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 智能系统学报  2017, Vol. 12 Issue (6): 899-905  DOI: 10.11992/tis.201605012 0

### 引用本文

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.

### 文章历史

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

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。

1 问题描述

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 $\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}$

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

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

3.3 递推CQLF矩阵

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

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

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

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

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

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

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

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

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

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

3.4 鲁棒二次Lyapunov函数矩阵寻找算法

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

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

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

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

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

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

4 应用仿真

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

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

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

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

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

（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）

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

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

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

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

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

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

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

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 ${{\mathit{\boldsymbol{P}}}_3} = \left[ {\begin{array}{*{20}{c}} {{\rm{39}}{\rm{.144 \,\, 9}}} & {{\rm{2}}{\rm{.714 \,\, 1}}}\\ {{\rm{2}}{\rm{.714 \,\, 1}}} & {{\rm{1}}{\rm{.541 \,\, 1}}} \end{array}} \right]$

 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

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