西南师范大学学报(自然科学版)   2019, Vol. 44 Issue (7): 46-51.  DOI: 10.13718/j.cnki.xsxb.2019.07.007
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  • 具有非匹配不确定性的离散时滞复杂系统的变结构控制    [PDF全文]
    姚合军 , 严谦泰 , 杨恒     
    安阳师范学院 数学与统计学院, 河南 安阳 455000
    摘要:针对一类具有非匹配不确定性的离散时滞复杂系统,利用极点配置方法设计了渐近稳定的滑模面,并在此基础上设计了变结构控制器,此过程不再要求系统不确定性满足匹配条件.最后通过仿真算例说明了该方法的有效性.
    关键词变结构控制    时滞系统    离散    非匹配不确定性    

    近年来,对时滞系统的分析与设计问题引起了众多学者的广泛关注和深入研究[1-7].众所周知,基于不连续控制的变结构控制理论是处理动力系统鲁棒镇定问题的有效工具[8].传统的变结构控制理论主要针对无时滞的动力系统,而时滞的出现会使得系统变结构控制器设计变得异常复杂,容易使得系统出现高频抖振现象.文献[9]针对具有非线性输入的不确定时滞系统,设计了系统的变结构控制器.文献[10]基于Lyapunov稳定性理论,利用线性矩阵不等式方法得到了不确定时滞系统渐近稳定的充分条件.近年来,出现了若干有关离散时滞系统的变结构控制问题的研究成果[11-13].文献[11]利用线性矩阵不等式方法设计了不确定离散网络控制系统的保成本变结构控制器.文献[12]考虑了具有输入时滞的多输入离散系统的变结构控制设计问题.文献[13]利用变结构控制方法,设计了离散时滞系统的准变结构控制.

    然而,上述结果所涉及的时滞系统中的不确定性均满足匹配条件,有关非匹配的不确定性的处理却不多见,尤其是对时滞大系统的变结构控制方面的研究鲜见报道.本文在前人研究的基础上,研究了针对带有非匹配不确定性和时滞的大系统,利用线性矩阵不等式方法得到了滑模面的设计的充分条件,并在此基础上利用极点配置方法设计了使系统状态在有限时间内到达并保持在滑模面上的变结构控制器.

    1 问题描述

    考虑下面有N个关联子系统的不确定离散时滞大系统

    $ s_{i} :(i=1,2, \cdots, N) $
    $ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{x}}_i}(k + 1) = \left( {{\mathit{\boldsymbol{A}}_i} + \Delta {\mathit{\boldsymbol{A}}_i}(k)} \right){\mathit{\boldsymbol{x}}_i}(k) + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{u}}_i}(k) + \sum\limits_{j = 1}^N {\left( {{\mathit{\boldsymbol{A}}_{ij}} + \Delta {\mathit{\boldsymbol{A}}_{ij}}(k)} \right)} {\mathit{\boldsymbol{x}}_j}\left( {k - {h_{ij}}} \right)}\\ {{\mathit{\boldsymbol{x}}_i}(k) = {\psi _i}(k)\quad - h \le k \le 0} \end{array} $ (1)

    其中:$\boldsymbol{x}_{i}(k) \in \mathbb{R}^{n_{i}}$是系统状态;$\boldsymbol{u}_{i}(k) \in \mathbb{R}^{m_{i}}$是控制输入;正整数hij代表系统时滞,$h = \mathop {\max }\limits_{i,j = 1,2, \cdots ,N} \left\{ {{h_{ij}}} \right\}$ψi(k)是定义在[-h,0]上的初始状态;AiBiAij是具有适当维数的常数矩阵;ΔAi(k),ΔAij(k)是代表时变不确定性的未知矩阵.对系统(1)做如下假设:

    假设 1    Bi是列满秩的.

    假设 2     (AiBi)是可控的.

    对系统(1),存在非奇异变换$z_{i}(k)=\boldsymbol{T}_{i} \boldsymbol{x}_{i}(k)(i=1, 2, \cdots, N)$使系统(1)等价于

