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目录摘要··········································································································1关键词·······································································································1Abstract······································································································1Keywords···································································································1引言··········································································································11LMI工具箱介绍和使用·············································································21．1系统描述·····················································································21．2信息检························································································31．3问题求解·····················································································31．4结果验证·····················································································32LMI基础知识·························································································32．1LMI的一般形式···········································································42．2描述LMI的相关术语·····································································42．3LMI应用的基本问题·····································································52．4LMI不等式问题···········································································52．5三类标准的LMI控制问题及其相应的求解器······································62．5．1可行性问题·······································································62．5．2具有线性矩阵不等式约束的一个线性目标函数的最小化问题·······62．5．3广义特征值最小化问题························································63LMI工具箱函数详解················································································83．1LMI工具箱函数列表·····································································83．2确定LMI系统的函数·····································································93．3对LMI变量的操作······································································103．4LMI求解器命令·········································································103．5结果验证和修改··········································································123．5．1结果验证········································································123．5．2结果修改········································································123．6LMI系统信息提取······································································124线性矩阵不等式在控制理论中的应用·························································134．1LMI用于控制问题常用的技术·······················································144．1．1矩阵操作技术··································································144．1．2LFT(LinearFractionTransformation)和多胞(polytopy)表示方法·····144．2在控制理论中的应用举例······························································144．3用线性矩阵不等式求解控制问题的实例···········································154．4基于LMI的鲁棒控制器设计····························································165结论和展望··························································································19致谢········································································································20参考文献··································································································211Matlab中的LMI工具箱的研究与使用摘要：由于内点算法可以利用计算机求解高阶矩阵不等式，使得线性矩阵不等式（LMI）在控制理论得以应用，因而线性矩阵不等式在控制系统分析、设计中扮演着越来越重的角色。在毕业设计中，我学习了LMI理论，Matlab中LMI工具箱的使用，研究了常见的控制问题与LMI关系以及其表达式，并研究了基于LMI方法的鲁棒控制器设计问题，推导了如何将鲁棒控制器设计问题转化为LMI形式，给出了通过求解LMI方程构造控制率的算法。LMI是一种较新的方法，可以在控制系统的许多领域得以应用，如鲁棒控制、非线性控制预测控制等有着巨大的潜力。关键词：线性矩阵不等式；LMI工具箱；控制系统TheresearchanduseoftheMatlbaLMItoolboxStudentmajoringinAutomationSUNPeng-kunTutorAbstract：Duetotheinteriorpointalgorithmcanusecomputertosolvehighordermatrixinequality,Itmakesthelinearmatrixinequality(LMI)inthecontroltheorytoapplicationandlinearmatrixinequality(LMI)playedmoreandmoreheavyroleinthecontrolsystemanalysisanddesign.Atthegraduationdesign,IstudiedtheLMItheory,intheuseofMatlabLMItoolbox,andstudiestheproblemofcommoncontrolwithLMIrelationsanditsexpression,andIstudiedtheproblemoftherobustcontoellerdesignthatbasedonlmimethod,howtotransformtherobustcontrollerdesignproblemintotheLMIform,andgivenbysolvingtheLMIequationoftectoniccontrolalgorithm.LMIisanewmethod,itcanbeusedinmanycontrolsystemfields,suchasrobustcontrol,nonlinearcontrolpredictivecontrol,andithasgreatpotential.Keywords:linearmatrixinequality;LMItoolbox;controlsystem;robustcontrol引言线性矩阵不等式（LMI）工具箱是求解一般线性矩阵不等式问题的一个高性能软件包。由于其面向结构的线性矩阵不等式表示方式，使得各种线性矩阵不等式能够以自然块矩阵的形式加以描述。一个线性矩阵不等式问题一旦确定，就可以通过调用适当的线2性矩阵不等式求解器来对这个问题进行数值求解。LMI工具箱提供了确定、处理和数值求解线性矩阵不等式的一些工具，它们主要用于：①以自然块矩阵形式来直接描述线性矩阵不等式；②获取关于现有的线性矩阵不等式系统的信息；③修改现有的线性矩阵不等式系统；④求解三个一般的线性矩阵不等式问题；⑤验证结果。在60年代，已经提出了线性矩阵不等式，但由于当时的线性矩阵不等式的算法还不够成熟。再加上求解量大，因而线性矩阵不等式在实际中未得到充分应用。近几年来，由于线性矩阵不等式的理论不断完善，求解算法也不断成熟，加上计算机的广泛应用，线性矩阵不等式的求解变得很方便，因此线性矩阵不等式在实际工程中尤其在控制工程理论中得到广泛的应用。由于用线性矩阵不等式求解控制理论中的问题是当今控制理论发展的一个重要方向，因此出现了许多计算机应用软件，其中以美国MathsWorks.Inc公司用C语言开发的MATLAB软件最为流行。在MATLAB5.1版本中，就增加了用于求解线性矩阵不等式的线性矩阵不等式控制工具