﻿ 冠状动脉系统的微分积分终端滑模混沌抑制
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 智能系统学报  2019, Vol. 14 Issue (4): 650-654  DOI: 10.11992/tis.201801022 0

### 引用本文

QIAN Dianwei, XI Yafei. Chaos suppression in coronary artery systems using differential-integral terminal sliding mode[J]. CAAI Transactions on Intelligent Systems, 2019, 14(4): 650-654. DOI: 10.11992/tis.201801022.

### 文章历史

Chaos suppression in coronary artery systems using differential-integral terminal sliding mode
QIAN Dianwei , XI Yafei
School of Control and Computer Engineering, North China Electric Power University, Beijing 102206, China
Abstract: The chaos phenomenon of coronary artery systems can lead to serious health problems. Taking nonlinear coronary artery systems as the research object, a dynamics model of uncertainty in coronary artery systems was established, and the differential-integral terminal sliding mode control (DI-SMC) was investigated for the uncertain coronary artery system. The disturbance observer was designed for the uncertainty of the model. Stability of the designed control system was proven according to the Lyapunov theory. The feasibility and effectiveness of the proposed chaos suppression method was verified using simulation results.
Key words: chaos suppression    coronary artery system    non-matching uncertainty    sliding mode control    disturbance observer

1 数学模型及问题描述

 $\begin{array}{l} {{\dot x}_1}= - b{x_1} - c{x_2} \\ {{\dot x}_2}= - {\textit{λ}} (1 + b){x_1} - {\textit{λ}} \left( {1 + c} \right){x_2} + {\textit{λ}} x_1^3 + E\cos \omega \tau \\ \end{array}$ (1)

 Download: 图 1 状态随着参数c变化的分岔图 Fig. 1 Bifurcation diagrams of the states with respect to the change of c
 Download: 图 2 冠状动脉系统的相平面图 Fig. 2 Phase plane of the coronary artery system

 $\begin{array}{l} {{\dot {\bar x}}_1}= - b{{\bar x}_1} - c{{\bar x}_2} + {d_1} \\ {{\dot {\bar x}}_2}= - {\textit{λ}} (1 + b){{\bar x}_1} - {\textit{λ}} \left( {1 + c} \right){{\bar x}_2} + {\textit{λ}} \bar x_1^3 +\\ \qquad E\cos \omega \tau + u + {d_2} \\ \end{array}$ (2)

 $\mathop {\lim }\limits_{t \to \infty } {{e}}={{\bf{{0}}}_{2 \times 1}}$ (3)

 $\begin{array}{l} {{\dot e}_1}= - b{e_1} - c{e_2} + {d_1} \\ {{\dot e}_2}= - \left( {{\textit{λ}} + b{\textit{λ}} } \right){e_1} - \left( {{\textit{λ}} + c{\textit{λ}} } \right){e_2} +{\textit{λ}} e_1^3 \\ \;\;\;\; + 3{x_1}{{\bar x}_1}{e_1} + u + {d_2} \\ \end{array}$ (4)

 $\begin{array}{l} {{\dot {\tilde {x}}}_1}={{{\tilde x}}_2} \\ {{\dot {\tilde {x}}}_2}={\textit{λ}} \left( {c - b} \right){{{\tilde x}}_1} - \left( {b +{\textit{λ}} + c{\textit{λ}} } \right){{{\tilde x}}_2} - {\textit{λ}} c{\tilde x}_1^3 -\\ \qquad 3c{x_1}{{\bar x}_1}{{{\tilde x}}_1} - cu - c{d_2} + {{{\dot d}}_1} + {\textit{λ}} \left( {1 + c} \right){d_1} \\ \end{array}$ (5)

 $\begin{array}{l} {\dot {\tilde {{{x}}}}}={{F}}\left( {{\tilde {x}},d} \right) + {{B}}u \\ {{y}}={{H}}\left( {{\tilde x}} \right) \\ \end{array}$ (6)

 $\ddot {{y}}=L_F^2{{H}}\left( {{\tilde {x}},d} \right) + {L_B}{L_F}{{H}}\left( {{\tilde {x}},d} \right){{u}}$ (7)

