Page 165 - Mathematical Techniques of Fractional Order Systems

Fractional Order Time-Varying-Delay Systems Chapter | 5 153

2.5

2

1.5

1

0.5

0
0 10 20 30 40 50
FIGURE 5.2 Evolution of the forced system pseudo-state vector xðtÞ under a stabilizing
control low.

5.6.3 Example 2

For a comparison between the two approaches summarized in the Theorems
1 and 2. In this example, the following fractional order time-varying delay
system given by

21 2 0:50 2
D 0:75 xðtÞ 5 xðtÞ 1 xðt 2 τðtÞÞ 1 uðtÞ ð5:66Þ
0 21 0 1 1
is considered.
The pseudo-state time-varying delay function is chosen as

τðtÞ 5 τ 1 sinðϖt 2 ϕÞ 1 γ;
where τ 1 5 0:5, ϖ 5 50 rad/s, γ 5 1 and ϕ 5 0.
The LMI constraint (5.22) given in the Theorem 1 was found infeasible.
For the same parameters of the system and the same solver, the LMI con-
straint (5.40) given in the theorem 2 was found feasible, and the solution is
given by the following matrices

4:1969 2:0055 0:65502 20:53683
P 5 ; ϒ 11 5 ;
2:0055 14:953 20:53683 2:8218

20:31046 20:15357 5:0806 0:52675
ϒ 12 5 ; ϒ 22 5 ;
20:15357 22:6286 0:52675 7:8508
20:0039364 20:067655 20:059691 0:029161

T 5 ; Y 5 ;
20:067655 20:042041 0:029161 20:078605

0:21828 0:2245
Z 5 :
0:2245 0:66927