Mixed robustness: analysis of systems with uncertain deterministic and random parameters by the example of linear systems

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.