A Detection Algorithm for Bifurcations in Dynamical Systems Using Reduced Order Models

Finite element or finite volume discretizations of distributed parameter systems (DPS) typically lead to high order finite dimensional systems. Model approximation is then an important first step towards the construction of optimal controllers. However, model reduction methods hardly take model uncertainties and parameter variations into account. As such, reduced order models are not well equipped when uncertain system parameters vary in time. This is particularly true when system behavior does not depend continuously on the parameters. It is shown in this paper that the performance of reduced order models inferred from Galerkin projections and proper orthogonal decompositions can deteriorate considerable when system parameters vary over bifurcation points. Motivated by these observations, we propose a detection mechanism based on reduced order models and proper orthogonal decompositions that allows to characterize the influence of parameter variations around a bifurcation value. for this, a hybrid model structure is proposed. The ideas are applied on the example of a tubular reactor. In particular, this paper discusses the difficulties in approximating the transition from extinction to ignited state in a tubular reactor.