Identification of nonlinear dynamic systems: Time-frequency filtering and skeleton curves

The nonlinear behavior varying with the instantaneous response was analyzed through the joint time-frequency analysis method for a class of S. D. O. F nonlinear system. A masking operator an definite regions is defined and two theorems are presented. Based on these, the nonlinear system is modeled with a special time-varying linear one, called the generalized skeleton linear system (GSLS). The frequency skeleton curve and the damping skeleton curve are defined to describe the main feature of the non-linearity as well. Moreover, an identification method is proposed through the skeleton curves and the time-frequency filtering technique.

Asymptotic solutions to a class of nonlinear oscillation systems

In this paper, we present an asymptotic method for the analysis of a class of strongly nonlinear oscillators, derive second-order approximate solutions to them expressed in terms of their amplitudes and phases, and obtain the equations governing the amplitudes and phases, by which the amplitudes of the corresponding limit cycles and their behaviour can be determined. As an example, we investigate the modified van der Pol oscillator and give the second-order approximate analytical solution of its limit cycle. The comparison with the numerical solutions shows that the two results agree well with each other.

An adaptive controller for a class of nonlinear system using direction basis function

For a class of nonlinear dynamical systems, the adaptive controllers are investigated using direction basis function (DBF) in this paper. Based on the criterion of Lyapunov' stability, DBF is designed which guarantees that the output of the controlled system asymptotically tracks the reference signals. Finally, the simulation shows the good tracking effectiveness of the adaptive controller.

Relationship between strain localization and catastrophic rupture

In order to explore a prior warning to catastrophic rupture of heterogeneous media, like rocks, the present study investigates the relationship between surface strain localization and catastrophic rupture. Instrumented observations on the evolution of surface strain field and the catastrophic rupture of a rock under uniaxial compression were carried out. It is found that the evolution of surface strain field displays two phases: at the early stage, the strain field keeps nearly uniform with weak fluctuations increasing slowly; but at the stage prior to catastrophic rupture, a certain accelerating localization develops and a localized zone emerges. Based on the measurements, an analysis was performed with local mean-field approximation. More importantly, it is found that the scale of localized zone is closely related to the catastrophic rupture strain and the rupture strain can be calculated in accord with the local-mean-field model satisfactorily. This provides a possible clue to the forecast of catastrophic rupture. (c) 2007 Elsevier Ltd. All rights reserved.

Predicting chaotic time-series with wavelet networks

A new technique, wavelet network, is introduced to predict chaotic time series. By using this technique, firstly, we make accurate short-term predictions of the time series from chaotic attractors. Secondly, we make accurate predictions of the values and bifurcation structures of the time series from dynamical systems whose parameter values are changing with time. Finally we predict chaotic attractors by making long-term predictions based on remarkably few data points, where the correlation dimensions of predicted attractors are calculated and are found to be almost identical to those of actual attractors.

Asymptotic periodicity of the Volterra equation with infinite delay

For some species, hereditary factors have great effects on their population evolution, which can be described by the well-known Volterra model. A model developed is investigated in this article, considering the seasonal variation of the environment, where the diffusive effect of the population is also considered. The main approaches employed here are the upper-lower solution method and the monotone iteration technique. The results show that whether the species dies out or not depends on the relations among the birth rate, the death rate, the competition rate, the diffusivity and the hereditary effects. The evolution of the population may show asymptotic periodicity, provided a certain condition is satisfied for the above factors. (c) 2006 Elsevier Ltd. All rights reserved.