    $ \begin{array}{l} {\mathit{\boldsymbol{z}}_i}\left( {k + 1} \right) = \left[ {\begin{array}{*{20}{l}} {{z_{i1}}(k + 1)}\\ {{z_{i2}}(k + 1)} \end{array}} \right] = {\mathit{\boldsymbol{T}}_i}{\mathit{\boldsymbol{x}}_i}(k + 1) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{T}}_i}\left( {{\mathit{\boldsymbol{A}}_i} + \Delta {\mathit{\boldsymbol{A}}_i}\left( k \right)} \right){\mathit{\boldsymbol{x}}_i}\left( k \right) + {\mathit{\boldsymbol{T}}_i}{\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{u}}_i}\left( k \right) + \sum\limits_{j = 1}^N {\left( {{\mathit{\boldsymbol{T}}_i}{\mathit{\boldsymbol{A}}_{ij}} + {\mathit{\boldsymbol{T}}_i}\Delta {\mathit{\boldsymbol{A}}_{ij}}\left( k \right)} \right){\mathit{\boldsymbol{x}}_j}\left( {k - {h_{ij}}} \right)} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{T}}_i}{\mathit{\boldsymbol{A}}_i}\mathit{\boldsymbol{T}}_i^{ - 1}{\mathit{\boldsymbol{z}}_i}\left( k \right) + {\mathit{\boldsymbol{T}}_i}\Delta {\mathit{\boldsymbol{A}}_i}\left( k \right)\mathit{\boldsymbol{T}}_i^{ - 1}{\mathit{\boldsymbol{z}}_i}\left( k \right) + {\mathit{\boldsymbol{T}}_i}{\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{u}}_i}\left( k \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{j = 1}^N {\left( {{\mathit{\boldsymbol{T}}_i}{\mathit{\boldsymbol{A}}_{ij}}\mathit{\boldsymbol{T}}_i^{ - 1} + {\mathit{\boldsymbol{T}}_i}\Delta {\mathit{\boldsymbol{A}}_{ij}}\left( k \right)\mathit{\boldsymbol{T}}_i^{ - 1}} \right){\mathit{\boldsymbol{z}}_j}\left( {k - {h_{ij}}} \right)} \end{array} $ (2)

    其中

    $ {\mathit{\boldsymbol{\bar A}}_i} = {\mathit{\boldsymbol{T}}_i}{\mathit{\boldsymbol{A}}_i}\mathit{\boldsymbol{T}}_i^{ - 1} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{A}}_{i11}}}&{{\mathit{\boldsymbol{A}}_{i12}}}\\ {{\mathit{\boldsymbol{A}}_{i21}}}&{{\mathit{\boldsymbol{A}}_{i22}}} \end{array}} \right] $ (3)
    $ \Delta \overline{\boldsymbol{A}}_{i}(k)=\boldsymbol{T}_{i} \Delta \boldsymbol{A}_{i}(\boldsymbol{k}) \boldsymbol{T}_{i}^{-1}=\left[\begin{array}{c}{\Delta \boldsymbol{A}_{i1}(\boldsymbol{k})} \\ {\Delta \boldsymbol{A}_{i 2}(\boldsymbol{k})}\end{array}\right] $ (4)
    $ \overline{\boldsymbol{B}}_{i}=\boldsymbol{T}_{i} \boldsymbol{B}_{i}=\left[\begin{array}{c}{\bf{0}} \\ {\boldsymbol{B}_{i 2}}\end{array}\right] $ (5)
    $ \overline{\boldsymbol{A}}_{i j}=\boldsymbol{T}_{i} \boldsymbol{A}_{i j} \boldsymbol{T}_{i}^{-1}=\left[\begin{array}{cc}{\boldsymbol{A}_{i j 11}} & {\boldsymbol{A}_{i j 12}} \\ {\boldsymbol{A}_{i j 21}} & {\boldsymbol{A}_{i j 22}}\end{array}\right] $ (6)
    $ \Delta {\mathit{\boldsymbol{\overline A}} _{ij}}(k) = {\mathit{\boldsymbol{T}}_i}\Delta {\mathit{\boldsymbol{A}}_{ij}}(k)\mathit{\boldsymbol{T}}_i^{ - 1} = \left[ {\begin{array}{*{20}{c}} {\Delta {\mathit{\boldsymbol{A}}_{ij1}}(k)}\\ {\Delta {\mathit{\boldsymbol{A}}_{ij2}}(k)} \end{array}} \right] $ (7)

    由式(2)-(7),得到

    $ {\mathit{\boldsymbol{z}}_i}(k + 1) = {\mathit{\boldsymbol{\bar A}}_i}{\mathit{\boldsymbol{z}}_i}(k) + \Delta {\mathit{\boldsymbol{\overline A}} _i}(k){\mathit{\boldsymbol{z}}_i}(k) + {\mathit{\boldsymbol{\overline B}} _i}{\mathit{\boldsymbol{u}}_i}(k) + \sum\limits_{j = 1}^N {\left( {{{\mathit{\boldsymbol{\overline A}} }_{ij}} + \Delta {{\mathit{\boldsymbol{\overline A}} }_{ij}}(k)} \right)} {\mathit{\boldsymbol{z}}_j}\left( {k - {h_{ij}}} \right) $ (8)

