﻿ 一类分数阶冠状动脉系统的混沌同步控制
 文章快速检索 高级检索
 山东大学学报(工学版)  2018, Vol. 48 Issue (4): 55-60  DOI: 10.6040/j.issn.1672-3961.0.2016.463 0

### 引用本文

MENG Xiaoling, WANG Jianjun. Chaos synchronization of a class of fractional-order coronary artery systems[J]. Journal of Shandong University (Engineering Science), 2018, 48(4): 55-60. DOI: 10.6040/j.issn.1672-3961.0.2016.463.

### 文章历史

Chaos synchronization of a class of fractional-order coronary artery systems
MENG Xiaoling, WANG Jianjun
College of Science, Zhengzhou University of Aeronautics, Zhengzhou 450015, Henan, China
Abstract: The problem of chaos synchronization for a class of fractional-order coronary artery systems was studied based on Lyapunov stability theory and fractional-order calculus. Three sufficient conditions were arrived that the fractional order systems was chaos synchronized under appropriate controller. The research conclusion illustrated that systems was chaos synchronization under proper conditions.
Key words: fractional-order systems    coronary artery    sliding mode    chaos synchronization
0 引言

1 系统描述及预备知识

 $\left\{ \begin{array}{l} \dot x = - bx - cy\\ \dot y = - \lambda \left( {1 + b} \right)x - \lambda \left( {1 + c} \right)y + \lambda {x^3} + E\cos \omega t' \end{array} \right.$ (1)

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

 ${E_\alpha }\left( z \right) = \sum\limits_{k = 1}^\infty {\frac{{{z^k}}}{{{\rm{\Gamma }}\left( {k\alpha + 1} \right)}}} ,\alpha > 0,z \in C。$

 ${E_{\alpha ,\beta }}\left( z \right) = \sum\limits_{k = 1}^\infty {\frac{{{z^k}}}{{{\rm{\Gamma }}\left( {k\alpha + \beta } \right)}}} ,\alpha ,\beta > 0,有\;{E_\alpha }\left( z \right) = {E_{\alpha ,1}}\left( z \right),{E_{1,1}}\left( z \right) = {e^z}。$

 $\left| {{E_{\alpha ,\beta }}\left( z \right)} \right| \le \frac{C}{{1 + \left| z \right|}},\;\;\;\rho \le \left| {\arg \left( z \right)} \right| \le {\rm{ \mathsf{ π} }},\left| z \right| > 0。$

2 控制方案一

 $\left\{ \begin{array}{l} {\rm{D}}_t^\alpha {x_1} = - b{x_1} - c{y_1}\\ {\rm{D}}_t^\alpha {y_1} = - \lambda \left( {1 + b} \right){x_1} - \lambda \left( {1 + c} \right){y_1} + \lambda x_1^3 + E\cos \omega t \end{array} \right.,$ (2)

 $\left\{ \begin{array}{l} {\rm{D}}_t^\alpha {x_2} = - b{x_2} - c{y_2}\\ {\rm{D}}_t^\alpha {y_2} = - \lambda \left( {1 + b} \right){x_2} - \lambda \left( {1 + c} \right){y_2} + \lambda x_2^3 + E\cos \omega t + u\left( t \right) \end{array} \right.,$ (3)

 ${e_1} = {x_2} - {x_1},{e_2} = {y_2} - {y_1},$ (4)

 $\left\{ \begin{array}{l} {\rm{D}}_t^\alpha {e_1} = - b{e_1} - c{e_2}\\ {\rm{D}}_t^\alpha {e_2} = - \lambda \left( {1 + b} \right){e_1} - \lambda \left( {1 + c} \right){e_2} + \lambda x_2^3 - \lambda x_1^3 + u\left( t \right) \end{array} \right.。$ (5)

 $- b\varepsilon {e_1} - c\varepsilon {e_2} - \lambda \left( {1 + b} \right){e_1} - \lambda \left( {1 + c} \right){e_2} + \lambda \left( {x_2^3 - x_1^3} \right) + u\left( t \right) = 0,$

 ${u_{{\rm{eq}}}} = b\varepsilon {e_1} + c\varepsilon {e_2} + \lambda \left( {1 + b} \right){e_1} + \lambda \left( {1 + c} \right){e_{2 }} - \lambda \left( {x_2^3 - x_1^3} \right),$

 $\begin{array}{*{20}{c}} {{u_{{\rm{sw}}}} = - \eta {\mathop{\rm sgn}} \left( {s\left( t \right)} \right),}&{u\left( t \right) = {u_{{\rm{eq}}}} + {u_{{\rm{sw}}}}} \end{array},$

