Control Systems
===============



Feedback Control System
-----------------------

Consider a system where we heat the water in a tank. We set a
temperature, say 70 degrees. We compare the set temperature (set
value) with the actual temperature (the process value). The error is
used to control the heater. In case the set value minus the process
value is positive we need to heat the water, in case the difference
(error) is negative we should cool the water.

In a process diagram this leads to a block diagram like:

.. tikz:: A Heated Water Tank

   \bXInput{A}
   \bXComp[4]{B}{A}
   \bXLink[\$q_\text{set}\$]{A}{B}
   \bXBloc[4]{C}{Controller}{B}
   \bXLink[\$\Delta q\$]{B}{C}
   \bXBloc{D}{Tank}{C}
   \bXLink[]{C}{D}
   \bXBloc{E}{Temp. Sensor}{D}
   \bXLink[]{D}{E}
   \bXOutput[4]{F}{E}
   \bXLink[\$q\$]{E}{F}
   \bXReturn{E-F}{B}{}


We assume that all blocks in such a scheme represent linear time
invariant systems. In that case each system can be represented with
its transfer function $H_i(s)$. Representing all blocks in the $s$
domain we get:

.. tikz:: A Heated Water Tank (s-domain)

   \bXInput{A}
   \bXComp[4]{B}{A}
   \bXLink[\$Q_\text{set}(s)\$]{A}{B}
   \bXBloc[4]{C}{\$H_{\text{C}\$}(s)}{B}
   \bXLink[\$\Delta q\$]{B}{C}
   \bXBloc{D}{\$H_{\text{T}(s)\$}{C}
   \bXLink[]{C}{D}
   \bXBloc{E}{\$H_{\text{S}}(s)\${D}
   \bXLink[]{D}{E}
   \bXOutput[4]{F}{E}
   \bXLink[\$Q\$]{E}{F}
   \bXReturn{E-F}{B}{}




Stability
---------


Root Locus Analysis
-------------------



PID Controller
--------------