强流加速器中束晕—混沌的延迟反馈控制研究
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摘要
本文在综述混沌控制和回顾束晕—混沌形成机制及其非线性控制的基础上，提出了应用延迟反馈控制束晕—混沌的方法。该法不仅控制效果显著，而且由于采用线性控制律，控制器形式简单，施加的反馈较小，因此该法非常有利于工程技术实现和降低控制代价，可为实用强流加速器的研制提供有价值的参考。
在综述混沌控制部分，选取了OGY方法、外力反馈法、延迟反馈法、正比于系统变量的脉冲反馈法、自适应控制方法等一些有代表性的控制方法进行了阐述。
束晕—混沌的控制是新一代强流加速器研制的关键问题，随着强流离子束应用前景的日趋广阔而日益成为研究的热点。传统机械限束器因无法解决束晕的再生而收效甚微，因为束晕的形成有着其内在动力学机制—非线性共振以及混沌等。基于此，中国原子能科学院研究员方锦清将混沌控制的理论和方法开创性的运用于束晕—混沌的控制上，提出了控制束晕—混沌的非线性控制策略，即在粒子径向所受束自生场力方程的右边加上非线性控制函数G：
并选取一些非线性函数如等进行了控制的模拟研究，将束晕强度控制在0.1078左右，取得了初步的控制效果。
文献[72]在选取更有效的非线性函数上作了进一步的探索并且取得了进展。该
文选取小波函数构造控制器将束晕强
度控制到零，达到较好的控制结果。
但从实用角度来看，一些简单的非线性函数如的控制效果距实际应用的最低限制仍有一定差距；小波函数法虽可取得较好的控制效果，但控制器形式复杂，控制参数有两个，在工程实现上有一定难度。本文在保证控制效果的前提下，致力于寻求更为简单易行的控制方法。经过大量的理论分析和模拟研究，提出了应用延迟反馈控制束晕—混沌的方法。
本文对延迟反馈法运用于束晕混沌作了简单的理论分析。经分析，确定了延迟反馈控制器的延迟参数为S(一个聚焦磁场通道周期)；而后，计算了系统的最大李雅普诺夫指数。结果表明，系统在延迟反馈控制下最大李雅普诺夫指数由正值转为负值，
说明受控系统是稳定的，混浊得到了有效地抑制。在此基础上，选择a为控制变量，
构造延迟反馈控制器G：
G（S＝glrrms（－S）一 rms（S）（2）
同（l）式一致，将此控制器函数加在粒子径向所受束自生场力方程的右边。利
用多粒子数值模拟程序（PIC）进行了控制试验。模拟结果表明，用同一个控制器和
同一个控制参数即可实现五种不同初始分布情况下的束运一混饨的有效控制。对K一
V分布、水袋分布、抛物分布均能达到束晕强度为零的理想控制效果，对3—sigma
分布、全高斯分布的控制结果也基本能满足强流加速器中 10‘数量级的实际最低限
制。同时研究中还发现，较小的反馈即可实现束流的稳定。
由此可见，该法可达到小波函数法相同的控制效果，而控制器采用线性控制律，
仅有一个调节参数，形式简单，施加的反馈较小，因而应用前景较好。
对该法以多周期间隔形式控制也作了探讨。运用混饨控制理论，分析了两种多周
期间隔控制形式并以K—V分布为例进行模拟试验。分析和实验结果均表明，先连续
控制而后再间隔控制能够保证较好的控制效果和较低的控制代价。比如，对于K—V
束，一开始连续控制50个周期然后再每间隔200周期的情况下仍可将束晕强度控制
为零，较大的突破了文献［71」间隔周期为 16的上限。从而可将控制代价大幅度降
三 低。
In this paper,we give a summary of controlling chaos and a review of the mechanism of beam halo and its nonlinear control.Then delayed self-controlling feedback method is presented to control beam halo-chaos.This method,not only can control beam halo-chaos well ,but also is easy to work out because of the linearity and the simpleness of its controller.Considering the advantage in technology realizing and cost saving,this method is recommended to the development of high intensity accelerator.
In the part giving a summary of controlling chaos ,OGY control and delayed self-controlling feedback and adaptive control etc.are selected as typical examples to give a further description.
