Environmental Engineering Reference
In-Depth Information
This nominal plant G(s) (Eq. 5.11 ) is represented by the state space matrices
A PP, B PP, C PP and D PP and it has 55 states. The differences of the family of linear
plants compared to the nominal plant are considered as additive uncertainties.
These differences appear due to their non-linear behavior of the plant which relates
the pitch angle and the generator speed.
_ X ðÞ¼ A PP X ðÞþ B PP b ðÞ
w g ð t Þ
a Tfa ð t Þ
¼ C PP X ðÞþ D PP b ðÞ
ð 5 : 11 Þ
The nominal plant is generalized including the performance output channels
and the scale constants D u , D d1 , D d2 , D e1 , D e2 (Eq. 5.12 ) to scale the different
channels of the MISO mixed sensitivity scenario.
D u ¼ 1
D e1 ¼ 10; D e2 ¼ 0 : 1
D d1 ¼ 10; D d2 ¼ 0 : 1
ð 5 : 12 Þ
Five weight functions are included to create the generalized plant. In this mixed
sensitivity problem, the W 11 (s) , W 12 (s) , W 2 (s) are only used (see Fig. 5.10 ).
W 11 (s) is an inverted high pass filter and it is used to define the closed loop output
sensitivity performance, W 12 (s) is an inverted notch filter centered at the first tower
fore-aft mode to mitigate the wind effect in this mode and W 2 (s) is an inverted low
pass filter to reduce the controller activity in high frequencies. Some inverted
notch filters are included in W 2 (s) to include notch filters in the controller
dynamics. These filters are centered at the rotational frequencies 1P (0.2 Hz) and
3P (0.6 Hz) and at other structural modes.
The controller obtained by using the MATLAB Robust toolbox has to be re-
scaled to adapt the inputs and output to the real non-scaled plant. The designed H ?
Pitch Controller has two inputs (generator speed error in rad/s and tower top fore-
aft acceleration in m/s 2 ) and one output (collective pitch angle b H? in rad). This
designed controller is state space represented and its order is 45. Finally, the
controller is reduced to order 24 without losing important information in its
dynamics. After reducing, the last step is the controller discretization using a
sample time of 0.01 s. The Bode diagram of the discrete state space controller
(Eq. 5.13 ) appears in Fig. 5.9 .
ew g ðÞ
a Tfa ðÞ
Xk þ 1
ð
Þ¼ A BD X ðÞþ B BD
ð 5 : 13 Þ
ew g ðÞ
a Tfa ðÞ
b H 1 ðÞ¼ C BD X ðÞþ D BD
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