Systems with sector bound nonlinearities: A behavioral approach

The class of nonlinear dynamical systems obtained by interconnection of a linear, time-invariant dynamical system and a memoryless, time-invariant, sector-bound nonlinearity has been widely studied in literature. In this paper it is shown that many important results for this class of systems can be elegantly unified using behavioral theory of dynamical systems. Systems with slope-restricted nonlinearities are also considered.     

Instantaneous impulse response of nonlinear system

Mathematical models for nonlinear systems, which admit both standard integrable and impulsive inputs are developed. It is shown that instantaneous response of a nonlinear system to an impulse is not, in general, single-valued, whereas a particular response can be selected through the construction of an auxiliary system. The complete integrability of the auxiliary system is proven to guarantee uniqueness of the impulse response     

Analysis of nonlinear circuits by using differential Taylor transform

An effective but not widely used method, differential Taylor transform, is introduced for the analysis of the nonlinear electrical circuits. To apply the method the differential transform of the mathematical model of the system is obtained first, and then the response function is evaluated by using the inverse transform of the differential spectra. The inverse transform can be written in the form of Taylor series. The method is described with two examples for nonlinear electrical circuits.     

Robust approximation-free prescribed performance control for nonlinear systems and its application

This paper presents a robust prescribed performance control approach and its application to nonlinear tail-controlled missile systems with unknown dynamics and uncertainties. The idea of prescribed performance function (PPF) is incorporated into the control design, such that both the steady-state and transient control performance can be strictly guaranteed. Unlike conventional PPF-based control methods, we further tailor a recently proposed systematic control design procedure (i.e. approximation-free control) using the transformed tracking error dynamics, which provides a proportional-like control action. Hence, the function approximators (e.g. neural networks, fuzzy systems) that are widely used to address the unknown nonlinearities in the nonlinear control designs are not needed. The proposed control design leads to a robust yet simplified function approximation-free control for nonlinear systems. The closed-loop system stability and the control error convergence are all rigorously proved. Finally, comparative simulations are conducted based on nonlinear missile systems to validate the improved response and the robustness of the proposed control method.     

Functional expansions for the response of nonlinear differential systems

This paper is concerned with representing the response of nonlinear differential systems by functional expansions. An abstract theory of variational expansions, similar to that of L. M. Graves (1927), is developed. It leads directly to concrete expressions (multilinear integral operators) for the functionals of the expansions and sets conditions on the differential systems which insure that the expansions give reasonable approximations of the response. Similarly, it is shown that the theory of analytic functions in Banach spaces leads directly to conditions which imply uniform convergence of functional series. The main results on differential systems are summarized in a set of theorems, some of which overlap and extend the recent results of Brockett on Volterra series representations for the response of linear analytic differential systems. Other theorems apply to more general nonlinear differential systems. They provide a rigorous foundation for a large body of previous research on Volterra series expansions. The multilinear integral operators are obtained from systems of differential equations which characterize exactly the variations. These equations are of much lower order than those obtained by the technique of Carleman. A nonlinear feedback system serves as an example of an application of the theory.     

Bifurcations in the frequency response of nonlinear feedback control systems

The bifurcation theory is applied to bifurcations and accompanying jump phenomena which appear in the frequency response of nonlinear feedback control systems. It is also shown by a numerical example that such complicated structures exist in the frequency response of a simple nonlinear feedback control system.     

Parameter estimation for control systems based on impulse responses

The impulse signal is an instant change signal in very short time. It is widely used in signal processing, electronic technique, communication and system identification. This paper considers the parameter estimation problems for dynamical systems by means of the impulse response measurement data. Since the cost function is highly nonlinear, the nonlinear optimization methods are adopted to derive the parameter estimation algorithms to enhance the estimation accuracy. By using the iterative scheme, the Newton iterative algorithm and the gradient iterative algorithm are proposed for estimating the parameters of dynamical systems. Also, a damping factor is introduced to improve the algorithm stability. Finally, using simulation examples, this paper analyzes and compares the merit and weakness of the proposed algorithms.     

