Update Content - 2022-08-24

content/zettels/discrete_transfer_functions.md

@@ -8,6 +8,100 @@ Tags
 : [Digital Filters]({{< relref "digital_filters.md" >}})
 
 
+## Continuous to discrete transfer function {#continuous-to-discrete-transfer-function}
+
+In order to convert an analog filter (Laplace domain) to a digital filter (z-domain), the `c2d` command can be used ([doc](https://fr.mathworks.com/help/control/ref/lti.c2d.html)).
+
+<div class="exampl">
+
+Let's define a simple first order low pass filter in the Laplace domain:
+
+```matlab
+s = tf('s');
+G = 1/(1 + s/(2*pi*10));
+```
+
+To obtain the equivalent digital filter:
+
+```matlab
+Ts = 1e-3; % Sampling Time [s]
+Gz = c2d(G, Ts, 'tustin');
+```
+
+</div>
+
+There are several methods to go from the analog to the digital domain, `Tustin` is the one I use the most as it ensures the stability of the digital filter provided that the analog filter is stable.
+
+
+## Obtaining analytical formula of filter {#obtaining-analytical-formula-of-filter}
+
+The Matlab [Symbolic Toolbox](https://fr.mathworks.com/help/symbolic/) can be used to obtain analytical formula for discrete transfer functions.
+
+Let's consider a notch filter:
+
+\begin{equation}
+  G(s) = \frac{s^2 + 2 g\_c \xi \omega\_n s + \omega\_n^2}{s^2 + 2 \xi \omega\_n s + \omega\_n^2}
+\end{equation}
+
+with:
+
+-   \\(\omega\_n\\): frequency of the notch
+-   \\(g\_c\\): gain at the notch frequency
+-   \\(\xi\\): damping ratio (notch width)
+
+First the symbolic variables are declared (`Ts` is the sampling time, `s` the Laplace variable and `z` the "z-transform" variable).
+
+```matlab
+%% Declaration of the symbolic variables
+syms gc wn xi Ts s z
+```
+
+The symbolic formula of the notch filter is defined:
+
+```matlab
+%% Notch Filter - Symbolic representation
+Ga = (s^2 + 2*xi*gc*s*wn + wn^2)/(s^2 + 2*xi*s*wn + wn^2);
+```
+
+Then the bi-linear transformation is performed to go from continuous to discrete:
+
+```matlab
+%% Bilinear Transform
+s = 2/Ts*(z - 1)/(z + 1);
+```
+
+Finally, the numerator and denominator coefficients can be extracted:
+
+```matlab
+%% Get numerator and denominator
+[N,D] = numden(Ga);
+
+%% Extract coefficients (from z^0 to z^n)
+num = coeffs(N, z);
+den = coeffs(D, z);
+```
+
+```text
+num = (Ts^2*wn^2 - 4*Ts*gc*wn*xi + 4) + (2*Ts^2*wn^2 - 8) * z + (Ts^2*wn^2 + 4*Ts*gc*wn*xi + 4) * z^2
+```
+
+```text
+den = (Ts^2*wn^2 - 4*Ts*wn*xi + 4) + (2*Ts^2*wn^2 - 8) * z + (Ts^2*wn^2 + 4*Ts*wn*xi + 4) * z^2
+```
+
+
+## Variable Discrete Filter {#variable-discrete-filter}
+
+Once the analytical formula of a discrete transfer function is obtained, it is possible to vary some parameters in real time.
+
+This is easily done in Simulink (see Figure [1](#figure--fig:variable-controller-simulink)) where a `Discrete Varying Transfer Function` block is used.
+The coefficients are simply computed with a Matlab function.
+
+<a id="figure--fig:variable-controller-simulink"></a>
+
+{{< figure src="/ox-hugo/variable_controller_simulink.png" caption="<span class=\"figure-number\">Figure 1: </span>Variable Discrete Filter in Simulink" >}}
+
+
 ## Typical Transfer functions {#typical-transfer-functions}