Mixed robustness: Analysis of systems with uncertain deterministic and random parameters using the example of linear systems

TL;DR

This paper introduces the concept of mixed robustness for linear systems with both deterministic and random uncertainties, proposing new methods for stability analysis that bridge classical deterministic and probabilistic approaches.

Contribution

It defines mixed parametric robustness, develops a graphical stability estimation method, and generalizes robustness analysis to systems with combined uncertainties.

Findings

Proposes a new mixed robustness framework for systems with combined uncertainties.

Introduces a graphical approach for stability probability estimation.

Provides examples and generalizations to other system types.

Abstract

Robustness of linear systems with constant coefficients is considered. There exist methods and tools for analyzing the stability of systems with random or deterministic uncertainties. At the same time, there are no approaches for the analysis of systems containing both types of parametric uncertainty. The types of robustness are reviewed and new type of "mixed parametric robustness" is introduced. It includes several variations. The proposed formulations of mixed robustness problems can be considered as intermediate type between the classical deterministic and probabilistic approaches to robustness. Several cases are listed in which the tasks are easily solved. In general, tests of the stability of robust systems using the scenario approach are applicable, but these tests can be computationally complex. To calculate the desired stability probability, a simple graphical approach based on a robust D-partition is proposed. This method is suitable for the case of a small number of random parameters. The final estimate of the probability of stability is calculated in a deterministic way and can be found with arbitrary precision. Approximate ways of solving the assigned tasks are described. Examples and generalization of mixed robustness to other types of systems are given.