A detection algorithm for bifurcations in dynamical systems using reduced order models

Abstract 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.

[1]  P. Astrid,et al.  Reduction of process simulation models : a proper orthogonal decomposition approach , 2004 .

[2]  Karlene A. Hoo,et al.  Low-order model identification for implementable control solutions of distributed parameter systems , 2002 .

[3]  W. Harmon Ray,et al.  The Bifurcation Behavior of Tubular Reactors. , 1982 .

[4]  Martin Mönnigmann,et al.  A method for robustness analysis of controlled nonlinear systems , 2004 .

[5]  Hanns Hofmann,et al.  Modeling of chemical reactors — XVII Steady state axial heat and mass transfer in tubular reactors Numerical investigation of multiplicity , 1970 .

[6]  Steffen Heinze.  Traveling waves in combustion processes with complex chemical networks , 1987 .

[7]  S. Shvartsman,et al.  Nonlinear model reduction for control of distributed systems: A computer-assisted study , 1998 .

[8]  R. Aris.  On stability criteria of chemical reaction engineering , 1969 .

[9]  Lucia Russo,et al.  On POD reduced models of tubular reactor with periodic regimes , 2008, Comput. Chem. Eng..

[10]  Siep Weiland,et al.  Singular value decompositions and low rank approximations of multi-linear functionals , 2007, 2007 46th IEEE Conference on Decision and Control.

[11]  P. Holmes,et al.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 1996 .

[12]  Wolfgang Marquardt,et al.  A GREY-BOX MODELING APPROACH FOR THE REDUCTION OF NONLINEAR SYSTEMS 1 1This work has been supported by the European Union within the Marie-Curie Training Network PROMATCH under the grant number MRTN-CT-2004-512441. , 2007 .