Risk Analysis in Robust Control -- Making the Case for Probabilistic Robust Control

TL;DR

This paper advocates for probabilistic robust control over worst-case methods, arguing that probabilistic approaches can be safer and more reliable by accounting for modeling uncertainties and inherent risks.

Contribution

It challenges the supremacy of worst-case robust control, demonstrating that probabilistic methods can better handle uncertainties and reduce risks in control system design.

Findings

Worst-case control may exclude feasible cases due to conservative assumptions.

Probabilistic control can lead to lower risk of failure compared to worst-case methods.

Probabilistic approaches are computationally feasible and potentially more reliable.

Abstract

This paper offers a critical view of the "worst-case" approach that is the cornerstone of robust control design. It is our contention that a blind acceptance of worst-case scenarios may lead to designs that are actually more dangerous than designs based on probabilistic techniques with a built-in risk factor. The real issue is one of modeling. If one accepts that no mathematical model of uncertainties is perfect then a probabilistic approach can lead to more reliable control even if it cannot guarantee stability for all possible cases. Our presentation is based on case analysis. We first establish that worst-case is not necessarily "all-encompassing." In fact, we show that for some uncertain control problems to have a conventional robust control solution it is necessary to make assumptions that leave out some feasible cases. Once we establish that point, we argue that it is not uncommon for the risk of unaccounted cases in worst-case design to be greater than that of the accepted risk in a probabilistic approach. With an example, we quantify the risks and show that worst-case can be significantly more risky. Finally, we join our analysis with existing results on computational complexity and probabilistic robustness to argue that the deterministic worst-case analysis is not necessarily the better tool.