Abstract

The robustness of linear systems with constant coefficients is considered. There are methods and tools for analyzing the stability of systems with random or deterministic uncertainties. At the same time, there are no approaches to the analysis of systems containing both types of parametric uncertainty. The article classifies the types of robustness and introduces a new type—“mixed parametric robustness”—which includes several variations. The proposed statements of mixed robustness problems can be viewed as intermediate versions between the classical deterministic and probabilistic approaches to robustness. Several cases are listed in which the problems are easy to solve. In the general case, stability tests based on the scenario approach can be applied to robust systems; however, these tests can be computationally costly. A simple graph-analytical approach based on robust \(D \)-decomposition (robust \(D \)-partition) is proposed to calculate the desired stability probability. This method is suitable for the case of a small number of random parameters. The final stability probability estimate is calculated in a deterministic way and can be found with arbitrary precision. Approximate methods for solving the above problems are described. Examples and a generalization of mixed robustness to other types of systems are given.

中文翻译：

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混合鲁棒性：具有不确定性和随机参数的系统的线性系统分析

摘要

考虑具有恒定系数的线性系统的鲁棒性。存在用于分析具有随机或确定性不确定性的系统的稳定性的方法和工具。同时，还没有分析包含两种类型参数不确定性的系统的方法。本文对鲁棒性的类型进行了分类，并介绍了一种新类型“混合参数鲁棒性”，其中包括多种变体。提出的关于混合鲁棒性问题的陈述可以看作是鲁棒性的经典确定性和概率方法之间的中间版本。列出了几种容易解决的问题。在一般情况下，基于场景方法的稳定性测试可以应用于健壮的系统。但是，这些测试的计算量很大。\（d \） -decomposition（鲁棒\（d \） -partition）提出了计算所需的稳定性概率。此方法适用于少量随机参数的情况。最终稳定性概率估计以确定性方式计算，并且可以任意精度找到。描述了解决上述问题的近似方法。给出了混合鲁棒性对其他类型系统的示例和概括。