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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.

A very effective implementation of this is using a combined sliding mode control (SMC) and adaptive control law. This combines the resistance to uncertainties that SMC offers, with the reduction in uncertainties that results from adaptive control. Keep in mind though that this is of limited practical use. A quick example could is detailed as follows:

Consider the system of the form, $$ x^{(n)} = sum_{i = 1}^{m}gamma_iphi_i(textbf{x}) + butag{1}$$

$begin{align}text{where, }\&textbf{x} = begin{bmatrix}x dot{x} ... x^{(n-1)}end{bmatrix}^Ttext{ is the state vector} \ & phi_i(textbf{x}) text{ are some known functions} \ & gamma_i, btext{ are constant unknown parameters}end{align}$

It is assumed that the sign of $b$ is known and is assumed positive. Now consider, as in all SMC formulations, a Hurwitz stable linear combination of the system error. Here I will take a combination as suggested in Slotine and Li, $$s = left(dfrac{d}{dt} + lambdaright)^{n - 1}e$$

where, $e = x - x_d$ given some suitably differentiable desired trajectory $x_d$

Now if we take derivative of $s$, $$dot{s} = sum_{i=1}^{m}gamma_iphi_i(textbf{x}) + bu - vtag{2}$$
where, $v = x_d^{(n)} - lambda x^{(n-1)} - text{ ....}$

Using parameter estimates, let $u = hat{b}^{-1}(v - Ksgn(s)) - sum_{i=1}^{m}hat{gamma_i}phi_i$, where $K > 0$

Substituting this in (2) and rearranging we get,
$$dot{s} = sum_{i=1}^{m}(gamma_i - bhat{gamma_i})phi_i + (bhat{b}^{-1} -1)(v - Ksgn(s)) - Ksgn(s)tag{3}$$

Consider the Lyapunov function, $ V = frac{1}{2}(s^2 + b^{-1}(bhat{b}^{-1} -1)^2 + b^{-1}sum_{i=1}^{m}(gamma_i - bhat{gamma_i})^2phi_i) $. Taking its derivative,

$$begin{align}& dot{V} = sdot{s} + (bhat{b}^{-1} -1)dot{hat{b}}^{-1} - sum_{i=1}^{m}(gamma_i - bhat{gamma_i})dot{hat{gamma_i}} \ & dot{V} = (bhat{b}^{-1} - 1)(vs - Kmid smid + dot{hat{b}}^{-1})
+ sum_{i=1}^{m}(sphi_i - dot{hat{gamma_i}})(gamma_i - bhat{gamma_i}) - Kmid smidtag{from (3)}end{align}$$

Choosing adaptation laws as:
$$begin{align}&dot{hat{b}}^{-1} = Kmid smid - vs \ & dot{hat{gamma_i}} = sphi_iend{align}$$

we are ensured that $dot{V} < 0$ and hence we have global asymptotic stability.

As is obvious, to use above formulation the dynamics of the system must be linear in unknown parameters. If that is not the case, one can make use of adaptive fuzzy control (as direct or indirect adaptive fuzzy control) to solve the problem. This again can be coupled with sliding mode control.

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.