A Semi-active Control System for Wind Turbines - Wind Turbine Control and Monitoring - page 378

In this case, the free vibrations (Eq. 13.7 ) becomes:

z ¼ Az þ Bf d ¼ Az þ B Gz

ð Þ¼ A BG ð Þ A CL z : ð 13 : 15 Þ

Eigenvalues and eigenvectors of A CL (summarizing the properties of the closed
loop controlled system) are different from those of A or, in other words, fre-
quencies, damping ratios, and modal shapes of the controlled system are different
from those of the uncontrolled one. Therefore, a question arises if modal param-
eters can be modified in a more favorable way. An answer to this question was
probably first introduced by Moore [ 11 ] and then explored by many scholars, but
the authors are unaware of any application to the particular case of SA controlled
wind turbines. In the latter case, in order to reduce stresses in the supporting tower,
it would be desirable to have a first modal shape of the controlled structure
dominated by a highly damped rigid motion around the hinged base and higher
modes with mass participation factors close to 0.
Let us assume that a matrix G does exist so that s d,i and w d,i are the desired
eigenvalue and eigenvector of the i-th mode of the closed loop (CL) system. When
the CL system vibrates according to that mode, the system state varies proportionally
to the displacements and velocities described by the corresponding eigenvector:

e s d ; i t

z ð t Þ¼ w d ; i ð 13 : 16 Þ

and in this case the desired control forces f ui can be expressed as:

f ui ðÞ¼ u i e s d ; i t ¼ Gz ð t Þ¼ G w d ; i e s d ; i t : ð 13 : 17 Þ

The product A CL w d,i can be written as:

¼ A w d ; i þ Bu i :

A CL w d ; i ¼ð A BG Þ w d ; i ¼ A w d ; i þ B G w d ; i ð 13 : 18 Þ

Being s d,i and w d,i an eigenvalue and the corresponding eigenvector of A CL , the
same product is also equal to:

A CL w d ; i ¼ s d ; i w d ; i : ð 13 : 19 Þ

By combining Eqs. 13.18 and 13.19 ,

A CL w d ; i ¼ A w d ; i þ Bu i ¼ s d ; i w d ; i

ð 13 : 20 Þ

or

w d ; i

Bu i ¼ s d ; i I A ð 13 : 21 Þ