Rigorous probability bounds analysis for nonlinear process dynamics

Mark A. Stadtherr, Department of Chemical and Biomolecular Engineering, University of Notre Dame

29 October 2009 at 10:30

Location: JHE 326H

Models of process dynamics often involve uncertainties in parameters, inputs and/or initial states. Determining how these uncertainties propagate through a model to affect its outputs, and doing so rigorously, can be a very challenging problem, especially for nonlinear systems. The problem is further complicated by the fact that the probability distributions describing the uncertainties may not be known precisely, if they are known at all. If there is no known probability distribution for an uncertain quantity, but only bounds, then the uncertainty can be modeled using an interval. If some knowledge of the probability distribution is available, but it is imprecise, then this can be modeled using a probability box (p-box), which provides upper and lower bounds on the cumulative probability distribution function for the uncertain quantity. We will consider both of these cases, but concentrate on the latter case, in which uncertain quantities in the process model are characterized by imprecise probabilities represented by p-boxes.

A common approach for studying the effect of uncertainties in this and other contexts is the use of repeated sampling (e.g., Monte Carlo methods). In general, however, it is not possible to investigate the complete space of uncertainties in a finite number of samples. Thus, sampling methods such as Monte Carlo cannot produce rigorous bounds, as they may fail to capture all system behaviors, especially when there are nonlinearities. Sampling methods have other limitations as well, especially in dealing with uncertainties that have unknown dependencies or that cannot be characterized by a precise probability distribution. We will describe here a much different approach, not based on sampling, for the analysis of uncertainty in nonlinear dynamic systems. Given interval bounds on the uncertain quantities, this approach, which makes use of Taylor models, can be employed to rigorously bound all possible trajectories of an initial value problem for an uncertain dynamic model. Furthermore, given bounds (p-boxes) on the probability distributions of the uncertain quantities, this approach can be used to rigorously bound the probability distributions of model outputs at specified points in time. Thus, we can obtain rigorous bounds on the probabilities that desired outcomes are achieved. Examples involving reactor dynamics will be used to demonstrate the approach, with comparison to results obtained from Monte Carlo simulations.

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