    Bi列满秩,易知不确定性ΔAi2,ΔAij2满足匹配条件

    $ \Delta {\mathit{\boldsymbol{A}}_{i2}}(k) = {\mathit{\boldsymbol{B}}_{i2}}\Delta {\mathit{\boldsymbol{L}}_{i2}}(k)\quad \Delta {\mathit{\boldsymbol{A}}_{ij2}}(k) = {\mathit{\boldsymbol{B}}_{i2}}\Delta {\mathit{\boldsymbol{M}}_{ij2}}(k) $

    其中ΔLi2(k),ΔMij2(k)满足

    $ \left\|\Delta \boldsymbol{L}_{i 2}(k)\right\| \leqslant L_{i 2} \quad\left\|\Delta \boldsymbol{M}_{i j 2}(k)\right\| \leqslant M_{i j 2} $ (9)

    其中Li2Mij2是常数.

    假设非匹配不确定性ΔAi1(k),ΔAij1(k)满足

    $ \Delta \boldsymbol{A}_{i 1}(k)=\boldsymbol{D}_{i} \boldsymbol{F}(k) \boldsymbol{E}_{i}, \Delta \boldsymbol{A}_{i j 1}(k)=\boldsymbol{D}_{i j d} \boldsymbol{F}(k) \boldsymbol{E}_{i j d} $ (10)

    其中DiEiDijdEijd是具有适当维数的已知常数矩阵,EiEijd满足

    $ {\mathit{\boldsymbol{E}}_i} = {\mathit{\boldsymbol{E}}_{ia}}{\mathit{\boldsymbol{c}}_i}\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{E}}_{ijd}} = {\mathit{\boldsymbol{E}}_{ijad}}{\mathit{\boldsymbol{c}}_j} $ (11)

    F(k)是未知的时变矩阵函数,满足

    $ \boldsymbol{F}^{\mathrm{T}}(k) \boldsymbol{F}(k) \leqslant \boldsymbol{I} $

    对不确定离散时滞大系统(2),选择如下滑模面

    $ {\mathit{\boldsymbol{s}}_i}(k) = {\mathit{\boldsymbol{c}}_i}{\mathit{\boldsymbol{z}}_i}(k) = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{c}}_{i1}}}&{{\mathit{\boldsymbol{c}}_{i2}}} \end{array}} \right]{\mathit{\boldsymbol{z}}_i}(k)\quad i = 1,2, \cdots ,N $

    其中ci1ci2具有合适维数.选取ci使得(ciBi)是非奇异的,从而滑模面si(k)是稳定的[12].

    到达条件如下[13]

    $ \boldsymbol{S}^{\mathrm{T}}(k) \Delta \boldsymbol{S}(k)=\sum\limits_{i=1}^{N} \boldsymbol{s}_{i}^{\mathrm{T}}(k) \Delta \boldsymbol{s}_{i}(k)<0, s_{i}(k) \neq 0 $

    其中S(k)=[s1(k),s2(k),…,sN(k)]T.

    2 主要结果

    引理 1[13]     对任意的$ \mathit{\boldsymbol{x}}, \mathit{\boldsymbol{y}} \in \mathbb{R}^{n}$和矩阵函数F(k)满足

    $ \boldsymbol{F}^{\mathrm{T}}(k) \boldsymbol{F}(k) \leqslant \boldsymbol{I} $

    不等式

    $ 2 \boldsymbol{x}^{\mathrm{T}} \boldsymbol{F}(k) y \leqslant \boldsymbol{x}^{\mathrm{T}} \boldsymbol{x}+\boldsymbol{y}^{\mathrm{T}} \boldsymbol{y}^{\mathrm{T}} $