On identification of parallel block-cascade nonlinear models

The class of nonlinear systems studied in this paper is assumed to be modelled by parallel block-cascades. Such models are composed of parallel branches where each branch has a linear block in cascade with a zero-memory nonlinear block followed by another linear block. These types of models are extensively used to represent nonlinear dynamic systems and are known in the literature as Wiener-Hammerstein models. Using a zero-mean stationary white gaussian sequence as an input to such models, a structure identification criterion is developed, utilizing the bispectrum estimate of the output sequence only. The application of this criterion is shown by several simulation examples. Also, impulse response estimation of an example of such a model is considered to show the effectiveness of the proposed identification technique.     

Testing and evaluation of digital control systems

Testing procedures for analogue instruments are described as well as for controllers and control systems based on the use of microprocessors. The paper compares the difference between system testing and testing of system elements. The difference between digital and analogue evaluations are also described. The dynamic response to various and nonlinear algorithms are both shown and explained.     

胞映射方法及其在非线性随机动力学中的应用

介绍了与非线性随机动力学研究密切相关的几类胞映射方法的研究和进展,主要有广义胞映射图方法、基于短时高斯逼近的胞映射方法、并行胞映射方法等.简述了胞映射方法在随机动力学中的应用情况,重点关注随机响应、分岔、离出和碰撞振动系统.给出了胞映射方法面临的挑战,以及未来研究可能的发展方向.     

Identification and frequency domain analysis of non-stationary and nonlinear systems using time-varying NARMAX models

This paper introduces a new approach for nonlinear and non-stationary (time-varying) system identification based on time-varying nonlinear autoregressive moving average with exogenous variable (TV-NARMAX) models. The challenging model structure selection and parameter tracking problems are solved by combining a multiwavelet basis function expansion of the time-varying parameters with an orthogonal least squares algorithm. Numerical examples demonstrate that the proposed approach can track rapid time-varying effects in nonlinear systems more accurately than the standard recursive algorithms. Based on the identified time domain model, a new frequency domain analysis approach is introduced based on a time-varying generalised frequency response function (TV-GFRF) concept, which enables the analysis of nonlinear, non-stationary systems in the frequency domain. Features in the TV-GFRFs which depend on the TV-NARMAX model structure and time-varying parameters are investigated. It is shown that the high-dimensional frequency features can be visualised in a low-dimensional time–frequency space.     

Response of nonlinear sampled-data systems with nonzero initial conditions and response for in-between sampling via Volterra series

Nonzero initial conditions are included in the Volterra series solution of a class of nonlinear sampled-data systems that is described by ordinary difference equations with constant coefficients. Also, it is shown how one can find an upper bound on the intersampling response of the system under consideration. These developments are made through the use of the Banach fixed-point theorem.     

Asymptotic behavior of the extended Kalman filter as a parameter estimator for linear systems

The extended Kalman filter is an approximate filter for nonlinear systems, based on first-order linearization. Its use for the joint parameter and state estimation problem for linear systems with unknown parameters is well known and widely spread. Here a convergence analysis of this method is given. It is shown that in general, the estimates may be biased or divergent and the causes for this are displayed. Some common special cases where convergence is guaranteed are also given. The analysis gives insight into the convergence mechanisms and it is shown that with a modification of the algorithm, global convergence results can be obtained for a general case. The scheme can then be interpreted as maximization of the likelihood function for the estimation problem, or as a recursive prediction error algorithm.     

基于未建模动态补偿的非线性自适应切换控制方法

针对一类不确定的离散时间零动态不稳定的单输入-单输出(Single-input single-output, SISO)非线性系统,提出了一种基于未建模动态补偿的非线性控制器. 采用自适应神经模糊推理系统(Adaptive-network-based fuzzy inference system, ANFIS)和一一映射相结合的方法估计未建模动态.在此基础上,提出了由线性自 适应控制器、非线性自适应控制器以及切换机制组成的自适应切换控制方法.该方法通过对上述两种控制器的切换, 保证闭环系统输入输出信号有界的同时,改善系统性能.本文将要求未建模动态全局有界的条件放宽为线性增长, 建立了所提自适应控制方法的稳定性和收敛性分析.通过仿真比较和水箱的液位控制实验,验证了所提方法的有效性.     

