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  智能系统学报  2017, Vol. 12 Issue (6): 899-905  DOI: 10.11992/tis.201605012
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引用本文  

张晓宇, 李平. 切换系统的鲁棒二次公共Lyapunov函数矩阵寻找算法[J]. 智能系统学报, 2017, 12(6), 899-905. DOI: 10.11992/tis.201605012.
ZHANG Xiaoyu, LI Ping. Matrix search algorithm of robust common quadratic Lyapunov function for switched systems[J]. CAAI Transactions on Intelligent Systems, 2017, 12(6), 899-905. DOI: 10.11992/tis.201605012.

基金项目

国家自然科学基金项目(61304024);河北省科技计划项目(15272118);中央高校基本科研业务费基金项目(3142017046,3142016022,3142015101).

通信作者

张晓宇. E-mail:ysuzxy@aliyun.com.

作者简介

张晓宇,男,1978年生,教授,博士,主要研究方向为滑模变结构控制、非线性系统智能自适应控制、复杂动态系统分析、综合及应用。主持完成国家自然科学基金青年基金项目1项、河北省自然科学基金项目1项、河北省科技计划项目1项、各类纵向科研项目5项、横向科研项目3项。已发表学术论文70余篇,其中有20余篇被SCI、EI检索;
李平,女,1992年生,硕士研究生,主要研究方向为安全生产自动化和信息化

文章历史

收稿日期:2016-05-16
网络出版日期:2017-11-28
切换系统的鲁棒二次公共Lyapunov函数矩阵寻找算法
张晓宇, 李平    
华北科技学院 电子信息工程学院,北京 101601
摘要:为了获得不确定线性切换系统稳定性判别的公共二次Lyapunov函数寻找方法,提出了鲁棒公共二次Lyapunov函数的概念,运用矩阵不等式分析,得到了在鲁棒稳定矩阵集对合和不对合的情况下,鲁棒公共二次Lyapunov函数存在的充分性条件以及LMI形式的递推搜寻算法。获得的结果便于计算机实现,对不确定切换系统鲁棒稳定性判别具有一定价值。应用仿真测试验证了其正确性。
关键词切换系统    不确定    公共Lyapunov函数    二次Lyapunov函数    控制    鲁棒    稳定    LMI    
Matrix search algorithm of robust common quadratic Lyapunov function for switched systems
ZHANG Xiaoyu, LI Ping    
School of Electronics and Information Engineering, North China Institute of Science and Technology, Beijing 101601, China
Abstract: To obtain a searching algorithm for the common quadratic Lyapunov function (CQLF) of an uncertain switched system (SS), the concept of the common robust quadratic Lyapunov function (CRQLF) is proposed. In addition, sufficient conditions for the CRQLF, and its corresponding recurrence search algorithm method in LMI forms, are obtained using a matrix inequality analysis when the stable matrix set is both involuntary and voluntary. These results enable easy computer implementation, and are valuable for making robust stability judgments of uncertain SSs. Furthermore, the application simulation test certificates their validity.
Key words: switched system    uncertain    common Lyapunov function    quadratic Lyapunov function    control    robust    stability    LMI    

线性切换系统稳定性判断有几种方法,其中公共Lyapunov函数(common Lyapunov function, CLF)方法是在多Lyapunov函数方法之后被提出来的。其出发点是若切换系统所有子系统存在一个单Lyapunov函数,并且这个Lyapunov函数在整个状态空间中沿着特定的切换序列或者是任意切换都能递减,则整个系统稳定[1-2]。Beldiman等[3]指出通过对原来稳定的非线性切换系统线性化后,其线性化的系统是渐近稳定的。目前研究焦点是如何构造CLF,或者如何判断存在CLF。Dogruel首先提出了CLF方法,证明了切换系统如果存在一个Lyapunov函数 $V(x(t)) > 0$ ,使得所有的子系统满足 $\dot V(x(t)) < 0$ 则对于任意的切换信号切换系统都全局渐近稳定[2]。之后,围绕着CLF存在的代数条件,学者们展开了一系列的研究。Ooba等[4]提出了一对不可交换系统CLF存在的条件。文献[5]证明了若子系统均渐近稳定,且各个子系统的状态矩阵两两相乘时满足交换条件,则系统存在公共二次Lyapunov函数(common quadratic Lyapunov function, CQLF)。Liberzon利用Lie代数研究了线性切换系统存在CLF的代数条件[6],证明了如果由 ${{\mathit{\boldsymbol{A}}}_i},\;i = 1,2, \cdots ,N$ 生成的Lie代数可解,则切换系统存在CQLF。在此基础上,Margaliota进一步研究了非线性切换系统的稳定性[7]

显然,CLF只是切换系统稳定的充分条件,反之,如果切换系统在任意切换信号下全局渐近稳定,是否存在CLF?针对这一问题,Dayawansa证明了若线性切换系统在任意切换信号下全局指数稳定,则线性切换系统存在CLF[8]。Cheng等[9]应用CLF分析了几类切换系统的稳定性,提出了确保闭环切换系统稳定的CLF。

CQLF的存在性必然有一定条件,而且和切换系统的分析和控制器设计密切相关。对CLF存在的充分和必要条件讨论可以参考文献[10]。CLF的构造方法也已经取得了许多成果。基本都是假设 ${{\mathit{\boldsymbol{A}}}_i},\;i = \{ 1,2, \cdots ,N\} $ 是渐近稳定的,即 $\{ {{\mathit{\boldsymbol{A}}}_1}, {{\mathit{\boldsymbol{A}}}_2}, \cdots {{\mathit{\boldsymbol{A}}}_N}\} $ 构成稳定矩阵集。文献[5]考虑了一组可交换稳定矩阵,提出了一种构造CQLF的方法。文献[11]对寻找CLF方法进行了讨论,并且给出了几个CQLF存在的条件。文献[11]给出了稳定矩阵集 ${\mathit{\boldsymbol{A}}}$ 中,矩阵两两不能互换但满足对合条件时,其CQLF的相应构造方法。

本文将讨论不确定线性切换系统的稳定性判定CQLF问题。如果单独考虑带有不确定性的线性切换系统稳定性,即鲁棒稳定性问题,其CQLF的构造将会更加困难。为了克服这个困难,本文提出了公共鲁棒稳定矩阵集的概念,并进一步扩展推出鲁棒稳定矩阵集的CQLF矩阵的判定定理和构造定理。本文的结果对于任意切换规则下的不确定线性切换系统鲁棒控制问题具有以下重要意义:1)有了一套实用的搜寻CQLF的具体LMI算法;2)有了一个判断任意切换规则下系统鲁棒二次稳定的充分性条件。

1 问题描述

考虑如下的不确定切换系统

$\dot x(t) = ({{\mathit{\boldsymbol{A}}}_\sigma } + \Delta {{\mathit{\boldsymbol{A}}}_\sigma })x(t)$ (1)

式中: $x\left( t \right) \in {{\bf{R}}^n}$ 为系统状态变量, $\Delta {{\mathit{\boldsymbol{A}}}_\sigma }$ 表示参数不确定性, $\sigma \left( t \right):{\bf{R}} \to {\bf{N}} \cong \left\{ {1,2, \cdots ,N} \right\}$ 是关于时间t的分段常值函数,称作切换信号(规则)。定义切换序列

$\begin{array}{c}{\mathit{\boldsymbol{Q}}}: = {x_0};\left( {{i_0},{t_0}} \right),\left( {{i_1},{t_1}} \right), \cdots ,\left( {{i_N},{t_N}} \right), \cdots ,\\\forall {i_k} \in {\bf{N}},k \in {{\bf{Z}}^ + },\end{array}$