    成立

    定理 1     对不确定离散时滞大系统(1),选取如下控制器,则系统状态在有限时间内到达滑模面

    $ \begin{array}{l} {\mathit{\boldsymbol{u}}_i}(k) = - \left( {{\mathit{\boldsymbol{c}}_{i2}}{\mathit{\boldsymbol{B}}_{i2}}} \right) - 1\left[ {{\mathit{\boldsymbol{c}}_{i1}}\left( {{\mathit{\boldsymbol{A}}_{i11}}{\mathit{\boldsymbol{z}}_{i1}}(k) + {\mathit{\boldsymbol{A}}_{i12}}{\mathit{\boldsymbol{z}}_{i2}}(k) + \sum\limits_{j = 1}^N {\left( {{\mathit{\boldsymbol{A}}_{ij11}}{\mathit{\boldsymbol{z}}_{j1}}\left( {k - {h_{ij}}} \right) + } \right.} } \right.} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left. {\left. {{\mathit{\boldsymbol{A}}_{ij12}}{\mathit{\boldsymbol{z}}_{j2}}\left( {k - {h_{ij}}} \right)} \right)} \right) + {\mathit{\boldsymbol{c}}_{i2}}\left( {{\mathit{\boldsymbol{A}}_{i21}}{\mathit{\boldsymbol{z}}_{i1}}(k) + {\mathit{\boldsymbol{A}}_{i22}}{\mathit{\boldsymbol{z}}_{i2}}(k)} \right) + {\mathit{\boldsymbol{c}}_{i2}}\left( {\sum\limits_{j = 1}^N {\left( {{\mathit{\boldsymbol{A}}_{ij21}}{\mathit{\boldsymbol{z}}_{j1}}\left( {k - {h_{ij}}} \right) + } \right.} } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left. {\left. {{\mathit{\boldsymbol{A}}_{ij22}}{\mathit{\boldsymbol{z}}_{j2}}\left( {k - {h_{ij}}} \right)} \right)} \right) - {\mathit{\boldsymbol{c}}_{i1}}{z_{i1}}(k) - {\mathit{\boldsymbol{c}}_{i2}}{\mathit{\boldsymbol{z}}_{i2}}(k) + \frac{1}{2}\mathit{\boldsymbol{s}}_i^{\rm{T}}(k){\mathit{\boldsymbol{c}}_{i1}}{\mathit{\boldsymbol{D}}_i}\mathit{\boldsymbol{D}}_i^{\rm{T}}\mathit{\boldsymbol{c}}_{i1}^{\rm{T}}{\mathit{\boldsymbol{s}}_i}(k) + \\ \;\;\;\;\;\;\;\;\;\;\;\frac{1}{2}\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right)\mathit{\boldsymbol{E}}_{ia}^{\rm{T}}\mathit{\boldsymbol{E}}_{ia}^{\rm{T}}{\mathit{\boldsymbol{s}}_i}\left( k \right) + \sum\limits_{j = 1}^N {\frac{1}{2}\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right){\mathit{\boldsymbol{c}}_{i1}}{\mathit{\boldsymbol{D}}_{id}}\mathit{\boldsymbol{D}}_{id}^{\rm{T}}\mathit{\boldsymbol{c}}_{i1}^{\rm{T}}{\mathit{\boldsymbol{s}}_i}\left( k \right)} + \\ \;\;\;\;\;\;\;\;\;\;\;\frac{1}{2}\frac{{{\mathit{\boldsymbol{s}}_i}}}{{{{\left\| {{\mathit{\boldsymbol{s}}_i}} \right\|}^2}}}\sum\limits_{j = 1}^N {\mathit{\boldsymbol{s}}_j^{\rm{T}}\left( {k - {h_{ij}}} \right)\mathit{\boldsymbol{E}}_{ijad}^{\rm{T}}{\mathit{\boldsymbol{E}}_{ijad}}{\mathit{\boldsymbol{s}}_j}\left( {k - {h_{ij}}} \right)} + \frac{{{\mathit{\boldsymbol{s}}_i}}}{{{{\left\| {{\mathit{\boldsymbol{s}}_i}} \right\|}^2}}}\left\| {\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right){\mathit{\boldsymbol{c}}_{i2}}{\mathit{\boldsymbol{B}}_{i2}}} \right\|{L_{i2}} + \\ \;\;\;\;\;\;\;\;\;\;\;\left. {\frac{{{\mathit{\boldsymbol{s}}_i}}}{{{{\left\| {{\mathit{\boldsymbol{s}}_i}} \right\|}^2}}}\sum\limits_{j = 1}^N {\left\| {\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right){\mathit{\boldsymbol{c}}_{i2}}{\mathit{\boldsymbol{B}}_{i2}}} \right\|{M_{ij}}\left\| {{\mathit{\boldsymbol{z}}_j}\left( {k - {h_{ij}}} \right)} \right\|} + \left( {{k_{i1}}{\mathop{\rm sgn}} \left( {{\mathit{\boldsymbol{s}}_i}\left( k \right)} \right) + {k_{i2}}{\mathit{\boldsymbol{s}}_i}\left( k \right)} \right)} \right] \end{array} $ (12)

