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时滞是动态系统中无法避免的一类现象，它广泛存在于工业控制系统中，如信号传输、网络控制系统、神经网络以及混沌系统之间的同步等[1]。时滞常常是控制系统振荡和性能下降的根源，在某些情况下，甚至会导致系统的不稳定[2]。T-S模糊模型可以把模糊逻辑理论和线性系统理论结合成一个统一的框架去近似复杂的非线性系统，被广泛用于非线性系统的稳定性分析与控制器设计当中。因此，T-S模糊时滞系统的稳定性分析与控制器综合，无论在理论上还是在实际应用中都具有重要意义，进而越来越受到人们关注。
对于时滞系统的稳定性分析，如何减少保守性一直是研究的热点。通常选择合适的Lyapunov-Krasovskii泛函及其导数中积分项的边界处理是减少保守性的主要途径。文献[1]构造了一个包含线性积分Lyapunov函数的增广Lyapunov-Krasovskii泛函，运用Wirtinger不等式处理积分项，进而给出了T-S模糊时滞系统的稳定性准则。文献[3]提出了一个改进的凸不等式，并且证明了一些已有的不等式是其特殊形式。该不等式通过引入较少的松弛变量得到更大的时滞上界。同时，构造一个基于二阶Bessel-Legendre不等式的增广Lyapunov-Krasovskii泛函，给出了时滞系统保守性较小的稳定性准则。文献[4]构造一个模糊Lyapunov-rasovskii泛函，运用时滞分割技术和Jensen不等式处理积分项，同时引入松弛矩阵，给出了T-S模糊时滞系统时滞依赖的稳定性准则。值得注意的是，对于时滞系统的处理，每种方法都不是独立的，通常需要结合多种方法以达到降低保守性的目的。
对于T-S模糊时滞系统的控制问题，当前研究成果主要是基于并行分布补偿(parallel distributed compensation, PDC)技术提出的，PDC算法要求模糊控制器与模糊模型拥有相同的前提隶属函数和模糊规则数目。然而，PDC技术可能会增加模糊控制器的复杂度、降低控制器的设计自由度[5]。近年来，前提不匹配技术的提出克服了上述局限性，此方法不要求模糊控制器与模糊模型分享相同的隶属函数和规则数目，极大地提高了控制器设计的灵活性。文献[6]考虑隶属函数的信息，结合平方和方法和Lyapunov稳定性理论，基于前提不匹配技术，设计了T-S模糊时滞系统的隶属函数依赖的多项式模糊状态反馈控制器。文献[7]构造了一个新的有限和不等式，给出了新的时滞依赖的稳定性条件。基于前提不匹配技术，考虑参数不确定性和数据包丢失的情况，给出了二型模糊时滞系统的状态反馈控制器设计方法。另一方面，对于时滞系统的镇定问题，就控制输入中有无时滞而言，时滞系统的控制器可分为记忆控制器和无记忆控制器。所谓记忆控制器，就是控制输入中含有时滞；而控制输入没有时滞，称之为无记忆控制器。控制输入中加入时滞可以对系统中的时滞项加以控制，减少系统中时滞的影响[8]。文献[9]构造了一个包含模糊线性积分Lyapunov函数的Lyapunov-asovskii泛函，运用Wirtinger不等式处理积分项，引入松弛变量，基于PDC算法，给出了T-S模糊时滞系统的记忆状态反馈控制器的设计方法。在文献[10]中，Lyapunov- Krasovskii泛函的被积函数不仅取决于积分变量，还取决于隶属函数，同时考虑隶属函数导数的信息，给出了保守性较小的T-S模糊时滞系统的稳定性准则，基于PDC技术，提出了记忆状态反馈控制器的设计方法。
但文献[6-7]基于前提不匹配技术设计的无记忆模糊状态反馈控制器，都只考虑了隶属函数不同的情况，并未考虑模糊规则数目不一致的情况。此外，模糊记忆控制器的设计也是有待进一步探讨的课题，前提不匹配的记忆控制器的设计仍是一个开放性的问题。受这些启发，本文研究了T-S模糊时滞系统的稳定性分析与控制器设计问题。首先，构造一个简单而有效的Lyapunov-Krasovskii泛函，选择一个近期提出的基于辅助函数的积分不等式处理积分项，给出了保守性较小的稳定性条件。基于前提不匹配技术，提出了模糊记忆状态反馈控制器的设计方法，这是前提不匹配的模糊记忆状态反馈控制器第一次被提出。该模糊记忆控制器不要求和系统拥有相同的隶属函数和模糊规则数目。通过选择更为简单的隶属函数和较小的规则数目，给出了更为灵活的模糊记忆控制器设计方法。
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考虑具有$p$个模糊规则数的T-S模糊模型描述的时滞非线性系统：
规则$i$：如果${\delta _1}(\mathit{\boldsymbol{x}}(t))$是${F_1}^i$且${\delta _2}(\mathit{\boldsymbol{x}}(t))$是${F_2}^i, \cdots , $ ${\delta _\chi }(\mathit{\boldsymbol{x}}(t))$是${F_\chi }^i$，则：
$$\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot x}}(t) = {\mathit{\boldsymbol{A}}_i}\mathit{\boldsymbol{x}}(t) + {\mathit{\boldsymbol{A}}_{\tau i}}\mathit{\boldsymbol{x}}(t - \tau ) + {\mathit{\boldsymbol{B}}_i}\mathit{\boldsymbol{u}}(t)}\\ {\mathit{\boldsymbol{x}}(t) = \boldsymbol{\phi} (t){\rm{ }}t \in [ - \tau , 0]} \end{array}} \right.$$ (1) 式中，${F_\alpha }^i$($\alpha = 1, 2, \cdots , \chi $，$i = 1, 2, 3, \cdots , p$)是模糊集；时滞$\tau $是已知且固定的；$\boldsymbol{\phi} (t)$是$[ - \tau , 0]$上的连续初始函数；$\boldsymbol{x}(t) \in {\mathbb{R}^n}$是系统状态，$\boldsymbol{u}(t) \in {\mathbb{R}^m}$是控制输入；${\boldsymbol{A}_i}$，$\boldsymbol{A}_{\tau i}$，${\boldsymbol{B}_i}$是具有适当维数的已知矩阵。
动态模糊模型(1)可表示为：
$$\mathit{\boldsymbol{\dot x}}(t) = \mathop \sum \limits_{i = 1}^p {m_i}(\mathit{\boldsymbol{x}}(t))({\mathit{\boldsymbol{A}}_i}x(t) + {\mathit{\boldsymbol{A}}_{\tau i}}\mathit{\boldsymbol{x}}(t - \tau ) + {\mathit{\boldsymbol{B}}_i}\mathit{\boldsymbol{u}}(t))$$ (2) 式中，${m_i}(\mathit{\boldsymbol{x}}(t)) = {\omega _{\rm{i}}}(\mathit{\boldsymbol{x}}(t))\left( {\sum\limits_{i = 1}^p {{\omega _i}(\mathit{\boldsymbol{x}}(t))} } \right)$，${\omega _{\rm{i}}}(\mathit{\boldsymbol{x}}(t)) = $ $\mathop \prod \limits_{\alpha = 1}^\chi {\mu _{{F_\alpha }^i({\delta _\alpha }(\mathit{\boldsymbol{x}}(t)))}}$，${\mu _{{F_\alpha }^i({\delta _\alpha }(\mathit{\boldsymbol{x}}(t)))}}$是${\delta _\alpha }(\mathit{\boldsymbol{x}}(t))$对应的隶属度函数。由定义可知${m_i}(\mathit{\boldsymbol{x}}(t)) \geqslant 0$，$\mathop \sum \limits_{i = 1}^p {m_i}(\mathit{\boldsymbol{x}}(t)) = 1$。
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基于前提不匹配技术，设计具有$c$个模糊规则数目的记忆状态反馈控制器：
规则$j$：如果${\delta _1}(\mathit{\boldsymbol{x}}(t))$是${F_1}^j$且${\delta _2}(\mathit{\boldsymbol{x}}(t))$是${F_2}^j, \cdots , $ ${\delta _\chi }(\mathit{\boldsymbol{x}}(t))$是${F_\chi }^j$，则：