意味着当 $t \in \left[ {{t_k}} \right.,\left. {{t_{k + 1}}} \right)$ 时运行第ik个子系统。对于切换信号 $\sigma \left( t \right) = i$ $i \in {\bf{N}}$ ,记第i个子系统的参数为

${{\mathit{\boldsymbol{A}}}_\sigma } \buildrel \Delta \over = {{\mathit{\boldsymbol{A}}}_i},\Delta {{\mathit{\boldsymbol{A}}}_\sigma } \buildrel \Delta \over = \Delta {{\mathit{\boldsymbol{A}}}_i}$

因此,在第k次切换,对于 ${t_k} \leqslant t < {t_{k + 1}}$ ,设 $\sigma \left( t \right) = $ i,即 ${i_k} = i \in {\bf{N}}$ 。然后根据式(1),系统描述为

$\dot x(t) = ({{\mathit{\boldsymbol{A}}}_i} + \Delta {{\mathit{\boldsymbol{A}}}_i})x(t)$ (2)

切换系统式(2)满足以下假设。

假设1[14]  不确定参数 $\Delta {{\mathit{\boldsymbol{A}}}_i}$ 满足

$\Delta {{\mathit{\boldsymbol{A}}}_i} = {{\mathit{\boldsymbol{H}}}_{a,i}}{{\mathit{\boldsymbol{F}}}_{a,i}}\left( t \right){{\mathit{\boldsymbol{E}}}_{a,i}}$ (3)

式中: ${{\mathit{\boldsymbol{H}}}_{a,i}} \in {{\bf{R}}^{n \times {r_a}}}$ ${{\mathit{\boldsymbol{E}}}_{a,i}} \in {{\bf{R}}^{{r_a} \times n}}$ 均为已知常数矩阵,未知时变矩阵 ${{\mathit{\boldsymbol{F}}}_{a,i}}\left( t \right)$ 满足

${\mathit{\boldsymbol{F}}}_{a,i}^{\rm{T}}\left( t \right){{\mathit{\boldsymbol{F}}}_{a,i}}\left( t \right) \leqslant {\mathit{\boldsymbol{I}}}$

接下来给出本文用到的常用引理。

引理1[13]  (Schur补引理) 对于给定对称矩阵 ${\mathit{\boldsymbol{S}}} = \left[ {\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{S}}}_{11}}} & {{{\mathit{\boldsymbol{S}}}_{12}}}\\ {{{\mathit{\boldsymbol{S}}}_{21}}} & {{{\mathit{\boldsymbol{S}}}_{22}}} \end{array}} \right]$ ,以下3个条件是等价的:

1) ${\mathit{\boldsymbol{S}}} < 0$

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

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

引理2[14]  设HE是具有适当维数的实常数矩阵, ${\mathit{\boldsymbol{F}}}\left( t \right)$ 满足 ${{\mathit{\boldsymbol{F}}}^{\rm{T}}}\left( t \right){\mathit{\boldsymbol{F}}}\left( t \right) \leqslant {\mathit{\boldsymbol{I}}}$ 。那么对于任意常数 $\varepsilon > 0$ ,有

${\mathit{\boldsymbol{HF}}}\left( t \right){\mathit{\boldsymbol{E}}} + {{\mathit{\boldsymbol{E}}}^{\rm{T}}}{{\mathit{\boldsymbol{F}}}^{\rm{T}}}\left( t \right){{\mathit{\boldsymbol{H}}}^{\rm{T}}} \leqslant {\varepsilon ^{ - 1}}{\mathit{\boldsymbol{H}}}{{\mathit{\boldsymbol{H}}}^{\rm{T}}} + \varepsilon {{\mathit{\boldsymbol{E}}}^{\rm{T}}}{\mathit{\boldsymbol{E}}}$
2 已有结果

假设稳定矩阵集 ${\mathit{\boldsymbol{A}}} = \left\{ {{{\mathit{\boldsymbol{A}}}_1},{{\mathit{\boldsymbol{A}}}_2}, \cdots ,{{\mathit{\boldsymbol{A}}}_N}} \right\}$ (矩阵集中每一个矩阵对应的线性子系统都是稳定的集合,成为稳定矩阵集)并且 ${\mathit{\boldsymbol{A}}}$ 是对合的, $\left[ {{{\mathit{\boldsymbol{A}}}_N},{{\mathit{\boldsymbol{A}}}_i}} \right] \!=\! \sum\nolimits_{k = 1}^N {\gamma _i^k{{\mathit{\boldsymbol{A}}}_k}} $ $\forall i \in {\bf{N}}$ 。这里 $\gamma _i^k > 0$ 是标量系数参数。任选 ${{\mathit{\boldsymbol{P}}}_{N - 1}} > 0$ ,并设定

${{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} = - {{\mathit{\boldsymbol{P}}}_{N - 1}}$

引理3[12]  对于稳定矩阵集 ${\mathit{\boldsymbol{A}}}$ ,如果 $\forall i = 1, 2, \cdots ,$ N – 1 满足以下条件

$\left\{ \begin{array}{l}{\rm{max}}\left( {\gamma _i^i} \right) < 2{\rm{min}}\left| {{\rm{Re}}\lambda \left( {{{\mathit{\boldsymbol{A}}}_N}} \right)} \right|\\ - {{\mathit{\boldsymbol{P}}}_{i,N - 1}} + \gamma _i^N{{\mathit{\boldsymbol{P}}}_{N - 1}} - \sum\limits_{k = 1,k \ne i}^{N - 1} {\gamma _i^k{{\mathit{\boldsymbol{P}}}_{k,N}} > 0} \end{array} \right.$ (4)

式中: ${{\mathit{\boldsymbol{P}}}_{i,j}} = {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_j} + {{\mathit{\boldsymbol{P}}}_j}{{\mathit{\boldsymbol{A}}}_i}$ $\forall i,j \in {\bf{N}}$ ,则 ${{\mathit{\boldsymbol{P}}}_N}$ ${\mathit{\boldsymbol{A}}}$ 的CQLF。

当稳定矩阵集 ${\mathit{\boldsymbol{A}}}$ 不是对合的,也即关系式 $\left[ {{{\mathit{\boldsymbol{A}}}_N},{{\mathit{\boldsymbol{A}}}_i}} \right] = \sum\nolimits_{k = 1}^N {\gamma _i^k{{\mathit{\boldsymbol{A}}}_k}} $ $\forall i \in {\bf{N}}$ 不再成立时, ${\mathit{\boldsymbol{A}}}$ 的CQLF如何构造呢?仍然首先任选 ${{\mathit{\boldsymbol{P}}}_{N - 1}} > 0$ ,并仍然设定 ${{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} = - {{\mathit{\boldsymbol{P}}}_{N - 1}}$ 。而且,

$\left[ {{{\mathit{\boldsymbol{A}}}_N},{{\mathit{\boldsymbol{A}}}_j}} \right] = {{\mathit{\boldsymbol{C}}}_{N,j}},\;\forall j \in {\bf{N}}$ (5)

引理4[12]  对于稳定矩阵集 ${\mathit{\boldsymbol{A}}}$ ,定义 ${{\mathit{\boldsymbol{C}}}_{N,j}},\;\forall j \in {\bf{N}}$ 如式(5)。如果 $\forall i = 1, 2, \cdots ,N - 1$ ,满足

${{\mathit{\boldsymbol{P}}}_{i,N - 1}} + {{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{C}}}_{N,i}} + {\mathit{\boldsymbol{C}}}_{N,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} < 0$ (6)

${{\mathit{\boldsymbol{P}}}_N}$ 构成 ${\mathit{\boldsymbol{A}}}$ 的一个CQLF。

3 主要结果 3.1 鲁棒二次稳定

引理5 若 $\forall {\varepsilon _i} > 0$ ,线性矩阵不等式(Linear matrix inequality, LMI)

