mathematics//dynamical systems//integrator
An integrator is a system whose output is the running total of its input: the level of a tank adding up the difference between inflow and outflow, a position adding up velocity, a bank balance adding up deposits and withdrawals. It is the simplest dynamical system with memory, and spotting one inside a plant tells an engineer at once how the plant behaves under a constant input and what kind of controller it needs.
An integrator is a system whose output is the running total of its input: the level of a tank adding up the difference between inflow and outflow, a position adding up velocity, a bank balance adding up deposits and withdrawals. It is the simplest dynamical system with memory, and spotting one inside a plant tells an engineer at once how the plant behaves under a constant input and what kind of controller it needs.
x˙=u,x(t)=x(0)+∫0tu(s) ds\dot x=u,\qquad x(t)=x(0)+\int_0^t u(s)\,dsx˙=u,x(t)=x(0)+∫0tu(s)ds
Its eigenvalue is zero, so it neither returns nor runs away by itself: left alone it keeps its value, and under a constant input it ramps for ever without settling. That is what sets it apart from a lag, which also accumulates but leaks, and so settles at a level proportional to its input (exponential decay).
Plants that integrate are common and need care. A tank emptied by a pump at fixed flow, the position of any motor, the heading of a vehicle under a constant turn rate: none of them settles under a constant input other than an exact balance, so a step test shows a ramp instead of an S-curve, and the model of the plant is a rate (units per second per unit of input) rather than a gain (plant).
Two in a row make a double integrator, force to velocity to position: the model of any mass pushed by a force, and of the drone's altitude, whose two open-loop eigenvalues are both zero (vertical drone). A constant force error makes it drift quadratically, which is why such plants are never left without feedback.
Its memory is the property controllers borrow. An integrator's output stops changing only when its input is zero, so a stable loop that contains one, on the error, must end with zero error (steady-state error).
A numerical integrator is a different thing with the same name: an algorithm that advances a differential equation in time steps (discretization).