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火箭发动机燃烧过程的鲁棒非脆弱H控制

吴玉彬 张合新 朱开锐 李国梁 惠俊军

引用本文:
Citation:

火箭发动机燃烧过程的鲁棒非脆弱H控制

    作者简介: 吴玉彬(1984- ),男,博士,工程师, wuyubinxa@126.com;
    通讯作者: 张合新, 59153220@qq.com
  • 中图分类号: O381; TP13

The robust non-fragile H control for the combustion process in rocket motor

    Corresponding author: ZHANG Hexin, 59153220@qq.com ;
  • CLC number: O381; TP13

  • 摘要: 针对某液体火箭发动机燃烧室的燃烧过程,设计了鲁棒非脆弱 H 状态反馈控制器。首先,基于一种新型的时滞分割法和交互式凸组合技术,借助于构造一个包含四重积分项的Lyapunov-Krasovskii泛函(LKF),并利用新的积分不等式方法给出了LMI形式的时滞相关有界实判据;其次,在有界实判据的基础上,采用矩阵的合同变换以及变量替代技巧将燃烧过程非线性矩阵不等式线性化,通过求解线性矩阵不等式得到相应的非脆弱H控制器的可行解。模拟结果验证了本文设计方法的有效性。
  • 图 1  燃烧室燃烧过程的自由运动曲线

    Figure 1.  Free motion of combustion in chambers

    图 2  非脆弱H控制器作用下系统的状态响应

    Figure 2.  Response of system under non-fragile H controller

    图 3  不同控制器作用下系统状态${x_2}(t)$的响应

    Figure 3.  Response of ${x_2}(t)$ under different controller

    图 4  控制器对时滞参数$\delta $的鲁棒性

    Figure 4.  Robust performance of controller to variation of $\delta $

    表 1  不同的H${_\infty }$性能指标ρ,模拟给出的MADB值hM

    Table 1.  The maximum allowable delay bound hM for a given ρ

    来源hM
    ρ=2.0ρ=2.5ρ=3.0ρ=3.5ρ=4.0
    文献[16]0.405 70.466 00.504 70.531 60.551 5
    文献[17]0.405 70.466 00.504 60.531 60.551 5
    文献[18]0.420 30.477 90.514 60.540 10.558 9
    文献[4]0.473 40.523 70.554 50.575 40.590 4
    文献[19]0.662 00.704 00.730 00.747 00.759 5
    定理10.957 11.013 61.056 51.081 21.092 7
    下载: 导出CSV

    表 2  不同的MADB值hM,仿真给出的MAPI值ρ

    Table 2.  The minimum allowable performance index ρ for a given hM

    来源ρ
    hM=0.1hM=0.2hM=0.3hM=0.4hM=0.5
    文献[16]1.071 41.242 61.506 71.963 42.298 1
    文献[17]1.071 41.242 51.506 71.963 42.298 1
    文献[18]1.057 71.211 21.451 51.873 32.775 7
    文献[19]0.933 10.952 51.021 61.120 41.284 3
    定理10.815 60.853 20.924 51.042 81.123 9
    下载: 导出CSV
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  • 收稿日期:  2017-06-30
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火箭发动机燃烧过程的鲁棒非脆弱H控制

    作者简介:吴玉彬(1984- ),男,博士,工程师, wuyubinxa@126.com
    通讯作者: 张合新, 59153220@qq.com
  • 1. 火箭军工程大学控制工程系,陕西 西安 710025
  • 2. 中国人民解放军陆军重庆军代局,重庆 400060
  • 3. 中国人民解放军96037部队,陕西 宝鸡 721013

摘要: 针对某液体火箭发动机燃烧室的燃烧过程,设计了鲁棒非脆弱 H 状态反馈控制器。首先,基于一种新型的时滞分割法和交互式凸组合技术,借助于构造一个包含四重积分项的Lyapunov-Krasovskii泛函(LKF),并利用新的积分不等式方法给出了LMI形式的时滞相关有界实判据;其次,在有界实判据的基础上,采用矩阵的合同变换以及变量替代技巧将燃烧过程非线性矩阵不等式线性化,通过求解线性矩阵不等式得到相应的非脆弱H控制器的可行解。模拟结果验证了本文设计方法的有效性。

English Abstract

  • 时滞现象常存在于导弹的制导、飞行器的控制与航天系统当中,例如某液体火箭发动机燃烧室内的燃烧过程就包含了非常明显的时滞,其动态模型可简化为一个不稳定的时滞系统[1-2]。在系统的分析与控制器的设计中,如果忽略时滞的影响,往往导致系统不稳定以及性能的下降。因而近年来,时滞系统的稳定性分析及控制问题成为一个热点的研究领域[3-7]。而对于液体火箭发动机而言,燃烧不稳定性问题一直是困扰其技术发展的难题。当发生不稳定燃烧时,燃烧室内压力振荡会给系统的敏感制导元件以及系统结构带来破坏性的影响[8]。因此,如何设计反馈控制器以改善燃烧过程的稳定性能,成为学者们关注的热点问题[9-11]

    文献[12-14]讨论了在状态反馈控制器作用下系统的镇定问题,但所给出的控制器都要求能够精确实现,不具有鲁棒性。在控制器的设计实现中,由于硬件和软件等原因,不同程度上或多或少都存在一定的不确定性[15]。当控制器参数存在摄动时,常规的鲁棒控制器表现出高度的脆弱性,从而造成闭环系统的性能下降甚至控制器失效。非脆弱控制[16]的提出,弥补了这一不足,它不仅考虑被控对象的不确定性,而且考虑控制器增益本身的不确定性,因而更具适应性。文献[17-19]和文献[20-21]分别针对时滞系统的非脆弱H控制问题和非脆弱保性能控制问题进行了深入研究。在这些研究中,主要围绕如何降低所得结论的保守性和满足一定的性能指标而展开。因此,如何选取合适的L-K泛函和界定条件,进一步得到保守性更小的时滞相关条件从而设计有效的控制器便成为目前时滞系统稳定性分析与控制综合的首要问题。

    本文中针对某液体火箭发动机的动态燃烧模型,在假定控制器增益存在加性不确定的条件下,提出一个形式简单、保守性更低的时滞相关有界实判据。该判据借助于一种新型的时滞分割法,把时滞区间分割成不均匀的两部分,针对每一分割区间构造新的L-K泛函,并采用新的积分不等式和交互式凸组合技术给出不包含任何多余参量的LMI形式结论。在此基础上设计了鲁棒非脆弱${H_\infty }$控制器。模拟结果表明,该控制器允许控制器增益存在一定范围内的摄动,改善了传统鲁棒控制器的镇定效果,具有更强的鲁棒性。

    首先给出以下标记:${\mathbb{R}^n}$$n$维欧氏空间,${\mathbb{R}^{n \times m}}$$n \times m$维实矩阵,*为对称矩阵中的对称项,I为适当维数的单位矩阵。${{M}} = {{M}}^{\rm{T}} > 0$表示矩阵${{M}}$为对称矩阵,${{e}_i}$表示适当维数的块输入矩阵,例如${{e}}_6^{\rm{T}} = [{array}{*{20}{c}}0&0&0&0&0&1&0&0&0&0{array}]$

    • 考虑带有反馈伺服机构的单一燃料液体火箭发动机推进系统,系统描述见文献[9-10]。在假定非恒稳流动并考虑一致滞后情况下,整个系统的动态方程[1-2]为:

      $\left\{ {\begin{array}{*{20}{l}} {\dot \phi (t) = (\gamma - 1)\phi (t) - \gamma \phi (t - \delta ) + \mu (t - \delta )} \\ {{{\dot \mu }_1}(t) = \displaystyle\frac{1}{{\xi J}}\left[ { - \psi (t) + \frac{{{p_0} - {p_1}(t)}}{{2\Delta p}} + \omega (t)} \right]} \\ {\dot \mu (t) = \displaystyle\frac{1}{{(1 - \xi )J}}\left[ { - \mu (t) + \psi (t) - P\phi (t)} \right]} \\ {\dot \psi (t) = \displaystyle\frac{1}{E}\left[ {{\mu _1}(t) - \mu (t)} \right]} \end{array}} \right.$

      式中:$\phi (t)$表示燃烧室内的无量纲瞬时压力,${\mu _1}(t)$表示无量纲瞬时质逆流容量,$\mu (t)$表示喷射推进剂的无量纲瞬时速率;$\psi (t)$表示流路上的无量纲瞬时压力,$\omega (t)$表示无量纲扰动输入,选取${{u}} = ({p_0} - {p_1}(t))/(2\Delta p)$为控制变量,$\gamma = 1$, $\xi = 0.5$, $P = 1$, $J = 2$, $E = 1$,令${{x}}(t) = {\left[ {{array}{*{20}{c}} {\phi (t)}&{{\mu _1}(t)}&{\mu (t)}&{\psi (t)} {array}} \right]^{\rm T}}$,那么系统可化为:

      $\dot {{x}}(t) = {{Ax}}(t) + {{{A}}_1}{{x}}(t - \delta ) + {{Bu}}(t) + {{{B}}_1}{{\omega}} (t)$

      $ {\text{其中,}}{{A}} = \left[ {\begin{array}{*{20}{c}} 0&0&0&0 \\ 0&0&0&{ - 1} \\ { - 1}&0&{ - 1}&1 \\ 0&1&{ - 1}&0 \end{array}} \right],\;{{{A}}_1} = \left[ {\begin{array}{*{20}{c}} { - 1}&0&1&0 \\ 0&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \end{array}} \right],\;{{B}} = {{{B}}_1} = \left[ {\begin{array}{*{20}{c}} 0 \\ 1 \\ 0 \\ 0 \end{array}} \right] $

      假设初始函数取为:${{x}}(\theta ) = {\left[ {{array}{*{20}{c}} 1&1&1&1 {array}} \right]^{\rm T}}$$\theta \in [ - 1,0]$,时滞$\delta = 1$,则系统不加控制(${{u}}(t) = 0$)时的状态曲线如图1所示,此时系统的运动是不稳定的。

      图  1  燃烧室燃烧过程的自由运动曲线

      Figure 1.  Free motion of combustion in chambers

      针对系统(1)定义如下性能指标

      $ \Lambda (\omega ) = \int_0^\infty {[z{{(t)}^{\rm T}}z(t) - {\rho ^2}{\omega ^{\rm T}}(t)\omega (t)]} {\rm d}t\quad\quad\quad(3)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

      式中:$\rho \text{>} 0$为给定标量,$\omega (t)$为系统扰动输入,$z(t)$为系统受控输出。

      本文主要目标是在外部干扰作用下,设计状态反馈非脆弱${H_\infty }$控制器${{u}}(t) = \left[ {{{K}} + \Delta {{K}}(t)} \right]{{x}}(t)$,其中${{K}}$为控制器增益,$\Delta {{K}}(t)$为增益摄动且具有加性不确定性,满足:$\Delta {{K}}(t) = {{{D}}_{\rm c}}{{{F}}_{\rm c}}(t){{{E}}_{\rm c}}$${{F}}_{\rm c}^{\rm T}(t){{{F}}_{\rm c}}(t) \text{≤} {{I}}$${{{D}}_{\rm c}}$${{{E}}_{\rm c}}$为具有适当维数的常数矩阵,${{{F}}_{\rm c}}(t)$为未知时变矩阵。

      非脆弱${H_\infty }$控制器${{u}}(t)$使得所构造的闭环系统不仅内部稳定,而且在零初始条件下具有给定的${H_\infty }$扰动抑制水平$\rho $,即满足:$||{{z}}(t)|{|_2} \text{<} \rho ||{{\omega}} (t)|{|_2}$, ${{\omega}} (t) \in {L_2}[0,\infty )$,${{\omega}} (t) \ne 0$

      为了方便稳定性判据的证明,现将下一步需用到的引理归纳如下:

      引理1[4]。假定任意的正定矩阵${{M}} = {{{M}}^{\rm T}} \text{>} 0$,标量$h \text{>} 0$和向量函数:${\dot{{x}}}(t):[0,h] \to {{\mathbb{R}}^n}$,则有以下不等式成立:

