Page 88 - Mathematical Techniques of Fractional Order Systems

76 Mathematical Techniques of Fractional Order Systems

behavior and can conveniently be considered separately. To this purpose, ref-
erence is made to the fairly numerous class of inputs with rational order trans-
form. Following a path similar to that taken in Casagrande et al. (2017) for
integer order systems and based on Dorato et al. (1994), the forced response
of a stable fractional order system to a persistent input of this kind is decom-
posed into the sum of two components: (1) a component with the same
pseudo-polynomial denominator as the system transfer function; and (2) a
component with the same pseudo-polynomial denominator as the input trans-
form. The first is called the system component of the forced response and the
second is called the input component because the first is characterized by the
same evolution modes of the impulse response, which depends on the system
only, and the second by elementary functions that exhibit similar structure
(Mittag Leffler functions) but depend only on the input and can therefore be
called “input modes.” If there are common modes between the input and the
system, a third resonant component is also present. However, for the sake of
simplicity, this possibility is ruled out (which is necessarily true when the sys-
tem is stable, so that all of its modes tend to zero as time tends to infinity, and
the input is antistable, so that all of its modes are persistent).
To ascertain the stability of the rational order system, resort can be made
either to efficient numerical algorithms or to Routh Hurwitz-like criteria for
polynomials with real and complex coefficients. A section of this chapter is
dedicated to this problem. As is known, it entails determining the distribution
of the roots of a characteristic polynomial with respect to two radii delimit-
ing a sector of the right half-plane (instead of the entire right half-plane, as
is the case for integer order systems).
Robust stability issues are outside the scope of the present contribution
and, therefore, are not treated in the sequel. Let us only observe, in this
regard, that many results concerning the so-called D-stability (Levkovich
et al., 1996; Tempo, 1989) can be extended to fractional order systems.
The rest of this chapter is organized as follows. Section 3.2 introduces
some essential notation and specifies the families of fractional order systems
and inputs to which the aforementioned decomposition of the forced
response applies. Section 3.3 shows how such a decomposition can uniquely
be obtained from the Laplace transform of the forced response and, for
stable systems, defines its transient and steady-state components. Section 3.4
presents some simple stability conditions. Section 3.5 shows how the decom-
position can be used to find simplified models that reproduce the asymptotic
response of original complex systems while still approximating well the tran-
sient behavior. Some illustrative examples are worked out in Section 3.6.
The results are discussed in Section 3.7 where the relationship between the
suggested response decomposition and the model-matching problem, strictly
related to controller synthesis, is also pointed out. Possible directions of
future research are indicated in Section 3.8.