I'd like to determine the lowest practical control frequency of a LQR controller needed for a double integrator-like system. What I do is to build the LTI representation
$$sys \triangleq A, B, C, D$$ of the system, and convert it into its discrete counterparts Ad Bd by doing $$[A_d,B_d] = c2d(A,B,Ts)$$ where $Ts$ is the sampling frequency of the controller, that is assigned and equal to 20 Hz. Then, I compute the lqr gains by using the function dlqr. $$K = dlqr(A,B,Q,R,Ts)$$ I tuned the controller by penalizing state errors (through $Q$) and control efforts (through $R$) and looking at the simulation results. The controller has to counteract errors on position and velocity, as well as parameters uncertainties.
What is the analytical tool I can use to determine the lowest control frequency? As far as I know, with the $R$ matrix I penalize the control magnitude, but I read somewhere that it also controls the closed-loop bandwidth of the system. How do I determine the minimum frequency that stabilizes the system?