Page 107 - Mathematical Techniques of Fractional Order Systems

Fractional Order System Chapter | 3 95

hand, as outlined in the next section, some difficulties arise in achieving a
simplification in terms of number of model parameters. A similar problem
arises in the use of the suggested decomposition for solving the model-
matching problem, strictly related to controller synthesis, as the example in
the next section will show.

3.7 DISCUSSION AND EXTENSIONS

In the previous Section 3.5 the decomposition of the forced response has
profitably been applied to the derivation of “simplified” models of fractional
order systems. It has been observed, in this regard, that the definition of
model complexity is to some extent arbitrary. It may be related to the (finite)
dimension of the integer order models associated with the fractional order
systems via the variable transformation (3.3), or to the “compactness” of the
fractional order transfer functions, in particular, the number of nonzero para-
meters that appear in them, or to the maximum degree of the denominator of
the transfer functions. The models obtained in the previous section can be
considered simpler from all of these points of view. It should be observed,
however, that the results strongly depend on the input whose asymptotic
component of the forced response must be retained. If the minimum common
denominator (mcd) of the fractional exponents of the input transform does
not coincide with the mcd of fractional exponents of the original transfer
function, forcing UðwÞ and GðwÞ to have a common q might entail a consid-
erable increase of the order of the integer order model obtained via (3.3)
from the fractional order transforms.
Also the reduction criterion adopted to approximate the system compo-
nent of the forced response is rather arbitrary. Even if its choice is outside
the scope of the present contribution, it should be noted that not all methods
cannot be applied. In fact, most reduction methods suggested in the literature
for integer order systems are directly applicable only to (stable) systems with
poles in the open LHP. To overcome this problem, it is sometimes suggested
to preliminarily separate the stable and unstable parts of the systems with
RHP poles and then apply the reduction procedure only to the first. Further
difficulties arise in the case of fractional order systems, because their stabil-
ity is compatible with the presence of RHP poles in the integer order func-
tion derived from the fractional one via (3.3), provided these poles are
outside the instability sector.
In this book chapter attention has been focused on the model reduction
problem, but the relevance of the response decomposition goes beyond
model simplification. Suffice it to recall, in this regard, the interpolation
problem, strictly related to the model matching problem (Doyle et al., 1992)
or, more generally, the moment matching problem (Astolfi, 2010). Indeed,
forcing the coincidence of the input components of two different systems in
the response to a given input entails interpolating the values taken by the