals with the behavior of fermions moving in a static magnetic induction and a time-harmonic electric field, both oriented along Oz. For the ultra-relativistic particles described by a Heun double confluent equation, we derive the corresponding wave functions and the conserved current density components.

Analytically determining frequency and amplitude of spontaneous alpha oscillation in Jansen's neural mass model using the describing function method

Spontaneous alpha oscillations are a ubiquitous phenomenon in the brain and play a key role in neural information processing and various cognitive functions.Jansen＇s neural mass model（NMM） was initially proposed to study the origin of alpha oscillations.Most of previous studies of the spontaneous alpha oscillations in the NMM were conducted using numerical methods.In this study,we aim to propose an analytical approach using the describing function method to elucidate the spontaneous alpha oscillation mechanism in the NMM.First,the sigmoid nonlinear function in the NMM is approximated by its describing function,allowing us to reformulate the NMM and derive its standard form composed of one nonlinear part and one linear part.Second,by conducting a theoretical analysis,we can assess whether or not the spontaneous alpha oscillation would occur in the NMM and,furthermore,accurately determine its amplitude and frequency.The results reveal analytically that the interaction between linearity and nonlinearity of the NMM plays a key role in generating the spontaneous alpha oscillations.Furthermore,strong nonlinearity and large linear strength are required to generate the spontaneous alpha oscillations.

Multi-range controller design for highly nonlinear systems with application to unmanned aerial vehicles

A new procedure for a design of multi-range controllers for use with highly nonlinear systems is developed.The procedure involves obtaining the describing function models of the nonlinear plant by software followed by designing a controller at nominal conditions.Then,the controller parameters are optimized to yield a satisfactory closed-loop response at all operating regimes.Finally,the performance and stability of the closed-loop system comprised of the designed controller and the nonlinear plant are verified.The procedure and the associated software are applied to a nonlinear control problem of the sort encountered in aerospace,and the results are compared with two other approaches.

Design of Robust Controller for Flexible Joint Robot with Nonlinear Friction Characteristics

A robust controller method for flexible joint robot considering the effect caused by nonlinear friction was presented.The nonlinear friction was denoted as inverse additive output uncertainty relative to the nominal model in our work,based on which the describing function was analyzed in frequency domain,and the weighting function of nonlinear friction was further calculated as well. By combining the friction uncertainty,the mixed sensitivity H∞optimization was proposed as the benchmark for controller design, which also leaded to good performance of robustness. Furthermore,unstructured perturbation to the system was analyzed so that the stability was guaranteed. Simulation results show that the proposed controller can provide excellent tracking and regulation performance.

Exact Dynamic Modeling of PWM DC-to-DC Power Converters—Part I: Continuous Conduction Mode

A general approach is presented by which the exact frequency response of any transfer function of switched linear networks can be determined. This is achieved with a describing function approach using a state space equation formulation. This work presents a somewhat simplified set of equations to one previously given by one of the authors. To demonstrate application of the general formulation, the frequency responses of switched networks used as PWM DC-to-DC converters operating in continuous conduction mode (CCM) under voltage mode control are derived. (The accompanying paper, Part II, will present results for converters operating in discontinuous conduction mode (DCM)). From the general sets of equations developed here, both the control to output and input source variation to output frequency responses are derived. The describing function approach enables exact frequency response determination, even at high frequencies where the accuracy using average models may be compromised. Confirmation of the accuracy of the derived models is provided by comparing the responses with those obtained using the commercial simulator PSIM on a PWM boost converter. The magnitude and phase responses are shown to match perfectly over the full range of frequencies up to close to half the switching frequency. Matlab code that implements the models is given such that the user can easily adapt for use with other PWM converter topologies.

