Say we have SISO system (e.g. the velocity of a permanent-magnet motor) that we're controlling with proportional feedback, with the condition that our measurements of the process variable are delayed by some nominal period $\tau$. For the sake of simplicity, we'll account for this by instead considering the system composed with a single-pole IIR filter with time constant $\tau$.
Is there a way to calculate an equivalent of a "deadbeat gain" for such a system - that is, the largest proportional gain possible without introducing oscillation?
Some background: I'm working on a system identification tool that estimates optimal feedback gains for proportional control of certain motors. The tool currently uses LQR, and generally works quite well - however, there are problems with aliasing when the sample rate and the measurement delay are of similar size, which can sometimes result in LQR-optimal gains that are in practice large stable oscillations rather than quick exponential transients. I'm wondering if there's a more direct way than LQR to get the "largest stable feedback gain."