1、附录A 英文文献原文出处：http:/trixie.eecs.berkeley.edu/chaos/chaos.htmlStimulation with hardware of Chuas circuit1. Introduction Chaos is a fascinating nonlinear phenomenon. Dr. Leon Chua invented Chuas circuit (circa 1983), a simple nonlinear circuit capable of producing strange attractors. Before you can get
2、 started on Chuas circuit, it would be instructive to understand the basic concept of nonlinear circuits: the DP (or driving-point) plot. This term was coined in the classic book Linear and Nonlinear Circuits by Chua, Leon O., Desoer, Charles A. and Kuh, Ernest S. 1987. McGraw-Hill. ISBN 0070108986.
3、 unfortunely this book is out of print. Here is the Amazon link to where you may find used copies of the book. But, here is a document on introductory nonlinear circuit analysis that I wrote for sophomore electronics students. This should be enough background for you to understand the implementation
4、 of Chuas circuit. Once you understand the basics of nonlinear circuit analysis, here are some links and papers to get you started with chaos and Chuas circuit: Experiment aim:(1)To learn some concepts of chaos(2)Mearsure V-A charicteristc of nonlinear resistor of source (3)To learn about chaos and
5、how it produces throngh a simple nonlinear circuit.2. Working with Chaos: Simulating Chuas Circuit First we need to simulate Chuas circuit. The simulation tool we use is MultiSim. Two points: First, this version of the circuit uses the LMC6482 which is more robust and easier to obtain than the JFET
6、Dynamics used in Michael Peter Kennedys paper. A more subtle point is the series resistance of the inductor. You have to take this into account while building Chuas Circuit. 3. Datasheets Here I provide links and datasheets to the two most important items in the circuit: the dual op-amp and inductor
7、 LMC6482AIN, Datasheet. Cost: $2.29. T1105 - Toko 8 mH Variable inductor, Datasheet. Cost: $4.74. Note: This part is pretty difficult to get. The rest of the components in the circuit are standard: a 2k pot, a 100 nF capacitor, a 4.7 nF capacitor and 22k x 2, 220 ohm x 2, 2.2k and 3.3k resistors. Y
8、ou can find these at a local Radioshack. Hence, the total cost of the circuit is around $10! 4. Working with Chaos: Building the circuit The hardest part in building the circuit is getting the correct value of the inductance（电感）. I used a simple RL filter to tune the inductance. I used a known R and
9、 applied a sinusoid at the input. Since I know the frequency and amplitude of the sinusoid, I can use the frequency response of the circuit to obtain the value of the inductance I want. In order to measure the series resistance of the inductor, use a simple ohm-meter. I even used an ohm-meter to fig
10、ure out across which pins in the T1105 is the coil actually connected. Screenshots: 5. Other possible component values for Chuas circuit The list below shows some other possible component values for Chuas circuit. Please note that the nonlinear resistor (Chua Diode) is the same as shown in the schem
11、atic from the Simulation section. You can refer to the schematic shown at the banner on top of the page. L=8mH, C2=47nF, C1=3nF, R=1.85k L=18mH, C2=50nF, C1=4.7nF, R=2.1k 6. Applications of Chaos Believe it or not, there are tons of applications for Chaos. Here are a few: The stock market (finance)
12、Power systems (electrical engineering) ,Population Dynamics (biology) ,Communication Systems (electrical engineering) There are also very interesting chaotic processes in the human brain. Here are two excellent papers by French scientists on this topic (pubmed links to both articles): 出处：Control Th
13、eory & Applications Vol.20No.5Stabilization of unstable equilibria of chaotic systemsand its applications to Chuas circuitAbstract: Based on the ergodicity of chaos and the state PI regulator approach, a new method was proposed for stabilizing unstable equilibria and for tracking set point targets f
14、or a class of chaotic systems with nonlinearities satisfying a specific condition. A criterion was derived for designing the controller gains, in which control parameters could be selected by solving a Lyapunov matrix inequality. In particular, for piecewise linear chaotic systems, such as Chuas cir
15、cuit, the control parameters can be selected via the pole placement technique in linear control theory. More importantly, this method has high robustness to system parametric variations and strong rejection to external constant disturbances. For verification and demonstration, the design method is a
16、pplied to the chaotic Chuas circuit, showing satisfactory simulation results.Key words: Chuas circuit; unstable equilibrium point; stabilization; PI regulator1.IntroductionIn the past decade, much attention has been paid to chaos control, and many methods have been proposed for suppressing chaos1,2.
