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'''Digital control''' is a branch of [[control theory]] that uses [[Digital data|digital]] [[computer]]s to act as system controllers.
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Depending on the requirements, a digital control system can take the form of a [[microcontroller]] to an [[Application-specific integrated circuit|ASIC]] to a standard desktop computer.
Since a digital computer is a [[discrete mathematics|discrete]] system, the [[Laplace transform]] is replaced with the [[Z-transform]].
Also since a digital computer has finite precision (''See [[Quantization (signal processing)|quantization]]''), extra care is needed to ensure the error in coefficients, [[Analog to digital converter|A/D conversion]], [[digital to analog converter|D/A conversion]], etc. are not producing undesired or unplanned effects.


The application of digital control can readily be understood in the use of [[feedback]].
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Since the creation of the [[Atanasoff–Berry Computer|first digital computer]] in the early 1940s the price of digital computers has dropped considerably, which has made them key pieces to control systems for several reasons:
*Inexpensive: under $5 for many microcontrollers
*Flexible: easy to configure and reconfigure through software
*Scalable: programs can scale to the limits of the memory or storage space without extra cost
*Adaptable: parameters of the program can change with time (''See [[adaptive control]]'')
*Static operation: digital computers are much less prone to environmental conditions than [[capacitor]]s, [[inductor]]s, etc.


== Digital Controller Implementation ==
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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:
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*[[Analog to digital converter|A/D conversion]] to convert analog inputs to machine readable (digital) format
*[[Digital to analog converter|D/A 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 ===
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*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
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*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 which 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>
 
We must never forget that the 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
 
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
 
http://mtc-m05.sid.inpe.br/col/sid.inpe.br/deise/1999/09.14.15.39/doc/homepage.pdf
 
http://www.sae.org/technical/papers/2002-01-3468
 
*Yutaka Yamamoto and his "lifting function space model" http://wiener.kuamp.kyoto-u.ac.jp/~yy/Papers/yamamoto_cwi96.pdf
 
*Alexander Sesekin and his studies about impulsive systems. http://www.amazon.com/dp/0792343948
 
and
 
*M.U. Akhmetov and his studies about impulsive and pulse control. http://portal.acm.org/author_page.cfm?id=81100182444&coll=GUIDE&dl=GUIDE&trk=0&CFID=27536832&CFTOKEN=71744014
 
== See also ==
*[[Sampled data systems]]
*[[Adaptive control]]
*[[Analog control]]
*[[Control theory]]
*[[Digital data|Digital]]
*[[Feedback]], [[Negative feedback]], [[Positive feedback]]
*[[Laplace transform]]
*[[Real-time control]]
*[[Z-transform]]
 
==References==
{{reflist}}
*FRANKLIN, G.F.; POWELL, J.D. Digital control of dynamical systems. USA, California: Addison-Wesley. 1981. ISBN 0-201-82054-4
 
*KATZ, P. Digital control using microprocessors. Englewood Cliffs: Prentice-Hall, 293p. 1981.
*OGATA, K. Discrete-time control systems. Englewood Cliffs: Prentice-Hall,984p. 1987.
*PHILLIPS, C.L.; NAGLE, H. T. Digital control system analysis and design. Englewood Cliffs, New Jersey: Prentice Hall International. 1995.
*M. Sami Fadali, Antonio Visioli, (2009) "Digital Control Engineering", Academic Press, ISBN 978-0-12-374498-2.
*JURY, E.I. Sampled-data control systems. New-York: John Wiley. 1958.
 
{{DEFAULTSORT:Digital Control}}
[[Category:Control theory]]
 
[[de:Regelungstechnik#Zeitdiskrete_Regelung]]

Latest revision as of 12:33, 7 January 2015

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