﻿ Delta算子系统的状态量化H∞控制 H∞ Control for Delta Operator Systems with State Quantization

Vol. 11  No. 02 ( 2022 ), Article ID: 48585 , 9 pages
10.12677/AAM.2022.112069

Delta算子系统的状态量化H控制

Delta算子系统，状态量化，H控制

H Control for Delta Operator Systems with State Quantization

Congyi Liu

College of Mathematics and Statistics, Fujian Normal University, Fuzhou Fujian

Received: Jan. 7th, 2022; accepted: Feb. 3rd, 2022; published: Feb. 10th, 2022

ABSTRACT

This paper mainly studies the H control for Delta operator systems with state quantization. Firstly, by designing state feedback controller, the Delta operator systems are asymptotically stable and satisfy the H performance index. In addition, considering the dynamic quantizer, the delta operator systems can satisfy the same performance index under it. Based on a Lyapunov function, the target is designed by using linear matrix inequality. Finally, a numerical example is given to verify the feasibility and effectiveness of the design method.

Keywords:Delta Operator Systems, State Quantization, H Control

1. 引言

Delta算子是用来描述离散系统的一种方法 [1]，相较于传统前向移位算子来说，其不仅可以拥有前向移位算子的全部优点，且在高速采样的情况下，Delta算子离散化方法可以避免传统前向移位算子离散化方法引起的数值不稳定问题。当采样周期无限趋于零时，Delta算子公式化模型趋近相应的离散化前的连续时间模型，故连续时间系统的设计方法可以直接应用于离散时间系统，可以获得更好的控制结果。近年来，国内外学者对于Delta算子系统的研究逐渐深入，例如，Zhang等人 [2] 研究了Delta算子系统的故障检测问题。肖民卿 [3] 研究了一系列Delta算子系统的控制问题。向峥嵘等人 [4] 研究了Delta算子描述的区间系统的鲁棒性能稳定分析与综合设计问题。胡号 [5] 研究了几类Delta算子切换系统的容错控制问题。

1) $S<0$

2) ${S}_{11}<0$${S}_{22}-{S}_{21}{S}_{11}^{-1}{S}_{12}<0$

3) ${S}_{22}<0$${S}_{11}-{S}_{12}{S}_{22}^{-1}{S}_{21}<0$

1) $T+{S}^{\text{T}}{F}^{\text{T}}+FS<0$

2) $\left[\begin{array}{cc}T& *\\ \xi {F}^{\text{T}}+US& -\xi U-\xi {U}^{\text{T}}\end{array}\right]<0$

2. 问题描述

Delta算子的定义为：

$\delta =\frac{q-1}{h}$

$\left\{\begin{array}{l}\delta x\left(k\right)=Ax\left(k\right)+Bu\left(k\right)+E\omega \left(k\right)\\ y\left(k\right)=Cx\left(k\right)\end{array}$ (1)

$u\left(k\right)=Kx\left(k\right)$ (2)

$\left\{\begin{array}{l}\delta x\left(k\right)=\left(A+BK\right)x\left(k\right)+E\omega \left(k\right)\\ y\left(k\right)=Cx\left(k\right)\end{array}$ (3)

1) 当 $\omega \left(k\right)=0$ 时，闭环系统(3)是渐近稳定的；

2) 当 $\omega \left(k\right)\ne 0$ 时，对于标量 $\gamma >0$，系统测量输出和外部扰动输入应满足 $\underset{k=0}{\overset{\infty }{\sum }}{‖y\left(k\right)‖}^{2}<{\gamma }^{2}\underset{k=0}{\overset{\infty }{\sum }}{‖\omega \left(k\right)‖}^{2}$

3. 主要结论

$\left[\begin{array}{cccc}{\stackrel{^}{\Omega }}_{\text{}2}& *& *& *\\ {\stackrel{^}{\Omega }}_{\text{}1}& -P& *& *\\ {\Omega }_{\text{}3}& 0& -I& *\\ {\Omega }_{\text{}4}& \xi {\left(hPB-QU\right)}^{\text{T}}& 0& {\Omega }_{\text{}5}\end{array}\right]<0$ (4)

${\stackrel{^}{\Omega }}_{\text{}2}=\left[\begin{array}{cc}-P+{R}_{11}& *\\ {R}_{21}& -{\gamma }^{2}I+{R}_{22}+{E}^{\text{T}}PE\end{array}\right]$

