OPTIMAL INTERIOR PARTIAL REGULARITY FOR NONLINEAR ELLIPTIC SYSTEMS UNDER THE NATURAL GROWTH CONDITION:THE METHOD OF A-HARMONIC APPROXIMATION

In this article, the authors consider the nonlinear elliptic systems under the natural growth condition. They use a new method introduced by Duzaar and Grotowski, for proving partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation. And directly establish the optimal Holder exponent for the derivative of a weak solution.

On the Existence of Harmonic Solutions for Higher-Order Lienard Systems

In this paper, by using of the theory of coincidence degree ,we obtain the new conditions which guarantee the existence of harmonic solutions for Lienard Systems our resuls do not require that the damping must be positire.

New trajectory linearization control for nonlinear systems undergoing harmonic disturbance

This paper presents a new trajectory linearization control scheme for a class of nonlinear systems subject to harmonic disturbance. It is supposed that the frequency of the disturbance is known, but the amplitude and the phase are unknown. A disturbance observer dynamics is constructed to estimate the harmonic disturbance, and then the estimation is used to implement a compensation control law to cancel the disturbance. By Lyapunov＇s direct method, a rigorous poof shows that the composite error of the closed-loop system can approach zero exponentially. Finally, the proposed method is illustrated by the application to control of an inverted pendulum. Compared with two existing methods, the proposed method demonstrates better performance in tracking error and response time.

Harmonic Analysis in Discrete Dynamical Systems

In this paper we review several contributions made in the field of discrete dynamical systems, inspired by harmonic analysis. Within discrete dynamical systems, we focus exclusively on quadratic maps, both one-dimensional (1D) and two-dimensional (2D), since these maps are the most widely used by experimental scientists. We first review the applications in 1D quadratic maps, in particular the harmonics and antiharmonics introduced by Metropolis, Stein and Stein (MSS). The MSS harmonics of a periodic orbit calculate the symbolic sequences of the period doubling cascade of the orbit. Based on MSS harmonics, Pastor, Romera and Montoya (PRM) introduced the PRM harmonics, which allow to calculate the structure of a 1D quadratic map. Likewise, we review the applications in 2D quadratic maps. In this case both MSS harmonics and PRM harmonics deal with external arguments instead of with symbolic sequences. Finally, we review pseudoharmonics and pseudoantiharmonics, which enable new interesting applications.

Harmonic Propagation from the Low Voltage Four-Wire Delta Systems

The electric networks for the distribution to low voltage costumers can be configured in different layouts. Two main approaches are used: the European system composed by three-phase distribution transformers or the North American system composed by single-phase distribution transformers and three-phase transformer banks of single-phase transformers. With respect to harmonic analysis, much more attention has been focused on the three-phase balanced systems arrangements than on the unbalanced four-wire delta system extensively used to supply low voltage loads of 120/240 V. Different authors have shown the three-phase power systems modeling on a phase-coordinates frame. However, the presence of significant asymmetries in the network forces the need of adding a new phase-coordinates model to represent the three-phase transformers banks of two or three single-phase transformers in its various connections. Several papers treat the use of harmonic analysis programs based on a phase-coordinates frame to study the Wye or Delta connected three-phase systems. However, the commonly used four-wire delta connected systems are not fully treated in literature. This paper presents a phase-coordinates model for the representation of the commonly used three-phase transformer banks of three or two single-phase transformers, and single-phase distribution transformers for the harmonic analysis of the four-wire delta connected systems. The harmonic analysis method based on the presented model is used to examine the characteristics of this kind of distribution system with respect to the penetration of harmonics currents from loads to the primary system.

Analysing chaos in fractional-order systems with the harmonic balance method

In this paper, the fractional-order Genesio-Tesi system showing chaotic behaviours is introduced, and the corresponding one in an integer-order form is studied intensively. Based on the harmonic balance principle, which is widely used in the frequency analysis of nonlinear control systems, a theoretical approach is used to investigate the conditions of system parameters under which this fractional-order system can give rise to a chaotic attractor. Finally, the numerical simulation is used to verify the validity of the theoretical results.

The Jaffa Transform for Hessian Matrix Systems and the Laplace Equation

Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.

Stochastic resonance in an under-damped bistable system driven by harmonic mixing signal

Stochastic resonance（SR） is studied in an under-damped bistable system driven by the harmonic mixing signal and Gaussian white noise. Using the linear response theory（LRT）, the expressions of the spectral amplification at fundamental and higher-order harmonic are obtained. The effects of damping coefficient, noise intensity, signal amplitude, and frequency on spectral amplifications are explored. Meanwhile, the power spectral density（PSD） and signal-to-noise ratio（SNR） are calculated to quantify SR and verify the theoretical results. The SNRs at the first and second harmonics exhibit a minimum first and a maximum later with increasing noise intensity. That is, both of the noise-induced suppression and resonance can be observed by choosing proper system parameters. Especially, when the ratio of the second harmonic amplitude to the fundamental one takes a large value, the SNR at the fundamental harmonic is a monotonic function of noise intensity and the SR phenomenon disappears.

