The Linear Quadratic Regulator (LQR) design method is well known to provide the optimal static state feedback with respect to a specific cost function.
It is also well known that LQR can be extended to include an integrator term to remove steady state errors. This is known by the name Linear Quadratic Integral (LQI) control.
However, I have (so far) never heard of, for example, Linear Quadratic Derivative ("LQD") control which would include an extra derivative term (similar to the integral term in LQI).
Or, if generalized, any additional dynamics in the controler (integral/derivative of higher orders).
Question: Does such an extension of the standard LQR exist and how is it called? If yes, why is it not used more frequently?
Since you mentioned static state feedback I will only consider the LQR problem for infinite horizon of linear time invariant systems with constant cost matrices $Q$, $R$ and $N$. I believe the same can also be shown for time varying matrices, but the expressions will probably become a bit more messy. Therefore, under these assumptions the LQR problem can thus be formulated as
\begin{gather} J(u(t)) = \int_0^\infty x^\top\!(t)\, Q\,x(t) + 2\,x^\top\!(t)\, N\,u(t) + u^\top\!(t)\, R\,u(t)\,dt, \tag{1} \\ \min_{u(t)} J(u(t)), \tag{2} \\ \text{s.t.}\quad \dot{x}(t) = A\,x(t) + B\,u(t). \tag{3} \end{gather}
One can add terms containing time derivatives of the state to the cost function from $(1)$ and for example define a new cost function as
$$ J(u(t)) = \int_0^\infty x^\top\!(t)\, Q_p\,x(t) + 2\,x^\top\!(t)\, N_p\,u(t) + \dot{x}^\top\!(t)\, Q_d\,\dot{x}(t) + 2\,\dot{x}^\top\!(t)\, N_d\,u(t) + u^\top\!(t)\, R_p\,u(t)\,dt. $$
However, by using the system dynamics from $(3)$ this would just be equivalent to using
\begin{align} Q &= Q_p + A^\top Q_d\,A, \\ R &= R_p + B^\top Q_d\,B + B^\top N_d + N_d^\top B, \\ N &= N_p + A^\top Q_d\,B + A^\top N_d. \end{align}
It can be noted that when adding cross terms between $x(t)$ and $\dot{x}(t)$ one can still find equivalent $Q$, $R$ and $N$ matrices by using $(3)$. So the problem remains of the same type as the original LQR problem and no extension similar to LQI is needed.
It can be noted that when including terms containing the time derivatives of the input this does change, since those time derivatives are not well defined at the initial conditions. This could for example also happen when terms are added to the cost function containing higher order time derivatives of the state $x(t)$, since for example $\ddot{x} (t)$ also contains a $B\,\dot{u}(t)$ term. One way this could be solved is to add all lower time derivatives to the state (so including $u(t)$ itself as well) and use the highest time derivative of the input as the new virtual input. However, I am not aware of how this might be called in academia. However, the problem does still essentially remain the same, only the considered input is a virtual one.