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(M, beta)-Stability of Positive Linear Systems

机译：（M，beta）-正线性系统的稳定性

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摘要

The main purpose of this work is to show that the Perron-Frobenius eigenstructure of a positive linear system is involved not only in the characterization of long-term behavior (for which well-known results are available) but also in the characterization of short-term or transient behavior. We address the analysis of the short-term behavior by the help of the "(M,beta)-stability" concept introduced in literature for general classes of dynamics. Our paper exploits this concept relative to Holder vector P-norms, 1 <= p <= infinity, adequately weighted by scaling operators, focusing on positive linear systems. Given an asymptotically stable positive linear system, for each 1 <= p <= infinity, we prove the existence of a scaling operator (built from the right and left Perron-Frobenius eigenvectors, with concrete expressions depending on p) that ensures the best possible values for the parameters M and beta, corresponding to an "ideal" short-term (transient) behavior. We provide results that cover both discrete- and continuous-time dynamics. Our analysis also captures the differences between the cases where the system dynamics is defined by matrices irreducible and reducible, respectively. The theoretical developments are applied to the practical study of the short-term behavior for two positive linear systems already discussed in literature by other authors.
机译：这项工作的主要目的是证明正线性系统的Perron-Frobenius本征结构不仅涉及长期行为的表征（可获得著名的结果），而且还涉及短期行为的表征。长期或短暂行为。我们借助一般动力学类别的文献中引入的“（M，β）-稳定性”概念来解决短期行为的分析问题。本文采用相对于Holder向量P范数（1 <= p <=无限）的这一概念，并通过缩放算符对它们进行了适当加权，重点放在正线性系统上。给定一个渐近稳定的正线性系统，对于每个1 <= p <=无穷大，我们证明了存在比例缩放算子（由左右Perron-Frobenius特征向量构建，具体表达式取决于p）可以确保最大可能参数M和beta的值，对应于“理想”的短期（瞬态）行为。我们提供的结果涵盖了离散时间和连续时间动力学。我们的分析还捕获了系统动力学分别由不可约矩阵和可约矩阵定义的情况之间的差异。理论发展被应用于其他作者已经在文献中讨论过的两个正线性系统的短期行为的实践研究。

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