Advanced system identification techniques for wind turbine structures Page: 7 of 10
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teger. We then solve for the observer Markov parame-
Y=Di CB: CABL... :CA B] (12)
using a least-squares algorithm.
The observer Markov parameters in equation (12) in-
clude the system Markov parameters and the observer
gain Markov parameters. The system Markov parame-
ters are used to compute the system matrices A, B, C,
and D, whereas the observer gain Markov parameters
are used to determine the observer gain matrix G. The
proper algorithm for obtaining these Markov parame-
ters has been introduced by Phan et al.  and is also
discussed by Juang . Software implementation of
this identification algorithm was developed at NASA
Langley and is known as the Matlab function OKID.
Finally, the state-space representation (A, B, C, D) of
the system is obtained using the Eigensystem Reali-
zation Algorithm (ERA), based on system realization
It can be proven that the truncated observer model
(12), obtained as a result of the dead beat approxima-
tion of equation (10), produces the same input-output
map as a Kalman filter if the data length is sufficient
so that the truncation error is negligible. In this case,
G, when computed from the combined Markov pa-
rameters of equation (12), gives the steady-state Kal-
man filter gain K = -G.
5 IDENTIFICATION PROCEDURE
The identification of the Observer/Kalman Filter
model of a wind turbine is performed by the MATLAB
program fiokuy.m which uses the MATLAB function
okid. The initial estimate of the number of observer
Markov parameters is specified considering that the
maximum system order that can be identified equals
the product p.m where p is the number of Markov
parameters considered and m is the number of meas-
urements (or outputs). Using the measurement ma-
trix, the Hankel matrix is formed and a plot of its sin-
gular values is displayed to aid in selecting the correct
system order. After selecting system order, the per-
centage of data realized by the model is computed. It
is recommended to choose the lowest system order
resulting in 100% realization of the measurement
data. The corresponding modal parameters are also
displayed on the screen in a tabular form showing the
mode singular values (SV) and modal amplitude co-
herence (MAC) factors. This provides additional
evaluation of the quality of the identified model. Exam-
ining this table, the user can determine the modes
whose contribution to the system dynamics is insig-
nificant. Such modes can be classified as the noise
The identified system matrices A, B, C, D, generated
by the program for the structure model of a selected
order, are available as MATLAB variables Af, Bf, Cf
Df. The identification error is displayed in the Figure
The next step is to run other identification programs
for all input-output data files. They return the list of
identified eigenvalues and corresponding modal fre-
quencies in [rad/s and [Hz], as well as system zeros
related to the selected output. The frequency re-
sponse plot is also displayed in the Figure Window. A
special program can be used to enlarge a selected
portion of this plot.
All the outlined steps of the above procedure are illus-
trated in the Appendix, where a case study is pre-
sented using simulation data obtained from the
ADAMS model of the Micon 65/13 wind turbine.
The input-output time-series obtained from the virtual
wind turbine were used to develop and to validate the
identification procedure presented above. It was
found that to identify all vibration modes, we have to
process, repeating the same procedure, the in-
put/output time-series for both in-plane and out-of-
plane excitations applied at various points of the
wind-turbine structure. This has been done for three
data files generated by out-of-plane excitations, collo-
cated with the measurements near the tips of two
blades and at 2/3 of the height of the tower, and for
one data file generated by the in-plane excitation col-
located with the measurement at 2/3 of the height of
For each of the four above listed data files, each con-
taining five measurements, the Observer/Kalman Fil-
ter state-space model was identified interactively in
order to determine the model order providing the best
fit for the measurement data. The corresponding set
of modal parameters was generated. Then, for each of
the five input-output pairs, the frequency response
was plotted and the corresponding set of system zeros
and their frequencies determined.
The Appendix presents the scope of the tests per-
formed. It also gives the numerical results of modal
parameter identification, graphically illustrated by
frequency response plots. This graphical illustration
is most distinct on the frequency response plot for the
system output (measurement) collocated with the exci-
tation used to obtain the analyzed data file.
Examining all the capabilities of the developed identi-
fication software tools, it seems that the scope of ap-
plied research this software could support is very
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Bialasiewicz, J.T. & Osgood, R.M. Advanced system identification techniques for wind turbine structures, article, March 1, 1995; Golden, Colorado. (https://digital.library.unt.edu/ark:/67531/metadc688470/m1/7/: accessed April 26, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.