PID Control Fundamentals

Introduction:
PID control is the most well-known controller but is probably the most poorly implemented control method  when it comes to applying the control algorithm.
Most likely due to very little control theory is needed to apply the control method (plant transfer function left-hand pole-zero frequency/s-domain stability) and documentation is widely available to the layman or laywoman.
PID control can only be applied for linear plant dynamics (99% of applications), linear in the sense that if I apply an input into the actuator-plant transfer function block, the plant output responds in a deterministic manner (not random) and the output is linear.
For instance, if I spin the motor on a conveyor belt, the conveyor will linearly move faster, if I spin the motor faster then the conveyor will move faster in a proportional linear manner.  
Orbital vehicle maneuvers in space for example are highly angular dependent which are based on the space object reference to central point and thus non-linear.
A non-linear sliding controller may be a better method for controlling an object in space.
For other instances, PID control with feed-forward which predicts the plant dynamics based on some non-linear step condition can change immediately the performance of the PID control based on this step/threshold condition before controls become too erratic. In this case, the PID controls function for these known non-linear conditions and "one is shaping the pole-zero location of the plant transfer function."

Basic Applications:
PID control is excellent for a variety of linear applications including pump/valve pressure and flow control, fluid level control using pumps or valves, HVAC/Chiller temperature control, chemical pump concentration control, crane motion control, drilling feed and weight controls among other application. For whatever the application it is used in, the PID control must be running 2x as fast as the plant dynamics, this can be tested in the time domain by putting the PID controller in manual mode or bypassing it, putting a step input into the actuator that you are controlling and seeing how the fast the output of the plant dynamics responds. This determines how fast your PLC task times need to be set. Note, running the PID controls too quickly or sampling IO too quickly, specifically the feedback sensor used for control can cause noise to enter the PID control. When a Derivative term is used in the PID control, the noise can be amplified.

Elementary Theory:
Most PID control is implemented with negative feedback, where the feedback is subtracted from the set-point to give the error. The  PID control acts on this error to create an actuator input (error action/correction) which is used to control the plant dynamics (flow, pressure, etc.).  Positive feedback is where the sensor feedback is added to the set-point.
The Proportional term determines the amount of action that must taken based on the difference between the set-point and feedback. The Integral term is the action based on the difference between set-point and feedback over time, this is why integral windup is large when the PID control is forgotten to be turned off or the control action is held back (i.e. I hold a drone back with my hand when it is commanded to go to a height of 1000 ft).  The Derivative term is the control action based on how fast the feedback has changed compared to the set-point. A great table describing the affects of changing Kp, Ki, and Kd terms independently given by the PID controller wiki is shown below with overall PID equation:
 https://en.wikipedia.org/wiki/PID_controller

Fundamental Tuning:
Typical implementations of PID control do not use the Derivative term as this can cause instability of the control when sensor noise is entered into the feedback loop as the derivative of a sinusoidal signal with noise is noise times the sinusoidal signal.
Most PID controls are tuned manually but the closed loop Ziegler-Nichols method can be implemented if the process cannot be turned off while tuning.
​Open loop tuning can be implemented such as the Cohen-Coon method or Integrated Absolute Error method using a step input and monitoring the response. 
​More tuning discussions will come later.