$\left[ {\begin{array}{*{20}{c}}{{{\mathit{\boldsymbol{P}}}_i}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{\mathit{\boldsymbol{P}}} + {{\mathit{\boldsymbol{L}}}_i}} & {{{\mathit{\boldsymbol{P}}}_i}{{\mathit{\boldsymbol{H}}}_{a,i}}}\\* & { - {\varepsilon _i}{\mathit{\boldsymbol{I}}}}\end{array}} \right] < 0,\;\forall i \in {\bf{N}}$ (7)

有正定对称阵解 ${{\mathit{\boldsymbol{P}}}_i}$ ,则每个子系统(2)是鲁棒二次稳定的。其中 ${{\mathit{\boldsymbol{G}}}_i}$ ${{\mathit{\boldsymbol{I}}}_i}$ 是系统不确定性引起的Lyapunov方程矩阵项

${{\mathit{\boldsymbol{G}}}_i} = \varepsilon _i^{ - 1}{{\mathit{\boldsymbol{H}}}_{a,i}}{\mathit{\boldsymbol{H}}}_{a,i}^{\rm{T}},{\rm{ }}{{\mathit{\boldsymbol{L}}}_i} = {\varepsilon _i}{\mathit{\boldsymbol{E}}}_{a,i}^{\rm{T}}{{\mathit{\boldsymbol{E}}}_{a,i}}$ (8)

证明 选取各子系统(2)的Lyapunov函数:

${V_i}\left( {{\mathit{\boldsymbol{x}}}(t)} \right) = {{\mathit{\boldsymbol{x}}}^{\rm{T}}}\left( t \right){{\mathit{\boldsymbol{P}}}_i}{\mathit{\boldsymbol{x}}}\left( t \right)$

沿子系统(2)求其时间导数:

${\dot V_i}\left( {{\mathit{\boldsymbol{x}}}(t)} \right) = {{\mathit{\boldsymbol{x}}}^{\rm{T}}}\left[ {{{\mathit{\boldsymbol{P}}}_i}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_i} + {{\mathit{\boldsymbol{P}}}_i}\Delta {{\mathit{\boldsymbol{A}}}_i} + \Delta {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_i}} \right]{\mathit{\boldsymbol{x}}}$

根据假设1和引理2,对于任意 ${\varepsilon _i} > 0$ 有不等式:

${{\mathit{\boldsymbol{P}}}_i}\Delta {{\mathit{\boldsymbol{A}}}_i} + \Delta {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_i} \leqslant {{\mathit{\boldsymbol{G}}}_i} = \varepsilon _i^{ - 1}{{\mathit{\boldsymbol{P}}}_i}{{\mathit{\boldsymbol{H}}}_{a,i}}{\mathit{\boldsymbol{H}}}_{a,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_i} + {\varepsilon _i}{\mathit{\boldsymbol{E}}}_{a,i}^{\rm{T}}{{\mathit{\boldsymbol{E}}}_{a,i}}$

成立。将其代入 ${\dot V_i}\left( {x(t)} \right)$ 有不等式:

${\dot V_i}\left( {{\mathit{\boldsymbol{x}}}(t)} \right) \leqslant {{\mathit{\boldsymbol{x}}}^{\rm{T}}}\left[ {{{\mathit{\boldsymbol{P}}}_i}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_i} + {{\mathit{\boldsymbol{P}}}_i}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_i} + {{\mathit{\boldsymbol{L}}}_i}} \right]{\mathit{\boldsymbol{x}}}$

成立。根据引理7,显然若满足,则有

${\dot V_i}\left( {{\mathit{\boldsymbol{x}}}(t)} \right) < - {{\mathit{\boldsymbol{x}}}^{\rm{T}}}\left( t \right){{\mathit{\boldsymbol{Q}}}_i}{\mathit{\boldsymbol{x}}}\left( t \right)$

成立,其中 ${{\mathit{\boldsymbol{Q}}}_i}$ 是某一正定矩阵。那么每个子系统(2)是鲁棒二次稳定的。

切换系统(2)的系数矩阵满足如下假设。

假设2 稳定矩阵集 ${\mathit{\boldsymbol{A}}} = \left\{ {{{\mathit{\boldsymbol{A}}}_1},{{\mathit{\boldsymbol{A}}}_2}, \cdots ,{{\mathit{\boldsymbol{A}}}_N}} \right\}$ 的每个稳定矩阵是鲁棒二次稳定的,即满足引理5。此时我们称 ${\mathit{\boldsymbol{A}}}$ 是一个鲁棒稳定矩阵集。

定义

${{\mathit{\boldsymbol{P}}}_{i,j}} = {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_j} + {{\mathit{\boldsymbol{P}}}_j}{{\mathit{\boldsymbol{A}}}_i},\;\forall i,j \in {\bf{N}}$ (9)

以及

${{\mathit{\boldsymbol{U}}}_{i,N}} = {{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{L}}}_i}$ (10)
3.2 鲁棒CLF矩阵

若鲁棒稳定矩阵集 ${\mathit{\boldsymbol{A}}}$ 中的矩阵是对合的,即仍然认为 $\left[ {{{\mathit{\boldsymbol{A}}}_N},{{\mathit{\boldsymbol{A}}}_i}} \right] = \sum\nolimits_{k = 1}^N {\gamma _i^k{{\mathit{\boldsymbol{A}}}_k}} $ $\forall i \in {\bf{N}}$ ,这里 $\gamma _i^k > 0$ 是标量系数参数。对于任选 ${{\mathit{\boldsymbol{P}}}_{N - 1}} > 0$ ,仍然设定

${{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N}{\rm{ + }}{{\mathit{\boldsymbol{U}}}_{N,N}} = - {{\mathit{\boldsymbol{P}}}_{N - 1}}$ (11)

根据上述CQLF引理3,我们得到以下推论。

推论1 对于满足引理5的鲁棒稳定矩阵集 ${\mathit{\boldsymbol{A}}} \!=\! \left\{ {{{\mathit{\boldsymbol{A}}}_1},{{\mathit{\boldsymbol{A}}}_2}, \cdots \! ,{{\mathit{\boldsymbol{A}}}_N}} \right\}$ ,如果 $\forall i \!=\! 1, 2, \cdots \! ,N - 1$ 满足以下条件

$\left\{ \begin{array}{l}{\rm{max}}\left( {\gamma _i^i} \right) < 2{\rm{min}}\left| {{\rm{Re}}\lambda \left( {{A_N}} \right)} \right|{\rm{}}\\ - {{\mathit{\boldsymbol{P}}}_{i,N - 1}} + \gamma _i^N{{\mathit{\boldsymbol{P}}}_{N - 1}} - \sum\limits_{k = 1,k \ne i}^{N - 1} {\gamma _i^k{{\mathit{\boldsymbol{P}}}_{k,N}}} + \gamma _i^i{{\mathit{\boldsymbol{U}}}_{i,N}} + \\\quad\quad{\rm{ }}\gamma _i^N{{\mathit{\boldsymbol{U}}}_{N,N}} + {{\mathit{\boldsymbol{U}}}_{i,N}}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,N}} - {\rm{ }}{{\mathit{\boldsymbol{U}}}_{N,N}}{{\mathit{\boldsymbol{A}}}_i} - \\[5pt]\quad\quad{\rm{ }}{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{N,N}} > 0{\rm{}}\end{array} \right.$ (12)

${{\mathit{\boldsymbol{P}}}_N}$ ${\mathit{\boldsymbol{A}}}$ 的鲁棒二次CLF矩阵,即 $\forall i \in {\bf{N}}$ 满足Riccati不等式:

${{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{L}}}_i} < 0$ (13)