      $ - h\int_{t - h}^0 {{{{{\dot x}}}^{\rm T}}} (s){{M}}{{\dot x}}(s){\rm d}s \text{≤} {\left[ {\begin{array}{*{20}{c}} {{{x}}(t)}\\ {{{x}}(t - h)} \end{array}} \right]^{\rm T}}\left[ {\begin{array}{*{20}{c}} { - {{M}}}&{{M}}\\ {{M}}&{ - {{M}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{x}}(t)}\\ {{{x}}(t - h)} \end{array}} \right]$

      引理2[6]。假定任意的正定矩阵${{M}} = {{{M}}^{\rm T}} \text{>} 0$,标量h>0和向量函数:${{x}}(t):[0,h] \to {{\mathbb{R}}^n}$,则有以下不等式成立:

      $\begin{aligned} &- h\int_{t - h}^t {{{{{{x}}}}^{\rm{T}}}(s)} {{{{Mx}}}}(s){\rm d}s \text{≤} - \int_{t - h}^t {{{{{{x}}}}^{\rm{T}}}(s)} {\rm d}s{{{{M}}}}\int_{t - h}^t {{{{{x}}}}(s)} {\rm d}s\\ & - \displaystyle\frac{{{h^2}}}{2}\int_{ - h}^0 {\int_{t + \beta }^t {{{{x}}^{\rm T}}(s)} } {{Mx}}(s){\rm d}s{\rm d}\beta \text{≤} - \int_{ - h}^0 {\int_{t + \beta }^t {{{{x}}^{\rm T}}(s)} } {\rm d}s{\rm d}\beta {{M}}\int_{ - h}^0 {\int_{t + \beta }^t {{{x}}(s)} } {\rm d}s{\rm d}\beta \\ &- \displaystyle\frac{{{h^3}}}{6}\int_{ - h}^0 {\int_\beta ^0 {\int_{t + \lambda }^t {{{{x}}^{\rm T}}(s)} {{Mx}}(s){\rm d}s{\rm d}\beta d\lambda } } \text{≤} - \int_{ - h}^0 {\int_\beta ^0 {\int_{t + \lambda }^t {{{{x}}^{\rm T}}(s)}{\rm d}s{\rm d}\beta {\rm d}\lambda } } {{M}}\int_{ - h}^0 {\int_\beta ^0 {\int_{t + \lambda }^t {{{x}}(s)} {\rm d}s{\rm d}\beta {\rm d}\lambda } } \end{aligned}$

      引理3[7]。假定任意的正定矩阵${{M}} = {{{M}}^{\rm T}}\text{>} 0$,标量$0 \text{≤} \alpha ,\varepsilon \text{≤} 1$${h_m} \text{≤} h(t) \text{≤} {h_M}$,向量函数:${{x}}(t):[0,h] \to {{\mathbb{R}}^n}$,则有以下不等式成立:

      $ \begin{aligned} &- ({h_M} - {h_m})\displaystyle\int_{t - {h_M}}^{t - {h_m}} {{{{x}}^{\rm T}}(s)} {{Mx}}(s){\rm d}s \text{≤} - {\zeta ^{\rm T}}(t)({{{e}}_7}{{Me}}_7^{\rm T} + {{{e}}_6}{{Me}}_6^{\rm T})\zeta (t) - \alpha {\zeta ^T}(t){{{e}}_7}{{Me}}_7^{\rm T}\zeta (t) - (1 - \alpha ){\zeta ^{\rm T}}(t){{{e}}_6}{{Me}}_6^{\rm T}\zeta (t)\\ &- \displaystyle\frac{{(h_M^2 - h_m^2)}}{2}\int_{ - {h_M}}^{ - {h_m}} {\int_{t + \beta }^t {{{{x}}^{\rm T}}(s){{Mx}}(s)} } {\rm d}s{\rm d}\beta \text{≤} - {\zeta ^T}(t)({{{e}}_{10}}{{Me}}_{10}^{\rm T} + {{{e}}_9}{{Me}}_9^T)\zeta (t) - \varepsilon {\zeta ^{\rm T}}(t){{{e}}_{10}}{{Me}}_{10}^{\rm T}\zeta (t) - (1 - \varepsilon ){\zeta ^{\rm T}}(t){{{e}}_9}{{Me}}_9^{\rm T}\zeta (t) \end{aligned}$

      式中:

      $ \begin{aligned} \zeta (t) =& \left[ {\begin{array}{*{20}{c}} {{{x}}(t)}&{{{x}}(t - h(t))}&{{{x}}(t - {h_m})}&{{{x}}(t - {h_M})}&{\displaystyle\int_{t - {h_m}}^t {{{x}}(s){\rm d}s} }&{\displaystyle\int_{t - h(t)}^{t - {h_m}} {{{x}}(s){\rm d}s} }&{\displaystyle\int_{t - {h_M}}^{t - h(t)} {{{x}}(s){\rm d}s} } \end{array}} \right.\\ &\!\!\!\!\!\!\!\!\left. {\begin{array}{*{20}{c}} {\displaystyle\int_{ - {h_m}}^0 {\displaystyle\int_{t + \beta }^t {{{x}}(s)} {\rm d}s} {\rm d}\beta }&{\displaystyle\int_{ - h(t)}^{ - {h_m}} {\displaystyle\int_{t + \beta }^t {{{x}}(s)} {\rm d}s} {\rm d}\beta }&{\displaystyle\int_{ - {h_M}}^{ - h(t)} {\displaystyle\int_{t + \beta }^t {{{x}}(s)} {\rm d}s} {\rm d}\beta } \end{array}} \right] \end{aligned} $

      引理4[21]。给定适当维数的矩阵${{Q}} = {{{Q}}^{\rm T}}$,${{H}}$${{E}}$,则有${{Q}} + {{HF}}(t){{E}} + {{{E}}^{\rm T}}{{F}}{(t)^{\rm T}}{{{H}}^{\rm T}} \text{<} 0$,对任意满足${{F}}{(t)^{\rm T}}{{F}}(t) \text{≤} {{I}}$${{F}}(t)$成立的充要条件是存在标量$\vartheta \text{>} 0$,使得:

      ${{Q}} + {\vartheta ^{ - 1}}{{H}}{{{H}}^{\rm T}} + \vartheta {{{E}}^{\rm T}}{{E}} \text{<} 0$

    • 针对系统(1)~(2),考虑如下一类具有区间变时滞的线性系统:

      $\left\{ \begin{array}{l} \dot {{x}}(t) = {{Ax}}(t) + {{{A}}_1}{{x}}(t - h(t)) + {{{B}}_u}u(t) + {{{B}}_\omega }{{\omega}} (t)\\ {{z}}(t) = {{Cx}}(t) + {{{C}}_d}{{x}}(t - h(t)){\rm{ + }}{{{D}}_u}{{u}}(t) + {{{D}}_\omega }{{\omega}} (t)\\ {{x}}(t) = {{\varphi}} (t)\quad\forall t \in [ - {h_M},0] \end{array} \right.$

      式中:${{x}}(t) \in {\mathbb{R}^n}$${{u}}(t) \in {\mathbb{R}^m}$${{\omega}} (t) \in {\mathbb{R}^p}$${{z}}(t) \in {\mathbb{R}^l}$分别为系统状态向量、控制输入向量、扰动输入向量以及受控输出向量,且${{\omega}} (t) \in {L_2}[0,\infty )$${{A}}$${{{A}}_1}$${{{B}}_u}$${{{B}}_\omega }$${{C}}$${{{C}}_d}$${{{D}}_u}$${{{D}}_\omega }$为适当维数的常数实矩阵。$h(t)$为时变连续的函数且满足:$0 \text{≤} {h_m} \text{≤} h(t) \text{≤} {h_M}$$\dot h(t) \text{≤} \mu $${h_m}$${h_M}$$\mu $为常数,${{\varphi}} (t)$$[ - {h_M},0]$上的连续可微初始函数。

      将非脆弱${H_\infty }$控制器${{u}}(t)$代入系统(4)中,可得闭环系统为:

      $\left\{ \begin{array}{l} \dot {{x}}(t) = {{{A}}_k}{{x}}(t) + {{{A}}_1}{{x}}(t - h(t)) + {{{B}}_\omega }{{\omega}} (t)\\ {{z}}(t) = {{{C}}_k}{{x}}(t) + {{{C}}_d}{{x}}(t - h(t)) + {{{D}}_\omega }{{\omega}} (t)\\ {{x}}(t) = {{\varphi}} (t)\quad\forall t \in [ - {h_2},0] \end{array} \right.$

      式中:${{{A}}_k} = {{A}} + {{{B}}_u}{{K}} + {{{B}}_u}\Delta {{K}}$${{{C}}_k} = {{C}} + {{{D}}_u}{{K}} + {{{D}}_u}\Delta {{K}}$

      定理1。对于给定的标量${h_m}$${h_M}$${\lambda _1}$${\lambda _2}$${\lambda _1} \text{>} {\lambda _2}$),且若存在正定对称矩阵${{{P}}_i}$$i = 1,2{\rm{,}}3{\rm{,}}4,5$)、${{{Q}}_1}$${{{Q}}_2}$${{{U}}_1}$${{{U}}_2}$${{{X}}_j}$${{{R}}_j}$($j = 1,2{\rm{,}}3{\rm{,}}4$),使得如下LMIs成立:

      ${{\varPhi}} = {\left( {{{{\varPhi}} _{i,j}}} \right)_{10 \times 10}} \text{<} 0$

      则系统(5)在非脆弱控制器(3)的作用下不仅渐近稳定,而且在零初始条件下具有给定的${H_\infty }$扰动抑制水平$\rho $

      式(6)中:

