﻿ 纠缠混沌系统的比例积分滑模同步
 文章快速检索 高级检索
 山东大学学报(工学版)  2018, Vol. 48 Issue (4): 50-54,87  DOI: 10.6040/j.issn.1672-3961.0.2017.553 0

### 引用本文

MAO Beixing. Ratio integral sliding mode synchronization control of entanglement chaotic systems[J]. Journal of Shandong University (Engineering Science), 2018, 48(4): 50-54,87. DOI: 10.6040/j.issn.1672-3961.0.2017.553.

### 文章历史

Ratio integral sliding mode synchronization control of entanglement chaotic systems
MAO Beixing
College of Science, Zhengzhou University of Aeronautics, Zhengzhou 450015, Henan, China
Abstract: The problem of sliding mode and ratio integral sliding mode synchronization of a class of entanglement chaotic systems were studied based on sliding mode control in the paper. The surfaces and controllers were designed using sliding mode and ratio integral sliding mode approach. And sliding mode uniform speed reaching law was adopted.Two cases for system trajectory on sliding mode surface and not on sliding mode surface were analyzed based on Lyapunov stability theory.The systems errors could approach to coordinate zero under the corporate action of surfaces and controllers. Two sufficient conditions were arrived for entanglement chaotic systems acquire sliding mode synchronization and integral sliding mode synchronization.The research conclusion illustrated that the master-slave systems of entanglement chaotic systems were sliding mode and ratio integral sliding mode synchronization if proper controllers and sliding mode surfaces was chosen.
Key words: chaos    synchronization    entanglement systems    integral sliding mode    sliding mode control
0 引言

1 系统描述

 $\left\{ \begin{array}{l} {{\dot x}_1} = a{x_1} + b{y_1}\\ {{\dot y}_1} = b{x_1} + c{y_1} \end{array} \right.,$ (1)
 ${\dot z_1} = - m{z_1},$ (2)

 $\left\{ \begin{array}{l} {{\dot x}_1}{\rm{ = }}a{x_1} + b{y_1} + l\sin {y_1}\\ {{\dot y}_1} = - b{x_1} + c{y_1} + l\sin {z_1}\\ {{\dot z}_1} = - m{z_1} + l\sin {x_1} \end{array} \right.,$ (3)

 图 1 系统吸引子相图 Figure 1 System attractor of phase diagram

 $\left\{ \begin{array}{l} {{\dot x}_2}{\rm{ = }}a{x_2} + b{y_2} + l\sin {y_2}\\ {{\dot y}_2} = - b{x_2} + c{y_2} + l\sin {z_2} + u\left( t \right)\\ {{\dot z}_2} = - m{z_2} + l\sin {x_2}, \end{array} \right.,$ (4)

 $\left\{ \begin{array}{l} {{\dot e}_1} = a{e_1} + b{e_2} + l\left( {\sin {y_2} - \sin {y_1}} \right)\\ {{\dot e}_2} = - b{e_1} + c{e_2} + l\left( {\sin {z_2} - \sin {z_1}} \right) + u\left( t \right)\\ {{\dot e}_3} = - m{e_3} + l\left( {\sin {x_2} - \sin {x_1}} \right) \end{array} \right.。$ (5)
2 滑模同步

 $\begin{array}{l} \dot V = s\dot s = s\left[ {{{\dot e}_1} + {{\dot e}_2} + {{\dot e}_3}} \right]\\ \;\;\; = s\left\{ {\left( {a - b} \right){e_1} + \left( {b + c} \right){e_2} - m{e_3} + l\left[( {\sin {z_2} - \sin {z_1} ) +\\ \left( {\sin {y_2} - \sin {y_1}} \right) + \left( {\sin {x_2} - \sin {x_1}} \right)} \right] + u} \right\}\\ \;\;\; \le l\left| s \right|\sum\limits_{i = 1}^3 {\left| {{e_i}} \right| - } l\sum\limits_{i = 1}^3 {\left| {{e_i}} \right|s \cdot {\mathop{\rm sgn}} \;s - \eta s \cdot {\mathop{\rm sgn}} \;s。} \end{array}$

3 比例积分滑模同步

 $s\left( t \right) = {e_2}\left( t \right) + \int_0^t {\left[ {k{e_2}\left( \tau \right) + b{e_1}\left( \tau \right) - l\left( {\sin {z_2}\left( \tau \right) - \sin {z_1}\left( \tau \right)} \right)} \right]d\tau , }$

 $\left\{ \begin{array}{l} {{\dot e}_1} = a{e_1} + b{e_2} + l\left( {\sin {y_2} - \sin {y_1}} \right)\\ {{\dot e}_2} = - b{e_1} + k{e_2} + l\left( {\sin {z_2} - \sin {z_1}} \right)\\ {{\dot e}_3} = - m{e_3} + l\left( {\sin {x_2} - \sin {x_1}} \right) \end{array} \right.。$ (6)

 $\begin{array}{l} \dot V\left( t \right){\rm{ = }}s\dot s = s\left[ {{{\dot e}_2}\left( t \right) + k{e_2}\left( t \right) + b{e_1}\left( t \right) - l\left( {\sin {z_2}\left( t \right) - \sin {z_1}\left( t \right)} \right)} \right]\\ \;\;\;\;\;\;\; = s\left[ { - b{e_1} + c{e_2} + l\left( {\sin {z_2} - \sin {z_1}} \right) + u\left( t \right) +\\ k{e_2}\left( t \right) + b{e_1}\left( t \right) - l\left( {\sin {z_2}\left( t \right) - \sin {z_1}\left( t \right)} \right) + u\left( t \right)} \right]\\ \;\;\;\;\;\;\; = - \eta s \cdot {\mathop{\rm sgn}} \left( s \right) = - \eta \left| {s\left( t \right)} \right| < 0, \end{array}$

4 数值仿真

 $s\left( t \right) = {e_2}\left( t \right) + \int_0^t {\left[ {k{e_2}\left( \tau \right) + b{e_1}\left( \tau \right) - l\left( {\sin {z_2}\left( \tau \right) - \sin {z_1}\left( \tau \right)} \right)} \right]} {\rm{d}}\tau 。$

 图 2 定理1中的系统误差曲线 Figure 2 The system errors of theorem 1
 图 3 定理2中的系统误差曲线 Figure 3 The system errors of theorem 2
5 结论