证明

$\begin{array}{c}\left( {{{\mathit{\boldsymbol{P}}}_{i,N}} + {{\mathit{\boldsymbol{U}}}_{i,N}}} \right)\left( {{{\mathit{\boldsymbol{A}}}_N} + \gamma _i^i{\mathit{\boldsymbol{I}}}/2} \right) + {\left( {{{\mathit{\boldsymbol{A}}}_N} + \gamma _i^i{\mathit{\boldsymbol{I}}}/2} \right)^{\rm{T}}} \times \left( {{{\mathit{\boldsymbol{P}}}_{i,N}} + {{\mathit{\boldsymbol{U}}}_{i,N}}} \right) = \\[5pt]\left( {{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{U}}}_{i,N}}} \right){{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}\left( {{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_i} + } \right.\end{array}$
$\begin{array}{c}\left. {{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{U}}}_{i,N}}} \right) + \gamma _i^i\left( {{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{U}}}_{i,N}}} \right) = \\[5pt]{{\mathit{\boldsymbol{P}}}_N}\left( {{{\mathit{\boldsymbol{A}}}_N}{{\mathit{\boldsymbol{A}}}_i} - \sum\limits_{k = 1}^N {\gamma _i^k} {{\mathit{\boldsymbol{A}}}_k}} \right) + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_i} + \\[5pt]\left( {{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{\mathit{\boldsymbol{A}}}_N^{\rm{T}} - \sum\limits_{k = 1}^N {\gamma _i^k} {\mathit{\boldsymbol{A}}}_k^{\rm{T}}} \right){{\mathit{\boldsymbol{P}}}_N} + \gamma _i^i\left( {{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N}} \right. + \\[5pt]\left. {{{\mathit{\boldsymbol{U}}}_{i,N}}} \right) + {{\mathit{\boldsymbol{U}}}_{i,N}}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,N}} = \\[5pt]\left( {{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N}} \right){{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}\left( {{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N}} \right) - \\[5pt]\gamma _i^N\left( {{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N}} \right) - \sum\limits_{k = 1,k \ne i}^{N - 1} {\gamma _i^k} \left( {{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_k} + } \right.\\[5pt]\left. {{\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N}} \right) + {{\mathit{\boldsymbol{U}}}_{i,N}}\left( {{{\mathit{\boldsymbol{A}}}_N} + \gamma _i^i} \right) + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,N}} = \\[5pt] - {{\mathit{\boldsymbol{P}}}_{i,N - 1}} + \gamma _i^N{{\mathit{\boldsymbol{P}}}_{N - 1}} - \sum\limits_{k = 1,k \ne i}^{N - 1} {\gamma _i^k} {{\mathit{\boldsymbol{P}}}_{k,N}} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,N}} + \\[5pt]{{\mathit{\boldsymbol{U}}}_{i,N}}\left( {{{\mathit{\boldsymbol{A}}}_N} + \gamma _i^i} \right) - {\rm{ }}{{\mathit{\boldsymbol{U}}}_{N,N}}\left( {{{\mathit{\boldsymbol{A}}}_i} - \gamma _i^N} \right) - {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{N,N}}\end{array}$

若式(12)第2个条件 $\forall i = 1,2, \cdots ,N - 1$ 成立,表明两个矩阵 ${{\mathit{\boldsymbol{P}}}_{i,N}} + {{\mathit{\boldsymbol{U}}}_{i,N}}$ ${{\mathit{\boldsymbol{A}}}_N} + \displaystyle\frac{{\gamma _i^i}}{2}{\mathit{\boldsymbol{I}}}$ 转置正定。由式(12)的第1个不等式知 ${{\mathit{\boldsymbol{A}}}_N} + \displaystyle\frac{{\gamma _i^i}}{2}{\mathit{\boldsymbol{I}}}$ 是稳定的,因此 $\forall i = 1,$ $2, \cdots \! , N - 1$

${{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{L}}}_i} < 0$ (14)

成立,即 ${{\mathit{\boldsymbol{P}}}_N}$ 对于每一个矩阵 ${{\mathit{\boldsymbol{A}}}_i}$ 都满足包含不确定项 ${{\mathit{\boldsymbol{U}}}_{i,N}}$ 的Riccati不等式(14),因此 ${{\mathit{\boldsymbol{P}}}_N}$ ${\mathit{\boldsymbol{A}}}$ 的鲁棒二次CLF。

若鲁棒稳定矩阵集 ${\mathit{\boldsymbol{A}}}$ 中的矩阵不是对合的,根据CQLF引理4,我们得到以下推论。

推论2 对于鲁棒稳定矩阵集 ${\mathit{\boldsymbol{A}}}$ ,定义 ${C_{N,j}},\forall j \in$ $ {\bf{N}}$ ,如(5)。若 $\forall i = 1, 2, \cdots ,N - 1$ 满足

$\begin{array}{c}{{\mathit{\boldsymbol{P}}}_{i,N - 1}} + {{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{C}}}_{N,i}} + {\mathit{\boldsymbol{C}}}_{N,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{U}}}_{N,N}}{{\mathit{\boldsymbol{A}}}_i} + \\{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{N,N}} - {{\mathit{\boldsymbol{U}}}_{i,N}}{{\mathit{\boldsymbol{A}}}_N} - {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,N}} < 0\end{array}$ (15)

${{\mathit{\boldsymbol{P}}}_N}$ 构成 ${\mathit{\boldsymbol{A}}}$ 的一个鲁棒二次CLF矩阵。即

${{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{L}}}_i} < 0$ (16)

证明

$\begin{array}{c}\left( {{{\mathit{\boldsymbol{P}}}_{i,N}} + {{\mathit{\boldsymbol{U}}}_{i,N}}} \right){{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}\left( {{{\mathit{\boldsymbol{P}}}_{i,N}} + {{\mathit{\boldsymbol{U}}}_{i,N}}} \right) = \\[5pt]\left( {{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{U}}}_{i,N}}} \right){{\mathit{\boldsymbol{A}}}_N} + \\[5pt]{\mathit{\boldsymbol{A}}}_N^{\rm{T}}\left( {{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{U}}}_{i,N}}} \right) = \\[5pt]{{\mathit{\boldsymbol{P}}}_N}\left( {{{\mathit{\boldsymbol{A}}}_N}{{\mathit{\boldsymbol{A}}}_i} - {{\mathit{\boldsymbol{C}}}_{N,i}}} \right) + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_i} + \\[5pt]\left( {{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{\mathit{\boldsymbol{A}}}_N^{\rm{T}} - {\mathit{\boldsymbol{C}}}_{N,i}^{\rm{T}}} \right){{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{U}}}_{i,N}}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,N}} = \\[5pt]\left( {{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N}} \right){{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}\left( {{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N}} \right) - \\[5pt]{{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{C}}}_{N,i}} - {\mathit{\boldsymbol{C}}}_{N,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{U}}}_{i,N}}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,N}} = \\[5pt]- \left( {{{\mathit{\boldsymbol{P}}}_{i,N - 1}} + {{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{C}}}_{N,i}} + {\mathit{\boldsymbol{C}}}_{N,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{U}}}_{N,N}}{{\mathit{\boldsymbol{A}}}_i} + } \right.\\[5pt]\left. {{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{N,N}} - {{\mathit{\boldsymbol{U}}}_{i,N}}{{\mathit{\boldsymbol{A}}}_N} - {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,N}}} \right)\end{array}$

若不等式(15)条件成立,则 $\forall i = 1,2, \cdots ,N - 1$

$\left( {{{\mathit{\boldsymbol{P}}}_{i,N}} + {{\mathit{\boldsymbol{U}}}_{i,N}}} \right){{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}\left( {{{\mathit{\boldsymbol{P}}}_{i,N}} + {{\mathit{\boldsymbol{U}}}_{i,N}}} \right) > 0$

