(508f) Computing Fuzzy Trajectories for Nonlinear Dynamic Systems

Nonlinear dynamic systems arise in many areas of process modeling and analysis. In many situations, model parameters and initial conditions are imprecise and contain confidence bounds, which need to be propagated throughout calculations. In this presentation, we consider uncertainty propagation over nonlinear dynamic systems with fuzzy set inputs.

Based on Zadeh's possibility theory (Zadeh, 1999), fuzzy sets are mathematical constructs for representing uncertain information. Fuzzy sets are characterized by a membership function μ, interpreted as the compatibility of the set's components with the observable information, or how likely a certain crisp value is to capture the real behavior of the system. Fuzzy sets are not to be confused with probability distributions of information, although possibility and probability are weakly connected by the possibility/probability consistency principle (PPCP) (Gupta, 1993).

In this presentation, we demonstrate a general method for computing verified fuzzy trajectories of nonlinear dynamic systems with fuzzy parameters and/or initial conditions. The uncertainty in the state variable transient behavior over a given time horizon is calculated as a sequence of fuzzy set outputs; the collection of these fuzzy output snapshots is termed a fuzzy trajectory. The notion of fuzzy time series, or trajectories, has been described by Möller and Reuter (2008), who use this information for stochastic analysis of uncertain structural responses in civil engineering problems. Our emphasis is the development of a rigorous approach for the actual computation of fuzzy trajectories, a problem that has not been explored elsewhere. The technique is based on a method (Lin and Stadtherr, 2007) for the verified solution of parametric ODEs (implemented in the code VSPODE) with subsequent coupling to fuzzy arithmetic operations on the resulting Taylor models of the outputs. VSPODE uses an interval Taylor series (ITS) to represent dependence on time, and uses Taylor models (TM) to represent dependence on uncertain quantities.

We applied our approach to several nonlinear bioreactor models (e.g., Monod and Haldane kinetics), obtaining fuzzy trajectories for the outputs of interest. We assumed simple symmetric fuzzy sets inputs for the specific growth rate (μ), the Monod constant (K_S), the inhibition constant (K_I), and/or the initial concentration of biomass (X₀), and we monitored the evolution of each fuzzy state variable over a given time horizon. One major advantage of this method is that fuzzy outputs incorporate all combinations of crisp and different-likelihood uncertainties concomitantly, thus allowing for quick decisions in applications where safety regimes are important. Moreover, this technique borrows from the flexibility of VSPODE, and as such may be used in various contexts, for example sensitivity studies, control, etc.

References

Gupta, C., CP (1993). A note on the transformation of possibilistic information into probabilistic information for investment decisions. Fuzzy Sets and Systems, 56, 175–182.

Lin, Y. & Stadtherr, M. A. (2007). Validated solutions of initial value problems for parametric ODEs. Applied Numerical Mathematics, 57, 1145–1162.

Möller, B. & Reuter, U. (2008). Prediction of uncertain structural responses using fuzzy time series. Computers & Structures, 86, 1123–1139.

Zadeh, L. (1999). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 100, 9–34.