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
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
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:
$\begin{array}{c}{\mathit{\boldsymbol{Q}}}: = {x_0};\left( {{i_0},{t_0}} \right),\left( {{i_1},{t_1}} \right), \cdots ,\left( {{i_N},{t_N}} \right), \cdots ,\\\forall {i_k} \in {\bf{N}},k \in {{\bf{Z}}^ + },\end{array}$ |
意味着当
${{\mathit{\boldsymbol{A}}}_\sigma } \buildrel \Delta \over = {{\mathit{\boldsymbol{A}}}_i},\Delta {{\mathit{\boldsymbol{A}}}_\sigma } \buildrel \Delta \over = \Delta {{\mathit{\boldsymbol{A}}}_i}$ |
因此,在第k次切换,对于
$\dot x(t) = ({{\mathit{\boldsymbol{A}}}_i} + \Delta {{\mathit{\boldsymbol{A}}}_i})x(t)$ | (2) |
切换系统式(2)满足以下假设。
假设1[14] 不确定参数
$\Delta {{\mathit{\boldsymbol{A}}}_i} = {{\mathit{\boldsymbol{H}}}_{a,i}}{{\mathit{\boldsymbol{F}}}_{a,i}}\left( t \right){{\mathit{\boldsymbol{E}}}_{a,i}}$ | (3) |
式中:
${\mathit{\boldsymbol{F}}}_{a,i}^{\rm{T}}\left( t \right){{\mathit{\boldsymbol{F}}}_{a,i}}\left( t \right) \leqslant {\mathit{\boldsymbol{I}}}$ |
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
1)
2)
3)
引理2[14] 设H和E是具有适当维数的实常数矩阵,
${\mathit{\boldsymbol{HF}}}\left( t \right){\mathit{\boldsymbol{E}}} + {{\mathit{\boldsymbol{E}}}^{\rm{T}}}{{\mathit{\boldsymbol{F}}}^{\rm{T}}}\left( t \right){{\mathit{\boldsymbol{H}}}^{\rm{T}}} \leqslant {\varepsilon ^{ - 1}}{\mathit{\boldsymbol{H}}}{{\mathit{\boldsymbol{H}}}^{\rm{T}}} + \varepsilon {{\mathit{\boldsymbol{E}}}^{\rm{T}}}{\mathit{\boldsymbol{E}}}$ |
假设稳定矩阵集
${{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} = - {{\mathit{\boldsymbol{P}}}_{N - 1}}$ |
引理3[12] 对于稳定矩阵集
$\left\{ \begin{array}{l}{\rm{max}}\left( {\gamma _i^i} \right) < 2{\rm{min}}\left| {{\rm{Re}}\lambda \left( {{{\mathit{\boldsymbol{A}}}_N}} \right)} \right|\\ - {{\mathit{\boldsymbol{P}}}_{i,N - 1}} + \gamma _i^N{{\mathit{\boldsymbol{P}}}_{N - 1}} - \sum\limits_{k = 1,k \ne i}^{N - 1} {\gamma _i^k{{\mathit{\boldsymbol{P}}}_{k,N}} > 0} \end{array} \right.$ | (4) |
式中:
当稳定矩阵集
$\left[ {{{\mathit{\boldsymbol{A}}}_N},{{\mathit{\boldsymbol{A}}}_j}} \right] = {{\mathit{\boldsymbol{C}}}_{N,j}},\;\forall j \in {\bf{N}}$ | (5) |
引理4[12] 对于稳定矩阵集
${{\mathit{\boldsymbol{P}}}_{i,N - 1}} + {{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{C}}}_{N,i}} + {\mathit{\boldsymbol{C}}}_{N,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} < 0$ | (6) |
则
引理5 若
$\left[ {\begin{array}{*{20}{c}}{{{\mathit{\boldsymbol{P}}}_i}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{\mathit{\boldsymbol{P}}} + {{\mathit{\boldsymbol{L}}}_i}} & {{{\mathit{\boldsymbol{P}}}_i}{{\mathit{\boldsymbol{H}}}_{a,i}}}\\* & { - {\varepsilon _i}{\mathit{\boldsymbol{I}}}}\end{array}} \right] < 0,\;\forall i \in {\bf{N}}$ | (7) |
有正定对称阵解
${{\mathit{\boldsymbol{G}}}_i} = \varepsilon _i^{ - 1}{{\mathit{\boldsymbol{H}}}_{a,i}}{\mathit{\boldsymbol{H}}}_{a,i}^{\rm{T}},{\rm{ }}{{\mathit{\boldsymbol{L}}}_i} = {\varepsilon _i}{\mathit{\boldsymbol{E}}}_{a,i}^{\rm{T}}{{\mathit{\boldsymbol{E}}}_{a,i}}$ | (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
