mathematics//dynamical systems//feedback loop
There is **feedback** when a perturbation can travel through the dependencies and come back to influence the variable it started from. It can be an internal structure of the process or an interconnection designed by a controller.
There is feedback when a perturbation can travel through the dependencies and come back to influence the variable it started from. It can be an internal structure of the process or an interconnection designed by a controller.
Feedback depends on the state and returns to it. The other way a rate can be pushed is an external forcing, a term that depends on time or on an input and not on the state; a rumor or an infection add a third shape, the interaction term, which needs two quantities to meet.
The product of the connection signs characterizes the local loop: positive, reinforcing; negative, compensating. If sensitivities change sign with the state, this classification can change too.
The sign does not decide stability. Take
x˙1=−ax1+k12x2,x˙2=k21x1−bx2,a,b>0.\dot x_1=-ax_1+k_{12}x_2,\qquad \dot x_2=k_{21}x_1-bx_2,\qquad a,b>0.x˙1=−ax1+k12x2,x˙2=k21x1−bx2,a,b>0.
The trace is −(a+b)-(a+b)−(a+b) and the determinant ab−k12k21ab-k_{12}k_{21}ab−k12k21. In this two-state case the origin is asymptotically stable exactly when ab>k12k21ab>k_{12}k_{21}ab>k12k21. A positive loop can be dominated by the decays. A negative loop is stable in this particular example, but in higher order, with delays or enough phase lag, it can destabilize. A property of this example is not a universal rule.
Feedforward does not wait for the error. In control, feedforward is an action computed from the reference or a known disturbance; feedback uses output or state information. They coexist: a drone can receive a nominal weight compensation and, on top of it, correct its altitude error. In a dependency graph "feedforward" also describes a chain with no return; context says whether topology or controller function is meant.
Two meanings of saturation. In x˙=rx−γx2\dot x=rx-\gamma x^2x˙=rx−γx2, with positive parameters and x≥0x\ge0x≥0, the nonlinear brake compensates growth around a finite level: an emergent dynamic saturation, developed in exponential decay. In an actuator,
ureal=sat(ucmd;umin,umax),u_{\mathrm{real}}=\operatorname{sat}(u_{\mathrm{cmd}};u_{\min},u_{\max}),ureal=sat(ucmd;umin,umax),
the output is limited by actuation bounds: a static saturation nonlinearity. It needs no state of its own, though it can change the stability of the loop. With integral action the accumulated error keeps growing while the actuator is pinned; that is windup.
The connections themselves are the subject of coupling; the timing of the return is explained in delay and lag. The collective outcome, not the intention of the loop, is what stability analysis studies.