Abstract
The desire to increase fuel efficiency and reduce carbon dioxide emissions of
vehicles has led to an increased use of vehicles equipped with leanburn engines,
such as diesel and leanburn gasoline engines. This type of engine uses excess
oxygen when compared to the amount required to stoichiometrically combust
fuel. A serious disadvantage of this type of engine, however, is the increased
emission levels of nitrogen oxides. These airborne pollutants are toxic for hu
mans when inhaled and are harmful to the environment. They are therefore
subject to increasingly strict environmental regulations, which motivate the au
tomotive industry to innovate in order to decrease the nitrogen oxide emission
levels of new engines and vehicle designs. An important method to accomplish
this is the aftertreatment of exhaust gas with the help of catalytic converters.
Owing to the composition of the exhaust gas of leanburn engines, however, the
reduction of nitrogen oxide emission obtained with standard threeway catalysts
(TWC) is nevertheless insufficient.
One type of catalytic converter that might be used to reduce nitrogen oxide
emissions of vehicles with leanburn engines below the regulated maximum is
the Lean NOx Trap (LNT). In addition to the platinum used in TWC’s, LNT
catalytic converters use barium as active catalytic material. This gives the
LNT the ability to store a certain amount of nitrogen oxides on its catalytic
surface. The operating principle of a LNT catalyst is the temporary storage
of the nitrogen oxides emitted during a time period of lean operation and a
brief switchover of the engine to rich operation which produces exhaust gas
containing excess reductants the reduce the stored nitrogen oxides to nitrogen,
thereby regenerating the trap. This periodic operation requires active control
of the engine to determine how and at which time instant to switch to rich
operation. Modelbased design methods can be used to derive estimation and
control algorithms for LNT in a systematic manner. These methods require a
dynamic model of the LNT of limited complexity, which describes the dynamics
of the LNT that are essential to estimation and control and which is referred to
as a controloriented LNT model.
Such a model can be obtained by two routes. The first route is to empirically
determine the structure of the controloriented LNT model from the observed
behavior and tune the parameters of the model using vehicle test data. This
approach has two disadvantages: it requires knowledge of the LNT dynamics to
determine the right model structure and the vehicle tests required to tune the
parameters are expensive and timeconsuming.
There is an alternative route provided we have a LNT model at our dis
posal based on physical laws, physical constants and the reaction constants of
the elementary reaction steps which determine the kinetics of the chemicals
reactions in the LNT. This type of model is referred to as a firstprinciples mo
del. Such a model can be obtained by combining existing physical knowledge
with laboratory tests designed to determine the kinetics of elementary reacti
on steps. This alternative route means systematically deriving controloriented
models from the firstprinciple model using model reduction methods. These
methods determine which parts of the firstprinciple model are relevant for esti
mation and control and help to arrive at a controloriented model by reducing
the firstprinciple model in such a way that it retains only these parts.
This thesis examines the reduction of the complexity of LNT models using
model reduction methods. First, the relation between detailed firstprinciple
models and existing controloriented models is explored. The essential simplifi
cation steps that link both model types are 1) the simplification of the kinetics
and 2) the compression of the information on the axial spatial concentration
profiles for gasphase species and species on the catalyst surface into a limited
number of states. A link can be established between parts of both models using
a remodeling approach in which assumptions with a physical interpretation are
used to mathematically derive the controloriented model from the firstprinciple
model. The existing controloriented model also contains relations that are em
pirical, however, and these cannot directly be related to the firstprinciple model
using a remodeling approach.
The compression of the axial spatial concentration profiles was approached
using existing model reduction methods known as projection methods. Two
types of projection methods were used. The Balanced Truncation method uses
an energy norm to quantify the energy transfer from model inputs to model
states and from model states to model outputs. States which are associated
with limited energy transfer from the input to the output are then truncated,
removing them from the model. The Proper Orthogonal Decomposition (POD)
method exploits correlations between the states of a dynamic model that occur
if the model is simulated to redefine the statespace so that only states with
limited correlation are retained in the model. It was found, however, that these
methods perform poorly for models that are similar to the LNT model. The
reason for this poor performance is that the dynamics of the LNT model are
characterized by nonlinearity of the kinetics, the occurrence of traveling wave
phenomena and large changes in time in the magnitude of species concentrations.
One contribution of this thesis is the adaptation of POD algorithms to impro
ve their performance for systems which exhibit wave phenomena on a bounded
domain, such as the LNT catalyst model. This was accomplished by inclu
ding phase information in the basisfunctions used in the POD method. Data
correlation methods and extrapolation were used to gather information on the
dominant features of the traveling wave from simulation data. This information
was then used to construct the basisfunctions.
Next, we have considered simplification of the kinetic part of the LNT mo
del. A large number of different schemes exist to accomplish simplification of
complex kinetic models. In this thesis lumping methods are examined, in parti
cular an approach whereby QuasiSteadyState (QSS) and Partial Equilibrium
(PE) assumptions are used to lump species. The effect of lumping on mass
conservation and on the positive invariance of kinetic models is discussed. A
requirement for the derivation of reduced order models using lumping methods
the definition of an expansion relation. We determine under which conditions
such a relation exists. Furthermore, in this thesis we take two steps towards a
practical implementation of lumping methods. First, a method is developed to
retain positive invariance of species concentrations in lumped models, when the
expansion relation is approximated using polynomials. This method is based on
existing results of function approximation using SumofSquares polynomials.
Next, a greedytype algorithm is proposed to select QSS and PE relations so
that the expansion relation can be obtained as the symbolic solution to a set of
algebraic equations.
A possible application of LNT models is the goaloriented optimization of
LNT operation. The application of model reduction as a part of goaloriented
optimization procedure is investigated. The techniques used for simplification of
kinetic mechanisms are extended to spatially distributed models and applied to
a testcase model similar to LNT models. It was found that application of model
reduction methods during optimization is possible although it requires careful
tuning of the model reduction accuracy versus the accuracy of the optimization
algorithm.
This thesis finally illustrates the difficulties with a straightforward applica
tion of existing model reduction methods to LNT models. The essential part
of LNT models are convectionreaction equations required to model changes in
species concentrations along the catalyst axis. A number of steps have been
taken towards making these methods more suitable in practice for reduction
of convectionreaction equations. However, a number of fundamental theore
tical problems remain, such as error control, and closely related to this, the
preservation of stability and positive invariance of concentrations.
Original language  English 

Qualification  Doctor of Philosophy 
Awarding Institution 

Supervisors/Advisors 

Award date  15 Sep 2008 
Place of Publication  Eindhoven 
Publisher  
Print ISBNs  9789038613765 
DOIs  
Publication status  Published  2008 