How Is a Two-Stage Model Constructed?

Local models find the best fit of a curve to the data in each test. Each test in this case is a sweep of torque against spark angle, with speed, load, and cams held at a constant value for each sweep. The following illustrates a single sweep with a local model fitted.

The local models provide the coefficients to generate global models. The equations describing those local model curves have certain coefficients such as max and knot, which for this data are peak torque and MBT spark (the spark angle that generates maximum brake torque).

Local models are fitted to each test, in different places across the global space, as illustrated above. Each local model has coefficients for MBT and peak torque, etc. These coefficients become the data to which the global models are fitted. Coefficients such as peak torque and MBT are used to make the second stage of modeling more intuitive; an engineer will have a much better understanding of how a feature such as MBT spark varies through the global factor space than some esoteric curve fit parameter. Familiar variables like these are helpful to engineers trying to decide how well a model describes engine behavior. Better intuitive understanding allows much greater confidence in your models.

Global models are the best fit of a curve to the values of, for example, MBT for each test. This is repeated for each coefficient, producing several global models fitted to different coefficients of the local models. These coefficients are referred to as response features of the local models. The following example shows a global model for maximum torque across the speed/load global space.

The two-stage model is a surface fitted across all the global models, to describe the behavior across all global variables.

It can be useful to think of local and global models as a series of 2-D slices, while the two-stage model fits a 3-D surface across the curves of the global model slices. It is difficult to visualize more dimensions! The following example shows a variety of 3-D plots of global models for properties of the local torque/spark curves (such as MBT, peak torque, and torque a number of degrees before and after MBT), showing how these properties vary across the speed/load global space. The 2-D plot of the global MBT model (on the right) demonstrates how MBT varies with engine speed.

The two-stage model can take values of each coefficient at a certain value of, say, speed, to generate a new curve of torque against spark. This is a slice through the two-stage model surface.

In other words, you can test your two-stage model by comparing it with the local fit and with the data. For example, you can reconstruct a local torque/spark curve at an operating point by taking the values of MBT and peak torque and the curvature from the two-stage model, and then validate this reconstructed curve against the original fit and the data. The two-stage model can also predict responses between tests, for new sweeps at intermediate values for which there is no data. If the two-stage model shows an accurate fit when compared to the local sweeps, this is a good sign that the engine behavior is well described by the model across the global variables.

For more details on two-stage modeling, see Two-Stage Models, and see Two-Stage Models for Engines in the Model Browser documentation for more statistical depth. (In the Help Browser you can right-click and select Back to return to previous pages).

Importing and Filtering Data

Building the Models

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