        由si(k)沿系统(2)的前向差分,易知

    $ \begin{array}{l} {\mathit{\boldsymbol{S}}^{\rm{T}}}\left( k \right)\Delta S\left( k \right) = \sum\limits_{i = 1}^N {\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right)} \left[ {{\mathit{\boldsymbol{c}}_{i1}}\left( {{\mathit{\boldsymbol{A}}_{i11}}{\mathit{\boldsymbol{z}}_{i1}}\left( k \right) + {\mathit{\boldsymbol{A}}_{i12}}{\mathit{\boldsymbol{z}}_{i2}}\left( k \right) + \Delta {\mathit{\boldsymbol{A}}_{i1}}{\mathit{\boldsymbol{z}}_i}\left( k \right)} \right. + \sum\limits_{j = 1}^N {\left( {{\mathit{\boldsymbol{A}}_{ij11}}{\mathit{\boldsymbol{z}}_{j1}}\left( {k - {h_{ij}}} \right) + } \right.} } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left. {{\mathit{\boldsymbol{A}}_{ij12}}{\mathit{\boldsymbol{z}}_{j2}}\left( {k - {h_{ij}}} \right) + \Delta {\mathit{\boldsymbol{A}}_{ij1}}{\mathit{\boldsymbol{z}}_j}\left( {k - {h_{ij}}} \right)} \right)} \right) + {\mathit{\boldsymbol{c}}_{i2}}\left( {{\mathit{\boldsymbol{A}}_{i21}}{\mathit{\boldsymbol{z}}_{i1}}\left( k \right) + {\mathit{\boldsymbol{A}}_{i22}}{\mathit{\boldsymbol{z}}_{i2}}\left( k \right) + } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Delta {\mathit{\boldsymbol{A}}_{i2}}{\mathit{\boldsymbol{z}}_i}\left( k \right) + {\mathit{\boldsymbol{B}}_{i2}}{u_i}\left( k \right) + \sum\limits_{j = 1}^N {\left( {{\mathit{\boldsymbol{A}}_{ij21}}{\mathit{\boldsymbol{z}}_{j1}}\left( {k - {h_{ij}}} \right) + {\mathit{\boldsymbol{A}}_{ij22}}{\mathit{\boldsymbol{z}}_{j2}}\left( {k - {h_{ij}}} \right) + } \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left. {\left. {\Delta {\mathit{\boldsymbol{A}}_{ij2}}{\mathit{\boldsymbol{z}}_j}\left( {k - {h_{ij}}} \right)} \right)} \right) - {\mathit{\boldsymbol{c}}_{i1}}{\mathit{\boldsymbol{z}}_{i1}}(k) - {\mathit{\boldsymbol{c}}_{i2}}{\mathit{\boldsymbol{z}}_{i2}}(k)} \right] = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{i = 1}^N {\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right)\left[ {{\mathit{\boldsymbol{c}}_{i1}}\left( {{\mathit{\boldsymbol{A}}_{i11}}{\mathit{\boldsymbol{z}}_{i1}}\left( k \right) + {\mathit{\boldsymbol{A}}_{i12}}{\mathit{\boldsymbol{z}}_{i2}}\left( k \right) + \sum\limits_{j = 1}^N {\left( {{\mathit{\boldsymbol{A}}_{ij11}}{\mathit{\boldsymbol{z}}_{j1}}\left( {k - {h_{ij}}} \right) + } \right.} } \right.} \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left. {{\mathit{\boldsymbol{A}}_{ij12}}{\mathit{\boldsymbol{z}}_{j2}}\left( {k - {h_{ij}}} \right)} \right)} \right) + {\mathit{\boldsymbol{c}}_{i2}}\left( {{\mathit{\boldsymbol{A}}_{i21}}{\mathit{\boldsymbol{z}}_{i1}}\left( k \right) + {\mathit{\boldsymbol{A}}_{i22}}{\mathit{\boldsymbol{z}}_{i2}}\left( k \right)} \right) + {\mathit{\boldsymbol{c}}_{i2}}{\mathit{\boldsymbol{B}}_{i2}}{u_i}\left( k \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {{\mathit{\boldsymbol{c}}_{i2}}\left( {\sum\limits_{j = 1}^N {\left( {{\mathit{\boldsymbol{A}}_{ij21}}{\mathit{\boldsymbol{z}}_{j1}}\left( {k - {h_{ij}}} \right) + {\mathit{\boldsymbol{A}}_{ij22}}{\mathit{\boldsymbol{z}}_{j2}}\left( {k - {h_{ij}}} \right)} \right)} } \right) - {\mathit{\boldsymbol{c}}_{i1}}{\mathit{\boldsymbol{z}}_{i1}}\left( k \right) - {\mathit{\boldsymbol{c}}_{i2}}{\mathit{\boldsymbol{z}}_{i2}}\left( k \right) + {\mathit{\Pi }_i}\left( k \right)} \right] \end{array} $ (13)

