# How to plot the Eular Transformation?

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Hardik on 22 Dec 2022
hello below is my question I have derive the bilinear TF but when I try to do the Euler transformation part I am not able to do that. I want to implement forward type but i dont know where to start.
below is my Question
Use sampling period of 0.10 Sec Euler transformations to convert the compensated systems into digital (discrete). Provide the overall closed loop transfer function (indiscrete domain) andraw the step responses of the systems obtained

Paul on 22 Dec 2022
The Control System Toolbox, for unknown reasons, doesn't support the forward or backward Euler transformations. I'm not aware of any other toolboxes that do either.
You can implement them with a bit of work, see this answer for one approach illustrating the forward method.
Or, you can go back and forth with the Symbolic Math Toolbox, which would be pretty straightforward.
Also, instead of using bilinear, you can use c2d and save a few keystrokes:
num = [1.5 60 600];
den = [1 15 83 210.5 300 700];
Qc = tf(num,den);
Fs = 10; % 1/0.1=10
Qp = c2d(Qc,1/Fs,'tustin')
Qp = 0.0003777 z^5 + 0.001133 z^4 + 0.001133 z^3 + 0.0003777 z^2 ----------------------------------------------------------- z^5 - 3.739 z^4 + 5.558 z^3 - 4.098 z^2 + 1.5 z - 0.218 Sample time: 0.1 seconds Discrete-time transfer function.
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Paul on 23 Dec 2022
That expression for for Hf is incorrect, it should use the tf command.
There is no such function c2d_eular.
There is a function c2d in the Control System Toolbox and in the System Identification Toolbox, but neither supports forward or backward Euler tranformation.
So you'll have to implement something yourself.
Do you know the relationships between s and z for forward and backward Euler?

Sulaymon Eshkabilov on 24 Dec 2022
num = [1 2];
den = [1 2 3 4];
Qc = tf(num,den);
ts = 0.1; % Sampling time
Qd_zoh = c2d(Qc,ts,'zoh') % Continuous - to - Discrete with zero-order-hold
Qd_zoh = 0.004988 z^2 + 0.0009038 z - 0.004083 ------------------------------------- z^3 - 2.79 z^2 + 2.612 z - 0.8187 Sample time: 0.1 seconds Discrete-time transfer function.
Qd_foh = c2d(Qc,ts,'foh') % Continuous - to - Discrete with zero-order-hold
Qd_foh = 0.001664 z^3 + 0.005284 z^2 - 0.003778 z - 0.001362 --------------------------------------------------- z^3 - 2.79 z^2 + 2.612 z - 0.8187 Sample time: 0.1 seconds Discrete-time transfer function.
step(Qc), hold on
step(Qd_zoh)
step(Qd_foh)
xlim([0, 15])
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