非线性机电系统的鲁棒及预测控制

近年来，机器人、数控机床等机电系统在国民生产及生活中得到了越来越广泛的应用，与之相对应的，机电系统的控制无形中也逐渐成为机电一体化和自动控制的研究热点。另一方面，非线性特性是任何实际系统普遍存在的现象，在机电系统中尤其如此。本文从控制器设计的角度研究非线性机电系统的两个典型问题：系统闭环优化性能的改善；控制器鲁棒性的增强。这也是当前自动控制研究领域的两个重点及热点问题。 最优性是闭环系统性能最实用的评价指标之一。最优控制及预测控制是试图实现控制性能优化的两种典型方法。但前者因本身不构成闭环而严重缺乏鲁棒性，在实践中很少能得到应用；后者作为前者理念的推广，在实践中已经得到了较为广泛地应用，并取得了不错的效果，但在实现某种程度的鲁棒性时依然面临困难。除最优性外，闭环控制的鲁棒性也是非线性系统控制中亟待解决的问题之一，这是由于实际的系统几乎不可能避免模型不确定性。 现有的非线性预测控制及鲁棒控制方法无论在方法的广泛适用性还是在可行性方面都还远非完备。据此，本论文沿鲁棒性和最优性两条主线，以典型的机电系统（无人直升机模型）为研究对象，分别进行了深入的理论和实验研究，并最终形成一种同时兼顾最优性和鲁棒性的控制器设计框架，以期在一定程度上解决现有方法中存在的问题，并为以后更深入的研究工作奠定基础。 鉴于此，本论文分别针对鲁棒控制和预测控制展开讨论，其中前者主要解决基于加速度反馈实现鲁棒控制的方法，内容为第二章和第三章；后者则旨在解决基于控制Lyapunov函数方法实现实时稳定预测控制，主要内容为第四章和第五章。本论文的具体内容安排如下： 论文的第一章综述了控制理论在鲁棒性与最优性两个方向的发展概况（主要针对非线性系统），包括其发展历史，现存方法的局限性等。从而引出本论文的研究内容及研究意义。 第二章，研究了基于加速度反馈的控制器鲁棒增强方法。在深入分析常规加速度反馈控制方法的基础上，指出其存在的三方面主要问题：代数环问题；高增益实现问题和不能用于欠驱动非线性系统等。并针对两种典型的非线性系统（以无人直升机模型为代表）将新的加速度反馈控制方法与H∞控制相结合，得到了一种能够保证输入输出稳定的扰动抑制方法。大量的仿真结果验证了方法的可行性及有效性。 随后，在第三章研究了加速度的估计问题。基于加速度反馈的鲁棒控制器增强技术得以实现的前提是加速度信号的获取，本章在分析了现有加速度估计方法存在严重的滞后问题的同时，提出了将Kalman滤波方法同牛顿预测方法相结合以改善相位滞后问题的方法。实验及仿真结果验证了方法的有效性。 第四章提出了基于控制Lyapunov函数的稳定闭环控制器设计框架。本章利用集值分析理论研究了控制Lyapunov函数具有的一些性质及其在控制器设计中的应用。随后，介绍了两种典型的根据控制Lyapunov函数设计控制器的方法。接着，将引导函数的概念引入到Freeman的逐点最小范数控制方法中，形成了一种新的利用控制Lyapunov函数设计非线性控制器的方法—广义逐点最小范数控制器。最后指出，在这种框架下，鲁棒控制器设计也可以实现，并针对三种不同的不确定性系统给出了鲁棒广义逐点最小范数控制器设计方法。 最后，在第五章，将前面提出的广义逐点最小范数控制引入到非线性预测控制中去，以期利用控制Lyapunov函数保证闭环稳定性，同时利用控制器中的参数化变量作为优化对象以减轻预测控制算法的计算负担，从而达到实时稳定预测控制的目的。另外，在这一章我们还在第二章和第四章的基础上，结合加速度反馈思想和鲁棒控制Lyapunov函数的概念，提出了一种用于扰动抑制的鲁棒实时预测控制算法。同样，仿真实验验证了方法的有效性和可行性。

Identification of Nonlinear Systems Through Time-Frequency Filtering Technique

In the previous paper, a class of nonlinear system is mapped to a so-called skeleton linear model (SLM) based on the joint time-frequency analysis method. Behavior of the nonlinear system may be indicated quantitatively by the variance of the coefficients of SLM versus its response. Using this model we propose an identification method for nonlinear systems based on nonstationary vibration data in this paper. The key technique in the identification procedure is a time-frequency filtering method by which solution of the SLM is extracted from the response data of the corresponding nonlinear system. Two time-frequency filtering methods are discussed here. One is based on the quadratic time-frequency distribution and its inverse transform, the other is based on the quadratic time-frequency distribution and the wavelet transform. Both numerical examples and an experimental application are given to illustrate the validity of the technique.