      $\begin{aligned} {{{\varPhi}} _{1,1}} =& {{{P}}_1}{{A}}{\rm{ + }}{{{A}}^{\rm T}}{{{P}}_1} + {{{Q}}_1} + h_m^2{{{X}}_1} + h_m^2{{{A}}^{\rm T}}{{{X}}_2}{{A}} - {{{X}}_2} + {({h_M} - {h_m})^2}{{{X}}_3} + {({h_M} - {h_m})^2}{{{A}}^{\rm T}}{{{X}}_4}{{A}} + \displaystyle\frac{{h_m^4}}{4}{{{R}}_1} + \displaystyle\frac{{h_m^4}}{4}{{{A}}^{\rm T}}{{{R}}_2}{{A}} - h_m^2{{{R}}_2}+\\ & \displaystyle\frac{{{{(h_M^2 - h_m^2)}^2}}}{4}{{{R}}_3} + \displaystyle\frac{{{{(h_M^2 - h_m^2)}^2}}}{4}{{{A}}^{\rm T}}{{{R}}_4}{{A}} - 3{({h_M} - {h_m})^2}{{{R}}_4} + \displaystyle\frac{{h_m^6}}{{36}}{{{A}}^{\rm T}}{{{U}}_1}{{A}} - \displaystyle\frac{{h_m^4}}{4}{{{U}}_1} + \displaystyle\frac{{{{(h_M^3 - h_m^3)}^2}}}{{36}}{{{A}}^{\rm T}}{{{U}}_2}{{A}} + \displaystyle\frac{{{{(h_M^2 - h_m^2)}^2}}}{4}{{{U}}_2},\\ {{{\varPhi }}_{1,2}} =& h_m^2{{{A}}^{\rm T}}{{{X}}_2}{{B}} + {({h_M} - {h_m})^2}{{{A}}^{\rm T}}{{{X}}_4}{{B}} + \displaystyle\frac{{h_m^4}}{4}{{{A}}^{\rm T}}{{{R}}_2}{{B}} + \displaystyle\frac{{{{(h_M^2 - h_m^2)}^2}}}{4}{{{A}}^{\rm T}}{{{R}}_4}{{B}} + \displaystyle\frac{{h_m^6}}{{36}}{{{A}}^{\rm T}}{{{U}}_1}{{B}} + \displaystyle\frac{{{{(h_M^3 - h_m^3)}^2}}}{{36}}{{{A}}^{\rm T}}{{{U}}_2}{{B}},\\ {{{\varPhi}} _{1,3}} =& {{{X}}_2},{{{\varPhi}} _{1,4}} = 0,{{{\varPhi}} _{1,5}} = 2{{{P}}_2} + {h_m}{{{R}}_2},{{{\varPhi}} _{1,6}} = (2 - \varepsilon )({h_M} - {h_m}){{{R}}_4},{{{\varPhi}} _{1,7}} = (1 + \varepsilon )({h_M} - {h_m}){{{R}}_4},\\ {{\varPhi} _{1,8}} = &2{h_m}{{{P}}_4} + \displaystyle\frac{{h_m^2}}{2}{{{U}}_1},{{{\varPhi}} _{1,9}} = {{{\varPhi}} _{1,10}} = 2({h_M} - {h_m}){{{P}}_5} + \displaystyle\frac{{(h_M^2 - h_m^2)}}{2}{{{U}}_2},\\ {{{\varPhi}} _{2,2}} =& h_m^2{{{B}}^{\rm T}}{{{X}}_2}{{B}} + {({h_M} - {h_m})^2}{{{B}}^{\rm T}}{{{X}}_4}{{B}} - {{{X}}_4} + \displaystyle\frac{{h_m^4}}{4}{{{B}}^{\rm T}}{{{R}}_2}{{B}} + \displaystyle\frac{{{{(h_M^2 - h_m^2)}^2}}}{4}{{{B}}^{\rm T}}{{{R}}_4}{{B}} + \displaystyle\frac{{h_m^6}}{{36}}{{{B}}^{\rm T}}{{{U}}_1}{{B}}+ \displaystyle\frac{{{{(h_M^3 - h_m^3)}^2}}}{{36}}{{{B}}^{\rm T}}{{{{{U}}}}_2}{{B}},\\ {{{\varPhi}} _{2,3}} =& - (\alpha - 2){{{X}}_4},{{{\varPhi}} _{2,4}} = (1 + \alpha ){{{X}}_4},{{{\varPhi}} _{2,5}} = {{{\varPhi}} _{2,6}} = {{{\varPhi}} _{2,7}} = {{{\varPhi}} _{2,8}} = {{{\varPhi}} _{2,9}} = {{{\varPhi}} _{2,10}} = 0,\\ {{{\varPhi}} _{3,3}} =& {{{Q}}_2} - {{{Q}}_1} - {{{{{X}}}}_2} + (\alpha - 2){{{X}}_4},{{{\varPhi}} _{3,4}} = 0,{{{\varPhi}} _{3,5}} = - 2{{{P}}_2},{{{\varPhi}} _{3,6}} = {{{\varPhi}} _{3,7}} = 2{{{P}}_3},{{{\varPhi}} _{3,8}} = {{{\varPhi}} _{3,9}} = {{{\varPhi}} _{3,10}} = 0,\\ {{{\varPhi}} _{4,4}} =& - {{{Q}}_2} - (1 + \alpha ){{{X}}_4},{ {{\varPhi}} _{4,5}} = 0,{ {{\varPhi}} _{4,6}} = { {{\varPhi}} _{4,7}} = - 2{ {{P}}_3},{ {{\varPhi}} _{4,8}} = { \varPhi _{4,9}} = { {{\varPhi}} _{4,10}} = 0,{ {{\varPhi}} _{5,5}} = - { {{X}}_1} - { {{R}}_2},\\ { {{\varPhi}} _{5,6}} =& { {{\varPhi}} _{5,7}} = 0,{ {{\varPhi}} _{5,8}} = - 2{ {{P}}_4},{ {{\varPhi}} _{5,9}} = { {{\varPhi}} _{5,10}} = 0,{ {{\varPhi}} _{6,6}} = (\alpha - 2){ {{X}}_3} - (2 - \varepsilon ){ {{R}}_4},{ {{\varPhi}} _{6,7}} = { {{\varPhi}} _{6,8}} = 0,\\ { {{\varPhi}} _{6,9}} =& { {{\varPhi}} _{6,10}} = - 2{ {{P}}_5},{ {{\varPhi}} _{7,7}} = - (\alpha + 1){ {{X}}_3} - (1 + \varepsilon ){ {{R}}_4},{ {{\varPhi}} _{7,8}} = 0,{ {{\varPhi}} _{7,9}} = { {{\varPhi}} _{7,10}} = - 2{ {{P}}_5},{ {{\varPhi}} _{8,8}} = - { {{R}}_1} - { {{U}}_1},\\ { {{\varPhi}} _{8,9}} =& { {{\varPhi}} _{8,10}} = 0,{ {{\varPhi}} _{9,9}} = - (2 - \varepsilon ){ {{R}}_3} - { {{U}}_2},{ {{\varPhi}} _{9,10}} = - { {{U}}_2},{ {{\varPhi}} _{10,10}} = (1 + \varepsilon ){ {{R}}_3} - { {{U}}_2},\alpha = \displaystyle\frac{{h(t) - {h_m}}}{{{h_M} - {h_m}}},\varepsilon = \displaystyle\frac{{h{{(t)}^2} - h_m^2}}{{h_M^2 - h_m^2}},\end{aligned}$

      证明:令${h_\Delta }{\rm{ = }}\chi {h_m} + (1 - \chi ){h_M}$,$(0 \text{<} \chi \text{<} 1)$,则有${h_m}\text{<} {h_\Delta } \text{<} {h_M}$。基于${h_\Delta }$把时滞区间分成不均匀的两部分,即$[{h_m},{h_\Delta }]$$[{h_\Delta },{h_M}]$,下面分两种情况讨论。

      情形1:当${h_\Delta } \text{≤} h(t) \text{≤} {h_M}$时,设计如下LKF:

      ${{V}}({{x}}(t)) = {{{V}}_1}({{x}}(t)) + {{{V}}_2}({{x}}(t)) + {{{V}}_3}({{x}}(t)) + {{{V}}_4}({{x}}(t)) + {{{V}}_5}({{x}}(t))$

      式中:

      $\begin{aligned} {{{V}}_1}({{x}}(t)) =& {{{x}}^{\rm T}}(t){{{P}}_1}{{x}}(t) + \int_{t - {h_\Delta }}^t {{{{x}}^{\rm T}}(s){\rm d}s} {{{P}}_2}\int_{t - {h_\Delta }}^t {{{x}}(s){\rm d}s} + \int_{t - {h_M}}^{t - {h_\Delta }} {{{{x}}^{\rm T}}(s)} {\rm d}s{{{P}}_3}\int_{t - {h_M}}^{t - {h_\Delta }} {{{x}}(s)} {\rm d}s +\\ & \int_{ - {h_\Delta }}^0 {\int_{t + \beta }^t {{{{x}}^{\rm T}}(s){\rm d}s{\rm d}\beta } } {{{P}}_4}\int_{ - {h_\Delta }}^0 {\int_{t + \beta }^t {{{x}}(s){\rm d}s{\rm d}\beta } } + \int_{ - {h_M}}^{ - {h_\Delta }} {\int_{t + \beta }^t {{{{x}}^{\rm T}}(s){\rm d}s{\rm d}\beta } } {{{P}}_5}\int_{ - {h_M}}^{ - {h_\Delta }} {\int_{t + \beta }^t {{{x}}(s){\rm d}s{\rm d}\beta } } \\ {{{V}}_2}({{x}}(t)) =& \int_{t - {h_\Delta }}^t {{{{x}}^{\rm T}}(s)} {{{Q}}_1}{{x}}(s){\rm d}s + \int_{t - {h_M}}^{t - {h_\Delta }} {{{{x}}^{\rm T}}(s)} {{{Q}}_2}x(s){\rm d}s \\ {{{V}}_3}({{x}}(t)) =& {h_\Delta }\int_{ - {h_\Delta }}^0 {\int_{t + \beta }^t {{{{x}}^{\rm T}}(s)} } {{{X}}_1}{{x}}(s){\rm d}s{\rm d}\beta + {h_\Delta }\int_{ - {h_\Delta }}^0 {\int_{t + \beta }^t {{{\dot {{x}}}^{\rm T}}(s)} } {{{X}}_2}\dot {{x}}(s){\rm d}s{\rm d}\beta +\\ & ({h_M} - {h_\Delta })\int_{ - {h_M}}^{ - {h_\Delta }} {\int_{t + \beta }^t {{{{x}}^{\rm T}}(s)} } {{{X}}_3}{{x}}(s){\rm d}s{\rm d}\beta + ({h_M} - {h_\Delta })\int_{ - {h_M}}^{ - {h_\Delta }} {\int_{t + \beta }^t {{{\dot {{x}}}^{\rm T}}(s)} } {{{X}}_4}\dot {{x}}(s){\rm d}s{\rm d}\beta \\ {{{V}}_4}({{x}}(t)) =& \frac{{h_\Delta ^2}}{2}\int_{ - {h_\Delta }}^0 {\int_\beta ^0 {\int_{t + \lambda }^t {{{{x}}^{\rm T}}(s)} } {{{R}}_1}{{x}}(s){\rm d}s{\rm d}\lambda } d\beta + \frac{{h_\Delta ^2}}{2}\int_{ - {h_\Delta }}^0 {\int_\beta ^0 {\int_{t + \lambda }^t {{{\dot {{x}}}^{\rm T}}(s)} } {{{R}}_2}\dot {{x}}(s){\rm d}s{\rm d}\lambda } {\rm d}\beta + \\ & \frac{{(h_M^2 - h_\Delta ^2)}}{2}\int_{ - {h_M}}^{ - {h_\Delta }} {\int_\beta ^0 {\int_{t + \lambda }^t {{{{x}}^{\rm T}}(s)} } {{{R}}_3}{{x}}(s){\rm d}s{\rm d}\lambda } {\rm d}\beta + \frac{{(h_M^2 - h_\Delta ^2)}}{2}\int_{ - {h_M}}^{ - {h_\Delta }} {\int_\beta ^0 {\int_{t + \lambda }^t {{{\dot {{x}}}^{\rm T}}(s)} } {{{R}}_4}\dot {{x}}(s){\rm d}s{\rm d}\lambda } {\rm d}\beta \\ {{{V}}_5}({{x}}(t)) =& \frac{{h_\Delta ^3}}{6}\int_{ - {h_\Delta }}^0 {\int_\beta ^0 {\int_\lambda ^0 {\int_{t + \varphi }^t {{{\dot {{x}}}^{\rm T}}(s){{{U}}_1}\dot {{x}}(s)} } {\rm d}s{\rm d}\varphi } } {\rm d}\lambda {\rm d}\beta + \frac{{(h_M^3 - h_\Delta ^3)}}{6}\int_{ - {h_M}}^{ - {h_\Delta }} {\int_\beta ^0 {\int_\lambda ^0 {\int_{t + \varphi }^t {{{\dot {{x}}}^T}(s){{{U}}_2}\dot {{x}}(s)} } {\rm d}s{\rm d}\varphi } }{\rm d}\lambda {\rm d}\beta \end{aligned}$

      计算L-K泛函${{V}}({{x}}(t))$沿系统(4)的导数,可得:

      $\dot {{V}}({{x}}(t)) = {\dot {{V}}_1}(t) + {\dot {{V}}_2}(t) + {\dot {{V}}_3}(t) + {\dot {{V}}_4}(t) + {\dot {{V}}_5}(t)$