$$\mathit{\boldsymbol{u}}(t) = {\mathit{\boldsymbol{K}}_j}\mathit{\boldsymbol{x}}(t){\rm{ + }}{\mathit{\boldsymbol{K}}_{\tau j}}\mathit{\boldsymbol{x}}(t - \tau )~~~j = 1, 2, \cdots , c$$ (3) 此时，模糊控制律可描述为：
$$\mathit{\boldsymbol{u}}(t) = \mathop \sum \limits_{j = 1}^c {h_j}(\mathit{\boldsymbol{x}}(t))({\mathit{\boldsymbol{K}}_j}\mathit{\boldsymbol{x}}(t){\rm{ + }}{\mathit{\boldsymbol{K}}_{\tau j}}\mathit{\boldsymbol{x}}(t - \tau ))$$ (4) 这里，${\mathit{\boldsymbol{K}}_j}$、${\mathit{\boldsymbol{K}}_{\tau j}}$是待求解的控制增益矩阵。
由式(2)和式(4)，得到T-S模糊时滞系统的闭环系统：
$$\begin{array}{c} \mathit{\boldsymbol{\dot x}}(t) = \mathop \sum \limits_{i = 1}^p \sum\limits_{j = 1}^c {{m_i}(\mathit{\boldsymbol{x}}(t)){h_j}(\mathit{\boldsymbol{x}}(t))} (({\mathit{\boldsymbol{A}}_i} + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{K}}_j})\mathit{\boldsymbol{x}}(t) + \\ ({\mathit{\boldsymbol{A}}_{\tau i}} + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{K}}_{\tau j}})\mathit{\boldsymbol{x}}(t - \tau )) \end{array}$$ (5) 式(5)的紧缩形式为:
$$\begin{array}{c} \mathit{\boldsymbol{\dot x}}(t){\rm{ = }}(\mathit{\boldsymbol{A}}(t) + \mathit{\boldsymbol{B}}(t)\mathit{\boldsymbol{K}}(t))\mathit{\boldsymbol{x}}(t) + \\ ({\mathit{\boldsymbol{A}}_\tau }(t) + \mathit{\boldsymbol{B}}(t){\mathit{\boldsymbol{K}}_\tau }(t))\mathit{\boldsymbol{x}}(t - \tau ) \end{array} $$ 不同于传统的PDC技术，本文设计的前提不匹配的模糊记忆状态反馈控制器不要求与模糊模型分享相同的前提隶属函数和模糊规则数目，即不要求$p \equiv c$，且${m_i}(\mathit{\boldsymbol{x}}(t)) \equiv {h_i}(\mathit{\boldsymbol{x}}(t))$，$i = 1, 2, \cdots , p$。在下文可以发现，本文所提的模糊控制器的设计方法更为灵活，适用范围更加广泛。
本文的目标是得到基于前提不匹配技术的模糊记忆状态反馈增益${\mathit{\boldsymbol{K}}_j}$、${\mathit{\boldsymbol{K}}_{\tau j}}$，保证闭环系统(5)是渐进稳定的。为此首先分析系统(2)在$\mathit{\boldsymbol{u}}(t) = 0$情况下的稳定性。
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引理1[11] (基于辅助函数的积分不等式)对正定矩阵$\mathit{\boldsymbol{R}}( \in {\mathbb{R}^{n \times n}}) > 0$和任意连续可微函数$\mathit{\boldsymbol{x}}$：$[a, b] \to {\mathbb{R}^n}$，有如下不等式成立：
$$\begin{array}{c} \int_a^b {{{\mathit{\boldsymbol{\dot x}}}^T}(s)\mathit{\boldsymbol{R\dot x}}(s)} {\rm{ d}}s \ge \frac{1}{{b - a}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_1}^T\mathit{\boldsymbol{R}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_1} + \frac{3}{{b - a}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_2}^T\mathit{\boldsymbol{R}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_2} + \\ \frac{5}{{b - a}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_3}^T\mathit{\boldsymbol{R}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_3} \end{array}$$ 式中，
$$\begin{array}{c} {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_1} = \mathit{\boldsymbol{x}}(b) - \mathit{\boldsymbol{x}}(a)\\ {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_2} = \mathit{\boldsymbol{x}}(b) + \mathit{\boldsymbol{x}}(a) - \frac{2}{{b - a}}\int_{{\rm{ }}a}^{{\rm{ }}b} {\mathit{\boldsymbol{x}}(s)} {\rm{ d}}s\\ {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_3} = \mathit{\boldsymbol{x}}(b) - \mathit{\boldsymbol{x}}(a) + \frac{6}{{b - a}}\int_{{\rm{ }}a}^{{\rm{ }}b} {\mathit{\boldsymbol{x}}(s)} {\rm{ d}}s - \\ \frac{{12}}{{{{(b - a)}^2}}}\int_{{\rm{ }}a}^{{\rm{ }}b} {\int_{{\rm{ }}u}^{{\rm{ }}b} {\mathit{\boldsymbol{x}}(s)} } {\rm{ d}}s{\rm{d}}u \end{array}$$ 引理2[12] (Finsler引理)设$\mathit{\boldsymbol{\varsigma}} \in {\mathbb{R}^n}$是一个随机向量，$\mathit{\boldsymbol{ \boldsymbol{\varGamma}}} \in {\mathbb{R}^{n \times m}}$是一个对称矩阵，$\mathit{\boldsymbol{B}} \in {\mathbb{R}^{m \times n}}$是任意矩阵且${\rm{rank}}(\mathit{\boldsymbol{B}}) = r < n$。${\mathit{\boldsymbol{B}}^ \bot } \in {\mathbb{R}^{n \times (n - r)}}$是矩阵$\mathit{\boldsymbol{B}}$的右正交补，满足$\mathit{\boldsymbol{BB}}^ \bot = 0$且${\mathit{\boldsymbol{B}}^ \bot }{\mathit{\boldsymbol{B}}^{ \bot T}} > 0$。则如下命题是等价的：
1) ${\mathit{\boldsymbol{\varsigma}} ^T}\mathit{\boldsymbol{ \boldsymbol{\varGamma} \boldsymbol{\varsigma}}} < 0$，$\forall \mathit{\boldsymbol{\varsigma}} \ne 0$，$\mathit{\boldsymbol{B}}\mathit{\boldsymbol{\varsigma}} = 0$
