Editing Digital control (section)

=== Design of digital controller in s-domain ===
The digital controller can also be designed in the s-domain (continuous). The [[Arnold Tustin|Tustin]] transformation can transform the continuous compensator to the respective digital compensator. The digital compensator will achieve an output that approaches the output of its respective analog controller as the sampling interval is decreased.

<math> s = \frac{2(z-1)}{T(z+1)} </math>

==== Tustin transformation deduction ====
Tustin is the [[Padé table|Padé<sub>(1,1)</sub>]] approximation of the exponential function <math> \begin{align} z &= e^{sT} \end{align} </math> :

: <math>
\begin{align}
z &= e^{sT}   \\
  &= \frac{e^{sT/2}}{e^{-sT/2}} \\
  &\approx \frac{1 + s T / 2}{1 - s T / 2}
\end{align}
</math>

And its inverse

: <math>
\begin{align}
s &= \frac{1}{T} \ln(z)  \\
  &= \frac{2}{T} \left[\frac{z-1}{z+1} + \frac{1}{3} \left( \frac{z-1}{z+1} \right)^3  + \frac{1}{5} \left( \frac{z-1}{z+1} \right)^5  + \frac{1}{7} \left( \frac{z-1}{z+1} \right)^7 + \cdots \right] \\
  &\approx  \frac{2}{T} \frac{z - 1}{z + 1} \\
  &=  \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}}
\end{align}
</math>

Digital control theory is the technique to design strategies in discrete time, (and/or) quantized amplitude (and/or) in (binary) coded form to be implemented in computer systems (microcontrollers, microprocessors) that will control the analog (continuous in time and amplitude) dynamics of analog systems. From this consideration many errors from classical digital control were identified and solved and new methods were proposed:

*Marcelo Tredinnick and Marcelo Souza and their new type of analog-digital mapping<ref>{{Cite web |title=Referência Completa |url=http://mtc-m18.sid.inpe.br/col/sid.inpe.br/mtc-m18@80/2008/03.17.15.17.24/doc/mirrorget.cgi?languagebutton=pt-BR&metadatarepository=sid.inpe.br/mtc-m18@80/2009/02.09.14.45.33&index=0&choice=full |archive-url=https://web.archive.org/web/20120305110221/http://mtc-m18.sid.inpe.br/col/sid.inpe.br/mtc-m18@80/2008/03.17.15.17.24/doc/mirrorget.cgi?languagebutton=pt-BR&metadatarepository=sid.inpe.br/mtc-m18@80/2009/02.09.14.45.33&index=0&choice=full |archive-date=March 5, 2012 |website=mtc-m18.sid.inpe.br}}</ref><ref>{{cite web |title=Discrete attitude control of artificial satellites with flexible appendages |url=http://mtc-m05.sid.inpe.br/col/sid.inpe.br/deise/1999/09.14.15.39/doc/homepage.pdf |url-status=dead |archive-url=https://web.archive.org/web/20110706160612/http://mtc-m05.sid.inpe.br/col/sid.inpe.br/deise/1999/09.14.15.39/doc/homepage.pdf |archive-date=6 July 2011 |access-date=12 January 2022 |website=mtc-m05.sid.inpe.br}}</ref><ref>{{cite web |url=http://www.sae.org/technical/papers/2002-01-3468 |title=An Analytical Approach for Discrete Controllers Design Using a New S-Z Mapping with Two Tuning Parameters |website=www.sae.org |access-date=27 January 2022 |archive-url=https://archive.today/20130113082747/http://www.sae.org/technical/papers/2002-01-3468 |archive-date=13 January 2013 |url-status=dead}}</ref>
*Yutaka Yamamoto and his "lifting function space model"<ref>{{cite web |last=Yamamoto |first=Yutaka |date=1996 |title=A Retrospective View on Sampled-Data - Control Systems |url=http://wiener.kuamp.kyoto-u.ac.jp/~yy/Papers/yamamoto_cwi96.pdf |url-status=dead |archive-url=https://web.archive.org/web/20110722072133/http://wiener.kuamp.kyoto-u.ac.jp/~yy/Papers/yamamoto_cwi96.pdf |archive-date=22 July 2011 |access-date=12 January 2022 |website=[[Kyoto University]]}}</ref>
*Alexander Sesekin and his studies about impulsive systems.<ref>{{Cite book|isbn=0792343948|title=Dynamic Impulse Systems: Theory and Applications|last1=Zavalishchin|first1=S. T.|last2=Sesekin|first2=A. N.|date=28 February 1997|publisher=Springer }}</ref>
*M.U. Akhmetov and his studies about impulsive and pulse control<ref>{{Cite web |title=Author page |url=http://portal.acm.org/author_page.cfm?id=81100182444&coll=GUIDE&dl=GUIDE&trk=0&CFID=27536832&CFTOKEN=71744014 |url-status= |archive-url= |archive-date= |access-date=2009-03-20 |website=[[Association for Computing Machinery]]}}{{Dead link|date=November 2024|fix-attempted=yes}}</ref>