Delay Margin Analysis of Uncertain Linear Control Systems Using Probabilistic μ $$ \mu $$

Monte Carlo simulations have long been a widely used method in the industry for control system validation. They provide an accurate probability measure for sufficiently frequent phenomena but are often time-consuming and may fail to detect very rare events. Conversely, deterministic techniques such as $\mu$ or IQC-based analysis allow fast calculation of worst-case stability margins and performance levels, but in the absence of a probabilistic framework, a control system may be invalidated on the basis of extremely rare events. Probabilistic $\mu$-analysis has therefore been studied since the 1990s to bridge this analysis gap by focusing on rare but nonetheless possible situations that may threaten system integrity. The solution adopted in this paper implements a branch-and-bound algorithm to explore the whole uncertainty domain by dividing it into smaller and smaller subsets. At each step, sufficient conditions involving $\mu$ upper bound computations are used to check whether a given requirement–related to the delay margin in the present case–is satisfied or violated on the whole considered subset. Guaranteed bounds on the exact probability of delay margin satisfaction or violation are then obtained, based on the probability distributions of the uncertain parameters. The difficulty here arises from the exponential term $e^{-\tau s}$ classically used to represent a delay $\tau$, which cannot be directly translated into the Linear Fractional Representation (LFR) framework imposed by $\mu$-analysis. Two different approaches are proposed and compared in this paper to replace the set of delays $e^{-\tau s},\tau \in \left[0\varphi \right]$. First, an equivalent representation using a rational function with unit gain and phase variations that exactly cover those of the original delays, resulting in an LFR with frequency-dependent uncertainty bounds. Then, Padé approximations, whose order is chosen to handle the trade-off between conservatism and complexity. A constructive way to derive minimal LFR from Padé approximations of any order is also provided as an additional contribution. The whole method is first assessed on a simple benchmark, and its applicability to realistic problems with a larger number of states and uncertainties is then demonstrated.