${{\mathit{\boldsymbol{P}}}_i}\Delta {{\mathit{\boldsymbol{A}}}_i} + \Delta {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_i} \leqslant {{\mathit{\boldsymbol{G}}}_i} = \varepsilon _i^{ - 1}{{\mathit{\boldsymbol{P}}}_i}{{\mathit{\boldsymbol{H}}}_{a,i}}{\mathit{\boldsymbol{H}}}_{a,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_i} + {\varepsilon _i}{\mathit{\boldsymbol{E}}}_{a,i}^{\rm{T}}{{\mathit{\boldsymbol{E}}}_{a,i}}$ |
成立。将其代入
${\dot V_i}\left( {{\mathit{\boldsymbol{x}}}(t)} \right) \leqslant {{\mathit{\boldsymbol{x}}}^{\rm{T}}}\left[ {{{\mathit{\boldsymbol{P}}}_i}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_i} + {{\mathit{\boldsymbol{P}}}_i}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_i} + {{\mathit{\boldsymbol{L}}}_i}} \right]{\mathit{\boldsymbol{x}}}$ |
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
切换系统(2)的系数矩阵满足如下假设。
假设2 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) |
若鲁棒稳定矩阵集
${{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N}{\rm{ + }}{{\mathit{\boldsymbol{U}}}_{N,N}} = - {{\mathit{\boldsymbol{P}}}_{N - 1}}$ | (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
$\left\{ \begin{array}{l}{\rm{max}}\left( {\gamma _i^i} \right) < 2{\rm{min}}\left| {{\rm{Re}}\lambda \left( {{A_N}} \right)} \right|{\rm{}}\\ - {{\mathit{\boldsymbol{P}}}_{i,N - 1}} + \gamma _i^N{{\mathit{\boldsymbol{P}}}_{N - 1}} - \sum\limits_{k = 1,k \ne i}^{N - 1} {\gamma _i^k{{\mathit{\boldsymbol{P}}}_{k,N}}} + \gamma _i^i{{\mathit{\boldsymbol{U}}}_{i,N}} + \\\quad\quad{\rm{ }}\gamma _i^N{{\mathit{\boldsymbol{U}}}_{N,N}} + {{\mathit{\boldsymbol{U}}}_{i,N}}{{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,N}} - {\rm{ }}{{\mathit{\boldsymbol{U}}}_{N,N}}{{\mathit{\boldsymbol{A}}}_i} - \\[5pt]\quad\quad{\rm{ }}{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{N,N}} > 0{\rm{}}\end{array} \right.$ | (12) |
则
${{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{L}}}_i} < 0$ | (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
${{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{L}}}_i} < 0$ | (14) |
成立,即
若鲁棒稳定矩阵集
推论2 对于鲁棒稳定矩阵集
$\begin{array}{c}{{\mathit{\boldsymbol{P}}}_{i,N - 1}} + {{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{C}}}_{N,i}} + {\mathit{\boldsymbol{C}}}_{N,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{U}}}_{N,N}}{{\mathit{\boldsymbol{A}}}_i} + \\{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{N,N}} - {{\mathit{\boldsymbol{U}}}_{i,N}}{{\mathit{\boldsymbol{A}}}_N} - {\mathit{\boldsymbol{A}}}_N^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,N}} < 0\end{array}$ | (15) |
则
${{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{L}}}_i} < 0$ | (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
$\left( {{{\mathit{\boldsymbol{P}}}_{i,N}} + {{\mathit{\boldsymbol{U}}}_{i,N}}} \right){{\mathit{\boldsymbol{A}}}_N} + {\mathit{\boldsymbol{A}}}_N^{\rm{T}}\left( {{{\mathit{\boldsymbol{P}}}_{i,N}} + {{\mathit{\boldsymbol{U}}}_{i,N}}} \right) > 0$ |
成立,表明两个矩阵
