CAGE User's Guide

Filling Breakpoints

Filling breakpoints spaces the breakpoints in such a way as to place the breakpoints by reference to the model. For example, one method places the majority of the breakpoints where the curvature of the model is greatest.

This option is only available when you are performing Feature calibrations.

For example, a model of the spark angle that produces the maximum brake torque (MBT) has the following inputs: engine speed, N, relative air charge, L, and air-fuel ratio, A. You can space the breakpoints for engine speed and relative air charge over the range of these variables by referring to the model.

To space the breakpoints based on model curvature:

1. Click  or select Normalizer -> Fill.

1. The Breakpoint Fill Options dialog box opens.

   

2. Choose the appropriate method to space your breakpoints, from the drop-down menu in the dialog box.

1. For the preceding example, select ShareAveCurv. For more information about the methods for spacing the breakpoints, see Filling Methods.

3. Enter the ranges of the values for the normalizers.

1. In the preceding example, for Range N, enter 500 6500, and for Range L, enter 0.1 1.

4. Enter the ranges of the other model variables.

1. CAGE spaces the breakpoints by reference to the model. It does this at selected points of the other model variables. In the preceding example, enter 11 17 for the Range of A and enter 2 for the Number of points. This takes two slices through the model at A = 11 and A = 17. Each slice is a surface in N and L. That is, MBT(N, L, 11) and MBT(N, L, 17).

   CAGE computes the average value of these two surfaces to give an average model, MBT_AV(N, L).

5. Click OK.

Note    If any of the breakpoints are locked, each group of unlocked breakpoints is independently spaced according to the selected algorithm.

If you increase the number of slices through the model, you increase the computing time required to calculate where to place the breakpoints.

Filling Methods

This section describes in detail the methods for spacing the breakpoints of your normalizers in CAGE.

• For one-dimensional tables, the two fill methods are
• ReduceError
• ShareAveCurv
• For two-dimensional tables, the two fill methods are
• ShareAveCurv
• ShareCurvThenAve

ReduceError

Spacing breakpoints using ReduceError uses a greedy algorithm:

1. CAGE locks two breakpoints at the extremities of the range of values.
2. Then CAGE interpolates the function between these two breakpoints.
3. CAGE calculates the maximum error between the model and the interpolated function.
4. CAGE places a breakpoint where the error is maximum.
5. Steps 2, 3, and 4 repeat.
6. The algorithm ends when CAGE locates all the breakpoints.

ShareAveCurv and ShareCurvThenAve

Consider calibrating the normalizers for speed, N, and relative air-charge, L, in the preceding MBT model.

In both cases, CAGE approximates the MBT_AV(N, L) model, in this case, using a fine mesh.

The breakpoints of each normalizer are calibrated in turn. In this example, these routines calibrate the normalizer in N first.

Spacing breakpoints using ShareAveCurv or ShareCurvThenAve calculates the curvature, K, of the model MBT_AV(N, L)

as an approximation for

Both routines calculate the curvature for a number of slices of the model at various values of L. For example, the figure shown has a number of slices of a model at various values of L.

Then

• ShareAveCurv averages the curvature over the range of N, then spaces the breakpoints by placing the i^th breakpoint according to the following rule.
• ShareCurvThenAve places the i^th breakpoint according to the rule, then finds the average position of each breakpoint.

Rule for Placing Breakpoints..   If j breakpoints need to be placed, the i^th breakpoint, N_i, is placed where the average curvature so far is

Reference.   de Boor, C., A Practical Guide to Splines, New York, Springer-Verlag, 1978.

See Also

• Calibrating the Normalizers

Initializing Breakpoints

Optimizing Breakpoints