付景超,孫 敬,李鵬松

(東北電力大學(xué) 理學(xué)院,吉林 吉林 132012)

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一類簡(jiǎn)單二次非線性Sprott混沌系統(tǒng)的分析與控制

付景超,孫 敬,李鵬松

(東北電力大學(xué) 理學(xué)院,吉林 吉林 132012)

用自適應(yīng)反推方法考慮一類簡(jiǎn)單非線性Sprott混沌系統(tǒng)的控制問題,得到了平衡點(diǎn)的穩(wěn)定性及Hopf分岔存在性的條件,通過Lyapunov指數(shù)圖及混沌吸引子驗(yàn)證了系統(tǒng)的混沌現(xiàn)象,通過分岔圖分析得到了系統(tǒng)存在復(fù)雜動(dòng)力學(xué)行為,并設(shè)計(jì)自適應(yīng)反推控制器控制混沌系統(tǒng)到給定的平衡點(diǎn).數(shù)值仿真驗(yàn)證了所設(shè)計(jì)控制器的有效性.

Sprott混沌系統(tǒng)；Lyapunov指數(shù)；分岔；自適應(yīng)反推

0 引 言

自Lorenz[1]提出第一個(gè)混沌模型以來,關(guān)于混沌系統(tǒng)的研究得到了迅速發(fā)展[2-21].R?ssler構(gòu)造了數(shù)學(xué)結(jié)構(gòu)比Lorenz系統(tǒng)更簡(jiǎn)單的混沌系統(tǒng)[2],并進(jìn)一步給出了R?ssler原型-4混沌系統(tǒng)[3];Schot[4]發(fā)現(xiàn)了機(jī)械系統(tǒng)中的Jerk混沌系統(tǒng);Sprott[6]給出了具有五項(xiàng)和兩個(gè)非線性二次項(xiàng)或六項(xiàng)和一個(gè)非線性二次項(xiàng)的混沌系統(tǒng),稱為SQ系列混沌吸引子.

本文擴(kuò)展了SQN混沌系統(tǒng)[6],考慮如下系統(tǒng):

(1)

其中:x1,x2,x3是狀態(tài)變量;b,c∈+是常參數(shù).當(dāng)取b=2,c=1時(shí),系統(tǒng)(1)存在一個(gè)混沌吸引子[6].本文研究系統(tǒng)(1)的平衡點(diǎn)及其Hopf分岔的存在性.通過繪制Lyapunov指數(shù)圖和混沌吸引子驗(yàn)證了混沌現(xiàn)象的存在,通過繪制并分析分岔圖發(fā)現(xiàn)系統(tǒng)存在復(fù)雜動(dòng)力學(xué)行為.在系統(tǒng)參數(shù)確定和不確定的條件下,分別設(shè)計(jì)了自適應(yīng)反推控制器穩(wěn)定混沌系統(tǒng),仿真實(shí)驗(yàn)表明了控制器的有效性.

1 平衡點(diǎn)的穩(wěn)定性及Hopf分岔的存在性

系統(tǒng)(1)的平衡點(diǎn)滿足如下方程組:

(2)

顯然,系統(tǒng)(1)存在唯一的平衡點(diǎn)E(-c2/b2,0,c/b).

命題1平衡點(diǎn)E(-c2/b2,0,c/b)是穩(wěn)定的充要條件為b>0,c<0.

證明:系統(tǒng)(1)在E(-c2/b2,0,c/b)的Jacobian矩陣為

J(E)的特征方程為

(3)

令A(yù)1=b,A2=-(2c/b-2),A3=2b,由Routh-Hurwitz定理,當(dāng)滿足A1>0,A3>0,A1A2-A3>0時(shí),也即b>0,c<0時(shí),E(-c2/b2,0,c/b)是穩(wěn)定的.證畢.

命題2當(dāng)c=0時(shí),在平衡點(diǎn)E(-c2/b2,0,c/b)會(huì)發(fā)生Hopf分岔.

證明:令λ=iω(ω>0)是方程(3)的一個(gè)根,則有

(4)

分離實(shí)虛部,有

(5)

2 參數(shù)變化時(shí)系統(tǒng)的復(fù)雜動(dòng)力學(xué)分析

2.1Lyapunov指數(shù)譜及混沌吸引子

混沌吸引子的基本特征之一是系統(tǒng)對(duì)初值極其敏感.初值及其靠近的兩個(gè)軌道隨時(shí)間變化按指數(shù)分離,Lyapunov指數(shù)[21]是描述該現(xiàn)象的定量指標(biāo).若系統(tǒng)Lyapunov指數(shù)是正數(shù),則相鄰軌道分離,系統(tǒng)處于混沌狀態(tài)；若Lyapunov指數(shù)是負(fù)數(shù),則表明將收斂到不動(dòng)點(diǎn)或周期軌；若Lyapunov指數(shù)是零,則系統(tǒng)處于臨界狀態(tài),會(huì)有分岔發(fā)生.對(duì)三維自治混沌系統(tǒng),Lyapunov指數(shù)譜為(+,0,-).