${{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{P}}}_N}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_N} + {{\mathit{\boldsymbol{L}}}_i} < 0$ | (17) |
那么
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
$\left\{ \begin{array}{l}{{\mathit{\boldsymbol{P}}}_{k,k}} < 0\\{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_{k,k}} + {{\mathit{\boldsymbol{P}}}_{k,k}}{{\mathit{\boldsymbol{A}}}_i} - {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{C}}}_{k,i}} - {\mathit{\boldsymbol{C}}}_{k,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} > 0\end{array} \right.$ | (18) |
的正定对称阵
${{\mathit{\boldsymbol{C}}}_{i,j}} = {{\mathit{\boldsymbol{A}}}_i}{{\mathit{\boldsymbol{A}}}_j} - {{\mathit{\boldsymbol{A}}}_j}{{\mathit{\boldsymbol{A}}}_i}$ |
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
${{\mathit{\boldsymbol{P}}}_{i,k}} = {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_i} < 0,{\rm{ }}\forall i = 1, 2, \cdots ,k$ |
成立。因此
假设一个鲁棒稳定矩阵集
定理2 假设鲁棒稳定矩阵集
$\left\{ \begin{array}{l}{{\mathit{\boldsymbol{P}}}_{k,k}} + {{\mathit{\boldsymbol{U}}}_{k,k}} < 0,\\[5pt]{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_{k,k}} + {{\mathit{\boldsymbol{P}}}_{k,k}}{{\mathit{\boldsymbol{A}}}_i} - {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{C}}}_{k,i}} - {\mathit{\boldsymbol{C}}}_{k,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + \\[5pt]\quad\quad{{\mathit{\boldsymbol{U}}}_{i,k}}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,k}} > 0\end{array} \right.$ | (19) |
的正定对称阵
${{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_i} + {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + {{\mathit{\boldsymbol{U}}}_{i,k}} < 0$ | (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
${{\mathit{\boldsymbol{P}}}_{i,k}}{\rm{ + }}{{\mathit{\boldsymbol{U}}}_{i,k}} = {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_i}{\rm{ + }}{{\mathit{\boldsymbol{U}}}_{i,k}} < 0$ |
成立。因此
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
$\left\{ \begin{array}{l}\left[ {\begin{array}{*{20}{c}}{{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + {{\mathit{\boldsymbol{L}}}_k}} & {{{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{H}}}_{a,k}}}\\* & { - {\varepsilon _k}{\mathit{\boldsymbol{I}}}}\end{array}} \right] < 0,\\[9pt]\left[ {\begin{array}{*{20}{c}}{{\Xi _{i,k}}} & {\left( {{\mathit{\boldsymbol{I}}} - {\mathit{\boldsymbol{A}}}_k^{\rm{T}}} \right){{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{H}}}_{a,i}}}\\* & {{\varepsilon _i}{\mathit{\boldsymbol{I}}}}\end{array}} \right] > 0,{\rm{ }}i = 1, 2, \cdots ,k - 1\end{array} \right.$ | (21) |
那么
$\begin{array}{c}{\Xi _{i,k}} = {\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_{k,k}} + {{\mathit{\boldsymbol{P}}}_{k,k}}{{\mathit{\boldsymbol{A}}}_i} - {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{C}}}_{k,i}} - \\[4pt]{\mathit{\boldsymbol{C}}}_{k,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + {{\mathit{\boldsymbol{L}}}_i}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{L}}}_i}\end{array}$ | (22) |
