TRYING TO SOLVE SIMPLE SYSTEM OF DIFFERENTIAL EQUATIONS

Anyone knows how to stabilize the Systems of Differential Equations attached?? Breaking my head trying to figure it out but can't find the problem.

The equations are the ones corresponding to a very very simplified dynamic model of a motorcycle. Basically what I am trying to do its include a simple PID controller to simulate a "pseudo" virtual rider and hopefully solve the problem with that, as the system by itself it's naturally unstable.

[Geoff Hayes]

Jacobo - there is no attachment.

[Jacobo Puyoles]

Geoff, it should be there now.

[Jacobo Puyoles]

Sorry, I attached the wrong file in which the variables where not defined! I updated the model now.

Answer 1

[Arkadiy Turevskiy]

Open the PID Controller block dialog, and press the "tune" button.You get the PID Tuner app open and it says that it could not find the stabilizing controller.

Reduce the response time by clicking on the "slower" button in the top slider twice. You get this:

Press "Update block" to apply the gains.

You get these block parameters:

Click ok, run the simulation and look at the reference and feedback signals:

Looks like controller is doing a good job.

HTH

[Jacobo Puyoles]

Arkadiy, the only way of obtaining the same parameters for the PID controller as you did is to introduce them manually. Even so, the system still does not work properly. I am not sure whether or not I am doing something wrong.

Attached is the modified system with the new PID controller

[Arkadiy Turevskiy]

No, entering PID gains manually is not the only way. In the first answer I outlined the exact steps I followed to get those gains.

As for "the system not working properly", you need to clearly define what that means. As you have it, you just have one PID controller to control pisdot. With the gains PID Tuner comes up with, PID controller tracks your psidot reference very well. I am attaching the model with a viewer for psidot and the reference signal. Run the simulation and look at this viewer to see that the controller tracks the reference signal very well.

[Jacobo Puyoles]

Sorry,I explained myself incorrectly. I meant that I could not reach the controller parameters you showed by using the "slower" button in the PID tuner, that is why I type them myself.

Yes, I can see how the reference is being track very well indeed, thank you for that. My problem is that the overall solution which I am after, i.e. y, psi, beta and roll angle (the right most 4 input scope) does not produce the "theoretical" response, as it only produces a solution between 4.90 and 5 seconds approx, as shown in the figure. Also the graphs shapes are not well (attending to the results of the paper I got the equations from) so I deduced that the system is not working as I expected but can´t figure out why. The system is not commplex but still, I am missing something.

I really appreciate your help.