Derive Non-Linear Transfer Function

[CCLIUTW]

Hello engineers:

I have an question of derive transfer function in nonlinear control system, as title.
Generally, a linear system can easy derive transfer function like the picture, but it is difficult to derive transfer function when it include friction model.

I want to ask every excellent engineers can derive the transfer function?

Friction model is a velocity function : F(w) = (Fs-Fc)exp(-(w/ws)^2)+Fc*sgn(w)+q2*w
Fs, Fc, ws and q2 is known


Thank you!!

 

[xnuke]

Transfer functions only exist for linear time-invariant systems.

xnuke
"Live and act within the limit of your knowledge and keep expanding it to the limit of your life." Ayn Rand, Atlas Shrugged.
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[CCLIUTW]

Thank you for your answer, xnuke.

 

[PNachtwey]

Why can't a transfer function be expressed as a non-linear differential equation or system of non-linear differential equations? I do this all the time for hydraulic simulations. It looks like CCLIUTW is trying to add Stribeck friction to his simple motor system in velocity mode.

This is an example of why I say that serious simulations use differential equations instead of Laplace transforms or state space.


Peter Nachtwey
Delta Computer Systems

http://www.deltamotion.com

http://forum.deltamotion.com/

 

[xnuke]

A transfer function (in the continuous-time domain) is the ratio of the output of a linear time-invariant system to the input of the system, in the s-domain, when all initial conditions are zero. I believe the requirement for linearity comes from the application of the Laplace transform to convert from the time domain to the complex frequency domain. Typically, some method is used to linearize a nonlinearity like sgn(ω) (e.g., using the describing function of ideal relay for the sgn function in this case) when an approximate transfer function is required so that analytical methods can be used.

I don't disagree with you about simulations, Peter. Simulations using differential (really, difference) equations allow systems that cannot be solved analytically to be solved numerically. I

xnuke
"Live and act within the limit of your knowledge and keep expanding it to the limit of your life." Ayn Rand, Atlas Shrugged.
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[CCLIUTW]

Thanks for all response.

Yes, this friction model is Stribeck friction model, and I tried to derive W_out(t)'s differential equations, but it is too difficult to get W_out(t).

I will try to linearize Stribeck friction model, and wish it can get good result.

 

[PNachtwey]

CCLIUTW, use differential equations then integrate with Runge Kutta.
I was once hired by a construction company that was replacing the dam gates on a dam just down stream from Pittsburg, PA.
The army corp of engineers ( ACE ) didn't want the dam gates to drop due to leakage so they insisted that the hydraulic cylinders lifting the gates had no leak ( read too tight to move easily ) seals. The dam gates would vibrate at they moved which could damage the dam since the gates weighed about 100 tons a piece. The ACE accused the construction company of putting in the gates wrong. We monitored to the gates with position, acceleration and pressure sensors and used this data to make a model. The Stribeck friction was severe and causing the slip-stick motion. I was able to show it was the seal friction from the no leak seals that was the problem not the construction company's installation that was at fault. This got the construction off the hook and the ACE had to admit it was the seals

If you want to get the job done and model the system accurately then use differential equations.
Start with the two differential equations shown in equations 3 and 4 here

http://ctms.engin.umich.edu/CTMS/index.php?example=MotorSpeed&section=SystemModeling

Convert your force(omega) to Torque(omega) and insert it into equation 3.



Peter Nachtwey
Delta Computer Systems

http://www.deltamotion.com

http://forum.deltamotion.com/

 

[CCLIUTW]

Hello Nachtwey,

Thank you for your sharing, it is an interesting experience, and RK method is very useful.
I used RK method calculated the results of W, and it is right result compare with Simulink.
I learned a lot of knowledge! Thank you so much. Today is very lucky! You are my Santa this year.

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CCLIUTW

 

[xnuke]

Not surprising. I think the first solver Simulink tries (when VariableStepAuto is selected - the default for continuous-time models) is ode45 - which is a variable time-step version of the RK(4,5) integration method. You've essentially repeated what Simulink did using a slightly different numerical integrator. I'm happy you learned something, though.

xnuke
"Live and act within the limit of your knowledge and keep expanding it to the limit of your life." Ayn Rand, Atlas Shrugged.
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.