Feedback Control of Limit Cycle Amplitudes from A Frequency Domain Approach

In this paper, a problem in feedback control of limit cycle amplitudes is studied. A graphical approach for this bifurcation control problem is developed by means of higher-order harmonic balance approximations for both amplitude and frequency of the system oscillatory outputs. The approach starts with the familiar Hopf bifurcation mechanism, and employs the second, fourth, and sixth-order harmonic balance approximations to generate a sequence of graphical tests for convergence analysis of the system oscillatory outputs. This sequential graphical testing leads to accurate approximations of limit cycles of small amplitudes in the system outputs. Degenerate Hopf bifurcation theory is used to formulate an appropriate control objective of capturing small-amplitude limit cycles, which can avoid reaching unstable equilibria or other undesirable limit sets. A rich cubic planar model is presented for illustration of the proposed control method.     

Approximate expressions of a fractional order Van der Pol oscillator by the residue harmonic balance method

Although the Van der Pol oscillator, which was originally proposed as a model of vacuum tube circuits, has been widely used in electronics, biology and acoustics, its characteristics in fractional order formulations are not clearly explained even now. This paper is interested in gaining insights of approximate expressions of the periodic solutions in a fractional order Van der Pol oscillator. The presence of fractional derivatives requires the use of suitable criteria, which usually makes the analytical work much hard. Most existing methods for studying the nonlinear dynamics fail when applied to such a class of fractional order systems. In this paper, based on the residue harmonic balance method, a detailed analysis on approximations to the periodic oscillations of the fractional order Van der Pol equation is investigated. The relations that express the frequency and amplitude of the generated oscillations as functions of the orders and parameters are shown. Moreover, some examples are provided for comparing approximations with numerical solutions of the periodic oscillations. Numerical results reveal that the residue harmonic balance method is very effective for obtaining approximate solutions of fractional oscillations.     

Nonlinear identification of systems with parametric excitation

In this paper, the incremental harmonic balance nonlinear identification (IHBNID) is presented for modelling and parametric identification of nonlinear systems. The effects of harmonic balance nonlinear identification (HBNID) and IHBNID are also studied and compared by using numerical simulation. The effectiveness of the IHBNID is verified through the Mathieu-Duffing equation as an example. With the aid of the new method, the derivation procedure of the incremental harmonic balance method is simplified. The...     

Generalized Fourier series for non-linear systems: application to the study of limit cycles in second-order approximation

The approximate solution of non-linear differential equations is studied to second-order using the method of harmonic balance with generalized Fourier series and Jacobian elliptic functions. As an interesting use of the series, very good analytic approximations to the limit cycles of Liénard's ordinary differential equation, [Xdot] + g(X) = f(X)[Xdot] are presented. In the generalized van der Pol equation with f(X)= ε(1 ? X²) and g(X) = AX + 2BX₃ a very good second-order approximation is given that depends on the values of A/B and ε.     

Nonlinear circuit analysis using the method of harmonic balance—A review of the art. Part I. Introductory concepts

The harmonic balance method is a technique for the numerical solution of nonlinear analog circuits operating in a periodic, or quasi-periodic, steady-state regime. The method can be used to efficiently derive the continuous-wave response of numerous nonlinear microwave components including amplifiers, mixers, and oscillators. Its efficiency derives from imposing a predetermined steady-state form for the circuit response onto the nonlinear equations representing the network, and solving for the set of unknown coefficients in the response equation. Its attractiveness for nonlinear microwave applications results from its speed and ability to simply represent the dispersive, distributed elements that are common at high frequencies. The last decade has seen the development and application of harmonic balance techniques to model analog circuits, particularly microwave circuits. The first part of this paper reviews the fundamental achievements made during this time. The second part covers the extension of the method to quasi-periodic regimes, optimization analysis, and practical application. A critical assessment of the various types of harmonic balance techniques is given. The different sampling and Fourier transform methods are compared, and numerical speed and precision results are given enabling a quantitative analysis of the merits of the major variants of the harmonic balance technique. Examples of designs which have been modeled using the harmonic balance technique and built both in hybrid and MMIC form are presented.     

A harmonic balance approach to robust neural control of MIMO nonlinear processes applied to a distillation column

Robust stability and performance are the two most basic features of feedback control process. The harmonic balance analysis based on the describing function technique enables to analyze the stability of limit cycles arising from a closed loop control process operating over nonlinear plants. In this work a robust stability analysis based on the harmonic balance is presented and applied to a neural network controller in series with a dynamic multivariable nonlinear plant under generic Lur’e configuration. The neural controller is replaced by its sinusoidal input describing function while a linearized model is derived to represent the nonlinear plant dynamics. The uncertainty induced by the high harmonics effect for the neural controller, together with the neglected nonlinear dynamics due to plant linearization are incorporated in the robustness analysis as structured norm bounded uncertainties. Stability and robustness conditions for the neural closed loop control system are discussed using the harmonic balance equation together with the structured singular values of the uncertainty. The application to a multivariable binary distillation column under feedback neurocontrol illustrates the usefulness of the robustness approach here developed to predict the absence of limit cycles, which of course is subject to the usual restrictions of the describing function method.     

