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No 8-2 Finite time 1

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Finitetimesynchronizationofchaoticsystems

ShihuaLi*,Yu-PingTian

DepartmentofAutomaticControl,SoutheastUniversity,Nanjing210096,PRChina

Accepted29April2002

Abstract

Usingfinitetimecontroltechniques,continuousstatefeedbackcontrollawsaredevelopedtosolvethesynchro-

nizationproblemoftwochaoticsystems.Wedemonstratethatthesetwochaoticsystemscanbesynchronizedinfinite

time.ExamplesofDuffingsystems,Lorenzsystemsarepresentedtoverifyourmethod.

Ó2002ElsevierScienceLtd.Allrightsreserved.

1.Introduction

Chaossynchronizationhasbeenofbroadinterestinrecentyears[1–7].In[1–5],activecontrolmethodswereusedto

solvethesynchronizationproblemofchaoticsystemssuchasRosslersystems,Lorenzsystems,Duffingsystems,Chen

systems.Themainideaofsynchronizationistousethestatesofthemastersystemtocontroltheslavesystemsothat

thestatesoftheslavesystemfollowsthestatesofthemastersystemasymptotically.Themastersystemandtheslave

systemmayhaveidenticalorcompletelydifferentstructures.Theconvergenceofthesynchronizationprocedurein[1–5]

isexponentialwithinfinitesettlingtime.

Toachievefasterconvergenceincontrolsystems,aneffectivemethodisusingfinitetimecontroltechniques.Finite

timestabilitymeanstheoptimalityinsettlingtime[8].Moreover,thefinitetimecontroltechniqueshavedemonstrated

betterrobustnessanddisturbancerejectionproperties[9].Theproblemwewanttodiscusshereisthatcanthesechaotic

systemsmentionedabovebesynchronizedinfinitesettlingtime?Theansweriscertain,whichwewilldemonstrateinthe

followingsections.

So,ourgoalinthispaperistodevelopfeedbackcontrollawstomakethesynchronizationprocedureconvergingin

finitetime,i.e.,thestatesoftheslavesystemfollowthestatesofthemastersysteminfinitetime.Discontinuousoropen-

loopcontroltechniquesfromfinitetimecontroltheorymaybeavailabletosolvethisproblem.However,considering

theconvenienceofcontinuouscontrollersinpracticalimplementationandtherobustnessofclosed-loopfeedbacksto

systemuncertainties,thefinitetimecontroltechniquewediscussedhereisbasedoncontinuousstatefeedbacks[8,9].

2.Finitetimesynchronizationofsecondorderchaoticsystems

Someresultsoffinitetimecontroltechniquesusingcontinuousfeedbacksaregivenin[8,9],andarerephrasedbythe

followingtwolemmas.

Lemma1([8,9]).Thesystem

dx

dt¼uð1Þ

*Tel.:+86-25-3794168;fax:+86-25-7712719.E-mailaddress:lsh@seu.edu.cn(S.Li).

0960-0779/03/$-seefrontmatterÓ2002ElsevierScienceLtd.Allrightsreserved.PII:S0960-0779(02)00100-

5Chaos,SolitonsandFractals15(2003)

303–310www.elsevier.com/locate/chaoscanbegloballystabilizedinfinitetimeunderthefeedbackcontrollaw

u¼ÀkÁsignðxÞjxjað2Þ

withk>0,a2ð0;1Þ.

Thesolutiontrajectoryof(1)and(2)is

xðtÞ1Àa¼xð0Þ1ÀaÀð1ÀaÞkt;xð0ÞP0

½ÀxðtÞ󰀉1Àa¼½Àxð0Þ󰀉1ÀaÀð1ÀaÞkt;xð0Þ<0󰀁

ð3Þ

ForanyinitialvalueofstatexðtÞatt¼0,i.e.,xð0Þ,itiseasilycomputedthatthesolutiontrajectoryof(1)and(2)will

reachx¼0infinitetimetsdeterminedbyts¼jxð0Þj1Àa=ð1ÀaÞk.

Lemma2([9]).Thesystem

dxðtÞ

dt¼vðtÞ;

dvðtÞ

dt¼uðtÞð4Þ

canbegloballystabilizedinfinitetimeunderthefeedbackcontrollaw

u¼Àk1signðxÞjxja1Àk2signðvÞjvja2ð5Þ

withk1,k2>0,a12ð0;1Þ,a2¼2a1=ð1þa1Þ.

