Page 155 - Mathematical Techniques of Fractional Order Systems

Fractional Order Time-Varying-Delay Systems Chapter | 5 143

T
T
A simple proof of the equality of X ðtÞΩXðtÞ 5 X ðtÞΩ 1 XðtÞ can be found
in Boroujeni and Momeni (2012a,b).
Then, the sufficient stability condition of the system (5.21) is given in a
LMI formulation (5.22). That means the system (5.21) is asymptotically
stable if the matrix Ω is negative definite. It completes the proof.

5.4.2 Time-Delay-Dependent Stability

The time-delay-dependent stability analysis for time-varying delay systems
has been attracting increasing researcher attention. In this case, the stability
analysis typically requires the use of the Lyapunov Krasovskii functional
method in the time domain. In addition, some useful inequalities and the
Newton Leibnitz formula (5.9) are used to overcome the existence of
delayed pseudo-state in the Lyapunov Krasovskii functional dynamics, and
replace it by pseudo-states without delay and its integral. This approach
allows us to obtain time-delay-dependent stability criterion which will be
used hereafter to synthesize a controller law.
In the following theorem, a sufficient time-delay-dependent stability
condition for the system (5.21) is derived. The obtained condition is formu-
lated in LMI expression. New decision variables are added to the Lyapunov
matrices of the functional candidate functional, which add more degrees of
freedom in the optimization problem.

Theorem 2: The unforced fractional order time-delay system (5.21) is
asymptotically stable if there exist four symmetric positive definite matrices
P; Q; Z and Γ, and two matrices T and Y with appropriate dimensions
such that the following LMIs
2 ϒ 11 ϒ 12 ϒ 13 3
ϒ 5 ϒ 22 ϒ 23 5 , 0 ð5:40aÞ
4
ϒ 33

2 Γ 11 Γ 12 Y 3
Ω 5 Γ 22 T 5 $ 0 ð5:40bÞ
4
Z
where
T T T
ϒ 11 5 PA 0 1 A P 1 Y 1 Y 1 τ m Γ 11 1 Q
0
T
ϒ 12 5 PA τ 2 Y 1 T 1 τ m Γ 12
T
ϒ 12 5 τ m A Z
0
T
ϒ 22 52 Λ 1 Q 2 T 25 T 1 τ m Γ 22
T
ϒ 23 5 τ m A Z
τ
ϒ 33 52 τ m Z