Exact Dynamic Modeling of PWM DC-to-DC Power Converters—Part II: Discontinuous Conduction Mode

This paper follows on from the first paper, Part I, where a general formulation of a describing function approach to frequency response determination of switched linear networks, such as PWM converters, was simplified and updated. The models assume a piecewise linear state space equation description of the system and results in a closed form solution for the sought after frequency response. In Part I, model derivation was demonstrated for the case of PWM converters operating in the continuous conduction mode (CCM). This operating mode does not feature any state dependent switching times. In this paper, Part II, frequency response models for any transfer function for PWM converters operating in discontinuous conduction mode (DCM) are derived based on the theory presented in Part I. This operating model features state dependent switching times. The describing function models developed are exact and therefore, in terms of accuracy, are to be preferred over averaged models which are widely used. The example of a boost dc-to-dc converter operating in DCM is simulated to obtain the control to output and input to output frequency responses and are compared with the models derived here. Excellent agreement between the simulated and model responses was found. Matlab code implementing the analytical models is also presented which the user can adapt for any other PWM converter topology. The models derived here may be used as a basis from which simplified models may be derived while still preserving required accuracy.

Low-Frequency Stability Analysis of Power Electronic Traction Transformer Based Train-Network System

Power electronic traction transformers(PETTs)will be increasingly applied to locomotives in the future for their small volume and light weight.However,similar to conventional trains,PETTs behave as constant power loads and may cause low-frequency oscillation(LFO)to the train-network system.To solve this issue,a mathematical model of the PETT is firstly proposed and verified based on the extended describing function(EDF)method in this paper.In the proposed model,the LLC converter is simplified to an equivalent circuit consisting of a capacitor and a resistor in parallel.It is further demonstrated that the model can apply to various LLC converters with different topologies and controls.Particularly,when the parameter differences between cells are not obvious,the PETT can be simplified to a single-phase rectifier(i.e.,conventional train)by equivalent transformation.Based on the model of PETT,the system low-frequency stability and influential factors are analyzed by using the generalized Nyquist criterion.Lastly,the correctness and accuracy of theoretical analyses are validated by off-line and hardware-in-the-loop simulation results.

On limit cycle chattering in sliding mode control systems under the influence of sampling intervals based on describing function approach

In this paper, the influence of sampling intervals on the chattering in sliding mode （SM） control systems is considered. The describing function （DF） approach is employed to analyze the chattering characteristics in the sampling SM control. By the DF calculations and limit cycle existence conditions, an unstable limit cycle and two stable limit cycles are found in the SM control system. The frequencies and amplitudes of the two limit cycles can also be estimated by graphical calculations. The estimation accuracy of chattering parameters is evaluated by the simulations. The results of simulations show that the system could converge to a large and a small limit cycle from different initial conditions.

Similarity phenomena of lean swirling flames at different bulk velocities with acoustic disturbances

In this study,flame responses to acoustic disturbances with different frequencies and amplitudes were experimentally investigated in a lean premixed swirl-stabilized combustor operating at different bulk velocities.The total heat release rate fluctuations and spatial CH*chemiluminescence distributions were captured using a photomultiplier tube and high-speed camera,respectively.The results indicate that the heat release rate exhibits a relatively drastic oscillation and high-order harmonics for low-frequency disturbances.When the bulk velocity and forcing frequency were doubled simultaneously,similar flame structures were observed in the CH*chemiluminescence distributions.As the bulk velocity increases,the gain of the Flame Describing Function(FDF)extends toward the higher frequencies,and the delay time of the flame response decreases.The similarity among FDFs at different bulk velocities was effectively captured by introducing a non-dimensional parameter,defined as the ratio of the flame response delay to the forcing time scale,to replace the dimensional forcing frequency.Furthermore,the availability of the newly defined non-dimensional parameter was verified for flames with different swirl numbers,as this played an important role in determining the flame structures and associated unsteady heat release rate.

