# TransferFunction

```block TransferFunction "Discrete Transfer Function block"
parameter Real b[:] = {1} "Numerator coefficients of transfer function.";
parameter Real a[:] = {1} "Denominator coefficients of transfer function.";

extends Interfaces.DiscreteSISO;

output Real x[size(a, 1) - 1](each start = 0, each fixed = true) "State of transfer function from controller canonical form";
protected
parameter Integer nb = size(b, 1) "Size of Numerator of transfer function";
parameter Integer na = size(a, 1) "Size of Denominator of transfer function";
Real x1(start = 0, fixed = true);
Real xext[size(a, 1)](each start = 0, each fixed = true);
equation
when sampleTrigger then
x1 = (u - a[2:size(a, 1)] * pre(x)) / a[1];
xext = vector([x1; pre(x)]);
x = xext[1:size(x, 1)];
y = vector([zeros(na - nb, 1); b]) * xext;
end when;

annotation (
Documentation(
info = "<html>\n<p>The <strong>discrete transfer function</strong> block defines the\ntransfer function between the input signal u and the output\nsignal y. The numerator has the order nb-1, the denominator\nhas the order na-1.</p>\n<pre>\n          b(1)*z^(nb-1) + b(2)*z^(nb-2) + ... + b(nb)\n   y(z) = -------------------------------------------- * u(z)\n          a(1)*z^(na-1) + a(2)*z^(na-2) + ... + a(na)\n</pre>\n<p>State variables <strong>x</strong> are defined according to\n<strong>controller canonical</strong> form. Initial values of the\nstates can be set as start values of <strong>x</strong>.</p>\n<p>Example:</p>\n<pre>     Blocks.Discrete.TransferFunction g(b = {2,4}, a = {1,3});\n</pre>\n<p>results in the following transfer function:</p>\n<pre>        2*z + 4\n   y = --------- * u\n         z + 3\n</pre>\n\n</html>",
revisions = "<html>\n<p><strong>Release Notes:</strong></p>\n<ul>\n<li><em>November 15, 2000</em>\n    by <a href=\"http://www.dynasim.se\">Hans Olsson</a>:<br>\n    Converted to when-semantics of Modelica 1.4 with special\n    care to avoid unnecessary algebraic loops.</li>\n<li><em>June 18, 2000</em>\n    by <a href=\"http://www.robotic.dlr.de/Martin.Otter/\">Martin Otter</a>:<br>\n    Realized based on a corresponding model of Dieter Moormann\n    and Hilding Elmqvist.</li>\n</ul>\n</html>"),
Icon(
coordinateSystem(
preserveAspectRatio = true,
extent = {
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graphics = {
Line(
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color = {0, 0, 127}),
Text(
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textString = "b(z)"),
Text(
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textString = "a(z)")}),
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end TransferFunction;

```