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."


1 Answer 1


A lowpass filter is typically a poor approximation of a delay, have you considered a Pade approximation instead? If you model the system in discrete time, modeling the delay becomes trivial (if it's an integer multiple of the sample time).

Having said that, you can easily calculate the gain margin for your system, this will indicate how much you can increase the gain of the loop-transfer function without the closed-loop system becoming unstable. With dead-beat control, we typically refer to a controller that steers the system to zero in as few samples as possible, which is quite different from what you're asking here (and also not applicable in continuous time).

I must ask, is the largest possible gain really what you want? The largest gain such that the system is stable will by definition make the closed-loop system marginally stable, i.e., it will have terrible performance. Are you perhaps really looking for a feedback gain matrix such that the system reaches zero as quickly as possible? This would be what is typically referred to as a dead-beat controller, but different from

The largest proportional gain possible without introducing oscillation?

A deadbeat controller can be found in discrete time by placing all poles in the origin.

  • $\begingroup$ I was interested in the gain mostly as a sort of "measuring stick" by which to compare a number of related systems, not as an actual gain to be used in practice. I chose it because it seems easy to interpret, more than because it seems itself useful. The context of the problem is in modeling the amount of useful feedback control that is lost when a known signal delay is introduced into the measurements. $\endgroup$ Jun 16, 2022 at 21:02

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