next up previous
Next: QuaSi Up: Previous Approaches Previous: Previous Approaches

Interval Arithmetic Approach

The coefficients of the Taylor's series for a numeric differential system are numeric coefficients. Representing fuzzy values by $\alpha$-cuts makes it necessary to extend the Taylor's series to intervals at different \ensuremath{\alpha}-cut levels.
[ymin,ymax](t)= [y0min,y0max]+
  $\frac{(t-[t_{0_{min}},t_{0_{max}}])*[y_{0_{min}},y_{0_{max}}]'}{1!}+$
  $\frac{(t-[t_{0_{min}},t_{0_{max}}])^2*[y_{0_{min}},y_{0_{max}}]''}
{2!}+...+$
  $\frac{(t-[t_{0_{min}},t_{0_{max}}])^n*[y_{0_{min}},y_{0_{max}}]^{(n)}}
{n!}+...$
Using interval arithmetic the width of the result is always equal to the sum of the widths of the two expressions. As a result, the width of [ymin,ymax](t) becomes wider with t increasing. In the case of a dynamic system, the system output would be affected by an always increasing width, independent of stability of other properties of the system. Most likely the simulation result will reach regions of the phase space where no numerical solution of the differential system pass through, contradicting with the physical reality described by the model.



Christoph Reich
1999-12-22