H-Analysis Based Algorithms for Probabilistic Finite-Frequency H2-Performance Analysis of Uncertain Linear Systems

Current industrial validation and verification (V & V) activities mostly rely on time-consuming Monte Carlo simulations. They are able to quantify the probability of sufficiently frequent phenomena, but they may fail in detecting rare but critical combinations of uncertainties. In recent years, deterministic model-based worst-case analysis methods have reached a good level of maturity. Without the need of simulations, they explore the entire space of all possible combinations of uncertain parameters, and provide guaranteed mathematical bounds on robust stability margins and worst-case performance levels. Identified problematic parameter configurations can then be used to guide the Monte Carlo campaigns and shorten the V & V process. But a limitation of these deterministic methods is that the probability of occurrence of critical parameter combinations is not measured, which means that a control system can be rejected based on an extremely unlikely scenario. Probabilistic µ-analysis has therefore been studied since the 1990s to bridge this analysis gap between traditional simulation-based and deterministic worst-case analysis techniques, by focusing on rare but nonetheless possible situations that may threaten system integrity. In this context, this paper introduces three novel µ-based probabilistic
$H$
2
-analysis algorithms, which provide guaranteed upper and lower bounds on the probability that an uncertain system violates a desired
$H$
2
-performance requirement on a fixed frequency range.