Background
For beginners to PID there is often confusion: why does a position error processed by PID become a desired velocity? Is there a physical relationship between them? And for drones, why is the controller output sometimes called desired lift, sometimes desired acceleration, and sometimes desired throttle? What are the actual units of these outputs?
The root cause of this confusion is a failure to distinguish open-loop control from closed-loop control. In practical control systems we commonly combine open-loop (feedforward) and closed-loop (feedback) control. The following simple example illustrates the relationship between these approaches.
Open-loop vs Closed-loop Control
Consider a block sliding on a plane. Assume its acceleration can be set instantly, and we want the block to reach a target velocity. What acceleration should be applied?
Open-loop control: Using basic kinematics, one can calculate the acceleration profile that achieves the target velocity after a given time. In an ideal environment this works, but it is impractical in real conditions. Actuation noise, imperfect execution, and small forces such as air resistance will cause the integrated velocity to deviate from the target. Keeping the velocity exactly at the target for long periods becomes nearly impossible with pure open-loop control.
Closed-loop control: A straightforward solution is real-time feedback. If there is a velocity error, apply an acceleration proportional to that error, for example a = Kp * (v_target - v), where Kp is a positive constant. If the error is positive, accelerate; if negative, decelerate. This proportional feedback maintains the desired velocity over long periods despite disturbances.
Comparing open-loop and closed-loop control: open-loop control follows explicit physical relationships where acceleration and time determine velocity. A feedback controller, however, does not represent a fixed physical equation mapping error to actuation; it provides a rule that is tuned to achieve the desired behavior. The controller gain can be adjusted freely. For example, replacing the proportional law with a nonlinear control rule such as a = K * sign(error) or another nonlinear mapping will still reach the desired velocity but does not correspond to a single simple physical equality.
It is important to distinguish the controller from the plant. The plant (the physical system) still obeys physical laws: acceleration integrates to velocity. If the plant lacks a definable physical relationship between its variables, the system is not controllable and controller design is not meaningful.
Units and Dimensions
Now assume acceleration cannot be controlled directly, but external force can be controlled. Since F = m * a, an open-loop mapping between desired acceleration and force exists. Under closed-loop control, the controller might output a value proportional to the error; whether the output is interpreted as acceleration or force matters only insofar as a unit conversion is required. In practice, PID gains absorb that conversion, so whether the controller output is called acceleration, force, lift, or throttle is often not critical: the scaling is handled by the gains.
This explains why in drones a desired lift can be expressed as a desired throttle in implementation. If lift and throttle are approximately proportional, adjusting PID gains implements the unit conversion. The specific naming follows industry convention.
Note that this unit-equivalence applies to linear, proportional relationships. If the mapping between a controller output and the plant is nonlinear, simple scaling of PID gains is insufficient; a corresponding nonlinear control law may be required. If the relationship involves integration or differentiation, the controller structure must be adjusted accordingly.