成立,表明两个矩阵 ${{\mathit{\boldsymbol{P}}}_{i,N}} + {{\mathit{\boldsymbol{U}}}_{i,N}}$ ${{\mathit{\boldsymbol{A}}}_N}$ 的转置积正定。已知 ${{\mathit{\boldsymbol{A}}}_N}$ 是稳定的,因此 ${{\mathit{\boldsymbol{P}}}_{i,N}} + {{\mathit{\boldsymbol{U}}}_{i,N}} < 0$ 成立,也即 $\forall i = 1,2, \cdots ,N - 1$ 式(17)成立

${{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{L}}}_i} < 0$ (17)

那么 ${{\mathit{\boldsymbol{P}}}_N}$ ${\mathit{\boldsymbol{A}}}$ 的一个CQLF,且 ${{\mathit{\boldsymbol{P}}}_N}$ 对于每一个矩阵 ${{\mathit{\boldsymbol{A}}}_i}$ 都满足包含不确定项 ${{\mathit{\boldsymbol{U}}}_{i,N}}$ 的Riccati不等式(17),因此 ${{\mathit{\boldsymbol{P}}}_N}$ ${\mathit{\boldsymbol{A}}}$ 的鲁棒CQLF。

3.3 递推CQLF矩阵

根据引理4,我们进一步得到稳定矩阵集的如下CQLF的构造算法定理。

定理1  若稳定矩阵集 ${\mathit{\boldsymbol{A}}} = \left\{ {{{\mathit{\boldsymbol{A}}}_1},{{\mathit{\boldsymbol{A}}}_2}, \cdots ,{{\mathit{\boldsymbol{A}}}_N}} \right\}$ 存在CLF,则 $\forall i = 1,2, \cdots ,k - 1,k \in {\bf{N}}$ 满足

$\left\{ \begin{array}{l}{{\mathit{\boldsymbol{P}}}_{k,k}} < 0\\{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_{k,k}} + {{\mathit{\boldsymbol{P}}}_{k,k}}{{\mathit{\boldsymbol{A}}}_i} - {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{C}}}_{k,i}} - {\mathit{\boldsymbol{C}}}_{k,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} > 0\end{array} \right.$ (18)

的正定对称阵 ${{\mathit{\boldsymbol{P}}}_k}$ ,即 ${{\mathit{\boldsymbol{A}}}_k} = \left\{ {{{\mathit{\boldsymbol{A}}}_1},{{\mathit{\boldsymbol{A}}}_2}, \cdots ,{{\mathit{\boldsymbol{A}}}_k}} \right\}$ 的CQLF矩阵。其中 ${{\mathit{\boldsymbol{C}}}_{i,j}}$ 是矩阵集两两矩阵交换差:

${{\mathit{\boldsymbol{C}}}_{i,j}} = {{\mathit{\boldsymbol{A}}}_i}{{\mathit{\boldsymbol{A}}}_j} - {{\mathit{\boldsymbol{A}}}_j}{{\mathit{\boldsymbol{A}}}_i}$

证明

$\begin{array}{c}{{\mathit{\boldsymbol{P}}}_{i,k}}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{P}}}_{i,k}} =\left( {{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k}} \right){{\mathit{\boldsymbol{A}}}_k} + \\[5pt] {\mathit{\boldsymbol{A}}}_k^{\rm{T}}\left( {{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k}} \right) = {{\mathit{\boldsymbol{P}}}_k}\left( {{{\mathit{\boldsymbol{A}}}_k}{{\mathit{\boldsymbol{A}}}_i} - {{\mathit{\boldsymbol{C}}}_{k,i}}} \right) +\\[5pt] {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_i} +{\rm{ }}{\left( {{{\mathit{\boldsymbol{A}}}_k}{{\mathit{\boldsymbol{A}}}_i} - {{\mathit{\boldsymbol{C}}}_{k,i}}} \right)^{\rm{T}}}{{\mathit{\boldsymbol{P}}}_k} = \\[5pt]\left( {{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k}} \right){{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}\left( {{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k}} \right) - \\[5pt]{\rm{ }}{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{C}}}_{k,i}} - {\mathit{\boldsymbol{C}}}_{k,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} ={\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_{k,k}} + {{\mathit{\boldsymbol{P}}}_{k,k}}{{\mathit{\boldsymbol{A}}}_i} - \\[5pt] {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{C}}}_{k,i}} - {\mathit{\boldsymbol{C}}}_{k,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k}\end{array}$

如果不等式(18)满足,则显然有

${{\mathit{\boldsymbol{P}}}_{i,k}}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{P}}}_{i,k}} > 0$

因为 ${{\mathit{\boldsymbol{A}}}_k}$ 是稳定的,则有

${{\mathit{\boldsymbol{P}}}_{i,k}} = {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_i} < 0,{\rm{ }}\forall i = 1, 2, \cdots ,k$

成立。因此 ${{\mathit{\boldsymbol{P}}}_k}$ ${{\mathit{\boldsymbol{A}}}_k} = \left\{ {{{\mathit{\boldsymbol{A}}}_1},{{\mathit{\boldsymbol{A}}}_2}, \cdots ,{{\mathit{\boldsymbol{A}}}_k}} \right\}$ 的CQLF。

假设一个鲁棒稳定矩阵集 ${\mathit{\boldsymbol{A}}}$ ,其中的每个稳定矩阵是鲁棒稳定的,即满足Lyapunov方程(7)。仍然定义(9),(10),但是不定义(11)。而且矩阵集 ${\mathit{\boldsymbol{A}}}$ 不是对合的。根据上述CQLF定理1,我们得到以下鲁棒二次CLF的构造算法定理。

定理2 假设鲁棒稳定矩阵集 ${\mathit{\boldsymbol{A}}} \!=\! \left\{ {{{\mathit{\boldsymbol{A}}}_1},{{\mathit{\boldsymbol{A}}}_2},} \right. \cdots \left. {,{{\mathit{\boldsymbol{A}}}_N}} \right\}$ 存在CLF。若存在任意正数 ${\varepsilon _1},{\varepsilon _2}, \cdots ,{\varepsilon _k}$ 以及 $\forall i = $ $ 1, 2, \cdots ,k - 1,k \in {\bf{N}}$ 满足不等式

$\left\{ \begin{array}{l}{{\mathit{\boldsymbol{P}}}_{k,k}} + {{\mathit{\boldsymbol{U}}}_{k,k}} < 0,\\[5pt]{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_{k,k}} + {{\mathit{\boldsymbol{P}}}_{k,k}}{{\mathit{\boldsymbol{A}}}_i} - {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{C}}}_{k,i}} - {\mathit{\boldsymbol{C}}}_{k,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + \\[5pt]\quad\quad{{\mathit{\boldsymbol{U}}}_{i,k}}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,k}} > 0\end{array} \right.$ (19)

的正定对称阵 ${{\mathit{\boldsymbol{P}}}_k}$ ,则 ${{\mathit{\boldsymbol{P}}}_k}$ ${{\mathit{\boldsymbol{A}}}_k} = \left\{ {{{\mathit{\boldsymbol{A}}}_1},{{\mathit{\boldsymbol{A}}}_2}, \cdots ,{{\mathit{\boldsymbol{A}}}_k}} \right\}$ 的鲁棒二次CLF矩阵。即 $\forall i = 1,2, \cdots ,k$ 满足Riccati不等式

${{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + {{\mathit{\boldsymbol{U}}}_{i,k}} < 0$ (20)

