In the simplest case of Linear Feedback Control, the corrections are proportion to the magnitude of the difference, or error. This may not work very well for nonlinear plants, so it's common to multiply these control inputs by a gain matrix. The desired state acts as an attractor, and the system behaves as a simple spring. Springs are notorious for exhibiting oscillations, so it is common to include some damping terms. E.g. where is the control input, the state of the system, and the target state.
Dots represent derivatives with respect to time.
The damping coefficient can be set to give critical damping. PID stands for proportional, integrative, damped, which should be fairly obvious from the above. Feedback Control is, by definition, an error-driven process, so we need errors for it to be doing anything. This means that it is very likely that we will get a path that lags continuously behind the desired path. Also, feedback systems usually take a finite amount of time to respond to errors, so perturbing oscillations above a certain frequency will not by corrected.