2) ${\mathit{\boldsymbol{B}}^{ \bot T}}\mathit{\boldsymbol{ \boldsymbol{\varGamma}}} {\mathit{\boldsymbol{B}}^ \bot } < 0 $
3) $\exists \kappa \in \mathbb{R}$：$\mathit{\boldsymbol{ \boldsymbol{\varGamma}}} - \kappa {\mathit{\boldsymbol{B}}^T}\mathit{\boldsymbol{B}} < 0$
4) $\exists \mathit{\boldsymbol{R}} \in {\mathbb{R}^{n \times m}}$：$\mathit{\boldsymbol{ \boldsymbol{\varGamma}}} + \mathit{\boldsymbol{RB}} + {\mathit{\boldsymbol{B}}^T}{\mathit{\boldsymbol{R}}^T} < 0$
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定理1 对于给定时滞$\tau > 0$，T-S模糊时滞系统(2)在$\mathit{\boldsymbol{u}}(t) = 0$下是渐近稳定的，如果存在以下具有合适维数的矩阵$0 < \mathit{\boldsymbol{P}} = {\mathit{\boldsymbol{P}}^T}$，$0 < \mathit{\boldsymbol{Q}} = {\mathit{\boldsymbol{Q}}^T}$，$0 < \mathit{\boldsymbol{R}} = {\mathit{\boldsymbol{R}}^T}$，使LMIs满足：
$${\mathit{\boldsymbol{ \boldsymbol{\varXi}}} _i} < 0, ~~\forall i = 1, 2, \cdots , p$$ (6) 其中，
$$\begin{array}{c} {\mathit{\boldsymbol{ \boldsymbol{\varXi} }}_i} = {\rm{Sym}}\{ {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_1}^T\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_{2i}}\} + {\mathit{\boldsymbol{e}}_1}^T\mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{e}}_1} - {\mathit{\boldsymbol{e}}_2}^T\mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{e}}_2} + \tau {\mathit{\boldsymbol{U}}_i}^T\mathit{\boldsymbol{R}}{\mathit{\boldsymbol{U}}_i} + \\ \frac{1}{\tau }( - {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_3}^T\mathit{\boldsymbol{R}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_3} - 3{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_4}^TR{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_4} - 5{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_5}^T\mathit{\boldsymbol{R}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_5}) \end{array}$$ $$\begin{array}{c} {\mathit{\boldsymbol{U}}_i} = {\mathit{\boldsymbol{A}}_i}{\mathit{\boldsymbol{e}}_1} + {\mathit{\boldsymbol{A}}_{\tau i}}{\mathit{\boldsymbol{e}}_2}\\ {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_1} = {[\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{e}}_1}^T}&{\tau {\mathit{\boldsymbol{e}}_3}^T}&{{\tau ^2}{\mathit{\boldsymbol{e}}_4}^T} \end{array}]^T}\\ {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_{2i}} = {[\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{U}}_i}^T}&{{\mathit{\boldsymbol{e}}_1}^T - {\mathit{\boldsymbol{e}}_2}^T}&{\tau ({\mathit{\boldsymbol{e}}_1}^T - {\mathit{\boldsymbol{e}}_3}^T)} \end{array}]^T}\\ {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_3} = {\mathit{\boldsymbol{e}}_1} - {\mathit{\boldsymbol{e}}_2}\\ {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_4} = {\mathit{\boldsymbol{e}}_1} + {\mathit{\boldsymbol{e}}_2} - 2{\mathit{\boldsymbol{e}}_3}\\ {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_5} = {\mathit{\boldsymbol{e}}_1} - {\mathit{\boldsymbol{e}}_2} + 6{\mathit{\boldsymbol{e}}_3} - 12{\mathit{\boldsymbol{e}}_4}\\ {\mathit{\boldsymbol{e}}_i} = [\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{0}}_{n \times (i - 1)n}}}&{{\mathit{\boldsymbol{I}}_n}}&{{\mathit{\boldsymbol{0}}_{n \times (4 - i)n}}} \end{array}]{\rm{ }}i = 1, 2, 3, 4 \end{array}$$ 证明： 考虑如下的Lyapunov-Krasovskii泛函：
$$V(t) = {V_1}(t){\rm{ + }}{V_2}(t){\rm{ + }}{V_3}(t)$$ (7) 其中，
$$\begin{array}{c} {V_1}(t) = {\mathit{\boldsymbol{\eta }}^T}(t)\mathit{\boldsymbol{P\eta }}(t)\\ {V_2}(t) = \int_{{\rm{ }}t - \tau }^{{\rm{ }}t} {{\mathit{\boldsymbol{x}}^T}(s)\mathit{\boldsymbol{Qx}}(s){\rm{ }}} {\rm{d}}s\\ {V_3}(t) = \int_{{\rm{ }} - \tau }^{{\rm{ }}0} {\int_{{\rm{ }}t + \theta }^{{\rm{ }}t} {{{\mathit{\boldsymbol{\dot x}}}^T}(s)\mathit{\boldsymbol{R\dot x}}(s)} {\rm{ d}}s} {\rm{d}}\theta \\ \mathit{\boldsymbol{\eta }}(t) = {\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{x}}^T}(t)}&{\int_{{\rm{ }}t - \tau }^{{\rm{ }}t} {{\mathit{\boldsymbol{x}}^T}(s)} {\rm{ d}}s}&{\int_{{\rm{ }}t - \tau }^{{\rm{ }}t} {\int_{{\rm{ }}s}^{{\rm{ }}t} {{\mathit{\boldsymbol{x}}^T}(u)} } {\rm{ d}}u{\rm{d}}s} \end{array}} \right]^T}\\ \mathit{\boldsymbol{\xi }}(t) = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{x}}^T}(t)}&{{\mathit{\boldsymbol{x}}^T}(t - \tau )} \end{array}}&{\frac{1}{\tau }\int_{{\rm{ }}t - \tau }^{{\rm{ }}t} {{\mathit{\boldsymbol{x}}^T}(s){\rm{ }}} {\rm{d}}s} \end{array}} \right.