當(dāng)b=2,c=1時(shí),系統(tǒng)(1)Lyapunov指數(shù)隨時(shí)間的變化曲線如圖1所示.圖2是系統(tǒng)(1)的混沌吸引子.由圖1可見,系統(tǒng)3個(gè)Lyapunov指數(shù)一個(gè)大于0,一個(gè)等于0,一個(gè)小于0.經(jīng)計(jì)算LE1=0.089 082,LE2=0,LE3=-2.095.可見系統(tǒng)(1)是混沌的.

2.2改變參數(shù)b的系統(tǒng)動(dòng)力學(xué)行為

圖3為系統(tǒng)(1)的狀態(tài)x1當(dāng)參數(shù)c=1,b∈[1.8,2.5]時(shí)的分岔圖.由圖3可見,系統(tǒng)軌道從混沌最終經(jīng)逆倍周期分岔到達(dá)周期狀態(tài).圖4是圖3的放大圖,由圖4可見,系統(tǒng)除混沌外還有更復(fù)雜的動(dòng)力學(xué)行為.

圖1 當(dāng)b=2,c=1時(shí)系統(tǒng)(1)的Lyapunov指數(shù)Fig.1 Lyapunov exponents of system (1)for b=2,c=1

圖2 當(dāng)b=2,c=1時(shí)系統(tǒng)(1)的混沌吸引子Fig.2 Chaotic attractor of system (1)for b=2,c=1

圖3 當(dāng)c=1,b∈[1.8,2.5]時(shí)系統(tǒng)(1)的分岔圖Fig.3 Bifurcation diagram of system (1)for c=1,b∈[1.8,2.5]

圖4 當(dāng)c=1,b∈[2,2.2]時(shí)系統(tǒng)(1)的分岔圖Fig.4 Bifurcation diagram of system (1)for c=1,b∈[2,2.2]

2.3改變參數(shù)c的系統(tǒng)動(dòng)力學(xué)行為

圖5為系統(tǒng)(1)的狀態(tài)x1當(dāng)參數(shù)b=2,c∈[0.5,1,3]時(shí)的分岔圖.由圖5可見,系統(tǒng)經(jīng)由倍周期分岔最終到達(dá)混沌狀態(tài).圖6是對(duì)圖5中倍周期分岔過程的放大圖.由上述結(jié)果可見,系統(tǒng)隨著參數(shù)的變化存在復(fù)雜的倍周期分岔及混沌等動(dòng)力學(xué)行為.

圖5 當(dāng)b=2,c∈[0.5,1.3]時(shí)系統(tǒng)(1)的分岔圖Fig.5 Bifurcation diagram of system (1)for b=2,c∈[0.5,1.3]

圖6 當(dāng)b=2,c∈[0.75,0.98]時(shí)系統(tǒng)(1)的分岔圖Fig.6 Bifurcation diagram of system (1)for b=2,c∈[0.75,0.98]

3 自適應(yīng)反推控制器設(shè)計(jì)

3.1參數(shù)已知的自適應(yīng)反推控制器

當(dāng)參數(shù)b=2,c=1時(shí),對(duì)系統(tǒng)(1)施加控制,可得下列受控系統(tǒng)方程:

(6)

設(shè)α1=0,α2=c1e1,α3=c2e1+c3e2,定義誤差信號(hào)為e1=x1-α1,e2=x2-α2,e3=x3-α3,把e1,e2,e3代入式(6),得到誤差系統(tǒng)如下:

(7)

(8)

(9)

(10)

定理1對(duì)于參數(shù)已知的系統(tǒng),當(dāng)c1>0,k>0時(shí),控制器(8)-(10)能將系統(tǒng)(6)控制到給定的平衡點(diǎn)(α1,α2,α3),這里:α1=0;α2=c1e1;α3=c2e1+c3e2.

3.2參數(shù)不確定的自適應(yīng)反推控制器

當(dāng)參數(shù)b,c未知時(shí),系統(tǒng)(1)的受控系統(tǒng)為

(11)

設(shè)α1=0,α2=c1e1,α3=c2e1+c3e2,定義誤差信號(hào)為e1=x1-α1,e2=x2-α2,e3=x3-α3,把e1,e2,e3代入式(11),得誤差系統(tǒng)如下:

(12)

選擇如下Lyapunov函數(shù):

這里:γ1,γ2>0是自適應(yīng)增益系數(shù);b1,d1分別是b,c的參數(shù)估計(jì).則有

整理,得

令

(17)

定理2對(duì)參數(shù)未知系統(tǒng),當(dāng)c1>0,k1>0,k2>0且式(16),(17)成立時(shí),控制器(13)-(15)能夠被控制到點(diǎn)(α1,α2,α3),這里:α1=0;α2=c1e1;α3=c2e1+c3e2.

