# Seeking examples where PID control fails [closed]

I seek examples of systems that are difficult to control using standard PID control. In particular, examples where the output eventually becomes unstable despite being stable initially.

• Nonlinear systems, certainly. Commented Jun 23 at 1:47
• I am looking for the dynamics explicitly. Commented Jun 23 at 2:01
• Classic examples are systems with backlash (hysteresis) and/or dead time. I see Fabio's already mentioned the Smith predictor. Backlash compensation in motion control is also outside of PID. Commented Jun 23 at 5:10
• @DSPinfinity Variable and long delays are notoriously hard for PID. So are processes with undamped oscillatory poles (especially bad in this case with load disturbances, less a problem with setpoint changes.) When you say dynamics explicitly are you asking for a time domain case, such as this system with dead time: $\frac{\text{d}}{\text{d}t}y=-\frac12 y+\frac12u(t-6)$? Or what, exactly? Commented Jun 23 at 8:08
• @periblepsis, yes I mean the mathematical equations for the system dynamics. I will really extremely be happy if I know such a system where PID performance is poor. Commented Jun 24 at 1:58

PID controllers struggle in cases where the process being controlled:

1. has significant time delays from control input to output response.
2. is very non-linear, & particularly non-monotonic.
3. changes significantly over time.

Examples:
1. The Process has time delays:
Many processes are known to have significant time delays from control action occurring to the measured response, eg:

1. Heating tanks of liquids.
2. Release of a liquid from a reservoir and its arrival time at a given location (eg: releasing water from dams).
3. An electrical pulse sent from one location and its arrival time at another location; the time delay is highly dependent upon the nature of the transmission medium, which can be dependent upon ambient temperature.

These time delays can be highly dependent upon unknowns, for example, in the case of heating tanks of liquids, the time delay is dependent upon the temperature and flow rate of the incoming unheated product, both of which are often not being measured or controlled.

The Smith Predictor was proposed back in the 1950s as a means of using standard PID control theory to obtain good control over such processes, however, it requires the time delay to be known, predictable and stable.

Control of processes with time-delays is a highly active area of research, here is just one example:
https://www.sciencedirect.com/science/article/abs/pii/S0959152498000365

2. The Process has non-linear response:
PID control is particularly bad when the response is non-monotonic with control input, eg: if the variable being controlled does not always increase as the control input increases. One example of this is a Maximum Power-Point Tracker for a solar panel; this is the classic "hill seeking" problem; we are trying to control the voltage across the PV panel such that it is delivering maximum power.

There are many examples in chemistry and biology, such as the rate of fermentation vs temperature in wine and beer making. The relationship between fermentation rate and temperature is highly non-linear and actually reverses beyond a certain temperature; controlling the temperature of this process can be difficult because the process itself generates heat.

Other examples of very non-linear responses in chemistry and biology are presented here:
https://neurophysics.ucsd.edu/courses/physics_173_273/BZ_Epstein_Review.pdf

3. The process changes significantly over time
This could be due to wear and tear of an actuator, for example, a flow control valve may develop increasing friction and backlash during its life. A PID-based controller that may be stable and effective on day one may become unstable after a period of time due to the degraded performance of this valve.

• I will be happy if I know the mathematical equations for the system dynamics. Commented Jun 24 at 1:59