    其中

    $ \begin{array}{l} {\mathit{\Pi }_\mathit{i}}\left( k \right) = {\mathit{\boldsymbol{c}}_{i1}}{\mathit{\boldsymbol{D}}_i}F\left( k \right){\mathit{\boldsymbol{E}}_i}{\mathit{\boldsymbol{z}}_i}\left( k \right) + \sum\limits_{j = 1}^N {\left( {{\mathit{\boldsymbol{c}}_{i1}}{\mathit{\boldsymbol{D}}_{ijd}}F\left( k \right){\mathit{\boldsymbol{E}}_{ijd}}{\mathit{\boldsymbol{z}}_j}\left( {k - {h_{ij}}} \right)} \right)} + \\ \;\;\;\;\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{c}}_{i2}}{\mathit{\boldsymbol{B}}_{i2}}\Delta {\mathit{\boldsymbol{L}}_{i2}}\left( k \right) + \sum\limits_{j = 1}^N {\left( {{\mathit{\boldsymbol{c}}_{i2}}{\mathit{\boldsymbol{B}}_{i2}}\Delta {\mathit{\boldsymbol{M}}_{ij2}}\left( k \right){\mathit{\boldsymbol{z}}_j}\left( {k - {h_{ij}}} \right)} \right)} \end{array} $ (14)

    由式(9),(10),(11)和(14),并利用引理1,得到

    $ \begin{array}{l} \mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right){\mathit{\Pi }_\mathit{i}}\left( k \right) = \mathit{\boldsymbol{s}}_i^{\rm{T}}{\mathit{\boldsymbol{c}}_{i1}}{\mathit{\boldsymbol{D}}_i}F\left( k \right){\mathit{\boldsymbol{E}}_i}{\mathit{\boldsymbol{z}}_i}\left( k \right) + \sum\limits_{j = 1}^N {\left( {\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right){\mathit{\boldsymbol{c}}_{i1}}{\mathit{\boldsymbol{D}}_{ijd}}\mathit{\boldsymbol{F}}\left( k \right){\mathit{\boldsymbol{E}}_{ijd}}{\mathit{\boldsymbol{z}}_j}\left( {k - {h_{ij}}} \right) + } \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right){\mathit{\boldsymbol{c}}_{i2}}{\mathit{\boldsymbol{B}}_{i2}}\Delta {L_{i2}}\left( k \right) + \sum\limits_{j = 1}^N {\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right){\mathit{\boldsymbol{c}}_{i2}}{\mathit{\boldsymbol{B}}_{i2}}\Delta {\mathit{\boldsymbol{M}}_{ij2}}\left( k \right){\mathit{\boldsymbol{z}}_j}\left( {k - {h_{ij}}} \right)} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{2}\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right){\mathit{\boldsymbol{c}}_{i1}}{\mathit{\boldsymbol{D}}_i}\mathit{\boldsymbol{D}}_i^{\rm{T}}\mathit{\boldsymbol{c}}_{i1}^{\rm{T}}{\mathit{\boldsymbol{s}}_i}\left( k \right) + \frac{1}{2}\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right)\mathit{\boldsymbol{E}}_{ia}^{\rm{T}}\mathit{\boldsymbol{E}}_{ia}^{\rm{T}}{\mathit{\boldsymbol{s}}_i}\left( k \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{j = 1}^N {\left( {\frac{1}{2}\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( {k - {h_{ij}}} \right)\mathit{\boldsymbol{E}}_{ijad}^{\rm{T}}{\mathit{\boldsymbol{E}}_{ijad}}{\mathit{\boldsymbol{s}}_j}\left( {k - {h_{ij}}} \right) + } \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right){\mathit{\boldsymbol{c}}_{i2}}{\mathit{\boldsymbol{B}}_{i2}}\Delta {L_{i2}}\left( k \right) + \sum\limits_{j = 1}^N {\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right){\mathit{\boldsymbol{c}}_{i2}}{\mathit{\boldsymbol{B}}_{i2}}\Delta {\mathit{\boldsymbol{M}}_{ij2}}\left( k \right){\mathit{\boldsymbol{z}}_j}\left( {k - {h_{ij}}} \right)} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{2}\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right){\mathit{\boldsymbol{c}}_{i1}}{\mathit{\boldsymbol{D}}_i}\mathit{\boldsymbol{D}}_i^{\rm{T}}\mathit{\boldsymbol{c}}_{i1}^{\rm{T}}{\mathit{\boldsymbol{s}}_i}\left( k \right) + \frac{1}{2}\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right)\mathit{\boldsymbol{E}}_{ia}^{\rm{T}}\mathit{\boldsymbol{E}}_{ia}^{\rm{T}}{\mathit{\boldsymbol{s}}_i}\left( k \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{j = 1}^N {\left( {\frac{1}{2}\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right){\mathit{\boldsymbol{c}}_{i1}}{\mathit{\boldsymbol{D}}_{ijd}}\mathit{\boldsymbol{D}}_{ijd}^{\rm{T}}\mathit{\boldsymbol{c}}_{i1}^{\rm{T}}{\mathit{\boldsymbol{s}}_i}\left( k \right) + } \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\frac{1}{2}\mathit{\boldsymbol{s}}_j^{\rm{T}}\left( {k - {h_{ij}}} \right)\mathit{\boldsymbol{E}}_{ijad}^{\rm{T}}{\mathit{\boldsymbol{E}}_{ijad}}{\mathit{\boldsymbol{s}}_j}\left( {k - {h_{ij}}} \right)} \right) + \left\| {\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right){\mathit{\boldsymbol{c}}_{i2}}{\mathit{\boldsymbol{B}}_{i2}}} \right\|{L_{i2}} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{j = 1}^N {\left\| {\mathit{\boldsymbol{s}}_i^{\rm{T}}\left( k \right){\mathit{\boldsymbol{c}}_{i2}}{\mathit{\boldsymbol{B}}_{i2}}} \right\|{M_{ij}}\left\| {{\mathit{\boldsymbol{z}}_j}\left( {k - {h_{ij}}} \right)} \right\|} \end{array} $ (15)

