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  山东大学学报(工学版)  2017, Vol. 47 Issue (4): 31-36  DOI: 10.6040/j.issn.1672-3961.0.2016.327
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引用本文 

毛北行, 程春蕊. 分数阶Victor-Carmen混沌系统的自适应滑模控制[J]. 山东大学学报(工学版), 2017, 47(4): 31-36. DOI: 10.6040/j.issn.1672-3961.0.2016.327.
MAO Beixing, CHENG Chunrui. Self-adaptive sliding mode control of fractional-order Victor-Carmen chaotic systems[J]. Journal of Shandong University (Engineering Science), 2017, 47(4): 31-36. DOI: 10.6040/j.issn.1672-3961.0.2016.327.

基金项目

国家自然科学青年基金资助项目(NSFC11501525);河南省科技厅软科学资助项目(142400411192);河南省高等学校青年骨干教师资助计划项目(2013GGJS-142);河南省高等学校重点科研资助项目(15B110011)

作者简介

毛北行(1976—), 男, 河南洛阳人, 副教授, 硕士, 主要研究方向为复杂网络与混沌同步.E-mail:bxmao329@163.com

文章历史

收稿日期:2016-10-18
网络出版时间:2017-05-05 21:54:59
分数阶Victor-Carmen混沌系统的自适应滑模控制
毛北行, 程春蕊     
郑州航空工业管理学院理学院, 河南 郑州 450015
摘要:根据分数阶微积分的相关理论利用自适应滑模控制方法研究分数阶Victor-Carmen混沌系统的滑模同步控制问题, 设计分数阶滑模函数并给出控制器的构造, 利用Lyapunov稳定性理论给出严格的数学证明, 得到系统取得滑模同步的两个充分性条件。研究结果表明:选取适当的控制律以及滑模面下, 分数阶Victor-Carmen系统取得混沌同步。数值算例表明该方法有效。
关键词分数阶    Victor-Carmen系统    滑模    混沌同步    
Self-adaptive sliding mode control of fractional-order Victor-Carmen chaotic systems
MAO Beixing, CHENG Chunrui     
College of Science, Zhengzhou University of Aeronautics, Zhengzhou 450015, Henan, China
Abstract: The problem of sliding mode synchronization of fractional-order Victor-Carmen systems was studied using self-adaptive sliding mode control approach based on fractional-order calculus theory. The fractional-order slding mode function was designed, the controllers and the strict proof in mathematics using Lyapunov stability theory were given. Two sufficient conditions were arrived for the fractional order systems getting sliding model synchronization. The research conclusion illustrated that fractional-order multi-scroll systems was sliding mode chaos synchronization under proper controllers and sliding mode surface.The numerical simulations demonsrrated the effectiveness of the proposed method.
Key words: fractional-order    Victor-Carmen systems    sliding mode    chaos synchronization    
0 引言

分数阶微分对一些实际系统的描述更为准确, 因而备受关注。随着混沌控制的发展, 分数阶混沌系统的控制与同步问题引起了广泛关注[1-11], 文献[12]研究了不确定分数阶混沌系统的滑模投影同步问题, 主从系统实现了投影同步。文献[13]研究了一类不确定分数阶混沌系统的自适应滑模混沌同步问题, 能够使驱动系统与响应系统达到同步。文献[14]基于主动滑模控制方法实现了分数阶混沌系统的同步控制, 文献[15]研究了分数阶Rayleigh-Duffling-like系统的自适应追踪广义投影同步问题。Victor Grigoras和Carmen Grigoras通过在线性震荡环节基础上引入非线性动态得到了一个新型的三维混沌系统[16], 文献[17]研究了一类不确定混沌系统的自适应滑模终端控制问题。在上述研究的基础上, 本研究分析了分数阶Victor-Carmen混沌系统的滑模控制问题, 根据分数阶微积分的相关理论给出系统取得同步的充分性条件, 结果表明:选取适当的控制律以及滑模面下, 分数阶Victor-Carmen系统取得混沌同步。

1 主要结果

定义1[18]    Caputo分数阶导数定义为:

$_c{\rm{D}}_{{t_0},{\rm{ }}t}^\alpha = {\rm{D}}_{{t_0},{\rm{ }}t}^{ - (n - \alpha )}\frac{{{{\rm{d}}^n}}}{{{\rm{d}}{t^n}}}x\left( t \right) = \frac{1}{{\Gamma \left( {n - \alpha } \right)}}\int_{{t_0}}^t {} {\left( {t - \tau } \right)^{n - \alpha - 1}}{x^{(n)}}(\tau ){\rm{d}}\tau ,{\rm{ }}n - 1 < \alpha < n \in {Z^ + },$