Nonlinear analysis methods for the simulation of digital wireless communication systems

Adoption of digital modulation for many wireless communication systems has resulted in significant performance improvements over systems based on analog modulation. Concomitant with this have been changes in methods of characterizing and simulating system performance, particularly with respect to amplifier linearity. Digitally modulated signals are best represented by a power spectral density. In contrast, analog-modulated signals are adequately represented by discrete spectra. Consequently, many of the common discrete spectra nonlinear microwave analysis techniques are ill-suited to simulating systems characterized by signals with power spectral density representations. This study examines some of the common types of signals used in digital wireless communication systems, and looks at their representation and characterization. The applicability of various nonlinear analysis methods to microwave power amplifier simulation in the context of digital modulation is explored. © 1996 John Wiley & Sons, Inc.     

Chattering in sliding mode control systems with boundary layer approximation of discontinuous control

It has been a widely accepted notion that approximation of discontinuous control by certain continuous function in a boundary layer results in chattering elimination in sliding mode control systems. It is shown through three different types of analysis that in the presence of parasitic dynamics, this approach to chattering elimination would work only if the slope of the continuous nonlinear function within the boundary layer is low enough, which may result in the deterioration of performance of the system. A few examples are provided. An approach to robust stability of linear systems from the consideration of the saturating control is proposed.     

The parametric characteristic of frequency response functions for nonlinear systems

The characteristic of the frequency response functions of nonlinear systems can be revealed and analyzed by analyzing of the parametric characteristics of these functions. To achieve these objectives, a new operator is defined, and several fundamental and important results about the parametric characteristics of the frequency response functions of nonlinear systems are developed. These theoretical results provide a significant and novel insight into the frequency domain characteristics of nonlinear systems and circumvent a large amount of complicated integral and symbolic calculations which have previously been required to perform nonlinear system frequency domain analysis. Several new results for the analysis and synthesis of nonlinear systems are also developed. Examples are included to illustrate potential applications of the new results.     

A general criterion for jump resonance of nonlinear control systems

A new criterion for jump resonance of nonlinear control systems is presented. The jump resonance phenomenon is classified into various types according to the form of the frequency response curve, some of which are new and not found in the previous publications. A simple graphical method to predict each type of jump resonance is studied. Experimental results by analog simulation are shown.     

Nonlinear manifold representation in natural systems: The SOMersault

Natural systems often contain rhythmically fluctuating individual components which, when combined, can result in nonlinear patterns such as cycles, helixes, and parabolas. The self-organizing map (SOM) is a widely used artificial neural network for exploratory data analysis of high dimensional, multivariate data sets, however it encounters limitations when dealing with such highly nonlinear patterns. The SOMersault method is an expansion of the SOM, effective for gaining an understanding of patterns and clusters in natural data sets containing a low dimensional nonlinear manifold set amongst complex high dimensional data measurements. Data clusters become ordered with respect to the nonlinear degrees of freedom in the data, and patterns extracted are closely related to the data they represent. Results are shown on synthetic and real world data, involving a global set of river basins, with clustering and pattern extraction improvements displayed visually and quantified through a new set of geodesic error measures.     

Construction of bode envelopes using REP based range finding algorithms

The frequency domain analysis of systems is an important topic in control theory. Powerful graphical tools exist in classic control, such as the Nyquist plot, Bode plots, and Nichols chart. These methods have been widely used to evaluate the frequency domain behavior of system. A literature survey shows that various approaches are available for the computation of the frequency response of control systems under different types of parametric dependencies, such as affine, multi-linear, polynomial, etc. However, there is a lack of tools in the literature to construct the Bode envelopes for the general nonlinear type of parametric dependencies. In this paper, we address the problem of computation of the envelope of Bode frequency response of a non-rational transfer function with nonlinear parametric uncertainties varying over a box. We propose two techniques to compute the Bode envelopes:first, based on the natural interval extensions (NIE) combined with uniform subdivision and second, based on the existing Taylor model combined with subdivision strategy. We also propose the algorithms to further speed up both methods through extrapolation techniques.