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## Section SKALIERUNG

A proper scaling of all dependent and independent variables and functions may effect the efficiency and robustness of any numerical algorithm used on computers with limited arithmetic precision.

It is therefore recommendable to scale the state variables y(t) and v(t), the control variables u(t) and also the objective function properly by applying suitable linear transformations.

As a rule of thumb, the range of the transformed variables , and should be approximately [-1,+1] (after scaling). If the original range of the variables is not known a priori one may get a first idea by looking at the initial trajectories.

For the purpose of scaling by linear transformation the formulas

are used. The constants , , , , , and have to be selected properly in order to transform the ranges of the state variables and the control variables onto [-1,+1] (as a rule of thumb).

In the same manner, the objective can be transformed by

As a rule of thumb, the minimum value of the transformed objective should be approximately 1000 and the range of variation of the transformed objective should be between zero and 1000.

The scalings by linear transformations are optionally. The constants defining the linear transformations can be provided in this section using the syntax

variable = value of value of factor

As variable one may use

The linear transformations are then applied internally. The user does not have to deal with the transformed variables. Input, output and problem dependent subroutines are handled in the original units of all variables and functions.

Example: Looking at the initial trajectory of an optimal control problem from flight mechanics one may find that that the differential state variable (altitude) does have a range of approximately  (meters) to  (meters). In this case, it may be useful to select and .

These constants for the internal linear transformation of are now supplied to the program by the line Y( 1) = 240.0E+3 210.0E+3

in section SKALIERUNG of the input file.

Next: Section STARTWERTE Up: Input File of PAREST Previous: Section GRENZEN

Oskar von Stryk
Tue Feb 1 13:50:42 CET 2000