Nonlinear circuit analysis using the method of harmonic balance—a review of the art. II. Advanced concepts

The harmonic balance method is a technique for the numerical solution of nonlinear analog circuits operating in a periodic, or quasi-periodic, steady-state regime. The method can be used to efficiently derive the continuous-wave response of numerous nonlinear microwave components including amplifiers, mixers, and oscillators. Its efficiency derives from imposing a predetermined steady-state form for the circuit response onto the nonlinear equations representing the network, and solving for the set of unknown coefficients in the response equation. Its attractiveness for nonlinear microwave applications results from its speed and ability to simply represent the dispersive, distributed elements that are common at high frequencies. The last decade has seen the development and application of harmonic balance techniques to model analog circuits, particularly microwave circuits. The first part of this article reviewed the fundamental achievements made during this time. In this part, the extension of the method to quasi-periodic regimes, optimization analysis, oscillator analysis, studies of various convergence strategies, and practical applications are discussed. A critical assessment of the various types of harmonic balance techniques is given. Examples of designs which have been modeled using the harmonic balance technique and built both in hybrid and MMIC form are presented.     

Computation of the 1‐dB compression point in radio frequency amplifier circuits using moments analysis

Nonlinear distortion analysis is one of the most computationally challenging aspects in the simulation of radio frequency circuits. Recently, an efficient moments‐based approach to determine the third‐order intercept point from the general harmonic balance equations, without the need to perform a harmonic balance simulation, was presented. In this article, the efficient computation of another important performance figure of merit, that of the 1‐dB compression point, using moments analysis is presented. In addition, the computations of the second‐ and fifth‐order intercept points are also presented. As a result, moments analysis becomes a comprehensive approach for quantifying nonlinear distortion using several key distortion figures of merit with a significant speed‐up over traditional harmonic balance methods. © 2014 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:10–20, 2015.     

Harmonic analysis of the vibrations of a cantilevered beam with a closing crack

In this article, the vibrational response of a cracked cantilevered beam to harmonic forcing is analysed. The study has been performed using a finite element model of the beam, in which a so-called closing crack model, fully open or fully closed, is used to represent the damaged element. Undamaged parts of the beam are modelled by Euler-type finite elements with two nodes and 2 d.f. (transverse displacement and rotation) at each node. Recently the harmonic balance method has been employed by other researchers to solve the resulting non-linear equations of motion. Instead, in this study, the analysis has been extended to employ the first and higher order harmonics of the response to a harmonic forcing in order to characterize the non-linear behaviour of the cracked beam. Correlating the higher order harmonics of the response with the forcing term the so-called higher order frequency response function (FRFs), defined from the Volterra series representation of the dynamics of non-linear systems, can be determined by using the finite element model to simulate the time domain response of the cracked beam. Ultimately the aim will be to employ such a series of FRFs, an estimate of which in practice could be measured in a stepped sine test on the beam to indicate both the location and depth of the crack, thus forming the basis of an experimental structural damage identification procedure.     

用谐波-能量平衡法求解单摆方程

应用谐波—能量平衡法求解了强非线性单摆方程,谐波-能量平衡法与经典的摄动法和谐波平衡法不同,不是把微分方程和初始条件分离处理;而是把微分方程和初始条件同时处理.用谐波平衡,将描述动力系统的二阶常微分方程,化为以角频率、振幅为变量的非线性代数方程组,考虑能量平衡,构成角频率、振幅为变量的封闭方程组求得解析解.谐波-能量平衡法将谐波平衡与能量平衡相结合,克服了二者的缺点吸取了二者的优点.实例表明,谐波-能量平衡法方法简单,取较少谐波就可以达到较高的精度.     

Approximate computation of the parameters of self-excited oscillations in the case of non-uniform convergence of the Fourier series

An approach to the analysis of self-excited oscillations by using the method of harmonic linearization for controlled systems not satisfying the harmonic balance conditions in the first approximation, as to a problem of non-linear mathematical programming, is proposed.     

Harmonic Balance Nonlinear Identification of a Capacitive Dual-Backplate MEMS Microphone

This paper describes the application of a nonlinear identification method to extract model parameters from the steady-state response of a capacitive dual-backplate microelectromechanical systems microphone. The microphone is modeled as a single-degree-of-freedom second-order system with both electrostatic and mechanical nonlinearities. A harmonic balance approach is applied to the nonlinear governing equation to obtain a set of algebraic equations that relate the unknown system parameters to the steady-state response of the microphone. Numerical simulations of the governing equation are also performed, using theoretical system parameters, to validate the accuracy of the harmonic balance solution for a weakly nonlinear microphone system with low damping. Finally, the microphone is experimentally characterized by extracting the system parameters from the response amplitude and phase relationships of the experimental data.     

