Page 14 - Mathematical Techniques of Fractional Order Systems

4 Mathematical Techniques of Fractional Order Systems

1.2 VARIABLE ORDER DERIVATIVES
This section introduces a variable order differential operator D αðtÞ fðtÞ,
where fðtÞ is the differentiated function, and αðtÞ is a time-variable differen-
α
tiation order. The reader is presumed to already know what D fðtÞ; αAR
is. The variable order operator is introduced informally in Section 1.2.1 for
integer orders only, and then more formally for arbitrary real orders in
Section 1.2.2.

1.2.1 Intuitive Results for Integer Orders
If αðtÞAZ; ’ t then its variations with time must be steps, since it can only
assume values in a discrete set. Still, this simple case illustrates an impor-
tant issue when dealing with variable order derivatives: the existence or not
of a memory of past values of the order. In fact, considering for instance
the case
fðtÞ 5 t 2 ð1:1Þ

2if tA½0; 1½ , ½2; 3½ , ½4; 5½ , ...
αðtÞ 5 ð1:2Þ
1if tA½1; 2½ , ½3; 4½ , ½5; 6½ , ...
it can intuitively be seen that the reasonable result of operator D αðtÞ fðtÞ,in
this case, when orders are positive, should be
2if tA½0; 1½ , ½2; 3½ , ½4; 5½ , ...

D αðtÞ fðtÞ 5 ð1:3Þ
2t if tA½1; 2½ , ½3; 4½ , ½5; 6½ , ...
In other words, the result simply jumps from one derivative to another.
But things are different if there are negative orders. Consider now the case
fðtÞ 5 t ð1:4Þ

2 1if tA½0; 1½ , ½2; 3½ , ½4; 5½ , ...

αðtÞ 5 ð1:5Þ
0if tA½1; 2½ , ½3; 4½ , ½5; 6½ , ...
If the result should again merely jump between the derivative of order
21 (the integral of the function) and the derivative of order 0 (the function
itself), then
1 2
8
t if tA½0; 1½ , ½2; 3½ , ½4; 5½ , ...
<
D αðtÞ fðtÞ 5 2 ð1:6Þ
:
t if tA½1; 2½ , ½3; 4½ , ½5; 6½ , ...
Notice, consequently, that the integration goes on even when it is not
being used. The question naturally arises of whether it should, or not.
Without such a memory, the result would be