| 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 equationand some nonlinear function are selected to control beam halos in simulations .Inpaper[69],controllerG = -0.15sin(rmax -am)2 was used and the halo intensity wasdecreased to 0.1078,the halos are removed partly.Paper [72] goes further in finding more effctive nonlinear function and waveletfunction was selected to construct the controller.Underthe 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'tcontrol 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 controlledeffectively. 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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