Page 117 - Mathematical Techniques of Fractional Order Systems

Exact Solution of Linear Fractional Distributed Order Systems Chapter | 4 105

describes anomalous nonexponential relaxation processes (Kochubei, 2009)
for exponential weight functions.
By the advent of distributed order elements, distributed order calculus is
hoped to soon appear in different problem contexts by opening paths in vari-
ous engineering fields, such as electrical circuits, signal processing, and con-
trol systems. Therefore, the results of this chapter provide a small step
towards initial contributions in studying the behaviors of various systems
containing such elements by presenting the exact solution of a class of sys-
tems of distributed order differential equations. In order to emphasize this
notion, a simple electrical circuit containing a distributed order capacitor is
considered in this chapter whose exact response in the time domain is imme-
diately obtained by using the results of the chapter. In addition, stability
analysis comes first in studying an emerging system or designing a circuit
performing a particular task which could be controlling a wheeled robot for
instance. In this respect, a stability discussion is provided at the end of this
chapter giving some insights about the poles of distributed order systems,
stability and the relationship that exists between them. Some important
results on this matter are brought from the literature which introduces a gen-
eral graphical approach towards the stability problem of distributed order
systems. Then it is shown how stability analysis of distributed order systems
considered in this chapter shares some common traits with that of retarded
systems with time delays. In particular, it is shown that the characteristic
equations of both systems are made up of Lambert W functions and both
generate infinitely many poles on the complex plane. It is also noticed that
both sets of poles are located on the left side of a vertical line. This is actu-
ally an important property to observe, since it allows one to determine the
stability of the systems by merely checking the location of characteristic
equations roots associated with them.
The rest of this chapter is organized into six sections. At first, we provide
a review on distributed order calculus and some other essential preliminaries
in Section 4.2. The main results are then presented in Section 4.3, which are
followed by a numerical simulation in Section 4.4 to test the obtained results.
In addition, an electrical circuit with a distributed order element is presented
in Section 4.5 to give some insights about the physical applications of the
results. Stability analysis of LTI distributed order systems is briefly dis-
cussed in Section 4.6 and the chapter is finally concluded in Section 4.7.We
end this section by providing a table containing definitions of all the symbols
used in the chapter for the reader’s convenience (Table 4.1).

4.2 PRELIMINARIES
This section starts with some essential definitions and continues with a brief
review on fractional order and distributed order operators, which will come
in useful in deriving the main results in the next section. We commence by