What are systematic ways of approximating a non-smooth (non-continuously differentiable) system dynamic to be n-smooth?

Hendrik Christoph Lorenz @Hendrik_Lorenz2

13 June 2019 3 7K Report

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.

Richard Epenoy

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