Applying Constraints :: Tutorial: Design of Experiment (Model-Based Calibration Toolbox™)

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Applying Constraints

In many cases, designs might not coincide with the operating region of the system to be tested. For example, a conventional stoichiometric AFR automobile engine normally does not operate with high exhaust gas recirculation (EGR) in a region of low speed (n) and low load (l). You cannot run 15% EGR at 800 RPM idle with a homogeneous combustion process. There is no point selecting design points in impractical regions, so you can constrain the candidate set for test point generation. Only optimal designs have candidate sets of points; classical designs have set points, and space-filling designs distribute points between the coded values of (1, -1).

You would usually set up constraints before making designs. Applying constraints to classical and space-filling designs simply removes points outside the constraint. Constraining the candidate set for optimal designs ensures that design points are optimally chosen within the area of interest only.

Designs can have any number of geometric constraints placed upon them. Each constraint can be one of four types: an ellipsoid, a hyperplane, a 1-D lookup table, or a 2-D lookup table.

To add a constraint to your currently selected design:

Select Edit > Constraints from the Design Editor menus.
The Constraints Manager dialog appears. Click Add.
The Constraint Editor dialog with available constraints appears. The default 1D Table is selected in the Constraint Type drop-down menu.
You can select the appropriate factors to use. For this example, choose speed (N) and air/fuel ratio (A) for the X and Y factors.
Move the large dots (click and drag them) to define a boundary. The Constraint Editor should look something like the following.
Click OK.
Your new constraint appears in the Constraint Manager list box. Click OK to return to the Design Editor. A dialog appears because there are points in the design that fall outside your newly constrained candidate set.
- You can click Continue to delete the points outside the constraint, or cancel the constraint. Note that fixed points are not deleted by this process.
- For optimal designs you see the following dialog, where you also have the option to replace the points with new ones chosen (optimally if possible) within the new candidate set.
The default if you are constraining your space-filling design is to Continue and remove the points outside the new constraint area; choose this.

If you examine the 2-D projection of the hypercube, you will notice the effects of the new constraint on the shape of the design, as shown in the preceding example.
Right-click the display pane to reach the context menu, and select Current View > 3D Constraints.

These views are intended to give some idea of the region of space that is currently available within the constraint boundaries.
Return to the Constraint Editor, choose Edit > Constraint, and click Add in the Constraint Manager.
Add an ellipsoid constraint. Choose Ellipsoid from the drop-down menu of constraint types.

Enter 0 as the value for the L diagonal in the table, as shown. This will leave L unconstrained (a cylinder). The default ellipsoid constraint is a sphere. To constrain a factor, if you want a radius of r in a factor, enter 1/(r^2). For this example, leave the other values at the defaults. Click OK to apply the constraint.
Click OK, click OK again in the Constraint Manager, and click Continue to remove design points outside the new candidate set (or Replace if you are constraining an optimal design). Examine the new constraint 3-D plot illustrated.

Both constraints are applied to this design.