Time-Frequency Analysis of Nonlinear Systems: The Skeleton Linear Model and the Skeleton Curves

The joint time-frequency analysis method is adopted to study the nonlinear behavior varying with the instantaneous response for a class of S.D.O.F nonlinear system. A time-frequency masking operator, together with the conception of effective time-frequency region of the asymptotic signal are defined here. Based on these mathematical foundations, a so-called skeleton linear model (SLM) is constructed which has similar nonlinear characteristics with the nonlinear system. Two skeleton curves are deduced which can indicate the stiffness and damping in the nonlinear system. The relationship between the SLM and the nonlinear system, both parameters and solutions, is clarified. Based on this work a new identification technique of nonlinear systems using the nonstationary vibration data will be proposed through time-frequency filtering technique and wavelet transform in the following paper.

Kramers Escape Rate in Nonlinear Diffusive Media

In this paper, we study nonlinear Kramers problem by investigating overdamped systems ruled by the one-dimensional nonlinear Fokker-Planck equation. We obtain an analytic expression for the Kramers escape rate under quasistationary conditions by employing

Optimal bounded control of first-passage failure of quasi-integrable Hamiltonian systems with wide-band random excitation

The optimal bounded control of quasi-integrable Hamiltonian systems with wide-band random excitation for minimizing their first-passage failure is investigated. First, a stochastic averaging method for multi-degrees-of-freedom (MDOF) strongly nonlinear quasi-integrable Hamiltonian systems with wide-band stationary random excitations using generalized harmonic functions is proposed. Then, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximizinig reliability and maximizing mean first-passage time are formulated based on the averaged It$\ddot{\rm o}$ equations by applying the dynamical programming principle. The optimal control law is derived from the dynamical programming equations and control constraints. The relationship between the dynamical programming equations and the backward Kolmogorov equation for the conditional reliability function and the Pontryagin equation for the conditional mean first-passage time of optimally controlled system is discussed. Finally, the conditional reliability function, the conditional probability density and mean of first-passage time of an optimally controlled system are obtained by solving the backward Kolmogorov equation and Pontryagin equation. The application of the proposed procedure and effectiveness of control strategy are illustrated with an example.

Micro-systems for Optical Protein-Chip

Protein-Chip as micro-assays for the determination of protein interaction, the analysis, the identification and the purification of proteins has large potential applications. The Optical Protein-Chip is able to detect the multi-interaction of proteins and multi-bio-activities of molecules directly and simultaneously with no labeling. The chip is a small matrix on solid substrate containing multi-micro-area prepared by microfabrication with photolithography or soft lithography for surface patterning, and processed with surface modification which includes the physical, chemical, and bio-chemical modifications, etc. The ligand immobilization, such as protein immobilization, especially the oriented immobilization with low steric hindrance and high bio-specific binding activity between ligand and receptor is used to form a sensing surface. Each area of the pattern is corresponding to only one bioactivity. The interval between the areas is non-bioactive and optically extinctive. The affinity between proteins is used to realize non-labeling microassays for the determination of protein identification and protein interaction. The sampling of the chip is non-disturbing, performed with imaging ellipsometry and image processing on a database of proteins.

First-passage failure of quasi-integrable Hamiltonian systems

The first-passage failure of quasi-integrable Hamiltonian si-stems (multidegree-of-freedom integrable Hamiltonian systems subject to light dampings and weakly random excitations) is investigated. The motion equations of such a system are first reduced to a set of averaged Ito stochastic differential equations by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Then, a backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function, and the conditional probability density and moments of first-passage time are obtained by solving these equations with suitable initial and boundary conditions. Two examples are given to illustrate the proposed procedure and the results from digital simulation are obtained to verify the effectiveness of the procedure.

Is there chaotic synchronization in space extend systems

Based on coupled map lattice (CML), the chaotic synchronous pattern in space extend systems is discussed. Making use of the criterion for the existence and the conditions of stability, we find an important difference between chaotic and nonchaotic movements in synchronization. A few numerical results are presented.

Non-linear theory of a positive-column in a magnetic-field

A nonlinear theory of an intermediate pressure discharge column in a magnetic field is presented. Motion of the neutral gas is considered. The continuity and momentum transfer equations for charged particles and neutral particles are solved by numerical methods. The main result obtained is that the rotating velocities of ionic gas and neutral gas are approximately equal. Bohm's criterion and potential inversion in the presence of neutral gas motion are also discussed.