Linearization: d_x != A*x+B*u, y != C*x+D*u

Hi,

with FAST 8 linearization, the result is four matrices and operational state conditions u_op, x_op, y_op. With these, I am expecting to find

xdot_op = Ax_op + Bu_op
y_op = Cx_op + Du_op

However, this is not the case:

[code]x_op’ = Ax_op + Bu_op
0.0000
0.0107

x_op’ FAST
1.0e-05 *

0.4259

-0.9195

y_op = Cx_op + Du_op
1.0e+03 *

     0
     0
4.8246

-0.0825
-0.0005
-0.0006

y_op FAST
1.0e+03 *

-0.0098
-0.0001
5.7420
0.5519
0.0002
0.0003[/code]

I attached my input files, which are the basic 5MW reference turbine with very few degrees of freedom. I use Matlab to structure linearization and linear model. It should be possible to run main.m and reproduce my results (setting FAST8 directory name might be required, but is straight forward).

Where’s the error in my understanding of FAST 8’s linearization?

Thanks,
Tobi
linearization.zip (93.4 KB)

Dear Tobias,

I have not looked at your attachment, but your expectation is incorrect. The linearized system of equations output from FAST v8 apply to perturbations about the operating point (OP), not to the OP themselves. That is, let:

dx = x - x_op
du = u - u_op
dy = y - y_op

Then, the FAST linearization gives the A, B, C, and D matrices for:

dxdot = Adx + Bdu
dy = Cdx + Ddu

More information is available in the following publications:
nrel.gov/docs/fy17osti/67015.pdf
nrel.gov/docs/fy13osti/57228.pdf

I hope that helps.

Best regards,

It does indeed. Thanks for the quick reply!

Tobi