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首页> 外文期刊>IEEE Transactions on Automatic Control >Parameter-Dependent LMIs in Robust Analysis: Characterization of Homogeneous Polynomially Parameter-Dependent Solutions Via LMI Relaxations
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Parameter-Dependent LMIs in Robust Analysis: Characterization of Homogeneous Polynomially Parameter-Dependent Solutions Via LMI Relaxations

机译：稳健分析中依赖参数的LMI：通过LMI弛豫描述均质多项式参数依赖的解决方案

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摘要

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 $g$, 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 $g$ 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.
机译：本文利用参数不确定性表示提供的一些代数性质，通过参数依赖线性矩阵不等式（PD-LMI）条件研究了多主题域中不确定线性时不变系统的鲁棒稳定性。提供了构建有限维LMI松弛族的系统过程。鲁棒稳定性是通过存在Lyapunov函数，更具体而言是任意阶的均一多项式参数相关Lyapunov（HPPDL）函数来评估的。对于给定的度数$ g $，如果存在HPPDL解决方案，则基于实数代数性质的一系列弛豫可提供足够的LMI条件，以提高精度和确定变量的恒定数目，这对于HPPDL函数的存在是必要的。或者，如果存在度数为$ g $的HPPDL解决方案，则一系列的松弛会增加变量的数量和LMI的数量，从而会提供较大度数的HPPDL解决方案。所提出的方法可以应用于通过变换有限维LMI条件序列中属于单位单纯形的不确定参数描述的无限维LMI问题，来确定各种PD-LMI的均质参数相关矩阵解。它收敛到存在任意阶均值多项式参数相关解的必要条件。说明性示例显示了与文献中的其他方法相比，所提出条件的功效。

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