Page 160 - Mathematical Techniques of Fractional Order Systems

148 Mathematical Techniques of Fractional Order Systems

5.5 FEEDBACK STABILIZATION
The main objective of this section is to investigate how the system (5.19)
can be asymptotically stabilized by using a pseudo-state feedback controller.
In Boukal et al. (2016b), a delay-independent stability condition has been
proposed. On the basis of the results given in Boukal et al. (2016b), a simple
pseudo-state feedback has been used to stabilize the considered class of
linear systems with time-varying delay. Therefore, the delay-dependent stabi-
lization may be directly extended form the stabilization methodology and the
delay-dependent stability given in Theorem 2.

Assumption 2: In the following, all the pseudo-states are assumed to be
available in order to establish the control law.

Over the past decade, the problem of controller synthesis to ensure the
stabilization of time-delay systems has been considered by many researchers
in a large number of papers concerning time-delay systems with integer
order. In this work, a simple linear pseudo-state feedback control law of the
form

uðtÞ 5 K 0 xðtÞ ð5:61Þ
is considered to prove the stabilization of FOS in the form (5.19), where
K 0 AR m 3 n is the feedback gain matrix.
In fact, the closed-loop given by the system (5.19) and (5.61) is
described by
α
D xðtÞ 5 ðA 0 1 B u K 0 ÞxðtÞ 1 A τ xðt 2 τðtÞÞ ð5:62aÞ
xðtÞ 5 ψðtÞ; tA½ 2 τ m ; 0Š; 0 , α , 1 ð5:62bÞ

Definition 2: The control law (5.61) is an asymptotic pseudo-state feedback
control of the system (5.19) if the closed loop system (5.62) is asymptotically
stable.

5.5.1 Feedback Stabilization Based on Time-Delay-Independent
Stability Condition
The Theorem 3 will lead us to an optimization problem, which provides as a
solution the parameters of the considered control law (5.61). The following
Theorem 3 gives a sufficient condition on the stabilization of the system
(5.19) by the pseudo-state feedback controller gain matrix, expressed in
terms of LMIs, which can be solved easily by using any free solvers based,