${\stackrel{^}{\Omega }}_{\text{1}}=\left[\begin{array}{cc}P+hPA+QG& hPE\end{array}\right]$${\Omega }_{\text{}3}=\left[\begin{array}{cc}C& 0\end{array}\right]$${\Omega }_{\text{}4}=\left[\begin{array}{cc}G& 0\end{array}\right]$${\Omega }_{\text{}5}=-\xi U-\xi {U}^{\text{T}}$

$V\left(x\left(k\right)\right)={\text{x}}^{\text{T}}\left(k\right)\text{Px}\left( k \right)$

$\begin{array}{l}\text{x}\left(k+1\right)=h\delta \text{x}\left(k\right)+\text{x}\left(k\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=h\left[\left(\text{A}+\text{BK}\right)\text{x}\left(k\right)+\text{Eω}\left(k\right)\right]+\text{x}\left(k\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=\left[\text{I}+h\left(\text{A}+\text{BK}\right)\right]\text{x}\left(k\right)+h\text{Eω}\left( k \right)\end{array}$

$\begin{array}{l}\Theta ={\text{y}}^{\text{T}}\left(k\right)\text{y}\left(k\right)-{\gamma }^{2}{\text{ω}}^{\text{T}}\left(k\right)\text{ω}\left(k\right)+{\text{ω}}^{\text{T}}\left(k\right){\text{E}}^{\text{T}}\text{PEω}\left(k\right)+\Delta V\left(\text{x}\left(k\right)\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }={\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\end{array}\right]}^{\text{T}}\left({\left[\begin{array}{cc}\text{C}& 0\end{array}\right]}^{\text{T}}\left[\begin{array}{cc}\text{C}& 0\end{array}\right]-\left[\begin{array}{cc}\text{P}& 0\\ 0& {\gamma }^{2}\text{I}-{\text{E}}^{\text{T}}\text{PE}\end{array}\right]+\text{ }{\left[\begin{array}{cc}\text{I}+h\left(\text{A}+\text{BK}\right)& h\text{E}\end{array}\right]}^{\text{T}}\text{P}\left[\begin{array}{cc}\text{I}+h\left(\text{A}+\text{BK}\right)& h\text{E}\end{array}\right]\right)\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\end{array}\right]\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }={\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\end{array}\right]}^{\text{T}}\left({\left[\begin{array}{cc}\text{I}+h\left(\text{A}+\text{BK}\right)& h\text{E}\end{array}\right]}^{\text{T}}\text{P}\left[\begin{array}{cc}\text{I}+h\left(\text{A}+\text{BK}\right)& h\text{E}\end{array}\right]+\text{ }\left[\begin{array}{cc}-\text{P}+{\text{C}}^{\text{T}}\text{C}& 0\\ 0& -{\gamma }^{2}\text{I+}{\text{E}}^{\text{T}}\text{PE}\end{array}\right]\right)\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\end{array}\right]\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }={\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\end{array}\right]}^{\text{T}}\left({\Omega }_{\text{}1}+{\Omega }_{\text{}2}\right)\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\end{array}\right]\end{array}$

${\Omega }_{\text{}1}={\left[\begin{array}{cc}\text{I}+h\left(\text{A}+\text{BK}\right)& h\text{E}\end{array}\right]}^{\text{T}}\text{P}\left[\begin{array}{cc}\text{I}+h\left(\text{A}+\text{BK}\right)& h\text{E}\end{array}\right]$

${\Omega }_{\text{}2}=\left[\begin{array}{cc}-\text{P}+{\text{C}}^{\text{T}}\text{C}& 0\\ 0& -{\gamma }^{2}\text{I}+{\text{E}}^{\text{T}}\text{PE}\end{array}\right]$

${\text{y}}^{\text{T}}\left(k\right)\text{y}\left(k\right)-{\gamma }^{2}{\text{ω}}^{\text{T}}\left(k\right)\text{ω}\left(k\right)+\Delta V\left(\text{x}\left(k\right)\right)<-{\text{ω}}^{\text{T}}\left(k\right){\text{E}}^{\text{T}}\text{PEω}\left(k\right)<0$