\\ {\left. {\frac{1}{{{\tau ^2}}}\int_{{\rm{ }}t - \tau }^{{\rm{ }}t} {\int_{{\rm{ }}s}^{{\rm{ }}t} {{\mathit{\boldsymbol{x}}^T}(u){\rm{ }}} } {\rm{d}}u{\rm{d}}s} \right]^T} \end{array}$$ 矩阵$\mathit{\boldsymbol{P}} = {\mathit{\boldsymbol{P}}^T} > 0$，$\mathit{\boldsymbol{Q}} = {\mathit{\boldsymbol{Q}}^T} > 0$，$\mathit{\boldsymbol{R}} = {\mathit{\boldsymbol{R}}^T} > 0$具有适合的维数。
对Lyapunov-Krasovskii泛函(7)沿着轨线(2)对时间$t$求导可得：
$$\begin{array}{c} {{\dot V}_1}(t){\rm{ = }}{{\mathit{\boldsymbol{\dot \eta }}}^T}(t)\mathit{\boldsymbol{P\eta }}(t) + {\mathit{\boldsymbol{\eta }}^T}(t)\mathit{\boldsymbol{P\dot \eta }}(t){\rm{ = }}\\ \mathop \sum \limits_{i = 1}^r {m_i}(\mathit{\boldsymbol{x}}(t)){\mathit{\boldsymbol{\xi }}^T}(t){\rm{Sym}}\{ {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_1}^T\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_{2i}}\} \mathit{\boldsymbol{\xi }}(t) \end{array}$$ (8) $$\begin{array}{c} {{\dot V}_2}(t){\rm{ = }}{\mathit{\boldsymbol{x}}^T}(t)\mathit{\boldsymbol{Qx}}(t) - {\mathit{\boldsymbol{x}}^T}(t - \tau )\mathit{\boldsymbol{Qx}}(t - \tau ){\rm{ = }}\\ \mathop \sum \limits_{i = 1}^p {m_i}(\mathit{\boldsymbol{x}}(t)){\mathit{\boldsymbol{\xi }}^T}(t)({\mathit{\boldsymbol{e}}_1}^T\mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{e}}_1} - {\mathit{\boldsymbol{e}}_2}^T\mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{e}}_2})\mathit{\boldsymbol{\xi }}(t) \end{array}$$ (9) $$\begin{array}{c} {{\dot V}_3}(t) = \tau {{\mathit{\boldsymbol{\dot x}}}^T}(t)\mathit{\boldsymbol{R\dot x}}(t) - \int_{{\rm{ }} - \tau }^{{\rm{ }}0} {{{\mathit{\boldsymbol{\dot x}}}^T}(t + \theta )\mathit{\boldsymbol{R\dot x}}(t + \theta ){\rm{ d}}\theta } \le \\ \tau \mathop \sum \limits_{i = 1}^p {m_i}(\mathit{\boldsymbol{x}}(t)){\mathit{\boldsymbol{\xi }}^T}(t){\mathit{\boldsymbol{U}}_i}^T\mathit{\boldsymbol{R}}{\mathit{\boldsymbol{U}}_i}\mathit{\boldsymbol{\xi }}(t) - \\ \int_{t - \tau }^t {{{\mathit{\boldsymbol{\dot x}}}^T}(s)\mathit{\boldsymbol{R\dot x}}(s)} {\rm{ d}}s \end{array}$$ (10) 由式(8)，(9)和(10)得
$$\begin{array}{c} \dot V(t) = {{\dot V}_1}(t){\rm{ + }}{{\dot V}_2}(t){\rm{ + }}{{\dot V}_3}(t) \le \\ \mathop \sum \limits_{i = 1}^p {m_i}(\mathit{\boldsymbol{x}}(t)){\mathit{\boldsymbol{\xi }}^T}(t){\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}_i}\mathit{\boldsymbol{\xi }}(t) - \int_{t - \tau }^t {{{\mathit{\boldsymbol{\dot x}}}^T}(s)\mathit{\boldsymbol{R\dot x}}(s)} {\rm{ d}}s \end{array}$$ (11) 其中
$${\mathit{\boldsymbol{ \boldsymbol{\varDelta} }}_i} = {\rm{Sym}}\{ {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_1}^T\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_{2i}}\} + {\mathit{\boldsymbol{e}}_1}^T\mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{e}}_1} - {\mathit{\boldsymbol{e}}_2}^T\mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{e}}_2} + \tau {\mathit{\boldsymbol{U}}_i}^T\mathit{\boldsymbol{R}}{\mathit{\boldsymbol{U}}_i}$$ 运用引理1中基于辅助函数的不等式对积分项对进行放缩：
$$\begin{array}{C} - \int_{t - \tau }^t {{{\mathit{\boldsymbol{\dot x}}}^T}(s)\mathit{\boldsymbol{R\dot x}}(s){\rm{ }}} {\rm{d}}s \le \mathop \sum \limits_{i = 1}^p {m_i}(\mathit{\boldsymbol{x}}(t)){\mathit{\boldsymbol{\xi }}^T}(t)\frac{1}{\tau }( - {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_3}^T\mathit{\boldsymbol{R}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_3} - \\ 3{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_4}^T\mathit{\boldsymbol{R}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_4} - 5{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_5}^T\mathit{\boldsymbol{R}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}_5})\mathit{\boldsymbol{\xi }}(t) \end{array}$$ (12) 将式(12)带入(11)得