    由控制器(12)和式(13),(15)易知

    $ {\mathit{\boldsymbol{S}}^{\rm{T}}}\left( k \right)\Delta \mathit{\boldsymbol{S}}\left( k \right) = - \sum\limits_{i = 1}^N {\left( {{k_{i1}}\left\| {{s_i}\left( k \right)} \right\| + {k_{i2}}s_i^2\left( k \right)} \right)} < 0\;\;\;\;\;\;\;\;\;\;{s_i}\left( k \right) \ne 0 $

    其中ki1ki2是常数并满足ki1>0,ki2>0(i=1,2,…,N).

    3 仿真算例

    考虑形如式(8)的不确定时滞大系统,其中

    $ {\mathit{\boldsymbol{\bar A}}_1} = \left[ {\begin{array}{*{20}{c}} 0&{0.5}\\ 0&{0.1} \end{array}} \right]\;\;\;\;\;\;\;\;\;\;\;\Delta {\mathit{\boldsymbol{\bar A}}_1}\left( k \right) = \left[ {\begin{array}{*{20}{c}} 0&{0.4\cos \left( k \right)}\\ {0.2\sin \left( k \right)}&0 \end{array}} \right] $
    $ {\mathit{\boldsymbol{\bar B}}_1} = \left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right]\;\;\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{\bar A}}_{12}} = \left[ {\begin{array}{*{20}{c}} 0&{0.1}\\ {0.1}&0 \end{array}} \right] $
    $ \Delta \overline{\boldsymbol{A}}_{12}(k)=\left[\begin{array}{cc}{0} & {0} \\ {0} & {0.05 \cos (k)}\end{array}\right] \quad \overline{\boldsymbol{A}}_{13}=\left[\begin{array}{cc}{0.1} & {0} \\ {0} & {0.1}\end{array}\right] $
    $ \Delta \overline{\boldsymbol{A}}_{13}(k)=\left[\begin{array}{ccc}{0} & {0.04 \cos (k)} \\ {0} & {0.04 \sin (k)}\end{array}\right] \quad \overline{\boldsymbol{A}}_{2}=\left[\begin{array}{cc}{0} & {1} \\ {0.5} & {-1.4}\end{array}\right] $
    $ \Delta \overline{\boldsymbol{A}}_{2}(k)=\left[\begin{array}{cc}{0} & {0.09 \cos (k)} \\ {0.09 \sin (k)} & {0}\end{array}\right] \quad \overline{\boldsymbol{B}}_{2}=\left[\begin{array}{l}{0} \\ {2}\end{array}\right] $
    $ {\mathit{\boldsymbol{\overline A}} _{21}} = \left[ {\begin{array}{*{20}{c}} {0.05}&0\\ 0&{0.05} \end{array}} \right]\;\;\;\;\;\;\;\Delta {\mathit{\boldsymbol{\overline A}} _{21}}(k) = \left[ {\begin{array}{*{20}{c}} 0&{0.04\cos (k)}\\ 0&0 \end{array}} \right] $
    $ \overline{\boldsymbol{A}}_{23}=\left[\begin{array}{cc}{0} & {0.09} \\ {0.09} & {0}\end{array}\right] \quad \Delta \overline{\boldsymbol{A}}_{23}(k)=\left[\begin{array}{ccc}{0} & {0.05 \cos (k)} \\ {0} & {0.05 \sin (k)}\end{array}\right] $
    $ {\mathit{\boldsymbol{\overline A}} _3} = \left[ {\begin{array}{*{20}{c}} { - 0.5}&0\\ 0&{0.3} \end{array}} \right]\quad \Delta {\mathit{\boldsymbol{\overline A}} _3}(k) = \left[ {\begin{array}{*{20}{c}} 0&{0.1\cos \left( k \right)}\\ 0&{0.2\sin \left( k \right)} \end{array}} \right] $
    $ \overline{\boldsymbol{B}}_{3}=\left[\begin{array}{l}{0} \\ {3}\end{array}\right] \quad \overline{\boldsymbol{A}}_{31}=\left[\begin{array}{cc}{0} & {0.1} \\ {0.02} & {0.1}\end{array}\right] $
    $ \Delta \overline{\boldsymbol{A}}_{31}(k)=\left[\begin{array}{cc}{0} & {0.04 \cos (k)} \\ {0.04 \sin (k)} & {0}\end{array}\right] \quad \overline{\boldsymbol{A}}_{32}=\left[\begin{array}{cc}{0} & {0} \\ {0.1} & {0.1}\end{array}\right] $
    $ \Delta {\mathit{\boldsymbol{\overline A}} _{32}}(k) = \left[ {\begin{array}{*{20}{c}} 0&{0.04\cos (k)}\\ {0.04\sin (k)}&0 \end{array}} \right]\quad {\mathit{\boldsymbol{z}}_i}(k) = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{z}}_{i1}^{\rm{T}}(k)}&{\mathit{\boldsymbol{z}}_{i2}^{\rm{T}}(k){]^{\rm{T}}}} \end{array}} \right. $
    $ {\mathit{\boldsymbol{u}}_i}(k) = {\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{u}}_{i1}^{\rm{T}}(k)}&{\mathit{\boldsymbol{u}}_{i2}^{\rm{T}}(k)} \end{array}} \right]^{\rm{T}}}\;\;\;\;\;\;\;\;\;\;i = 1,2,3. $

