﻿ 分数阶Victor-Carmen混沌系统的自适应滑模控制
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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 0

### 引用本文

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.

### 文章历史

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

1 主要结果

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 \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}

 \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)

 \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}

 \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}

 \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.

 \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}

 \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}

2 数值仿真

 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

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