Page 161 - Mathematical Techniques of Fractional Order Systems

Fractional Order Time-Varying-Delay Systems Chapter | 5 149

e.g., on Interior Point techniques. So, the computed pseudo-state feedback
controller ensures the stabilization of an unstable system given by (5.62).

Theorem 3: There exists a pseudo-state feedback controller (5.61) that
stabilizes an unstable fractional order time-delay system in the form (5.19) if
there exists a positive definite matrix P and a matrix Y 0 which satisfy the
following LMI
T T T A τ P

0
Ω 5 A 0 P 1 PA 1 B 0 Y 0 1 Y B 1 2Λ 1 , 0 ð5:63Þ
0
0

2Λ 2
where
1 1
Λ 1 5 I; Λ 2 5 I:
2ð1 2 @τ max Þ ð1 2 @τ max Þ
The pseudo-state feedback controller gain matrix is given as K 0 5 Y 0 P.
&
Proof 3: The proof is similar to the one of Theorem 1 and is omitted.
Indeed, its suffice to replace A 0 in the proof of Theorem 1 by A 0 1 B u K 0 .
Then, the inequality (5.63) is obtained by pre- and postmultiplication of
21
P 0
the above inequality with the following matrices and
0 I
21T
P 0 21 21T
, respectively, with P 5 P . Thus, using this change of
0 I
21
variable Y 0 5 K 0 P , the obtained expression becomes (5.63). It completes
the proof.

5.5.2 Feedback Stabilization Based on Time-Delay-Dependent
Stability Condition
In this part, the procedure given in the subsection 5.1 is extended to the case
where the time-delay-dependent stability condition is used.

Theorem 4: There exists a pseudo-state feedback controller (5.61) that sta-
bilizes an unstable fractional order time-delay system in the form (5.19) if
~
there exist four symmetric positive definite matrices P; Q; Z and Γ, and
three matrices T ; Y 0 and Y with appropriate dimensions such that the fol-
lowing matrices inequalities
2 3
ϒ 11 ϒ 12 ϒ 13
ϒ 5 4 ϒ 22 ϒ 23 5 , 0 ð5:64aÞ

ϒ 33