Now,letusfirstconsiderthesynchronizationproblemofsecondorderchaoticsystemswhichcanbewritteninthe

followingform[3]

dx1ðtÞ

dt¼v1ðtÞ;

dv1ðtÞ

dt¼f1x1ðtÞ;v1ðtÞ;tðÞð6Þ

wheref1ðx1ðtÞ;v1ðtÞ;tÞisanonlinearfunction.Thissystemiscalledthemastersystem.Theequationsthatdescribethe

slavesystemare

dx2ðtÞ

dt¼v2ðtÞþu1ðtÞ;

dv2ðtÞ

dt¼f2x2ðtÞ;v2ðtÞ;tðÞþu2ðtÞð7Þ

wheref2ðx2ðtÞ;v2ðtÞ;tÞisanonlinearfunctionandu1ðtÞ,u2ðtÞarethecontrolsignalstobedesigned.

Thedifferencebetweenthetwodistancesandthetwovelocitiesisdescribedby

x3¼x2Àx1;

v3¼v2Àv1:ð8Þ

Thus,weget

dx3ðtÞ

dt¼v3ðtÞþu1ðtÞ;

dv3ðtÞ

dt¼f2x2ðtÞ;v2ðtÞ;tðÞÀf1x1ðtÞ;v1ðtÞ;tðÞþu2ðtÞð9Þ

Severalpossiblechoicesforthecontrolsignalsu1ðtÞ,u2ðtÞcanbetakentosynchronizetheslavesystemtothemaster

system.304S.Li,Y.-P.Tian/Chaos,SolitonsandFractals15(2003)303–3102.1.Controlstrategy1

Onepossiblechoiceisgivenby

u1ðtÞ¼Àv3ðtÞÀk1signðx3Þjx3ja;

u2ðtÞ¼Àf2x2ðtÞ;v2ðtÞ;tðÞþf1x1ðtÞ;v1ðtÞ;tðÞ

Àk2signðv3Þjv3jb;k1;k2>0;a;b;2ð0;1Þ:ð10Þ

Bothsignalsu1ðtÞ,u2ðtÞareavailableinthesynchronizationprocedure.With(10),(9)canberewritteninthefollowingnotation:

dx3ðtÞ

dt¼Àk1signðx3Þjx3ja;

dv3ðtÞ

dt¼Àk2signðv3Þjv3jb;k1;k2>0;a;b2ð0;1Þ:ð11Þ

DuetoLemma1,(11)impliesthatthetwochaoticsystemsaresynchronizedwithcontinuousstatefeedbacksinfinite

time.

2.2.Controlstrategy2

Anotherpossiblechoiceisgivenby

u1ðtÞ¼0;

u2ðtÞ¼Àf2x2ðtÞ;v2ðtÞ;tðÞþf1ðx1ðtÞ;v1ðtÞ;tÞÀk1signðx3Þjx3ja1

Àk2signðv3Þjv3ja2;k1;k2>0;a12ð0;1Þ;a2¼2a11þa1:ð12Þ

Here,onlycontrolsignalu2ðtÞismadeavailablewhileu1ðtÞ󰀋0.

With(12),thesystem(9)canberewritteninthefollowingnotation:

dx3ðtÞ

dt¼v3;

dv3ðtÞ

dt¼Àk1signðx3Þjx3ja1Àk2signðv3Þjv3ja2;k1;k2>0;a12ð0;1Þ;a2¼2a11þa1:ð13Þ

AccordingtoLemma2,thedifferenceswillconvergetozeroinfinitetime.

Example1(Duffingsystems).Asanexample,letusconsiderthesynchronizationoftwoDuffingsystems[3].The

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新标准大学英语3(第二版)(综合教程)参考答案

新标准大学英语3(第二版)(综合教程)参考答案

Unit 1

Active reading 1

First reading Task 1

C

First reading Task 2

BCCDCA

Reading in detail: Vocabulary exercises Task 1

1.attendance

2.ambitious

3.productive

4.resistance

5.script

6.acceptance

Reading in detail: Vocabulary exercises Task 2

1.mortgage

2.deck

3.surf

4.(1)coastal (2) defy

5.clamped

6.criticized

7.hauled

8.preceding

Reading in detail: Vocabulary exercises Task 3

BABBB BABB

Active reading 2

First reading Task

Reading in detail: Vocabulary exercises Task 1

1. (1)elapsed (2) cemetery 2. (1)rear (2) crammed

3triple

4. (1)budding (2) biography

5.finite

6.dwellers

7.invest

8.skip

9.inevitable

Reading in detail: Vocabulary exercises Task 2

ABAAA BAAAB BAABB BA

Guided writing

Writing Task 1

1-b,2-c,3-a

Writing Task 2

1-c,2-a,3-b

Unit test

Part I: Vocabulary and Structure

Section A

1.ingenious

综合英语教程第六册课后翻译参考答案

综合英语教程第六册课后翻译参考答案

综英第六册课后翻译参考答案

Unit 1

1.这项计划为智力迟钝者提供长期的照顾。

The program offers long-term care for the mentally retarded.