3.3數(shù)值仿真

3.3.1 參數(shù)已知的控制系統(tǒng)仿真 參數(shù)b=2,c=1時(shí)系統(tǒng)是混沌的,施加控制(8)-(10),取c1=0.5,c2=0.6,c3=0.8,k=0.4,此時(shí)受控系統(tǒng)的波形圖和相平面圖分別如圖7和圖8所示.由圖7和圖8可見,受控系統(tǒng)漸近穩(wěn)定到原點(diǎn)((α1,α2,α3)=(0,0,0)).

圖7 參數(shù)b=2,c=1時(shí)受控系統(tǒng)(6)的波形圖Fig.7 Waveform diagram of system (6)for b=2,c=1

圖8 參數(shù)b=2,c=1時(shí)受控系統(tǒng)(6)的相平面圖Fig.8 Phase diagram of system (6)for b=2,c=1

3.3.2 參數(shù)未知的控制系統(tǒng)仿真 設(shè)參數(shù)b的估計(jì)值是b1,參數(shù)c的估計(jì)值是d1.對(duì)系統(tǒng)(11)施加控制器(13)-(15),取c1=0.5,c2=0.8,c3=0.7,k1=0.5,k2=0.5,γ1=2;γ2=1.8,則受控系統(tǒng)的波形圖如圖9所示.由圖9可見,受控系統(tǒng)(11)漸近穩(wěn)定到原點(diǎn)((α1,α2,α3)=(0,0,0)).圖10為參數(shù)b,c的估計(jì)值b1,d1隨時(shí)間的變化曲線.

圖9 參數(shù)未知時(shí)受控系統(tǒng)(11)的波形圖Fig.9 Waveform of system (11)with unknown parameters

圖10 參數(shù)未知時(shí)受控系統(tǒng)(11)的參數(shù)估計(jì)曲線Fig.10 Parameter estimation curves of system (11)with unknown parameters

綜上,本文研究了一類具有簡(jiǎn)單二次非線性項(xiàng)Sprott混沌系統(tǒng)的復(fù)雜行為及混沌控制問題,給出了該系統(tǒng)平衡點(diǎn)的穩(wěn)定性及其Hopf分岔的存在性,并通過給出系統(tǒng)Lyapunov指數(shù)圖及混沌吸引子驗(yàn)證了系統(tǒng)存在混沌,通過繪制和分析系統(tǒng)隨參數(shù)變化的分岔圖發(fā)現(xiàn)系統(tǒng)存在復(fù)雜動(dòng)力學(xué)行為.最后,設(shè)計(jì)自適應(yīng)反推控制器將混沌系統(tǒng)控制到給定的平衡點(diǎn)上.數(shù)值仿真結(jié)果驗(yàn)證了所設(shè)計(jì)控制器的有效性.

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(責(zé)任編輯：趙立芹)

AnalysisandControlofaClassofSprottChaoticSystemwithSimpleQuadraticNonlinearities

FU Jingchao,SUN Jing,LI Pengsong

(CollegeofScience,NortheastDianliUniversity,Jilin132012,JilinProvince,China)

The control problems of a class of nonlinear Sprott simple chaotic systems were solved via an adaptive backstepping method.Firstly,the conditions of stability of the equilibria point and the existence of Hopf bifurcation were obtained.Secondly,chaotic phenomenon of the system was verified by means of presenting Lyapunov exponent and chaotic attractor,and complex dynamic behaviors were found via analyzing bifurcation diagram.Finally,an adaptive backstepping controller was designed to control chaotic system to a given equilibrium point.Numerical simulation has shown the effectiveness of the controller designed.

Sprott chaotic system;Lyapunov exponent;bifurcation;adaptive backstepping

10.13413/j.cnki.jdxblxb.2015.03.09

2014-09-03.

付景超(1977—),男,漢族,博士,副教授,從事非線性混沌動(dòng)力系統(tǒng)分析與控制的研究,E-mail:neufujingchao@163.com.

國(guó)家自然科學(xué)基金青年基金(批準(zhǔn)號(hào)：11201057)、吉林省自然科學(xué)基金(批準(zhǔn)號(hào)：20130101065JC)和吉林省教育廳“十二五”科學(xué)技術(shù)研究項(xiàng)目(批準(zhǔn)號(hào)：2013429).

O231.2

：A

：1671-5489(2015)03-0395-06