I have a system dynamic that is non-smooth because it has several signum and absolute value functions in it (three-tank level control).
I can obviously choose different sigmoid functions to approximate the dynamics in a way that they become n-times continuously differentiable.
Can somebody point me out to a systematic approach for doing this?
I can think of many ways to approximate the functions but it is difficult for me to decide on the "best" one.
Right now I would base my decision on the following criteria:
- smoothness (differentiability class)
- numerical computation effort (especially concerning integration)
An example term that I would want to be smooth would be:
$$
sign(x_1 - x_2) \sqrt{abs(x_1-x_2)}
$$
The context of my question is nonlinear model predictive control.
Hi, you are right sigmoid functions do the job. More generally, a convolution between the non-smooth function and a probability density function leads to a smooth approximation of the original function. We used this ifor solving optimal control problems with bang-bang controls in the following paper:
R. BERTRAND, R. EPENOY “New Smoothing Techniques for Solving Bang-bang Optimal Control Problems - Numerical Results and Statistical Interpretation”, Optimal Control Applications and Methods, Vol. 23, No. 4, pp. 171-197, 2002. doi:10.1002/oca.709
Best
Hi,
Please take a look at this attached paper it's on subject of approximatin a non-smooth non-continuously differentiable.
Best regards
Google "splines" and all the technical terms that you have - and see what links come out! :-)
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