Structural Nonlinear Flutter Characteristics Analysis for an Actuator-fin System with Dynamic Stiffness

The flutter characteristics of an actuator-fin system are investigated with structural nonlinearity and dynamic stiffness of the electric motor. The component mode substitution method is used to establish the nonlinear governing equations in time domain and frequency domain based on the fundamental dynamic equations of the electric motor and decelerator. The existing describing function method and a proposed iterative method are used to obtain the flutter characteristics containing preload freeplay nonlinearity when the control command is zero. A comparison between the results of frequency domain and those of time domain is studied. Simulations are carried out when the control command is not zero and further analysis is conducted when the freeplay angle is changed. The results show that structural nonlinearity and dynamic stiffness have a significant influence on the flutter characteristics. Limit cycle oscillations （LCOs） are observed within linear flutter boundary. The response of the actuator-fin system is related to the initial disturbance. In the nonlinear condition, the amplitude of the control command has an influence on the flutter characteristics.

Third harmonic distortion calculation of a self-oscillating power amplifier

It is difficult to analyze the harmonic distortion of a self-oscillating power amplifier (SOPA),because the SOPA is a hard nonlinear system without an external clock.The single or multiple sinusoidal inputs describing function (DF) method is commonly used to linearize a nonlinear element,but this method considers only the components at the same frequencies as the input signals (i.e.,fundamental components) at the nonlinear element's output.In this paper,besides the fundamental components,the third harmonic components are also calculated at the output of a comparator with three sinusoidal inputs,to create a linearized model of the comparator,and thus of the SOPA.The third harmonic distortion of the SOPA is calculated.The models of the zeroth and the first order SOPA are verified by behavioral simulation using MATLAB.

Multi-harmonic equivalent modeling for a planar repetitive structure with polynomial-nonlinear joints

How nonlinear joints affect the response of large space structures is an important problem to investigate.In this paper,a multi-harmonic equivalent modeling method is presented to establish a frequency-domain model of planar repetitive structures with nonlinear joints.First,at the local level,the nonlinear joint is modeled by the multi-harmonic describing function matrix.The element of the hybrid beam is obtained by the dynamic condensation of the beam-joint element.Second,at the global level,the displacement-equivalence method is used to model the multi-harmonic Euler continuum beam equivalent to the planar repetitive structure.Then,the pseudo-arc-length continuation method is applied to track the multi-harmonic trajectory of response.Afterwards,an experiment is conducted to validate the correctness of the modeling method,considering the effect of hanging rope and air damping.In the numerical studies,several simulation results indicate the similarity of response between a single-degree-of-freedom system with a single nonlinear joint and the system of the planar repetitive structure with a large number of nonlinear joints.Finally,the component of higher-order harmonics is shown to be important for predicting the resonance frequencies and amplitudes.

Dynamic model identification of a real-time simple level control system

Using a relay feedback approach,mathematical expressions for identification of real-time level control system in transfer function form are proposed in this paper.An ideal relay with hysteresis is feedback to a simple level control system for the generation of sustained oscillations.Subsequently,limit cycle information is utilised in deduced expressions for the estimation of unknown system parameters.Using describing function analysis,the gain of an ideal relay with hysteresis is approximated to yield simple mathematical expressions.Typical examples from literature are considered to show the effectiveness of the proposed identification scheme.Yokogawa distributed control system:CENTUM CS 3000 is considered as a platform for the realisation of both relay feedback and Ziegler-Nichols(ZN)method.The accuracy of derived transfer function models is compared with a model derived from ZN method and thereafter demonstrated through frequency response plots which illustrate the experimental validation of the identified real-time process model.

Review of General Modeling Approaches of Power Converters

The modeling approaches of power converters occupy an important position in power electronic systems and have made considerable progress over the past years.Continuous modeling approaches and linearization techniques are reviewed,including the state-space average model,generalized average model,averaged small-signal model,and describing function method.A Buck converter with PWM modulation and voltage-mode control is taken as an example to compare the advantages and disadvantages of different methods through simulation analysis.Moreover,the corresponding equivalent circuit with an intuitive physical meaning of state-space average model,generalized average model,and averaged small-signal model is given.The results point out that the generalized average model can improve the modeling accuracy based on the state-space average model.In the linearization techniques,the averaged small-signal model reflects accuracy at low frequencies,but introduces phase lag in the high-frequency region.The describing function method is derived from harmonic linearization,which takes into account the sideband effect and improves the modeling accuracy at high frequencies.