Control theory

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<Control theory> studies how to choose <control inputs> for a <dynamical system> so that its <plant output> follows a <reference signal>, despite <control disturbances> and imperfect knowledge of the <controlled plant>. A <controller> turns the available information into a <control signal>. The design must consider both the resulting response and <stability theory>: a rule that corrects a small mismatch can still produce growing deviations if the interconnection is badly chosen. https://www.fbswiki.org/wiki/index.php/Introduction[Åström and Murray, Feedback Systems, Chapter 1: Introduction]{ref}

The central distinction is informational. <Open-loop control> executes planned commands; <feedback control> uses the <measured output> of the actual <controlled plant> to update those commands. A <reference signal> can describe a destination, while <control disturbances> describe influences that the <controller> did not command. A model can predict where the <controlled plant> should be; measurement tells the <controller> what actually happened. The authors provide a free full-book PDF through https://www.fbswiki.org/wiki/index.php/Feedback_Systems:_An_Introduction_for_Scientists_and_Engineers[Feedback Systems, second edition: authors' book and download page]{ref}.

For a concrete comparison, all examples below use the same ideal <scalar> <controlled plant>:
$$
\dot x(t)=u(t)+w(t),\qquad y(t)=y_m(t)=x(t),\qquad r(t)=1,\qquad x(0)=0.
$$
Here $x$ is an illustrative <position>, $u$ the <control input>, $w$ a <control disturbance>, and $y_m$ the <measured output>. Both $x$ and $t$ are <dimensionless quantities>; the numbers are chosen to expose the mechanism, with no empirical interpretation. The <ordinary differential equation> says that the present <control input> and <control disturbance> accumulate into $x$. The <measured output> equals the entire state only in this ideal example; a general <plant output> need not reveal the entire state.

\Image[/feedback-reference-and-disturbance.png]
{title=Planned commands and <feedback control> for an ideal <controlled plant>}
{description=Top: the <reference signal> and <measured output> form the <tracking error>, which the <controller> converts into a <control signal>. Bottom left: with no <control disturbance>, the planned pulse reaches $x=1$ at $t=1$, while <proportional control> with <feedback gain> $k=2$ approaches it smoothly. Bottom right: a <control disturbance> $w=0.2$ starts at $t=2$; <open-loop control> drifts, while <proportional control> approaches $1.1$. All coordinates are <dimensionless quantities>.}


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