Thank you for this, this was really interesting, I think this effect is not known enough. It is similar to pilot-induced oscillation in aerospace https://en.wikipedia.org/wiki/Pilot-induced_oscillation but with several layers of pilots
I've never heard of it before, but pilot-induced oscillation seems to be an instance of more general phenomena in control theory called ringing, which happens when there is too much overshoot (when the system goes past the desired position when you're trying to control it). To avoid having the system oscillate due to ringing we often need to lower the amount of overshoot, but by doing so we inevitably make our own control response slower.
This is specially bad for systems with non-minimum phase. Such systems have a counter-intuitive property: once you give it an input, for example, to move in a certain direction, it first moves in the opposite direction (which is called undershoot) and then, after a delay, it moves to the desired position, and perhaps overshoot. The trouble is that as the system undershoots the control operator might be tempted to increase the signal to counteract this, but it makes things worse, increasing the undershoot and, after the relevant delay, the system will severely overshoot. There's a mathematical theorem that says that if the delay is too great it's impossible to control a system like this.
PS: since control theory and signal processing share an underlying theory of linear, time-invariant systems, there's an analogous phenomenon on signal processing, https://en.wikipedia.org/wiki/Ringing_artifacts - this kind of graph https://en.wikipedia.org/wiki/Ringing_artifacts#/media/File:... shows that ringing in signal processing and in control theory have the same mathematical treatment, and in both cases the cause is overshooting too much.
My understanding is that Overshoot and Ringing are mostly amplitude-related phenomena, they happen even if the transfer function has a phase of zero over the spectrum, as long as the gain is greater than one over the right part of the spectrum.
On the other hand, pilot induced oscillation, and the bullwhip effect seem to need some amount of phase-shift or delay to happen.
> Category 1 PIO: Characterized by oscillations with an underlying linear cause such as excessive time delay, phase loss, etc., which makes it easy to understand and study. Several criteria for manned aircraft focusing on excessive phase loss and time delay have already been developed. Certain criteria are based on open-loop analysis such as the Bandwidth/Pitch rate overshoot criteria [6,7], while criteria such as Neal-Smith [8] is a closed-loop analysis method with an assumed pilot model.
This one I think is kind like the non-minimum phase system I described and also like your description of delay being the root cause.
> Category 2 PIO: Characterized by nonlinear events which can be modeled as Quasi-linear events such as actuator rate limiting or amplitude limiting, etc. This is the most common type of PIO observed. Most PIOs associated with non-linear events were found to be “cliff-like” [5]; that is, the pilot reported the onset of the PIO as sudden and unexpected. Since control surface actuator rate limiting is a common non-linearity associated with modern flight control systems [9,10,11], most of the studies are focused on studying its influence on aircraft handling quality and PIO. Currently, Open Loop Onset Point (OLOP) developed by Holger Duda at Deutsches Zentrum für Luft-und Raumfahrt e.V. (DLR) is the only commonly accepted criterion for Category 2 PIO resulting from rate limited actuator in the fully rate saturated case [6,7].
This one goes way over my head
> Category 3 PIO: This category of PIO is caused by highly nonlinear events which involve transition in the control element of the aircraft or the human pilot behavioral dynamics. The non-linearities associated are more complex and cannot be modeled as quasi-linear effects. The PIOs associated with this category are also “cliff-like” [5]. Category 3 PIOs are difficult to recognize and are relatively rare, but could be highly dangerous when they do occur.
https://en.wikipedia.org/wiki/Overshoot_(signal)
https://en.wikipedia.org/wiki/Ringing_(signal)
This is specially bad for systems with non-minimum phase. Such systems have a counter-intuitive property: once you give it an input, for example, to move in a certain direction, it first moves in the opposite direction (which is called undershoot) and then, after a delay, it moves to the desired position, and perhaps overshoot. The trouble is that as the system undershoots the control operator might be tempted to increase the signal to counteract this, but it makes things worse, increasing the undershoot and, after the relevant delay, the system will severely overshoot. There's a mathematical theorem that says that if the delay is too great it's impossible to control a system like this.
https://ealizadeh.com/blog/non-minimum-phase-systems
PS: since control theory and signal processing share an underlying theory of linear, time-invariant systems, there's an analogous phenomenon on signal processing, https://en.wikipedia.org/wiki/Ringing_artifacts - this kind of graph https://en.wikipedia.org/wiki/Ringing_artifacts#/media/File:... shows that ringing in signal processing and in control theory have the same mathematical treatment, and in both cases the cause is overshooting too much.