      $\begin{aligned} {{\dot {{V}}}_1}(t)=& 2{{{x}}^{\rm T}}(t){{{A}}^{\rm T}}{{{P}}_1}{{x}}(t) + {{{x}}^{\rm T}}(t - h(t)){{{B}}^{\rm T}}{{{P}}_1}{{x}}(t) + 2{{{x}}^{\rm T}}(t){{{P}}_2}\int_{t - {h_\Delta }}^t {{{x}}(s)} {\rm d}s - 2{{{x}}^{\rm T}}(t - {h_\Delta }){{{P}}_2}\int_{t - {h_\Delta }}^t {{{x}}(s)} {\rm d}s +\\ & 2{{{x}}^{\rm T}}(t - {h_\Delta }){{{P}}_3}\int_{t - {h_M}}^{t - {h_\Delta }} {{{x}}(s)} {\rm d}s - 2{x^{\rm T}}(t - {h_M}){{{P}}_3}\int_{t - {h_M}}^{t - {h_\Delta }} {{{x}}(s)} {\rm d}s{\rm{ + }}2{h_\Delta }{{{x}}^{\rm T}}(t){{{P}}_4}\int_{ - {h_\Delta }}^0 {\int_{t{\rm{ + }}\beta }^t {{{x}}(s)} {\rm d}s} {\rm d}\beta -\\ & 2\int_{t - {h_\Delta }}^t {{{{x}}^{\rm T}}(s)} {\rm d}s{{{P}}_4}\int_{ \!-\! {h_\Delta }}^0 {\int_{t{\rm{ \!+\! }}\beta }^t {{{x}}(s)} } {\rm d}s{\rm d}\beta \!+\! 2({h_M} - {h_\Delta }){{{x}}^{\rm T}}(t){{{P}}_5}\int_{ - {h_M}}^{ - {h_\Delta }} {\int_{t{\rm{\! +\! }}\beta }^t {{{x}}(s)} {\rm d}s} {\rm d}\beta - 2\int_{t - {h_M}}^{t - {h_\Delta }} {{{{x}}^{\rm T}}(s)} {\rm d}s{{{P}}_5}\int_{ - {h_M}}^{ - {h_\Delta }} {\int_{t{\rm{ + }}\beta }^t {{{x}}(s)} {\rm d}s} {\rm d}\beta\\ {\dot {{V}}_2}(t) =& {{{x}}^{\rm T}}(t){{{Q}}_1}{{x}}(t) - {{{x}}^{\rm T}}(t - {h_\Delta }){{{Q}}_1}{{x}}(t - {h_\Delta }) + {{{x}}^{\rm T}}(t - {h_\Delta }){{{Q}}_2}{{x}}(t - {h_\Delta }) - {{{x}}^{\rm T}}(t - {h_M}){{{Q}}_2}{{x}}(t - {h_M}) \end{aligned}$

      $\begin{aligned} {{\dot {{V}}}_3}(t) =& h_\Delta ^2{{{x}}^{\rm T}}(t){{{X}}_1}{{x}}(t) - {h_\Delta }\int_{t - {h_\Delta }}^t {{{{x}}^{\rm T}}(s)} {{{X}}_1}{{x}}(s){\rm d}s + h_\Delta ^2{{\dot {{x}}}^{\rm T}}(t){{{X}}_2}\dot {{x}}(t) - {h_\Delta }\int_{t - {h_\Delta }}^t {{{\dot {{x}}}^{\rm T}}(s)} {{{X}}_2}\dot {{x}}(s){\rm d}s+\\ & {({h_M} - {h_\Delta })^2}{{{x}}^{\rm T}}(t){{{X}}_3}{{x}}(t) - ({h_M} - {h_\Delta })\int_{t - {h_M}}^{t - {h_\Delta }} {{{{x}}^{\rm T}}(s)} {{{X}}_3}{{x}}(s){\rm d}s + {({h_M} - {h_\Delta })^2}{{\dot {{x}}}^{\rm T}}(t){{{X}}_4}\dot {{x}}(t) - ({h_M} - {h_\Delta })\int_{t - {h_M}}^{t - {h_\Delta }} {{{\dot {{x}}}^{\rm T}}(s)} {{{X}}_4}\dot {{x}}(s){\rm d}s \\ {{\dot {{V}}}_4}(t) =& \frac{{h_\Delta ^4}}{4}{{{x}}^{\rm T}}(t){{{R}}_1}{{x}}(t) - \frac{{h_\Delta ^2}}{2}\int_{ - {h_\Delta }}^0 {\int_{t + \beta }^t {{{{x}}^{\rm T}}(s)} {{{R}}_1}{{x}}(s)} {\rm d}s{\rm d}\beta + \frac{{h_\Delta ^4}}{4}{{\dot {{x}}}^{\rm T}}(t){{{R}}_2}\dot {{x}}(t) - \frac{{h_\Delta ^2}}{2}\int_{ - {h_\Delta }}^0 {\int_{t + \beta }^t {{{\dot {{x}}}^T}(s){{{R}}_2}\dot {{x}}(s)} }{\rm d}s{\rm d}\beta + \\ & \frac{{{{(h_M^2 - h_\Delta ^2)}^2}}}{4}{{{x}}^{\rm T}}(t){{{R}}_3}{{x}}(t) - \frac{{(h_M^2 - h_\Delta ^2)}}{2}\int_{ - {h_M}}^{ - {h_\Delta }} {\int_{t + \beta }^t {{{{x}}^{\rm T}}(s){{{R}}_3}{{x}}(s)} } {\rm d}s{\rm d}\beta + \frac{{{{(h_M^2 - h_\Delta ^2)}^2}}}{4}{{\dot {{x}}}^{\rm T}}(t){{{R}}_4}\dot {{x}}(t) -\\ & \frac{{(h_M^2 - h_\Delta ^2)}}{2}\int_{ - {h_M}}^{ - {h_\Delta }} {\int_{t + \beta }^t {{{\dot {{x}}}^{\rm T}}(s){{{R}}_4}\dot {{x}}(s)} } {\rm d}s{\rm d}\beta \\ {{\dot {{V}}}_5}(t) =& \frac{{h_\Delta ^6}}{{36}}{{\dot {{x}}}^{\rm T}}(t){{{U}}_1}\dot {{x}}(t) - \frac{{h_\Delta ^3}}{6}\int_{ - {h_\Delta }}^0 {\int_\beta ^0 {\int_{t + \lambda }^t {{{\dot {{x}}}^{\rm T}}(s)} {{{U}}_1}\dot {{x}}(s){\rm d}s} {\rm d}\lambda } {\rm d}\beta + \frac{{{{(h_M^3 - h_\Delta ^3)}^2}}}{{36}}{{\dot {{x}}}^{\rm T}}(t){{{U}}_2}\dot {{x}}(t)- \\ &\frac{{(h_M^3 - h_\Delta ^3)}}{6}\int_{ - {h_M}}^{ - {h_\Delta }} {\int_\beta ^0 {\int_{t + \lambda }^t {{{\dot {{x}}}^{\rm T}}(s)} {{{U}}_2}\dot {{x}}(s){\rm d}s} {\rm d}\lambda } {\rm d}\beta \end{aligned}$

      由引理1与引理2可得:

      $ - {h_\Delta }\int_{t - {h_\Delta }}^t {{{{x}}^{\rm T}}(s)} {{{X}}_1}{{x}}(s){\rm d}s \text{≤} - {\zeta ^{\rm T}}(t){{{e}}_5}{{{X}}_1}{{e}}_5^{\rm T}\zeta (t)$

      $ - {h_\Delta }\int_{t - {h_\Delta }}^t {{{\dot {{x}}}^{\rm T}}(s)} {{{X}}_2}\dot {{x}}(s){\rm d}s \text{≤} - {\zeta ^{\rm T}}(t)({{{e}}_1} - {{{e}}_3}){{{X}}_2}({{e}}_1^{\rm T} - {{e}}_3^{\rm T})\zeta (t)$

      式中:

      $\begin{aligned} \zeta (t) =& \left[ {\begin{array}{*{20}{c}} {{{x}}(t)}&{{{x}}(t - h(t))}&{{{x}}(t - {h_\Delta })}&{{{x}}(t - {h_M})}&{\displaystyle\int_{t - {h_\Delta }}^t {{{x}}(s){\rm d}s} }&{\displaystyle\int_{t - h(t)}^{t - {h_\Delta }} {{{x}}(s){\rm d}s} }&{\displaystyle\int_{t - {h_M}}^{t - h(t)} {{{x}}(s){\rm d}s} } \end{array}} \right. \\ & \!\!\!\!\!\!\!\left. {\begin{array}{*{20}{c}} {\displaystyle\int_{ - {h_\Delta }}^0 {\displaystyle\int_{t + \beta }^t {{{x}}(s)} {\rm d}s} {\rm d}\beta }&{\displaystyle\int_{ - h(t)}^{ - {h_\Delta }} {\displaystyle\int_{t + \beta }^t {{{x}}(s)} {\rm d}s} {\rm d}\beta }&{\displaystyle\int_{ - {h_M}}^{ - h(t)} {\displaystyle\int_{t + \beta }^t {{{x}}(s)} {\rm d}s} {\rm d}\beta } \end{array}} \right] \\ \end{aligned} $

      由引理3可得:

      $ - ({h_M} - {h_\Delta })\int_{t - {h_M}}^{t - {h_\Delta }} {{{{x}}^{\rm T}}(s)} {{{X}}_3}{{x}}(s){\rm d}s \text{≤} - {\zeta ^{\rm T}}(t)({{{e}}_7}{{{X}}_3}{{e}}_7^{\rm T} + {{{e}}_6}{{{X}}_3}{{e}}_6^{\rm T})\zeta (t) - \alpha {\zeta ^{\rm T}}(t){{{e}}_7}{{{X}}_3}{{e}}_7^{\rm T}\zeta (t) - (1 - \alpha ){\zeta ^{\rm T}}(t){{{e}}_6}{{{X}}_3}{{e}}_6^{\rm T}\zeta (t)$

      同样可以得到:

      $\begin{aligned} - ({h_M} - &{h_\Delta })\displaystyle\int_{t - {h_M}}^{t - {h_\Delta }} {{{\dot x}^{\rm T}}(s)} {{{X}}_4}\dot {{x}}(s){\rm d}s \text{≤} - {\zeta ^{\rm T}}(t)({{{e}}_2} - {{{e}}_4}){{{X}}_4}({{e}}_2^{\rm T} - {{e}}_4^{\rm T})\zeta (t) - {\zeta ^{\rm T}}(t)({{{e}}_3} - {{{e}}_2}){{{X}}_4}({{e}}_3^{\rm T} - {{e}}_2^{\rm T})\zeta (t) -\\ &\quad \alpha {\zeta ^{\rm T}}(t)({{{e}}_2} - {{{e}}_4}){{{X}}_4}({{e}}_2^{\rm T} - {{e}}_4^{\rm T})\zeta (t) - (1 - \alpha ){\zeta ^{\rm T}}(t)({{{e}}_3} - {{{e}}_2}){{{X}}_4}({{e}}_3^T - {{e}}_2^{\rm T})\zeta (t) \\ \end{aligned} $

      $ - \frac{{h_\Delta ^2}}{2}\int_{ - {h_\Delta }}^0 {\int_{t + \beta }^t {{{{x}}^{\rm T}}(s)} {{{R}}_1}{{x}}(s)} {\rm d}s{\rm d}\beta \text{≤} - {\zeta ^{\rm T}}(t){{{e}}_8}{{{R}}_1}{{e}}_8^{\rm T}\zeta (t)$

      $ - \frac{{h_\Delta ^2}}{2}\int_{ - {h_\Delta }}^0 {\int_{t + \beta }^t {{{\dot {{x}}}^{\rm T}}(s){{{R}}_2}\dot {{x}}(s)} } {\rm d}s{\rm d}\beta \text{≤} - {\zeta ^{\rm T}}(t)({h_\Delta }{{{e}}_1} - {{{e}}_5}){{{R}}_3}({h_\Delta }e_1^{\rm T} - {{e}}_5^{\rm T})\zeta (t)$

      $\begin{aligned} - \displaystyle\frac{{(h_M^2 - h_\Delta ^2)}}{2}\displaystyle\int_{ - {h_M}}^{ - {h_\Delta }} {\int_{t + \beta }^t {{{{x}}^{\rm T}}(s){{{R}}_3}{{x}}(s)} } {\rm d}s{\rm d}\beta \text{≤} & - {\zeta ^{\rm T}}(t)({{{e}}_{10}}{{{R}}_3}e_{10}^{\rm T} + {{{e}}_9}{{{R}}_3}{{e}}_9^{\rm T})\zeta (t) - \varepsilon {\zeta ^{\rm T}}(t){{{e}}_{10}}{{{R}}_3}{{e}}_{10}^{\rm T}\zeta (t) - \\&\quad(1 - \varepsilon ){\zeta ^{\rm T}}(t){{{e}}_9}{{{R}}_3}{{e}}_9^{\rm T}\zeta (t) \end{aligned} $