17、 For instance, the delayed feedback control (DFC) method3is based on the difference between the current system output and the time_delayed output signals, which does not require any knowledge of the target points.However, this approach in general cannot specify the target setting point and is subjec
18、t to the so_called odd number eigenvalue limitation46. On the other hand, the OGY method7,which is a local control scheme, and the methods8,9that are based on precise state feedback control usually fail with system parameters variation and are inconvenient for practical engineering systems. In this
19、paper, based on the ergodicity of chaos and state PIregulator approach10, a feedback control design method is proposed for stabilizing unstable equilibria and for set-point tracking for a class of chaotic systems with nonlinearities satisfying a specific condition. The proposed method combines a sta
20、te feedback and an integral of the difference between the target output and the current output signals. The output signal is a simple function (e.g., linear combination) of the state variables of the chaotic system. In particular, if a suitable linear combination is selected and used as the output f
21、eedback, the target output signal can become zero, and then no information about the target equilibrium is needed in the integral part of the controller. Moreover, this control method has satisfactory control performance and robustness. It will also be demonstrated that this control method can rejec
22、t external bounded constant-disturbances asymptotically. Based on the Lyapunov stabilization theory, a criterion is derived for choosing the proportional and integral gains. The control parameters can be selected via solving a Lyapunov matrix inequality. In particular, for piece-wise linear chaotic
23、systems, such as Chuas circuit, the control parameters can be chosen via the pole placement technique in linear control theory2Stabilizing unstable equilibria of a class of chaotic systems Consider a controlled chaotic system of the formAx+g(x)+u (1) Where is the state vector, is the control input t
24、o be designed, Aa constant matrix, and g(x) is a continuous nonlinear function satisfying the following condition11:,where is a bounded matrix that depends on both and .Remark1Many chaotic systems can be described by (1) and (2), such as the classic Chuas circuit12,the modified Chuas circuit with a