 $V\left( t \right) = \frac{1}{2}{s^2}\left( t \right),$

 $\begin{array}{l} \dot V\left( t \right) = s\dot s = s\left[ { - b\varepsilon {e_1} - c\varepsilon {e_2} - \lambda \left( {1 + b} \right){e_1} - \lambda \left( {1 + c} \right){e_2} + \lambda \left( {x_2^3 - x_1^3} \right) + u\left( t \right)} \right] = \\ \;\;\;\;\;\;\;\;\;\;\;\; - \eta s\left( t \right){\mathop{\rm sgn}} \left( {s\left( t \right)} \right) = - \eta \left| {s\left( t \right)} \right| < 0。\end{array}$
3 控制方案二

 $V\left( t \right) = \left| {{e_1}\left( t \right)} \right| + \left| {{e_2}\left( t \right)} \right|,$

 $\begin{array}{l} {\rm{D}}_t^\alpha V\left( t \right) = {\rm{D}}_t^\alpha \left[ {\left| {{e_1}\left( t \right)} \right| + \left| {{e_2}\left( t \right)} \right|} \right] = {\mathop{\rm sgn}} \left( {{e_1}\left( t \right)} \right){\rm{D}}_t^\alpha {e_1}\left( t \right) + \\{\mathop{\rm sgn}} \left( {{e_2}\left( t \right)} \right){\rm{D}}_t^\alpha {e_2}\left( t \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{\mathop{\rm sgn}} \left( {{e_1}} \right)\left( { - b{e_1} - c{e_2}} \right) + {\mathop{\rm sgn}} \left( {{e_2}} \right)\left[ {1 - \lambda \left( {1 + b} \right){e_1} - \lambda \left( {1 + c} \right){e_2} +\\ \lambda \left( {x_2^3 - x_1^3} \right) + u} \right]。\end{array}$

 $\mathit{V}\left( t \right) \le \mathit{V}\left( {{t_0}} \right){E_\alpha }\left( { - b{{\left( {t - {t_0}} \right)}^\alpha }} \right),$
 $\mathit{V}\left( t \right) = \left| {{e_1}\left( t \right)} \right| + \left| {{e_2}\left( t \right)} \right|,V\left( {{t_0}} \right) = \left| {{e_1}\left( {{t_0}} \right)} \right| + \left| {{e_2}\left( {{t_0}} \right)} \right|。$

z=-b(t-t0)α, |arg(z)|=π, 根据引理1, 存在常数C使得:

 $\left\| {\mathit{V}\left( t \right)} \right\| \le \frac{{C\left\| {\mathit{V}\left( {{t_0}} \right)} \right\|}}{{1 + \left| { - b{{\left( {t - {t_0}} \right)}^\alpha }} \right|}} \Rightarrow t \to \infty ,\left\| {\mathit{V}\left( t \right)} \right\| \to 0 \Rightarrow \left\| {{e_i}\left( t \right)} \right\|\left( {i = 1,2} \right) \to 0。$
4 控制方案三

 $\left\{ \begin{array}{l} {\rm{D}}_t^\alpha {x_1} = - b{x_1} - c{x_2}\\ {\rm{D}}_t^\alpha {x_2} = - \lambda \left( {1 + b} \right){x_1} - \lambda \left( {1 + c} \right){x_2} + \lambda x_1^3 + E\cos \omega t \end{array} \right.,$ (6)

 $\left\{ \begin{array}{l} {\rm{D}}_t^\alpha {y_1} = - b{y_1} - c{y_2}\\ {\rm{D}}_t^\alpha {y_2} = - \lambda \left( {1 + b} \right){y_1} - \lambda \left( {1 + c} \right){y_2} + \lambda y_1^3 + E\cos \omega t + \Delta g\left( {{y_1},{y_2}} \right) + d\left( t \right) + u\left( t \right) \end{array} \right.。$ (7)

 $\left\{ \begin{array}{l} {\rm{D}}_t^\alpha {e_1} = - b{e_1} - c{e_2}\\ {\rm{D}}_t^\alpha {e_2} = - \lambda \left( {1 + b} \right){e_1} - \lambda \left( {1 + c} \right){e_2} + \lambda y_1^3 - \lambda x_1^3 + \Delta g\left( {{y_1},{y_2}} \right) + d\left( t \right) + u\left( t \right) \end{array} \right.。$ (8)

 $s\left( t \right) = {\rm{D}}_t^{2\alpha - 1}{e_1}\left( t \right) + \lambda \int_0^t {{{\left| {{\rm{D}}_t^{\alpha - 1}{e_2}\left( \tau \right)} \right|}^r}{\mathop{\rm sgn}} \left( {{\rm{D}}_t^{\alpha - 1}{e_2}\left( \tau \right)} \right){\rm{d}}\tau } 。$ (9)

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

 ${\rm{D}}_t^{2\alpha }{e_1}\left( t \right) = {\rm{D}}_t^\alpha {e_2}\left( t \right) = - \lambda {\left| {{\rm{D}}_t^{\alpha - 1}{e_2}\left( t \right)} \right|^r}{\mathop{\rm sgn}} \left( {{\rm{D}}_t^{\alpha - 1}{e_2}\left( t \right)} \right)。$