$$\dot V(t) \le \sum\limits_{i = 1}^p {{m_i}(\mathit{\boldsymbol{x}}(t)){\mathit{\boldsymbol{\xi }}^T}(t){\mathit{\boldsymbol{ \boldsymbol{\varXi} }}_i}\mathit{\boldsymbol{\xi }}(t)} $$ (13) 如果LMIs (6)成立，则有$\dot V(t) < 0$，由Lyapunov稳定性定理可知，系统(2)是渐近稳定的，证明完毕。
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定理2 给定时滞$\tau > 0$和参数$\varepsilon $，闭环系统(5)是渐近稳定的，如果存在以下具有合适维数的矩阵$0 < \mathit{\boldsymbol{\tilde P}} = {\mathit{\boldsymbol{\tilde P}}^T}$，$0 < \mathit{\boldsymbol{\tilde Q}} = {\mathit{\boldsymbol{\tilde Q}}^T}$，$0 < \mathit{\boldsymbol{\tilde R}} = {\mathit{\boldsymbol{\tilde R}}^T}$，及任意矩阵$\mathit{\boldsymbol{X}}$，${\mathit{\boldsymbol{G}}_j}$，${\mathit{\boldsymbol{G}}_{\tau j}}$使如下的LMIs满足
$$ {\mathit{\boldsymbol{ \boldsymbol{\tilde \varXi}}} _{ij}} < 0~~~\forall i = 1, 2, \cdots , p~~~j = 1, 2, \cdots , c $$ (14) 此时，控制器增益为${\mathit{\boldsymbol{K}}_j} = {\mathit{\boldsymbol{G}}_j}{\mathit{\boldsymbol{X}}^{ - 1}}$，${\mathit{\boldsymbol{K}}_{\tau j}} = {\mathit{\boldsymbol{G}}_{\tau j}}{\mathit{\boldsymbol{X}}^{ - 1}}$，$j = 1, 2, \cdots , c$。
其中，
$$\begin{array}{c} {{\mathit{\boldsymbol{ \boldsymbol{\tilde \varXi} }}}_{ij}} = {\rm{Sym}}\{ {{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_1}^T\mathit{\boldsymbol{\tilde P}}{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_2}\} + {{\mathit{\boldsymbol{\tilde e}}}_1}^T\mathit{\boldsymbol{\tilde Q}}{{\mathit{\boldsymbol{\tilde e}}}_1} - {{\mathit{\boldsymbol{\tilde e}}}_2}^T\mathit{\boldsymbol{\tilde Q}}{{\mathit{\boldsymbol{\tilde e}}}_2} + \tau {{\mathit{\boldsymbol{\tilde e}}}_5}^T\mathit{\boldsymbol{\tilde R}}{{\mathit{\boldsymbol{\tilde e}}}_5} + \\ \frac{1}{\tau }( - {{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_3}^T\mathit{\boldsymbol{\tilde R}}{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_3} - 3{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_4}^T\mathit{\boldsymbol{\tilde R}}{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_4} - 5{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_5}^T\mathit{\boldsymbol{\tilde R}}{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_5}) + \\ {\rm{Sym}}\{ ({{\mathit{\boldsymbol{\tilde e}}}_1}^T + \varepsilon {{\mathit{\boldsymbol{\tilde e}}}_5}^T)({\mathit{\boldsymbol{A}}_i}\mathit{\boldsymbol{X}}{{\mathit{\boldsymbol{\tilde e}}}_1} + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{G}}_j}{{\mathit{\boldsymbol{\tilde e}}}_1} + {\mathit{\boldsymbol{A}}_{\tau i}}\mathit{\boldsymbol{X}}{{\mathit{\boldsymbol{\tilde e}}}_2} + \\ {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{G}}_{\tau j}}{{\mathit{\boldsymbol{\tilde e}}}_2} - \mathit{\boldsymbol{X}}{{\mathit{\boldsymbol{\tilde e}}}_5})\} \end{array}$$ $$\begin{array}{C} {{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_1} = {[\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\tilde e}}}_1}^T}&{\tau {{\mathit{\boldsymbol{\tilde e}}}_3}^T}&{{\tau ^2}{{\mathit{\boldsymbol{\tilde e}}}_4}^T} \end{array}]^T}\\ {{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_2} = {[\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\tilde e}}}_5}^T}&{{{\mathit{\boldsymbol{\tilde e}}}_1}^T - {{\mathit{\boldsymbol{\tilde e}}}_2}^T}&{\tau ({{\mathit{\boldsymbol{\tilde e}}}_1}^T - {{\mathit{\boldsymbol{\tilde e}}}_3}^T)} \end{array}]^T}\\ {{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_3} = {{\mathit{\boldsymbol{\tilde e}}}_1} - {{\mathit{\boldsymbol{\tilde e}}}_2}\\ {{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_4} = {{\mathit{\boldsymbol{\tilde e}}}_1} + {{\mathit{\boldsymbol{\tilde e}}}_2} - 2{{\mathit{\boldsymbol{\tilde e}}}_3}\\ {{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_5} = {{\mathit{\boldsymbol{\tilde e}}}_1} - {{\mathit{\boldsymbol{\tilde e}}}_2} + 6{{\mathit{\boldsymbol{\tilde e}}}_3} - 12{{\mathit{\boldsymbol{\tilde e}}}_4}\\ {{\mathit{\boldsymbol{\tilde e}}}_i} = [\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{0}}_{n \times (i - 1)n}}}&{{\mathit{\boldsymbol{I}}_n}}&{{\mathit{\boldsymbol{0}}_{n \times (5 - i)n}}} \end{array}], i = 1, \cdots , 5 \end{array}$$ 证明： 首先定义矩阵：