      $\begin{aligned} & - \displaystyle\frac{{(h_M^2 - h_\Delta ^2)}}{2}\int_{ - {h_M}}^{ - {h_\Delta }} {\int_{t + \beta }^t {{{\dot { x}}^{\rm T}}(s){{{R}}_4}\dot { x}(s)} } {\rm d}s{\rm d}\beta \text{≤} \\ &\quad - {\zeta ^{\rm T}}(t)\left( {({h_M} - {h_\Delta }){{{e}}_1} - {{{e}}_7}} \right){{{R}}_4}\left( {({h_M} - {h_\Delta }){{e}}_1^{\rm T} - {{e}}_7^{\rm T}} \right)\zeta (t) - \varepsilon {\zeta ^{\rm T}}(t)\left( {({h_M} - {h_\Delta }){{{e}}_1} - {{{e}}_7}} \right){{{R}}_4}\left( {({h_M} - {h_\Delta }){{e}}_1^{\rm T} - {{e}}_7^{\rm T}} \right)\zeta (t) - \\ &\quad {\zeta ^{\rm T}}(t)\left( {({h_M} - {h_\Delta }){{{e}}_1} - {{{e}}_6}} \right){{{R}}_4}\left( {({h_M} - {h_\Delta }){{e}}_1^{\rm T} - {{e}}_6^{\rm T}} \right)\zeta (t) - (1 - \varepsilon ){\zeta ^{\rm T}}(t)\left( {({h_M} - {h_\Delta }){{{e}}_1} - {{{e}}_6}} \right){{{R}}_4}\left( {({h_M} - {h_\Delta }){{e}}_1^{\rm T} - {{e}}_6^{\rm T}} \right)\zeta (t) \end{aligned} $

      $ - \frac{{h_\Delta ^3}}{6}\int_{ - {h_\Delta }}^0 {\int_\beta ^0 {\int_{t + \lambda }^t {{{\dot {{x}}}^{\rm T}}(s)} {{{U}}_1}\dot {{x}}(s){\rm d}s} {\rm d}\lambda } {\rm d}\beta \text{≤} - {\zeta ^{\rm T}}(t)\left(\frac{{h_\Delta ^2}}{2}{{{e}}_1} - {e_8}\right){{{U}}_1}\left(\frac{{h_\Delta ^2}}{2}{{e}}_1^{\rm T} - {{e}}_8^{\rm T}\right)\zeta (t)$

      $ - \frac{{(h_M^3 - h_\Delta ^3)}}{6}\int_{ - {h_M}}^{ - {h_\Delta }} {\int_\beta ^0 {\int_{t + \lambda }^t {{{\dot {{x}}}^{\rm T}}(s)} {{{U}}_2}\dot {{x}}(s){\rm d}s} {\rm d}\lambda } {\rm d}\beta \text{≤} - {\zeta ^{\rm T}}(t)\left(\frac{{(h_M^2 - h_\Delta ^2)}}{2}{{{e}}_1} - {{{e}}_9} - {{{e}}_{10}}\right){{{U}}_2}\left(\frac{{(h_M^2 - h_\Delta ^2)}}{2}{{e}}_1^{\rm T} - {{e}}_9^{\rm T} - {{e}}_{10}^{\rm T}\right)\zeta (t)$

      把式(9)~(18)代入式(8),则$\dot {{V}}({{x}}(t))$可表示为:

      $\dot {{V}}({{x}}(t)) \text{≤}{\zeta ^{\rm T}}(t)\left[ {\alpha {{{\varGamma}} _1} + (1 - \alpha ){{{\varGamma}} _2} + \varepsilon {{{\varGamma}} _3} + (1 - \varepsilon ){{{\varGamma}} _4}} \right]\zeta (t)$

      式中:

      $ \begin{aligned}{{{\varGamma}} _1}{\rm{ = }}& - {{{e}}_7}{{{X}}_3}{{e}}_7^{\rm T} - {\rm{(}}{{{e}}_2} - {{{e}}_4}{\rm{)}}{{{X}}_4}{\rm{(}}{{e}}_2^{\rm T} - {{e}}_4^{\rm T}{\rm{),}}{{{\Gamma}} _2}{\rm{ = }} - {{{e}}_6}{{{X}}_3}{{e}}_6^{\rm T} - {\rm{(}}{{{e}}_3} - {{{e}}_2}{\rm{)}}{{{X}}_4}{\rm{(}}{{e}}_3^{\rm T} - {{e}}_2^{\rm T}{\rm{),}}\\ {{{\varGamma}} _3}{\rm{ = }}& - {{{e}}_{10}}{{{R}}_3}{{e}}_{10}^{\rm T} - {\rm{((}}{h_M} - {h_\Delta }{\rm{)}}{{{e}}_1} - {{{e}}_7}{\rm{)}}{{{R}}_4}{\rm{((}}{h_M} - {h_\Delta }{\rm{)}}{{e}}_1^{\rm T} - {{e}}_7^{\rm T}{\rm{),}}{{{\Gamma}} _4}{\rm{ = }} - {{{e}}_9}{{{R}}_3}{{e}}_9^{\rm T} - {\rm{((}}{h_M} - {h_\Delta }{\rm{)}}{{{e}}_1} - {{{e}}_6}{\rm{)}}{{{R}}_4}{\rm{((}}{h_M} - {h_\Delta }{\rm{)}}{{e}}_1^{\rm T} - {{e}}_6^{\rm T}{\rm{)}}{\text{。}} \end{aligned}$

      因为$0 \text{≤} \alpha ,\varepsilon \text{≤} 1$,根据交互式凸组合技术,如下不等式成立:

      $\alpha ({{{\varGamma}} _1} + {\lambda _1}{{I}}){\rm{ + (1}} - \alpha {\rm{)(}}{{{\varGamma}} _2} + {\lambda _1}{{I}}{\rm{)}} \text{<}0$

      $\varepsilon ({{{\varGamma}} _3} - {\lambda _2}{{I}}){\rm{ + (1}} - \varepsilon {\rm{)(}}{{{\varGamma}} _4} - {\lambda _2}{{I}}{\rm{)}} \text{<} 0$

      $\alpha {{{\varGamma}} _1}{\rm{ + (1}} - \alpha {\rm{)}}{{{\varGamma}} _2} \text{<} - {\lambda _1}{{I}}$

      $\varepsilon {{{\varGamma}} _3}{\rm{ + (1}} - \varepsilon {\rm{)}}{{{\varGamma}} _4} \text{<}{\lambda _2}{{I}}$

      由于${\lambda _1} \text{>} {\lambda _2}$,合并式(22)~(23),可得

      $\alpha {{{\varGamma}} _1}{\rm{ + (1}} - \alpha {\rm{)}}{{{\varGamma}} _2} + \varepsilon {{{\varGamma}} _3}{\rm{ + (1}} - \varepsilon {\rm{)}}{{{\varGamma}} _4} \text{<}({\lambda _2} - {\lambda _1}){{I}} \text{<} 0$

      根据L-K稳定性定理,如果$\alpha {{{\varGamma}} _1}{\rm{ + (1}} - \alpha {\rm{)}}{{{\varGamma}} _2} + \varepsilon {{{\varGamma}} _3}{\rm{ + (1}} - \varepsilon {\rm{)}}{{{\varGamma}} _4} \text{<} 0$,则存在充分小正数${\delta _0}$使得$\dot {{V}}({{x}}(t)) \text{<} - {\delta _0}{\left\| {{{x}}(t)} \right\|^2}\,$成立,进而可知系统(5)渐近稳定。

      对于给定的${{\kappa}} $,考虑性能指标$\Lambda (\omega )$,则把${{z}}{(t)^{\rm T}}{{z}}(t) - {\rho ^2}{{{\omega}} ^{\rm T}}(t){{\omega}} (t)$加到不等式(19)两边,可得:

      $\dot {{V}}({{x}}(t)) + {{z}}{(t)^{\rm T}}{{z}}(t) - {\rho ^2}{{{\omega}} ^{\rm T}}(t){{\omega}} (t) \text{≤} {\zeta ^{\rm T}}(t)({{\varOmega}} + {{{\varPsi}} ^{\rm T}}{{\varPsi}} + \alpha {{{\varGamma}} _1} + (1 - \alpha ){{{\varGamma}} _2} + \varepsilon {{{\Gamma}} _3} + (1 - \varepsilon ){{\varGamma _4}})\zeta (t)$

      式中:${{\Omega}} = {\rm{diag}}\left\{ {{array}{*{20}{c}}0&0&0&0&0&0&0&{ - {\rho ^2}{{I}}}{array}} \right\},$${{\varPsi}} = \left[ {{array}{*{20}{c}}{{{{C}}_k}} &0 &0 &0 &0 &0 &0 &{{{{D}}_\omega }}{array}} \right]$

      如果

      ${{\varOmega}} + {{{\varPsi}} ^{\rm T}}{{\varPsi}} + \alpha {{{\varGamma}} _1} + (1 - \alpha ){{{\varGamma}} _2} + \varepsilon {{{\varGamma}} _3} + (1 - \varepsilon ){{{\varGamma}} _4} \text{<} 0$

      那么

      $ \dot {{V}}({{x}}(t)) + {{z}}{(t)^{\rm T}}z(t) - {\rho ^2}{{{\omega}} ^{\rm T}}(t){{\omega}} (t) \text{≤} 0 $

      ${{\omega}} (t) = 0$时,$\dot {{V}}({{x}}(t)) \text{<} 0$,同样可得系统(5)是渐近稳定的;当${{\omega}} (t) \ne 0$时,式(27)两边对$t$从0到∞积分,并注意到在零初始条件下,有${{V}}({{x}}(t)){|_{t = 0}}$,得到

      $\int_0^\infty {\left[ {{{{z}}^{\rm T}}(t){{z}}(t) - {\rho ^2}{{{\omega}} ^{\rm T}}(t)\omega (t)} \right]{\rm d}t} \text{<} - {{V}}({{x}}(t)){|_{t = \infty }} + {{V}}({{x}}(t)){|_{t = 0}} \text{<} 0$

      $||{{z}}(t)|{|_2} \text{<} \rho ||{{\omega}} (t)|{|_2}$,从而闭环系统在零初始条件下具有给定的${H_\infty }$扰动抑制水平$\rho $

      情形2:当${h_m} \text{≤} h(t) \text{≤} {h_\Delta }$时,设计如下L-K泛函:

      ${{{V}}_0}({{x}}(t)) = {{{V}}_{01}}({{x}}(t)) + {{{V}}_{02}}({{x}}(t)) + {{{V}}_{03}}({{x}}(t)) + {{{V}}_{04}}({{x}}(t)) + {{{V}}_{05}}({{x}}(t))$

      式中:

      $ \begin{aligned} {{{V}}_{01}}({{x}}(t)) =& {{{x}}^{\rm T}}(t){{{P}}_1}{{x}}(t) + \int_{t - {h_m}}^t {{{{x}}^{\rm T}}(s){\rm d}s} {{{P}}_2}\int_{t - {h_m}}^t {{{x}}(s){\rm d}s} + \int_{t - {h_\Delta }}^{t - {h_m}} {{{{x}}^{\rm T}}(s)} {\rm d}s{{{P}}_3}\int_{t - {h_\Delta }}^{t - {h_m}} {{{x}}(s)} {\rm d}s +\\ & \int_{ - {h_m}}^0 {\int_{t + \beta }^t {{{{x}}^{\rm T}}(s){\rm d}s{\rm d}\beta } } {{{P}}_4}\int_{ - {h_m}}^0 {\int_{t + \beta }^t {{{x}}(s){\rm d}s{\rm d}\beta } } + \int_{ - {h_\Delta }}^{ - {h_m}} {\int_{t + \beta }^t {{{{x}}^{\rm T}}(s){\rm d}s{\rm d}\beta } } {{{P}}_5}\int_{ - {h_\Delta }}^{ - {h_m}} {\int_{t + \beta }^t {{{x}}(s){\rm d}s{\rm d}\beta } } \\ {{{V}}_{02}}({{x}}(t)) =& \int_{t - {h_{\rm{m}}}}^t {{{{x}}^{\rm T}}(s)} {{{Q}}_1}{{x}}(s){\rm d}s + \int_{t - {h_\Delta }}^{t - {h_{\rm{m}}}} {{{{x}}^{\rm T}}(s)} {{{Q}}_2}{{x}}(s){\rm d}s\\ {{{V}}_{03}}({{x}}(t)) =& {h_{\rm{m}}}\int_{ - {h_{\rm{m}}}}^0 {\int_{t + \beta }^t {{{{x}}^{\rm T}}(s)} } {{{X}}_1}{{x}}(s){\rm d}s{\rm d}\beta + {h_{\rm{m}}}\int_{ - {h_{\rm{m}}}}^0 {\int_{t + \beta }^t {{{\dot {{x}}}^{\rm T}}(s)} } {{{X}}_2}\dot {{x}}(s){\rm d}s{\rm d}\beta +\\ & ({h_\Delta } - {h_{\rm{m}}})\int_{ - {h_\Delta }}^{ - {h_{\rm{m}}}} {\int_{t + \beta }^t {{{{x}}^{\rm T}}(s)} } {{{X}}_3}{{x}}(s){\rm d}s{\rm d}\beta + ({h_\Delta } - {h_{\rm{m}}})\int_{ - {h_\Delta }}^{ - {h_{\rm{m}}}} {\int_{t + \beta }^t {{{\dot {{x}}}^{\rm T}}(s)} } {{{X}}_4}\dot x(s){\rm d}s{\rm d}\beta \\ {{{V}}_{04}}(x(t)) =& \frac{{h_{\rm{m}}^2}}{2}\int_{ - {h_{\rm{m}}}}^0 {\int_\beta ^0 {\int_{t + \lambda }^t {{{{x}}^{\rm T}}(s)} } {{{R}}_1}{{x}}(s){\rm d}s{\rm d}\lambda } {\rm d}\beta + \frac{{h_{\rm{m}}^2}}{2}\int_{ - {h_{\rm{m}}}}^0 {\int_\beta ^0 {\int_{t + \lambda }^t {{{\dot {{x}}}^{\rm T}}(s)} } {{{R}}_2}\dot {{x}}(s){\rm d}s{\rm d}\lambda } {\rm d}\beta \\ &+ \frac{{(h_\Delta ^2 - h_{\rm{m}}^2)}}{2}\int_{ - {h_\Delta }}^{ - {h_{\rm{m}}}} {\int_\beta ^0 {\int_{t + \lambda }^t {{{{x}}^{\rm T}}(s)} } {{{R}}_3}{{x}}(s){\rm d}s{\rm d}\lambda } {\rm d}\beta + \frac{{(h_\Delta ^2 - h_{\rm{m}}^2)}}{2}\int_{ - {h_\Delta }}^{ - {h_{\rm{m}}}} {\int_\beta ^0 {\int_{t + \lambda }^t {{{\dot {{x}}}^{\rm T}}(s)} } {{{R}}_4}\dot {{x}}(s){\rm d}s{\rm d}\lambda } {\rm d}\beta \end{aligned} $

      $ \begin{aligned} {{{V}}_{05}}({{x}}(t)) = \frac{{h_{\rm{m}}^3}}{6}\int_{ - {h_\Delta }}^0 {\int_\beta ^0 {\int_\lambda ^0 {\int_{t + \varphi }^t {{{\dot {{x}}}^{\rm T}}(s){{{U}}_1}\dot {{x}}(s)} } {\rm d}s{\rm d}\varphi } } {\rm d}\lambda {\rm d}\beta + \frac{{(h_\Delta ^3 - h_{\rm{m}}^3)}}{6}\int_{ - {h_\Delta }}^{ - {h_{\rm{m}}}} {\int_\beta ^0 {\int_\lambda ^0 {\int_{t + \varphi }^t {{{\dot {{x}}}^{\rm T}}(s){{{U}}_2}\dot {{x}}(s)} } {\rm d}s{\rm d}\varphi } } {\rm d}\lambda {\rm d}\beta \end{aligned} $

      式中:

      $\begin{split} {\zeta _0}(t) = &\left[ {\begin{array}{*{20}{c}} {{{x}}(t)}&{{{x}}(t - h(t))}&{{{x}}(t - {h_{\rm{m}}})}&{{{x}}(t - {h_\Delta })}&{\displaystyle\int_{t - {h_{\rm{m}}}}^t {{{x}}(s){\rm d}s} }&{\displaystyle\int_{t - h(t)}^{t - {h_{\rm{m}}}} {{{x}}(s){\rm d}s} }&{\displaystyle\int_{t - {h_\Delta }}^{t - h(t)} {{{x}}(s){\rm d}s} } \end{array}} \right.\\ &\!\!\!\!\!\!\!\left. {\begin{array}{*{20}{c}} {\displaystyle\int_{ - {h_{\rm{m}}}}^0 {\displaystyle\int_{t + \beta }^t {{{x}}(s)} {\rm d}s} {\rm d}\beta }&{\displaystyle\int_{ - h(t)}^{ - {h_{\rm{m}}}} {\displaystyle\int_{t + \beta }^t {{{x}}(s)} {\rm d}s} {\rm d}\beta }&{\displaystyle\int_{ - {h_\Delta }}^{ - h(t)} {\displaystyle\int_{t + \beta }^t {{{x}}(s)} {\rm d}s} {\rm d}\beta } \end{array}} \right], \end{split}$

      ${{{P}}_i}\left( {i = 1,2{\rm{,}}3{\rm{,}}4,5} \right),{{{Q}}_1},{{{Q}}_2},{{{U}}_1},{{{U}}_2},$${{{X}}_j},{{{R}}_j}\left( {j = 1,2{\rm{,}}3{\rm{,}}4} \right)$,同式(6)中所定义的矩阵。利用同样的方法,可得

      ${\dot {{V}}_0}({{x}}(t)) \text{≤} \zeta _0^{\rm T}(t)\left[ {\alpha {{{\varGamma}} _{01}} + (1 - \alpha ){{{\varGamma}} _{02}} + \varepsilon {{{\varGamma}} _{03}} + (1 - \varepsilon ){{{\varGamma}} _{04}}} \right]{\zeta _0}(t)$

      式中:

      $\begin{split} &{{{\varGamma}} _{01}}{\rm{ = }}{{{\varGamma}} _1}{\rm{ = }} - {{{e}}_7}{{{X}}_3}{{e}}_7^{\rm T} - {\rm{(}}{{{e}}_2} - {{{e}}_4}{\rm{)}}{{{X}}_4}{\rm{(}}{{e}}_2^{\rm T} - e_4^{\rm T}{\rm{),}}\;{{{\varGamma}} _{02}}{\rm{ = }}{{{\varGamma}} _2}{\rm{ = }} - {{{e}}_6}{{{X}}_3}{{e}}_6^{\rm T} - {\rm{(}}{{{e}}_3} - {{{e}}_2}{\rm{)}}{{{X}}_4}{\rm{(}}{{e}}_3^{\rm T} - {{e}}_2^{\rm T}{\rm{),}}\\ &{{{\varGamma}} _{03}}{\rm{ = }} - {{{e}}_{10}}{{{R}}_3}{{e}}_{10}^{\rm T} - {\rm{((}}{h_\Delta } - {h_{\rm{m}}}{\rm{)}}{{{e}}_1} - {{{e}}_7}{\rm{)}}{{{R}}_4}{\rm{((}}{h_\Delta } - {h_{\rm{m}}}{\rm{)}}{{e}}_1^{\rm T} - {{e}}_7^{\rm T}{\rm{),}}\;{{{\varGamma}} _{04}}{\rm{ = }} - {{{e}}_9}{{{R}}_3}e_9^{\rm T} - {\rm{((}}{h_\Delta } - {h_{\rm{m}}}{\rm{)}}{{{e}}_1} - {{{e}}_6}{\rm{)}}{{{R}}_4}{\rm{((}}{h_\Delta } - {h_{\rm{m}}}{\rm{)}}{{e}}_1^{\rm T} - {{e}}_6^{\rm T}{\rm{)}}\text{。} \end{split}$

      根据Lyapunov稳定性定理,如果$\alpha {{{\varGamma}} _{01}} + (1 - \alpha ){{{\varGamma}} _{02}} + \varepsilon {{{\varGamma}} _{03}} + (1 - \varepsilon ){{{\varGamma}} _{04}} \text{<} 0$,则存在充分小正数${\delta _{01}}$使得${\dot {{V}}_0}({{x}}(t)) \text{<} - {\delta _{01}}{\left\| {x(t)} \right\|^2}\,$成立,进而保证系统(5)渐近稳定。用同样的处理方法,可以得到

      ${\dot {{V}}_0}({{x}}(t)) + {{z}}{(t)^{\rm T}}{{z}}(t) - {\rho ^2}{{{\omega}} ^{\rm T}}(t)\omega (t) \text{≤} \zeta _0^T(t)({{\varOmega}} + {{{{\varPsi}}} ^{\rm T}}{{{\varPsi}}} + \alpha {{{\varGamma}} _{01}} + (1 - \alpha ){{{\varGamma}} _{02}} + \varepsilon {{{\varGamma}} _{03}} + (1 - \varepsilon ){{{\varGamma}} _{04}}){\zeta _0}(t)$

      如果

      ${{\varOmega}} + {{{{\varPsi}}} ^{\rm T}}{{{\varPsi}}} + \alpha {{{\varGamma}} _{01}} + (1 - \alpha ){{{\varGamma}} _{02}} + \varepsilon {{{\varGamma}} _{03}} + (1 - \varepsilon ){{{\varGamma}} _{04}} \text{<} 0$

      那么

      ${\dot {{V}}_0}({{x}}(t)) + {{z}}{(t)^{\rm T}}{{z}}(t) - {\rho ^2}{{{\omega}} ^{\rm T}}(t)\omega (t) \text{≤} 0$

      从而闭环系统在零初始条件下具有给定的${H_\infty }$扰动抑制水平$\rho $

      对式(19)或式(30)应用引理3,则其等价于式(6)。证毕。

    • 针对式(1)~(2)描述的系统,考虑如下一类不确定时滞系统:

      $\left\{ \begin{array}{l} \dot {{x}}(t) = ({{A}} + \Delta {{A}}(t)){{x}}(t) + ({{{A}}_1} + \Delta {{{A}}_1}(t)){{x}}(t - h) + {{{B}}_u}{{u}}(t) + {{{B}}_\omega }{{\omega}} (t)\\ {{z}}(t) = {{Cx}}(t) + {{{C}}_d}{{x}}(t - h(t)) + {{{D}}_u}{{u}}(t) + {{{D}}_\omega }{{\omega}} (t)\\ {{x}}(t) = {{\varphi}} (t)\quad\forall t \in [ - {h_M},0] \end{array} \right.$

      式中:$\Delta {{A}}(t)$$\Delta {{{A}}_1}(t)$为具有结构不确定性的未知矩阵且满足:$[{array}{*{20}{c}} {\Delta {{A}}(t)}&{\Delta {{{A}}_1}(t)} {array}] = {{HF}}(t)[{array}{*{20}{c}} {{{{E}}_0}}&{{{{E}}_1}} {array}]$$H$${{{E}}_0}$${{{E}}_1}$为适当维数的常数矩阵,${{F}}(t)$定义同${{{F}}_{\rm c}}(t)$一样,也是未知时变矩阵。其他参数与系统(4)中定义一致。

      针对系统(34),本节在上一节有界实判据的基础上,给出非脆弱${H_\infty }$控制器的设计方法。

      定理2对于给定的标量$0 \text{<} {h_m} \text{<} {h_M}$$\mu $${\lambda _1}$${\lambda _2}$${\lambda _1} \text{>} {\lambda _2}$)、$\vartheta \text{>} 0$且若存在正定对称矩阵${\tilde {{P}}_i}\left( {i = 1,2{\rm{,}}3{\rm{,}}4,5} \right)$${\tilde {{Q}}_1}$${\tilde {{Q}}_2}$${\tilde {{Q}}_3}$${\tilde {{U}}_1}$${\tilde {{U}}_2}$${\tilde {{X}}_j}5{\tilde {{R}}_j}\left( {j = 1,2{\rm{,}}3{\rm{,}}4} \right)$,适当维数的自由矩阵${{\Xi}} $${{Y}}$,使得如LMIs成立:

