Page 87 - Mathematical Techniques of Fractional Order Systems

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Chapter 3

Fractional Order System

Forced-response
Decomposition and Its

Application

1
2
Daniele Casagrande , Wiesław Krajewski and Umberto Viaro 1
1 2
University of Udine, Udine, Italy, Systems Research Institute, Polish Academy of Sciences,
Warsaw, Poland
3.1 INTRODUCTION

The theory of fractional order systems has already attained sufficient maturity
to allow its systematic presentation in several books (Azar et al., 2017;
Tepljakov, 2017; Kaczorek, 2011; Caponetto et al., 2010; Monje et al., 2010)
and to be the subject of many special journal issues (Caponetto et al., 2016;
Ionescu et al., 2016; Psychalinos et al., 2016). Despite its increasing popular-
ity, however, some important aspects need further investigation, among them
the detailed analysis of the system forced response to inputs with rational
order transform, of which the harmonic and singularity inputs (integrals of the
impulse) are distinctive cases. In particular, relatively little attention has been
paid to the separate study of the transient and asymptotic responses with some
notable exceptions limited to canonical inputs (Monje et al., 2010; Trigeassou
et al., 2012; Jakubowska and Walczak, 2016; Semary et al., 2016; Kesarkar
and Selvaganesan Narayanasamy, 2016). A more thorough characterization of
the system dynamic behavior requires the consideration of both the short- and
long-term behavior of the responses to more general inputs. Such an analysis
is particularly meaningful in the derivation of simplified models that retain
essential properties of the original system, such as stability and performance
indices, and in the synthesis of controllers that satisfy both transient and
asymptotic specifications, e.g., on overshoot and steady-state error.
The present contribution aims at a more systematic study of the constituent
parts of the forced-response that characterize different aspects of the system

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