Page 164 - Mathematical Techniques of Fractional Order Systems

152 Mathematical Techniques of Fractional Order Systems

5.6.2 Example 1
Consider an unstable fractional order time-varying delay system given by

1 2 20:2 0:8 2
0:5
D xðtÞ 5 xðtÞ 1 xðt 2 τðtÞÞ 1 uðtÞ ð5:65Þ
1 22 20:75 0:5 1
The pseudo-state time-varying delay function is chosen as
τðtÞ 5 τ 1 sinðϖt 2 ϕÞ 1 γ;
where τ 1 5 0:3, ϖ 5 50rad/s, γ 5 0:5 and ϕ 5 0.
The instability of the considered system (5.65) is shown in Fig. 5.1,
where the evolution of the pseudo-state vector xðtÞ in open loop with uðtÞ 5 0
is drawn.
The solvability of the LMI problem (5.22) given in Theorem 1 can be
checked by using any (SDP solver. Therefore, the pseudo-state feedback con-
troller gain matrix K 0 can be deduced from the LMI (5.63) constraint
solution.
A feasible solution of the LMI (5.63) is given by

1:2275 0:45528
P 5 ; Y 0 522:1154 20:40352
0:45528 1:0637
Then, from the matrices P and Y 0 , the controller gain matrix K 0

K 0 5 Y 0 P 522:7805 21:3923
is deduced.
By adding the pseudo-state feedback control law, in Fig. 5.2, the evolu-
tion of the pseudo-state vector xðtÞ converges asymptotically to zero, which
is in agreement with what is proved in Theorem 3.

300

250
200

150

100

50
0
0 1 2 3 4 5
FIGURE 5.1 Evolution of the unforced system pseudo-state vector xðtÞ.