  Fractional order controllers become increasingly popular due to their
versatility and superiority in various performance. However, the bottleneck in
deploying these tools in practice is related to their analog or numerical
implementation. Numerical approximations are usually employed in which the
approximation of fractional differintegrator is the foundation. Generally, the
following three identical equations always hold, i.e.,
$\frac{1}{s^\alpha}\frac{1}{s^{1-\alpha}} = \frac{1}{s}$, $s^\alpha
\frac{1}{s^\alpha} = 1$ and $s^\alpha s^{1-\alpha} = s$. However, for the
approximate models of fractional differintegrator $s^\alpha$,
$\alpha\in(-1,0)\cup(0,1)$, there usually exist some conflicts on the mentioned
equations, which might enlarge the approximation error or even cause fallacies
in multiple orders occasion. To overcome the conflicts, this brief develops a
piecewise approximate model and provides two procedures for designing the model
parameters. The comparison with several existing methods shows that the
proposed methods do not only satisfy the equalities but also achieve high
approximation accuracy. From this, it is believed that this work can serve for
simulation and realization of fractional order controllers more friendly.