式中:cDt0, tα表示Caputo定义下, α阶分数阶导数, 设计如下一类分数阶Victor-Carmen混沌系统作为主系统:

$\left\{ \begin{array}{l} {\rm{D}}_t^q{x_1} = - {x_1} - \alpha {x_2}{x_3},\\ {\rm{D}}_t^q{x_2} = - {x_2} + a{x_3} - \beta {x_1}{x_3},\\ {\rm{D}}_t^q{x_3} = - b{x_1} - a{x_2} + {x_3} + \gamma {x_1}{x_2}, \end{array} \right.$ (1)

式中:x1, x2, x3R3为系统的状态变量; a, b, α, β, γ为系统参数; Dtq表示q阶微分算子, 当α=50, β=20, γ=4.1, a=5, b=9, q=0.873时出现混沌吸引子, 其对应的从系统为:

$\left\{ \begin{array}{l} {\rm{D}}_t^q{y_1} = - {y_1} - \alpha {y_2}{y_3} + \Delta {f_1}\left( y \right) + {d_1}\left( t \right) + {u_1},\\ {\rm{D}}_t^q{y_2} = - {y_2} + a{y_3} - \beta {y_1}{y_3} + \Delta {f_2}\left( y \right) + {d_2}\left( t \right) + {u_2},\\ {\rm{D}}_t^q{y_3} = - b{y_1} - a{y_2} + {y_3} + \gamma {y_1}{y_2} + \Delta {f_3}\left( y \right) + {d_3}\left( t \right) + {u_3}。\end{array} \right.$ (2)

假设1    设不确定项Δfi(y)和外部扰动di(t)有界, 即存在mi, ni>0, 其中mi, ni为未知正常数, 使得:

$\begin{array}{*{20}{c}} {|\Delta {f_i}\left( y \right)| < {m_i},}&{\left| {{d_i}\left( t \right)} \right| < {n_i}} \end{array}。$

假设2    mi, ni(i=1, 2, 3) 未知。

定义系统误差:e1=y1-x1, e2=y2-x2, e3=y3-x3, 很容易得到误差方程

$\left\{ \begin{array}{l} {\rm{D}}_t^q{e_1} = - {e_1} - \alpha {y_2}{y_3} + \alpha {x_2}{x_3} + \Delta {f_1}\left( y \right) + {d_1}\left( t \right) + {u_1}\left( t \right),\\ {\rm{D}}_t^q{e_2} = - {e_2} + a{y_3} - \beta {y_1}{y_3} + \beta {x_1}{x_3} + \Delta {f_2}\left( y \right) + {d_2}\left( t \right) + {u_2}\left( t \right),\\ {\rm{D}}_t^q{e_3} = - b{e_1} - a{e_2} + {e_3} + \gamma {y_1}{y_2} - \gamma {x_1}{x_2} + \Delta {f_3}\left( y \right) + {d_3}\left( t \right) + {u_3}\left( t \right)。\end{array} \right.$ (3)

引理1[19]    假设存在连续正定函数V(t)满足微分不等式

$\dot V\left( t \right) \le - p{V^n}\left( t \right),\forall t \ge {t_0},{\rm{ }}V({t_0}) \ge 0,$

式中:pη是两个正常数, p>0, 0<η<1。则对于任意给定的t0, V(t)满足:

$\begin{array}{l} {V^{1 - \eta }}\left( t \right) \le {V^{1 - \eta }}({t_0}) - p\left( {1 - \eta } \right)(t - {t_0}),{\rm{ }}{t_0} \le t \le T,\\ \quad \quad \quad \quad \quad \quad V\left( t \right) \equiv 0,{\rm{ }}t \ge T, \end{array}$

式中:$T = {t_0} + \frac{{{V^{1 - \eta }}({t_0})}}{{p\left( {1 - \eta } \right)}}$

引理2[20]    设有实数a1, a2, …, an, 0<β<2, 则有:

$|{a_1}{|^\beta } + |{a_2}{|^\beta } + \cdots + |{a_n}{|^\beta } \ge {({a_1}^2 + {a_2}^2 + \cdots + a_n^2)^{\beta 2}}。$