 $\ddot {{y}}={{f}}\left( {{\tilde x}} \right) + {{d}} - c{{u}}$ (8)
2 控制设计

 $s={e_{D_1}} + \alpha {e_{I_1}}$ (9)

 ${T_{D_1}}=\frac{{{{\left| {{e_{{D_1}}}\left( 0 \right)} \right|}^{1 - {q_{21}}/{p_{21}}}}}}{{\alpha \left( {1 - {q_{21}}/{p_{21}}} \right)}} + \frac{{{{\left| {{e_{{D_0}}}\left( {{t_{11}}} \right)} \right|}^{1 - {q_{11}}/{p_{11}}}}}}{{\beta \left( {1 - {q_{11}}/{p_{11}}} \right)}}$ (10)

 ${{\dot {e}}_{I_1}}= - {\alpha ^{{q_{21}}/{p_{21}}}}{e_{I_1}}^{{q_{21}}/{p_{21}}}\left( t \right)$ (11)

 ${t_{11}}=\frac{{{{\left| {{e_{{D_1}}}\left( 0 \right)} \right|}^{1 - {q_{21}}/{p_{21}}}}}}{{\alpha \left( {1 - {q_{21}}/{p_{21}}} \right)}}$ (12)

 ${{\dot {e}}_{{D_0}}}\left( {{t_{11}}} \right)= - {\beta ^{{q_{11}}/{p_{11}}}}e_{{D_0}}^{{q_{11}}/{p_{11}}}\left( {{t_{11}}} \right)$ (13)

 ${t_{01}}=\frac{{{{\left| {{e_{{D_0}}}\left( {{t_{11}}} \right)} \right|}^{1 - {q_{11}}/{p_{11}}}}}}{{\beta \left( {1 - {q_{11}}/{p_{11}}} \right)}}$ (14)

 ${\dot s}=\frac{{{p_{11}}}}{{{q_{11}}}}{\dot e}_{D_0}^{\left( {{p_{11}}/{q_{11}} - 1} \right)}\left( {f\left( {{\tilde x}} \right) + d - cu} \right) + \psi$ (15)

 $u\!=\!\frac{{ - 1}}{c}\left[ \!\!{ - {k_{\rm{e}}}\left| \psi \right|\left(\! {\frac{{{q_{11}}}}{{{p_{11}}}}}\! \right){{{A}}^{ - 1}}\frac{s}{{\left| s \right|}} - f\left( {{\tilde x}} \right) - \left| {k\operatorname{sgn} (s) + \eta s} \right| - d_0^ {*} } \!\right]$ (16)

 $\left\| {\left( {{{A}} - {\dot e}_{D_0}^{\left( {{p_{11}}/{q_{11}} - 1} \right)}} \right){{{A}}^{ - 1}}} \right\| \leqslant \phi < 1$ (17)

 $\begin{array}{l} {\dot {\textit{z}}}= - {{L}}{{{B}}_2}{{P}} - {{L}}\left( {{{{B}}_2}{{L}}{\tilde x} + {{A}}{\tilde x} + {{{B}}_1}u + {{M}}} \right) \\ {\hat d}={\textit{z}} + {{L}}{\tilde x} \\ \end{array}$ (18)

 $\begin{array}{l} {{A}}\!=\!\left[ \!{\begin{array}{*{20}{c}} 0&1 \\ {{\textit{λ}} (c - b)}&{ - (b + {\textit{λ}} + c{\textit{λ}} )} \end{array}}\! \right]; \;{{{B}}_1}=\left[ {\begin{array}{*{20}{c}} 0 \\ { - c} \end{array}} \right];\; {{{B}}_2}=\left[ {\begin{array}{*{20}{c}} 0 \\ 1 \end{array}} \right];\\ {{M}}=\left[ {\begin{array}{*{20}{c}} 0 \\ { - {\textit{λ}} c{\tilde x}_1^3 - 3c{x_1}{{\bar x}_1}{{{\tilde x}}_1}} \end{array}} \right]{\text{。}} \end{array}$

