# Advanced system identification techniques for wind turbine structures Page: 6 of 10

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each of the three driving points. We also applied one

in-plane excitation at 2/3 of the tower height. To see

the difference in the frequency response, i.e., to estab-

lish that the excitation and consequently the identifi-

cation of different modes depends on both the driving

point and the set of measurement points, we per-

formed the identification twice. We used the set of

out-of-plane measurement points the first time and

the properly selected set of in-plane measurement

points the second time. Corresponding pairs of meas-

urement points, related to measuring acceleration in

two different directions, have the same locations on

the wind turbine structure.

The important conclusion for modal testing on a real

wind turbine structure is that the number of meas-

urement points can be substantially reduced without

loss of the modal information. Such a properly se-

lected driving point-measurement set leads to accu-

rate identification. This statement is supported by

comparing frequency responses for a given excitation

point and different outputs. It can be seen from fre-

quency responses for any driving point that the fre-

quency response for a collocated excitation-

measurement pair gives the best resolution of the sys-

tem's resonance modes.

4 WiD TURBINE STATE-SPACE MODEL

IDENTIFICATION

The state-space model is generated using a proper

input-output sequence, generated as discussed in

Section 2 and Section 3. The discrete-time state-

space model (A,B,C,D) to be identified, defines the fol-

lowing relation between the scalar driving excitation

u(k) and the measurement m-vector (or output) y(k):

x(k+1) = Ax(k)+ Bu(k)

y(k) = Cx(k) + Du) (5)

Note that this state space model depends on the

choice of the state vector x4t) and the sampling interval

T. Assuming that x(0)= 0 and solving for the system

output, we obtainy(k) = CA -1 Buk- i) + Du(k)

(6)

Equation (6) represents the convolution of the sys-

tem's input sequence u(k) and the sequence Y(k) with

the following elements:

Yo=D,YI=CB,Y2=CAB, ...,Yk=CAk-1B (7)

Therefore, these elements represent consecutive sam-

ples of the system's pulse response and are known as

Markov parameters. Assuming that our input-output

sequence has a length of 1, we can write l equations of

the type of (6) with the number of terms on the rightside increasing as the new input-output pairs become

available. This set of i equations can be represented

by the following equation:

[y(o:yy);:.. - =[D CB: CAB: ... iCA1-2BI

u(0) u(1) uj2) ... u(l - )

u(0) u(I) ... u(-2)

u(0) ... u(- 3)

U(O)(8)

The wind turbine structure is a flexible structure with

lightly damped low-frequency modes. For such a

system and a sufficiently large p,

Ak =0 for k > p

This signifies that to solve for the Markov parameters

as an adequate system representation, a sufficiently

large l is required.

As alternative possible approach is to artificially in-

crease system damping to solve for Markov parame-

ters. The observer model of the system is used in this

approach. The state equation (5) can be manipulated

as follows:x(k +1) = Ax(k) + Bu(k) + Gy(k) - Gy(k)

= (A + GC)x(k) + (B + GD)u(k) - Gy(k)(9)

where G is an nxm matrix chosen to make A+GC as

stable as desired. Equation (9) can be rewritten in a

standard compact form:x(k + 1) =Ax(k) + B()

where

A=A+GC, B=[B+GD - G],(10)

v(k) =

y(k)Now, we can write an equation, similar to (8),. but one

which involves observer Markov parameters:

-p-1 1-

Y = [D: CB: CAB: ... :CA B... :CAl 2](1

For an observable system, we can assign the eigenval-

ues of A arbitrarily through a proper choice of G. In

the case of the dead beat observer, i.e., when all the

eigenvalues of A are placed at the origin,

-k -

CA B = 0Ofor k ? p where p is a sufficiently large in-

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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/m1/6/: accessed April 22, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.