针对误差系统(3) 设计非奇异终端滑模面

${s_i}\left( t \right) = {\rm{D}}_t^{q - 1}{e_i}\left( t \right) + \lambda \int_0^t {{{\left| {D_t^{q - 1}{e_i}\left( \tau \right)} \right|}^r}} {\rm{sgn}}\left( {{\rm{D}}_t^{q - 1}{e_i}\left( \tau \right)} \right){\rm{d}}\tau 。$ (4)

定理1    误差系统(3) 在非奇异滑模面(4) 上, 系统的轨迹在有限时间ts内到达平衡点, 其中

${t_s} \le \frac{{{{\left( {\sum\limits_{i = 1}^3 {{{\left( {{\rm{D}}_t^{q - 1}{e_i}\left( 0 \right)} \right)}^2}} } \right)}^{\left( {1 - r} \right)/2}}}}{{\left( {1 - r} \right)\mu }},{\rm{ }}\mu = {\rm{min}}\left\{ {{\lambda _1},{\rm{ }}{\lambda _2},{\rm{ }}{\lambda _3}} \right\}。$ (5)

证明    误差系统满足滑模面方程

${s_i}\left( t \right) = 0,{{\dot s}_i}\left( t \right) = 0,$

于是有

${\rm{D}}_t^q{e_i}\left( t \right) = - {\lambda _i}|{\rm{D}}_t^{q - 1}{e_i}\left( t \right){|^r}{\rm{sgn}}\left( {{\rm{D}}_t^{q - 1}{e_i}\left( t \right)} \right)。$

选取Lyapunov函数

$V\left( t \right) = \frac{1}{2}\sum\limits_{i = 1}^3 {{{\left( {{\rm{D}}_t^{q - 1}{e_i}\left( t \right)} \right)}^2},} $

$\dot V = \sum\limits_{i = 1}^3 {{\rm{D}}_t^{q - 1}{e_i}\cdot{\rm{D}}_t^q{e_i}} = - \sum\limits_{i = 1}^3 { - {\lambda _i}|{\rm{D}}_t^{q - 1}{e_i}\left( t \right){|^{r + 1}}} \le - \mu \sum\limits_{i = 1}^3 {|{\rm{D}}_t^{q - 1}{e_i}\left( t \right){|^{1 + r}}} ,$

由引理2得:

$\dot{V}\le -{{2}^{\frac{1+r}{2}}}\mu {{\left( 1/2\sum\limits_{i=1}^{3}{{{\left( \text{D}_{t}^{q-1}{{e}_{i}}\left( t \right) \right)}^{2}}} \right)}^{\frac{1+r}{2}}}=-{{2}^{\frac{1+r}{2}}}\mu {{V}^{\frac{1+r}{2}}}。$

又由引理1易得, 误差轨迹会在有限时间ts内达到平衡点, ${{t}_{s}}\le \frac{{{\left( \sum\limits_{i=1}^{3}{{{\left( \text{D}_{t}^{q-1}{{e}_{i}}\left( 0 \right) \right)}^{2}}} \right)}^{\left( 1-r \right)/2}}}{\left( 1-r \right)\mu },\text{ }\mu =\text{min}\left\{ {{\lambda }_{1}},\text{ }{{\lambda }_{2}},\text{ }{{\lambda }_{3}} \right\}$

在假设2下设计自适应控制器, 使得系统的所有状态到达并永远保持在滑模面上, 设计控制律

$\begin{align} &\quad \quad {{u}_{1}}={{e}_{1}}+\alpha {{y}_{2}}{{y}_{3}}-\alpha {{x}_{2}}{{x}_{3}}-{{\lambda }_{1}}|\text{D}_{t}^{q-1}{{e}_{1}}{{|}^{r}}\text{sgn}(\text{D}_{t}^{q-1}~{{e}_{1}})-({{{\hat{m}}}_{1}}+{{{\hat{n}}}_{1}}+{{k}_{1}})\text{sgn}({{s}_{1}}), \\ &\quad {{u}_{2}}={{e}_{2}}-a{{y}_{3}}+\beta {{y}_{1}}{{y}_{3}}-\beta {{x}_{1}}{{x}_{3}}-{{\lambda }_{2}}|\text{D}_{t}^{q-1}{{e}_{2}}{{|}^{r}}\text{sgn}(D_{t}^{q-1}{{e}_{2}})-({{{\hat{m}}}_{2}}+{{{\hat{n}}}_{2}}+{{k}_{2}})\text{sgn}({{s}_{2}}), \\ &{{u}_{3}}=b{{e}_{1}}+a{{e}_{2}}-{{e}_{3}}-\gamma {{y}_{1}}{{y}_{2}}+\gamma {{x}_{1}}{{x}_{2}}-{{\lambda }_{3}}|\text{D}_{t}^{q-1}{{e}_{3}}{{|}^{r}}\text{sgn}(D_{t}^{q-1}{{e}_{3}})-({{{\hat{m}}}_{3}}+{{{\hat{n}}}_{3}}+{{k}_{3}})\text{sgn}({{s}_{3}}), \\ \end{align}$

