control//feedback control

Feedback control is a form of control in which a system's inputs are computed from its measurements in order to modify how it evolves, and it is used to stabilize a system, regulate an output or track a reference. Because it decides from the measured output, it corrects what the model did not know without needing to know what it was: with enough loop gain the output stops depending on the plant's details, which is how Harold Black's vacuum-tube amplifiers stopped drifting in 1927. Its analysis keeps plant and controller apart before any eigenvalue is quoted (what control buys in general is the subject of control). The **plant** receives the actuation, the **controller** computes it, and the **closed-loop dynamics** belongs to their interconnection.

Feedback control is a form of control in which a system's inputs are computed from its measurements in order to modify how it evolves, and it is used to stabilize a system, regulate an output or track a reference. Because it decides from the measured output, it corrects what the model did not know without needing to know what it was: with enough loop gain the output stops depending on the plant's details, which is how Harold Black's vacuum-tube amplifiers stopped drifting in 1927. Its analysis keeps plant and controller apart before any eigenvalue is quoted (what control buys in general is the subject of control). The plant receives the actuation, the controller computes it, and the closed-loop dynamics belongs to their interconnection.

For a linear plant x˙=Fx+Bu\dot x=\cd{F}x+Bux˙=Fx+Bu, y=Cxy=Cxy=Cx, a state feedback u=−Kx+Nru=-Kx+N\cc{r}u=−Kx+Nr produces

x˙=(F−BK)x+BNr.\dot x=(\cd{F}-BK)x+BN\cc{r}.x˙=(F−BK)x+BNr.

The open plant's matrix in violet, the reference in copper. The eigenvalues of F\cd{F}F describe the open plant, those of F−BK\cd{F}-BKF−BK the closed loop, and they must never be attributed to the system without naming the configuration (closed-loop stability, state feedback).

Stabilizing, regulating and tracking are different goals. Stabilizing makes a desired equilibrium stable, regulating holds an output at a constant reference, tracking drives the error e=r−ye=\cc{r}-ye=r−y of a varying reference to zero. Stability implies neither zero steady-state error nor a fast transient, and proves nothing about delays, noise or actuator limits.

A vertical drone makes this concrete. With mq¨=T−mgm\ddot q=T-mgmq¨​=T−mg, one proposal is

T=mg−kp(q−r)−kdq˙.T=mg-k_p(q-\cc{r})-k_d\dot q.T=mg−kp​(q−r)−kd​q˙​.

mgmgmg nominally cancels the weight (feedforward); the other terms correct position and velocity. The rule orders an actuation the motor still has to deliver; the tracking version and the full case are in vertical drone. When the state is not all measured, an observer rebuilds it from the outputs, provided the system is observable (controllability and observability).

Feedback forgives bad models, within its bandwidth.

A PID with a crude model works because it measures and corrects, and its integrator erases biases nobody modelled; but it corrects only what is slower than the loop and reaches it in time. Past the bandwidth, behind a delay or against a saturated actuator, the forgiveness ends, and that is where robust and adaptive control begin.

Feedback has three prices. It needs a sensor and inherits its errors (a biased sensor gives an output held off target, perfectly regulated, sensitivity function); proportional action alone leaves a residual error; and with delay and lag, too much gain makes the loop oscillate (loop delay). The error in a closed loop lives in those places: the sensor's bias and noise, the delay and jitter that eat margin, the actuator that saturates or sticks and hunts, the half period that discretization adds, the feedforward model that is wrong.

The family runs from the parts to the structures. The control loop names the parts as a technician draws them, open-loop control is the case with the loop cut, and the PID controller is the law most loops run. Nested loops are cascade control, and an estimate of the unknown push subtracted from the command is the disturbance observer. The interconnection as a mathematical object is a feedback loop, its timing delay and lag, and guarantees beyond the linear test come from a Lyapunov function.