Saturation limit compared to constrained limit

Question

I have a simple question. What's the difference in behaviour between saturation limit and constrained limit in control theory?

We say that we got this objective function:

$$J_{min} = \frac{1}{2}x^THx + c^Tx $$ With subject to:

$$lb <= x <= ub$$

And then we got this saturation limit, where $u$ is input signal:

$$x = sat(u, lb, ub)$$

Will this be the same in practice or does the quadratic optimazation give a more smoother behaviour in SISO case?

Answer 1

No, it will not be the same except in trivial cases. If it was the same, we would not spend so much effort on developing efficient quadratic programming solvers.

They will have the same smoothness properties (piecewise affine over polytopic regions, i.e. not smooth)