和自适应律

$\left\{ \begin{align} &{{{\dot{\hat{m}}}}_{i}}=|{{s}_{i}}|,~{{{\hat{m}}}_{i}}\left( 0 \right)={{{\hat{m}}}_{i0}}, \\ &{{{\dot{\hat{n}}}}_{i}}=|{{s}_{i}}|,~{{{\hat{n}}}_{i}}\left( 0 \right)={{{\hat{n}}}_{i0}}, \\ \end{align} \right.$ (7)

式中:${{{\hat{m}}}_{i}},{{{\hat{n}}}_{i}}$分别为mi, ni的估计值; ki>0, i=1, 2, 3。

定理2    在控制器(6) 和自适应律(7) 的作用下, 误差系统(3) 的状态轨迹能达到滑模面。

证明    选择Lyapunov函数$V\left( t \right)=\frac{1}{2}\sum\limits_{i=1}^{3}{(s_{i}^{2}+{{({{{\hat{m}}}_{i}}-{{m}_{i}})}^{2}}+{{({{{\hat{n}}}_{i}}-{{n}_{i}})}^{2}})}$, 求导得:

$\begin{align} &\dot{V}=\sum\limits_{i=1}^{3}{\left\{ {{s}_{i}}\left[ \text{D}_{t}^{q}{{e}_{i}}\left( t \right)+{{\lambda }_{i}}|\text{D}_{t}^{q-1}{{e}_{i}}\left( t \right){{|}^{r}}\text{sgn}(\text{D}_{t}^{q-1}{{e}_{i}}) \right]+\left[ ({{{\hat{m}}}_{i}}-{{m}_{i}})|{{s}_{i}}|+({{{\hat{n}}}_{i}}-{{n}_{i}})|{{s}_{i}}| \right] \right\}}= \\ &\quad \quad {{s}_{1}}\left[ -{{e}_{1}}-\alpha {{y}_{2}}{{y}_{3}}+\alpha {{x}_{2}}{{x}_{3}}+\Delta {{f}_{1}}\left( y \right)+{{d}_{1}}\left( t \right)+{{u}_{1}}\left( t \right)+{{\lambda }_{1}}|\text{D}_{t}^{q-1}{{e}_{1}}{{|}^{r}}\text{sgn}(\text{D}_{t}^{q-1}{{e}_{1}}) \right]+ \\ &\quad \quad {{s}_{2}}\left[ -{{e}_{2}}+a{{y}_{3}}-\beta {{y}_{1}}{{y}_{3}}+\beta {{x}_{1}}{{x}_{3}}+\Delta {{f}_{2}}\left( y \right)+{{d}_{2}}\left( t \right)+{{u}_{2}}\left( t \right)+{{\lambda }_{2}}|\text{D}_{t}^{q-1}{{e}_{2}}{{|}^{r}}\text{sgn}(\text{D}_{t}^{q-1}{{e}_{2}}) \right]+ \\ &\quad \quad {{s}_{3}}\left[ -b{{e}_{1}}-a{{e}_{2}}+{{e}_{3}}+\gamma {{y}_{1}}{{y}_{2}}-\gamma {{x}_{1}}{{x}_{2}}+\Delta {{f}_{3}}\left( y \right)+{{d}_{3}}\left( t \right)+{{u}_{3}}\left( t \right)+{{\lambda }_{3}}|\text{D}_{t}^{q-1}{{e}_{3}}{{|}^{r}}\text{sgn}(\text{D}_{t}^{q-1}{{e}_{3}}) \right]+ \\ &\quad \quad \sum\limits_{i=1}^{3}{\left[ ({{{\hat{m}}}_{i}}-{{m}_{i}}+{{{\hat{n}}}_{i}}-{{n}_{i}})|{{s}_{i}}| \right]}。\\ \end{align}$

