Uncertainty Propagation in Models for Dynamic Nonlinear Systems: Methods and Applications

In the context of engineering and science, nonlinear dynamic models often involve uncertain quantities. For example, in an initial value problem (IVP) described by a system of ordinary differential equations (ODEs), the initial conditions may be uncertain, and there may be uncertain parameters in the ODE model. Determining the effect of such uncertainties on the model outputs is clearly an important issue. To address this problem requires first that an appropriate representation of the uncertain quantities be chosen, and then that these be propagated through the nonlinear ODE model to determine the corresponding uncertainties in the model outputs. One approach for representing uncertainty is the use of fuzzy sets or fuzzy numbers. A new approach is described for the solution of nonlinear dynamic systems with parameters and/or initial states that are uncertain and represented by fuzzy sets or fuzzy numbers. Unlike current methods, which address this problem through the use of sampling techniques and do not account rigorously for the effect of the uncertain quantities, the new approach is not based on sampling and provides mathematically and computationally rigorous results. These goals are achieved through the use of explicit analytic representations (Taylor models) of state variable bounds in terms of the uncertain quantities. Examples are given that demonstrate the use of this new approach and its computational performance. Alternative representations of uncertainty (i.e., probability boxes), and related deterministic methods are also discussed. Applications of our approach are drawn from ecological, bioreactor, and political science modeling, where initial populations, interaction parameters, apparent reaction rates, and other control variables are uncertain, and may be translated into either fuzzy numbers, or probability boxes. When these deterministic tools are not sufficient, we implement additional stochastic methods (i.e., first- and second-order Monte Carlo) to study global sensitivity for design and optimization. Such is the case of the transient regime of an open-loop absorption-refrigeration system. This comprehensive model advances the state of the art of simulating ionic liquid-water refrigeration machines during startup and shutdown, and can be used to fine-tune operating conditions and desired thermophysical properties for the absorbent.