    时滞和初始条件如下

    $ {h_{ij}} = j,j = 1,2,3,\quad \left[ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{z}}_1^{\rm{T}}(0)}&{\mathit{\boldsymbol{z}}_2^{\rm{T}}(0)}&{\mathit{\boldsymbol{z}}_3^{\rm{T}}(0)}&{\mathit{\boldsymbol{z}}_4^{\rm{T}}(0){]^{\rm{T}}} = {{\left[ {\begin{array}{*{20}{c}} 1&{ - 0.5}&2&1 \end{array}} \right]}^{\rm{T}}}} \end{array}} \right. $

    并有

    $ {L_{12}} = 0.2,{\mathit{L}_{22}} = 0.09,{L_{32}} = 0.2,{M_{122}} = 0.05,{M_{132}} = 0.04,\\{M_{212}} = 0,{M_{232}} = 0.05,{M_{312}} = 0.04, $
    $ {M_{322}} = 0.04,{D_1} = {D_2} = {D_3} = {D_{12d}} = {D_{13d}} = {D_{21d}} = {D_{23d}} = {D_{31d}} = {D_{32d}},F(k) = \cos (k) $

    选取$c_{1}=\left[\begin{array}{ll}{0} & {1}\end{array}\right], c_{2}=\left[\begin{array}{ll}{0} & {\frac{1}{2}}\end{array}\right], \boldsymbol{c}_{3}=\left[\begin{array}{ll}{0} & {\frac{1}{3}}\end{array}\right]$,得到$E_{1 a}=0.4, E_{2 a}=0.09, E_{3 a}=0.1, E_{12 \alpha d}=0$$E_{13 a d}=0.12, E_{21 a d}=0.04, E_{23 a d}=0.15, E_{31 a d}=0.04, E_{32 a d}=0.08$.

    设计变结构控制器(12),系统状态的仿真图如图 1-3.

    图 1 子系统1的状态曲线图
    图 2 子系统2的状态曲线图
    图 3 子系统3的状态曲线图

    图 1-3可看出,系统状态是渐近稳定的.

    4 结论

    本文研究了具有非匹配不确定性的离散时滞大系统的变结构控制问题.设计了系统的变结构控制器,使系统状态在有限时间内到达并保持在滑模面上.与传统变结构控制设计相比,克服了不确定性满足匹配条件的缺点.最后,仿真算例说明了该方法的有效性.

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    Variable Structure Control for a Class of Uncertain Discrete-Time Complex Systems with Unmatched Uncertainty and Delays
    YAO He-jun , YAN Qian-tai , YANG Heng     
    School of Mathematics and Statistics, Anyang Normal University, Anyang Henan 455000, China
    Abstract: An asymptotically stable sliding mode surface has been designed for a class of uncertain discrete-time complex large-scale systems with unmatched uncertainty and delays by using the pole assignment approach. The variable structure controller is designed subsequently. The matched uncertainty have not been needed in this design approach. Finally, a numerical example is given to illustrate the feasibility of the proposed design approach.
    Key words: variable structure control    time-delay systems    discrete    unmatched uncertainty    
    X