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<!--------------------------------------------------------------
-- file    : nonlinear.html
-- purpose : Page for website Piet Lammertse
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--   history :
--     2026-03-01 new
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<head>
  <link rel="stylesheet" href="plammertse.css">
  <title>  Piet Lammertse on nonlinear control  </title>
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<ul>
  <li> <a href="index.html">             Home         </a> </li>
  <li> <a style="background-color:#0000C0;"
          href="nonlinear.html">   Nonlinear control  </a> </li>
  <li> <a href="phase_plane.html">     Phase plane    </a> </li>
</ul>

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<h2> Nonlinear control </h2>

<h4> bananas and non-bananas </h4>

<p> Virtually all of control theory is about linear systems.
 Virtualy all of the world is not.
</p>
<p>&emsp;
As one teacher described it to me : "the world consists of
 bananas and non-bananas". Linear systems are the bananas.
They are not in the majority.
They may even be next to impossible, because linearity
of cause and effect implies immediate response.
This is not likely in a world where the speed of light is finite.
</p>

<h4> linearizing nonlinear systems </h4>

<p> Linear systems have been analyzed to death,
 and they are "all the same",
while all nonlinear systems
 tend to be different in different ways.
 But this is not the only reason that linear systems
are so popular in  textbooks.
</p>
<p>&emsp;
Linear systems are ubiquitous in a more practical sense.
Most linearities in the physical world around us
are relatively smooth.
 Their graphs are gentle curves, not sharp blocks and pulses.
 And a gentle curve is almost straight if you travel
 only a very short distance along it.
 Most engineering control systems move only over a short
 range around a desired setpoint.
This means that they we can approximate these curves
 quite well by their straight tangent lines at the setpoint.
 This amounts to taking the local derivatives of the curves,
and this is usually easy to do.
 The process is called linearization.
</p>
<p>&emsp;
In the engineering sense, most physical systems are linear
in the vicinity of their setpoint.
</p>

<h4> everything is bananas </h4>

<p> Almost all control systems belong to the  bananas.
 This makes it so useful to study the bananas extensively.
</p>
<p>
But occasionally it is still useful to study a particular
nonlinear system for its own good.
We will discuss one of these in the sections on the
<a href="phase_plane.html"> phase plane</a> and the
<a href="dual_control_law.html"> dual control law</a>.
</p>


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