2.他有一台又粗笨有庞大的旧电脑，速度慢，使用麻烦。

He’s got a cumbersome, bulky, old computer—it’s slow and

complicated to use.

3. 他沿着房间后部慢慢移动，尽量不引起别人注意。

He tried not to look conspicuous and moved slowly along the

back of the room.

4. 安装一个不同的计算机系统将会导致巨大的变化。

It would cause a tremendous upheaval to install a different

computer system.

5. 她再次与金牌失之交臂。

The gold medal continues to elude her.

6. 你真是个傻瓜，竟然没抓住这么好的机会。

You’d be a fool not to embrace an opportunity as good as that.

7.她的薪水将猛增10%。

Her salary will go up by a hefty 10%.

8.我草草的留了张纸条给希拉里，塞在门缝底下。

I scrawled a quick note to Hilary and put it under her door. 9.她的办公室视野极佳。

There’s a smashing view from her office.

10 去那儿旅行很愉快，只是旅馆有些脏兮兮的。

The trip out there was swell, but the hotel was a bit crummy.

HFSS-IE反射面仿真

HFSS-IE反射面仿真

2013/11/28

1反射面天线设计新方法

向胜昭

ANSYS高级应用工程师天线设计指标

①天线扫描方式

②天线工作频率

③天线增益

④天线尺寸限制

⑤天线重量要求

⑥天线波束覆盖

⑦天线副瓣电平

⑧天线极化形式

⑨天线驻波

……决定天

线采用

什么样

的形式

具体

实现反射

面？

反射面天线应用分类反射面天线解决方案

•“高增益、低副瓣、相控阵”反射面天线

•难点：

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HFIE （高频积

分方程法）混合全波分析PO混合……

Dish antennaSingle Cell-ADKHigh Accuracy

Unit Cell-Floquet

System-cosimulationFinite Array-HPC……

High complexity

------Variety of methods

High complexity

•解决策略：

–HFSS

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–HFSS-IE

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大学英语综合教程3第三版答案

大学英语综合教程3第三版答案

大学英语综合教程3第三版答案

【篇一：新标准大学英语综合教程3答案(全版)】

>unit1

active reading(1)

4. b c c d c a

5.productive attendance resistance ambitious acceptance

script impressive

6.attendance ambitious productive impressive resistance

script acceptance

7.mortgage deck surf coastal;defy lengthy

8.b a b b b a b b

active reading(2)

4.triple cemetery rear biography cram budding finite

elapse

5.elapsed;cemetery rear;crammed triple budding;biography

finite

6.a b a a b b a a

7.a b b a a b b b a

language in use

6.(1)我们都觉得在校时间不多了，以后再也不会有这样的学习机会了，所以都下定决心不再虚度光阴。当然，下一年四五月份的期末考试最为重要。我们谁都不想考全班倒数第一，那也太丢人了，因此同学们之间的竞争压力特别大。以前每天下午5点以后，图书馆就空无

一人了，现在却要等到天快亮时才会有空座，小伙子们熬夜熬出了眼袋，他们脸色苍白，睡眼惺忪，却很自豪，好像这些都是表彰他们勤奋好学的奖章。

（2）明天行吗？明天只是个谎言；根本就没有什么明天，只有一张我们常常无法兑现的期票。明天甚至压根儿就不存在。你早上醒来时又是另一个今天了，同样的规则又可以全部套用。明天只是现在的另一种说法，是一块空地，除非我们开始在那里播种，否则它永远都是空地。你的时间会流逝（时间就在我们说话的当下滴答滴答地走着，每分钟顺时针走60秒，如果你不能很好地利用它，它就会走得更快些），而你没有取得任何成就来证明它的存在，唯独留下遗憾，留下一面后视镜，上面写满了“本可以做”“本应该做”“本来会做”的事情。 7.(1)students differ about whether they should have their