Explicit Uncertainty Quantification for Systems with Parametric Uncertainty

Systems with known statistical models of parametric uncertainty are studied. It is assumed that model parameters with uncertain values have known probability density functions and this knowledge can be used to estimate control metrics of interest. One benefit to this approach is that it allows the closed-loop control design to be robust to the most probable parametric variations, while still providing control metric guarantees that allow for less conservative designs. Conversely, control designs that satisfy traditional robustness measures, such as gain and phase margins, require the system to be robust to worst case model variations, resulting in more conservative designs. In this paper, the control metric of interest is the probability of closed-loop instability. The expectation formula is used to directly compute this metric, which involves multi-dimensional integration over the joint distribution for the parametric uncertainty models. With simplistic integration schemes, this approach will have difficulty scaling to real-world, higher dimensional dynamic models with large numbers of uncertain parameters. Current approaches that address this problem include Monte Carlo methods, which can require millions of time-based simulation runs to provide the data needed to estimate control metric probabilities. Further, the Monte Carlo approach randomly samples the parametric distributions, which can potentially miss problematic combinations of parameter variations. Recent open-source integration software packages offer very fast and highly accurate computation of multi-dimensional integral formulas. Using these tools, the expectation formula can be directly evaluated and shows promise that real-world systems with parametric uncertainty models can be studied in this manner. Numerical results for two case studies are compared to Monte Carlo methods for determining the probability of instability.