$\text{R}=\left[\begin{array}{cc}{\text{R}}_{11}& *\\ {\text{R}}_{21}& {\text{R}}_{22}\end{array}\right]>0$，若能证明 $\Theta <-\text{R}$，则 $\Theta <0$ 成立，即证明 $\Theta +\text{R}<0$

${\stackrel{¯}{\Omega }}_{\text{}2}={\Omega }_{\text{}2}+\text{R}=\left[\begin{array}{cc}-\text{P}+{\text{C}}^{\text{T}}\text{C}+{\text{R}}_{11}& *\\ {\text{R}}_{21}& -{\gamma }^{2}\text{I}+{\text{E}}^{\text{T}}\text{PE}+{\text{R}}_{22}\end{array}\right]$ (5)

$\left[\begin{array}{cc}{\stackrel{¯}{\Omega }}_{\text{}2}& *\\ {\stackrel{¯}{\Omega }}_{\text{}1}& -{\text{P}}^{-1}\end{array}\right]<0$ (6)

$\left[\begin{array}{cc}{\stackrel{¯}{\Omega }}_{\text{}2}& *\\ {\stackrel{˜}{\Omega }}_{\text{}1}& -\text{P}\end{array}\right]<0$ (7)

$\left[\begin{array}{ccc}{\stackrel{^}{\Omega }}_{\text{2}}& *& *\\ {\stackrel{˜}{\Omega }}_{\text{}1}& -\text{P}& *\\ {\Omega }_{\text{3}}& 0& -\text{I}\end{array}\right]<0$ (8)

$\text{K}={\text{U}}^{-1}\text{G}$，则(8)式可以写为

$\left[\begin{array}{ccc}{\stackrel{^}{\Omega }}_{\text{2}}& *& *\\ {\stackrel{^}{\Omega }}_{\text{1}}& -\text{P}& *\\ {\Omega }_{\text{3}}& 0& -\text{I}\end{array}\right]+{\left(\text{M}{\text{U}}^{-1}\text{GN}\right)}^{\text{T}}+\text{M}{\text{U}}^{-1}\text{GN}<0$ (9)

$\stackrel{˜}{u}\left(k\right)=K\stackrel{˜}{x}\left(k\right)$ (10)

$M>\frac{1}{\eta }$

$\frac{1}{M}|\text{x}\left(k\right)|\le \mu \left(k\right)\le \eta |\text{x}\left(k\right)|$ (11)

$\eta =\frac{{\lambda }_{\mathrm{min}}\left(\text{R}\right)}{\phi \left(\varsigma +\sqrt{{\varsigma }^{2}+\tau \left({\lambda }_{\mathrm{min}}\left(\text{R}\right)}\right)}$ (12)

$‖\left(h\text{I}+{h}^{2}{\left(\text{A}+\text{BK}\right)}^{\text{T}}\right)\text{PBK}‖<\varsigma$ (13)

$‖2{h}^{2}{\left(\text{BK}\right)}^{\text{T}}\text{PBK}‖<\tau$ (14)

$\stackrel{˜}{x}\left(k\right)={q}_{\mu }\left(\text{x}\left(k\right)\right)=\mu \left(k\right)q\left(\frac{\text{x}\left(k\right)}{\mu \left( k \right)}\right)$

$\begin{array}{l}\delta \text{x}\left(k\right)=A\text{x}\left(k\right)+BK\stackrel{˜}{x}\left(k\right)+E\text{ω}\left(k\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=A\text{x}\left(k\right)+BK\mu \left(k\right)q\left(\frac{\text{x}\left(k\right)}{\mu \left(k\right)}\right)+E\text{ω}\left(k\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=A\text{x}\left(k\right)+BK\mu \left(k\right)q\left(\frac{\text{x}\left(k\right)}{\mu \left(k\right)}\right)+BK\text{x}\left(k\right)-\mu \left(k\right)BK\frac{\text{x}\left(k\right)}{\mu \left(k\right)}+E\text{ω}\left(k\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=\left(A+BK\right)\text{x}\left(k\right)+E\text{ω}\left(k\right)+BK\left[\mu \left(k\right)\left(q\left(\frac{\text{x}\left(k\right)}{\mu \left(k\right)}\right)-\frac{\text{x}\left(k\right)}{\mu \left(k\right)}\right)\right]\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=\left(A+BK\right)\text{x}\left(k\right)+E\text{ω}\left(k\right)+BK\text{e}\left(k\right)\end{array}$ (15)