其中是矩阵集中两两矩阵交换差。

证明

$\begin{array}{c}\left( {{{\mathit{\boldsymbol{P}}}_{i,k}}{\rm{ + }}{{\mathit{\boldsymbol{U}}}_{i,k}}} \right){{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}\left( {{{\mathit{\boldsymbol{P}}}_{i,k}} + {{\mathit{\boldsymbol{U}}}_{i,k}}} \right) = \\[5pt]\left( {{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + {{\mathit{\boldsymbol{U}}}_{i,k}}} \right){{\mathit{\boldsymbol{A}}}_k} + \\[5pt]{\rm{ }}{\mathit{\boldsymbol{A}}}_k^{\rm{T}}\left( {{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + {{\mathit{\boldsymbol{U}}}_{i,k}}} \right) = \\[5pt]{{\mathit{\boldsymbol{P}}}_k}\left( {{{\mathit{\boldsymbol{A}}}_k}{{\mathit{\boldsymbol{A}}}_i} - {{\mathit{\boldsymbol{C}}}_{k,i}}} \right) + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_i} + \\[5pt]{\rm{ }}{\left( {{{\mathit{\boldsymbol{A}}}_k}{{\mathit{\boldsymbol{A}}}_i} - {{\mathit{\boldsymbol{C}}}_{k,i}}} \right)^{\rm{T}}}{{\mathit{\boldsymbol{P}}}_k} + {{\mathit{\boldsymbol{U}}}_{i,k}}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,k}}=\end{array}$
$\begin{array}{c}\left( {{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k}} \right){{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}\left( {{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k}} \right) - \\[4pt]{\rm{ }}{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{C}}}_{k,i}} - {\mathit{\boldsymbol{C}}}_{k,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + {{\mathit{\boldsymbol{U}}}_{i,k}}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,k}} = \\[4pt]{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_{k,k}} + {{\mathit{\boldsymbol{P}}}_{k,k}}{{\mathit{\boldsymbol{A}}}_i} - {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{C}}}_{k,i}} - {\mathit{\boldsymbol{C}}}_{k,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + \\[4pt]{\rm{ }}{{\mathit{\boldsymbol{U}}}_{i,k}}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,k}}\end{array}$

如果不等式(19)满足,则显然有

$\left( {{{\mathit{\boldsymbol{P}}}_{i,k}}{\rm{ + }}{{\mathit{\boldsymbol{U}}}_{i,k}}} \right){{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}\left( {{{\mathit{\boldsymbol{P}}}_{i,k}}{\rm{ + }}{{\mathit{\boldsymbol{U}}}_{i,k}}} \right) > 0$

因为 ${A_k}$ 是稳定的,则 $\forall i = 1, 2, \cdots ,k - 1$

${{\mathit{\boldsymbol{P}}}_{i,k}}{\rm{ + }}{{\mathit{\boldsymbol{U}}}_{i,k}} = {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_i}{\rm{ + }}{{\mathit{\boldsymbol{U}}}_{i,k}} < 0$

成立。因此 ${P_k}$ ${{\mathit{\boldsymbol{A}}}_k} = \left\{ {{{\mathit{\boldsymbol{A}}}_1},{{\mathit{\boldsymbol{A}}}_2}, \cdots ,{{\mathit{\boldsymbol{A}}}_k}} \right\}$ 的鲁棒二次CLF。

由于实际系统矩阵往往不容易两两可交换,或者说构成对合矩阵集。因此,在实际控制应用中,引理3、推论1并不实用。而引理4和本文给出的定理1、定理2、推论2满足大多数实际应用计算情况。

3.4 鲁棒二次Lyapunov函数矩阵寻找算法

定理2在实际应用中更加广泛,因此我们进一步给出实用的鲁棒二次CLF寻找算法。

推论3 给定系统(2)的鲁棒稳定矩阵集 ${\mathit{\boldsymbol{A}}} = \left\{ {{{\mathit{\boldsymbol{A}}}_1},{{\mathit{\boldsymbol{A}}}_2}, \cdots ,{{\mathit{\boldsymbol{A}}}_N}} \right\}$ 。若 ${\mathit{\boldsymbol{A}}}$ 存在CQLF,则 $\forall k \in {\bf{N}}\left( {k > 1} \right)$ ,若有矩阵 ${{\mathit{\boldsymbol{P}}}_k} = {\mathit{\boldsymbol{P}}}_k^{\rm{T}},{{\mathit{\boldsymbol{P}}}_k} > 0$ 及任意 ${\varepsilon _i} > 0$ 满足LMIs

$\left\{ \begin{array}{l}\left[ {\begin{array}{*{20}{c}}{{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + {{\mathit{\boldsymbol{L}}}_k}} & {{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{H}}}_{a,k}}}\\* & { - {\varepsilon _k}{\mathit{\boldsymbol{I}}}}\end{array}} \right] < 0,\\[9pt]\left[ {\begin{array}{*{20}{c}}{{\Xi _{i,k}}} & {\left( {{\mathit{\boldsymbol{I}}} - {\mathit{\boldsymbol{A}}}_k^{\rm{T}}} \right){{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{H}}}_{a,i}}}\\* & {{\varepsilon _i}{\mathit{\boldsymbol{I}}}}\end{array}} \right] > 0,{\rm{ }}i = 1, 2, \cdots ,k - 1\end{array} \right.$ (21)

那么 ${{\mathit{\boldsymbol{P}}}_k}$ 是矩阵集 ${{\mathit{\boldsymbol{A}}}_k} = \left\{ {{{\mathit{\boldsymbol{A}}}_1},{{\mathit{\boldsymbol{A}}}_2}, \cdots ,{{\mathit{\boldsymbol{A}}}_k}} \right\}$ 的鲁棒二次CLF矩阵。其中

$\begin{array}{c}{\Xi _{i,k}} = {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_{k,k}} + {{\mathit{\boldsymbol{P}}}_{k,k}}{{\mathit{\boldsymbol{A}}}_i} - {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{C}}}_{k,i}} - \\[4pt]{\mathit{\boldsymbol{C}}}_{k,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + {{\mathit{\boldsymbol{L}}}_i}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{L}}}_i}\end{array}$ (22)

  证明 式(21)中第1个LMI证明 ${{\mathit{\boldsymbol{P}}}_k}$ ${{\mathit{\boldsymbol{A}}}_k}$ 的鲁棒二次Lyapunov函数矩阵,等价于式(19)的第1个不等式。若式(21)第二个LMI满足,有

${\Xi _{i,k}} - \left( {{\mathit{\boldsymbol{I}}} - {\mathit{\boldsymbol{A}}}_k^{\rm{T}}} \right){{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_k}\left( {{\mathit{\boldsymbol{I}}} - {{\mathit{\boldsymbol{A}}}_k}} \right) > 0$

将式(22)代入,有不等式:

$\begin{array}{c}{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_{k,k}} + {{\mathit{\boldsymbol{P}}}_{k,k}}{{\mathit{\boldsymbol{A}}}_i} - {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{C}}}_{k,i}} - {\mathit{\boldsymbol{C}}}_{k,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + \\[4pt]{{\mathit{\boldsymbol{L}}}_i}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{L}}}_i} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_k} + \\[4pt]{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_k} - {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_k} - {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_k} > 0\end{array}$

再根据,上式即

$\begin{array}{c}{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_{k,k}} + {{\mathit{\boldsymbol{P}}}_{k,k}}{{\mathit{\boldsymbol{A}}}_i} - {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{C}}}_{k,i}} - {\mathit{\boldsymbol{C}}}_{k,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + {{\mathit{\boldsymbol{U}}}_{i,k}}{{\mathit{\boldsymbol{A}}}_k} + \\[4pt]{\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,k}} - {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_k} - {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_k} > 0\end{array}$

