Abstract

Robust stability conditions are established for a family of linear discrete-time systems subjected to uncertainties. The traditional approach, which involves the construction of a common quadratic Lyapunov function for the entire family of systems with uncertainty, often leads to the problem of conservatism. In this connection, constructing parametric quadratic Lyapunov functions seems promising. The main tools of the proposed approach are the apparatus of linear matrix inequalities and a modification, presented here, of the well-known Petersen’s lemma. A simple approach to finding the robust quadratic stability radius of the family in question is proposed in the paper as well. The corresponding optimization problems have the form of semidefinite programming and one-dimensional minimization, which can be easily solved numerically. The efficiency of our approach is demonstrated via a numerical example. The results obtained can be generalized to design problems for linear discrete-time systems subjected to uncertainties, to other robust statements, and to the case of exogenous disturbances.

中文翻译：

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具有不确定性的线性离散时间系统族的鲁棒稳定性条件

摘要

鲁棒稳定性条件是为一系列受到不确定性影响的线性离散时间系统建立的。传统方法涉及为具有不确定性的整个系统族构建公共二次李雅普诺夫函数，通常会导致保守性问题。在这方面，构造参数二次李雅普诺夫函数似乎很有希望。所提出的方法的主要工具是线性矩阵不等式的装置和这里介绍的著名彼得森引理的修改。论文中还提出了一种简单的方法来寻找相关族的稳健二次稳定性半径。相应的优化问题有半定规划和一维极小化的形式，很容易用数值求解。我们的方法的效率通过一个数值例子来证明。获得的结果可以推广到受不确定性影响的线性离散时间系统的设计问题、其他鲁棒性陈述以及外生扰动的情况。