European Journal of Scientific Research - EuroJournals

800 S. Gherbi, S. Yahmedi and M. Sedraoui
W p
⎡ ( 0.
005 jw + 1)
⎤
⎢
0
0.
05 jw
⎥
( jw)
= ⎢
⎥
(19)
⎢
( 0.
005 jw + 1)
0
⎥
⎢⎣
0.
05 jw ⎥⎦
• The fig.6 represent the singular values of WP ( jw)
in the frequency plan, one notice that the
specifications on the performances are bigger in low frequencies (integrator frequency
behaviour), this guaranty no static error.
Then the standard problem of H∞ Control theory is then:
T(
jw)
⋅Wt
( jw)
min (20)
K stabili sin g S(
jw)
⋅W
p ( jw)
∞
i.e.: to find a stabilising controller K that minimise the norm (20).
IV. Application
The minimisation problem (20) is solved by using two Riccati equations [10] or with the linear matrix
inequalities approach [4]. For our system, we use the linear matrix inequalities solution with the
Matlab instruction lmi
hinf available at ‘LMI/ Toolbox’ of Matlab ® Mathworks inc (for more details
see [12]).
The controller obtained (in the state space form) is presented in appendix.
• The fig.7. and the fig.8. show the satisfaction of the stability and performances robustness
•
(conditions (14) and (17)).
We choose the desired outputs R = ( Vds
_ ref
and V qs of the nominal case.
= 1
0
Vqs
_ ref = 0 ) , and the fig.9.show the outputs V
1
ds
Singular values
10 -0.1
10 -0.2
10 -0.3
10 -0.4
10 -0.5
10 -0.6
10 -4
Figure 4: The system uncertainties maximum singular values
10 -3
10 -2
σ
[ Δ ( jw)
]
m
10 -1
10 0
Pulsations
10 1
10 2
10 3
10 4