Page 97 - Mathematical Techniques of Fractional Order Systems

Fractional Order System Chapter | 3 85

2Uπ=ð2qÞ, as shown in Fig. 3.2. Therefore, if all the n roots are outside the
instability sector, the overall phase variation is (3.20), which proves neces-
sity. The sufficiency of (3.20) can be proved by contradiction. To this pur-
pose, assume that (3.20) holds true but that a real root, say p j , lies inside the
instability sector. The phase variation of the factor associated with p j , i.e.,
j π
ρe 2 p j ,is 2½π 2 π=ð2qފ (see Fig. 3.2) so that the overall phase variation
2q
is less than (3.20), which contradicts the assumption that (3.20) holds true. A
similar reasoning can be used for a pair of conjugate roots inside the instabil-
ity sector.

Once system stability has been ascertained, efficient approximation meth-
ods can be applied to simplify an original complex model. The next section
is devoted to such a problem.

3.5 MODEL REDUCTION
The separate consideration of the two components (3.15) of (3.9) can be
used for analysis, synthesis, and approximation purposes. In this section, it is
shown how to obtain reduced order models that retain the original asymp-
totic response along the lines of Casagrande et al. (2017). Essentially, the
suggested procedure operates as follows.

3.5.1 Approximation Procedure

1. Find the fractional order system transfer function (3.1) and determine the
Laplace transform (3.2) of the input whose response is of interest.
2. Via the change of variable (3.3), convert the Laplace transform of the
related response into the rational function YðwÞ (see (3.9)).
3. Decompose YðwÞ according to (3.11) into a system component Y Σ ðwÞ and
an input component Y U ðwÞ (see (3.15)).
ν
4. Find a rational function Y ðwÞ of order ν , n (usually, ν{n) approximat-
Σ
ing the original system component Y Σ ðwÞ according to any criterion for
rational approximation.
5. Form the reduced order w-domain transfer function G r ðwÞ of the reduced
order model in such a way that the reduced order model response Y r ðwÞ
ν
to UðwÞ admits Y U ðwÞ as its input component and Y ðwÞ as its system
Σ
component up to an auxiliary additive term of negligible importance
a
Y ðwÞ, namely:
Σ
ν
a
Y r ðwÞ 5 G r ðwÞUðwÞ 5 Y U ðwÞ 1 Y ðwÞ 1 Y ðwÞ: ð3:22Þ
Σ
Σ
6. Construct the simplified transfer function G r ðsÞ from G r ðwÞ using again
b
(3.3).
To clarify step (v), some remarks are opportune.