$\left\{\begin{array}{l}\delta \text{x}\left(k\right)=\left(A+BK\right)\text{x}\left(k\right)+E\text{ω}\left(k\right)+BK\text{e}\left(k\right)\\ \text{y}\left(k\right)=\text{Cx}\left( k \right)\end{array}$

$\begin{array}{l}\text{x}\left(k+1\right)=h\delta \text{x}\left(k\right)+\text{x}\left(k\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=h\left[\left(\text{A}+\text{BK}\right)\text{x}\left(k\right)+\text{Eω}\left(k\right)+\text{BKe}\left(k\right)\right]+\text{x}\left(k\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=\left[\text{I}+h\left(\text{A}+\text{BK}\right)\right]\text{x}\left(k\right)+h\text{Eω}\left(k\right)+h\text{BKe}\left( k \right)\end{array}$

$\begin{array}{l}\stackrel{˜}{\Theta }={\text{y}}^{\text{T}}\left(k\right)\text{y}\left(k\right)-{\gamma }^{2}{\text{ω}}^{\text{T}}\left(k\right)\text{ω}\left(k\right)+\Delta V\left(\text{x}\left(k\right)\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }={\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\\ \text{e}\left(k\right)\end{array}\right]}^{\text{T}}\left({\left[\begin{array}{ccc}\text{C}& 0& 0\end{array}\right]}^{\text{T}}\left[\begin{array}{ccc}\text{C}& 0& 0\end{array}\right]-{\left[\begin{array}{ccc}0& \gamma \text{I}& 0\end{array}\right]}^{\text{T}}\left[\begin{array}{ccc}0& \gamma \text{I}& 0\end{array}\right]\underset{}{\overset{}{\underset{}{\overset{}{}}}}\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }+{\left[\begin{array}{ccc}\text{I}+h\left(\text{A}+\text{BK}\right)& h\text{E}& h\text{BK}\end{array}\right]}^{\text{T}}\text{P}\left[\begin{array}{ccc}\text{I}+h\left(\text{A}+\text{BK}\right)& h\text{E}& h\text{BK}\end{array}\right]-\left[\begin{array}{ccc}\text{P}& 0& 0\\ *& *& 0\\ *& *& 0\end{array}\right]\right)\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\\ \text{e}\left(k\right)\end{array}\right]\end{array}$

$\begin{array}{l}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }={\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\\ \text{e}\left(k\right)\end{array}\right]}^{\text{T}}\left(\left[\begin{array}{ccc}-\text{P}+{\text{C}}^{\text{T}}\text{C}& 0& 0\\ *& -{\gamma }^{2}\text{I}& 0\\ *& *& 0\end{array}\right]\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }+\left[\begin{array}{ccc}\left(\text{I}+h{\left(\text{A}+\text{BK}\right)}^{\text{T}}\right)\text{P}\left(\text{I}+h\left(\text{A}+\text{BK}\right)\right)& h\left(\text{I}+h{\left(\text{A}+\text{BK}\right)}^{\text{T}}\right)\text{PE}& h\left(\text{I}+h{\left(\text{A}+\text{BK}\right)}^{\text{T}}\right)\text{PBK}\\ \text{*}& {h}^{2}{\text{E}}^{\text{T}}\text{PE}& {h}^{2}{\text{E}}^{\text{T}}\text{PBK}\\ \text{*}& \text{*}& {h}^{2}{\left(\text{BK}\right)}^{\text{T}}\text{PBK}\end{array}\right]\right)\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\\ \text{e}\left(k\right)\end{array}\right]\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }={\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\\ \text{e}\left(k\right)\end{array}\right]}^{\text{T}}\left(\left[\begin{array}{cc}{\stackrel{˙}{\Omega }}_{\text{2}}& \begin{array}{c}0\\ 0\end{array}\\ \begin{array}{cc}*& *\end{array}& 0\end{array}\right]+\left[\begin{array}{cc}{\Omega }_{\text{}1}& \begin{array}{c}h\left(\text{I}+h{\left(\text{A}+\text{BK}\right)}^{\text{T}}\right)\text{PBK}\\ {h}^{2}{\text{E}}^{\text{T}}\text{PBK}\end{array}\\ \begin{array}{cc}\text{*}& \text{*}\end{array}& {h}^{2}{\left(\text{BK}\right)}^{\text{T}}\text{PBK}\end{array}\right]\right)\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\\ \text{e}\left(k\right)\end{array}\right]\end{array}$

