I have a qurius question! Is it possible to design a robust controller for a system by using algoritms and system identification, which are adaptive control + robust control?
I know there is a lot of math to do this, but is it possible? For example, I create an algorithm which identify the system and then creates a transfer function. With that transfer function, the algorithm designs a $H_{\infty}$ controller with integral action. It would be like a PI-controller with guaranteed stability margins and autotuning.
Yes it is.
The idea of indirect adaptive control, often called certainty-equivalence, is to estimate parameters in real time, and design a controller for the estimated plant model as if they were the real plant parameters. The control design method is left open - robust control is a possibility, as are many others. The resulting controller is unlikely to be a PI-controller because 1) it is adaptive, thus nonlinear and time-varying; and 2) H infinity controllers are most often of high order.
Caveat: adaptive controllers tend to be somewhat complex, and their performance in practice is very much dependent on the prior knowledge available about the plant. It is not realistic to expect good behavior if your initial estimates are far off the reality. More complicated methods such as adaptive neural networks and model-free controllers, as suggested in the comments, make even more stringent requirements on prior knowledge and controllers training, otherwise their performance is even more pitiful.