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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 tex2html_wrap_inline1197 properly by applying suitable linear transformations.

As a rule of thumb, the range of the transformed variables tex2html_wrap_inline1711 , tex2html_wrap_inline1713 and tex2html_wrap_inline1715 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 tex2html_wrap_inline1719 , tex2html_wrap_inline1721 , tex2html_wrap_inline1723 , tex2html_wrap_inline1725 , tex2html_wrap_inline1727 , and tex2html_wrap_inline1729 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 tex2html_wrap_inline1735 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 tex2html_wrap_inline1737 value of factor tex2html_wrap_inline1739

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 tex2html_wrap_inline1767 (altitude) does have a range of approximately tex2html_wrap_inline1769  (meters) to tex2html_wrap_inline1771  (meters). In this case, it may be useful to select tex2html_wrap_inline1773 and tex2html_wrap_inline1775 .

These constants for the internal linear transformation of tex2html_wrap_inline1767 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 up previous contents
Next: Section STARTWERTE Up: Input File of PAREST Previous: Section GRENZEN

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