轮-带驱动系统稳态周期响应谐波平衡分析

研究含有单向离合器、两滑轮及附件的轮-带驱动系统稳定稳态周期响应.通过单向离合器连接从动轮与附属系统,并计入传送带的横向振动的影响,导出了由偏微分-积分方程与分段常微分方程组成的连续-离散型非线性耦合方程组.利用Galerkin方法将连续非线性方程组截断为一组非线性常微分方程组,再运用谐波平衡法得到轮-带驱动系统耦合非线性振动的稳态响应.通过比较有无单向离合器装置的系统稳定稳态幅频响应曲线,研究了单向离合器对驱动系统以及轮-带系统非线性动态特性的影响.并首次研究了高频激励下轮-带系统的稳态响应.最后,运用Runge-Kutta方法对比验证了基于谐波平衡法得到的稳态响应.     

Optimal continuous-discrete nonlinear finite memory filter with a discrete predictions

We consider the following problem: using the discrete time measurements of the state variables of a continuous stochastic object and obtain the most precise estimate of these variables. To accelerate the estimating process, we synthesize the optimal structure of a discrete final-dimensional nonlinear filter with a piecewise-constant prediction, remembering the last few measurements in its state vector. The dimension of this vector (i.e., the filter memory volume) can be selected arbitrarily to balance the desired measurement precision and available processing speed for the measurements. We obtain the mean-square optimal structure functions of the filter expressed via the corresponding probability distributions and a chain of Fokker—Planck—Kolmogorov equations to find those distributions. We describe the procedure to obtain the structural functions of the filter numerically by means of the Monte-Carlo method. Also, we provide simple numeric-analytic approximations to the proposed filters: they are compared with approximations to the known filters. An example of the construction of such approximations is considered.     

Generalized Hermite Approximations and Spectral Method for Partial Differential Equations in Multiple Dimensions

In this paper, we consider a spectral method based on generalized Hermite functions in multiple dimensions. We first introduce three normed spaces and prove their equivalence, which enables us to develop and to analyze generalized Hermite approximations efficiently. We then establish some basic results on generalized Hermite orthogonal approximations in multiple dimensions, which play important roles in the relevant spectral methods. As examples, we consider an elliptic equation with a harmonic potential and a class of nonlinear wave equations. The spectral schemes are proposed, and the convergence is proved. Numerical results demonstrate the spectral accuracy of this approach.     

利用谐波平衡法对Buck变换器中倍周期分岔的仿真研究

利用谐波平衡法对Buck变换器中的倍周期分岔进行了仿真研究，首先给出了连续控制模式下电压控制型Buck变换器的动力学模型，然后采用谐波平衡法进行分析，获得了产生倍周期分岔的充要条件，同时也得到了分岔的准确位置。基于这个分岔条件，可以设计一个前馈控制来避免倍周期分岔的发生。此控制法有利于输入电压工作范围的大幅度扩大，以及较好的输出电压校准。     

有源电力滤波器多内模复合控制策略控制分析

为消除单一通过矢量谐振(VR)进行控制时面临的缺陷,采用重复控制(RC)并合理设置校正因子,设计了多内模控制器,通过对RC进行充分控制的方式达到跟踪与控制谐波分量的作用.采用MATLAB软件建立了上述系统模型并完成仿真测试.研究结果表明:电压与电流总谐波失真(THD)都低于1％,并没有发生明显畸变,呈现正弦波的特征.以...     

牵引变电所所用变压器二次侧故障分析

通过对牵引变电所所用变压器实测数据的分析,其结果表明电压高次谐波含量超标。本文从高次谐波的集肤效应、热损耗、高频谐振等多方面对所用变压器二次侧设备故障进行分析,提出使用高通滤波器来滤除高次谐波的措施,以消除高次谐波对二次侧设备的影响,并通过仿真对滤波效果进行了验证。     

Stability analysis of nonlinear multivariable Takagi-Sugeno fuzzycontrol systems

Points out how the nonlinearities involved in multivariable Takagi-Sugeno (T-S) fuzzy control systems could originate complex behavior phenomena, such as multiple equilibrium points or limit cycles, that cannot be detected using conventional stability analysis techniques. In the paper, the application of MIMO frequency-domain methods to predict the existence of multiple equilibria and of limit cycles are presented. The proposed method is based on the formulation of a Lur'e problem from the original structure of a T-S fuzzy system with a fuzzy controller. Furthermore, this technique makes straightforward the application of input-output stability techniques such as the multivariable circle criterion, also called the conicity criterion, and the harmonic balance method. Moreover, in the paper, the application of the harmonic balance method has been generalized to the case of a MIMO fuzzy system with asymmetric nonlinearities and improved by the decreasing conservatism. A new and more general stability index which could be used to perform a bifurcation analysis of fuzzy control systems is presented. The paper includes a collection of examples where the advantages of the proposed approach are made explicit comparing it to the input-output conicity criterion and the Lyapunov direct method     

Nonlinear models of dynamic processes in impulse control systems of brushless DC motors

Nonlinear mathematical and simulation models of the servo system of pulse width control of brushless direct current motors in machine drives are developed. Results of the analysis of the influence in the nonlinear pulse-width converter on time and frequency characteristics of the control system are presented. The analysis is performed using the harmonic balance, slowly varying amplitudes methods, and numerical simulation.