$\begin{array}{l}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }={\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\end{array}\right]}^{\text{T}}\left({\stackrel{˙}{\Omega }}_{\text{2}}+{\Omega }_{\text{}1}\right)\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\end{array}\right]+h{\text{e}}^{\text{T}}\left(k\right){\left(\text{BK}\right)}^{\text{T}}\text{P}\left(\text{I}+h\left(\text{A}+\text{BK}\right)\right)\text{x}\left(k\right)+{h}^{2}{\text{e}}^{\text{T}}\left(k\right){\left(\text{BK}\right)}^{\text{T}}\text{PEω}\left(k\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }+{h}^{2}{\text{e}}^{\text{T}}\left(k\right){\left(\text{BK}\right)}^{\text{T}}\text{PBKe}\left(k\right)+h{\text{x}}^{\text{T}}\left(k\right)\left(\text{I}+h{\left(\text{A}+\text{BK}\right)}^{\text{T}}\right)\text{PBKe}\left(k\right)+{h}^{2}{\text{ω}}^{\text{T}}\left(k\right){\text{E}}^{\text{T}}\text{PBKe}\left(k\right)\end{array}$ (16)

$\begin{array}{l}\stackrel{˜}{\Theta }\le {\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\end{array}\right]}^{\text{T}}\left({\stackrel{˙}{\Omega }}_{\text{2}}+{\Omega }_{\text{}1}\right)\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\end{array}\right]+h{\text{e}}^{\text{T}}\left(k\right){\left(\text{BK}\right)}^{\text{T}}\text{P}\left(\text{I}+h\left(\text{A}+\text{BK}\right)\right)\text{x}\left(k\right)+{h}^{2}{\text{ω}}^{\text{T}}\left(k\right){\text{E}}^{\text{T}}\text{PEω}\left(k\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }+{h}^{2}{\text{e}}^{\text{T}}\left(k\right){\left(\text{BK}\right)}^{\text{T}}\text{PBKe}\left(k\right)+h{\text{x}}^{\text{T}}\left(k\right)\left(\text{I}+h{\left(\text{A}+\text{BK}\right)}^{\text{T}}\right)\text{PBKe}\left(k\right)+{h}^{2}{\text{e}}^{\text{T}}\left(k\right){\left(\text{BK}\right)}^{\text{T}}\text{PBKe}\left(k\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }={\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\end{array}\right]}^{\text{T}}\left({\Omega }_{\text{2}}+{\Omega }_{\text{1}}\right)\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\end{array}\right]+h{\text{e}}^{\text{T}}\left(k\right){\left(\text{BK}\right)}^{\text{T}}\text{P}\left(\text{I}+h\left(\text{A}+\text{BK}\right)\right)\text{x}\left(k\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }+h{\text{x}}^{\text{T}}\left(k\right)\left(\text{I}+h{\left(\text{A}+\text{BK}\right)}^{\text{T}}\right)\text{PBKe}\left(k\right)+2{h}^{2}{\text{e}}^{\text{T}}\left(k\right){\left(\text{BK}\right)}^{\text{T}}\text{PBKe}\left(k\right)\end{array}$ (17)

${\Omega }_{\text{}1}+{\Omega }_{\text{2}}<-\text{R}$

$\begin{array}{l}\stackrel{˜}{\Theta }<{\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\end{array}\right]}^{\text{T}}\left(-\text{R}\right)\left[\begin{array}{c}\text{x}\left(k\right)\\ \text{ω}\left(k\right)\end{array}\right]+h{\text{e}}^{\text{T}}\left(k\right){\left(\text{BK}\right)}^{\text{T}}\text{P}\left(\text{I}+h\left(\text{A}+\text{BK}\right)\right)\text{x}\left(k\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }+h{\text{x}}^{\text{T}}\left(k\right)\left(\text{I}+h{\left(\text{A}+\text{BK}\right)}^{\text{T}}\right)\text{PBKe}\left(k\right)+2{h}^{2}{\text{e}}^{\text{T}}\left(k\right){\left(\text{BK}\right)}^{\text{T}}\text{PBKe}\left(k\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }<-{\lambda }_{\mathrm{min}}\left(\text{R}\right){|\text{x}\left(k\right)|}^{2}+2|\text{x}\left(k\right)|{\alpha }_{1}|\text{e}\left(k\right)|+{\alpha }_{2}{|\text{e}\left(k\right)|}^{2}\end{array}$

