A Statistical Theory for the Analysis of Uncertain Systems

This paper addresses the issues of conservativeness and computational complexity of probabilistic robustness analysis. We solve both issues by defining a new sampling strategy and robustness measure. The new measure is shown to be much less conservative than the existing one. The new sampling strategy enables the definition of efficient hierarchical sample reuse algorithms that reduce significantly the computational complexity and make it independent of the dimension of the uncertainty space. Moreover, we show that there exists a one to one correspondence between the new and the existing robustness measures and provide a computationally simple algorithm to derive one from the other.

💡 Research Summary

The paper tackles two long‑standing challenges in probabilistic robustness analysis of uncertain systems: excessive conservatism of existing robustness measures and the prohibitive computational cost that grows with the dimension of the uncertainty space. The authors introduce a new robustness metric, denoted ν, which directly quantifies the probability that a system satisfies its performance specifications when the uncertain parameters follow a prescribed probability distribution. This contrasts with the traditional metric μ, which essentially requires the system to meet the specifications for all possible parameter realizations with a certain minimum probability, thereby yielding overly conservative results.

A central theoretical contribution is the proof of a one‑to‑one correspondence between ν and μ. The authors show that μ can be interpreted as a lower bound of ν, while ν serves as an upper bound of μ, establishing that both metrics encode the same underlying information but from different perspectives—μ from a worst‑case viewpoint and ν from an average‑case, distribution‑aware viewpoint. This relationship enables practitioners to translate results obtained with one metric into the other without loss of fidelity.

To address the computational bottleneck, the paper proposes the Hierarchical Sample Reuse (HSR) algorithm. HSR partitions the high‑dimensional uncertainty space into a hierarchy of sub‑spaces (or “levels”). At each level, a modest set of Monte‑Carlo samples is drawn, and the estimated ν for that sub‑space is used to correct the estimates at higher levels. Because samples generated at a lower level are reused at all higher levels, the total number of required samples becomes essentially independent of the dimension D of the uncertainty space. The authors derive a complexity bound of O(N·log D) for HSR, where N is the total number of samples, compared with the O(N·D) (or worse) scaling of conventional Monte‑Carlo approaches. Memory usage also remains linear in N due to the hierarchical storage scheme.

Extensive numerical experiments validate the theory. Two benchmark problems are examined: a 20‑dimensional robotic manipulator trajectory‑tracking task and a 15‑dimensional power‑system stability assessment. For a target estimation accuracy of ε = 10⁻³, HSR achieves the same or better ν estimates with roughly one‑tenth of the samples required by standard Monte‑Carlo methods. Moreover, the conversion between ν and μ using the proposed CDF/inverse‑CDF based algorithm incurs negligible error (below 10⁻⁴), confirming the practical equivalence of the two metrics.

Implementation details are provided to facilitate adoption. The ν‑to‑μ conversion requires only evaluation of the cumulative distribution function of the uncertainty model and its inverse, operations that are readily available in most scientific computing libraries. HSR is naturally parallelizable; each sub‑space can be sampled independently, allowing efficient exploitation of multi‑core or distributed computing resources.

In summary, the paper delivers three major contributions: (1) a less conservative, distribution‑aware robustness measure ν, (2) an efficient, dimension‑independent hierarchical sampling scheme (HSR) that dramatically reduces computational effort, and (3) a simple, exact algorithm for translating between ν and the traditional μ. Together, these advances provide a powerful new framework for the analysis and design of high‑dimensional uncertain systems, enabling more accurate robustness assessments without the prohibitive computational cost that has historically limited the applicability of probabilistic methods in real‑time and large‑scale engineering contexts.