The control of beam halo-chaos becomes a critical problem in the development of high intensity accelerator.Efforts to remove the halo by collimation have been largely unsuccessful since the halos almost always regenerate.The mechanisms of halos are complex,such as nonlinear resonances and chaotic behavior etc.Considering this,Professor Fang Jin-Qing who works in China Institute of Atomic Energy pointed out that the theory of chaos control can be used to control beam halos .He presented the method to control halos by using nonlinear functions,which means nonlinear function G is added to the right of ion radial self-edlctric force equation
and some nonlinear function are selected to control beam halos in simulations .In
paper[69],controllerG = -0.15sin(rmax -am)2 was used and the halo intensity was
decreased to 0.1078,the halos are removed partly.
Paper [72] goes further in finding more effctive nonlinear function and wavelet
function was selected to construct the controller
.Under
the controller G the halo intensity was decreased to zero,The result is quite good.
When consider the practicality,we will find that some simple nonlinear function can't
control halo well enough while wavelet controller maybe too complex to work out .What's more.there are two parameters in wavelet controller and it maybe difficult to modify them.Considerable effort is devoted to exploring easier way in this paper.After large amount of simulations and analysis,we presented the way to control halo by delayed self-controlling feedback.
we give a brief theoretic analysis of using delayed self-controlling feedback to control beam halo.S(the length of one period of focusing channel)is selected as the delay parameter of the controller and the maximal lyapunov exponent is calculated. We found that the maximal lyapunov exponent of the system transfer from positive to negative under the control,which means that the controlled system is stable and the chaos is controlled
effectively. On the basis of this,we selected rms as feedback variable and constructed the controller G:
Add this controller to the right of ion radial self-edlctric force equation like equation (1).We simulated the motion of ion beam by using muti-paticle code(partice -in -cell(PIC)code).The results demonstrated that the beam halo of five different initial distribution is eliminated well under the same controller.The halo intensity of k-v distribution,water-bag distribution and parabolic distribution and be reduced to zero.In the case of 3-sigma distribution and full gauss distribution,the result is agree with minimum limitation(10") of halo intensity of factual high intensity accelerator. Also we found that weak feedback is need to control the beam halo.
This method,not only can control beam halo-chaos well as wavelet had done,but also is easy to work out because of the linearity and the simpleness of its controller.Considering the advantage in technology realizing and cost saving,this method is a good reference to application.
We also studied multi-periodical interval control of beam halo-chaos by self-controlling feedback. The beam with k-v distribution is taken as a typical example for our simulation and two form of multi-periodical interval control is discussed .Both the analysis and the simulations indicated that good control results and low cost can be obtained by mul
在综述混沌控制部分，选取了OGY方法、外力反馈法、延迟反馈法、正比于系统变量的脉冲反馈法、自适应控制方法等一些有代表性的控制方法进行了阐述。
束晕—混沌的控制是新一代强流加速器研制的关键问题，随着强流离子束应用前景的日趋广阔而日益成为研究的热点。传统机械限束器因无法解决束晕的再生而收效甚微，因为束晕的形成有着其内在动力学机制—非线性共振以及混沌等。基于此，中国原子能科学院研究员方锦清将混沌控制的理论和方法开创性的运用于束晕—混沌的控制上，提出了控制束晕—混沌的非线性控制策略，即在粒子径向所受束自生场力方程的右边加上非线性控制函数G：
并选取一些非线性函数如等进行了控制的模拟研究，将束晕强度控制在0.1078左右，取得了初步的控制效果。
文献[72]在选取更有效的非线性函数上作了进一步的探索并且取得了进展。该
文选取小波函数构造控制器将束晕强
度控制到零，达到较好的控制结果。
但从实用角度来看，一些简单的非线性函数如的控制效果距实际应用的最低限制仍有一定差距；小波函数法虽可取得较好的控制效果，但控制器形式复杂，控制参数有两个，在工程实现上有一定难度。本文在保证控制效果的前提下，致力于寻求更为简单易行的控制方法。经过大量的理论分析和模拟研究，提出了应用延迟反馈控制束晕—混沌的方法。
本文对延迟反馈法运用于束晕混沌作了简单的理论分析。经分析，确定了延迟反馈控制器的延迟参数为S(一个聚焦磁场通道周期)；而后，计算了系统的最大李雅普诺夫指数。结果表明，系统在延迟反馈控制下最大李雅普诺夫指数由正值转为负值，
说明受控系统是稳定的，混浊得到了有效地抑制。在此基础上，选择a为控制变量，
构造延迟反馈控制器G：
G（S＝glrrms（－S）一 rms（S）（2）
同（l）式一致，将此控制器函数加在粒子径向所受束自生场力方程的右边。利
用多粒子数值模拟程序（PIC）进行了控制试验。模拟结果表明，用同一个控制器和
同一个控制参数即可实现五种不同初始分布情况下的束运一混饨的有效控制。对K一
V分布、水袋分布、抛物分布均能达到束晕强度为零的理想控制效果，对3—sigma
分布、全高斯分布的控制结果也基本能满足强流加速器中 10‘数量级的实际最低限
制。同时研究中还发现，较小的反馈即可实现束流的稳定。
由此可见，该法可达到小波函数法相同的控制效果，而控制器采用线性控制律，
