【作者】 禹东川；

【导师】 吴爱国；

【作者基本信息】 天津大学 ， 电力电子与电力传动， 2007， 博士

【摘要】 自从20世纪60年代E. N. Lorenz发现第一个混沌吸引子以来,混沌在电子装置、激光系统、化学反应、生化系统、生命系统、力学系统、神经网络等众多领域被相继观察到,混沌的发现被称为继相对论与量子力学问世以来物理学的第三次大革命,在许多邻域获得了巨大而深远的发展,是当今举世瞩目的前沿课题及学术热点。混沌控制与混沌同步目前已经成为有着巨大应用前景的实用技术被广泛关注,因此分析自然系统和人工系统中的混沌控制与混沌同步问题,最终力求对自然系统和人工系统中的混沌及系统间的同步加以控制和利用使之为人类服务就具有重大的应用价值。本文主要在混沌控制、混沌同步、基于完全同步的参数估计等方面进行了有益探讨,取得了如下创新结果:1、指出了自同步方法自其提出后就存在的理论分析错误,必须将混沌动力学本身的特点融入到现有的李雅普诺夫直接法中才能在理论上确保参数估计。此外,还在如下四个方面对该方法进行了有益的拓展:1)矩阵M为半正定的情况;2)如何利用输出微分信息以减小输出维数的问题;3)噪声干扰情况下的参数估计问题;4)该方法的适用条件和限制条件是什么。2、为解决驱动系统标量输出情况下响应系统的设计问题,原创性地提出了矩阵匹配法,该方法首先通过微分嵌入维理论获得静态重构模型,然后通过一些规则构建响应系统使得其与驱动系统完全同步,该方法不仅可以消除静态重构模型中可能存在的奇异问题,而且还可以降低微分嵌入的维数(即降低静态重构模型中微分估计器的阶次)、甚至完全不需要输出的微分。3、为解决标量输出情况下的系统参数估计问题,原创性地提出一种参数估计律的设计准则——参数估计律设计使得同步流形附近的局部线性化系统的特征多项式的最小阶系数几乎处处为正,Lorenz系统和统一混沌系统等作为实例来证明其有效性,同时也说明全部系统参数未知、且标量输出的情况下仍可能实现全部参数估计。4、为解决变结构混沌控制问题中的抖振问题,针对一类不确定时变混沌系统,首先构造该系统的增广系统,接下来对该增广系统设计常规的变结构控制律,最后再经过一些简单的数学变换处理就能获得一种自适应无抖振变结构控制方法。该方法不仅能减小滑模面附近的抖振现象并实现渐近跟踪,而且不需要“不确定项的上界已知”的先验知识。但该方法在实际工程应用中存在一些不足,如参数估计律在噪声和外部干扰存在时的病态问题,高频抖振不能完美去除等。为解决这些不足,进一步提出了改进的自适应无抖振变结构控制方法,该方法引入带死区和非线性学习律的参数估计器,理论分析与仿真结果都表明:只要选择合适的控制器参数,改进的方法不仅彻底消除抖振,并且能保证在有限时间内达到任意的跟踪精度。5、研究了一种在滑模控制中引入积分的新方法,该方法仍然采用在常规滑模控制设计中采用的滑动变量,但在控制律设计中引入滑动变量的非线性比例积分作用,由于非线性积分作用的存在使得控制系统具有良好的动静态性能。6、研究了一种简单的全状态渐近轨道控制方法实现统一混沌系统的全状态渐近跟踪控制,该方法只需要系统单一的状态信息,理论分析和仿真结果都证实所提的方法能实现渐近跟踪控制。7、提出了一种带有限时间不确定估计的降阶同步方法,该方法能在有限时间内准确估计系统不确定项;同时对降阶同步的同步准则进行了适当探讨,研究了完全降阶同步和一般降阶同步问题的设计,理论分析和数值计算表明该方法的正确性。更多还原

【Abstract】 Since E. N. Lorenz found the first chaotic attractor in 1960’s, chaos phenomenon was observed one after the other in many systems such as electrical devices, chemical reactions, biological systems, mechanical systems and neural networks, among others. The discovery of chaos was honored as the third revolution in physics following relative theory and quantum mechanics. Chaotic control and synchronization has now attracted much attention due to its dramatic potential applications. It is of great benefit for human beings to analyze and control chaos and coupling synchronization in natural and man-made systems.This Dissertation focuses on the research on chaotic control, chaotic synchronization and complete synchronization based parameter estimation. Its main contributions include following seven points.1. This dissertation points out the drawbacks of the autosynchronization approach since it was established. A new method is suggested to ensure parameter estimation by the properties of chaotic dynamics, merged into the conventional Lyapunov’s direct method. Then, the new approach is further extended to several useful cases: 1) when matrix M is semi-positive definite, 2) when output derivative is used for reducing the dimension of output, 3) in the present of noise. When a general system is considered, the scope applicable and some limitations of the new approach are also discussed.2. To design a response system from a scalar output of the driving system, a matrix match method is proposed, which firstly uses the differential embedding theory for achieving the static reconstruction model and then constructs a response model synchronized with the driving system by some rules. The method presented cannot only remove the potential singular problem of the static reconstruction model but also reduce the dimension of differential embedding (i.e. the order of derivative estimator in the static reconstruction model) and even remove completely the output differential.3. To estimate parameters from a scalar output, a guidance is presented for the design of parameter estimation law, namely parameter estimation is designed such that the coefficient of the lowest order of the local linearization around the synchronization manifold is positive almost everywhere. The Lorenz model and a unified chaotic system are illustrated to validate this method and to demonstrate also that it is possible to estimate all parameters from a scalar output time-series.4. Trying to solve the high-frequency chattering of conventional variable structure control, an adaptive chattering-free variable structure control method is proposed for controlling a class of uncertain time-variant chaotic systems by three steps: 1) constructing an augmented model of the selected system, 2) designing a conventional variable structure control law for this augmented model and 3) using some mathematical operation. The method not only eliminates the chattering phenomenon and tracks arbitrarily desired trajectory asymptotically, but also removes the prior knowledge about the upper bound of the system uncertainty. However this approach exhibits some drawbacks in practice such as the ill-pose problem in parameter estimation law happening in the present of noise and external disturbances, as well as imperfect avoidance of high-frequency chattering. An improved method is suggested to solve the drawbacks, which introduces a parameter adaptive estimator with dead-zone and nonlinearly learning rate, and in addition replaces the signum function by the saturation one as well. Theoretical analysis and numerical simulation validate that the improved method not only avoids the high-frequency chattering completely but also obtains high-accuracy tracking as desired in a finite time.5. A new approach is suggested to introduce an integral action in the sliding mode control, which uses the same sliding variable in the conventional sliding mode but adds a nonlinearly integral action of the sliding variable in the controller. Due to the nonlinear integral action applied in the new sliding mode controller, the system controlled can achieve good dynamical and static performances.6. A simple full-state asymptotic trajectory control method is suggested to asymptotically drive full states of a unified chaotic system to arbitrarily desired trajectories from only a scalar state. Theoretic analysis and numerical simulation validate that the method proposed can ensure the asymptotic trajectory control.7. An approach to reduced-order synchronization is suggested with a finite-time uncertainty estimation which evaluates correctly the system uncertainty in a finite time. In addition, the reduced-order synchronization criterions are discussed, and the design of complete and general reduced-order synchronizations is then investigated. Theoretical analysis and numerical simulation validate the effectiveness of the method proposed.更多还原