CAGE User's Guide

Inverting a Table

You can use CAGE to produce a table that is the inverse of another table. This involves swapping a table input with a table output, and you can invert 1-D or 2-D tables.

Inverting a table allows you to link a forward strategy to a backward strategy; that is, swapping inputs and outputs. This process is desirable when you have a "forward" strategy, for example predicting torque as a function of speed and load, and you want to reverse this relationship in a "backward strategy" to find out what value of load would give a particular torque at a certain speed.

Normally you fill tables in CAGE by comparing with data or models. Ideally you want to fill using the correct strategy, but that might not be possible to find or measure. If you only have a forward strategy but want a backward one, you can fill using the forward strategy (tables or model) and then invert the table.

For example, in order to fill a table normally from a model, you need the model response to be the table output, and the model inputs to be a function of the table inputs (or it should be possible to derive the input -- for example, air mass from manifold pressure). If the available model is "inverted" (the model response is a table input and the table output is a model input) and you cannot change the model, you can invert the table in CAGE.

In the diagram of a table shown, the x- and y-axes represent the normalizers (which you want to be spark and load) and the z-axis is the output at each breakpoint (torque). To fill this table correctly from the model is a two-step process. First you need to fill a table that has the same input and output as the model, and then fill a second table by inversion.

For the inversion to be deterministic and accurate, the table to be inverted must be monotonic; that is, always increasing or decreasing. This requirement is explained by the following one-dimensional example. Every point on the y-axis must correspond to a unique point on the x-axis. The same problem applies also to two-dimensional tables: for any given output in the first table there must be a unique input condition; that is, every point on the z-axis should correspond to a unique point in the x-y plane. Some table inversions have multiple values and so do not meet this requirement, just as the square root function can take either positive or negative values. You can use the inversion wizard in CAGE to handle this problem; you can control the inversion process and determine what to do in these cases.

The preceding example illustrates a table with multiple values. There are two solutions for a single value of torque. CAGE has a table inversion wizard that can help overcome this problem. You can specify whether you want to use the upper or lower values for filling certain parts of the the table; this allows you to successfully invert a multiple-valued function. See the inversion instructions for 1-D and 2-D tables in the next sections.

The process of inverting a one-dimensional table is different from the process of inverting a two-dimensional table.

Filling the Table by Extrapolation

Inverting One-Dimensional Tables