Parameter-Dependent LMIs in Robust Analysis: Characterization of Homogeneous Polynomially Parameter-Dependent Solutions Via LMI Relaxations
Parameter-Dependent LMIs in Robust Analysis: Characterization of Homogeneous Polynomially Parameter-Dependent Solutions Via LMI Relaxations
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DOI：
10.1109/tac.2007.900848
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发表时间：
2007-07
影响因子：
6.8
通讯作者：
R. Oliveira;P. Peres
中科院分区：
文献类型：
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作者：
R. Oliveira;P. Peres
利用不确定性表示的代数性质，通过参数依赖线性矩阵不等式（PD-LMI）条件，研究了多面体域上不确定线性定常系统的鲁棒稳定性.一个系统的程序来构建一个家庭的有限维LMI松弛。鲁棒稳定性的评估通过存在的李雅普诺夫函数，更具体地说，一个齐次多项式参数依赖李雅普诺夫（HPPDL）函数的任意程度。对于给定的次数，如果存在HPPDL解，基于真实的代数性质的一系列松弛为HPPDL函数的存在性提供了充分的线性矩阵不等式（LMI）条件，该条件具有精度增加和决策变量数目不变的优点，且趋于必要性.或者，如果存在度的HPPDL解，则增加变量的数量和LMI的数量的松弛序列将提供更大度的HPPDL解。所提出的方法可以适用于确定齐次参数依赖矩阵解决方案的各种PD-LMI通过转换的无限维LMI问题描述的不确定参数属于单位单纯形在一系列的有限维LMI条件收敛到一个齐次多项式参数依赖解决方案的存在的必要条件的任意程度。说明性的例子表明所提出的条件相比，从文献中的其他方法的有效性。
This note investigates the robust stability of uncertain linear time-invariant systems in polytopic domains by means of parameter-dependent linear matrix inequality (PD-LMI) conditions, exploiting some algebraic properties provided by the uncertainty representation. A systematic procedure to construct a family of finite-dimensional LMI relaxations is provided. The robust stability is assessed by means of the existence of a Lyapunov function, more specifically, a homogeneous polynomially parameter-dependent Lyapunov (HPPDL) function of arbitrary degree. For a given degree , if an HPPDL solution exists, a sequence of relaxations based on real algebraic properties provides sufficient LMI conditions of increasing precision and constant number of decision variables for the existence of an HPPDL function which tend to the necessity. Alternatively, if an HPPDL solution of degree exists, a sequence of relaxations which increases the number of variables and the number of LMIs will provide an HPPDL solution of larger degree. The method proposed can be applied to determine homogeneous parameter-dependent matrix solutions to a wide variety of PD-LMIs by transforming the infinite-dimensional LMI problem described in terms of uncertain parameters belonging to the unit simplex in a sequence of finite-dimensional LMI conditions which converges to the necessary conditions for the existence of a homogeneous polynomially parameter-dependent solution of arbitrary degree. Illustrative examples show the efficacy of the proposed conditions when compared with other methods from the literature.