﻿ 动态输出反馈系统不确定项容忍区间分析 Tolerance Interval Analysis of System Uncertainty for Dynamic Output Feedback

Dynamical Systems and Control
Vol.07 No.02(2018), Article ID:24722,8 pages
10.12677/DSC.2018.72011

Tolerance Interval Analysis of System Uncertainty for Dynamic Output Feedback

Yao Zhang1, Fuzhong Wang2, Bo Yao1

1College of Mathematics and System Science, Shenyang Normal University, Shenyang Liaoning

2Department of Preparatory Courses, Shenyang Institute of Engineering, Shenyang Liaoning

Received: Mar. 29th, 2018; accepted: Apr. 23rd, 2018; published: Apr. 30th, 2018

ABSTRACT

Considering linear time-invariable systems, based on the problem of pole assignment for the disk region of dynamic output feedback, we gave the concept of tolerance interval of single deviation model with uncertain parameters for control system. Through the analysis of the influence on the performance of the system by uncertain parameters fluctuation of linear time-invariable systems, we gave the algorithm of uncertain parameters tolerance interval in disk region by single deviation model of the system. Furthermore, by comparing the size of tolerance intervals for various uncertain parameters, we determine the level of influence on the performance of systems. The system designer can design a high quality control system by referring to the level of its influence. Finally, a vertical wing of an aircraft example is given to illustrate scientificalness and effectiveness of the conclusion.

Keywords:Dynamic Output Feedback, Pole Placement, Tolerance Interval

1沈阳师范大学，数学与系统科学学院，辽宁 沈阳

2沈阳工程学院，基础教学部，辽宁 沈阳

1. 引言

2. 问题描述

$\left\{\begin{array}{l}\stackrel{˙}{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)\\ y\left(t\right)=Cx\left(t\right)\end{array}$ (1)

$\left\{\begin{array}{l}\stackrel{˙}{\stackrel{˜}{x}}\left(t\right)={A}_{0}\overline{x}\left(t\right)+{B}_{0}y\\ u\left(t\right)={C}_{0}\overline{x}\left(t\right)+{D}_{0}y\end{array}$ (2)

$\left\{\begin{array}{l}\stackrel{˙}{\stackrel{˜}{x}}\left(t\right)={A}_{c}\stackrel{˜}{x}\left(t\right)\\ y\left(t\right)={C}_{c}\stackrel{˜}{x}\left(t\right)\end{array}$ (3)

$\left(\begin{array}{cc}-rX& qX+AX\\ qX+X{A}^{T}& -rX\end{array}\right)<0$ (4)

$\left[\begin{array}{cccc}-rX& -rI& XA+\overline{B}C+qX& \overline{A}+qI\\ \ast & -rY& A+qI& AY+B\overline{C}+qY\\ \ast & \ast & -rX& -rI\\ \ast & \ast & \ast & -rY\end{array}\right]<0$ (5)

$\left[\begin{array}{cc}X& I\\ I& Y\end{array}\right]>0$ (6)

$\begin{array}{l}{A}_{0}={M}^{-1}\left(\overline{A}-XAY-M{B}_{0}CY-XB{C}_{0}{N}^{\text{T}}\right){N}^{-\text{T}}\\ {B}_{0}={M}^{-1}\overline{B}\\ {C}_{0}=\overline{C}{N}^{-\text{T}}\end{array}$ (7)

$\left\{\begin{array}{l}\stackrel{˙}{\stackrel{˜}{x}}\left(t\right)={A}_{m}\stackrel{˜}{x}\left(t\right)\\ y\left(t\right)={C}_{c}\stackrel{˜}{x}\left(t\right)\end{array}$ (8)

1) $eig\left(A\right)$ 为矩阵A的特征值；

2) ${\lambda }_{ij}$ 为不确定因素第ij项波动时闭环系统的特征值；

3) $\mathrm{Re}\left[{\lambda }_{ij}\right]$${\lambda }_{ij}$ 的实部值；

4) $\mathrm{Im}\left({\lambda }_{ij}\right)$${\lambda }_{ij}$ 的虚部值

3. 主要结果

1) 必要性

${A}_{m}=\left[\begin{array}{cc}A+\Delta A+B{D}_{0}C& B{C}_{0}\\ {B}_{0}C& {A}_{0}\end{array}\right]$

2) 充分性

4. 仿真算例

Figure 1. Pole distribution graph of pole disk region

Figure 2. Tolerance interval of parameter f22

Figure 3. Tolerance interval of parameter f24

Figure 4. Tolerance interval of parameter f12

5. 结论

Tolerance Interval Analysis of System Uncertainty for Dynamic Output Feedback[J]. 动力系统与控制, 2018, 07(02): 100-107. https://doi.org/10.12677/DSC.2018.72011

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