由式(8)可见, ${\mathit{\boldsymbol{G}}}_i^{\rm{T}} = {{\mathit{\boldsymbol{G}}}_i}$ ${{\mathit{\boldsymbol{G}}}_i} \geqslant 0$ ${\mathit{\boldsymbol{L}}}_i^{\rm{T}} = {{\mathit{\boldsymbol{L}}}_i}$ ${{\mathit{\boldsymbol{L}}}_i} \geqslant 0$ 。由上不等式显而易见,不等式

$\begin{array}{c}{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_{k,k}} + {{\mathit{\boldsymbol{P}}}_{k,k}}{{\mathit{\boldsymbol{A}}}_i} - {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{C}}}_{k,i}} - {\mathit{\boldsymbol{C}}}_{k,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + \\[4pt]{{\mathit{\boldsymbol{U}}}_{i,k}}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,k}} > 0\end{array}$

成立。这样LMI(21)就等价于定理2中式(19)。那么推论3与定理2是等价的。

推论3给出了便于计算机计算寻找CQLF矩阵的递推算法。我们可以首先给出某一个子系统的鲁棒二次Lyapunov函数矩阵 ${{\mathit{\boldsymbol{P}}}_1}$ ,然后令 $k \!=\! 2,3, \cdots $ ,运用MATLAB的LMI工具箱求解LMIs,每次求出的 ${{\mathit{\boldsymbol{P}}}_k}$ 即是子系统 ${\mathit{\boldsymbol{A}}} = \left\{ {{\mathit{\boldsymbol{A}}_1},{\mathit{\boldsymbol{A}}_2}, \cdots ,{\mathit{\boldsymbol{A}}_k}} \right\}$ 的鲁棒二次CLF矩阵。

需要指出的是,定理1、定理2、推论3均是充分条件,如果这些定理不能满足,并不能说明鲁棒二次CLF矩阵不存在。

4 应用仿真

现代农业中,温室大棚提供了经济作物适宜的生长环境。其中温度和湿度是最为重要的因素,各类农作物的需求各不相同。因此,合理的温室温度和湿度控制成为智能温室大棚的主要和关键工程问题。文献[15]选择较为传统的近似线性化方法,在选取的温湿度工作点对非线性模型进行泰勒展开,这样就获得了所有工作点的线性化模型组。针对每个子模型设计相应的最优跟踪控制器,根据然后进行了跟踪切换控制。

本文依据文献[15],考虑大棚的温度 ${T_i}$ 和湿度 ${\omega _i}$ 为温室大棚的状态变量,对温室大棚建模为

$\begin{array}{c}{C_T}\displaystyle\frac{{{\rm{d}}{T_i}}}{{{\rm{d}}t}} = {C_1}\left( {{T_p} - {T_i}} \right) - \left( {{C_2}\left( {{C_3}G + {\varphi _1}} \right) + {C_4}} \right) \times \\[4pt]\left( {{T_i} - {T_o}} \right) + {C_5}\left( {{T_1} - {T_i}} \right) + {C_6}{T_{{\rm{rad}}}}\\[4pt]{C_H}\displaystyle\frac{{{\rm{d}}{\omega _i}}}{{{\rm{d}}t}} = {C_8}{C_9}\left( {{C_{10}}{C_{11}}{T_i} - {\omega _i}} \right) - {C_{12}}\left( {{T_p} - {T_i}} \right) - \\[4pt]\left( {{C_3}G + {\varphi _1}} \right)\left( {{\omega _i} - {\omega _o}} \right) + {C_H}\left( {{C_5}{T_1} + {C_{14}}} \right)\end{array}$

式中:管道加热温度G,通风率G,土壤表层温度T1,室外温度 ${\omega _o}$ ,室外湿度 ${\omega _o}$ ,太阳辐射能量 ${T_{rad}}$ 、泄漏风量 ${\varphi _1}$ 。选择状态变量GGG,输入变量G分别为GG,其余视作干扰 ${v_1},{v_2},{v_3},{v_4}$ ,且假定上述所有变量均可测,状态空间模型为

$\begin{array}{c}{{\dot x}_1} = - \left( {{b_1} + {b_3} + {b_4}} \right){x_1} + {b_1}{u_1} + \left( { - {b_2}{x_1} + {b_2}{v_2}} \right){u_2} + \\[2pt]{b_5}{v_3} + {b_3}{v_1} + {b_4}{v_4}\\[2pt]{{\dot x}_2} = {c_4}{x_1} + {c_1}{x_2} + {c_6}{v_2} + {c_8}{u_1} + {c_2}{u_2}{v_2} + {c_2}{u_2}{x_2} + \\[2pt]{c_5}{v_1} + {c_3}{v_4} + {c_7}\end{array}$

式中:

$\begin{array}{c}{b_1} = {C_1}/{C_T},\;{b_2} = {C_2}{C_3}/{C_T}\\[4pt]{b_3} = \left( {{C_2}_1\varphi + {C_4}} \right)/{C_T},\;{b_4} = {C_5}/{C_T}\\[4pt]{b_5} = {C_6}/{C_T}\\[4pt]{c_1} = \left( {{C_{15}} - {C_8}{C_9} - {\varphi _1}} \right)/{C_H},\;{c_2} = {C_3}/{C_H}\\[4pt]{c_4} = {C_8}{C_9}{C_{10}}/{C_H},\;{c_5} = {C_{15}}/{C_H}\\[4pt]{c_6} = {\varphi _1}/{C_H}\end{array}$

各个参数物理意义及数值参考文献[15]。将上述模型在工作点 $S\left( {{x_{10}},{x_{20}}} \right)$ 近似线性化,可得线性化后模型为

$\begin{array}{c}{{\dot x}_1} = - \left( {{b_1} + {b_3} + {b_4}} \right){x_1} + {b_1}{u_1} + \left( { - {b_2}{x_{10}} + {b_2}{v_2}} \right){u_2} + \\[0pt]{b_5}{v_3} + {b_3}{v_1} + {b_4}{v_4}\\[0pt]{{\dot x}_2} = {c_4}{x_1} + {c_1}{x_2} + {c_8}{u_1} + {c_2}{v_2}{u_2} + {c_2}{x_{20}}{u_2} + {c_6}{v_2} + \\[0pt]{c_5}{v_1} + {c_3}{v_4} + {c_7}\end{array}$

根据文献[15],我们简单以温度、湿度的3个工作点: $S\left( {26,20} \right)$ $S\left( {26,28} \right)$ $S\left( {28,28} \right)$ 来进行线性化(在文献[15]基础上增加一个工作点)。设计其反馈控制器,得到闭环控制后系统的3个参数矩阵:

${{\mathit{\boldsymbol{A}}}_1} = \left[ {\begin{array}{*{20}{c}} { - 0.038 \,\, 9} & 0\\ { - 0.082} & { - 0.163 \,\, 2} \end{array}} \right]$

(这里设计反馈控制系数与文献[15]不同)

${{\mathit{\boldsymbol{A}}}_2} = \left[ {\begin{array}{*{20}{c}} { - 0.158} & {0.002 \,\, 2}\\ { - 0.067 \,\, 4} & { - 0.339 \,\, 1} \end{array}} \right]$

(这里设计最优反馈控制系数与文献[15]相同)

${{\mathit{\boldsymbol{A}}}_3} = \left[ {\begin{array}{*{20}{c}} { - 0.158} & {0.002 \,\, 2}\\ { - 0.067 \,\, 4} & { - 0.339 \,\, 1} \end{array}} \right]$

这样,对于这个温室大棚的控制问题,我们看做是一个在不同工作点线性化后的线性不确定切换系统。在各个不同工作点进行了最优反馈控制设计后的闭环系统,是多个工作点附近的稳定子系统。这样,各个工作点闭环控制后的系统参数矩阵 ${{\mathit{\boldsymbol{A}}}_1}{\text{、}}{{\mathit{\boldsymbol{A}}}_2}{\text{、}}{{\mathit{\boldsymbol{A}}}_3}$ 就构成一个稳定矩阵集以及具有3个子系统的切换系统(1), ${\mathit{\boldsymbol{N}}} \buildrel \Delta \over = \left\{ {1,2,3} \right\}$

