[YZF]

Your example is really a classical real world PID where a motion system tracks a motion profile. The short answer is yes, PIDs work just as well when the target changes. The stability of a PID depends on the open loop response which command or error are not part of.

The theory isn't that simple (you can take a course in Control Theory which would cover that) but I think many/most people that use/tune PIDs in the real world just use rules of thumb and standard approaches to tune the different components of the loop. Somewhat more advanced, but still non-theoretical tools, are actually measuring the open loop response which lets you pretty much see the margins you have. Optimizing PIDs is really trying to get the loop to respond quickly without becoming unstable.

Another interesting thing that's used in the real world is feed-forward. That is if you have some estimate of how to perform the function of the PID directly you can just feed that to the output and add the PID in parallel. The motion control analogy of this is called acceleration feed forward. If you know the target acceleration you can basically multiply that by some constant and feed it directly to the output (which is generally current to the motor which is more or less linear to acceleration given a constant load). This dramatically improves the performance of your control system because the PID now just corrects the much smaller residual error.