仅有一个调节参数，形式简单，施加的反馈较小，因而应用前景较好。
对该法以多周期间隔形式控制也作了探讨。运用混饨控制理论，分析了两种多周
期间隔控制形式并以K—V分布为例进行模拟试验。分析和实验结果均表明，先连续
控制而后再间隔控制能够保证较好的控制效果和较低的控制代价。比如，对于K—V
束，一开始连续控制50个周期然后再每间隔200周期的情况下仍可将束晕强度控制
为零，较大的突破了文献［71」间隔周期为 16的上限。从而可将控制代价大幅度降
三 低。
In this paper,we give a summary of controlling chaos and a review of the mechanism of beam halo and its nonlinear control.Then delayed self-controlling feedback method is presented to control beam halo-chaos.This method,not only can control beam halo-chaos well ,but also is easy to work out because of the linearity and the simpleness of its controller.Considering the advantage in technology realizing and cost saving,this method is recommended to the development of high intensity accelerator.
In the part giving a summary of controlling chaos ,OGY control and delayed self-controlling feedback and adaptive control etc.are selected as typical examples to give a further description.
The control of beam halo-chaos becomes a critical problem in the development of high intensity accelerator.Efforts to remove the halo by collimation have been largely unsuccessful since the halos almost always regenerate.The mechanisms of halos are complex,such as nonlinear resonances and chaotic behavior etc.Considering this,Professor Fang Jin-Qing who works in China Institute of Atomic Energy pointed out that the theory of chaos control can be used to control beam halos .He presented the method to control halos by using nonlinear functions,which means nonlinear function G is added to the right of ion radial self-edlctric force equation
and some nonlinear function are selected to control beam halos in simulations .In
paper[69],controllerG = -0.15sin(rmax -am)2 was used and the halo intensity was
decreased to 0.1078,the halos are removed partly.
Paper [72] goes further in finding more effctive nonlinear function and wavelet
function was selected to construct the controller
.Under
the controller G the halo intensity was decreased to zero,The result is quite good.
When consider the practicality,we will find that some simple nonlinear function can't
control halo well enough while wavelet controller maybe too complex to work out .What's more.there are two parameters in wavelet controller and it maybe difficult to modify them.Considerable effort is devoted to exploring easier way in this paper.After large amount of simulations and analysis,we presented the way to control halo by delayed self-controlling feedback.
we give a brief theoretic analysis of using delayed self-controlling feedback to control beam halo.S(the length of one period of focusing channel)is selected as the delay parameter of the controller and the maximal lyapunov exponent is calculated. We found that the maximal lyapunov exponent of the system transfer from positive to negative under the control,which means that the controlled system is stable and the chaos is controlled
effectively. On the basis of this,we selected rms as feedback variable and constructed the controller G:
Add this controller to the right of ion radial self-edlctric force equation like equation (1).We simulated the motion of ion beam by using muti-paticle code(partice -in -cell(PIC)code).The results demonstrated that the beam halo of five different initial distribution is eliminated well under the same controller.The halo intensity of k-v distribution,water-bag distribution and parabolic distribution and be reduced to zero.In the case of 3-sigma distribution and full gauss distribution,the result is agree with minimum limitation(10") of halo intensity of factual high intensity accelerator. Also we found that weak feedback is need to control the beam halo.
This method,not only can control beam halo-chaos well as wavelet had done,but also is easy to work out because of the linearity and the simpleness of its controller.Considering the advantage in technology realizing and cost saving,this method is a good reference to application.
We also studied multi-periodical interval control of beam halo-chaos by self-controlling feedback. The beam with k-v distribution is taken as a typical example for our simulation and two form of multi-periodical interval control is discussed .Both the analysis and the simulations indicated that good control results and low cost can be obtained by mul
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