🗻 Integral Control

Integral control extends 🕹️ Controllability with an integrator that counteracts disturbances. Specifically, it can eliminate errors caused by constant disturbance.

First, a 🇺🇸 State Space Model with disturbance (and ignoring ) has the form

With state feedback that provides stabilization, the state transition becomes

which has ♻️ Transfer Function Model > Steady State Gain from disturbance to output

The goal of integral control is to bring this error to zero, so that constant disturbance does not affect the output.

To do so, we introduce an integrator

where is the target equilibrium output value. The motivation behind this is that at steady state, , and thus . Including the integrator in the feedback, we have

and thus the full system is

In matrix form, this is

and

First, we need to be Hurwitz, to maintain steady state. Given this, we can compute steady state gain as

showing that does not affect the output, as well as

showing that output is exactly the target .