根据假设条件1、2, 很容易得到:

$\begin{align} &\dot{V}\le \left| {{s}_{1}} \right|({{m}_{1}}+{{n}_{1}})-\left| {{s}_{1}} \right|\left[ ({{{\hat{m}}}_{1}}+{{{\hat{n}}}_{1}})+{{k}_{1}} \right]+\left| {{s}_{2}} \right|({{m}_{2}}+{{n}_{2}})-\left| {{s}_{2}} \right|\left[ ({{{\hat{m}}}_{2}}+{{{\hat{n}}}_{2}})+{{k}_{2}} \right]+ \\ &\quad \quad \left| {{s}_{3}} \right|({{m}_{3}}+{{n}_{3}})-\left| {{s}_{3}} \right|\left[ ({{{\hat{m}}}_{3}}+{{{\hat{n}}}_{3}})+{{k}_{3}} \right]+\sum\limits_{i=1}^{3}{\left[ ({{{\hat{m}}}_{i}}-{{m}_{i}})\left| {{s}_{i}} \right|+({{{\hat{n}}}_{i}}-{{n}_{i}})\left| {{s}_{i}} \right| \right]}\le \\ &\quad \quad \left. \sum\limits_{i=1}^{3}{\left[ ({{m}_{i}}+{{n}_{i}})-({{{\hat{m}}}_{i}}+{{{\hat{n}}}_{i}})-{{k}_{i}} \right]}\left| {{s}_{i}} \right| \right\}+\sum\limits_{i=1}^{3}{\left[ \left. ({{{\hat{m}}}_{i}}-{{m}_{i}})\left| {{s}_{i}} \right|+({{{\hat{n}}}_{i}}-{{n}_{i}})\left| {{s}_{i}} \right| \right\} \right]}\text{ }= \\ &\quad \quad \sum\limits_{i=1}^{3}{\left[ ({{m}_{i}}-{{{\hat{m}}}_{i}})+({{n}_{1}}-{{{\hat{n}}}_{i}})-{{k}_{i}} \right]\left| {{s}_{i}} \right|}+\sum\limits_{i=1}^{3}{\left[ ({{{\hat{m}}}_{i}}-{{m}_{i}})|{{s}_{i}}|+({{{\hat{n}}}_{i}}-{{n}_{i}})\left| {{s}_{i}} \right| \right]}= \\ &\quad \quad -\sum\limits_{i=1}^{3}{{{k}_{i}}\left| {{s}_{i}} \right|<0}。\\ \end{align}$

假设3    Δfi(y)+di(t)=fi(t), i=1, 2, 3。

假设4    |fi(t)|≤M|ei(t)|。

假设5    ei(t)=0时, fi(t)=0, ei(t)≠0时, fi(t)≠0。

假设6    0<M<1。

式(2) 与式(1) 相减得到误差系统

$\left\{ \begin{align} &\text{D}_{t}^{q}{{e}_{1}}=-{{e}_{1}}-\alpha {{y}_{2}}{{y}_{3}}+\alpha {{x}_{2}}{{x}_{3}}+{{f}_{1}}\left( t \right)+{{u}_{1}}\left( t \right), \\ &\text{D}_{t}^{q}{{e}_{2}}=-{{e}_{2}}+a{{y}_{3}}-\beta {{y}_{1}}{{y}_{3}}+\beta {{x}_{1}}{{x}_{3}}+{{f}_{2}}\left( t \right)+{{u}_{2}}\left( t \right), \\ &\text{D}_{t}^{q}{{e}_{3}}=-b{{e}_{1}}-a{{e}_{2}}+{{e}_{3}}+\gamma {{y}_{1}}{{y}_{2}}-\gamma {{x}_{1}}{{x}_{2}}+{{f}_{3}}\left( t \right)+{{u}_{3}}\left( t \right)~ 。\\ \end{align} \right.$

引理3[17]    对于一般的分数阶自治非线性微分方程Dtαx(t)=f(x(t)), 当系统的阶数0<α≤1时, 如果存在实对称正定矩阵P, 使得J(x(t))=xT(t)PDtαx(t)<0, 则上述分数阶系统渐近稳定。

