Editing Digital control (section)

== Digital controller implementation ==
A digital controller is usually cascaded with the plant in a feedback system. The rest of the system can either be digital or analog.

Typically, a digital controller requires:
*Analog-to-digital conversion to convert analog inputs to machine-readable (digital) format
*Digital-to-analog conversion to convert digital outputs to a form that can be input to a plant (analog)
*A program that relates the outputs to the inputs

=== Output program ===
*Outputs from the digital controller are functions of current and past input samples, as well as past output samples - this can be implemented by storing relevant values of input and output in registers. The output can then be formed by a weighted sum of these stored values.

The programs can take numerous forms and perform many functions
*A [[digital filter]] for [[low-pass filter]]ing
*A [[state space (controls)|state space]] model of a system to act as a [[state observer]]
*A [[telemetry]] system

=== Stability ===
Although a controller may be stable when implemented as an analog controller, it could be unstable when implemented as a digital controller due to a large sampling interval. During sampling the aliasing modifies the cutoff parameters. Thus the sample rate characterizes the transient response and stability of the compensated system, and must update the values at the controller input often enough so as to not cause instability.

When substituting the frequency into the z operator, regular stability criteria still apply to discrete control systems. [[Nyquist stability criterion|Nyquist criteria]] apply to z-domain transfer functions as well as being general for complex valued functions. Bode stability criteria apply similarly.
[[Jury stability criterion|Jury criterion]] determines the discrete system stability about its characteristic polynomial.

=== 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>

=== Design of digital controller in z-domain ===
The digital controller can also be designed in the z-domain (discrete). The [[Pulse-transfer function|Pulse Transfer Function]] (PTF) <math> G(z) </math> represents the digital viewpoint of the continuous process <math> G(s) </math> when interfaced with appropriate ADC and DAC, and for a specified sample time <math> T </math> is obtained as:<ref name=":0">{{Cite book|last1=Åström|first1=Karl J.|url=https://books.google.com/books?id=TynEAgAAQBAJ&dq=Computer-Controlled+Systems%3A+Theory+and+Design&pg=PP1|title=Computer-Controlled Systems: Theory and Design, Third Edition|last2=Wittenmark|first2=Björn|date=2013-06-13|publisher=Courier Corporation|isbn=978-0-486-28404-0|language=en}}</ref>

<math> G(z) =\frac{B(z)}{A(z)} = \frac{(z-1)}{z}Z\biggl(\frac{G(s)}{s}\Biggr) </math>

Where <math> Z() </math> denotes z-Transform for the chosen sample time <math> T </math>. There are many ways to directly design a digital controller <math> D(z) </math> to achieve a given specification.<ref name=":0" /> For a type-0 system under unity negative feedback control, [[Michael Short (engineer)|Michael Short]] and colleagues have shown that a relatively simple but effective method to synthesize a controller for a given ([[Monic polynomial|monic]]) closed-loop denominator polynomial <math> P(z) </math> and preserve the (scaled) zeros of the PTF numerator <math> B(z) </math> is to use the design equation:<ref name=":1">{{Cite journal|last1=Short|first1=Michael|last2=Abugchem|first2=Fathi|last3=Abrar|first3=Usama|date=2015-02-11|title=Dependable Control for Wireless Distributed Control Systems|journal=Electronics|language=en|volume=4|issue=4|pages=857–878|doi=10.3390/electronics4040857|doi-access=free}}</ref>

<math> D(z) =\frac{k_p A(z)}{P(z) - k_p B(z)} </math>

Where the scalar term <math> k_p = P(1)/B(1) </math> ensures the controller <math> D(z) </math> exhibits integral action, and a steady-state gain of unity is achieved in the closed-loop. The resulting closed-loop discrete transfer function from the z-Transform of reference input <math> R(z) </math> to the z-Transform of process output <math> Y(z) </math> is then given by:<ref name=":1" />

<math> \frac{Y(z)}{R(z)} =\frac{k_p B(z)}{P(z)} </math>

Since process time delay manifests as leading co-efficient(s) of zero in the process PTF numerator <math> B(z) </math>, the synthesis method above inherently yields a predictive controller if any such delay is present in the continuous plant.<ref name=":1" />