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Exact Solution of Linear Fractional Distributed Order Systems Chapter | 4 125

4.6 STABILITY
Stability analysis of the dynamic system represented by the set of linear dif-
ferential equations of distributed order (4.17) is of interest in various applica-
tions, including control systems. Among several definitions of stability used
in the literature, the following definitions are within the scope of this
section.
Considering the pseudo state as the output, system (4.17) is BIBO stable,
if a bounded output xtðÞ is generated for every bounded input utðÞ in case of
a zero initial condition x 0ðÞ 5 0.
The origin in system (4.17) is said to be asymptotically stable, if for
every ε . 0 there exists some δ . 0 such that if :x 0ðÞ: , δ holds, then
:xtðÞ: , ε holds for t $ 0, and there exists a neighborhood of the origin such
that if x 0ðÞ is located inside it, then the limit lim t- 1 N xtðÞ 5 0 is guaranteed.
It can be shown that the same conditions on matrix A and the weight
function w αðÞ would guarantee stability of system (4.17) with respect to both
of the above definitions. Therefore, the term “stability” is used to refer to
either of the two stability definitions in the rest of the discussion for
convenience.
In order to analyze stability, it is possible to consider system (4.17) as the
two Volterra integral Eq. (4.19) and use the Lyapunov based approaches
available for stability analysis of these equations (Messina et al., 2015).
However, in order to obtain necessary and sufficient conditions of stability,
considering the characteristic equation of (4.17) is essential. The exact stabil-
ity criterion of system (4.17) stems from the characteristic equation of (4.17)
which is obtained from (4.23) as
det W 2lnsÞI 2 AÞ 5 0 ð4:102Þ
ð
ð
System (4.17) is stable if and only if all the roots of (4.102) are located
on the open left half plane. Since the characteristic Eq. (4.102) is irrational
in general, the stability analysis of distributed order systems is significantly
more complicated than the classical systems. All the same, distributed order
systems are not the only systems with this trait. Delay differential equations
have also been known to possess irrational characteristic equations. Today,
the stability analysis approaches available for systems with time delays are
generally divided into two main categories: the methods based on
Lyapunov Krasovskii functionals and characteristic equations related
approaches. Since the former is notorious for excessive conservatism, han-
dling irrational characteristic equations of time-delay systems is inevitable.
This is often realized by using graphical approaches discriminating the
stable region from the unstable one in a parameter space. The same idea can
be utilized for distributed order systems. This would result in a standard
graphical method for stability analysis of system (4.17) as presented by Jiao
et al. (2012). In order to explain this approach, denote the eigenvalues of