Page 470 - Mathematical Techniques of Fractional Order Systems

456 Mathematical Techniques of Fractional Order Systems

FIGURE 15.2 Chaotic attractor in different planes for the Arneodo system with q 5 0:9 and
β 3 5 0:4.

15.4 DESIGN OF CONTROLLERS

Let the drive system be the Arneodo system
q
d x 1
5 x 2
q
dt
q
d x 2
5 x 3 ð15:9Þ
dt q
q
d x 3 3
52 β x 1 2 β x 2 2 β x 3 1 β x
3
1
2
4 1
dt q
And the response system defined as the driven Duffing oscillator
q
d y 1
5 y 2 1 u 1 ðtÞ
dt q
q ð15:10Þ
d y 2 3
5 y 1 2 y 2 αy 2 1 fcosωt 1 u 2 ðtÞ
1
dt q
where u 1 ðtÞ and u 2 ðtÞ are controllers to be designed.