# Validation of Linear System

Dear Jason,

I have few confusions regarding validation of linear system.

Background:

I have created MBC_A,MBC_B,MBC_C,MBC_D matrices through running getmats__f8 and mbc3 commands.
I took the avg of above mentioned matrices through “MBC_Avg(B,C,D) = mean(MBC_(B,C,D), 3)” commands respectively (is it correct to use this command in this particular case?).

Problem:

Now I created the state-space model in simulink and was trying to compare the outputs of open-loop linear system with open-loop nonlinear system but my outputs are not matching they are not even close.

Relevent Information:

I have gone through these two threads (http://forums.nrel.gov/t/fast-linearization-model-in-simulink/1381/1) thoroughly. The information i get from your posts there is as follows:

1: “linear model is defined in terms of perturbations about the operating point e.g. y = y_op + dy for system outputs, so, you should either add the operating point value (y_op) to the output of the linear model or subtract the operating point value from the output of the nonlinear for comparison.”

2: “The linearized model output by FAST is only valid for small perturbations about the operating point. It looks like you want your operating point to be based on steady 10-m/s wind. Thus, the wind-input disturbance represents a deviation from 10 m/s; you are setting this disturbance to 10 m/s, which actually implies a wind speed of 10 + 10 = 20 m/s, so, it is natural for the linearized model output to differ from the steady-state FAST solution. Likewise for the other input, state, and output perturbations.”

Questions:

1: As I have lineraized my system around 18 m/s, so If I apply 0.1 m/s wind speed as input to linear system then will it be considered 18.1 m/s for the system? (so on and so forth for every input)

2: In order to get my linear system validated, should I add my respective nonlinear output into my linear output? e.g. y = y_op + dy (as in my case: nonlinear Gen Speed (rpm)+linear Gen Speed (rpm) should be compared to nonlinear Gen Speed (rpm)).
Note: To compare rotating outputs I have to apply inverse MBC transform.

Best Regards
Syed Shah

Dear Syed,

Yes, I agree that MBC_AvgB can be calculated as mean(MBC_B,3) etc.

Regarding (1), “yes,” if the wind speed operating point is 18 m/s, then a perturbation of 0.1 m/s results an effective wind speed of 18.1 m/s.

Regarding (2), you should add the linearized output (dy) to the operating point output (y_op) for comparison to the nonlinear solution (y) i.e. y = y_op + dy.

I hope that helps.

Best regards,

Dear Jason,
Thank you for your quick help and clearing my doubts.
Best regards
SS

Dear Jason,

I have few confusions regarding validation of linear system.

Background:

I have created a state-space model in simulink in order to compare the outputs of nonlinear system with linear system.

Problem:

I have done the comparison with the above mentioned procedure. but having issues in comparing the results of the rotating blade loads.
I have done the inverse MBC transformation on rotating blade loads of linear system by taking the azimuth output from the state-space model. (I guess this is the issue because I am taking the linearized azimuth)

Questions

1: My question is how to perform the correct inverse MBC transformation to compare the results of rotating blade loads of both the linear and nonlinear systems??
2: which azimuth is to use for this purpose?

Best Regards
Syed Shah

Dear Syed,

As discussed in the following forum topic: http://forums.nrel.gov/t/fast-linearization-v7/1680/1, once you linearize the FAST model, apply MBC3, and azimuth-average the resulting matrices, you’ll end up with a linear state-space model without the azimuth state. Thus, when comparing the linear and nonlinear models, I would suggest applying MBC3 to the result of the nonlinear model in order to compare to the result of the linear model.

Best regards,

Dear Jason,
Thank you for your quick help.

1: If i want to apply MBC3 on the blade-loads of the non linear system how do i apply it? what is the correct way?

2: do i need to perform the MBC3 transformation to the individual blade pitch angle inputs before applying to the to the linear system ? because the system and the outputs are MBC3 transformed. if yes how to do this?

Dear Syed,

Regarding (1), look at the MBC3 manual (nwtc.nrel.gov/system/files/MBC3.pdf) and you’ll find matrices relating how to transform outputs in the rotating frame to outputs in the fixed frame through application of MBC3.

Regarding (2), yes, the linear system has been transformed through MBC3, so, the input blade-pitch angle perturbations are not perturbations of blades 1, 2, and 3; instead, they are perturbations of the MBC-transformed 0, C, and S terms. Again. see the MBC3 manual for more information.

