Editing Digital control (section)

=== 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" />