$$\begin{array}{c} \mathit{\boldsymbol{\tilde \xi }}(t) = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{x}}^T}(t)}&{{\mathit{\boldsymbol{x}}^T}(t - \tau )} \end{array}}&{\frac{1}{\tau }\int_{{\rm{ }}t - \tau }^{{\rm{ }}t} {{\mathit{\boldsymbol{x}}^T}(s)} {\rm{ d}}s} \end{array}} \right.\\ {\left. {\begin{array}{*{20}{c}} {\frac{1}{{{\tau ^2}}}\int_{{\rm{ }}t - \tau }^{{\rm{ }}t} {\int_{{\rm{ }}s}^{{\rm{ }}t} {{\mathit{\boldsymbol{x}}^T}(u)} {\rm{ }}} {\rm{d}}u{\rm{d}}s}&{{{\mathit{\boldsymbol{\dot x}}}^T}(t)} \end{array}} \right]^T} \end{array}$$ (15) $$\begin{array}{c} \mathit{\boldsymbol{\psi }}(t) = [\begin{array}{*{20}{c}} {\mathit{\boldsymbol{A}}(t) + \mathit{\boldsymbol{B}}(t)\mathit{\boldsymbol{K}}(t)}&{{\mathit{\boldsymbol{A}}_\tau }(t) + \mathit{\boldsymbol{B}}(t){\mathit{\boldsymbol{K}}_\tau }(t)} \end{array}\\ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} \mathit{\boldsymbol{0}}&\mathit{\boldsymbol{0}} \end{array}}&{ - \mathit{\boldsymbol{I}}} \end{array}] \end{array}$$ (16) 在定理1证明的基础上，由式(13)可得：
$$\dot V(t) \le \sum\limits_{i = 1}^p {\sum\limits_{j = 1}^c {{m_i}(\mathit{\boldsymbol{x}}(t)){h_j}(\mathit{\boldsymbol{x}}(t)){{\mathit{\boldsymbol{\tilde \xi }}}^T}(t)\mathit{\boldsymbol{ \boldsymbol{\varPi} \tilde \xi }}(t)} } $$ (17) 其中，
$$\begin{array}{c} \mathit{\boldsymbol{ \boldsymbol{\varPi} }} = {\rm{Sym}}\{ {{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_1}^T\mathit{\boldsymbol{P}}{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_2}\} + {{\mathit{\boldsymbol{\tilde e}}}_1}^T\mathit{\boldsymbol{Q}}{{\mathit{\boldsymbol{\tilde e}}}_1} - {{\mathit{\boldsymbol{\tilde e}}}_2}^T\mathit{\boldsymbol{Q}}{{\mathit{\boldsymbol{\tilde e}}}_2} + \mathit{\boldsymbol{\tau }}{{\mathit{\boldsymbol{\tilde e}}}_5}^T\mathit{\boldsymbol{R}}{{\mathit{\boldsymbol{\tilde e}}}_5} + \\ \quad \frac{1}{\tau }( - {{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_3}^T\mathit{\boldsymbol{R}}{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_3} - 3{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_4}^T\mathit{\boldsymbol{R}}{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_4} - 5{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_5}^T\mathit{\boldsymbol{R}}{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_5}) \end{array}$$ (18) 由式(15)和(16)得$\mathit{\boldsymbol{\psi}} (t)\mathit{\boldsymbol{\tilde \xi}} (t) = 0$，运用引理2的命题(1)和(4)可知，闭环T-S模糊时滞系统(7)是渐进稳定的，如果存在$\mathit{\boldsymbol{L}} \in {R^{5n \times 5n}}$满足：
$$\mathit{\boldsymbol{ \boldsymbol{\varPi} }} + \mathit{\boldsymbol{L\psi }}(t) + {\mathit{\boldsymbol{\psi }}^T}(t){\mathit{\boldsymbol{L}}^T} < 0$$ (19) 设，$\boldsymbol{\varpi} {\rm{ = diag}}(\mathit{\boldsymbol{XXXXX}})$，其中$\mathit{\boldsymbol{X}} \in {R^{n \times n}}$是任意可逆矩阵，$\varepsilon $是调整参数。将式(19)左边乘以${\boldsymbol{\varpi} ^T}$，右边乘以$\boldsymbol{\varpi} $得：
$$\begin{array}{c} {\boldsymbol{\varpi} ^T}(\mathit{\boldsymbol{ \boldsymbol{\varPi} }} + \mathit{\boldsymbol{L\psi }}(t) + {\mathit{\boldsymbol{\psi }}^T}(t){\mathit{\boldsymbol{L}}^T})\boldsymbol{\varpi} = \\ {\rm{Sym}}\{ {{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_1}^T\mathit{\boldsymbol{\tilde P}}{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_2}\} + {{\mathit{\boldsymbol{\tilde e}}}_1}^T\mathit{\boldsymbol{\tilde Q}}{{\mathit{\boldsymbol{\tilde e}}}_1} - {{\mathit{\boldsymbol{\tilde e}}}_2}^T\mathit{\boldsymbol{\tilde Q}}{{\mathit{\boldsymbol{\tilde e}}}_2} + \tau {{\mathit{\boldsymbol{\tilde e}}}_5}^T\mathit{\boldsymbol{\tilde R}}{{\mathit{\boldsymbol{\tilde e}}}_5} + \\ \frac{1}{\tau }( - {{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_3}^T\mathit{\boldsymbol{\tilde R}}{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_3} - 3{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_4}^T\mathit{\boldsymbol{\tilde R}}{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_4} - 