证明 式(21)中第1个LMI证明
${\Xi _{i,k}} - \left( {{\mathit{\boldsymbol{I}}} - {\mathit{\boldsymbol{A}}}_k^{\rm{T}}} \right){{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{G}}}_i}{{\mathit{\boldsymbol{P}}}_k}\left( {{\mathit{\boldsymbol{I}}} - {{\mathit{\boldsymbol{A}}}_k}} \right) > 0$ |
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,
$\begin{array}{c}{\mathit{\boldsymbol{A}}}_i^{\rm{T}}{{\mathit{\boldsymbol{P}}}_{k,k}} + {{\mathit{\boldsymbol{P}}}_{k,k}}{{\mathit{\boldsymbol{A}}}_i} - {{\mathit{\boldsymbol{P}}}_k}{{\mathit{\boldsymbol{C}}}_{k,i}} - {\mathit{\boldsymbol{C}}}_{k,i}^{\rm{T}}{{\mathit{\boldsymbol{P}}}_k} + \\[4pt]{{\mathit{\boldsymbol{U}}}_{i,k}}{{\mathit{\boldsymbol{A}}}_k} + {\mathit{\boldsymbol{A}}}_k^{\rm{T}}{{\mathit{\boldsymbol{U}}}_{i,k}} > 0\end{array}$ |
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
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
$\begin{array}{c}{C_T}\displaystyle\frac{{{\rm{d}}{T_i}}}{{{\rm{d}}t}} = {C_1}\left( {{T_p} - {T_i}} \right) - \left( {{C_2}\left( {{C_3}G + {\varphi _1}} \right) + {C_4}} \right) \times \\[4pt]\left( {{T_i} - {T_o}} \right) + {C_5}\left( {{T_1} - {T_i}} \right) + {C_6}{T_{{\rm{rad}}}}\\[4pt]{C_H}\displaystyle\frac{{{\rm{d}}{\omega _i}}}{{{\rm{d}}t}} = {C_8}{C_9}\left( {{C_{10}}{C_{11}}{T_i} - {\omega _i}} \right) - {C_{12}}\left( {{T_p} - {T_i}} \right) - \\[4pt]\left( {{C_3}G + {\varphi _1}} \right)\left( {{\omega _i} - {\omega _o}} \right) + {C_H}\left( {{C_5}{T_1} + {C_{14}}} \right)\end{array}$ |
式中:管道加热温度G,通风率G,土壤表层温度T1,室外温度
$\begin{array}{c}{{\dot x}_1} = - \left( {{b_1} + {b_3} + {b_4}} \right){x_1} + {b_1}{u_1} + \left( { - {b_2}{x_1} + {b_2}{v_2}} \right){u_2} + \\[2pt]{b_5}{v_3} + {b_3}{v_1} + {b_4}{v_4}\\[2pt]{{\dot x}_2} = {c_4}{x_1} + {c_1}{x_2} + {c_6}{v_2} + {c_8}{u_1} + {c_2}{u_2}{v_2} + {c_2}{u_2}{x_2} + \\[2pt]{c_5}{v_1} + {c_3}{v_4} + {c_7}\end{array}$ |
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
$\begin{array}{c}{{\dot x}_1} = - \left( {{b_1} + {b_3} + {b_4}} \right){x_1} + {b_1}{u_1} + \left( { - {b_2}{x_{10}} + {b_2}{v_2}} \right){u_2} + \\[0pt]{b_5}{v_3} + {b_3}{v_1} + {b_4}{v_4}\\[0pt]{{\dot x}_2} = {c_4}{x_1} + {c_1}{x_2} + {c_8}{u_1} + {c_2}{v_2}{u_2} + {c_2}{x_{20}}{u_2} + {c_6}{v_2} + \\[0pt]{c_5}{v_1} + {c_3}{v_4} + {c_7}\end{array}$ |
根据文献[15],我们简单以温度、湿度的3个工作点:
(这里设计反馈控制系数与文献[15]不同)
(这里设计最优反馈控制系数与文献[15]相同)
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
注:这里闭环控制后的参数矩阵
${{\mathit{\boldsymbol{A}}}_1} = \left[ {\begin{array}{*{20}{c}}{0.148 \,\, 2} & {0.374 \,\, 6}\\{0.374 \,\, 6} & {1.892 \,\, 7}\end{array}} \right],\;{{\mathit{\boldsymbol{A}}}_2} = \left[ {\begin{array}{*{20}{c}}{0.148 \,\, 2} & {0.374 \,\, 6}\\{0.374 \,\, 6} & {1.892 \,\, 7}\end{array}} \right]$ |
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
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
k=3时子系统1、2、3的公共Lyapunuov矩阵为
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
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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