View source for Exercise: Integral action produces zero steady state error*

We say that an system with a constant input reaches ``steady
state'' if the output of the system approaches a constant value as
time increases.  Show that a controller with integral action, such as
those given by the equations
<center><amsmath>
  u(t) = k_i \int\limits _0^t\,e(\tau)d\tau .
</amsmath></center>
and
<center><amsmath>
  \begin{aligned}
     u(t) &= ke(t)+k_i \int_0^t e(\tau) \, d\tau + k_d \frac{de(t)}{dt} \\
  \end{aligned}
</amsmath></center>
gives zero error if the closed loop 
system reaches steady state.

[[Category:Introduction Exercises|Integral action]]
[[Category:Printed Exercises|1 5]]