25、sine function, the modified Chuas circuit with nonlinear quadratic function x | x |13,and the MLC circuit.Let be an unstable equilibrium of (1) when u =0,that is, (3)The objective is to design a controller u such that the states of system (1) are stabilized to , which is a constant vector independen
26、t of time. Later, the objective will also be extended to tracking a constant set point. According to the state PI regulator theory, a controller is constructed as follows:(4)Where Bis a constant gain matrix, Kis the proportional state feedback gain vector, kR s the integral gain, y = is the output w
27、ith a constant matrix C, is the observation of the target equilibrium ,andWhere denotes the neighborhood of the unstable equilibrium Remark2Because of the ergodicity of chaos, the trajectory will visit or access x sat times. When the trajectory accesses , the controller (4) is turned on, and the tra
28、jectory will converge to asymptotically under the controller (4), in which the control parameters will be chosen to ensure the error dynamic system is asymptotically stable, as further described below.Remark3In control law (4), if we choose , where is a constant set point for tracking, then the outp
29、ut y can track this set point asymptotically.Remark 4 If there exists an external bounded constant disturbance w, whose value is unknown but bounded, in the system (1), then we can easily prove that the chaotic system can be stabilized at the targeted unstable equilibrium point by using the similar
30、procedure above.Conclusion and discussion In this paper, a new method for stabilizing unstable equilibria has been developed for a class of chaotic systems based on the state PI regulator method. The proposed method is robust to a certain level of external disturbances as well as system parameters v
31、ariation. Based on the Lyapunov stabilization theory, a precise criterion is derived to accomplish the stabilization of the target unstable equilibria of the chaotic system. The control parameters can be selected via solving a Lyapunov matrix inequality. Particularly, for piece wise linear chaotic s
32、ystems such as Chuas circuit, they can be selected via the simple pole placement technique. This new design method is better than the state feedback control method in the sense that even the given.附录B 英文文献翻译出处：http:/trixie.eecs.berkeley.edu/chaos/chaos.html蔡氏电路硬件仿真1简介混沌现象是一个奇特的非线性现象。在电路领域中，蔡氏电路是一个典型
33、的混沌电路它由Leon Chua博士在1983年提出，是一个能产生异常吸引子简单的非线性自治电路。在研究蔡氏电路之前，DP阀值域（或阀值点）将会对非线性电路的基础理论的理解有指导作用。这个术语由Chua, Leon O., Desoer, Charles A. 和 Kuh, Ernest S. 1987. McGraw-Hill.所著的线性和非线性电路中一书中所命名，版本号为 ISBN 0070108986。但目前这本书已经不再出版，以下是了解蔡氏电路的简单背景。为了展现混沌行为，一个自治的电子电路必须包含至少一个非线性元件，一个可变负阻，三个储能元件。蔡氏电路是包含这些元件的最简单的电子电路