      $\left[ {\begin{array}{*{20}{c}} {\tilde {\varPhi }}&{\tilde {{\varGamma}} _a^{\rm T}}&{\tilde {{\varGamma}} _E^{\rm T}} \\ *&{ - \vartheta {{I}}}&0 \\ *&*&{ - \vartheta {{I}}} \end{array}} \right] \text{<} 0$

      则不确定系统(34)在非脆弱控制器(3)的作用下不仅渐近稳定,而且在零初始条件下具有给定的${H_\infty }$扰动抑制水平$\rho $,且控制器增益${{K}} = {{Y}}{\Xi ^{ - {\rm T}}}$

      式中:

      $ \begin{aligned} {\tilde {{\varPhi}}} = &{\left( {{{\tilde {\varPhi} }_{i,j}}} \right)_{10 \times 10}}\\[-2pt] {{\tilde {{\varPhi}} }_{1,1}} =& {{\tilde {{P}}}_1}{{A}}{\rm{ + }}{{{A}}^{\rm T}}{{\tilde {{P}}}_1} + {{\tilde {{Q}}}_1} + h_m^2{{\tilde {{X}}}_1} + h_m^2{{{A}}^{\rm T}}{{\tilde {{X}}}_2}{{A}} - {{\tilde {{X}}}_2} + {({h_M} - {h_m})^2}{{\tilde {{X}}}_3} + {({h_M} - {h_m})^2}{{{A}}^{\rm T}}{{\tilde {{X}}}_4}{{A}} + \frac{{h_m^4}}{4}{{\tilde {{R}}}_1} + \frac{{h_m^4}}{4}{{{A}}^{\rm T}}{{\tilde {{R}}}_2}{{A}} - h_m^2{{\tilde {{R}}}_2}+\\[-2pt] & \frac{{{{(h_M^2 \!-\! h_m^2)}^2}}}{4}{{\tilde {{R}}}_3} \!+\! {{\tilde {{Q}}}_3} \!+\! \frac{{{{(h_M^2 - h_m^2)}^2}}}{4}{{{A}}^{\rm T}}{{\tilde {{R}}}_4}{{A}} \!-\! 3{({h_M} \!- \!{h_m})^2}{{\tilde {{R}}}_4} \!+\! \frac{{h_m^6}}{{36}}{{{A}}^{\rm T}}{{\tilde U}_1}{{A}}\! -\! \frac{{h_m^4}}{4}{{\tilde U}_1} \!+\! \frac{{{{(h_M^3 \!- \!h_m^3)}^2}}}{{36}}{{{A}}^{\rm T}}{{\tilde U}_2}{{A}} \!+\! \frac{{{{(h_M^2 \!-\! h_m^2)}^2}}}{4}{{\tilde U}_2}, \end{aligned} $

      $ \begin{aligned} {\tilde {{\varPhi}} _{1,2}} =& h_m^2{{{A}}^{\rm T}}{\tilde {{X}}_2}{{B}} + {({h_M} - {h_m})^2}{{{A}}^{\rm T}}{\tilde {{X}}_4}{{B}} + \frac{{h_m^4}}{4}{{{A}}^{\rm T}}{\tilde {{R}}_2}{{B}} + \frac{{{{(h_M^2 - h_m^2)}^2}}}{4}{{{A}}^{\rm T}}{\tilde {{R}}_4}{{B}} + \frac{{h_m^6}}{{36}}{{{A}}^{\rm T}}{\tilde {{U}}_1}{{B}} + \frac{{{{(h_M^3 - h_m^3)}^2}}}{{36}}{{{A}}^{\rm T}}{\tilde {{U}}_2}{{B}},\\ {{\tilde {{\varPhi}} }_{1,3}} =& {{\tilde X}_2},{{\tilde {{\varPhi}} }_{1,4}} = 0,{{\tilde {{\varPhi}} }_{1,5}} = 2{{\tilde {{P}}}_2} + {h_m}{{\tilde {{R}}}_2},{{\tilde {{\varPhi}} }_{1,6}} = (2 - \varepsilon )({h_M} - {h_m}){{\tilde {{R}}}_4},{{\tilde {{\varPhi}} }_{1,7}} = (1 + \varepsilon )({h_M} - {h_m}){{\tilde {{R}}}_4},\\ {{\tilde {{\varPhi}} }_{1,8}} =& 2{h_m}{{\tilde {{P}}}_4} + \frac{{h_m^2}}{2}{{\tilde {{U}}}_1},{{\tilde {{\varPhi}} }_{1,9}} = {{\tilde {{\varPhi}} }_{1,10}} = 2({h_M} - {h_m}){{\tilde {{P}}}_5} + \frac{{(h_M^2 - h_m^2)}}{2}{{\tilde {{U}}}_2}\\ {{\tilde {{\varPhi}} }_{2,2}} =& h_m^2{{{B}}^{\rm T}}{{\tilde X}_2}{{B}} + {({h_M} - {h_m})^2}{{{B}}^{\rm T}}{{\tilde X}_4}{{B}} - {{\tilde X}_4} + \frac{{h_m^4}}{4}{{{B}}^{\rm T}}{{\tilde {{R}}}_2}{{B}} + \frac{{{{(h_M^2 - h_m^2)}^2}}}{4}{{{B}}^{\rm T}}{{\tilde {{R}}}_4}{{B}} + \frac{{h_m^6}}{{36}}{{{B}}^{\rm T}}{{\tilde {{U}}}_1}{{B}} + \frac{{{{(h_M^3 - h_m^3)}^2}}}{{36}}{{{B}}^{\rm T}}{{\tilde {{U}}}_2}{{B}} - \mu {{\tilde Q}_3},\\ {{\tilde {{\varPhi}} }_{2,3}} = & - (\alpha - 2){{\tilde {{X}}}_4},{{\tilde {{\varPhi}} }_{2,4}} = (1 + \alpha ){{\tilde {{X}}}_4},{{\tilde {{\varPhi}} }_{2,5}} = {{\tilde {{\varPhi}} }_{2,6}} = {{\tilde {{\varPhi}} }_{2,7}} = {{\tilde {{\varPhi}} }_{2,8}} = {{\tilde {{\varPhi}} }_{2,9}} = {{\tilde {{\varPhi}} }_{2,10}} = 0,\\ {{\tilde {{\varPhi}} }_{3,3}} =& {{\tilde {{Q}}}_2} - {{\tilde {{Q}}}_1} - {{\tilde {{X}}}_2} + (\alpha - 2){{\tilde {{X}}}_4},{{\tilde {{\varPhi}} }_{3,4}} = 0,{{\tilde {{\varPhi}} }_{3,5}} = - 2{{\tilde P}_2},{{\tilde {{\varPhi}} }_{3,6}} = {{\tilde {{\varPhi}} }_{3,7}} = 2{{\tilde P}_3},{{\tilde {{\varPhi}} }_{3,8}} = {{\tilde {{\varPhi}} }_{3,9}} = {{\tilde {{\varPhi}} }_{3,10}} = 0,\\ {{\tilde {{\varPhi}} }_{4,4}} =& - {{\tilde {{Q}}}_2} - (1 + \alpha ){{\tilde {{X}}}_4},{{\tilde {{\varPhi}} }_{4,5}} = 0,{{\tilde {{\varPhi}} }_{4,6}} = {{\tilde {{\varPhi}} }_{4,7}} = - 2{{\tilde {{P}}}_3},{{\tilde {{\varPhi}} }_{4,8}} = {{\tilde {{\varPhi}} }_{4,9}} = {{\tilde {{\varPhi}} }_{4,10}} = 0,{{\tilde {{\varPhi}} }_{5,5}} = - {{\tilde {{X}}}_1} - {{\tilde {{R}}}_2},\\ {{\tilde {{\varPhi}} }_{5,6}} =& {{\tilde {{\varPhi}} }_{5,7}} = 0,{{\tilde {{\varPhi}} }_{5,8}} = - 2{{\tilde {{P}}}_4},{{\tilde {{\varPhi}} }_{5,9}} = {{\tilde {{\varPhi}} }_{5,10}} = 0,{{\tilde {{\varPhi}} }_{6,6}} = (\alpha - 2){{\tilde {{X}}}_3} - (2 - \varepsilon ){{\tilde {{R}}}_4},{{\tilde {{\varPhi}} }_{6,7}} = {{\tilde {{\varPhi}} }_{6,8}} = 0,\\ {{\tilde {{\varPhi}} }_{6,9}} =& {{\tilde {{\varPhi}} }_{6,10}} = - 2{{\tilde {{P}}}_5},{{\tilde {{\varPhi}} }_{7,7}} = - (\alpha + 1){{\tilde {{X}}}_3} - (1 + \varepsilon ){{\tilde {{R}}}_4},{{\tilde {{\varPhi}} }_{7,8}} = 0,{{\tilde {{\varPhi}} }_{7,9}} = {{\tilde {{\varPhi}} }_{7,10}} = - 2{{\tilde {{P}}}_5},{{\tilde {{\varPhi}} }_{8,8}} = - {{\tilde {{R}}}_1} - {{\tilde {{U}}}_1},\\ {{\tilde {{\varPhi}} }_{8,9}} =& {{\tilde {{\varPhi}} }_{8,10}} = 0,{{\tilde {{\varPhi}} }_{9,9}} = - (2 - \varepsilon ){{\tilde {{R}}}_3} - {{\tilde {{U}}}_2},{{\tilde {{\varPhi}} }_{9,10}} = - {{\tilde {{U}}}_2},{{\tilde {{\varPhi}} }_{10,10}} = (1 + \varepsilon ){{\tilde {{R}}}_3} - {{\tilde {{U}}}_2},\alpha = \frac{{h(t) - {h_m}}}{{{h_M} - {h_m}}},\varepsilon = \frac{{h{{(t)}^2} - h_m^2}}{{h_M^2 - h_m^2}},\\ {{{\tilde \varGamma }}_a} =& \left[ {\begin{array}{*{20}{c}} {{\vartheta}} &0&0&0&0&0&{{\vartheta }} \end{array}} \right]H,\;{{\tilde \varGamma }_E}{\rm{ = }}\left[ {\begin{array}{*{20}{c}} {{{{E}}_0}{\Xi ^{\rm T}}}&0&{{{{E}}_1}{\Xi ^{\rm T}}}&0&0&0&0 \end{array}} \right] \end{aligned} $

      证明:由于定理1中式(6)给出的条件为非线性矩阵不等式,不能直接得到控制器的解。下面给出控制器的设计方法,首先将式(6)中的不确定项(即含$\Delta {{K}}$项)分离,即

      ${{{\varPhi}} '} + {{{\varGamma}} _a}{{{F}}_a}(t){{{\varGamma}} _E} + {{\varGamma}} _E^{\rm T}{{F}}_a^{\rm T}(t){{\varGamma}} _a^{\rm T} \text{<} 0$

      式中: ${{{\varPhi}} '}$${{\varPhi}} $中分离不确定项(含$\Delta {{K}}$项)所得结果。由引理4可得

      ${{{\varPhi}} '} + {\vartheta ^{ - 1}}{{{\varGamma}} _a}{{\varGamma}} _a^{\rm T} + \vartheta {{\varGamma}} _E^{\rm T}{{{\varGamma}} _E} \text{<} 0$

      式中:${{{\varGamma}} _E} = \left[ {{array}{*{20}{c}} {{{{E}}_c}{{{\varXi}} ^{\rm T}}}&0&0&0&0&0&0&0&0&0 {array}} \right].$

      进而对式(37)应用Schur补可得

      $\left[ {\begin{array}{*{20}{c}} {{{{\varPhi}} '}}&{{{\varGamma}} _a^{\rm T}}&{\vartheta {{\varGamma}} _E^{\rm T}} \\ *&{ - \vartheta {{I}}}&0 \\ *&*&{ - \vartheta {{I}}} \end{array}} \right] \text{<} 0$