The Generalized Pythagorean Comma Harmonic Powers of a Fundamental Frequency Are Equivalent the Standing Wave Harmonic Fraction System

Purpose: The Pythagorean Comma refers to an ancient Greek musical, mathematical tuning method that defines an integer ratio of exponential coupling constant harmonic law of two frequencies and a virtual frequency. A Comma represents a physical harmonic system that is readily observable and can be mathematically simulated. The virtual harmonic is essential and indirectly measurable. The Pythagorean Comma relates to two discrete frequencies but can be generalized to any including infinite harmonics of a fundamental frequency, vF. These power laws encode the physical and mathematical properties of their coupling constant ratio, natural resonance, the maximal resonance of the powers of the frequencies, wave interference, and the beat. The hypothesis is that the Pythagorean power fractions of a fundamental frequency, vF are structured by the same harmonic fraction system seen with standing waves. Methods: The Pythagorean Comma refers to the ratio of (3/2)12 and 27 that is nearly equal to 1. A Comma is related to the physical setting of the maximum resonance of the powers of two frequencies. The powers and the virtual frequency are derived simulating the physical environment utilizing the Buckingham Π theorem, array analysis, and dimensional analysis. The powers and the virtual frequency can be generalized to any two frequencies. The maximum resonance occurs when their dimensionless ratio closest to 1 and the virtual harmonic closest to 1 Hz. The Pythagorean possible power arrays for a vF system or any two different frequencies are evaluated. Results: The generalized Pythagorean harmonic power law for any two different frequencies coupling constant are derived with a form of an infinite number of powers defining a constant power ratio and a single virtual harmonic frequency. This power system has periodic and fractal properties. The Pythagorean power law also encodes the ratio of logs of the frequencies. These must equal or nearly equal the power ratio. When all of the harmonics are powers of a vF the Pythagorean powers are defined by a consecutive integer series structured in the identical form as standard harmonic fractions. The ratio of the powers is rational, and all of the virtual harmonics are 1 Hz. Conclusion: The Pythagorean Comma power law method can be generalized. This is a new isomorphic wave perspective that encompasses all harmonic systems, but with an infinite number of possible powers. It is important since there is new information: powers, power ratio, and a virtual frequency. The Pythagorean relationships are different, yet an isomorphic perspective where the powers demonstrate harmonic patterns. The coupling constants of a vF Pythagorean power law system are related to the vFs raised to the harmonic fraction series which accounts for the parallel organization to the standing wave system. This new perspective accurately defines an alternate valid physical harmonic system.

Impulse Controller Design of a Harmonic Drive System with Friction

A friction model was established for impulse control design in a precision control system. First, the physical characteristics of the impulse in momentum, such as motion and energy, were analyzed and formulated. Then, experimental response to a new pulse with two harmonic expansions was studied. The first harmonic is the main pulse to drive the arm, and the second harmonic has two functions： its first half helps the main pulse eliminate the dead zone, and its second half, a negative pulse, stops the arm motion quickly. Finally, an impulse feedback controller was developed. Comparison between simulation and experiments shows the effectiveness of the proposed controller.

DC System Modeling and AC/DC System Harmonic Calculation under Asymmetric Faults

The accurate DC system model is the key to fault analysis and harmonic calculation of AC/DC system. In this paper, a frequency domain analysis model of DC system is established, and based on it a unified fundamental frequency and harmonic iterative calculation method is proposed. The DC system model is derived considering the dynamic switching characteristic of converter and the steady-state response features of dc control system synchronously. And the proposed harmonic calculation method fully considers the AC/DC harmonic interaction and fault interaction under AC asymmetric fault condition. The method is used to the harmonic analysis and calculation of CIGRE HVDC system. Compared with those obtained by simulation using PSCAD/EMTDC software, the results show that the proposed model and method are accurate and effective, and provides the analysis basis of harmonic suppression, filter configuration and protection analysis in AC/DC system.

New Possibilities of Harmonic Oscillator Basis Application for Quantum System Description.Two Particles with Coulomb Interaction

This paper is addressed to the problem of Galilei invariant basis construction for identical fermions systems. The recently introduced method for spurious state elimination from expansions in harmonic oscillator basis [1] 08D0C9EA79F9BACE118C8200AA004BA90B02000000080000000E0000005F005200650066003400310034003400340037003600360038000000 is adopted and applied to bound states of two particles system with Coulomb potential description. Traditional expansions in this case demonstrate the extremely well-known slow convergence, and hence this is the best problem with known exact solutions for the test of the method. Obtained results demonstrate the significant simplification of the problem and fast convergence of expansions. We show that the application of this general method is very efficient in a test case of the energy spectrum calculation problem of two particles with different masses interacting with Coulomb potential.

