control//feedback control//control loop//loop gain

The loop gain is the factor by which a signal is multiplied on one trip round a feedback loop, through the controller and the plant and back to the comparison, measured at each frequency as a complex number \(L(j\omega)\) with a size and a phase; it is the quantity on which both the benefits of feedback and its risk of oscillation depend. Large loop gain is what makes a loop accurate and indifferent to the plant's details; loop gain that is still large where the phase has reached 180 degrees is what makes it oscillate.

The loop gain is the factor by which a signal is multiplied on one trip round a feedback loop, through the controller and the plant and back to the comparison, measured at each frequency as a complex number L(jω)L(j\omega)L(jω) with a size and a phase; it is the quantity on which both the benefits of feedback and its risk of oscillation depend. Large loop gain is what makes a loop accurate and indifferent to the plant's details; loop gain that is still large where the phase has reached 180 degrees is what makes it oscillate.

A small oven shows the benefit. With yyy the temperature above ambient, uuu the heater power and ddd a disturbance such as an open door, y˙=−ay+bu+d\dot y=-ay+bu+dy˙​=−ay+bu+d, where aaa is how fast heat leaks and bbb how much each watt heats. Open loop, the right power u=ar/bu=ar/bu=ar/b works only if aaa and bbb are exact and the door stays shut. Closed with u=K(r−y)u=K(r-y)u=K(r−y), the oven settles where y˙=0\dot y=0y˙​=0:

y∞=bKa+bK r+1a+bK d.y_\infty=\frac{bK}{a+bK}\,r+\frac{1}{a+bK}\,d .y∞​=a+bKbK​r+a+bK1​d.

As the gain grows, the first fraction tends to one whatever aaa and bbb are, and the disturbance is divided by a+bKa+bKa+bK. The output stops depending on the plant's parameters. Harold Black used exactly this in 1927 at Bell Labs: his vacuum-tube telephone amplifiers drifted with temperature and age, and wrapped in high-gain negative feedback they stopped drifting, because their gain was then set by a passive resistor network.

The residual error is the price of finite gain. With proportional control the oven stops short by ar/(a+bK)ar/(a+bK)ar/(a+bK), which shrinks as KKK grows and never reaches zero; an integrator makes the loop gain infinite at zero frequency and removes it (proportional action, integral action).

The gain cannot be large everywhere. Every real plant lags more at higher frequencies, so the loop gain must fall below one before the total phase reaches 180 degrees. The frequency where ∣L∣=1|L|=1∣L∣=1 is the crossover, roughly the loop's bandwidth, and how much room is left there is read in the stability margins.

How the loop gain splits the error between disturbances and sensor noise at each frequency is the sensitivity function, S=1/(1+L)S=1/(1+L)S=1/(1+L).

It is distinct from the controller's gain (KpK_pKp​) and from the plant's steady-state gain: the loop gain is their product along the loop, frequency by frequency, and a slow sensor or a filter in the path changes it as much as a retuned controller does.