 ${e_d}=d - {\hat d}$ (19)

 $\begin{array}{c} {{\dot e}_d} \simeq {{L}}{{{B}}_2}P + {{L}}({{{B}}_2}{{L}}{\tilde x} + {{A}}{\tilde x} + {{{B}}_1}u + {{M}}) -\\ \qquad {{L}}({{A}}{\tilde x} + {{{B}}_1}u + {{{B}}_2}d + {{M}})= \\ \qquad {{L}}{{{B}}_2}({\hat d} - {{L}}{\tilde x}) + {{{L}}^2}{{{B}}_2}{\tilde x} - {{L}}{{{B}}_2}d =\\ \qquad {{L}}{{{B}}_2}({\hat d} - d)= - {{L}}{{{B}}_2}{e_d} \\ \end{array}$ (20)

 ${e_d}=\exp ( - {{L}}{{{B}}_2}){e_d}(0)$ (21)

 $u= - \frac{1}{c} - {k_{\rm{e}}}\left| \psi \right|\displaystyle\frac{{{q_{11}}}}{{{p_{11}}}}{{{A}}^{ - 1}}s/\left| s \right| - f\left( {{\tilde x}} \right) - [k\operatorname{sgn} (s) + \eta s] - {\hat d}$ (22)

 ${{\dot {{V}}}_0}=s\frac{{{p_{11}}}}{{{q_{11}}}}{\dot e}_{D_0}^{({p_{11}}/{q_{11}} - 1)}[f\left( {{\tilde x}} \right) + d - cu] + s\psi$ (23)

 $\begin{array}{l} {{\dot {{V}}}_0}=s\psi - s{\dot e}_{D_0}^{({p_{11}}/{q_{11}} - 1)}{k_{\rm{e}}}\left| \psi \right|{{{A}}^{ - 1}}s/\left| s \right| +\\ \qquad s\displaystyle\frac{{{p_{11}}}}{{{q_{11}}}}{\dot e}_{D_0}^{({p_{11}}/{q_{11}} - 1)}[d - {\hat d} - (k\operatorname{sgn} (s) + \eta s)] \\ \end{array}$ (24)

 $\begin{array}{l} s\psi - s{\dot e}_{D_0}^{({p_{11}}/{q_{11}} - 1)}{k_{\rm{e}}}\left| \psi \right|{{{A}}^{ - 1}}s/\left| s \right|= \\ s\psi - {k_{\rm{e}}}\left| \psi \right|s + {k_{\rm{e}}}\left| \psi \right|{{A}}{{{A}}^{ - 1}}s -\\ s{\dot e}_{D_0}^{({p_{11}}/{q_{11}} - 1)}{k_{\rm{e}}}\left| \psi \right|{{{A}}^{ - 1}}s/\left| s \right| \leqslant \\ \left| {\psi s} \right| - {k_{\rm{e}}}\left| \psi \right|s + {k_{\rm{e}}}\left| {\psi ({{A}} - {\dot e}_{D_0}^{({p_{11}}/{q_{11}} - 1)}){{{A}}^{ - 1}}s} \right| \leqslant \\ \left| {\psi s} \right| - {k_{\rm{e}}}(1 - \phi )\left| {\psi s} \right| \\ \end{array}$ (25)

 $\begin{array}{c} s\displaystyle\frac{{{p_{11}}}}{{{q_{11}}}}{\dot e}_{D_0}^{({p_{11}}/{q_{11}} - 1)}[d - {\hat d} - (k\operatorname{sgn} (s) + \eta s)] = \\ \displaystyle \frac{{{p_{11}}}}{{{q_{11}}}}{\dot e}_{D_0}^{({p_{11}}/{q_{11}} - 1)}[{e_d}s - k\left| s \right| - \eta {s^2}] \leqslant \\ \displaystyle\frac{{{p_{11}}}}{{{q_{11}}}}{\dot e}_{D_0}^{({p_{11}}/{q_{11}} - 1)}[(e_d^ {*} - k)\left| s \right| - \eta {s^2}] \\ \end{array}$ (26)

3 仿真结果