5{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_5}^T\mathit{\boldsymbol{\tilde R}}{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varOmega} }}}_5}) + \\ {\rm{Sym}}\{ ({{\mathit{\boldsymbol{\tilde e}}}_1}^T + \varepsilon {{\mathit{\boldsymbol{\tilde e}}}_5}^T)({\mathit{\boldsymbol{A}}_i}\mathit{\boldsymbol{X}}{{\mathit{\boldsymbol{\tilde e}}}_1} + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{G}}_j}{{\mathit{\boldsymbol{\tilde e}}}_1} + {\mathit{\boldsymbol{A}}_{\tau i}}\mathit{\boldsymbol{X}}{{\mathit{\boldsymbol{\tilde e}}}_2} + \\ {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{G}}_{\tau j}}{{\mathit{\boldsymbol{\tilde e}}}_2} - \mathit{\boldsymbol{X}}{{\mathit{\boldsymbol{\tilde e}}}_5})\} \end{array}$$ (20) 其中，
$$\begin{array}{c} \mathit{\boldsymbol{\tilde P}} = {\rm{dia}}{{\rm{g}}^T}(\mathit{\boldsymbol{XXX}})\mathit{\boldsymbol{P}}{\rm{diag}}(\mathit{\boldsymbol{XXX}})\\ \mathit{\boldsymbol{\tilde Q}} = {\mathit{\boldsymbol{X}}^T}\mathit{\boldsymbol{QX}}\\ \mathit{\boldsymbol{\tilde R}} = {\mathit{\boldsymbol{X}}^T}\mathit{\boldsymbol{RX}}\\ \mathit{\boldsymbol{G}}(t) = \mathit{\boldsymbol{K}}(t)\mathit{\boldsymbol{X}}\\ {\mathit{\boldsymbol{G}}_\tau }(t) = {\mathit{\boldsymbol{K}}_\tau }(t)\mathit{\boldsymbol{X}}\\ {\mathit{\boldsymbol{G}}_\tau }(t) = {\mathit{\boldsymbol{K}}_\tau }(t)\mathit{\boldsymbol{X}} \end{array}$$ 由式(20)可得：
$$\dot V(t) \le \sum\limits_{i = 1}^p {\sum\limits_{j = 1}^c {{m_i}(\mathit{\boldsymbol{x}}(t)){h_j}(\mathit{\boldsymbol{x}}(t)){{\mathit{\boldsymbol{\tilde \xi }}}^T}(t){{\mathit{\boldsymbol{ \boldsymbol{\tilde \varXi} }}}_{ij}}\mathit{\boldsymbol{\tilde \xi }}(t)} } $$ (21) 如果LMIs (14)成立，则有$\dot V(t) < 0$，由Lyapunov稳定性可知，闭环系统(5)是渐近稳定的，证明完毕。
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考虑文献[13-17]给出的$\mathit{\boldsymbol{u}}(t) = 0$，具有两个规则数目的T-S模糊时滞系统：
规则1：如果${x_1}(t)$是${F_1}^1$，则$\dot x(t) = {\mathit{\boldsymbol{A}}_1}x(t) + $ ${\mathit{\boldsymbol{A}}_{\tau 1}}x(t - \tau )$
规则2：如果${x_1}(t)$是${F_1}^2$，则$\dot x(t) = {\mathit{\boldsymbol{A}}_2}x(t) + $ ${\mathit{\boldsymbol{A}}_{\tau 2}}x(t - \tau )$
其中，
选取隶属函数：
${m_1}({x_1}) = \frac{1}{{1 + \exp ( - 2{x_1})}}$ ${m_2}({x_1}) = 1 - {m_1}({x_1})$
采用该实例与文献[13-17]提出的方法作比较，首先表 1列出了根据不同方法计算出的保证T-S模糊时滞系统(22)渐进稳定的最大允许时滞$\tau $。可以看出，由本文定理1得到的$\tau $是最大的，说明本文给出的稳定性准则比文献[13-17]提出的方法具有更小的保守性。另外还可以看到，本文需要的变量个数是最小的，说明本文提出的方法计算复杂度是最小的。
表 1 最大允许时滞$\tau $
为了说明本文提出的控制器设计方法的有效性，考虑文献[18-20]给出的一个T-S模糊时滞系统中常用的基准例子：搅拌反应釜(CSTR)系统。CSTR系统由以下具有3个模糊规则数目的T-S模糊时滞模型描述：
规则1：如果${x_2}(t)$为0.886 2 (温度低)，则$\delta \dot x(t) = {\mathit{\boldsymbol{A}}_1}\delta x(t) + {\mathit{\boldsymbol{A}}_{\tau 1}}\delta x(t - \tau ) + {\mathit{\boldsymbol{B}}_1}\delta u(t)$
规则2：如果${x_2}(t)$为2.7520 (温度适中)，则$\delta \dot x(t) = {\mathit{\boldsymbol{A}}_2}\delta x(t) + {\mathit{\boldsymbol{A}}_{\tau 2}}\delta x(t - \tau ) + $ ${\mathit{\boldsymbol{B}}_2}\delta u(t)$
规则3：如果${x_2}(t)$为4.705 2 (温度高)，则$\delta \dot x(t) = {\mathit{\boldsymbol{A}}_3}\delta x(t) + {\mathit{\boldsymbol{A}}_{\tau 3}}\delta x(t - \tau ) + {\mathit{\boldsymbol{B}}_3}\delta u(t)$
其中，$\delta x(t) = x(t) - {x_d}$，$\delta x(t - \tau ) = x(t - \tau ) - {x_d}$，$\delta u(t) = u(t) - {u_d}$，$({x_d}, \;{u_d})$是一个期望操作点，此外：
$${\mathit{\boldsymbol{A}}_1} = \left[ {\begin{array}{*{20}{c}} { - {\rm{1}}{\rm{.427 4}}}&{0.{\rm{075 7}}} \\ { - {\rm{1}}{\rm{.418 9}}}&{ - 0.944{\rm{ }}2} \end{array}} \right]$$ $${\mathit{\boldsymbol{A}}_2} = \left[ {\begin{array}{*{20}{c}} { - {\rm{2}}{\rm{.050 8}}}&{0.395{\rm{ }}8} \\ { - {\rm{6}}{\rm{.406 6}}}&{1.616{\rm{ }}8} \end{array}} \right]$$ $${\mathit{\boldsymbol{A}}_3} = \left[ {\begin{array}{*{20}{c}} { - {\rm{4}}{\rm{.527 9}}}&{0.316{\rm{ }}7} \\ { - 26.222{\rm{ }}8}&{{\rm{0}}{\rm{.983 7}}} \end{array}} \right]$$ $${\mathit{\boldsymbol{A}}_{\tau 1}} = {\mathit{\boldsymbol{A}}_{\tau 2}} = {\mathit{\boldsymbol{A}}_{\tau 3}} = \left[ {\begin{array}{*{20}{c}} {0.25}&0 \\ 0&{0.25} \end{array}} \right]$$ $${\mathit{\boldsymbol{B}}_1} = {\mathit{\boldsymbol{B}}_2} = {\mathit{\boldsymbol{B}}_3} = \left[ {\begin{array}{*{20}{c}} 0 \\ {0.3} \end{array}} \right]$$ $${m_1}({x_2}) = \left\{ {\begin{array}{*{20}{l}} {1\quad \quad \;\quad \quad \;\quad \quad \;\quad \;\quad {x_2} \leqslant 0.886{\rm{ }}2} \\ {1 - \frac{{{x_2} - 0.886{\rm{ }}2}}{{2.752{\rm{ }}0 - 0.886{\rm{ }}2}}\;\;\;0.886{\rm{ }}2 < {x_2} < 2.752} \\ {0\quad \quad \;\quad \quad \;\quad \quad \;\quad \quad {x_2} \geqslant 2.752} \end{array}} \right.