引理4[18](Barbalat引理)    若函数f(t)在[0, +∞)上一致连续, 并且广义积分$\int_{0}^{+\infty }{f\left( t \right)\text{d}t}$存在, 则有$\mathop {\lim }\limits_{t \to \infty } {\mkern 1mu} f\left( t \right) = 0$

定理3    在假设3~6条件下, 设计控制器

$\begin{align} &{{u}_{1}}\left( t \right)=\alpha {{y}_{2}}{{y}_{3}}-\alpha {{x}_{2}}{{x}_{3}},\text{ }{{u}_{2}}\left( t \right)=-a{{y}_{3}}+\beta {{y}_{1}}{{y}_{3}}-\beta {{x}_{1}}{{x}_{3}}, \\ &{{u}_{3}}\left( t \right)=-\gamma {{y}_{1}}{{y}_{2}}+\gamma {{x}_{1}}{{x}_{2}}-2{{e}_{3}}-Mb\left[ \left| {{e}_{1}} \right|+a\left| {{e}_{2}} \right|+|{{e}_{3}}| \right]\text{sgn}\left( s \right)-\eta \text{sgn}\left( s \right), \\ \end{align}$

选取滑模面s(t)=Dtα-1(-be1-ae2+e3), η>0, 则系统(1)(2) 是滑模混沌同步的。

证明    当状态轨迹位于滑模面上时, 必然满足条件:s(t)=0, $\dot{s}\left( t \right)=0$, 在滑模面上由$s=0\Rightarrow \text{D}_{t}^{\alpha }{{e}_{i}}=-{{e}_{i}}+{{f}_{i}}\left( t \right),t=1,2$, 则Dtq e3=-e3+f3(t)-M[b|e1|+a|e2|+|e3|]sgn(s)-ηsgn(s), 由于在滑模面上s=0, ⇒Dtq e3=-e3+f3(t)⇒Dtα ei=-ei+fi(t), i=1, 2, 3, 构造J=eiDtqei, i=1, 2, 3, 则有:ei(t)≠0时, J=ei(-ei+fi(t))=(M-1)|ei|2<0, 从而根据引理3, Dtq ei=-ei+fi(t)的解ei(t)→0, i=1, 2, 3。

当状态轨迹不位于滑模面上时, 选取Lyapunov函数$V\left( t \right)=\frac{1}{2}{{s}^{2}}\left( t \right)\Rightarrow \dot{V}\left( t \right)=s\left( t \right)\dot{s}\left( t \right)$, 则

$\begin{align} &s\left( t \right)=\text{D}_{t}^{q-1}\left( -b{{e}_{1}}-a{{e}_{2}}+{{e}_{3}} \right)\Rightarrow \dot{s}\left( t \right)=D_{t}^{q}\left( -b{{e}_{1}}-a{{e}_{2}}+{{e}_{3}} \right),~ \\ &\quad \quad \dot{V}\left( t \right)=s\left( t \right)\dot{s}\left( t \right)=s\left( t \right)[-b\text{D}_{t}^{\alpha }{{e}_{1}}-a\text{D}_{t}^{\alpha }{{e}_{2}}+\text{D}_{t}^{\alpha }{{e}_{3}}]= \\ &\quad \quad s\left( t \right)\left[ -b{{e}_{1}}-b{{f}_{1}}\left( t \right)-a{{e}_{2}}-a{{f}_{2}}\left( t \right)+{{e}_{3}}+{{f}_{3}}\left( t \right)-M\left[ b\left| {{e}_{1}} \right|+a\left| {{e}_{2}} \right|+|{{e}_{3}}| \right]\text{sgn}\left( s\left( t \right) \right)-\eta \text{sgn}\left( s\left( t \right) \right) \right]\le \\ &\quad \quad M\left[ b\left| {{e}_{1}} \right|+a\left| {{e}_{2}} \right|+\left| {{e}_{3}} \right| \right]\left| s\left( t \right) \right|-M\left[ b\left| {{e}_{1}} \right|+a\left| {{e}_{2}} \right|+\left| {{e}_{3}} \right| \right]s\left( t \right)\text{sgn}\left( s\left( t \right) \right)-\eta \left| s\left( t \right) \right|= \\ &\quad \quad M\left[ b\left| {{e}_{1}} \right|+a\left| {{e}_{2}} \right|+\left| {{e}_{3}} \right| \right]\left| s\left( t \right) \right|-M\left[ b\left| {{e}_{1}} \right|+a\left| {{e}_{2}} \right|+\left| {{e}_{3}} \right| \right]\left| s\left( t \right) \right|-\eta \left| s\left( t \right) \right|= \\ &\quad \quad -\eta \left| s\left( t \right) \right|<0。\\ \end{align}$