      由于${{\varXi}} $为非奇异矩阵,因而可以求得${{{\varXi}} ^{ - 1}}$,对式(38)两边左乘${{{\varPsi}}} $,右乘其转置即${{{{\varPsi}}} ^{\rm T}}$,其中:

      ${{\varPsi}} = {\rm{diag}}\left\{ \begin{array}{*{20}{c}} {\underbrace {\begin{array}{*{20}{c}} {{\varXi}} & \cdots &{{\varXi}} \end{array}}_7}&{{I}}&{{I}}&{{\varXi}} &{{\vartheta ^{ - 1}}{{I}}}&{{\vartheta ^{ - 1}}{{I}}} \end{array}\right\} ,$

      $ {{P}_i} = {{\varXi}} {{P}_i}{{{\varXi}} ^{\rm T}}\left( {i = 1,2,3,4,5} \right),$${\tilde {{Q}}_j} = {{\varXi}} {\tilde {{Q}}_j}{{{\varXi}} ^{\rm T}}\left( {j = 1,2,3} \right),$${U}_k = {{\varXi}} {U}_k{{{\varXi}} ^{\rm T}}\left( {k = 1,2} \right),$${\tilde {{X}}_l} = {{\varXi}} {\tilde {{X}}_l}{{{\varXi}} ^{\rm T}},$${\tilde {{R}}_l} = {{\varXi}} {\tilde {{R}}_l}{{{\varXi}} ^{\rm T}}\left( {l{\rm{ = }}1,2,3,4} \right),$ Y= T。通过替换容易得到定理2的条件,证毕。

    • 本节将上节提出的非脆弱${H_\infty }$控制器的设计方法运用到液体火箭发动机燃烧过程当中,模拟分析系统的稳定性能以及系统参数摄动的镇定性能。

    • MADB (maximum allowable delay bound) 定义为保证系统稳定的最大允许时滞上界值,是时滞系统稳定性结论保守性最普遍的衡量标准;MAPI (minimum allowable performance index) 定义为保证系统稳定的最小允许性能指标值,是时滞系统在零初始条件下所具有的${H_\infty }$扰动抑制水平的衡量标准。

      下面考虑一类具有区间变时滞的线性系统,形如式(1)所示,其系统参数如下:

      $\begin{array}{l} {{A}} = \left[ {\begin{array}{*{20}{c}} { - 0.623\;8}&{ - 1.013\;2}\\ {2.011\;6}&{ - 0.210\;6} \end{array}} \right],{{{A}}_1} = \left[ {\begin{array}{*{20}{c}} { - 0.501\;1}&{ - 0.787\;1}\\ { - 0.300\;2}&{0.523\;1} \end{array}} \right],{{{B}}_\omega } = \left[ {\begin{array}{*{20}{c}} { - 0.432\;6}&{0.125\;3}\\ { - 1.665\;6}&{0.287\;7} \end{array}} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{{C}} = \left[ {\begin{array}{*{20}{c}} {0.213\;4}&{ - 0.019\;1}\\ {0.111\;9}&{ - 0.166\;5} \end{array}} \right],{{{C}}_d} = \left[ {\begin{array}{*{20}{c}} {0.081\;6}&{0.129\;0}\\ {0.071\;2}&{0.066\;9} \end{array}} \right],{{{D}}_\omega } = \left[ {\begin{array}{*{20}{c}} 0&0\\ 0&0 \end{array}} \right] \end{array}$

      在该数值例子中,考虑两个性能指标,即${H_\infty }$性能指标$\rho $和MADB值${h_M}$。根据定理1,当时滞变化率$\mu {\rm{ = }}0$${h_m}{\rm{ = }}0$时,针对不同的${H_\infty }$性能指标$\rho $表1模拟给出相应的MADB值;针对不同的MADB值,表2模拟给出相应的${H_\infty }$性能指标$\rho $

      来源hM
      ρ=2.0ρ=2.5ρ=3.0ρ=3.5ρ=4.0
      文献[16]0.405 70.466 00.504 70.531 60.551 5
      文献[17]0.405 70.466 00.504 60.531 60.551 5
      文献[18]0.420 30.477 90.514 60.540 10.558 9
      文献[4]0.473 40.523 70.554 50.575 40.590 4
      文献[19]0.662 00.704 00.730 00.747 00.759 5
      定理10.957 11.013 61.056 51.081 21.092 7

      表 1  不同的H${_\infty }$性能指标ρ,模拟给出的MADB值hM

      Table 1.  The maximum allowable delay bound hM for a given ρ

      来源ρ
      hM=0.1hM=0.2hM=0.3hM=0.4hM=0.5
      文献[16]1.071 41.242 61.506 71.963 42.298 1
      文献[17]1.071 41.242 51.506 71.963 42.298 1
      文献[18]1.057 71.211 21.451 51.873 32.775 7
      文献[19]0.933 10.952 51.021 61.120 41.284 3
      定理10.815 60.853 20.924 51.042 81.123 9

      表 2  不同的MADB值hM,仿真给出的MAPI值ρ

      Table 2.  The minimum allowable performance index ρ for a given hM

      通过表12的比较可以发现,对于指定的${H_\infty }$性能指标$\rho $,由定理1可以得出相应的MADB值。相比文献[4,16-19],本文所提出的时滞相关有界实判据扩大了系统稳定的最大允许时滞上界范围,具有更低的保守性;另一方面,对于指定的MADB值${h_M}$,我们也可以求得相应的MAPI值。相比文献[16-19],本文所提出的判据可以获得保证系统稳定的更小更佳${H_\infty }$性能指标$\rho $值。

    • 由于压力指数$\lambda $和时滞参数$\delta $是系统中最难确定的两个参数,这里只考虑系统对这两个参数的鲁棒性,并以振荡最强的状态变量${{{x}}_2}(t)$为研究对象。

    • 当压力指数$\lambda $不确定时,系统矩阵AA1的摄动值如下:

      $\begin{aligned}\Delta {{A}} =& \left[ {\begin{array}{*{20}{c}} {\lambda - {\lambda _0}}&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\Delta {\lambda _{\max }}} \\ 0 \\ 0 \\ 0 \end{array}} \right]\frac{{\Delta \lambda }}{{\Delta {\lambda _{\max }}}}\left[ {\begin{array}{*{20}{c}} 1&0&0&0 \end{array}} \right],\\ \Delta {{{A}}_1} =& \left[ {\begin{array}{*{20}{c}} {\lambda - {\lambda _0}}&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\Delta {\lambda _{\max }}} \\ 0 \\ 0 \\ 0 \end{array}} \right]\frac{{\Delta \lambda }}{{\Delta {\lambda _{\max }}}}\left[ {\begin{array}{*{20}{c}} { - 1}&0&0&0 \end{array}} \right] \end{aligned} $

      式中:$\Delta {\lambda _{\max }} = \max \left\{ {\left| {{\lambda _{\min }} - {\lambda _0}} \right|,\left| {{\lambda _{\max }} - {\lambda _0}} \right|} \right\}$,由此可得${{H}} = {\left[ {{array}{*{20}{c}} {\Delta {\lambda _{\max }}}&0&0&0 {array}} \right]^{\rm T}}$${{{E}}_0} = \left[ {{array}{*{20}{c}} 1&0&0&0 {array}} \right]$${{{E}}_1} = - {{{E}}_0}$

      情形1:当$\Delta {{K}} = 0$时,则本文设计的非脆弱控制器就成为一般的${H_\infty }$反馈控制器,取$\delta = 0.8$${\lambda _0} = 1$$\Delta {\lambda _{\max }} = 0.2$,性能指标$\kappa = 0.6$${{{B}}_\omega } = {[{array}{*{20}{c}} 0&0&0&1 {array}]^{\rm T}}$${{C}} = [{array}{*{20}{c}} 1&0&0&0 {array}]$${{{C}}_d} = 1$${{{D}}_\omega } = 0$,则由定理1可求${H_\infty }$状态反馈控制器的增益矩阵为:

      ${{{K}}_1} = \left[ {\begin{array}{*{20}{c}} { - {\rm{0}}{\rm{.426\;8}}}&{{\rm{0}}{\rm{.060\;8}}}&{{\rm{0}}{\rm{.299\;4}}}&{ - {\rm{0}}{\rm{.047\;0}}} \end{array}} \right]$

      情形2:当$\Delta {{K}} \ne 0$时,控制器增益摄动参数为${{{D}}_c} = 1$${{{E}}_c} = [{array}{*{20}{c}} 1&0&0&0 {array}]$,取与情形1中相同的系统参数,则由定理1可求得非脆弱控制器的增益矩阵为:

      ${{{K}}_2} = \left[ {\begin{array}{*{20}{c}} { - {\rm{0}}{\rm{.381\;7}}}&{ - 1.051\;2}&{0.850\;2}&{ - 1.872\;6} \end{array}} \right]$

      将非脆弱控制器代入燃烧过程方程,系统的状态响应曲线如图2所示。

      图  2  非脆弱H控制器作用下系统的状态响应

      Figure 2.  Response of system under non-fragile H controller

      图2可知,当系统参数和控制器增益存在不确定性时,该系统不仅具有鲁棒性而且是非脆弱的,燃烧过程得到了较好的镇定,系统能满足一定的性能指标要求,控制器具有足够的调节余地,系统的可靠性更高。

      下面以状态变量${{{x}}_2}(t)$为研究对象,分析比较本文设计的非脆弱${H_\infty }$反馈控制器和一般${H_\infty }$反馈控制器在燃烧过程中的镇定效果。在同样的条件下,将控制器${{{K}}_1}$${{{K}}_2}$代入燃烧过程,可得这两种控制器作用下系统状态变量${{{x}}_2}(t)$的响应曲线,如图3所示。

      图  3  不同控制器作用下系统状态${x_2}(t)$的响应

      Figure 3.  Response of ${x_2}(t)$ under different controller

      图3可知,在相同的条件下,本文设计的非脆弱${H_\infty }$状态反馈控制器相比一般${H_\infty }$状态反馈控制器,允许控制器增益存在一定的摄动,具有更强的鲁棒性。应用于燃烧过程时,其系统状态较为平稳,振幅较小,镇定效果优于一般${H_\infty }$状态反馈控制器。

    • 当系统时滞参数$\delta $变化时,考察控制器的镇定性能,同样取性能指标$\kappa = 0.6$,当控制器存在增益摄动${{{D}}_{\rm c}} = 1$${{{E}}_c} = [{array}{*{20}{c}} 1&0&0&0 {array}]$时,图4给出了不同的时滞参数$\delta $下系统状态变量x2(t)的响应曲线图。

      图  4  控制器对时滞参数$\delta $的鲁棒性

      Figure 4.  Robust performance of controller to variation of $\delta $

      图4可以清晰地看到,在本文设计控制器作用下,当时滞参数δ逐渐增大时,燃烧过程中状态变量x2(t)逐渐趋于不稳定,但是在$\delta \in [{array}{*{20}{c}}0, {1.1}{array}]$内,燃烧过程状态变量x2(t)仍然是可镇定的;需要说明的是,上面的模拟是在${{{D}}_{\rm c}} = 1$并且${{{E}}_{\rm c}} = [{array}{*{20}{c}} 1&0&0&0 {array}]$的情形下进行的,也就是说本文所设计控制器增益值满足一定摄动性,因而提升了燃烧过程的可靠性和鲁棒性。

    • (1)通过构造包含四重积分项的L-K泛函以及采用新颖的积分不等式来界定交叉项,给出了保证燃烧过程稳定并具有给定${H_\infty }$性能指标的时滞相关有界实引理;

      (2)在有界实引理的基础上,通过求解线性矩阵不等式的可行解来获得控制器的参数化表达式,进而设计了一种无需参数调整并易于求解的鲁棒非脆弱${H_\infty }$控制器;

      (3)通过仿真实例对比验证了所提出的时滞相关有界实引理的有效性以及所设计的鲁棒非脆弱${H_\infty }$控制器的优越性。

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