Add state noise for discrete dynamic function

Hi :wave:

I am a new to acados, thanks for your nice frame. I now want to configure MHE for a wind turbine model in python. I already have continuous implicit dynamic equations like 0=f(x, x_dot, u, p) and want to add additional state noise term w. From your MHE example, you seem directly add the noise term behind the continuous function like 0=f(x, x_dot, u, p)+w.
But if it is possible that add this noise term to the discretized equations (after multiple-shooting or use some discrete method based on acados or casadi) to make the function looks like x_k+1 = g(x_k, u_k, p)+w_k
Thanks for your help!
Best regards

Hi Benjamin,

I’m pretty sure the two formulations are equivalent up to scaling. If you have an implicit ODE of the form

0 = \tilde{f}(x, \dot{x}, u, p, w)

where \tilde{f}(x, \dot{x}, u, p, w) = f(x, \dot{x}, u, p) + w the resulting discrete dynamics that you get from the Runge Kutta integrator are of the form

x^+ = x + h\sum_i b_i k_i

where h is the integration interval and

k_i = \tilde{f}(*, *, u, p, w) = f(*, *, u, p) + w.

As the b_i coefficients sum to 1, the discrete dynamics take the form x^+ = g(x, u, p) + hw. Does that help?

Best, Kaethe!

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Hi kaethe,
Thanks for your quick answer, you are right, I only need to do some scaling for the noise term in the continuous equation to get similar results as in the discrete case. Thanks for your answer, again :smiley:
Best Regards,

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