由于$\dot V < - \eta \left| {s\left( t \right)} \right|$,可以得到:$\int_{0}^{t}{\left| s\left( \tau \right) \right|\text{d}\tau }\le \frac{\int_{0}^{t}{\dot{V}\left( \tau \right)\text{d}\tau }}{-\eta }\le \frac{V\left( 0 \right)-V\left( \infty \right)}{-\eta }\le V\left( 0 \right)<\infty $所以s(t)是可积的且有界, 根据引理4(Barbalat引理)可知, s(t)→0⇒ei(t)→0。由以上分析可知, 误差系统可在有限时间内到达或趋近滑模面s(t)=0, 误差系统将收敛于0。

2 数值仿真

利用龙格-库塔法分别对不加和加上控制器两种情况进行仿真。α=50, β=20, γ=4.1, a=5, b=9, q=0.873时, 出现混沌吸引子, 设计不确定项Δf1(y)=cos(2πy2), Δf2(y)=0.5cos(2πy3), Δf3(y)=0.3cos(2πy2), 外部扰动d1(t)=0.2cost, d2(t)=0.6sint, d3(t)=cos(3t), 滑模面参数取λ1=3, λ2=4, λ3=7, μ=3, r=0.6, k1=9, k2=8, k3=5, (${{{\hat{m}}}_{1}},{{{\hat{m}}}_{2}},{{{\hat{m}}}_{3}}$)=(0.3, 0.5, 1), (${{{\hat{n}}}_{1}},{{{\hat{n}}}_{2}},{{{\hat{n}}}_{3}}$)=(0.8, 0.6, 0.3), 其不加控制器和加控制器下的系统状态以及定理2的系统误差曲线如图 1~3所示, 从图 1看出不加控制器系统不取得同步, 从图 2看出加入控制器系统快速同步, 从图 3看出系统的误差很快趋近于0, 表明系统取得快速同步。

图 1 无控制的主从系统状态 Figure 1 State of master-slave with no control
图 2 有控制的主从系统状态 Figure 2 State of master-slave with control
图 3 定理2的系统误差曲线 Figure 3 The system errors of theorem 2

定理3中f1(t)=cos(2πy2(t))+0.2cos(t), f2(t)=0.5cos(2πy3(t))+0.6sin(t), f3(t)=0.3cos(2πy2(t))+cos(3t), η=2.5, 其系统误差曲线如图 4所示。

图 4 定理3中的系统误差曲线 Figure 4 The system errors of theorem 3
3 结论

基于稳定性理论研究了分数阶Victor-Carmen系统的滑模控制及滑模终端控制问题, 研究表明:设计适当的控制器以及构造适当的切换函数能够使主从系统取得滑模控制混沌同步, 并给出了严格的证明, 数值仿真表明了方法的有效性。