 $V\left( t \right) = \frac{1}{2}{\left( {D_t^{\alpha - 1}{e_2}\left( t \right)} \right)^2},$

 $\dot V = {\rm{D}}_t^{\alpha - 1}{e_2} \cdot {\rm{D}}_t^\alpha {e_2} = - \lambda {\left| {{\rm{D}}_t^{\alpha - 1}{e_2}\left( t \right)} \right|^r} \cdot {\mathop{\rm sgn}} \left( {{\rm{D}}_t^{\alpha - 1}{e_2}\left( t \right)} \right){\rm{D}}_t^{\alpha - 1}{e_2} =\\ - \lambda {\left| {{\rm{D}}_t^{\alpha - 1}{e_2}\left( t \right)} \right|^{r + 1}} < 0。$

 $\begin{array}{l} u\left( t \right) = \lambda \left( {1 + b} \right){e_1} + \lambda \left( {1 + c} \right){e_2} - \lambda y_1^3 + \lambda x_1^3 - \left( {m + n} \right){\mathop{\rm sgn}} \left( s \right) - \\ \;\;\;\;\;\;\;\;\;\lambda {\left| {{\rm{D}}_t^{\alpha - 1}{e_2}\left( t \right)} \right|^r} \cdot {\mathop{\rm sgn}} \left( {{\rm{D}}_t^{\alpha - 1}{e_2}\left( t \right)} \right) - \left( {{k_1}{{\left| s \right|}^\gamma } + {k_2}{{\left| s \right|}^\mu }} \right){\mathop{\rm sgn}} \left( s \right), \end{array}$ (10)

 $\begin{array}{l} \dot V = s\left[ {{\rm{D}}_t^\alpha {e_2}\left( t \right) + \lambda {{\left| {{\rm{D}}_t^{\alpha - 1}{e_2}\left( t \right)} \right|}^r}{\mathop{\rm sgn}} \left( {{\rm{D}}_t^{\alpha - 1}{e_2}\left( t \right)} \right)} \right] = \\ \;\;\;\;\;\;s\left[ { - \lambda \left( {1 + b} \right){e_1} - \lambda \left( {1 + c} \right){e_2} + \lambda y_1^3 - \lambda x_1^3 + \Delta g\left( {{y_1},{y_2}} \right) + } \right.\\ \;\;\;\;\;\;\left. {d\left( t \right) + u\left( t \right) + \lambda {{\left| {{\rm{D}}_t^{\alpha - 1}{e_2}\left( t \right)} \right|}^r}{\mathop{\rm sgn}} \left( {{\rm{D}}_t^{\alpha - 1}{e_2}\left( t \right)} \right)} \right], \end{array}$

 $\begin{array}{l} \dot V \le s\left( {\Delta g\left( {{y_1},{y_2}} \right) + d\left( t \right)} \right) - \left( {m + n} \right)\left| {{s_1}} \right| - \left( {{k_1}{{\left| s \right|}^{\gamma + 1}} + {k_2}{{\left| s \right|}^{\mu + 1}}} \right) \le \\ \;\;\;\;\;\; - \left( {{k_1}{{\left| s \right|}^{\gamma + 1}} + {k_2}{{\left| s \right|}^{\mu + 1}}} \right) < 0。\end{array}$
5 数值仿真

 图 1 定理1中的系统误差曲线 Figure 1 The system curvers of theorem 1

 图 2 定理2中的系统误差曲线 Figure 2 The system curvers of theorem 2

 $s\left( t \right) = {\rm{D}}_t^{2\alpha - 1}{e_1}\left( t \right) + \lambda \int_0^t {{{\left| {{\rm{D}}_t^{\alpha - 1}{e_2}\left( t \right)} \right|}^r}{\mathop{\rm sgn}} \left( {{\rm{D}}_t^{\alpha - 1}{e_2}\left( t \right)} \right){\rm{d}}\tau } ,$

 $\begin{array}{l} u\left( t \right) = \lambda \left( {1 + b} \right){e_1} + \lambda \left( {1 + b} \right){e_2} - \lambda y_1^3 + \lambda x_1^3 - \left( {m + n} \right){\mathop{\rm sgn}} \left( s \right) - \\ \;\;\;\;\;\;\;\;\;\lambda {\left| {{\rm{D}}_t^{2\alpha - 1}{e_2}\left( t \right)} \right|^r} \cdot {\mathop{\rm sgn}} \left( {{\rm{D}}_t^{\alpha - 1}{e_2}\left( t \right)} \right) - \left( {{k_1}{{\left| s \right|}^\gamma } + {k_2}{{\left| s \right|}^\mu }} \right){\mathop{\rm sgn}} \left( s \right), \end{array}$

 图 3 系统的误差曲线 Figure 3 The errors of systems

6 结语