In order to tune the parameters of the model, one or more different experiments or tests available are employed. Moreover, for each experiment or test, there are usually multiple emphresponses, e.g. measurements of various species concentrations at the catalyst outlet, temperature at several points of the monolithic converter etc. Measured responses are usually available as a function of time for each experiment. Modeling provides an estimation for each one of the measured responses. The computation for each response depends on the values of the tunable parameter of the model. The tuning of the model requires that the tunable parameters be fitted in order to minimize the error between available measurements and the respective computations.
Thus, the problem of model tuning is a parameter-fitting problem, and it may be tackled as an optimization problem. This involves the development of two com- ponents:
• A performance measure, which qualitatively assesses the goodness-of-fit of the model for each possible set of parameter values, i.e. it assesses the error between measured and computed responses.
• An optimization procedure, which finds a set of tunable parameters giving an optimum value for the performance measure, i.e. yields in modeling results that are as close to the measured results as possible.
Some discussion about the above two components of the optimization methodology is given below.
4.2.1 Performance measure
All efforts in the field of tuning practice so far have used the above approach. Nev- ertheless, the formulation of a pertinent performance measure was not given much
Sec. 4.2 Formulation of the optimization problem 97 focus compared to the optimization procedure. The performance measure, though, links the optimization procedure with the model and its definition is important for the success of the optimization methodology. Furthermore, there are reasons that are related to modeling assessment (and unrelated to the parameter estima- tion/optimization problem) that call for a careful performance measure definition.
Specifically, despite the richness and diversity of the modeling efforts that may be found in the bibliography, it is somewhat surprising that the assessment of the success of the models has always been qualitative—more specifically, by inspection. That is, the usual practice is to plot together the measurements and results of one or more simulations, inpect the resulting graph and comment about the quality of simulation results. Although such a visualization procedure is absolutely necessary to gain insight about the behaviour of a model and make a rough evaluation of its success, the introduction of a quantitative criterion (or a set of criteria) may aid in several directions where qualitative inspection seems inadequate. There are two reasons for this:
• Inspection is dependent on the scale that results are viewed and may therefore be misleading. Today’s catalytic converters are very efficient and reach light- off very fast compared to previous generations systems. As a result, outlet emissions range within several orders of magnitude, depending on the mode of converter operation. Thus, comparison between calculation and measurement may be difficult to assess purely by inspection. Quantitative criteria could be helpful to better assess modeling accuracy, unbiased from system configuration.
• There is considerable difficulty to compare the performance of different models. In order to compare different models directly, the models should be tested in the same set of measurements and the results should be plotted together for direct comparison. If this is not possible (as is the case for modeling results presented in the bibliography from different researchers) a generally accepted, quantitative criterion, could give an idea for the comparative performance of different modeling approaches.
Consequently, the formulation of the performance measure is useful per se, for the quantitative assessment of modeling results. The ultimate goal of the perfor- mance measure formulation is to express in a quantitative manner what is perceived by human intuition as the quality of the fit of the catalytic converter model to experimental results.
4.2.2 Optimization method
A properly formulated performance measure may be combined with an optimization method, to provide a methodology for the catalytic converter model parameters tuning. Any optimization method chosen for this task should take into account the following points:
1. We have no analytical expression of the performance measure, because it in- volves the output of the catalytic converter model
2. The performance measure is a non-linear function of the tunable parameters.
3. The parameter space is n-dimensional, where n is the number of the parameters being tuned.
4. The parameter space is continuous, since the tunable parameters are continu- ous real variables.
5. The search on the parameter space must be constrained, the constraints de- pending on each tunable parameter.
6. The parameter space may be unimodal or multimodal; this depends on both the protocol of the experiments used for model tuning and the parameters being tuned.
7. For each function evaluation, a run of the model must be invoked. Since this is very demanding in terms of computational power, the method of choice should perform as few function evaluations as possible.
As Goldberg [5] summarizes, there are three main categories of optimization and search techniques: (a) conventional calculus methods, (b) enumerative methods and (c) randomized methods.
The calculus based methods are local methods (because they proceed exploiting information only from the neighbourhood of the current point), presume that the parameter space is continuous and usually require derivatives values or their numer- ical approximations (to detect the gradient of the neighbourhood and decide how to proceed).∗ Typically, they are efficient but fail to find a global minimum in a multimodal parameter space. A typical example is the conjugate gradients method, that was the first method we tested (Section 4.4 below).
Enumerative methods simply evaluate all the possible points of the search space in order to find an extremum. Enumerative methods are simple but inefficient and are completely useless for problems with large search spaces, especially spaces of many dimensions.
Randomized methods, on the other hand, employ random choice of evaluation points, in contrast with the deterministic choice of calculus based methods. This does not necessarily mean, however, that these methods are completely random; otherwise, they would be equivalent to enumerative techniques.
Two popular families of randomized methods for the optimization of multimodal functions is Simulated Annealing and Genetic Algorithms. Both procedures are not completely random; they choose search directions following certain rules. However, the term randomized implies that these rules are not deterministic but employ ran- dom choice of search points.
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In this work, we first attempt to define a performance measure that is suitable for use in driving cycle tests, typical for the experimental assessment of catalytic converter behaviour. The requirements and rationale for its formulation are given in Section 4.3.
Then, the development of the optimization procedure is discussed. A conju- gate gradients method was originally tested (Section 4.4). The success was limited apparently because the parameter space of the problem is multimodal. Its fail- ure motivated the development of a genetic algorithm, detailed in Section 4.5, that circumvented the difficulties of the previous attempts and resulted in a workable methodology for catalytic converter parameter tuning.