$\begin{array}{l}\stackrel{˜}{\Theta }<-{\lambda }_{\mathrm{min}}\left(\text{R}\right){|\text{x}\left(k\right)|}^{2}+2|\text{x}\left(k\right)|{\alpha }_{1}|\text{e}\left(k\right)|+{\alpha }_{2}{|\text{e}\left(k\right)|}^{2}\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }<-{\lambda }_{\mathrm{min}}\left(\text{R}\right){|\text{x}\left(k\right)|}^{2}+2|\text{x}\left(k\right)|\varsigma \mu \left(k\right)\phi +\tau \mu {\left(k\right)}^{2}{\phi }^{2}\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{=}-{\lambda }_{\mathrm{min}}\left(\text{R}\right)\left({|\text{x}\left(k\right)|}^{2}-\frac{2|\text{x}\left(k\right)|\varsigma \mu \left(k\right)\phi }{{\lambda }_{\mathrm{min}}\left(\text{R}\right)}-\frac{\tau \mu {\left(k\right)}^{2}{\phi }^{2}}{{\lambda }_{\mathrm{min}}\left(\text{R}\right)}\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=-{\lambda }_{\mathrm{min}}\left(\text{R}\right)\left(|\text{x}\left(k\right)|-\frac{\varsigma +\sqrt{{\varsigma }^{2}+\tau {\lambda }_{\mathrm{min}}\left(\text{R}\right)}}{{\lambda }_{\mathrm{min}}\left(\text{R}\right)}\mu \left(k\right)\phi \right)×\left(|\text{x}\left(k\right)|-\frac{\varsigma -\sqrt{{\varsigma }^{2}+\tau {\lambda }_{\mathrm{min}}\left(\text{R}\right)}}{{\lambda }_{\mathrm{min}}\left(\text{R}\right)}\mu \left(k\right)\phi \right)\end{array}$

$|\text{x}\left(k\right)|>\frac{\varsigma +\sqrt{{\varsigma }^{2}+\tau {\lambda }_{\mathrm{min}}\left(\text{R}\right)}}{{\lambda }_{\mathrm{min}}\left(\text{R}\right)}\mu \left(k\right)\phi$

4. 数值算例

$\text{A}=\left[\begin{array}{cc}-\text{0}\text{.0756}& -\text{0}\text{.0778}\\ \text{1}\text{.1022}& 0.1986\end{array}\right]$$\text{B}=\left[\begin{array}{cc}0.6077& 0.1\\ 0.9768& 0.0019\end{array}\right]=\text{Q}$$\text{E}=\left[\begin{array}{c}0.5721\\ 0.0211\end{array}\right]$$\text{C}=\left[\begin{array}{cc}1& 0\end{array}\right]$

$\text{P}=\left[\begin{array}{cc}\text{3}\text{.4848}& -\text{0}\text{.3974}\\ -\text{0}\text{.3974}& \text{93}\text{.0087}\end{array}\right]$$\text{G}=\left[\begin{array}{cc}-\text{5}\text{.5967}& -\text{10}\text{.7701}\\ -\text{4}\text{.5307}& \text{22}\text{.3639}\end{array}\right]$$\text{U}=\left[\begin{array}{l}\text{13}\text{.40253}\text{.8044}\\ \text{4}\text{.701417}\text{.5896}\end{array}\right]$

$\text{R}=\left[\begin{array}{ccc}\text{1}\text{.191}1& -\text{10}\text{.6067}& -0.0061\\ \text{11}\text{.9317}& \text{4}\text{.704}1& 0.0325\\ \ast & \ast & 0.8261\end{array}\right]$

$\text{K}={\text{U}}^{-1}\text{G}=\left[\begin{array}{cc}-\text{0}\text{.3727}& -\text{1}\text{.260}1\\ -\text{0}\text{.1579}& \text{1}\text{.6082}\end{array}\right]$

Figure 1. The state trajectories of the closed-loop system

H∞ Control for Delta Operator Systems with State Quantization[J]. 应用数学进展, 2022, 11(02): 621-629. https://doi.org/10.12677/AAM.2022.112069

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