Best regards,

Dear Jason,

so what I have concluded from your statement is, if following are the inputs to the state space system.

u1; horizontal wind speed (steady/uniform wind), m/s
u2; vertical power-law shear exponent, -
u3; IfW Extended input: propagation direction, rad
u7; Yaw moment, Nm
u8; Generator torque, Nm

the u4, u5 and u6 are considered by the system as 0, C, and S terms, instead of individual blade-pitch angle perturbations?

Best Regards
Syed Shah

Dear Syed,

After you apply MBC3 to the linear system matrices, I agree that inputs u4-u6 i.e. pitch angles for blades 1-3 are converted to pitch angle 0, C, and S terms.

Best regards,

Dear Jason,
Thank you for your help and clearing my doubts.
Best regards
Syed Shah

Dear Jason,
I have a problem in validating the above mentioned (in previous posts of this topic) MBC averaged system with the nonlinear system.
I have linearized system at 18m/s wind speed and if below are the input to the MBC averaged state space system;

u1; horizontal wind speed (steady/uniform wind), m/s
u2; vertical power-law shear exponent, -
u3; IfW Extended input: propagation direction, rad
u7; Yaw moment, Nm
u8; Generator torque, Nm

then after I apply the unit step input only at u4 = (1*pi/180), the SrvD GenTq, (kN·m), ED RotSpeed, (rpm), ED GenSpeed, (rpm) decreases. Rest of the inputs are zeros.

In my opinion due to the decrease in torque the rotspeed and genspeed is also decreasing. Torque should not decrease ideally. why torque is decreasing? kindly can you shed some light on the issue?

Best Regards
Syed Shah

Dear Syed,

What generator or torque-control model did you enable when you linearized the model? The choice of modeling and linearization operating point will explain why the drop in rotor speed is leading to a drop in generator torque.

Best regards,

Dear Jason,
I Have enabled “SIMPLE VARIABLE-SPEED TORQUE CONTROL” model.
Best regards
Syed Shah

Dear Syed,

The simple variable-torque controller has no region where a negative change in rotor speed leads to a positive change in generator torque.

Best regards,

Dear Jason,

Then why do the rotspeed decreases with the the step increase in input of pitch angle?

Best regards,
Syed Shah

Dear Syed,

Above rated wind speed, I would expect the power, torque, and thrust to drop with an increase in blade-pitch angle. The drop in torque should lead to drop in rotor speed.

Best regards,

Dear Jason,
Thank you for answering my previous questions.

I agree to what you have said. But Ideally both the linear and nonlinear systems should behave somewhat similar.

I want to validate my linear system with nonlinear system.
For this reason, when i see the behavior of nonlinear system. Whether i apply step in wind speed or blade pitch angle, the torque is constant. To achieve this the nonlinear system only make adjustments in rotor speed and generator speed.

But for the linear system when i apply the unit step wind in the input, the torque increases and linear system make adjustments accordingly in rotor speed and generator speed. vice versa if i apply the unit step in blade pitch angle.

so the question in my mind is that why the systems are behaving differently in this regard i.e. for nonlinear case torque is constant as i am applying constant torque to the system. but the linear system’s torque is changing after applying unit step in blade pitch or wind speed, Ideally i am apply constant torque for the linear system as well, why is this so?

can you shed some light on the issue? or correct me if i am wrong.

Best regards
Syed Shah

Dear Syed,

I agree that the linear and nonlinear system should have similar behavior for small perturbations about the operating point for models that are set up the same. Do you have the same torque control model enabled for your linear and nonlinear system? I would have to see the behavior you are describing to better understand what is happening and why you are not seeing the behavior you expect.

Best regards,

Dear Jason,

Please see the attachment for the comparison of above mentioned outputs.

For linearization, i have enabled “simple variable-speed torque control” in which I define rated generator torque and rated generator speed.

As far as the torque control model is concerned, I have tried 2 torque control models, but the problem discussed before is still there.

1: Same torque control model is enabled for both - linear and nonlinear - systems. (simple variable-speed torque control)

2: For linearization, i have enabled “simple variable-speed torque control” and for nonlinear system, i have enabled “user-defined variable-speed control mode from Simulink” in which i apply the rated generator torque to the nonlinear system.

In my opinion, (perhaps) FAST has implemented some sort of closed-loop control during the linearization procedure due to which the generator torque is dependent on generator speed. I have looked at the averaged system of matrices before and after applying MBC transform and verified that the generator torque output in the C matrix depends on the generator speed state alone. How to solve this issue?

Best regards
Syed Shah
Comparison of SrvD GenTq ED RotSpeed ED GenSpeed.zip (377 KB)

Dear Syed,