$$ $${m_2}({x_2}) = \left\{ {\begin{array}{*{20}{c}} {1 - {m_1}\quad \;\quad \;{x_2} \leqslant 2.752} \\ {1 - {m_3}\quad \;\quad \;{x_2} \geqslant 2.752} \end{array}} \right.$$ $${m_3}({x_2}) = \left\{ {\begin{array}{*{20}{l}} {0\quad \quad \;\quad \quad \;\quad \quad \;\;\;{x_2} \leqslant 2.752} \\ {\frac{{{x_2} - 2.752}}{{4.705{\rm{ }}2 - 2.752}}\quad \;2.752 < {x_2} < 4.705{\rm{ }}2} \\ {1\;\quad \quad \;\quad \quad \;\quad \quad \;\;\;{x_2} \geqslant 4.705{\rm{ }}2} \end{array}} \right.$$ 基于前提不匹配策略，设计一个具有两个模糊规则数目的记忆状态反馈控制器：
规则1：如果${x_2}(t)$为${F_1}^1$，则$u(t)=K_{1} x(t)+$ ${K_{\tau 1}}x(t - \tau )$
规则2：如果${x_2}(t)$为${F_1}^2$，则$u(t)=K_{2} x(t)+$ ${K_{\tau 2}}x(t - \tau )$
选择模糊控制器的前提隶属函数：
${h_1}({x_1}) = \frac{1}{{1 + \exp \left( { - \frac{{{x_1}}}{2}} \right)}}$ ${h_2}({x_1}) = 1 - {h_1}({x_1})$
特别地，取$\tau = 8.5$且调整参数$\varepsilon = 0.5$，由定理2可以得到基于前提不匹配策略的状态反馈控制器。其中，记忆状态反馈控制器为：
$${K_1} = [{\rm{31}}{\rm{.048 4 }} - {\rm{9}}{\rm{.716 3}}]$$ $${K_2} = [{\rm{32}}{\rm{.792 9 }} - {\rm{10}}{\rm{.434 1}}]$$ $${K_{\tau 1}} = [ - {\rm{0}}{\rm{.072 5 }} - {\rm{0}}{\rm{.117 6}}]$$ $${K_{\tau 2}} = [{\rm{0}}{\rm{.160 0 0}}{\rm{.021 0}}]$$ 无记忆状态反馈控制器为：
$${K_1} = [{\rm{36}}{\rm{.790 7 }} - {\rm{11}}{\rm{.679 0}}]$$ $${K_2} = [{\rm{37}}{\rm{.527 0 }} - {\rm{12}}{\rm{.262 4}}]$$ 为了说明定理2控制器设计的有效性，取初始条件为$\boldsymbol{\phi} = {[1{\rm{ }} - 1.5]^T}$，仿真得到闭环系统的状态响应曲线如图 1所示，随着时间的增长，状态轨迹都趋于零，说明闭环系统是渐进稳定的。
图 1 闭环系统状态响应
文献[18-20]也研究了例2中的CSTR系统的控制器设计问题，但是文献[18-20]只考虑了无记忆模糊状态反馈控制器的设计方法，并未考虑记忆控制器的设计情况。另一方面，文献[18-20]的模糊控制器是基于PDC技术设计的，即要求模糊控制器与模糊系统分享相同的前提隶属函数和模糊规则数目。而本文定理2中提出的基于前提不匹配策略的模糊记忆状态反馈控制器，不要求与模糊模型分享相同的前提隶属函数和模糊规则数目。因此，本文提出的方法是传统的PDC设计技术的推广和延伸，可以处理更为一般的T-S模糊时滞系统，显然，本文定理2所提的方法比文献[18-20]提出的方法具有更小的保守性。
本文提出的基于前提不匹配策略的模糊控制器不要求与模糊系统分享相同的隶属函数和模糊规则数目，通过选择较为简单的隶属函数和更小的规则数目，从而降低模糊控制器的设计复杂度和设计成本。由于其设计的灵活性，本文所提的方法具有更好的应用前景。
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针对一类基于T-S模糊模型描述的非线性时滞系统，运用Lyapunov稳定性理论，结合积分不等式法，给出了新的保守性较小的稳定性条件；首次提出了基于前提不匹配技术的模糊记忆状态反馈控制器的设计方法。在求解控制器时，使用Finsler引理处理非线性耦合项。通过选择更为简单的隶属函数和较小的规则数目，降低了模糊控制器的设计复杂度，增加了设计的自由度。仿真实例显示了所提方法的有效性和适用性。在实际控制系统中，状态有可能不是完全可测的，因此，对于T-S模糊时滞系统的记忆输出反馈控制的研究是很有价值的，尤其是基于前提不匹配策略的记忆输出反馈控制是下一步需要深入研究的课题。
Memory State Feedback Control of Fuzzy Time-Delay Systems
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摘要: 该文研究了Takagi-Sugeno（T-S）模糊时滞系统的稳定与镇定问题。首先，选择一个近期提出的基于辅助函数的积分不等式，以线性矩阵不等式（LMIs）形式给出了保守性较小的时滞依赖的稳定性准则。其次，结合Finsler引理，首次提出了基于前提不匹配技术的模糊记忆状态反馈控制器设计方法，该前提不匹配的记忆控制器不要求与模糊系统拥有相同的隶属函数和模糊规则数目。最后，给出两个仿真算例证明所提理论的先进性和有效性。
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关键词:
- 基于辅助函数的积分不等式 /
- 线性矩阵不等式(LMIs) /
- Finsler引理 /
- 前提不匹配 /
- Takagi-Sugeno(T-S)模糊时滞系统
Abstract: The problems of stability and stabilization of Takagi-Sugeno (T-S) fuzzy time-delay systems are investigated in this paper. Firstly, less conservative delay-dependent stability criterion in terms of linear matrix inequalities (LMIs) is obtained by choosing a recently developed auxiliary function-based integral inequality. Secondly, combining with Finsler lemma, a fuzzy memory state feedback control design method is first time presented under the imperfect premise matching technique, while the memory controller under the imperfect premise matching technique does not be required to employ the same premise membership functions and the number of fuzzy rules as the fuzzy model. Finally, two numerical examples are given to show the progressiveness and effectiveness of the presented theory. -
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