Page **1** of **1**

### nonlinear state-space model

Posted: **Fri Apr 26, 2019 12:54 am**

by **zarei.atefe**

hello

I want to design a nonlinear individual pitch controller for wind turbine load reduction in my master's thesis, but I don't have the nonlinear state-space model of the 1.5 MW wind turbine available in the FAST test.

It is essential for me to have those non-linear equations to design a nonlinear controller without linearization. How can I access them?

Thanks.

### Re: nonlinear state-space model

Posted: **Fri Apr 26, 2019 6:48 am**

by **Jason.Jonkman**

Dear Zarei.Atefe,

FAST / OpenFAST itself employs nonlinear equations and each module is written in state-space form (in terms of inputs, outputs, states, and parameters. See the FAST modularization framework for more information:

https://www.nrel.gov/docs/fy13osti/57228.pdf.

Best regards,

### Re: nonlinear state-space model

Posted: **Tue Jun 11, 2019 1:59 am**

by **zarei.atefe**

Hello

thanks for your answer

To design a non-linear controller, I must have a nonlinear equation of pitch actuator and blade root bending moment related to the 13th test of FAST8.16. Can you give me them?

In fact, I want the nonlinear relation of the attached equation related to the 13th test of FAST8.16.

thanks.

### Re: nonlinear state-space model

Posted: **Tue Jun 11, 2019 3:20 am**

by **Jason.Jonkman**

Dear Zarei,

Just a few comments:

- FAST v8.16 does not include pitch-actuator dynamics, unless these are implemented by the user within the controller.
- The nonlinear equations of motion used by FAST are very large and cannot be written down in a forum post.
- What is theta in your equation?

Best regards,

### Re: nonlinear state-space model

Posted: **Tue Jun 11, 2019 3:53 am**

by **zarei.atefe**

theta is the output pitch angle given into the model.

Is it possible to email The nonlinear equations of motion used by FAST to me ?

My work depends heavily on having nonlinear equations of motion and pitch actuator.

thanks for your help.

### Re: nonlinear state-space model

Posted: **Tue Jun 11, 2019 4:28 am**

by **Jason.Jonkman**

Dear Zarei,

As I said, pitch actuator dynamics are not intrinsically included in FAST.

Please see the following post on our forum for information on the FAST theory basis:

viewtopic.php?f=4&t=424. WBut while the structural nonlinearities can be clearly seen e.g. in the "unofficial FAST theory manual", the aerodynamic nonlinearities are more difficult to write down because e.g. the blade element / momentum (BEM) theory depends on the solution to nonlinear algebraic equations.

Best regards,

### Re: nonlinear state-space model

Posted: **Tue Jun 11, 2019 5:00 am**

by **zarei.atefe**

thanks

and So, my question is, a nonlinear relation between the pitch angle and the rotor speed is available?

### Re: nonlinear state-space model

Posted: **Tue Jun 11, 2019 6:33 am**

by **Jason.Jonkman**

Dear Zarei,

I guess your asking about the aerodynamic torque as function of tip-speed ratio and blade-pitch angle? Such a surface can be calculated by FAST (including structural deformations), or by standalone AeroDyn (without structural deformations)--as discussed in other topics on this forum--but the output would be a tabulated relationship. You could fit a function to this tabulated surface to have nonlinear relationship.

Best regards,

### Re: nonlinear state-space model

Posted: **Tue Jun 11, 2019 8:12 am**

by **zarei.atefe**

Thank you very much for your help

Now, my question is, as you say, since non-linear equations of region 3 wind turbines are not specifically available, so can not design nonlinear controllers which require nonlinear dynamics?

### Re: nonlinear state-space model

Posted: **Tue Jun 11, 2019 8:16 am**

by **Jason.Jonkman**

Dear Zarei,

I'm not familiar with nonlinear optimal control theory. What form do the equations need to be in? Do you need an analytical expression or are tabulated relationships and/or calls directly to FAST usable?

Best regards,

### Re: nonlinear state-space model

Posted: **Tue Jun 11, 2019 11:12 am**

by **zarei.atefe**

i need the nonlinear differential equations of the wind turbine for region 3 goals.

But I can not find them.

thanks.

### Re: nonlinear state-space model

Posted: **Tue Jun 11, 2019 2:26 pm**

by **Jason.Jonkman**

Dear Zarei,

I believe I've already answered your question.

Best regards,

### Re: nonlinear state-space model

Posted: **Wed Jun 12, 2019 12:53 am**

by **zarei.atefe**

thanks.