参考文献
[1] 丁金凤, 张毅. 基于按指数律拓展的分数阶积分的El-Nabulsi-Pfaff变分问题的Noether对称性[J]. 中山大学学报(自然科学版), 2014, 53(6): 150-154
DING Jinfeng, ZHANG Yi. Neother symmetries for El-Nabulsi-Pfaff variational problem for extended exponential fractional integral[J]. Journal of Zhongshan University (Science Edition), 2014, 53(6): 150-154
[2] 金世欣, 张毅. 基于Caputo分数阶导数的含时滞的非保守系统动力学的Noether对称性[J]. 中山大学学报(自然科学版), 2015, 54(5): 49-55
JIN Shixin, ZHANG Yi. Noether symmetries for non-conservative Lagrange systems with time delay based on Caputo fractional derivative[J]. Journal of Zhongshan University (Science Edition), 2015, 54(5): 49-55
[3] ZHANG Yi. Fractional differential equations of motion interms of combined riemann-liouville derivatives[J]. Chinese Physics B, 2012, 8(21): 302-306
[4] SALARIEH H, ALASTY A. Adaptive synchronization of two chaotic systems with stochastic unknown parameters[J]. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(2): 508-519 DOI:10.1016/j.cnsns.2007.09.002
[5] SUN Y P, LI J M, WANG J A, et al. Generalized projective synchronization of chaotic systems via adaptive learning control[J]. Chinese Physics B, 2010, 19(2): 502-505
[6] LIU P, LIU S. Robust adaptive full state hybrid synchronization of chaotic complex systems with unknown parameters and external disturbances[J]. Nonlinear Dynamics, 2012, 70(1): 585-599 DOI:10.1007/s11071-012-0479-y
[7] YANG L, YANG J. Robust finite-time convergence of chaotic systems via adaptive terminal sliding mode scheme[J]. Communications in Nonlinear Science and Numerical Simulation, 2011, 16(6): 2405-2413 DOI:10.1016/j.cnsns.2010.09.022
[8] 毛北行, 张玉霞. 具有非线性耦合复杂网络混沌系统的有限时间同步[J]. 吉林大学学报(理学版), 2015, 53(4): 757-761
MAO Beixing, ZHANG Yuxia. Finite-time chaos synchronization of complex networks systems with nonlinear coupling[J]. Journal of Jiling University (Science Edition), 2015, 53(4): 757-761
[9] MOHAMMAD P A. Robust finite-time stabilization of fractional-order chaotic susyems based on fractional Lyapunov stability theory[J]. Journal of Computation and Nonlinear Dynamics, 2012, 32(7): 1011-1015
[10] MILAD Mohadeszadeh, HADI Delavari. Synchronization of fractional order hyper-chaotic systems based on a new adaptive sliding mode control[J]. International Journal of Dynamics and Control, 2015, 10(7): 435-446
[11] WANG X, HE Y. Projective synchronization of fractional order chaotic system based on linear separation[J]. Phys Lett A, 2008, 37(12): 435-441
[12] 孙宁, 张化光, 王智良. 不确定分数阶混沌系统的滑模投影同步[J]. 浙江大学学报(工学版), 2010, 44(7): 1288-1291
SUN Ning, ZHANG Huaguang, WANG Zhiliang. Projective synchronization of uncertain fractional order chaotic system using sliding mode controller[J]. Journal of Zhejiang University (Engineering Science), 2010, 44(7): 1288-1291
[13] 余明哲, 张友安. 一类不确定分数阶混沌系统的滑模自适应同步[J]. 北京航空航天大学学报, 2014, 40(9): 1276-1280
YU Mingzhe, ZHANG Youan. Sliding mede adaptive synchronization for a class of fractional-order chaotic systems with uncertainties[J]. Journal of Beijing University of Aeronautics and Astronautics, 2014, 40(9): 1276-1280
[14] 仲启龙, 邵永辉, 郑永爱. 分数阶混沌系统的主动滑模同步[J]. 动力学与控制学报, 2012, 17(2): 46-49
ZHONG Qilong, SHAO Yonghui, ZHENG Yongai. Synchronization of the fractional order chaotic systems based on TS models[J]. Journal of Dynamics and Control, 2012, 17(2): 46-49
[15] 张燕兰. 分数阶Rayleigh-Duffling-like系统的自适应追踪广义投影同步[J]. 动力学与控制学报, 2014, 12(4): 348-352
ZHANG Yanlan. Adaptive tracking generalized projective synchronization of fractional Rayleigh-Duffling-like system[J]. Journal of Dynamics and Control, 2014, 12(4): 348-352 DOI:10.6052/1672-6553-2014-042
[16] GRIGORAS V, GRIGORAS C. A novel chaotic systems for random pulse generation[J]. Advanced in Electrical and Computer Engineering, 2014, 14(2): 109-112 DOI:10.4316/aece
[17] 徐瑞萍, 高明美. 自适应终端滑模控制不确定混沌系统的同步[J]. 控制工程, 2016, 23(5): 715-719
XU Ruiping, GAO Mingmei. Synchronization of chaotic susyems with uncertainty using adaptive terminal sliding mode controller[J]. Control Engineering of China, 2016, 23(5): 715-719
[18] PODLUBN Y. Fractional differential equation[M]. New York: Academic Press, 1999.
[19] BHAT S P, BERNSTEIN D S. Geometric homogeneity with applications to finite-time stability[J]. Mathematics of Control Signals and Systems, 2005, 17(2): 101-127 DOI:10.1007/s00498-005-0151-x
[20] MOHAMMAD P A, SOHRAB K, GHASSE Mhassem A. Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique[J]. Applied Mathematical Modelling, 2011, 35(6): 3080-3091 DOI:10.1016/j.apm.2010.12.020