Study and Design of a DC/AC Energy Converter for PV System Connected to the Grid Using Harmonic Selected Eliminated (HSE) Approach

Nowadays, distributing network-connected photovoltaic (PV) systems are expanded by merging a PV system and a Direct Current (DC)/Alternating Current (AC) energy converter. DC/AC conversion of PV energy is in great demand for AC applications. The supply of electrical machines and transfer energy to the distribution network is a typical case. In this work, we study and design a DC/AC energy converter using harmonic selective eliminated (HSE) method. To this end, we have combined two power stages connected in derivation. Each power stage is constituted of transistors and transformers. The connection by switching of the two rectangular waves, delivered by each of the stages, makes it possible to create a quasi-sinusoidal output voltage of the inverter. Mathematical equations based on the current-voltage characteristics of the inverter have been developed. The simulation model was validated using experimental data from a 25.2 kWp grid-coupled (PV) system, connected to Gridfit type inverters. The data were exported and implemented in programming software. A good agreement was observed and this shows all the robustness and the technical performances of the energy converter device. It emerges from this analysis that the inverter output voltage and the phase angle thus simulated are very important to control in order to orientate the transfer of the power flow from the continuous cell to cell to the alternating part. Simulated and field-testing results also show that increases in the value of the modulation factor (m) for low power output are highly significant. This study is an important tool for DC/AC inverter designers during initial planning stages. A short presentation of the design model of the inverter has been proposed in this article.

Detection and Elimination Harmonic in Power System

This paper shows the harm of harmonic in power system,compares the measures of normal digital filter and wavelet MARto afford reference to the detection and elimination in power system harmonic control.

Influence Caused by Harmonic and Filtering Methods of Synchronous Generator in Short-capacity System

Relative to the power grid, the short-capacity system has smaller inertia and weaker ability to bear disturbance. As a result, the synchronous generator in short-capacity system will be greatly influenced by harmonic. To reveal how harmonic influence the generator, this article analyzed how harmonic current will influence the output voltage. Deduced a formula that can describe the electromagnetic torque pulsation brought by the theory of Instantaneous Power, which can explain why generator’s shaft vibrates. Then this article evaluated the applicability of current filtering methods in view of characteristics of the small capacity of the system. As a result, it was demonstrated that active filtering method is best suited for small capacity system. At last, it conducted the experiment that diesel generator set supply power to non-liner load to demonstrate the conclusion of theoretical analysis.

Cross-spectral root-min-norm algorithm for harmonics analysis in electric power system

To avoid drawbacks of classic discrete Fourier transform(DFT)method,modern spectral estimation theory was introduced into harmonics and inter-harmonics analysis in electric power system.Idea of the subspace-based root-min-norm algorithm was described,but it is susceptive to noises with unstable performance in different SNRs.So the modified root-min-norm algorithm based on cross-spectral estimation was proposed,utilizing cross-correlation matrix and independence of different Gaussian noise series.Lots of simulation experiments were carried out to test performance of the algorithm in different conditions,and its statistical characteristics was presented.Simulation results show that the modified algorithm can efficiently suppress influence of the noises,and has high frequency resolution,high precision and high stability,and it is much superior to the classic DFT method.

PARTIAL REGULARITY FOR WEAK SOLUTIONS OF STATIONARY NAVIER-STOKES SYSTEMS

This article is concerned with the partial regularity for the weak solutions of stationary Navier-Stokes system under the controllable growth condition.By A-harmonic approximation technique,the optimal regularity is obtained.

Methodology Approach Using Harmonic Synthesis as Simulation Method for Investigation of Power Network with Non-Linear Loads

The paper deals with analysis and synthesis of non-harmonic and non-linear sources and appliances, and their interaction with harmonic power supply network. Basic idea is based on knowledge of harmonic spectrum of the sources and/or appliances, respectively. Obviously, one need to know voltage harmonic components of voltage sources （renewable with inverters,...）, and current harmonic components generated by non-linear appliances （rectifiers,...）. Method of investigation lies on decomposition of real electric circuit into n-harmonic separated equivalent schemes for each harmonic component. Then transient analysis will be done for each scheme separately using ＂impedance harmonic matrices＂. The important fact is that each equivalent scheme is now linearized and therefore easily calculated. Finally, the effects of each investigated schemes arc summed into resulting quantities of real non-linear electric circuit.

OPTIMAL INTERIOR PARTIAL REGULARITY FOR NONLINEAR ELLIPTIC SYSTEMS WITH DINI CONTINUOUS COEFFICIENTS

In this article, we consider nonlinear elliptic systems of divergence type with Dini continuous coefficients. The authors use a new method introduced by Duzaar and Grotowski, to prove partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation and directly establish the optimal HSlder exponent for the derivative of a weak solution on its regular set.

PARTIAL REGULARITY OF STATIONARY NAVIER-STOKES SYSTEMS UNDER NATURAL GROWTH CONDITION

In this article, we consider the partial regularity of stationary Navier-Stokes system under the natural growth condition. Applying the method of A-harmonic approximation,we obtain some results about the partial regularity and establish the optimal Holder exponent for the derivative of a weak solution on its regular set.