注:这里闭环控制后的参数矩阵 ${{\mathit{\boldsymbol{A}}}_1}$ 不再沿用文献[15]中的数值,是因为在文献[15]中声称 ${{\mathit{\boldsymbol{A}}}_1}$ ${{\mathit{\boldsymbol{A}}}_2}$ 闭环后参数为

${{\mathit{\boldsymbol{A}}}_1} = \left[ {\begin{array}{*{20}{c}}{0.148 \,\, 2} & {0.374 \,\, 6}\\{0.374 \,\, 6} & {1.892 \,\, 7}\end{array}} \right],\;{{\mathit{\boldsymbol{A}}}_2} = \left[ {\begin{array}{*{20}{c}}{0.148 \,\, 2} & {0.374 \,\, 6}\\{0.374 \,\, 6} & {1.892 \,\, 7}\end{array}} \right]$

的CLF是

${\mathit{\boldsymbol{P}}} = \left[ {\begin{array}{*{20}{c}}{0.148 \,\, 2} & {0.374 \,\, 6}\\{0.374 \,\, 6} & {1.892 \,\, 7}\end{array}} \right]$

但经过验证P不满足 ${\mathit{\boldsymbol{A}}}_1^{\rm{T}}{\mathit{\boldsymbol{P}}} + {\mathit{\boldsymbol{P}}}{{\mathit{\boldsymbol{A}}}_1} < 0$ 以及 ${\mathit{\boldsymbol{A}}}_2^{\rm{T}}{\mathit{\boldsymbol{P}}} + $ ${\mathit{\boldsymbol{A}}}_2^{\rm{T}}{\mathit{\boldsymbol{P}}} + $ ,因此P并不是CLF。

我们把本文的理论和方法,进行应用,目标是判断在各个工作点线性化、最优反馈控制后的各个子系统构成的整体是否是鲁棒稳定的。我们把各个干扰部分进行取近似化为

$\begin{array}{c}\Delta {{\mathit{\boldsymbol{A}}}_1} = \left[ {\begin{array}{*{20}{c}}{0.001\sin \, (0.02\pi t)} & 0\\0 & {0.014{{\rm{e}}^{ - 0.1t}}}\end{array}} \right],\\[8pt]\Delta {{\mathit{\boldsymbol{A}}}_2} = \left[ {\begin{array}{*{20}{c}}{0.000\;1\cos \, (0.01t)} & {0.000\;2\cos \, (0.01t)}\\0 & {0.005{{\rm{e}}^{ - 0.2t}}}\end{array}} \right],\\[8pt]\Delta {{\mathit{\boldsymbol{A}}}_3} = \left[ {\begin{array}{*{20}{c}}0 & 0\\0 & {0.026{{\rm{e}}^{ - 0.01{t^2}}}}\end{array}} \right].\end{array}$

根据假设1,将各个子系统不确定性分解。各个不确定性矩阵为

子系统1中:

$\begin{array}{c}{{\mathit{\boldsymbol{H}}}_{a,1}} = \left[ {\begin{array}{*{20}{c}}{0.1} & 0\\0 & 1\end{array}} \right],{{\mathit{\boldsymbol{F}}}_{a,1}} = \left[ {\begin{array}{*{20}{c}}{\sin \, (0.02\pi t)} & 0\\0 & {{{\rm{e}}^{ - 0.1t}}}\end{array}} \right]\\[9pt]{{\mathit{\boldsymbol{E}}}_{a,1}} = \left[ {\begin{array}{*{20}{c}}{0.01} & 0\\0 & {0.014}\end{array}} \right]\end{array}$

子系统2中:

$\begin{array}{c}{{\mathit{\boldsymbol{H}}}_{a,2}} = \left[ {\begin{array}{*{20}{c}}{0.01} & 0\\0 & {0.1}\end{array}} \right],{{\mathit{\boldsymbol{F}}}_{a,2}} = \left[ {\begin{array}{*{20}{c}}{\cos \, (0.01t)} & 0\\0 & {{{\rm{e}}^{ - 0.2t}}}\end{array}} \right]\\[9pt]{{\mathit{\boldsymbol{E}}}_{a,2}} = \left[ {\begin{array}{*{20}{c}}{0.01} & {0.02}\\0 & {0.05}\end{array}} \right]\end{array}$

子系统3中:

${{\mathit{\boldsymbol{H}}}_{a,3}} = \left[ {\begin{array}{*{20}{c}}0\\{0.1}\end{array}} \right],{{\mathit{\boldsymbol{F}}}_{a,3}} = {{\rm{e}}^{ - 0.01{t^2}}},{{\mathit{\boldsymbol{E}}}_{a,3}} = \left[ {\begin{array}{*{20}{c}}0 & {0.26}\end{array}} \right]$

k=2时子系统1、2的公共Lyapunuov矩阵为 ${{\mathit{\boldsymbol{P}}}_2} = \left[ {\begin{array}{*{20}{c}} {{\rm{99}}{\rm{.798 \,\, 9}}} & {{\rm{15}}{\rm{.509 \,\, 9}}}\\ {{\rm{15}}{\rm{.509 \,\, 9}}} & {{\rm{11}}{\rm{.897 \,\, 6}}} \end{array}} \right]$

k=3时子系统1、2、3的公共Lyapunuov矩阵为 ${{\mathit{\boldsymbol{P}}}_3} = \left[ {\begin{array}{*{20}{c}} {{\rm{39}}{\rm{.144 \,\, 9}}} & {{\rm{2}}{\rm{.714 \,\, 1}}}\\ {{\rm{2}}{\rm{.714 \,\, 1}}} & {{\rm{1}}{\rm{.541 \,\, 1}}} \end{array}} \right]$

这样,找到了温室大棚这3个工作点的鲁棒二次CLF,说明在任意切换规则下系统渐近稳定。

我们不考虑设计跟踪设定温度、湿度曲线问题,只考虑镇定问题。因此为了验证CQLF方法的有效和正确性,设系统的初始状态为 ${\mathit{\boldsymbol{x}}}\left( {{t_0}} \right) = $ $ {\left[ {\begin{array}{*{20}{c}} \!\!\!\! {30} \!&\! {40} \!\!\!\! \end{array}} \right]^{\rm{T}}}$ ,开始以工作点 ${\mathit{\boldsymbol{S}}}\left( {28,28} \right)$ 的模型 ${{\mathit{\boldsymbol{A}}}_3}$ 运行,20 s后以工作点 ${\mathit{\boldsymbol{S}}}\left( {26,28} \right)$ 的模型 ${{\mathit{\boldsymbol{A}}}_2}$ 运行,再经过20 s以工作点 ${\mathit{\boldsymbol{S}}}\left( {26,20} \right)$ 的模型 ${{\mathit{\boldsymbol{A}}}_1}$ 运行。随后我们任意切换。系统仿真结果如图12所示。从图中可以看出,在不确定参数存在的情况下切换系统状态在任意切换信号下状态都能镇定。

图 1 切换规则下系统状态曲线 Fig.1 The state curves under switching signal
图 2 切换信号曲线 Fig.2 The switching signal curve
5 结束语

对不确定线性切换系统的CLF问题进行了讨论。若系统矩阵构成鲁棒稳定矩阵集,给出了CQLF矩阵的递推判定方法和构造算法。构造算法是递推式的LMI形式,求解方便,实用性强。在温室大棚的温湿度控制问题的典型应用仿真验证了结果的可行性。

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