C* is easier to understand if you start with pure g-force control. The basic components are:
- The control stick, where the pilot commands a load factor in the -1.0 to 2.5 range.
- The accelerometer, which measures the actual load factor in the body-z-direction.
- A PID controller, which controls elevator deflection so that measured load factor always matches the pilot's command. The P, I, and D gains in this controller are not considered part of the C* law and must be tuned for a specific airframe.
When we go from g-force control to C*, we replace the feedback source of the PID controller with a calculated value C*:
$$ C^* = n_z + \frac{v_{co}}{g} \dot\theta$$
This "fake load factor" C* has the same units (none) as the measured load factor it replaced. It goes into the feedback input of the PID controller. The command input of the PID controller (from the pilot via control stick) remains unchanged.
$n_z$ is the measured load factor, $v_{co}$ is crossover speed in meters per second and $\dot\theta$ is pitch rate in radians per second. $g$ is specific gravity in meters per second squared, as usual. The second part of the formula is basically "what would be the centrifugal force if the aircraft was traveling at crossover speed." We add this value to our fake load factor regardless of actual speed.
The C* law is a compromise between pure g-force control and direct elevator control. If we look at both laws in isolation, we can clearly see the disadvantages that we want to cure:
With direct elevator control, pitch rate is approximately proportional to stick deflection. This feels natural at low speed, but a careless pull on the stick at high speed results in excessive g-forces.
Pure g-force control (through a PID controller) works well at high speed, but at low speed your aircraft will simply not reach high g-forces, period. If you pull on the stick anyway, the PID controller's "whatever it takes" behavior results in excessive elevator deflections and pitch excursions.
C* is mostly g-force control, but with an additional fake feedback term based on pitch rate. From the PID's point of view, the aircraft now always responds to more elevator deflection with more g-forces (fake g-forces at low speed, actual g-forces at high speed), thus ensuring that the "whatever it takes" behavior does not cause problems. The aircraft still pitches more at low speed because actual g-forces are lower, which means overall feedback is lower, which makes the PID controller deflect the elevator harder.