34、蔡氏电路独有的混沌物理现象在数学意义上证明了Shilnikov理论。蔡氏电路用普遍的混沌特性为混沌学的研究提供了一个很好的范例，这个电路设计，构造非常简单，而且有超过40个吸引子的变化范围。所有应该从三阶自治常微分方程描述的系统中得到的分岔和混沌现象都能够在蔡氏电路中通过计算机仿真观察到，它已于控制和同步，既可以控制它由混沌状态转变为周期性或定常轨道，也可以使相同的蔡氏电路同步工作与周期振荡或混沌状态，使混沌电路可能在广泛的领域中得到应用。实验目的(1)了解混沌的一些基本概念；(2)测量有源非线性电阻的伏安特性；(3)通过研究一个简单的非线性电路，了解混沌现象和产生混沌的原因。2馄饨现象的产
35、生：蔡氏电路的激励首先，我们需要激励蔡氏电路仿真。所用的仿真工具为MultiSim。需要注意两点：第一，这个电路所用的LMC6482它要比在 Michael Peter Kennedy的论文中所用的JFET 更强大且更容易获得。更精细的一点就是电感的系列电阻值。在建立蔡氏电路必须要列入计算的。3.数据在这里我对电路为两个重要的术语提供了链接和数据表，即双运算放大器以及电感。LMC6482AIN, Datasheet. Cost: $2.29. T1105 - Toko 8 mH的可变电感，成本为4.74美元，而这一部很难获得。其余的电路组件的标准为：一个100 nF的电容器，一个4.7 nF的
36、电容器，和22k x 2,220 欧姆x2，2.2k和3.3k的电阻。4.混沌工作:建立电路组成蔡氏电路最难的部分是获得电感的准确参数。用一个简单的RL滤波器来调试电感。用一个已知电阻，并在输入端加以正弦信号。因为已知正弦波的频率和振幅，可以用电路的频率响应来获得一个想要的电感值。可用一个普通电阻表来测量电感线圈的串联电阻。用电阻表来测量和线圈交叉相连的T1105的管脚的数据。5.蔡氏电路元件的其它可能参数值以下列出了其它可用的蔡氏电路元件参数值。可以注意到非线性电阻和从仿真部分所示的大概相同。 L=8mH, C2=47nF, C1=3nF, R=1.85k L=18mH, C2=50nF,
37、C1=4.7nF, R=2.1k 6.混沌的应用混沌理论是非线性动力学系统的重要组成部分,它揭示了非线性科学的共同属性有序与无序的统一,确定性与随机性的统一。混沌控制及其应用是非线性科学应用新的研究领域,其研究受到了非常广泛的重视。混沌有广泛的应用范围，股票市场（经济）,电力系统（电力工程）,人口动力学(生物), 通信系统,（电力工程）, 混沌信息处理、混沌细胞神经网络、混沌保密通讯等领域具有很高的应用价值，受到了广泛的关注。出处：控制理论与应用 2003年10月 第20卷第5期混沌系统不稳定平衡点的镇定及其在蔡氏电路中的应用摘要:基于混沌系统的遍历性和状态PI调节器理论,提出一类混沌系统不稳
38、定平衡点的镇定和设定点跟踪新方法,给出用于控制器参数设计的Lyapunov矩阵不等式.对于分段线性混沌系统,如蔡氏电路,可通过控制理论中的极点配置技术来设计控制器参数.该方法对系统参数变化具有很强的鲁棒性,能够消除外部定值扰动.将该方法用于蔡氏混沌电路不稳定平衡点的镇定,取得了满意的结果.关键词:蔡氏电路;不稳定平衡点;镇定; PI调节器1简介在过去的十年中，混沌控制受到了很大重视，提出了许多控制混沌的方法。例如，延时反馈控制法基于当前系统输出和延时输出信号的不同，它并不需要对目标点的了解。然而，一般来讲这种途径不能详细说明目标设置点，要受到所谓的奇异数值特征值限制。另一方面OGY法，它是一个
39、局域控制法，基于精确状态反馈控制的方法通常因为系统参数变化和不便于实际工程系统而失败。在本论文中，基于混沌各遍历和状态PI调节器方法，一个反馈控制设计方法用于在满足一个特定条件的非线性混沌系统中镇定和跟踪设置点。他提出的方法是在一个目标输出和电流输出信号间让一个微分的积分和状态反馈联系。输出信号是混沌系统状态变量的简单函数。特别的，如果选择合适的线性连接用于输出反馈，目标输出信号可以变为零，在控制器积分部分就没有关于目标均衡点。此外，这种控制方法具有满意的控制性能和强度。它也可以渐近地拒绝外部有界常量干扰。基于Lyapunov稳定理论，一个准则来源于选择正比和积分增益。通过解决Lyapunov
40、不平等矩阵，控制系数可以选定。尤其对与分段线性化混沌系统，例如蔡氏电路，控制系数可以通过在非线性控制理论中极点替换技术来选择。2 混沌系统不稳定平衡点的镇定思考一个控制混沌系统的形式Ax+g(x)+u (1) 其中 是状态向量, 控制输入, 是一个常量矩阵, g(x)是一个满足以下条件11的持续性非线性函数 ,其中 是一个同时依赖 and 的有界矩阵.论述1许多混沌系统可以通过（1）和（2）来描述，例如经典蔡氏电路12。加正弦函数的变形蔡氏电路， 还有MLC电路。让为当u =0时（1）的不平衡点，即 (3)设计控制器u的目标是让系统状态（1）稳定到，这是一个独立时间的常向量。接着，目标将继续跟
41、踪一个常量设定值 。根据PI调整器理论，控制器将如以下构成：(4)其中， B是常量矩阵，K是正比态反馈增益向量，kR是积分增益，y = Cx是具有常量矩阵C的输出。是目标均衡的表达。其中表示不稳定平衡点的相邻点。论述2由于混沌的各态遍历性，它的轨道将经过的落点。当轨道进入点，控制器(4)将被打开，然后在控制器(4)下轨道会渐近集中到，在控制器中控制系数的选择将确保误差动态系统是逐渐稳定的，在下面将进一步叙述。论述3 在控制法（4）中，如果我们选择，为跟踪的常量设置点，然后输出y可以渐近这个设置点。论述4 如果一个永恒的有界常量干扰w，它的值是未知但是有界的，在系统（1）中，我们可一轻松地证明混沌系统通过以上步骤可以镇定其不稳定点。
